Research ArticleA Comfort-Aware Energy Efficient HVAC SystemBased on the Subspace Identification Method

O Tsakiridis1 D Sklavounos2 E Zervas1 and J Stonham2

1Department of Electronics TEI of Athens Egaleo 12210 Athens Greece2Department of Engineering and Design Brunel University Kingston Lane Middlesex UB8 3PH UK

Correspondence should be addressed to O Tsakiridis odytsakteiathgr

Received 12 October 2015 Revised 12 January 2016 Accepted 8 February 2016

Academic Editor Aleksander Zidansek

Copyright copy 2016 O Tsakiridis et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A proactive heating method is presented aiming at reducing the energy consumption in a HVAC system while maintainingthe thermal comfort of the occupants The proposed technique fuses time predictions for the zonesrsquo temperatures based on adeterministic subspace identification method and zonesrsquo occupancy predictions based on a mobility model in a decision schemethat is capable of regulating the balance between the total energy consumed and the total discomfort cost Simulation results forvarious occupation-mobility models demonstrate the efficiency of the proposed technique

1 Introduction

As the control and limitation of the energy consumptionremain a field with an exceptional technological and eco-nomical interest areas that have been considered as highenergy consumers comprise very challenging research issuesAccording to the US Energy Information Administrationfrom 2013 through 2040 the electricity consumption in thecommercial and residential sectors will be increasing by05 and 08 per year [1] It is well established and widelyaccepted through research studies that the main energyconsumers in the commercial and residential buildings arethe Heat Ventilation and Air Conditioning (HVAC) systemsas well as the lighting systems

Buildings contributed a 41 (or 40 quadrillion btu) to thetotal US energy consumption in 2014 On an average about43 of the energy consumption in a commercial and resi-dential building is due to HVAC systems [2] Therefore dueto the high rate of energy consumption the necessity of thedemand-driven control in the HVAC systems has becomeinevitable Inmodern buildings several sophisticated systemshave been applied aiming at providing this type of controlin the HVAC systems The state-of-the-art technology of theHVAC control considers the occupancy of the zones a very

important parameter playing a key role inmethods aiming atreducing energy consumption The detection and predictionof the zonesrsquo occupancy are a very challenging research fieldand several techniques have been proposed based on historicstatistical data as well as on probabilistic models

Another equally important factor taken into account inadvanced HVAC control systems is the thermal comfort ofthe occupants The objective of modern HVAC control sys-tems is to reduce energy consumptionwithout compromisingthe comfort of the occupants Wireless sensor networksequipped with temperature humidity and occupancy detec-tion sensor nodes are nowadays the basic platform to buildautomated HVAC control systems A number of methodsaiming to maintain the thermal comfort while saving energyhave been proposed Some of them are described in Section 2Towards this direction a novel technique is proposed in thispaper which aims at balancing the comfort and energy costsin a multizone system The decisions on heating the zones ornot may be taken either centrally or in a distributed mannerbywireless sensor nodes scattered in themultizone system Inany case temperature and zone occupancy information mustbe exchanged between a node responsible for a zone and itsneighboring nodes The decision process itself relies on twokinds of predictions (a) temperature-time predictions for

Hindawi Publishing CorporationJournal of EnergyVolume 2016 Article ID 5074846 13 pageshttpdxdoiorg10115520165074846

2 Journal of Energy

the zones and (b) the zonesrsquo occupancy profile The emphasisin this paper is on the zonesrsquo temperature predictions andto this end a deterministic subspace identification methodis used for modelling the thermal dynamics of each zoneThat is each zone is modelled by a simple state-space modelcapable of producing accurate predictions based on thesurrounding temperatures the heating power of the zone andthe current state that summarizes the temperature history ofthe zone For the zonesrsquo occupancy predictions we consider asemi-Markov model where occupants (moving as a swarm)stay in a zone for a random period of time and then moveto adjacent zones with given probabilities What is needed bythe decision process is the distribution of the first entrancetime to unoccupied zones Aiming at a proactive action theproposedmethod periodically computes the risk of activatingthe heater or not and decides in favor of the action thatproduces the smaller risk The computation of the risks relieson the relative weights of the energy and discomfort costs sothat the balance between the total energy consumed and thetotal discomfort cost may be regulated

The structure of the paper is as follows In Section 2 priorand related work of the field of the demand-driven HVACsystems that utilize occupancy and prediction methods ispresented In Section 3 the proposed comfort-aware energyefficient mechanism is described along with the subspaceidentification method used for obtaining temperature-timepredictions and a discussion on the distribution of thefirst entrance time to unoccupied zones Section 4 containssimulation results and assesses the effectiveness of the pro-posed algorithm Finally Section 5 summarizes the paper andproposes research directions for future work

2 Prior Work

The necessity of demand-driven HVAC systems for energyefficient solutions has orientated the researchers towardsoccupancy-based activated systems In multizone spacesthe reactive and proactive activation of the heatingcoolingof the zones contributes to significant energy savings andimproves the thermal comfort for the occupants Severalresearch works with valuable results of energy savings haveutilized the occupancy detection and prediction in order tocontrol the HVAC system appropriately For the occupancydetection several types of sensors are used (CO

2 motion

etc) while for the prediction combinational systems areusually applied utilizing mathematical predictive models(eg Markov chains) alongside with detected real data

The authors in [3] proposed an automatic thermostatcontrol system which is based on an occupancy predictionscheme that predicts the destination and arrival time of theoccupants in the air-conditioned areas in order to providea comfortable environment For the occupantsrsquo mobilityprediction the cell tower information of the mobile tele-phony system is utilized and the arrival time prediction isbased on historical patterns and route classification For thedestination prediction of locations very close to each other(intracell) the time-aided order-Markov predictor is usedThe authors in [4] proposed a closed-loop system for opti-mally controllingHVACsystems in buildings based on actual

occupancy levels named the Power-Efficient Occupancy-Based Energy Management (POEM) System In order toaccurately detect occupantsrsquo transition they deployed a wire-less network comprising two parts the occupancy estimationsystem (OPTNet) consisting of 22 camera nodes and a passiveinfrared (PIR) sensors system (BONet) By fusing the sensingdata from the WSN (OPTNet and BONet) with the outputof an occupancy transition model in a particle filter moreaccurate estimation of the current occupancy in each room isachieved Then according to the current occupancy in eachroom and the predicted one from the transition model acontrol schedule of the HVAC system takes over the preheatof the areas to the target temperature In [5] the authorsused real world data gathered from a wireless network of16 smart cameras called Smart Camera Occupancy PositionEstimation System (SCOPES) and in this way they developedoccupancy models Three types of Markov chain (MC)occupancy models were tested and these are the single MCthe closest distance and finally the blended (BMC) Theauthors concluded that BMC is the most efficient and thusthey embodied it as the occupancy predictionmethod in theirproposed ldquoOBSERVErdquo algorithm which is a temperaturecontrol strategy for HVAC systems

The authors in [6] have developed an integrated systemcalled SENTINEL that is a control system for HVAC systemsutilizing occupancy information For the occupancy detec-tion and localisation the system utilizes the existing Wi-Fi network and the clientsrsquo smart phones The occupancylocalization mechanism is based on the access point (AP)communication with a clientrsquos smart phone so that if anoccupantrsquos phone sends packets to an AP then he is locatedwithin the range of the AP By classifying the building areasinto twomain categories namely the personal and the sharedspaces the proposed mechanism activates the HVAC systemwhen an owner of a personal space (eg office) has beendetectedwithin an areawhere hisher office is located orwhenoccupants are detected within shared spaces In [7] an inte-grated heating and cooling control systemof a building is pre-sented aiming to reduce energy consumptionThe occupancybehavior prediction as well as the weather forecast as inputsto a virtual (software based) building model determinesthe control of the HVAC system The occupancy detectiontechnique utilizes Gaussian Mixture Models (GMM) for thecategorization of selected features yielding the highest infor-mation gain according to the different number of occupantsThis categorization was used for observation to a HiddenMarkovModel for the estimation of the number of occupantsA semi-Markov model was developed based on patternscomprised by sensory data of CO

2 acoustics motion and

light changes to estimate the duration of occupants in thespace The work in [8] proposes a model predictive control(MPC) technique aiming to reduce energy consumption ina HVAC system while maintaining comfortable environmentfor the occupants The occupancy predictive model is basedon the two-state Markov chain with the states modelling theoccupied and occupied condition of the areas The authorsin [9] propose a feedback control algorithm for a variableair volume (VAV) HVAC system for full actuated (zonesconsisting of one room) and underactuated (zones consisting

Journal of Energy 3

of more than one room) zones The proposed algorithmis called MOBSua (Measured Occupancy-Based Setback forunderactuated zones) and it utilizes real-time occupancydata through a WSN for optimum energy efficiency andthermal comfort of the occupants Moreover the algorithmcan be applied on conventional control systems with no needof occupancy information and it is scalable to arbitrary sizedbuildings

3 Comfort-Aware Energy Efficient Mechanism

We consider a multizone HVAC system consisting of119885 zones119911119894 119894 = 1 2 119885 Each zone is equipped with a wireless

sensor node capable of conveying the zonersquos temperatureand occupancy information to its neighbors or to a centralprocessing unit The occupancy information may be verysimple that is binary information indicating the presence orabsence of individuals in the zone or more advanced like thenumber of occupants in the zoneTheobjective is tominimizethe total energy and discomfort cost defined as

E = Eenergy +Ediscomfort

=

119885

sum

119894=1

int

infin

minusinfin

119882119894119868 (119867119894= on) 119889119905

+

119885

sum

119894=1

int

infin

minusinfin

119862119894119868 (119879119894lt 119879119888) 119889119905

(1)

Note that only the heating problem is addressed in this paperThus the case of spending energy for cooling the zonesand keeping the temperature 119879

119894within a certain comfort

zone is not treated for simplicity In (1) 119868(119860) denotes theindicator function which takes the value 1 or 0 dependingon whether condition 119860 is true or false 119882

119894is the power

of a heater that covers zone 119911119894and 119867

119894is the state of the

heater that is the heater is on or off Thus the first termof (1) is the total energy consumed by the multizone systemFor the discomfort cost we define the cost per unit time 119862

119894

and the target comfort threshold 119879119888 As long as the zonersquos

temperature 119879119894is higher than 119879

119888the occupants do not feel

discomfort The parameter 119862119894may depend on the zone the

number of the occupants (eg the total systemrsquos discomfortcost is proportional to the number of people experiencingdiscomfort) and the difference of the zonersquos temperature 119879

119894

and the comfort threshold 119879119888 It is worth noting that there is

a tradeoff between energy savings and discomfort cost Wemay preheat all zones to the comfort level thus renderingthe discomfort cost equal to zero This policy is inefficientdue to the large amount of energy consumed In the otherextreme we could act reactively by heating a zone only upondetecting occupants in it In this case the energy cost wouldbe as small as possible but the discomfort cost could increasedramatically In the sequel we will describe a method thatacts proactively by taking periodically decisions on whetherto heat a zone or not If the entrance to a zone is delayedthen we postpone heating until the next decision epoch Ifon the contrary it is anticipated that a zone will be occupiedbefore the zonersquos temperature reaches the comfort level thenwe preheat the zone

The decision of whether or not the heater of an unoccu-pied zone 119911

119894should be turned on depends on (a) the current

state of the zone (b) the relative value of the energy cost (119882119894)

and discomfort cost (119862119894) per unit time (c) the temperatures

of the surrounding zones and (d) an estimate of the time thatthe zone will become occupied This decision may be takeneither centrally from the central processing unit that gathersinformation by all zones or in a distributed manner by eachzonersquos node after collecting the relative information There isalso the possibility of dividing the decision process function-ality between the central unit and the wireless sensor networknodes For example the prediction of the first entrancetimes to unoccupied zones may be taken by the central unitthat is aware of the location of people in the multizonesystem and then the predicted values may be communicatedback to the zonesrsquo nodes for the final decision The specificimplementation of the decision process is irrelevant to thispaper and it will not be further analyzed

We assume that time is discretized 119905 = 119898119879119904119879119904is the sam-

pling period and that each node takes its decisions periodi-cally every119863 sampling periods Note that there is no need forthe nodes to be synchronized to each other Let 119884

119894(119905) denote

the random variable thatmodels the remaining time from thecurrent time instant 119905 until entrance to the unoccupied zone119911119894 It is obvious that the randomvariables119884

119896(119905) for the various

unoccupied zones 119911119896 do not follow the same distributionWe

further assume that the node is capable ofmaking predictionsfor the time it takes to exceed the comfort threshold 119879

119888 To

this end let 119878on119894(119905) denote the time period that is required to

exceed 119879119888if the heater is turned on immediately Similarly

we define 119878off119894(119905) as the predicted time to reach the comfort

threshold if the heater remains off for a period 119863 and thenat the next decision epoch it is turned onThe relation of theaforementioned parameters is shown in Figure 1

Suppose now that at time 119905 the node of the unoccupiedzone 119911

119894has to take a decisionwhether to turn the zonersquos heater

on or remain in the off state At the decision epoch 119905 the riskto turn the heater on is119877on119894(119905)

= 119882119894DP 119878off

119894(119905) le 119884

119894(119905)

+ 119882119894(119884119894(119905) minus 119878

on119894(119905)) 119875 119878

on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905)

(2)

That is if 119878off119894(119905) le 119884

119894(119905) there is plenty of time to reach the

comfort threshold even if we start heating at the next decisionepoch and therefore we consume unnecessarily 119882

119894119863 units

of energy (this is the case depicted in Figure 1) Similarly if119878on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905) we waste 119882

119894(119884119894(119905) minus 119878

on119894(119905)) units of

energy Note that there is no risk of turning on the heater ifthe entrance to the zone happens sooner than it is predictedIn a similar fashion we calculate the risk of keeping the heateroff In this case

119877off119894(119905)

= 119862119894DP 119884

119894(119905) le 119878

on119894(119905)

+ 119862119894(119878

off119894(119905) minus 119884

119894(119905)) 119875 119878

on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905)

(3)

4 Journal of Energy

Yi(t)

Soni (t)

Soffi (t)

Ti

t

DTs

t + DTs

Tc

Figure 1 Predicted time intervals related to the decision process

Using (2) and (3) the decision criterion takes the form

119877on119894(119905)

on≶

off119877off119894(119905) (4)

Consider for the moment that there is no randomness on119884119894(119905) that is strict time scheduled occupation of zones

Typical examples are the classrooms in a university campusIn this case criterion (4) takes a very simple form since one ofthe involved probabilities is equal to one and the rest assumethe value of zero Thus if 119884

119894(119905) le 119878

on119894(119905) then the heater

is turned on if 119878off119894(119905) lt 119884

119894(119905) the heater remains off and

if 119878on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905) the heater will be turned onoff

depending on the values of the relative energy and discomfortcost The balance between these two costs is regulated by theweights119882

119894and 119862

119894

If 119884119894(119905) is random then criterion (4) needs to be slightly

modified as the value of 119884119894(119905) is unknown In this case we

replace 119884119894(119905) with its conditional expected value given that

119878on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905) Therefore in criterion (4) we

substitute 119884119894(119905) by 119864[119884

119894(119905) | 119878

on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905)]

From the previous discussion it becomes clear that thedecision process needs the estimates 119878on

119894(119905) 119878off119894(119905) and the

distribution of the process 119884119894(119905) The former is the subject of

the next subsection whereas a discussion on the distributionof 119884119894(119905) is left for Section 32

31 Temperature-Time Predictions Based on SID It is con-ceivable that in order to make predictions about a zonersquostemperature evolution we need a model able to capture thethermal dynamics of the tagged zone To this end we resort tothe deterministic subspace identification method Each zone

is treated separately and is represented by a discrete timelinear time invariant state-space model That is

119909119896+1

= 119860119909119896+ 119861119906119896+ 119908119896 (5)

119910119896= 119862119909119896+ 119863119906119896+ V119896 (6)

where 119910119896is the zone temperature at time 119896 and is in general

an ℓ-dimensional vector if ℓ temperature sensors scatteredin the zone have been deployed Assuming a uniform zonetemperaturewe use only one sensor node and thus119910

119896is scalar

V119896represents the measurement noise due to imperfections

of the sensor 119906119896is a vector of input measurements and its

dimensionality varies from zone to zoneThis vector containsthe measurable output of sources that influence zonersquos tem-perature Such sources are the power of a heater located inthe zone the air temperature at various positions outside thezone and so forth The vector 119909

119896is the state vector of the

process at the discrete time 119896 The states have a conceptualrelevance only and they are not assigned physical interpreta-tion Of course a similarity transform can convert the statesto physical meaningful onesThe unmeasurable vector signal119908119896represents noise due to the presence of humans in the

zone lampsrsquo switching on and off and so forth For the restof the paper we neglect the effects of state and output noiseand thus we deal with a deterministic identification problemwhich is the computation of the matrices 119860 119861 119862 and 119863

from the given input-output data The no noise assumptionmakes sense since we need temperature-time predictions forthe empty zones only which are ldquofreerdquo from occupants andother disturbances

In order to estimate the state-space matrices 119860 119861 119862 and119863 of each zone we need a training phase during which allthe nodes in the structure report their relative measurementsto a central computing system These measurements consistof the temperature of the zone and the current power of aheater if the sensor node is in charge of a heated zone oronly the temperature if the sensor node monitors an areaimmaterial to the decision process but relevant to other zonesthat is the exterior of a building The training phase shouldbe performed only once but under controlled conditions forexample no extra sources of heating and closed windowsMoreover the length of this period should be quite large inorder to exhibit several variations in the input signals to excitethemodes of the system In the simulation part we considereda training period of 24 h that is 86400 sampleswith samplingperiod 1 sec The sampling period of 1 sec is too small for allpractical reasons However as the dominant factor of perfor-mance degradation is the estimation of the residual arrivaltimes we consider an ideal temperature prediction model(high accuracy sensors and temperature noiseless zones) andwe access only the uncertainty of the occupation model

After receiving all the measurements the central systemorganizes them to input-output data for each zone Forexample if a zone is named A neighbors zones are namedB C and D and the exterior to the building zone is namedE then the temperature of zone A is the output of the systemto be identified whereas the temperatures of zones B C Dand E as well as the power profile of the heater of zone Aare the input data for the system A deterministic subspace

Journal of Energy 5

identification process described in the following paragraphsis then run for each zone The relevant matrices 119860 119861 119862 and119863 of the state-space model of each zone are communicatedback to the sensor node responsible for the monitoring of thezone

Following the notation and the derivation in [10] wedefine the block Hankel matrices

119880119901= 1198800|119894minus1

=(

(

1199060

1199061sdot sdot sdot 119906

119895minus1

1199061

1199062sdot sdot sdot 119906

119895

119906119894minus1

119906119894sdot sdot sdot 119906119894+119895minus2

)

)

119880119891= 119880119894|2119894minus1

=(

(

119906119894

119906119894+1

sdot sdot sdot 119906119894+119895minus1

119906119894+1

119906119894+2

sdot sdot sdot 119906119894+119895

1199062119894minus1

1199062119894

sdot sdot sdot 1199062119894+119895minus2

)

)

(7)

which represent the ldquopastrdquo and the ldquofuturerdquo input with respectto the present time instant 119894 Stacking119880

119901on top of119880

119891we have

the block Hankel matrix

1198800|2119894minus1

= (1198800|119894minus1

119880119894|2119894minus1

) = (

119880119901

119880119891

) (8)

which can also be partitioned as

1198800|2119894minus1

= (1198800|119894

119880119894+1|2119894minus1

) = (

119880+

119901

119880minus

119891

) (9)

by moving the ldquopresentrdquo time 119894 one step ahead to time 119894 + 1Similarly we define the output block Hankel matrices 119884

0|2119894minus1

119884119901 119884119891 119884+119901 119884minus119891and the input-output data matrices

119882119901= 1198820|119894minus1

= (1198800|119894minus1

1198840|119894minus1

) = (

119880119901

119884119901

)

119882+

119901= (

119880+

119901

119884+119901

)

(10)

The state sequence 119883119898is defined as

119883119898= (119909119898 119909

119898+1sdot sdot sdot 119909119898+119895minus1) (11)

and the ldquopastrdquo and ldquofuturerdquo state sequences are 119883119901= 1198830

and 119883119891= 119883119894 respectively With the state-space model (13)

(15) we associate the extended observability matrix Γ119894and the

reversed extended controllability matrix Δ119894 where

Γ119894=

((((

(

119862

119862119860

1198621198602

119862119860119894minus1

))))

)

Δ119894= (119860119894minus1119861 119860119894minus2119861 sdot sdot sdot 119860119861 119861)

(12)

The subspace identificationmethod relies on determining thestate sequence 119883

119891and the extended observability matrix Γ

119894

directly from the input-output data 119906119896119910119896and based on them

in a next step to extract the matrices 119860 119861 119862 and 119863 To thisend we define the oblique projection of the row space of 119884

119891

along the row space of 119880119891on the row space of119882

119901

O119894= 119884119891119880119891119882119901= [119884119891119880perp

119891] [119882119901119880perp

119891]dagger

119882119901 (13)

where dagger denotes the Moore-Penrose pseudoinverse of amatrix and 119860119861perp is a shorthand for the projection of the rowspace ofmatrix119860 onto the orthogonal complement of the rowspace of matrix 119861 that is

119860119861perp= 119868 minus 119860119861

119879(119861119861119879)dagger

119861 (14)

As it is proved in [10] the matrix O119894is equal to the product

of the extended observability matrix and the ldquofuturerdquo statevector

O119894= Γ119894119883119891 (15)

Having in our disposal the matrixO119894we use its singular value

decomposition

O119894= (1198801 119880

2) (

11987810

0 0)(

119881119879

1

119881119879

2

) = 11988011198781119881119879

1(16)

and we identify the extended observation matrix Γ119894and the

state vector119883119891as

Γ119894= 119880111987812

1119879

119883119891= 119879minus111987812

1119881119879

1= Γdagger

119894O119894

(17)

where 119879 is an 119899times119899 arbitrary nonsingular matrix representinga similarity transformation

One method [10] to extract the matrices 119860 119861 119862 and 119863uses in addition the oblique projection

O119894minus1

= 119884minus

119891119880minus119891119882dagger

119901= Γ119894minus1119883119894+1 (18)

where Γ119894minus1

is obtained from Γ119894by deleting the last ℓ rows (ℓ = 1

in our case) Similar to the previous derivation we have

119883119894+1

= Γdagger

119894minus1O119894minus1

(19)

and the matrices 119860 119861 119862 and 119863 are obtained by solving (inleast squares fashion) the system

(

119883119894+1

119884119894|119894

) = (

119860 119861

119862 119863)(

119883119894

119880119894|119894

) (20)

After completion of the training phase the state-spacemodel matrices 119860 119861 119862 and 119863 for each zone are commu-nicated back to the node responsible for the zone Uponreception of the matrices the nodes enter a ldquoconvergencerdquophase that is starting from an initial state 119909

0 they update

6 Journal of Energy

the state (5) for a predetermined period of time using themeasurements of the surrounding zones and the zonersquos heaterstatus Note that the training and the convergence phaseneed to be executed only once prior to the normal operationof the nodes After the convergence phase the nodes enterthe ldquodecisionrdquo phase During this phase the nodes updatethe state of the model (as in the convergence phase) andperiodically they take decisions on whether to turn the zonersquosheater on or not Suppose that at the decision epoch 119905 thestate of the tagged unoccupied node is 119909

119905 The node bases its

decision on the estimates 119878on119894(119905) and 119878off

119894(119905) Recall that 119878on

119894(119905)

is the time needed to reach the comfort level 119879119888if the heater

is on That is

119878on119894(119905) = min ℓ 119910

119905+ℓge 119879119888 (21)

The node ldquofreezesrdquo the input-vector 119906119896to its current value 119906

119905

(more recent temperatures of the surrounding zones and thetagged zonersquos heater is on) and starting from state 119909

119905repeats

(5) (6) until 119910119905+119896

exceeds the value 119879119888 Note that

119910119905+119896

= 119862119860119896119909119905+ 119862119860119896minus1

119861119906119905+ 119862119860119896minus2

119861119906119905+1

+ sdot sdot sdot

+ 119862119860119861119906119905+119896minus2

+ 119862119861119906119905+119896minus1

+ 119863119906119905+119896

(22)

and for 119906119905= 119906119905+1

= sdot sdot sdot = 119906119905+119896

we obtain

119910119905+119896

= 119862119860119896119909119905+ 119862 (119868 + 119860 + sdot sdot sdot + 119860

119896minus1) 119861119906119905+ 119863119906119905

= 119862119860119896119909119905+ 119862 (119868 minus 119860)

minus1(119868 minus 119860

119896) 119861119906119905+ 119863119906119905

(23)

If we set 119910119905+119896

= 119879119888and

119908 = 119879119888minus 119862 (119868 minus 119860)

minus1119861119906119905minus 119863119906119905 (24)

then (23) takes the form

119908 = 119862119860119896119909119905minus 119862 (119868 minus 119860)

minus1119860119896119861119906119905

= 119862119881Λ119896119881minus1119909119905minus 119862 (119868 minus 119860)

minus1119881Λ119896119881minus1119861119906119905

(25)

where we have considered the diagonalization of the matrix119860 as 119860 = 119881Λ119881

minus1 with Λ being the diagonal matrix of theeigenvalues and 119881 being the matrix of the correspondingeigenvectors Numerical methods can be used to solve (25)for 119896 In a similar manner we compute 119878off

119894(119905) In this case

however we first repeat (5) 119863 times with input-vector 119906119896

reflecting the fact that the heater is off and then starting fromthe new state 119909

119905+119863we estimate the time needed to reach 119879

119888if

the heater is on

32 On the Distribution of 119884119899(119905) The random variable 119884

119899(119905)

expresses the remaining time from the current time instant 119905until entrance to zone 119911

119899 That is we assume that occupants

move around as a swarm visiting zones either in a prede-termined order or in a random fashion and finally they endup in zone 119911

119899 Each time the occupants visit a zone 119911

ℓthey

znz120001

Figure 2 Movement of occupants from zone 119911ℓto the adjacent

unoccupied zone 119911119899

stay in it for a nonzero time period which is itself a ran-dom variable with distribution 119865

ℓ(119909) We use the variable 119905

to designate calendar time the time elapsed since the processstarted and 120591 to designate the time elapsed since entry to azone The zone occupation time distribution 119865

ℓ(119909) may be

given parametrically either a priori or inferred from exam-ination of historic data by suitable fitting of the distributionrsquosparameters We may use the empirical distribution func-tion (edf) instead which is the nonparametric estimate of119865ℓ(119909) That is if a sample of119873 occupation periods of zone 119911

ℓ

1199091 1199092 119909

119873 is available then

ℓ(119909) =

1

119873

119873

sum

119898=1

119868 (119909119898isin [0 119909]) (26)

where 119868(sdot) is the indicator functionLet us consider first the simple case of two adjusting zones

119911ℓ 119911119899with zone 119911

ℓbeing occupied and 119911

119899unoccupied as it is

shown in Figure 2 We assume that the occupants remain tozone 119911

ℓfor a random period of time 119883

ℓwhich is distributed

according to 119865ℓ(119909) and then upon departure they enter zone

119911119899 In this case 119884

119899(119905) expresses the remaining waiting time

to zone 119911ℓand follows the residual life distribution which is

defined by

119877119884119899(119905)

(119910) = 1 minus 119877119884119899(119905)

(119910) = 119875 119883ℓgt 120591 + 119910 | 119883

ℓgt 120591

=119865ℓ(120591 + 119910)

119865ℓ(120591)

(27)

Next we consider the case that occupants are currently ina zone (call it 119911

0) and upon departure they visit in cascade

zones 1199111 1199112 119911

119899as it is shown in Figure 3 Then

119884119899(119905) = 119884

0+ 1198831+ 1198832+ sdot sdot sdot + 119883

119899minus1 (28)

where 1198840is the remaining time in zone 119911

0and 119883

119894is the

occupation period of zone 119911119894 It is well known that the

distribution of 119884119899(119905) is the convolution of the distributions of

the random variables 1198840 1198831 119883

119899minus1 That is

119865119884119899(119905)

(119910) = 119875 119884119899(119905) le 119910

= 1198770(119910) ⋆ 119865

1(119910) ⋆ 119865

2(119910) ⋆ sdot sdot sdot ⋆ 119865

119899minus1(119910)

(29)

with 1198770(119910) = 1 minus 119865

0(120591 + 119910)119865

0(120591) A more compact form is

obtained by considering a transform of the distributions that

Journal of Energy 7

zn

z0 z1

Figure 3 Movement of occupants in cascade from zone 1199110to zone

119911119899

is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865

119883(119909) of a nonnega-

tive random variable119883 is defined by

119865119883(119904) = int

infin

0

119890119904119909119889119865119883(119909) (30)

(29) takes the form

119865119884119899(119905)

(119904) = 1198770(119904) sdot 119865

1(119904) sdot 119865

2(119904) sdot sdot 119865

119899minus1(119904) (31)

In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865

119884119899(119905)(119910)

For example if the occupation periods 119883119894 119894 = 0 119899 minus 1

are exponentially distributed with corresponding parameters120582119894 then 119884

0is also exponentially distributed with parameter

1205820due to the memoryless property of the exponential

distribution and the probability density function of 119884119899(119905) is

(assuming that the parameters 120582119894are all distinct)

119891119884119899(119905)

(119909) =

119899minus1

sum

119894=0

1205820sdot sdot sdot 120582119899minus1

prod119899minus1

119895=0119895 =119894(120582119895minus 120582119894)

exp (minus119909120582119894) (32)

(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891

119884119899(119905)(119909) does not exist or it is difficult to be

obtained we resort to the saddlepoint approximation [11] toinvert 119865

119884119899(119905)(119904) To this end for a random variable 119883 with

cumulative distribution function (cdf) 119865119883(119909) we define the

cumulant generating function (cgf) 119870119883(119904) as the logarithm

of the corresponding mfg 119865119883(119904) that is

119870119883(119904) = log119865

119883(119904) (33)

Then the saddlepoint approximation of the probability den-sity function (pdf) 119891

119883(119909) is

119891119883(119909)

asymp (1

212058711987010158401015840119909( (119909))

)

12

exp 119870119883( (119909)) minus (119909) 119909

(34)

where (119909) is the solution to the saddlepoint equation

1198701015840

119883(119904) = 119909 (35)

Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865

119883(119904) Note that wemay

use the empirical mgf 119883(119904) instead so that a nonparametric

estimation of the pdf 119891119883(119909) is possible

The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901

119894119895 Given a transition from zone 119911

119894to

zone 119911119895 the occupation time in 119911

119894has distribution function

119865119894119895(119909) The matrix Q(119909) with elements 119876

119894119895(119909) = 119901

119894119895119865119894119895(119909)

is the so called semi-Markov kernel The occupation timedistribution in zone 119911

119894independent of the state to which a

transition is made is 119865119894(119909) = sum

119895119876119894119895(119909) Consider now the

case that at time 119905 occupants exist in zone 119911119894and the node of

zone 119911119899has tomake a decision onwhether to start heating the

zone or not For the decision process the distribution of timeuntil entrance to zone 119911

119899 119884119899(119905) is needed 119884

119899(119905) is the sum

of two terms The first term is the residual occupation timeat zone 119911

119894and the second term is the time to reach 119911

119899upon

departure from zone 119911119894 The distribution 119877

119894(119909) of the first

term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911

119894

Thus if zone 119911119894neighbors119873

119894zones namely 119911

119896 119896 = 1 119873

119894

then the time to reach zone 119911119899upon departure is distributed

according to

119866119894119899(119909) =

119873119894

sum

119896=1

119901119894119896119867119896119899(119909) (36)

where119867119896119899(119909) is the distribution of the first passage from zone

119911119896to zone 119911

119899 Pyke [12] andMason [13 14] provided solutions

for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867

119894119895(119909) is given by

119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]

minus1)119889minus1

(37)

where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865

119894119895(119909) Notation 119872

119889denotes the

matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ

119894119895= (minus1)

119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of

119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get

1198671119899(119904) =

Δ1198991(119904)

Δ119899119899(119904)

(38)

By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899

8 Journal of Energy

I

III

II

IV

V

IXVI

VIII

VII

To

To

To

To

Figure 4 Multizone model

In summary

119865119884119899(119905)

(119904) = 119877119894(119904) sdot 119866

119894119899(119904) = 119877

119894(119904) sdot (

119873119894

sum

119896=1

119901119894119896119867119896119899(119904)) (39)

with 119867119896119899(119904) computed using (38) Next the saddlepoint

approximation technique can be used to calculate the density119891119884119899(119905)

(119909)We have tomention that the process of estimating the first

passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865

119894119895(119904)

is obtained Having in our disposal the transforms 119865119894119895(119904) we

can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867

119894119895(119904) using (38) Finally numerical

methods may be used to invert these transforms in order toobtain the necessary probability density functions

4 Simulation Results

We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])

Table 1 Training and testing parameters for outer temperaturemodel and heat gain

Walterrsquos model Heat gainPhase 119879max 119879min 119879

119898119902(119905)

Training 15 2 7 600Testing 16 1 6 700

Table 2 Training and testing zonesrsquo target temperatures

Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21

5 10 15 20 250Time of day (h)

0

5

10

15

Tem

pera

ture

To

(∘C)

Figure 5 Empirical estimation of 119879119900

In Figure 4 119879119900denotes the outside temperature which

is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879

119898

need to be provided in order to estimate 119879119900 As an example if

119879max = 15∘C 119879min = 2

∘C and 119879119898= 7∘C the daily variations

of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm

41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879

119904=

1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892

to those of Table 2 (first row) with a margin equal to119879119898119892

= 1∘C That is for a specific zone the heater is on until

temperature119879119905119892+119879119898119892

is reached After this point the heater isturned off until temperature hits the lower threshold119879

119905119892minus119879119898119892

and the whole process is repeated

Journal of Energy 9

0 1 2 3 4 5 6 7 8 9 10 11Order

Zone VZone I

Zone II

10minus15

10minus10

10minus5

100

Sing

ular

val

ues

Figure 6 Singular values of zones I II and V

Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2

Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase

We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

02

4

6

8

10

12

14

16

18

20

Time (sec)

Zone I (real)

Zone I (predicted)

Zone IV (real)

Zone IV (predicted)

Tem

pera

ture

Figure 7 Real and predicted temperature of zones I and IV

zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model

Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is

119891119883(119909) =

120573120572

Γ (120572)119909120572minus1

119890minus120573119909

(40)

with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862

119894

The results are averages over 200 runs of the simulation Thecase of 119862

119894= 0 is the full reactive case where heating of

zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862

119894 the decision process

acts proactively by spending energy in order to reduce thediscomfort level

Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone

IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases

10 Journal of Energy

0

100

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 8 Total comfort cost versus119862119894 The occupation time of zone

I is Gamma distributed

15

2

25

3

Tota

l ene

rgy

cost

(kW

h)

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 9 Total energy cost versus 119862119894 The occupation time of zone

I is Gamma distributed

In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty

The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is

119891119883(119909) =

120572119909120572

119898

119909120572+1 119909 ge 119909

119898

0 119909 lt 119909119898

(41)

Zone I

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Zone IV(C = 0)

Zone IV(C = 10000)

Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0

1000 2000 3000 4000 5000 6000 70000Time (sec)

0

05

1

15

2

25

3

35Pa

reto

pdf

times10

minus3

xm = 300

xm = 900

xm = 1800

xm = 3600

Figure 11 Pareto pdf for various values of 119909119898

with expected value

119864 [119883] =

120572119909119898

120572 minus 1 120572 gt 1

infin 120572 le 1

(42)

Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=

300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases

Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862

119894 As it is observed a 50 improvement on the

comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909

119898= 300) which is due

to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909

119898

Journal of Energy 11

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

xm = 300xm = 900

xm = 3600

Figure 12 Total comfort cost for Pareto distributed occupationperiod

xm = 300

xm = 900

xm = 1800

xm = 3600

1

15

2

25

Tota

l ene

rgy

cost

(kW

h)

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

Figure 13 Total energy cost for Pareto distributed occupationperiod

and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13

Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution

Table 3 Parameters used in cascade movement scenario

Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20

Table 4 Energy and comfort costs for constant occupancy times

Comfort weight 119862 500 1000 1500 2000

119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020

119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025

I

IV

V

II

III VI

VII

VIII

IX

Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)

The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on

Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess

12 Journal of Energy

Fixed proactivecomfort cost

Energy cost

Comfort cost

1000

1500

2000

2500

3000

3500

4000

4500

Tota

l com

fort

cost

600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)

8

85

9

95

10

105

11

Tota

l ene

rgy

cost

(kW

h)

proactiveFixed

energy cost

Figure 15 Total energy and comfort costs (cascade movement sce-nario)

Zone I

Zone IX

Zone II

Zone VIII

Zone V

Zone IV

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Figure 16 Temperature profiles of zones (cascade movement sce-nario)

Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process

5 Conclusions and Future Work

Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone

to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique

Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862

119894may depend

on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov

[2] US Energy Information Administration June 2015 httpwwweiagov

[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013

[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013

[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011

[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013

[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014

[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014

[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015

Journal of Energy 13

[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996

[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954

[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961

[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953

[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956

[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005

[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015

[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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2 Journal of Energy

the zones and (b) the zonesrsquo occupancy profile The emphasisin this paper is on the zonesrsquo temperature predictions andto this end a deterministic subspace identification methodis used for modelling the thermal dynamics of each zoneThat is each zone is modelled by a simple state-space modelcapable of producing accurate predictions based on thesurrounding temperatures the heating power of the zone andthe current state that summarizes the temperature history ofthe zone For the zonesrsquo occupancy predictions we consider asemi-Markov model where occupants (moving as a swarm)stay in a zone for a random period of time and then moveto adjacent zones with given probabilities What is needed bythe decision process is the distribution of the first entrancetime to unoccupied zones Aiming at a proactive action theproposedmethod periodically computes the risk of activatingthe heater or not and decides in favor of the action thatproduces the smaller risk The computation of the risks relieson the relative weights of the energy and discomfort costs sothat the balance between the total energy consumed and thetotal discomfort cost may be regulated

The structure of the paper is as follows In Section 2 priorand related work of the field of the demand-driven HVACsystems that utilize occupancy and prediction methods ispresented In Section 3 the proposed comfort-aware energyefficient mechanism is described along with the subspaceidentification method used for obtaining temperature-timepredictions and a discussion on the distribution of thefirst entrance time to unoccupied zones Section 4 containssimulation results and assesses the effectiveness of the pro-posed algorithm Finally Section 5 summarizes the paper andproposes research directions for future work

2 Prior Work

The necessity of demand-driven HVAC systems for energyefficient solutions has orientated the researchers towardsoccupancy-based activated systems In multizone spacesthe reactive and proactive activation of the heatingcoolingof the zones contributes to significant energy savings andimproves the thermal comfort for the occupants Severalresearch works with valuable results of energy savings haveutilized the occupancy detection and prediction in order tocontrol the HVAC system appropriately For the occupancydetection several types of sensors are used (CO

2 motion

etc) while for the prediction combinational systems areusually applied utilizing mathematical predictive models(eg Markov chains) alongside with detected real data

The authors in [3] proposed an automatic thermostatcontrol system which is based on an occupancy predictionscheme that predicts the destination and arrival time of theoccupants in the air-conditioned areas in order to providea comfortable environment For the occupantsrsquo mobilityprediction the cell tower information of the mobile tele-phony system is utilized and the arrival time prediction isbased on historical patterns and route classification For thedestination prediction of locations very close to each other(intracell) the time-aided order-Markov predictor is usedThe authors in [4] proposed a closed-loop system for opti-mally controllingHVACsystems in buildings based on actual

occupancy levels named the Power-Efficient Occupancy-Based Energy Management (POEM) System In order toaccurately detect occupantsrsquo transition they deployed a wire-less network comprising two parts the occupancy estimationsystem (OPTNet) consisting of 22 camera nodes and a passiveinfrared (PIR) sensors system (BONet) By fusing the sensingdata from the WSN (OPTNet and BONet) with the outputof an occupancy transition model in a particle filter moreaccurate estimation of the current occupancy in each room isachieved Then according to the current occupancy in eachroom and the predicted one from the transition model acontrol schedule of the HVAC system takes over the preheatof the areas to the target temperature In [5] the authorsused real world data gathered from a wireless network of16 smart cameras called Smart Camera Occupancy PositionEstimation System (SCOPES) and in this way they developedoccupancy models Three types of Markov chain (MC)occupancy models were tested and these are the single MCthe closest distance and finally the blended (BMC) Theauthors concluded that BMC is the most efficient and thusthey embodied it as the occupancy predictionmethod in theirproposed ldquoOBSERVErdquo algorithm which is a temperaturecontrol strategy for HVAC systems

The authors in [6] have developed an integrated systemcalled SENTINEL that is a control system for HVAC systemsutilizing occupancy information For the occupancy detec-tion and localisation the system utilizes the existing Wi-Fi network and the clientsrsquo smart phones The occupancylocalization mechanism is based on the access point (AP)communication with a clientrsquos smart phone so that if anoccupantrsquos phone sends packets to an AP then he is locatedwithin the range of the AP By classifying the building areasinto twomain categories namely the personal and the sharedspaces the proposed mechanism activates the HVAC systemwhen an owner of a personal space (eg office) has beendetectedwithin an areawhere hisher office is located orwhenoccupants are detected within shared spaces In [7] an inte-grated heating and cooling control systemof a building is pre-sented aiming to reduce energy consumptionThe occupancybehavior prediction as well as the weather forecast as inputsto a virtual (software based) building model determinesthe control of the HVAC system The occupancy detectiontechnique utilizes Gaussian Mixture Models (GMM) for thecategorization of selected features yielding the highest infor-mation gain according to the different number of occupantsThis categorization was used for observation to a HiddenMarkovModel for the estimation of the number of occupantsA semi-Markov model was developed based on patternscomprised by sensory data of CO

2 acoustics motion and

light changes to estimate the duration of occupants in thespace The work in [8] proposes a model predictive control(MPC) technique aiming to reduce energy consumption ina HVAC system while maintaining comfortable environmentfor the occupants The occupancy predictive model is basedon the two-state Markov chain with the states modelling theoccupied and occupied condition of the areas The authorsin [9] propose a feedback control algorithm for a variableair volume (VAV) HVAC system for full actuated (zonesconsisting of one room) and underactuated (zones consisting

Journal of Energy 3

of more than one room) zones The proposed algorithmis called MOBSua (Measured Occupancy-Based Setback forunderactuated zones) and it utilizes real-time occupancydata through a WSN for optimum energy efficiency andthermal comfort of the occupants Moreover the algorithmcan be applied on conventional control systems with no needof occupancy information and it is scalable to arbitrary sizedbuildings

3 Comfort-Aware Energy Efficient Mechanism

We consider a multizone HVAC system consisting of119885 zones119911119894 119894 = 1 2 119885 Each zone is equipped with a wireless

sensor node capable of conveying the zonersquos temperatureand occupancy information to its neighbors or to a centralprocessing unit The occupancy information may be verysimple that is binary information indicating the presence orabsence of individuals in the zone or more advanced like thenumber of occupants in the zoneTheobjective is tominimizethe total energy and discomfort cost defined as

E = Eenergy +Ediscomfort

=

119885

sum

119894=1

int

infin

minusinfin

119882119894119868 (119867119894= on) 119889119905

+

119885

sum

119894=1

int

infin

minusinfin

119862119894119868 (119879119894lt 119879119888) 119889119905

(1)

Note that only the heating problem is addressed in this paperThus the case of spending energy for cooling the zonesand keeping the temperature 119879

119894within a certain comfort

zone is not treated for simplicity In (1) 119868(119860) denotes theindicator function which takes the value 1 or 0 dependingon whether condition 119860 is true or false 119882

119894is the power

of a heater that covers zone 119911119894and 119867

119894is the state of the

heater that is the heater is on or off Thus the first termof (1) is the total energy consumed by the multizone systemFor the discomfort cost we define the cost per unit time 119862

119894

and the target comfort threshold 119879119888 As long as the zonersquos

temperature 119879119894is higher than 119879

119888the occupants do not feel

discomfort The parameter 119862119894may depend on the zone the

number of the occupants (eg the total systemrsquos discomfortcost is proportional to the number of people experiencingdiscomfort) and the difference of the zonersquos temperature 119879

119894

and the comfort threshold 119879119888 It is worth noting that there is

a tradeoff between energy savings and discomfort cost Wemay preheat all zones to the comfort level thus renderingthe discomfort cost equal to zero This policy is inefficientdue to the large amount of energy consumed In the otherextreme we could act reactively by heating a zone only upondetecting occupants in it In this case the energy cost wouldbe as small as possible but the discomfort cost could increasedramatically In the sequel we will describe a method thatacts proactively by taking periodically decisions on whetherto heat a zone or not If the entrance to a zone is delayedthen we postpone heating until the next decision epoch Ifon the contrary it is anticipated that a zone will be occupiedbefore the zonersquos temperature reaches the comfort level thenwe preheat the zone

The decision of whether or not the heater of an unoccu-pied zone 119911

119894should be turned on depends on (a) the current

state of the zone (b) the relative value of the energy cost (119882119894)

and discomfort cost (119862119894) per unit time (c) the temperatures

of the surrounding zones and (d) an estimate of the time thatthe zone will become occupied This decision may be takeneither centrally from the central processing unit that gathersinformation by all zones or in a distributed manner by eachzonersquos node after collecting the relative information There isalso the possibility of dividing the decision process function-ality between the central unit and the wireless sensor networknodes For example the prediction of the first entrancetimes to unoccupied zones may be taken by the central unitthat is aware of the location of people in the multizonesystem and then the predicted values may be communicatedback to the zonesrsquo nodes for the final decision The specificimplementation of the decision process is irrelevant to thispaper and it will not be further analyzed

We assume that time is discretized 119905 = 119898119879119904119879119904is the sam-

pling period and that each node takes its decisions periodi-cally every119863 sampling periods Note that there is no need forthe nodes to be synchronized to each other Let 119884

119894(119905) denote

the random variable thatmodels the remaining time from thecurrent time instant 119905 until entrance to the unoccupied zone119911119894 It is obvious that the randomvariables119884

119896(119905) for the various

unoccupied zones 119911119896 do not follow the same distributionWe

further assume that the node is capable ofmaking predictionsfor the time it takes to exceed the comfort threshold 119879

119888 To

this end let 119878on119894(119905) denote the time period that is required to

exceed 119879119888if the heater is turned on immediately Similarly

we define 119878off119894(119905) as the predicted time to reach the comfort

threshold if the heater remains off for a period 119863 and thenat the next decision epoch it is turned onThe relation of theaforementioned parameters is shown in Figure 1

Suppose now that at time 119905 the node of the unoccupiedzone 119911

119894has to take a decisionwhether to turn the zonersquos heater

on or remain in the off state At the decision epoch 119905 the riskto turn the heater on is119877on119894(119905)

= 119882119894DP 119878off

119894(119905) le 119884

119894(119905)

+ 119882119894(119884119894(119905) minus 119878

on119894(119905)) 119875 119878

on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905)

(2)

That is if 119878off119894(119905) le 119884

119894(119905) there is plenty of time to reach the

comfort threshold even if we start heating at the next decisionepoch and therefore we consume unnecessarily 119882

119894119863 units

of energy (this is the case depicted in Figure 1) Similarly if119878on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905) we waste 119882

119894(119884119894(119905) minus 119878

on119894(119905)) units of

energy Note that there is no risk of turning on the heater ifthe entrance to the zone happens sooner than it is predictedIn a similar fashion we calculate the risk of keeping the heateroff In this case

119877off119894(119905)

= 119862119894DP 119884

119894(119905) le 119878

on119894(119905)

+ 119862119894(119878

off119894(119905) minus 119884

119894(119905)) 119875 119878

on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905)

(3)

4 Journal of Energy

Yi(t)

Soni (t)

Soffi (t)

Ti

t

DTs

t + DTs

Tc

Figure 1 Predicted time intervals related to the decision process

Using (2) and (3) the decision criterion takes the form

119877on119894(119905)

on≶

off119877off119894(119905) (4)

Consider for the moment that there is no randomness on119884119894(119905) that is strict time scheduled occupation of zones

Typical examples are the classrooms in a university campusIn this case criterion (4) takes a very simple form since one ofthe involved probabilities is equal to one and the rest assumethe value of zero Thus if 119884

119894(119905) le 119878

on119894(119905) then the heater

is turned on if 119878off119894(119905) lt 119884

119894(119905) the heater remains off and

if 119878on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905) the heater will be turned onoff

depending on the values of the relative energy and discomfortcost The balance between these two costs is regulated by theweights119882

119894and 119862

119894

If 119884119894(119905) is random then criterion (4) needs to be slightly

modified as the value of 119884119894(119905) is unknown In this case we

replace 119884119894(119905) with its conditional expected value given that

119878on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905) Therefore in criterion (4) we

substitute 119884119894(119905) by 119864[119884

119894(119905) | 119878

on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905)]

From the previous discussion it becomes clear that thedecision process needs the estimates 119878on

119894(119905) 119878off119894(119905) and the

distribution of the process 119884119894(119905) The former is the subject of

the next subsection whereas a discussion on the distributionof 119884119894(119905) is left for Section 32

31 Temperature-Time Predictions Based on SID It is con-ceivable that in order to make predictions about a zonersquostemperature evolution we need a model able to capture thethermal dynamics of the tagged zone To this end we resort tothe deterministic subspace identification method Each zone

is treated separately and is represented by a discrete timelinear time invariant state-space model That is

119909119896+1

= 119860119909119896+ 119861119906119896+ 119908119896 (5)

119910119896= 119862119909119896+ 119863119906119896+ V119896 (6)

where 119910119896is the zone temperature at time 119896 and is in general

an ℓ-dimensional vector if ℓ temperature sensors scatteredin the zone have been deployed Assuming a uniform zonetemperaturewe use only one sensor node and thus119910

119896is scalar

V119896represents the measurement noise due to imperfections

of the sensor 119906119896is a vector of input measurements and its

dimensionality varies from zone to zoneThis vector containsthe measurable output of sources that influence zonersquos tem-perature Such sources are the power of a heater located inthe zone the air temperature at various positions outside thezone and so forth The vector 119909

119896is the state vector of the

process at the discrete time 119896 The states have a conceptualrelevance only and they are not assigned physical interpreta-tion Of course a similarity transform can convert the statesto physical meaningful onesThe unmeasurable vector signal119908119896represents noise due to the presence of humans in the

zone lampsrsquo switching on and off and so forth For the restof the paper we neglect the effects of state and output noiseand thus we deal with a deterministic identification problemwhich is the computation of the matrices 119860 119861 119862 and 119863

from the given input-output data The no noise assumptionmakes sense since we need temperature-time predictions forthe empty zones only which are ldquofreerdquo from occupants andother disturbances

In order to estimate the state-space matrices 119860 119861 119862 and119863 of each zone we need a training phase during which allthe nodes in the structure report their relative measurementsto a central computing system These measurements consistof the temperature of the zone and the current power of aheater if the sensor node is in charge of a heated zone oronly the temperature if the sensor node monitors an areaimmaterial to the decision process but relevant to other zonesthat is the exterior of a building The training phase shouldbe performed only once but under controlled conditions forexample no extra sources of heating and closed windowsMoreover the length of this period should be quite large inorder to exhibit several variations in the input signals to excitethemodes of the system In the simulation part we considereda training period of 24 h that is 86400 sampleswith samplingperiod 1 sec The sampling period of 1 sec is too small for allpractical reasons However as the dominant factor of perfor-mance degradation is the estimation of the residual arrivaltimes we consider an ideal temperature prediction model(high accuracy sensors and temperature noiseless zones) andwe access only the uncertainty of the occupation model

After receiving all the measurements the central systemorganizes them to input-output data for each zone Forexample if a zone is named A neighbors zones are namedB C and D and the exterior to the building zone is namedE then the temperature of zone A is the output of the systemto be identified whereas the temperatures of zones B C Dand E as well as the power profile of the heater of zone Aare the input data for the system A deterministic subspace

Journal of Energy 5

identification process described in the following paragraphsis then run for each zone The relevant matrices 119860 119861 119862 and119863 of the state-space model of each zone are communicatedback to the sensor node responsible for the monitoring of thezone

Following the notation and the derivation in [10] wedefine the block Hankel matrices

119880119901= 1198800|119894minus1

=(

(

1199060

1199061sdot sdot sdot 119906

119895minus1

1199061

1199062sdot sdot sdot 119906

119895

119906119894minus1

119906119894sdot sdot sdot 119906119894+119895minus2

)

)

119880119891= 119880119894|2119894minus1

=(

(

119906119894

119906119894+1

sdot sdot sdot 119906119894+119895minus1

119906119894+1

119906119894+2

sdot sdot sdot 119906119894+119895

1199062119894minus1

1199062119894

sdot sdot sdot 1199062119894+119895minus2

)

)

(7)

which represent the ldquopastrdquo and the ldquofuturerdquo input with respectto the present time instant 119894 Stacking119880

119901on top of119880

119891we have

the block Hankel matrix

1198800|2119894minus1

= (1198800|119894minus1

119880119894|2119894minus1

) = (

119880119901

119880119891

) (8)

which can also be partitioned as

1198800|2119894minus1

= (1198800|119894

119880119894+1|2119894minus1

) = (

119880+

119901

119880minus

119891

) (9)

by moving the ldquopresentrdquo time 119894 one step ahead to time 119894 + 1Similarly we define the output block Hankel matrices 119884

0|2119894minus1

119884119901 119884119891 119884+119901 119884minus119891and the input-output data matrices

119882119901= 1198820|119894minus1

= (1198800|119894minus1

1198840|119894minus1

) = (

119880119901

119884119901

)

119882+

119901= (

119880+

119901

119884+119901

)

(10)

The state sequence 119883119898is defined as

119883119898= (119909119898 119909

119898+1sdot sdot sdot 119909119898+119895minus1) (11)

and the ldquopastrdquo and ldquofuturerdquo state sequences are 119883119901= 1198830

and 119883119891= 119883119894 respectively With the state-space model (13)

(15) we associate the extended observability matrix Γ119894and the

reversed extended controllability matrix Δ119894 where

Γ119894=

((((

(

119862

119862119860

1198621198602

119862119860119894minus1

))))

)

Δ119894= (119860119894minus1119861 119860119894minus2119861 sdot sdot sdot 119860119861 119861)

(12)

The subspace identificationmethod relies on determining thestate sequence 119883

119891and the extended observability matrix Γ

119894

directly from the input-output data 119906119896119910119896and based on them

in a next step to extract the matrices 119860 119861 119862 and 119863 To thisend we define the oblique projection of the row space of 119884

119891

along the row space of 119880119891on the row space of119882

119901

O119894= 119884119891119880119891119882119901= [119884119891119880perp

119891] [119882119901119880perp

119891]dagger

119882119901 (13)

where dagger denotes the Moore-Penrose pseudoinverse of amatrix and 119860119861perp is a shorthand for the projection of the rowspace ofmatrix119860 onto the orthogonal complement of the rowspace of matrix 119861 that is

119860119861perp= 119868 minus 119860119861

119879(119861119861119879)dagger

119861 (14)

As it is proved in [10] the matrix O119894is equal to the product

of the extended observability matrix and the ldquofuturerdquo statevector

O119894= Γ119894119883119891 (15)

Having in our disposal the matrixO119894we use its singular value

decomposition

O119894= (1198801 119880

2) (

11987810

0 0)(

119881119879

1

119881119879

2

) = 11988011198781119881119879

1(16)

and we identify the extended observation matrix Γ119894and the

state vector119883119891as

Γ119894= 119880111987812

1119879

119883119891= 119879minus111987812

1119881119879

1= Γdagger

119894O119894

(17)

where 119879 is an 119899times119899 arbitrary nonsingular matrix representinga similarity transformation

One method [10] to extract the matrices 119860 119861 119862 and 119863uses in addition the oblique projection

O119894minus1

= 119884minus

119891119880minus119891119882dagger

119901= Γ119894minus1119883119894+1 (18)

where Γ119894minus1

is obtained from Γ119894by deleting the last ℓ rows (ℓ = 1

in our case) Similar to the previous derivation we have

119883119894+1

= Γdagger

119894minus1O119894minus1

(19)

and the matrices 119860 119861 119862 and 119863 are obtained by solving (inleast squares fashion) the system

(

119883119894+1

119884119894|119894

) = (

119860 119861

119862 119863)(

119883119894

119880119894|119894

) (20)

After completion of the training phase the state-spacemodel matrices 119860 119861 119862 and 119863 for each zone are commu-nicated back to the node responsible for the zone Uponreception of the matrices the nodes enter a ldquoconvergencerdquophase that is starting from an initial state 119909

0 they update

6 Journal of Energy

the state (5) for a predetermined period of time using themeasurements of the surrounding zones and the zonersquos heaterstatus Note that the training and the convergence phaseneed to be executed only once prior to the normal operationof the nodes After the convergence phase the nodes enterthe ldquodecisionrdquo phase During this phase the nodes updatethe state of the model (as in the convergence phase) andperiodically they take decisions on whether to turn the zonersquosheater on or not Suppose that at the decision epoch 119905 thestate of the tagged unoccupied node is 119909

119905 The node bases its

decision on the estimates 119878on119894(119905) and 119878off

119894(119905) Recall that 119878on

119894(119905)

is the time needed to reach the comfort level 119879119888if the heater

is on That is

119878on119894(119905) = min ℓ 119910

119905+ℓge 119879119888 (21)

The node ldquofreezesrdquo the input-vector 119906119896to its current value 119906

119905

(more recent temperatures of the surrounding zones and thetagged zonersquos heater is on) and starting from state 119909

119905repeats

(5) (6) until 119910119905+119896

exceeds the value 119879119888 Note that

119910119905+119896

= 119862119860119896119909119905+ 119862119860119896minus1

119861119906119905+ 119862119860119896minus2

119861119906119905+1

+ sdot sdot sdot

+ 119862119860119861119906119905+119896minus2

+ 119862119861119906119905+119896minus1

+ 119863119906119905+119896

(22)

and for 119906119905= 119906119905+1

= sdot sdot sdot = 119906119905+119896

we obtain

119910119905+119896

= 119862119860119896119909119905+ 119862 (119868 + 119860 + sdot sdot sdot + 119860

119896minus1) 119861119906119905+ 119863119906119905

= 119862119860119896119909119905+ 119862 (119868 minus 119860)

minus1(119868 minus 119860

119896) 119861119906119905+ 119863119906119905

(23)

If we set 119910119905+119896

= 119879119888and

119908 = 119879119888minus 119862 (119868 minus 119860)

minus1119861119906119905minus 119863119906119905 (24)

then (23) takes the form

119908 = 119862119860119896119909119905minus 119862 (119868 minus 119860)

minus1119860119896119861119906119905

= 119862119881Λ119896119881minus1119909119905minus 119862 (119868 minus 119860)

minus1119881Λ119896119881minus1119861119906119905

(25)

where we have considered the diagonalization of the matrix119860 as 119860 = 119881Λ119881

minus1 with Λ being the diagonal matrix of theeigenvalues and 119881 being the matrix of the correspondingeigenvectors Numerical methods can be used to solve (25)for 119896 In a similar manner we compute 119878off

119894(119905) In this case

however we first repeat (5) 119863 times with input-vector 119906119896

reflecting the fact that the heater is off and then starting fromthe new state 119909

119905+119863we estimate the time needed to reach 119879

119888if

the heater is on

32 On the Distribution of 119884119899(119905) The random variable 119884

119899(119905)

expresses the remaining time from the current time instant 119905until entrance to zone 119911

119899 That is we assume that occupants

move around as a swarm visiting zones either in a prede-termined order or in a random fashion and finally they endup in zone 119911

119899 Each time the occupants visit a zone 119911

ℓthey

znz120001

Figure 2 Movement of occupants from zone 119911ℓto the adjacent

unoccupied zone 119911119899

stay in it for a nonzero time period which is itself a ran-dom variable with distribution 119865

ℓ(119909) We use the variable 119905

to designate calendar time the time elapsed since the processstarted and 120591 to designate the time elapsed since entry to azone The zone occupation time distribution 119865

ℓ(119909) may be

given parametrically either a priori or inferred from exam-ination of historic data by suitable fitting of the distributionrsquosparameters We may use the empirical distribution func-tion (edf) instead which is the nonparametric estimate of119865ℓ(119909) That is if a sample of119873 occupation periods of zone 119911

ℓ

1199091 1199092 119909

119873 is available then

ℓ(119909) =

1

119873

119873

sum

119898=1

119868 (119909119898isin [0 119909]) (26)

where 119868(sdot) is the indicator functionLet us consider first the simple case of two adjusting zones

119911ℓ 119911119899with zone 119911

ℓbeing occupied and 119911

119899unoccupied as it is

shown in Figure 2 We assume that the occupants remain tozone 119911

ℓfor a random period of time 119883

ℓwhich is distributed

according to 119865ℓ(119909) and then upon departure they enter zone

119911119899 In this case 119884

119899(119905) expresses the remaining waiting time

to zone 119911ℓand follows the residual life distribution which is

defined by

119877119884119899(119905)

(119910) = 1 minus 119877119884119899(119905)

(119910) = 119875 119883ℓgt 120591 + 119910 | 119883

ℓgt 120591

=119865ℓ(120591 + 119910)

119865ℓ(120591)

(27)

Next we consider the case that occupants are currently ina zone (call it 119911

0) and upon departure they visit in cascade

zones 1199111 1199112 119911

119899as it is shown in Figure 3 Then

119884119899(119905) = 119884

0+ 1198831+ 1198832+ sdot sdot sdot + 119883

119899minus1 (28)

where 1198840is the remaining time in zone 119911

0and 119883

119894is the

occupation period of zone 119911119894 It is well known that the

distribution of 119884119899(119905) is the convolution of the distributions of

the random variables 1198840 1198831 119883

119899minus1 That is

119865119884119899(119905)

(119910) = 119875 119884119899(119905) le 119910

= 1198770(119910) ⋆ 119865

1(119910) ⋆ 119865

2(119910) ⋆ sdot sdot sdot ⋆ 119865

119899minus1(119910)

(29)

with 1198770(119910) = 1 minus 119865

0(120591 + 119910)119865

0(120591) A more compact form is

obtained by considering a transform of the distributions that

Journal of Energy 7

zn

z0 z1

Figure 3 Movement of occupants in cascade from zone 1199110to zone

119911119899

is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865

119883(119909) of a nonnega-

tive random variable119883 is defined by

119865119883(119904) = int

infin

0

119890119904119909119889119865119883(119909) (30)

(29) takes the form

119865119884119899(119905)

(119904) = 1198770(119904) sdot 119865

1(119904) sdot 119865

2(119904) sdot sdot 119865

119899minus1(119904) (31)

In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865

119884119899(119905)(119910)

For example if the occupation periods 119883119894 119894 = 0 119899 minus 1

are exponentially distributed with corresponding parameters120582119894 then 119884

0is also exponentially distributed with parameter

1205820due to the memoryless property of the exponential

distribution and the probability density function of 119884119899(119905) is

(assuming that the parameters 120582119894are all distinct)

119891119884119899(119905)

(119909) =

119899minus1

sum

119894=0

1205820sdot sdot sdot 120582119899minus1

prod119899minus1

119895=0119895 =119894(120582119895minus 120582119894)

exp (minus119909120582119894) (32)

(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891

119884119899(119905)(119909) does not exist or it is difficult to be

obtained we resort to the saddlepoint approximation [11] toinvert 119865

119884119899(119905)(119904) To this end for a random variable 119883 with

cumulative distribution function (cdf) 119865119883(119909) we define the

cumulant generating function (cgf) 119870119883(119904) as the logarithm

of the corresponding mfg 119865119883(119904) that is

119870119883(119904) = log119865

119883(119904) (33)

Then the saddlepoint approximation of the probability den-sity function (pdf) 119891

119883(119909) is

119891119883(119909)

asymp (1

212058711987010158401015840119909( (119909))

)

12

exp 119870119883( (119909)) minus (119909) 119909

(34)

where (119909) is the solution to the saddlepoint equation

1198701015840

119883(119904) = 119909 (35)

Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865

119883(119904) Note that wemay

use the empirical mgf 119883(119904) instead so that a nonparametric

estimation of the pdf 119891119883(119909) is possible

The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901

119894119895 Given a transition from zone 119911

119894to

zone 119911119895 the occupation time in 119911

119894has distribution function

119865119894119895(119909) The matrix Q(119909) with elements 119876

119894119895(119909) = 119901

119894119895119865119894119895(119909)

is the so called semi-Markov kernel The occupation timedistribution in zone 119911

119894independent of the state to which a

transition is made is 119865119894(119909) = sum

119895119876119894119895(119909) Consider now the

case that at time 119905 occupants exist in zone 119911119894and the node of

zone 119911119899has tomake a decision onwhether to start heating the

zone or not For the decision process the distribution of timeuntil entrance to zone 119911

119899 119884119899(119905) is needed 119884

119899(119905) is the sum

of two terms The first term is the residual occupation timeat zone 119911

119894and the second term is the time to reach 119911

119899upon

departure from zone 119911119894 The distribution 119877

119894(119909) of the first

term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911

119894

Thus if zone 119911119894neighbors119873

119894zones namely 119911

119896 119896 = 1 119873

119894

then the time to reach zone 119911119899upon departure is distributed

according to

119866119894119899(119909) =

119873119894

sum

119896=1

119901119894119896119867119896119899(119909) (36)

where119867119896119899(119909) is the distribution of the first passage from zone

119911119896to zone 119911

119899 Pyke [12] andMason [13 14] provided solutions

for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867

119894119895(119909) is given by

119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]

minus1)119889minus1

(37)

where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865

119894119895(119909) Notation 119872

119889denotes the

matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ

119894119895= (minus1)

119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of

119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get

1198671119899(119904) =

Δ1198991(119904)

Δ119899119899(119904)

(38)

By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899

8 Journal of Energy

I

III

II

IV

V

IXVI

VIII

VII

To

To

To

To

Figure 4 Multizone model

In summary

119865119884119899(119905)

(119904) = 119877119894(119904) sdot 119866

119894119899(119904) = 119877

119894(119904) sdot (

119873119894

sum

119896=1

119901119894119896119867119896119899(119904)) (39)

with 119867119896119899(119904) computed using (38) Next the saddlepoint

approximation technique can be used to calculate the density119891119884119899(119905)

(119909)We have tomention that the process of estimating the first

passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865

119894119895(119904)

is obtained Having in our disposal the transforms 119865119894119895(119904) we

can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867

119894119895(119904) using (38) Finally numerical

methods may be used to invert these transforms in order toobtain the necessary probability density functions

4 Simulation Results

We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])

Table 1 Training and testing parameters for outer temperaturemodel and heat gain

Walterrsquos model Heat gainPhase 119879max 119879min 119879

119898119902(119905)

Training 15 2 7 600Testing 16 1 6 700

Table 2 Training and testing zonesrsquo target temperatures

Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21

5 10 15 20 250Time of day (h)

0

5

10

15

Tem

pera

ture

To

(∘C)

Figure 5 Empirical estimation of 119879119900

In Figure 4 119879119900denotes the outside temperature which

is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879

119898

need to be provided in order to estimate 119879119900 As an example if

119879max = 15∘C 119879min = 2

∘C and 119879119898= 7∘C the daily variations

of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm

41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879

119904=

1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892

to those of Table 2 (first row) with a margin equal to119879119898119892

= 1∘C That is for a specific zone the heater is on until

temperature119879119905119892+119879119898119892

is reached After this point the heater isturned off until temperature hits the lower threshold119879

119905119892minus119879119898119892

and the whole process is repeated

Journal of Energy 9

0 1 2 3 4 5 6 7 8 9 10 11Order

Zone VZone I

Zone II

10minus15

10minus10

10minus5

100

Sing

ular

val

ues

Figure 6 Singular values of zones I II and V

Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2

Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase

We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

02

4

6

8

10

12

14

16

18

20

Time (sec)

Zone I (real)

Zone I (predicted)

Zone IV (real)

Zone IV (predicted)

Tem

pera

ture

Figure 7 Real and predicted temperature of zones I and IV

zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model

Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is

119891119883(119909) =

120573120572

Γ (120572)119909120572minus1

119890minus120573119909

(40)

with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862

119894

The results are averages over 200 runs of the simulation Thecase of 119862

119894= 0 is the full reactive case where heating of

zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862

119894 the decision process

acts proactively by spending energy in order to reduce thediscomfort level

Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone

IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases

10 Journal of Energy

0

100

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 8 Total comfort cost versus119862119894 The occupation time of zone

I is Gamma distributed

15

2

25

3

Tota

l ene

rgy

cost

(kW

h)

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 9 Total energy cost versus 119862119894 The occupation time of zone

I is Gamma distributed

In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty

The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is

119891119883(119909) =

120572119909120572

119898

119909120572+1 119909 ge 119909

119898

0 119909 lt 119909119898

(41)

Zone I

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Zone IV(C = 0)

Zone IV(C = 10000)

Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0

1000 2000 3000 4000 5000 6000 70000Time (sec)

0

05

1

15

2

25

3

35Pa

reto

pdf

times10

minus3

xm = 300

xm = 900

xm = 1800

xm = 3600

Figure 11 Pareto pdf for various values of 119909119898

with expected value

119864 [119883] =

120572119909119898

120572 minus 1 120572 gt 1

infin 120572 le 1

(42)

Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=

300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases

Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862

119894 As it is observed a 50 improvement on the

comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909

119898= 300) which is due

to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909

119898

Journal of Energy 11

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

xm = 300xm = 900

xm = 3600

Figure 12 Total comfort cost for Pareto distributed occupationperiod

xm = 300

xm = 900

xm = 1800

xm = 3600

1

15

2

25

Tota

l ene

rgy

cost

(kW

h)

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

Figure 13 Total energy cost for Pareto distributed occupationperiod

and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13

Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution

Table 3 Parameters used in cascade movement scenario

Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20

Table 4 Energy and comfort costs for constant occupancy times

Comfort weight 119862 500 1000 1500 2000

119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020

119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025

I

IV

V

II

III VI

VII

VIII

IX

Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)

The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on

Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess

12 Journal of Energy

Fixed proactivecomfort cost

Energy cost

Comfort cost

1000

1500

2000

2500

3000

3500

4000

4500

Tota

l com

fort

cost

600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)

8

85

9

95

10

105

11

Tota

l ene

rgy

cost

(kW

h)

proactiveFixed

energy cost

Figure 15 Total energy and comfort costs (cascade movement sce-nario)

Zone I

Zone IX

Zone II

Zone VIII

Zone V

Zone IV

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Figure 16 Temperature profiles of zones (cascade movement sce-nario)

Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process

5 Conclusions and Future Work

Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone

to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique

Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862

119894may depend

on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov

[2] US Energy Information Administration June 2015 httpwwweiagov

[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013

[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013

[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011

[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013

[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014

[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014

[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015

Journal of Energy 13

[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996

[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954

[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961

[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953

[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956

[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005

[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015

[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967

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Renewable Energy

Submit your manuscripts athttpwwwhindawicom

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of Energy 3

of more than one room) zones The proposed algorithmis called MOBSua (Measured Occupancy-Based Setback forunderactuated zones) and it utilizes real-time occupancydata through a WSN for optimum energy efficiency andthermal comfort of the occupants Moreover the algorithmcan be applied on conventional control systems with no needof occupancy information and it is scalable to arbitrary sizedbuildings

3 Comfort-Aware Energy Efficient Mechanism

We consider a multizone HVAC system consisting of119885 zones119911119894 119894 = 1 2 119885 Each zone is equipped with a wireless

sensor node capable of conveying the zonersquos temperatureand occupancy information to its neighbors or to a centralprocessing unit The occupancy information may be verysimple that is binary information indicating the presence orabsence of individuals in the zone or more advanced like thenumber of occupants in the zoneTheobjective is tominimizethe total energy and discomfort cost defined as

E = Eenergy +Ediscomfort

=

119885

sum

119894=1

int

infin

minusinfin

119882119894119868 (119867119894= on) 119889119905

+

119885

sum

119894=1

int

infin

minusinfin

119862119894119868 (119879119894lt 119879119888) 119889119905

(1)

Note that only the heating problem is addressed in this paperThus the case of spending energy for cooling the zonesand keeping the temperature 119879

119894within a certain comfort

zone is not treated for simplicity In (1) 119868(119860) denotes theindicator function which takes the value 1 or 0 dependingon whether condition 119860 is true or false 119882

119894is the power

of a heater that covers zone 119911119894and 119867

119894is the state of the

heater that is the heater is on or off Thus the first termof (1) is the total energy consumed by the multizone systemFor the discomfort cost we define the cost per unit time 119862

119894

and the target comfort threshold 119879119888 As long as the zonersquos

temperature 119879119894is higher than 119879

119888the occupants do not feel

discomfort The parameter 119862119894may depend on the zone the

number of the occupants (eg the total systemrsquos discomfortcost is proportional to the number of people experiencingdiscomfort) and the difference of the zonersquos temperature 119879

119894

and the comfort threshold 119879119888 It is worth noting that there is

a tradeoff between energy savings and discomfort cost Wemay preheat all zones to the comfort level thus renderingthe discomfort cost equal to zero This policy is inefficientdue to the large amount of energy consumed In the otherextreme we could act reactively by heating a zone only upondetecting occupants in it In this case the energy cost wouldbe as small as possible but the discomfort cost could increasedramatically In the sequel we will describe a method thatacts proactively by taking periodically decisions on whetherto heat a zone or not If the entrance to a zone is delayedthen we postpone heating until the next decision epoch Ifon the contrary it is anticipated that a zone will be occupiedbefore the zonersquos temperature reaches the comfort level thenwe preheat the zone

The decision of whether or not the heater of an unoccu-pied zone 119911

119894should be turned on depends on (a) the current

state of the zone (b) the relative value of the energy cost (119882119894)

and discomfort cost (119862119894) per unit time (c) the temperatures

of the surrounding zones and (d) an estimate of the time thatthe zone will become occupied This decision may be takeneither centrally from the central processing unit that gathersinformation by all zones or in a distributed manner by eachzonersquos node after collecting the relative information There isalso the possibility of dividing the decision process function-ality between the central unit and the wireless sensor networknodes For example the prediction of the first entrancetimes to unoccupied zones may be taken by the central unitthat is aware of the location of people in the multizonesystem and then the predicted values may be communicatedback to the zonesrsquo nodes for the final decision The specificimplementation of the decision process is irrelevant to thispaper and it will not be further analyzed

We assume that time is discretized 119905 = 119898119879119904119879119904is the sam-

pling period and that each node takes its decisions periodi-cally every119863 sampling periods Note that there is no need forthe nodes to be synchronized to each other Let 119884

119894(119905) denote

the random variable thatmodels the remaining time from thecurrent time instant 119905 until entrance to the unoccupied zone119911119894 It is obvious that the randomvariables119884

119896(119905) for the various

unoccupied zones 119911119896 do not follow the same distributionWe

further assume that the node is capable ofmaking predictionsfor the time it takes to exceed the comfort threshold 119879

119888 To

this end let 119878on119894(119905) denote the time period that is required to

exceed 119879119888if the heater is turned on immediately Similarly

we define 119878off119894(119905) as the predicted time to reach the comfort

threshold if the heater remains off for a period 119863 and thenat the next decision epoch it is turned onThe relation of theaforementioned parameters is shown in Figure 1

Suppose now that at time 119905 the node of the unoccupiedzone 119911

119894has to take a decisionwhether to turn the zonersquos heater

on or remain in the off state At the decision epoch 119905 the riskto turn the heater on is119877on119894(119905)

= 119882119894DP 119878off

119894(119905) le 119884

119894(119905)

+ 119882119894(119884119894(119905) minus 119878

on119894(119905)) 119875 119878

on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905)

(2)

That is if 119878off119894(119905) le 119884

119894(119905) there is plenty of time to reach the

comfort threshold even if we start heating at the next decisionepoch and therefore we consume unnecessarily 119882

119894119863 units

of energy (this is the case depicted in Figure 1) Similarly if119878on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905) we waste 119882

119894(119884119894(119905) minus 119878

on119894(119905)) units of

energy Note that there is no risk of turning on the heater ifthe entrance to the zone happens sooner than it is predictedIn a similar fashion we calculate the risk of keeping the heateroff In this case

119877off119894(119905)

= 119862119894DP 119884

119894(119905) le 119878

on119894(119905)

+ 119862119894(119878

off119894(119905) minus 119884

119894(119905)) 119875 119878

on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905)

(3)

4 Journal of Energy

Yi(t)

Soni (t)

Soffi (t)

Ti

t

DTs

t + DTs

Tc

Figure 1 Predicted time intervals related to the decision process

Using (2) and (3) the decision criterion takes the form

119877on119894(119905)

on≶

off119877off119894(119905) (4)

Consider for the moment that there is no randomness on119884119894(119905) that is strict time scheduled occupation of zones

Typical examples are the classrooms in a university campusIn this case criterion (4) takes a very simple form since one ofthe involved probabilities is equal to one and the rest assumethe value of zero Thus if 119884

119894(119905) le 119878

on119894(119905) then the heater

is turned on if 119878off119894(119905) lt 119884

119894(119905) the heater remains off and

if 119878on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905) the heater will be turned onoff

depending on the values of the relative energy and discomfortcost The balance between these two costs is regulated by theweights119882

119894and 119862

119894

If 119884119894(119905) is random then criterion (4) needs to be slightly

modified as the value of 119884119894(119905) is unknown In this case we

replace 119884119894(119905) with its conditional expected value given that

119878on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905) Therefore in criterion (4) we

substitute 119884119894(119905) by 119864[119884

119894(119905) | 119878

on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905)]

From the previous discussion it becomes clear that thedecision process needs the estimates 119878on

119894(119905) 119878off119894(119905) and the

distribution of the process 119884119894(119905) The former is the subject of

the next subsection whereas a discussion on the distributionof 119884119894(119905) is left for Section 32

31 Temperature-Time Predictions Based on SID It is con-ceivable that in order to make predictions about a zonersquostemperature evolution we need a model able to capture thethermal dynamics of the tagged zone To this end we resort tothe deterministic subspace identification method Each zone

is treated separately and is represented by a discrete timelinear time invariant state-space model That is

119909119896+1

= 119860119909119896+ 119861119906119896+ 119908119896 (5)

119910119896= 119862119909119896+ 119863119906119896+ V119896 (6)

where 119910119896is the zone temperature at time 119896 and is in general

an ℓ-dimensional vector if ℓ temperature sensors scatteredin the zone have been deployed Assuming a uniform zonetemperaturewe use only one sensor node and thus119910

119896is scalar

V119896represents the measurement noise due to imperfections

of the sensor 119906119896is a vector of input measurements and its

dimensionality varies from zone to zoneThis vector containsthe measurable output of sources that influence zonersquos tem-perature Such sources are the power of a heater located inthe zone the air temperature at various positions outside thezone and so forth The vector 119909

119896is the state vector of the

process at the discrete time 119896 The states have a conceptualrelevance only and they are not assigned physical interpreta-tion Of course a similarity transform can convert the statesto physical meaningful onesThe unmeasurable vector signal119908119896represents noise due to the presence of humans in the

zone lampsrsquo switching on and off and so forth For the restof the paper we neglect the effects of state and output noiseand thus we deal with a deterministic identification problemwhich is the computation of the matrices 119860 119861 119862 and 119863

from the given input-output data The no noise assumptionmakes sense since we need temperature-time predictions forthe empty zones only which are ldquofreerdquo from occupants andother disturbances

In order to estimate the state-space matrices 119860 119861 119862 and119863 of each zone we need a training phase during which allthe nodes in the structure report their relative measurementsto a central computing system These measurements consistof the temperature of the zone and the current power of aheater if the sensor node is in charge of a heated zone oronly the temperature if the sensor node monitors an areaimmaterial to the decision process but relevant to other zonesthat is the exterior of a building The training phase shouldbe performed only once but under controlled conditions forexample no extra sources of heating and closed windowsMoreover the length of this period should be quite large inorder to exhibit several variations in the input signals to excitethemodes of the system In the simulation part we considereda training period of 24 h that is 86400 sampleswith samplingperiod 1 sec The sampling period of 1 sec is too small for allpractical reasons However as the dominant factor of perfor-mance degradation is the estimation of the residual arrivaltimes we consider an ideal temperature prediction model(high accuracy sensors and temperature noiseless zones) andwe access only the uncertainty of the occupation model

After receiving all the measurements the central systemorganizes them to input-output data for each zone Forexample if a zone is named A neighbors zones are namedB C and D and the exterior to the building zone is namedE then the temperature of zone A is the output of the systemto be identified whereas the temperatures of zones B C Dand E as well as the power profile of the heater of zone Aare the input data for the system A deterministic subspace

Journal of Energy 5

identification process described in the following paragraphsis then run for each zone The relevant matrices 119860 119861 119862 and119863 of the state-space model of each zone are communicatedback to the sensor node responsible for the monitoring of thezone

Following the notation and the derivation in [10] wedefine the block Hankel matrices

119880119901= 1198800|119894minus1

=(

(

1199060

1199061sdot sdot sdot 119906

119895minus1

1199061

1199062sdot sdot sdot 119906

119895

119906119894minus1

119906119894sdot sdot sdot 119906119894+119895minus2

)

)

119880119891= 119880119894|2119894minus1

=(

(

119906119894

119906119894+1

sdot sdot sdot 119906119894+119895minus1

119906119894+1

119906119894+2

sdot sdot sdot 119906119894+119895

1199062119894minus1

1199062119894

sdot sdot sdot 1199062119894+119895minus2

)

)

(7)

which represent the ldquopastrdquo and the ldquofuturerdquo input with respectto the present time instant 119894 Stacking119880

119901on top of119880

119891we have

the block Hankel matrix

1198800|2119894minus1

= (1198800|119894minus1

119880119894|2119894minus1

) = (

119880119901

119880119891

) (8)

which can also be partitioned as

1198800|2119894minus1

= (1198800|119894

119880119894+1|2119894minus1

) = (

119880+

119901

119880minus

119891

) (9)

by moving the ldquopresentrdquo time 119894 one step ahead to time 119894 + 1Similarly we define the output block Hankel matrices 119884

0|2119894minus1

119884119901 119884119891 119884+119901 119884minus119891and the input-output data matrices

119882119901= 1198820|119894minus1

= (1198800|119894minus1

1198840|119894minus1

) = (

119880119901

119884119901

)

119882+

119901= (

119880+

119901

119884+119901

)

(10)

The state sequence 119883119898is defined as

119883119898= (119909119898 119909

119898+1sdot sdot sdot 119909119898+119895minus1) (11)

and the ldquopastrdquo and ldquofuturerdquo state sequences are 119883119901= 1198830

and 119883119891= 119883119894 respectively With the state-space model (13)

(15) we associate the extended observability matrix Γ119894and the

reversed extended controllability matrix Δ119894 where

Γ119894=

((((

(

119862

119862119860

1198621198602

119862119860119894minus1

))))

)

Δ119894= (119860119894minus1119861 119860119894minus2119861 sdot sdot sdot 119860119861 119861)

(12)

The subspace identificationmethod relies on determining thestate sequence 119883

119891and the extended observability matrix Γ

119894

directly from the input-output data 119906119896119910119896and based on them

in a next step to extract the matrices 119860 119861 119862 and 119863 To thisend we define the oblique projection of the row space of 119884

119891

along the row space of 119880119891on the row space of119882

119901

O119894= 119884119891119880119891119882119901= [119884119891119880perp

119891] [119882119901119880perp

119891]dagger

119882119901 (13)

where dagger denotes the Moore-Penrose pseudoinverse of amatrix and 119860119861perp is a shorthand for the projection of the rowspace ofmatrix119860 onto the orthogonal complement of the rowspace of matrix 119861 that is

119860119861perp= 119868 minus 119860119861

119879(119861119861119879)dagger

119861 (14)

As it is proved in [10] the matrix O119894is equal to the product

of the extended observability matrix and the ldquofuturerdquo statevector

O119894= Γ119894119883119891 (15)

Having in our disposal the matrixO119894we use its singular value

decomposition

O119894= (1198801 119880

2) (

11987810

0 0)(

119881119879

1

119881119879

2

) = 11988011198781119881119879

1(16)

and we identify the extended observation matrix Γ119894and the

state vector119883119891as

Γ119894= 119880111987812

1119879

119883119891= 119879minus111987812

1119881119879

1= Γdagger

119894O119894

(17)

where 119879 is an 119899times119899 arbitrary nonsingular matrix representinga similarity transformation

One method [10] to extract the matrices 119860 119861 119862 and 119863uses in addition the oblique projection

O119894minus1

= 119884minus

119891119880minus119891119882dagger

119901= Γ119894minus1119883119894+1 (18)

where Γ119894minus1

is obtained from Γ119894by deleting the last ℓ rows (ℓ = 1

in our case) Similar to the previous derivation we have

119883119894+1

= Γdagger

119894minus1O119894minus1

(19)

and the matrices 119860 119861 119862 and 119863 are obtained by solving (inleast squares fashion) the system

(

119883119894+1

119884119894|119894

) = (

119860 119861

119862 119863)(

119883119894

119880119894|119894

) (20)

After completion of the training phase the state-spacemodel matrices 119860 119861 119862 and 119863 for each zone are commu-nicated back to the node responsible for the zone Uponreception of the matrices the nodes enter a ldquoconvergencerdquophase that is starting from an initial state 119909

0 they update

6 Journal of Energy

the state (5) for a predetermined period of time using themeasurements of the surrounding zones and the zonersquos heaterstatus Note that the training and the convergence phaseneed to be executed only once prior to the normal operationof the nodes After the convergence phase the nodes enterthe ldquodecisionrdquo phase During this phase the nodes updatethe state of the model (as in the convergence phase) andperiodically they take decisions on whether to turn the zonersquosheater on or not Suppose that at the decision epoch 119905 thestate of the tagged unoccupied node is 119909

119905 The node bases its

decision on the estimates 119878on119894(119905) and 119878off

119894(119905) Recall that 119878on

119894(119905)

is the time needed to reach the comfort level 119879119888if the heater

is on That is

119878on119894(119905) = min ℓ 119910

119905+ℓge 119879119888 (21)

The node ldquofreezesrdquo the input-vector 119906119896to its current value 119906

119905

(more recent temperatures of the surrounding zones and thetagged zonersquos heater is on) and starting from state 119909

119905repeats

(5) (6) until 119910119905+119896

exceeds the value 119879119888 Note that

119910119905+119896

= 119862119860119896119909119905+ 119862119860119896minus1

119861119906119905+ 119862119860119896minus2

119861119906119905+1

+ sdot sdot sdot

+ 119862119860119861119906119905+119896minus2

+ 119862119861119906119905+119896minus1

+ 119863119906119905+119896

(22)

and for 119906119905= 119906119905+1

= sdot sdot sdot = 119906119905+119896

we obtain

119910119905+119896

= 119862119860119896119909119905+ 119862 (119868 + 119860 + sdot sdot sdot + 119860

119896minus1) 119861119906119905+ 119863119906119905

= 119862119860119896119909119905+ 119862 (119868 minus 119860)

minus1(119868 minus 119860

119896) 119861119906119905+ 119863119906119905

(23)

If we set 119910119905+119896

= 119879119888and

119908 = 119879119888minus 119862 (119868 minus 119860)

minus1119861119906119905minus 119863119906119905 (24)

then (23) takes the form

119908 = 119862119860119896119909119905minus 119862 (119868 minus 119860)

minus1119860119896119861119906119905

= 119862119881Λ119896119881minus1119909119905minus 119862 (119868 minus 119860)

minus1119881Λ119896119881minus1119861119906119905

(25)

where we have considered the diagonalization of the matrix119860 as 119860 = 119881Λ119881

minus1 with Λ being the diagonal matrix of theeigenvalues and 119881 being the matrix of the correspondingeigenvectors Numerical methods can be used to solve (25)for 119896 In a similar manner we compute 119878off

119894(119905) In this case

however we first repeat (5) 119863 times with input-vector 119906119896

reflecting the fact that the heater is off and then starting fromthe new state 119909

119905+119863we estimate the time needed to reach 119879

119888if

the heater is on

32 On the Distribution of 119884119899(119905) The random variable 119884

119899(119905)

expresses the remaining time from the current time instant 119905until entrance to zone 119911

119899 That is we assume that occupants

move around as a swarm visiting zones either in a prede-termined order or in a random fashion and finally they endup in zone 119911

119899 Each time the occupants visit a zone 119911

ℓthey

znz120001

Figure 2 Movement of occupants from zone 119911ℓto the adjacent

unoccupied zone 119911119899

stay in it for a nonzero time period which is itself a ran-dom variable with distribution 119865

ℓ(119909) We use the variable 119905

to designate calendar time the time elapsed since the processstarted and 120591 to designate the time elapsed since entry to azone The zone occupation time distribution 119865

ℓ(119909) may be

given parametrically either a priori or inferred from exam-ination of historic data by suitable fitting of the distributionrsquosparameters We may use the empirical distribution func-tion (edf) instead which is the nonparametric estimate of119865ℓ(119909) That is if a sample of119873 occupation periods of zone 119911

ℓ

1199091 1199092 119909

119873 is available then

ℓ(119909) =

1

119873

119873

sum

119898=1

119868 (119909119898isin [0 119909]) (26)

where 119868(sdot) is the indicator functionLet us consider first the simple case of two adjusting zones

119911ℓ 119911119899with zone 119911

ℓbeing occupied and 119911

119899unoccupied as it is

shown in Figure 2 We assume that the occupants remain tozone 119911

ℓfor a random period of time 119883

ℓwhich is distributed

according to 119865ℓ(119909) and then upon departure they enter zone

119911119899 In this case 119884

119899(119905) expresses the remaining waiting time

to zone 119911ℓand follows the residual life distribution which is

defined by

119877119884119899(119905)

(119910) = 1 minus 119877119884119899(119905)

(119910) = 119875 119883ℓgt 120591 + 119910 | 119883

ℓgt 120591

=119865ℓ(120591 + 119910)

119865ℓ(120591)

(27)

Next we consider the case that occupants are currently ina zone (call it 119911

0) and upon departure they visit in cascade

zones 1199111 1199112 119911

119899as it is shown in Figure 3 Then

119884119899(119905) = 119884

0+ 1198831+ 1198832+ sdot sdot sdot + 119883

119899minus1 (28)

where 1198840is the remaining time in zone 119911

0and 119883

119894is the

occupation period of zone 119911119894 It is well known that the

distribution of 119884119899(119905) is the convolution of the distributions of

the random variables 1198840 1198831 119883

119899minus1 That is

119865119884119899(119905)

(119910) = 119875 119884119899(119905) le 119910

= 1198770(119910) ⋆ 119865

1(119910) ⋆ 119865

2(119910) ⋆ sdot sdot sdot ⋆ 119865

119899minus1(119910)

(29)

with 1198770(119910) = 1 minus 119865

0(120591 + 119910)119865

0(120591) A more compact form is

obtained by considering a transform of the distributions that

Journal of Energy 7

zn

z0 z1

Figure 3 Movement of occupants in cascade from zone 1199110to zone

119911119899

is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865

119883(119909) of a nonnega-

tive random variable119883 is defined by

119865119883(119904) = int

infin

0

119890119904119909119889119865119883(119909) (30)

(29) takes the form

119865119884119899(119905)

(119904) = 1198770(119904) sdot 119865

1(119904) sdot 119865

2(119904) sdot sdot 119865

119899minus1(119904) (31)

In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865

119884119899(119905)(119910)

For example if the occupation periods 119883119894 119894 = 0 119899 minus 1

are exponentially distributed with corresponding parameters120582119894 then 119884

0is also exponentially distributed with parameter

1205820due to the memoryless property of the exponential

distribution and the probability density function of 119884119899(119905) is

(assuming that the parameters 120582119894are all distinct)

119891119884119899(119905)

(119909) =

119899minus1

sum

119894=0

1205820sdot sdot sdot 120582119899minus1

prod119899minus1

119895=0119895 =119894(120582119895minus 120582119894)

exp (minus119909120582119894) (32)

(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891

119884119899(119905)(119909) does not exist or it is difficult to be

obtained we resort to the saddlepoint approximation [11] toinvert 119865

119884119899(119905)(119904) To this end for a random variable 119883 with

cumulative distribution function (cdf) 119865119883(119909) we define the

cumulant generating function (cgf) 119870119883(119904) as the logarithm

of the corresponding mfg 119865119883(119904) that is

119870119883(119904) = log119865

119883(119904) (33)

Then the saddlepoint approximation of the probability den-sity function (pdf) 119891

119883(119909) is

119891119883(119909)

asymp (1

212058711987010158401015840119909( (119909))

)

12

exp 119870119883( (119909)) minus (119909) 119909

(34)

where (119909) is the solution to the saddlepoint equation

1198701015840

119883(119904) = 119909 (35)

Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865

119883(119904) Note that wemay

use the empirical mgf 119883(119904) instead so that a nonparametric

estimation of the pdf 119891119883(119909) is possible

The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901

119894119895 Given a transition from zone 119911

119894to

zone 119911119895 the occupation time in 119911

119894has distribution function

119865119894119895(119909) The matrix Q(119909) with elements 119876

119894119895(119909) = 119901

119894119895119865119894119895(119909)

is the so called semi-Markov kernel The occupation timedistribution in zone 119911

119894independent of the state to which a

transition is made is 119865119894(119909) = sum

119895119876119894119895(119909) Consider now the

case that at time 119905 occupants exist in zone 119911119894and the node of

zone 119911119899has tomake a decision onwhether to start heating the

zone or not For the decision process the distribution of timeuntil entrance to zone 119911

119899 119884119899(119905) is needed 119884

119899(119905) is the sum

of two terms The first term is the residual occupation timeat zone 119911

119894and the second term is the time to reach 119911

119899upon

departure from zone 119911119894 The distribution 119877

119894(119909) of the first

term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911

119894

Thus if zone 119911119894neighbors119873

119894zones namely 119911

119896 119896 = 1 119873

119894

then the time to reach zone 119911119899upon departure is distributed

according to

119866119894119899(119909) =

119873119894

sum

119896=1

119901119894119896119867119896119899(119909) (36)

where119867119896119899(119909) is the distribution of the first passage from zone

119911119896to zone 119911

119899 Pyke [12] andMason [13 14] provided solutions

for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867

119894119895(119909) is given by

119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]

minus1)119889minus1

(37)

where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865

119894119895(119909) Notation 119872

119889denotes the

matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ

119894119895= (minus1)

119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of

119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get

1198671119899(119904) =

Δ1198991(119904)

Δ119899119899(119904)

(38)

By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899

8 Journal of Energy

I

III

II

IV

V

IXVI

VIII

VII

To

To

To

To

Figure 4 Multizone model

In summary

119865119884119899(119905)

(119904) = 119877119894(119904) sdot 119866

119894119899(119904) = 119877

119894(119904) sdot (

119873119894

sum

119896=1

119901119894119896119867119896119899(119904)) (39)

with 119867119896119899(119904) computed using (38) Next the saddlepoint

approximation technique can be used to calculate the density119891119884119899(119905)

(119909)We have tomention that the process of estimating the first

passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865

119894119895(119904)

is obtained Having in our disposal the transforms 119865119894119895(119904) we

can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867

119894119895(119904) using (38) Finally numerical

methods may be used to invert these transforms in order toobtain the necessary probability density functions

4 Simulation Results

We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])

Table 1 Training and testing parameters for outer temperaturemodel and heat gain

Walterrsquos model Heat gainPhase 119879max 119879min 119879

119898119902(119905)

Training 15 2 7 600Testing 16 1 6 700

Table 2 Training and testing zonesrsquo target temperatures

Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21

5 10 15 20 250Time of day (h)

0

5

10

15

Tem

pera

ture

To

(∘C)

Figure 5 Empirical estimation of 119879119900

In Figure 4 119879119900denotes the outside temperature which

is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879

119898

need to be provided in order to estimate 119879119900 As an example if

119879max = 15∘C 119879min = 2

∘C and 119879119898= 7∘C the daily variations

of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm

41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879

119904=

1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892

to those of Table 2 (first row) with a margin equal to119879119898119892

= 1∘C That is for a specific zone the heater is on until

temperature119879119905119892+119879119898119892

is reached After this point the heater isturned off until temperature hits the lower threshold119879

119905119892minus119879119898119892

and the whole process is repeated

Journal of Energy 9

0 1 2 3 4 5 6 7 8 9 10 11Order

Zone VZone I

Zone II

10minus15

10minus10

10minus5

100

Sing

ular

val

ues

Figure 6 Singular values of zones I II and V

Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2

Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase

We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

02

4

6

8

10

12

14

16

18

20

Time (sec)

Zone I (real)

Zone I (predicted)

Zone IV (real)

Zone IV (predicted)

Tem

pera

ture

Figure 7 Real and predicted temperature of zones I and IV

zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model

Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is

119891119883(119909) =

120573120572

Γ (120572)119909120572minus1

119890minus120573119909

(40)

with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862

119894

The results are averages over 200 runs of the simulation Thecase of 119862

119894= 0 is the full reactive case where heating of

zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862

119894 the decision process

acts proactively by spending energy in order to reduce thediscomfort level

Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone

IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases

10 Journal of Energy

0

100

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 8 Total comfort cost versus119862119894 The occupation time of zone

I is Gamma distributed

15

2

25

3

Tota

l ene

rgy

cost

(kW

h)

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 9 Total energy cost versus 119862119894 The occupation time of zone

I is Gamma distributed

In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty

The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is

119891119883(119909) =

120572119909120572

119898

119909120572+1 119909 ge 119909

119898

0 119909 lt 119909119898

(41)

Zone I

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Zone IV(C = 0)

Zone IV(C = 10000)

Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0

1000 2000 3000 4000 5000 6000 70000Time (sec)

0

05

1

15

2

25

3

35Pa

reto

pdf

times10

minus3

xm = 300

xm = 900

xm = 1800

xm = 3600

Figure 11 Pareto pdf for various values of 119909119898

with expected value

119864 [119883] =

120572119909119898

120572 minus 1 120572 gt 1

infin 120572 le 1

(42)

Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=

300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases

Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862

119894 As it is observed a 50 improvement on the

comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909

119898= 300) which is due

to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909

119898

Journal of Energy 11

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

xm = 300xm = 900

xm = 3600

Figure 12 Total comfort cost for Pareto distributed occupationperiod

xm = 300

xm = 900

xm = 1800

xm = 3600

1

15

2

25

Tota

l ene

rgy

cost

(kW

h)

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

Figure 13 Total energy cost for Pareto distributed occupationperiod

and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13

Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution

Table 3 Parameters used in cascade movement scenario

Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20

Table 4 Energy and comfort costs for constant occupancy times

Comfort weight 119862 500 1000 1500 2000

119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020

119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025

I

IV

V

II

III VI

VII

VIII

IX

Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)

The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on

Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess

12 Journal of Energy

Fixed proactivecomfort cost

Energy cost

Comfort cost

1000

1500

2000

2500

3000

3500

4000

4500

Tota

l com

fort

cost

600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)

8

85

9

95

10

105

11

Tota

l ene

rgy

cost

(kW

h)

proactiveFixed

energy cost

Figure 15 Total energy and comfort costs (cascade movement sce-nario)

Zone I

Zone IX

Zone II

Zone VIII

Zone V

Zone IV

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Figure 16 Temperature profiles of zones (cascade movement sce-nario)

Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process

5 Conclusions and Future Work

Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone

to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique

Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862

119894may depend

on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov

[2] US Energy Information Administration June 2015 httpwwweiagov

[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013

[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013

[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011

[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013

[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014

[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014

[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015

Journal of Energy 13

[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996

[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954

[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961

[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953

[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956

[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005

[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015

[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967

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Renewable Energy

Submit your manuscripts athttpwwwhindawicom

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High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

4 Journal of Energy

Yi(t)

Soni (t)

Soffi (t)

Ti

t

DTs

t + DTs

Tc

Figure 1 Predicted time intervals related to the decision process

Using (2) and (3) the decision criterion takes the form

119877on119894(119905)

on≶

off119877off119894(119905) (4)

Consider for the moment that there is no randomness on119884119894(119905) that is strict time scheduled occupation of zones

Typical examples are the classrooms in a university campusIn this case criterion (4) takes a very simple form since one ofthe involved probabilities is equal to one and the rest assumethe value of zero Thus if 119884

119894(119905) le 119878

on119894(119905) then the heater

is turned on if 119878off119894(119905) lt 119884

119894(119905) the heater remains off and

if 119878on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905) the heater will be turned onoff

depending on the values of the relative energy and discomfortcost The balance between these two costs is regulated by theweights119882

119894and 119862

119894

If 119884119894(119905) is random then criterion (4) needs to be slightly

modified as the value of 119884119894(119905) is unknown In this case we

replace 119884119894(119905) with its conditional expected value given that

119878on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905) Therefore in criterion (4) we

substitute 119884119894(119905) by 119864[119884

119894(119905) | 119878

on119894(119905) lt 119884

119894(119905) le 119878

off119894(119905)]

From the previous discussion it becomes clear that thedecision process needs the estimates 119878on

119894(119905) 119878off119894(119905) and the

distribution of the process 119884119894(119905) The former is the subject of

the next subsection whereas a discussion on the distributionof 119884119894(119905) is left for Section 32

31 Temperature-Time Predictions Based on SID It is con-ceivable that in order to make predictions about a zonersquostemperature evolution we need a model able to capture thethermal dynamics of the tagged zone To this end we resort tothe deterministic subspace identification method Each zone

is treated separately and is represented by a discrete timelinear time invariant state-space model That is

119909119896+1

= 119860119909119896+ 119861119906119896+ 119908119896 (5)

119910119896= 119862119909119896+ 119863119906119896+ V119896 (6)

where 119910119896is the zone temperature at time 119896 and is in general

an ℓ-dimensional vector if ℓ temperature sensors scatteredin the zone have been deployed Assuming a uniform zonetemperaturewe use only one sensor node and thus119910

119896is scalar

V119896represents the measurement noise due to imperfections

of the sensor 119906119896is a vector of input measurements and its

dimensionality varies from zone to zoneThis vector containsthe measurable output of sources that influence zonersquos tem-perature Such sources are the power of a heater located inthe zone the air temperature at various positions outside thezone and so forth The vector 119909

119896is the state vector of the

process at the discrete time 119896 The states have a conceptualrelevance only and they are not assigned physical interpreta-tion Of course a similarity transform can convert the statesto physical meaningful onesThe unmeasurable vector signal119908119896represents noise due to the presence of humans in the

zone lampsrsquo switching on and off and so forth For the restof the paper we neglect the effects of state and output noiseand thus we deal with a deterministic identification problemwhich is the computation of the matrices 119860 119861 119862 and 119863

from the given input-output data The no noise assumptionmakes sense since we need temperature-time predictions forthe empty zones only which are ldquofreerdquo from occupants andother disturbances

In order to estimate the state-space matrices 119860 119861 119862 and119863 of each zone we need a training phase during which allthe nodes in the structure report their relative measurementsto a central computing system These measurements consistof the temperature of the zone and the current power of aheater if the sensor node is in charge of a heated zone oronly the temperature if the sensor node monitors an areaimmaterial to the decision process but relevant to other zonesthat is the exterior of a building The training phase shouldbe performed only once but under controlled conditions forexample no extra sources of heating and closed windowsMoreover the length of this period should be quite large inorder to exhibit several variations in the input signals to excitethemodes of the system In the simulation part we considereda training period of 24 h that is 86400 sampleswith samplingperiod 1 sec The sampling period of 1 sec is too small for allpractical reasons However as the dominant factor of perfor-mance degradation is the estimation of the residual arrivaltimes we consider an ideal temperature prediction model(high accuracy sensors and temperature noiseless zones) andwe access only the uncertainty of the occupation model

After receiving all the measurements the central systemorganizes them to input-output data for each zone Forexample if a zone is named A neighbors zones are namedB C and D and the exterior to the building zone is namedE then the temperature of zone A is the output of the systemto be identified whereas the temperatures of zones B C Dand E as well as the power profile of the heater of zone Aare the input data for the system A deterministic subspace

Journal of Energy 5

identification process described in the following paragraphsis then run for each zone The relevant matrices 119860 119861 119862 and119863 of the state-space model of each zone are communicatedback to the sensor node responsible for the monitoring of thezone

Following the notation and the derivation in [10] wedefine the block Hankel matrices

119880119901= 1198800|119894minus1

=(

(

1199060

1199061sdot sdot sdot 119906

119895minus1

1199061

1199062sdot sdot sdot 119906

119895

119906119894minus1

119906119894sdot sdot sdot 119906119894+119895minus2

)

)

119880119891= 119880119894|2119894minus1

=(

(

119906119894

119906119894+1

sdot sdot sdot 119906119894+119895minus1

119906119894+1

119906119894+2

sdot sdot sdot 119906119894+119895

1199062119894minus1

1199062119894

sdot sdot sdot 1199062119894+119895minus2

)

)

(7)

which represent the ldquopastrdquo and the ldquofuturerdquo input with respectto the present time instant 119894 Stacking119880

119901on top of119880

119891we have

the block Hankel matrix

1198800|2119894minus1

= (1198800|119894minus1

119880119894|2119894minus1

) = (

119880119901

119880119891

) (8)

which can also be partitioned as

1198800|2119894minus1

= (1198800|119894

119880119894+1|2119894minus1

) = (

119880+

119901

119880minus

119891

) (9)

by moving the ldquopresentrdquo time 119894 one step ahead to time 119894 + 1Similarly we define the output block Hankel matrices 119884

0|2119894minus1

119884119901 119884119891 119884+119901 119884minus119891and the input-output data matrices

119882119901= 1198820|119894minus1

= (1198800|119894minus1

1198840|119894minus1

) = (

119880119901

119884119901

)

119882+

119901= (

119880+

119901

119884+119901

)

(10)

The state sequence 119883119898is defined as

119883119898= (119909119898 119909

119898+1sdot sdot sdot 119909119898+119895minus1) (11)

and the ldquopastrdquo and ldquofuturerdquo state sequences are 119883119901= 1198830

and 119883119891= 119883119894 respectively With the state-space model (13)

(15) we associate the extended observability matrix Γ119894and the

reversed extended controllability matrix Δ119894 where

Γ119894=

((((

(

119862

119862119860

1198621198602

119862119860119894minus1

))))

)

Δ119894= (119860119894minus1119861 119860119894minus2119861 sdot sdot sdot 119860119861 119861)

(12)

The subspace identificationmethod relies on determining thestate sequence 119883

119891and the extended observability matrix Γ

119894

directly from the input-output data 119906119896119910119896and based on them

in a next step to extract the matrices 119860 119861 119862 and 119863 To thisend we define the oblique projection of the row space of 119884

119891

along the row space of 119880119891on the row space of119882

119901

O119894= 119884119891119880119891119882119901= [119884119891119880perp

119891] [119882119901119880perp

119891]dagger

119882119901 (13)

where dagger denotes the Moore-Penrose pseudoinverse of amatrix and 119860119861perp is a shorthand for the projection of the rowspace ofmatrix119860 onto the orthogonal complement of the rowspace of matrix 119861 that is

119860119861perp= 119868 minus 119860119861

119879(119861119861119879)dagger

119861 (14)

As it is proved in [10] the matrix O119894is equal to the product

of the extended observability matrix and the ldquofuturerdquo statevector

O119894= Γ119894119883119891 (15)

Having in our disposal the matrixO119894we use its singular value

decomposition

O119894= (1198801 119880

2) (

11987810

0 0)(

119881119879

1

119881119879

2

) = 11988011198781119881119879

1(16)

and we identify the extended observation matrix Γ119894and the

state vector119883119891as

Γ119894= 119880111987812

1119879

119883119891= 119879minus111987812

1119881119879

1= Γdagger

119894O119894

(17)

where 119879 is an 119899times119899 arbitrary nonsingular matrix representinga similarity transformation

One method [10] to extract the matrices 119860 119861 119862 and 119863uses in addition the oblique projection

O119894minus1

= 119884minus

119891119880minus119891119882dagger

119901= Γ119894minus1119883119894+1 (18)

where Γ119894minus1

is obtained from Γ119894by deleting the last ℓ rows (ℓ = 1

in our case) Similar to the previous derivation we have

119883119894+1

= Γdagger

119894minus1O119894minus1

(19)

and the matrices 119860 119861 119862 and 119863 are obtained by solving (inleast squares fashion) the system

(

119883119894+1

119884119894|119894

) = (

119860 119861

119862 119863)(

119883119894

119880119894|119894

) (20)

After completion of the training phase the state-spacemodel matrices 119860 119861 119862 and 119863 for each zone are commu-nicated back to the node responsible for the zone Uponreception of the matrices the nodes enter a ldquoconvergencerdquophase that is starting from an initial state 119909

0 they update

6 Journal of Energy

the state (5) for a predetermined period of time using themeasurements of the surrounding zones and the zonersquos heaterstatus Note that the training and the convergence phaseneed to be executed only once prior to the normal operationof the nodes After the convergence phase the nodes enterthe ldquodecisionrdquo phase During this phase the nodes updatethe state of the model (as in the convergence phase) andperiodically they take decisions on whether to turn the zonersquosheater on or not Suppose that at the decision epoch 119905 thestate of the tagged unoccupied node is 119909

119905 The node bases its

decision on the estimates 119878on119894(119905) and 119878off

119894(119905) Recall that 119878on

119894(119905)

is the time needed to reach the comfort level 119879119888if the heater

is on That is

119878on119894(119905) = min ℓ 119910

119905+ℓge 119879119888 (21)

The node ldquofreezesrdquo the input-vector 119906119896to its current value 119906

119905

(more recent temperatures of the surrounding zones and thetagged zonersquos heater is on) and starting from state 119909

119905repeats

(5) (6) until 119910119905+119896

exceeds the value 119879119888 Note that

119910119905+119896

= 119862119860119896119909119905+ 119862119860119896minus1

119861119906119905+ 119862119860119896minus2

119861119906119905+1

+ sdot sdot sdot

+ 119862119860119861119906119905+119896minus2

+ 119862119861119906119905+119896minus1

+ 119863119906119905+119896

(22)

and for 119906119905= 119906119905+1

= sdot sdot sdot = 119906119905+119896

we obtain

119910119905+119896

= 119862119860119896119909119905+ 119862 (119868 + 119860 + sdot sdot sdot + 119860

119896minus1) 119861119906119905+ 119863119906119905

= 119862119860119896119909119905+ 119862 (119868 minus 119860)

minus1(119868 minus 119860

119896) 119861119906119905+ 119863119906119905

(23)

If we set 119910119905+119896

= 119879119888and

119908 = 119879119888minus 119862 (119868 minus 119860)

minus1119861119906119905minus 119863119906119905 (24)

then (23) takes the form

119908 = 119862119860119896119909119905minus 119862 (119868 minus 119860)

minus1119860119896119861119906119905

= 119862119881Λ119896119881minus1119909119905minus 119862 (119868 minus 119860)

minus1119881Λ119896119881minus1119861119906119905

(25)

where we have considered the diagonalization of the matrix119860 as 119860 = 119881Λ119881

minus1 with Λ being the diagonal matrix of theeigenvalues and 119881 being the matrix of the correspondingeigenvectors Numerical methods can be used to solve (25)for 119896 In a similar manner we compute 119878off

119894(119905) In this case

however we first repeat (5) 119863 times with input-vector 119906119896

reflecting the fact that the heater is off and then starting fromthe new state 119909

119905+119863we estimate the time needed to reach 119879

119888if

the heater is on

32 On the Distribution of 119884119899(119905) The random variable 119884

119899(119905)

expresses the remaining time from the current time instant 119905until entrance to zone 119911

119899 That is we assume that occupants

move around as a swarm visiting zones either in a prede-termined order or in a random fashion and finally they endup in zone 119911

119899 Each time the occupants visit a zone 119911

ℓthey

znz120001

Figure 2 Movement of occupants from zone 119911ℓto the adjacent

unoccupied zone 119911119899

stay in it for a nonzero time period which is itself a ran-dom variable with distribution 119865

ℓ(119909) We use the variable 119905

to designate calendar time the time elapsed since the processstarted and 120591 to designate the time elapsed since entry to azone The zone occupation time distribution 119865

ℓ(119909) may be

given parametrically either a priori or inferred from exam-ination of historic data by suitable fitting of the distributionrsquosparameters We may use the empirical distribution func-tion (edf) instead which is the nonparametric estimate of119865ℓ(119909) That is if a sample of119873 occupation periods of zone 119911

ℓ

1199091 1199092 119909

119873 is available then

ℓ(119909) =

1

119873

119873

sum

119898=1

119868 (119909119898isin [0 119909]) (26)

where 119868(sdot) is the indicator functionLet us consider first the simple case of two adjusting zones

119911ℓ 119911119899with zone 119911

ℓbeing occupied and 119911

119899unoccupied as it is

shown in Figure 2 We assume that the occupants remain tozone 119911

ℓfor a random period of time 119883

ℓwhich is distributed

according to 119865ℓ(119909) and then upon departure they enter zone

119911119899 In this case 119884

119899(119905) expresses the remaining waiting time

to zone 119911ℓand follows the residual life distribution which is

defined by

119877119884119899(119905)

(119910) = 1 minus 119877119884119899(119905)

(119910) = 119875 119883ℓgt 120591 + 119910 | 119883

ℓgt 120591

=119865ℓ(120591 + 119910)

119865ℓ(120591)

(27)

Next we consider the case that occupants are currently ina zone (call it 119911

0) and upon departure they visit in cascade

zones 1199111 1199112 119911

119899as it is shown in Figure 3 Then

119884119899(119905) = 119884

0+ 1198831+ 1198832+ sdot sdot sdot + 119883

119899minus1 (28)

where 1198840is the remaining time in zone 119911

0and 119883

119894is the

occupation period of zone 119911119894 It is well known that the

distribution of 119884119899(119905) is the convolution of the distributions of

the random variables 1198840 1198831 119883

119899minus1 That is

119865119884119899(119905)

(119910) = 119875 119884119899(119905) le 119910

= 1198770(119910) ⋆ 119865

1(119910) ⋆ 119865

2(119910) ⋆ sdot sdot sdot ⋆ 119865

119899minus1(119910)

(29)

with 1198770(119910) = 1 minus 119865

0(120591 + 119910)119865

0(120591) A more compact form is

obtained by considering a transform of the distributions that

Journal of Energy 7

zn

z0 z1

Figure 3 Movement of occupants in cascade from zone 1199110to zone

119911119899

is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865

119883(119909) of a nonnega-

tive random variable119883 is defined by

119865119883(119904) = int

infin

0

119890119904119909119889119865119883(119909) (30)

(29) takes the form

119865119884119899(119905)

(119904) = 1198770(119904) sdot 119865

1(119904) sdot 119865

2(119904) sdot sdot 119865

119899minus1(119904) (31)

In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865

119884119899(119905)(119910)

For example if the occupation periods 119883119894 119894 = 0 119899 minus 1

are exponentially distributed with corresponding parameters120582119894 then 119884

0is also exponentially distributed with parameter

1205820due to the memoryless property of the exponential

distribution and the probability density function of 119884119899(119905) is

(assuming that the parameters 120582119894are all distinct)

119891119884119899(119905)

(119909) =

119899minus1

sum

119894=0

1205820sdot sdot sdot 120582119899minus1

prod119899minus1

119895=0119895 =119894(120582119895minus 120582119894)

exp (minus119909120582119894) (32)

(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891

119884119899(119905)(119909) does not exist or it is difficult to be

obtained we resort to the saddlepoint approximation [11] toinvert 119865

119884119899(119905)(119904) To this end for a random variable 119883 with

cumulative distribution function (cdf) 119865119883(119909) we define the

cumulant generating function (cgf) 119870119883(119904) as the logarithm

of the corresponding mfg 119865119883(119904) that is

119870119883(119904) = log119865

119883(119904) (33)

Then the saddlepoint approximation of the probability den-sity function (pdf) 119891

119883(119909) is

119891119883(119909)

asymp (1

212058711987010158401015840119909( (119909))

)

12

exp 119870119883( (119909)) minus (119909) 119909

(34)

where (119909) is the solution to the saddlepoint equation

1198701015840

119883(119904) = 119909 (35)

Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865

119883(119904) Note that wemay

use the empirical mgf 119883(119904) instead so that a nonparametric

estimation of the pdf 119891119883(119909) is possible

The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901

119894119895 Given a transition from zone 119911

119894to

zone 119911119895 the occupation time in 119911

119894has distribution function

119865119894119895(119909) The matrix Q(119909) with elements 119876

119894119895(119909) = 119901

119894119895119865119894119895(119909)

is the so called semi-Markov kernel The occupation timedistribution in zone 119911

119894independent of the state to which a

transition is made is 119865119894(119909) = sum

119895119876119894119895(119909) Consider now the

case that at time 119905 occupants exist in zone 119911119894and the node of

zone 119911119899has tomake a decision onwhether to start heating the

zone or not For the decision process the distribution of timeuntil entrance to zone 119911

119899 119884119899(119905) is needed 119884

119899(119905) is the sum

of two terms The first term is the residual occupation timeat zone 119911

119894and the second term is the time to reach 119911

119899upon

departure from zone 119911119894 The distribution 119877

119894(119909) of the first

term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911

119894

Thus if zone 119911119894neighbors119873

119894zones namely 119911

119896 119896 = 1 119873

119894

then the time to reach zone 119911119899upon departure is distributed

according to

119866119894119899(119909) =

119873119894

sum

119896=1

119901119894119896119867119896119899(119909) (36)

where119867119896119899(119909) is the distribution of the first passage from zone

119911119896to zone 119911

119899 Pyke [12] andMason [13 14] provided solutions

for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867

119894119895(119909) is given by

119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]

minus1)119889minus1

(37)

where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865

119894119895(119909) Notation 119872

119889denotes the

matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ

119894119895= (minus1)

119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of

119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get

1198671119899(119904) =

Δ1198991(119904)

Δ119899119899(119904)

(38)

By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899

8 Journal of Energy

I

III

II

IV

V

IXVI

VIII

VII

To

To

To

To

Figure 4 Multizone model

In summary

119865119884119899(119905)

(119904) = 119877119894(119904) sdot 119866

119894119899(119904) = 119877

119894(119904) sdot (

119873119894

sum

119896=1

119901119894119896119867119896119899(119904)) (39)

with 119867119896119899(119904) computed using (38) Next the saddlepoint

approximation technique can be used to calculate the density119891119884119899(119905)

(119909)We have tomention that the process of estimating the first

passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865

119894119895(119904)

is obtained Having in our disposal the transforms 119865119894119895(119904) we

can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867

119894119895(119904) using (38) Finally numerical

methods may be used to invert these transforms in order toobtain the necessary probability density functions

4 Simulation Results

We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])

Table 1 Training and testing parameters for outer temperaturemodel and heat gain

Walterrsquos model Heat gainPhase 119879max 119879min 119879

119898119902(119905)

Training 15 2 7 600Testing 16 1 6 700

Table 2 Training and testing zonesrsquo target temperatures

Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21

5 10 15 20 250Time of day (h)

0

5

10

15

Tem

pera

ture

To

(∘C)

Figure 5 Empirical estimation of 119879119900

In Figure 4 119879119900denotes the outside temperature which

is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879

119898

need to be provided in order to estimate 119879119900 As an example if

119879max = 15∘C 119879min = 2

∘C and 119879119898= 7∘C the daily variations

of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm

41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879

119904=

1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892

to those of Table 2 (first row) with a margin equal to119879119898119892

= 1∘C That is for a specific zone the heater is on until

temperature119879119905119892+119879119898119892

is reached After this point the heater isturned off until temperature hits the lower threshold119879

119905119892minus119879119898119892

and the whole process is repeated

Journal of Energy 9

0 1 2 3 4 5 6 7 8 9 10 11Order

Zone VZone I

Zone II

10minus15

10minus10

10minus5

100

Sing

ular

val

ues

Figure 6 Singular values of zones I II and V

Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2

Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase

We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

02

4

6

8

10

12

14

16

18

20

Time (sec)

Zone I (real)

Zone I (predicted)

Zone IV (real)

Zone IV (predicted)

Tem

pera

ture

Figure 7 Real and predicted temperature of zones I and IV

zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model

Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is

119891119883(119909) =

120573120572

Γ (120572)119909120572minus1

119890minus120573119909

(40)

with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862

119894

The results are averages over 200 runs of the simulation Thecase of 119862

119894= 0 is the full reactive case where heating of

zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862

119894 the decision process

acts proactively by spending energy in order to reduce thediscomfort level

Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone

IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases

10 Journal of Energy

0

100

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 8 Total comfort cost versus119862119894 The occupation time of zone

I is Gamma distributed

15

2

25

3

Tota

l ene

rgy

cost

(kW

h)

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 9 Total energy cost versus 119862119894 The occupation time of zone

I is Gamma distributed

In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty

The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is

119891119883(119909) =

120572119909120572

119898

119909120572+1 119909 ge 119909

119898

0 119909 lt 119909119898

(41)

Zone I

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Zone IV(C = 0)

Zone IV(C = 10000)

Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0

1000 2000 3000 4000 5000 6000 70000Time (sec)

0

05

1

15

2

25

3

35Pa

reto

pdf

times10

minus3

xm = 300

xm = 900

xm = 1800

xm = 3600

Figure 11 Pareto pdf for various values of 119909119898

with expected value

119864 [119883] =

120572119909119898

120572 minus 1 120572 gt 1

infin 120572 le 1

(42)

Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=

300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases

Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862

119894 As it is observed a 50 improvement on the

comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909

119898= 300) which is due

to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909

119898

Journal of Energy 11

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

xm = 300xm = 900

xm = 3600

Figure 12 Total comfort cost for Pareto distributed occupationperiod

xm = 300

xm = 900

xm = 1800

xm = 3600

1

15

2

25

Tota

l ene

rgy

cost

(kW

h)

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

Figure 13 Total energy cost for Pareto distributed occupationperiod

and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13

Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution

Table 3 Parameters used in cascade movement scenario

Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20

Table 4 Energy and comfort costs for constant occupancy times

Comfort weight 119862 500 1000 1500 2000

119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020

119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025

I

IV

V

II

III VI

VII

VIII

IX

Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)

The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on

Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess

12 Journal of Energy

Fixed proactivecomfort cost

Energy cost

Comfort cost

1000

1500

2000

2500

3000

3500

4000

4500

Tota

l com

fort

cost

600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)

8

85

9

95

10

105

11

Tota

l ene

rgy

cost

(kW

h)

proactiveFixed

energy cost

Figure 15 Total energy and comfort costs (cascade movement sce-nario)

Zone I

Zone IX

Zone II

Zone VIII

Zone V

Zone IV

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Figure 16 Temperature profiles of zones (cascade movement sce-nario)

Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process

5 Conclusions and Future Work

Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone

to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique

Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862

119894may depend

on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov

[2] US Energy Information Administration June 2015 httpwwweiagov

[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013

[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013

[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011

[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013

[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014

[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014

[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015

Journal of Energy 13

[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996

[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954

[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961

[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953

[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956

[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005

[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015

[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967

TribologyAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Renewable Energy

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporation httpwwwhindawicom

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International Journal ofPhotoenergy

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Nuclear InstallationsScience and Technology of

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Solar EnergyJournal of

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Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

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High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of Energy 5

identification process described in the following paragraphsis then run for each zone The relevant matrices 119860 119861 119862 and119863 of the state-space model of each zone are communicatedback to the sensor node responsible for the monitoring of thezone

Following the notation and the derivation in [10] wedefine the block Hankel matrices

119880119901= 1198800|119894minus1

=(

(

1199060

1199061sdot sdot sdot 119906

119895minus1

1199061

1199062sdot sdot sdot 119906

119895

119906119894minus1

119906119894sdot sdot sdot 119906119894+119895minus2

)

)

119880119891= 119880119894|2119894minus1

=(

(

119906119894

119906119894+1

sdot sdot sdot 119906119894+119895minus1

119906119894+1

119906119894+2

sdot sdot sdot 119906119894+119895

1199062119894minus1

1199062119894

sdot sdot sdot 1199062119894+119895minus2

)

)

(7)

which represent the ldquopastrdquo and the ldquofuturerdquo input with respectto the present time instant 119894 Stacking119880

119901on top of119880

119891we have

the block Hankel matrix

1198800|2119894minus1

= (1198800|119894minus1

119880119894|2119894minus1

) = (

119880119901

119880119891

) (8)

which can also be partitioned as

1198800|2119894minus1

= (1198800|119894

119880119894+1|2119894minus1

) = (

119880+

119901

119880minus

119891

) (9)

by moving the ldquopresentrdquo time 119894 one step ahead to time 119894 + 1Similarly we define the output block Hankel matrices 119884

0|2119894minus1

119884119901 119884119891 119884+119901 119884minus119891and the input-output data matrices

119882119901= 1198820|119894minus1

= (1198800|119894minus1

1198840|119894minus1

) = (

119880119901

119884119901

)

119882+

119901= (

119880+

119901

119884+119901

)

(10)

The state sequence 119883119898is defined as

119883119898= (119909119898 119909

119898+1sdot sdot sdot 119909119898+119895minus1) (11)

and the ldquopastrdquo and ldquofuturerdquo state sequences are 119883119901= 1198830

and 119883119891= 119883119894 respectively With the state-space model (13)

(15) we associate the extended observability matrix Γ119894and the

reversed extended controllability matrix Δ119894 where

Γ119894=

((((

(

119862

119862119860

1198621198602

119862119860119894minus1

))))

)

Δ119894= (119860119894minus1119861 119860119894minus2119861 sdot sdot sdot 119860119861 119861)

(12)

The subspace identificationmethod relies on determining thestate sequence 119883

119891and the extended observability matrix Γ

119894

directly from the input-output data 119906119896119910119896and based on them

in a next step to extract the matrices 119860 119861 119862 and 119863 To thisend we define the oblique projection of the row space of 119884

119891

along the row space of 119880119891on the row space of119882

119901

O119894= 119884119891119880119891119882119901= [119884119891119880perp

119891] [119882119901119880perp

119891]dagger

119882119901 (13)

where dagger denotes the Moore-Penrose pseudoinverse of amatrix and 119860119861perp is a shorthand for the projection of the rowspace ofmatrix119860 onto the orthogonal complement of the rowspace of matrix 119861 that is

119860119861perp= 119868 minus 119860119861

119879(119861119861119879)dagger

119861 (14)

As it is proved in [10] the matrix O119894is equal to the product

of the extended observability matrix and the ldquofuturerdquo statevector

O119894= Γ119894119883119891 (15)

Having in our disposal the matrixO119894we use its singular value

decomposition

O119894= (1198801 119880

2) (

11987810

0 0)(

119881119879

1

119881119879

2

) = 11988011198781119881119879

1(16)

and we identify the extended observation matrix Γ119894and the

state vector119883119891as

Γ119894= 119880111987812

1119879

119883119891= 119879minus111987812

1119881119879

1= Γdagger

119894O119894

(17)

where 119879 is an 119899times119899 arbitrary nonsingular matrix representinga similarity transformation

One method [10] to extract the matrices 119860 119861 119862 and 119863uses in addition the oblique projection

O119894minus1

= 119884minus

119891119880minus119891119882dagger

119901= Γ119894minus1119883119894+1 (18)

where Γ119894minus1

is obtained from Γ119894by deleting the last ℓ rows (ℓ = 1

in our case) Similar to the previous derivation we have

119883119894+1

= Γdagger

119894minus1O119894minus1

(19)

and the matrices 119860 119861 119862 and 119863 are obtained by solving (inleast squares fashion) the system

(

119883119894+1

119884119894|119894

) = (

119860 119861

119862 119863)(

119883119894

119880119894|119894

) (20)

After completion of the training phase the state-spacemodel matrices 119860 119861 119862 and 119863 for each zone are commu-nicated back to the node responsible for the zone Uponreception of the matrices the nodes enter a ldquoconvergencerdquophase that is starting from an initial state 119909

0 they update

6 Journal of Energy

the state (5) for a predetermined period of time using themeasurements of the surrounding zones and the zonersquos heaterstatus Note that the training and the convergence phaseneed to be executed only once prior to the normal operationof the nodes After the convergence phase the nodes enterthe ldquodecisionrdquo phase During this phase the nodes updatethe state of the model (as in the convergence phase) andperiodically they take decisions on whether to turn the zonersquosheater on or not Suppose that at the decision epoch 119905 thestate of the tagged unoccupied node is 119909

119905 The node bases its

decision on the estimates 119878on119894(119905) and 119878off

119894(119905) Recall that 119878on

119894(119905)

is the time needed to reach the comfort level 119879119888if the heater

is on That is

119878on119894(119905) = min ℓ 119910

119905+ℓge 119879119888 (21)

The node ldquofreezesrdquo the input-vector 119906119896to its current value 119906

119905

(more recent temperatures of the surrounding zones and thetagged zonersquos heater is on) and starting from state 119909

119905repeats

(5) (6) until 119910119905+119896

exceeds the value 119879119888 Note that

119910119905+119896

= 119862119860119896119909119905+ 119862119860119896minus1

119861119906119905+ 119862119860119896minus2

119861119906119905+1

+ sdot sdot sdot

+ 119862119860119861119906119905+119896minus2

+ 119862119861119906119905+119896minus1

+ 119863119906119905+119896

(22)

and for 119906119905= 119906119905+1

= sdot sdot sdot = 119906119905+119896

we obtain

119910119905+119896

= 119862119860119896119909119905+ 119862 (119868 + 119860 + sdot sdot sdot + 119860

119896minus1) 119861119906119905+ 119863119906119905

= 119862119860119896119909119905+ 119862 (119868 minus 119860)

minus1(119868 minus 119860

119896) 119861119906119905+ 119863119906119905

(23)

If we set 119910119905+119896

= 119879119888and

119908 = 119879119888minus 119862 (119868 minus 119860)

minus1119861119906119905minus 119863119906119905 (24)

then (23) takes the form

119908 = 119862119860119896119909119905minus 119862 (119868 minus 119860)

minus1119860119896119861119906119905

= 119862119881Λ119896119881minus1119909119905minus 119862 (119868 minus 119860)

minus1119881Λ119896119881minus1119861119906119905

(25)

where we have considered the diagonalization of the matrix119860 as 119860 = 119881Λ119881

minus1 with Λ being the diagonal matrix of theeigenvalues and 119881 being the matrix of the correspondingeigenvectors Numerical methods can be used to solve (25)for 119896 In a similar manner we compute 119878off

119894(119905) In this case

however we first repeat (5) 119863 times with input-vector 119906119896

reflecting the fact that the heater is off and then starting fromthe new state 119909

119905+119863we estimate the time needed to reach 119879

119888if

the heater is on

32 On the Distribution of 119884119899(119905) The random variable 119884

119899(119905)

expresses the remaining time from the current time instant 119905until entrance to zone 119911

119899 That is we assume that occupants

move around as a swarm visiting zones either in a prede-termined order or in a random fashion and finally they endup in zone 119911

119899 Each time the occupants visit a zone 119911

ℓthey

znz120001

Figure 2 Movement of occupants from zone 119911ℓto the adjacent

unoccupied zone 119911119899

stay in it for a nonzero time period which is itself a ran-dom variable with distribution 119865

ℓ(119909) We use the variable 119905

to designate calendar time the time elapsed since the processstarted and 120591 to designate the time elapsed since entry to azone The zone occupation time distribution 119865

ℓ(119909) may be

given parametrically either a priori or inferred from exam-ination of historic data by suitable fitting of the distributionrsquosparameters We may use the empirical distribution func-tion (edf) instead which is the nonparametric estimate of119865ℓ(119909) That is if a sample of119873 occupation periods of zone 119911

ℓ

1199091 1199092 119909

119873 is available then

ℓ(119909) =

1

119873

119873

sum

119898=1

119868 (119909119898isin [0 119909]) (26)

where 119868(sdot) is the indicator functionLet us consider first the simple case of two adjusting zones

119911ℓ 119911119899with zone 119911

ℓbeing occupied and 119911

119899unoccupied as it is

shown in Figure 2 We assume that the occupants remain tozone 119911

ℓfor a random period of time 119883

ℓwhich is distributed

according to 119865ℓ(119909) and then upon departure they enter zone

119911119899 In this case 119884

119899(119905) expresses the remaining waiting time

to zone 119911ℓand follows the residual life distribution which is

defined by

119877119884119899(119905)

(119910) = 1 minus 119877119884119899(119905)

(119910) = 119875 119883ℓgt 120591 + 119910 | 119883

ℓgt 120591

=119865ℓ(120591 + 119910)

119865ℓ(120591)

(27)

Next we consider the case that occupants are currently ina zone (call it 119911

0) and upon departure they visit in cascade

zones 1199111 1199112 119911

119899as it is shown in Figure 3 Then

119884119899(119905) = 119884

0+ 1198831+ 1198832+ sdot sdot sdot + 119883

119899minus1 (28)

where 1198840is the remaining time in zone 119911

0and 119883

119894is the

occupation period of zone 119911119894 It is well known that the

distribution of 119884119899(119905) is the convolution of the distributions of

the random variables 1198840 1198831 119883

119899minus1 That is

119865119884119899(119905)

(119910) = 119875 119884119899(119905) le 119910

= 1198770(119910) ⋆ 119865

1(119910) ⋆ 119865

2(119910) ⋆ sdot sdot sdot ⋆ 119865

119899minus1(119910)

(29)

with 1198770(119910) = 1 minus 119865

0(120591 + 119910)119865

0(120591) A more compact form is

obtained by considering a transform of the distributions that

Journal of Energy 7

zn

z0 z1

Figure 3 Movement of occupants in cascade from zone 1199110to zone

119911119899

is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865

119883(119909) of a nonnega-

tive random variable119883 is defined by

119865119883(119904) = int

infin

0

119890119904119909119889119865119883(119909) (30)

(29) takes the form

119865119884119899(119905)

(119904) = 1198770(119904) sdot 119865

1(119904) sdot 119865

2(119904) sdot sdot 119865

119899minus1(119904) (31)

In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865

119884119899(119905)(119910)

For example if the occupation periods 119883119894 119894 = 0 119899 minus 1

are exponentially distributed with corresponding parameters120582119894 then 119884

0is also exponentially distributed with parameter

1205820due to the memoryless property of the exponential

distribution and the probability density function of 119884119899(119905) is

(assuming that the parameters 120582119894are all distinct)

119891119884119899(119905)

(119909) =

119899minus1

sum

119894=0

1205820sdot sdot sdot 120582119899minus1

prod119899minus1

119895=0119895 =119894(120582119895minus 120582119894)

exp (minus119909120582119894) (32)

(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891

119884119899(119905)(119909) does not exist or it is difficult to be

obtained we resort to the saddlepoint approximation [11] toinvert 119865

119884119899(119905)(119904) To this end for a random variable 119883 with

cumulative distribution function (cdf) 119865119883(119909) we define the

cumulant generating function (cgf) 119870119883(119904) as the logarithm

of the corresponding mfg 119865119883(119904) that is

119870119883(119904) = log119865

119883(119904) (33)

Then the saddlepoint approximation of the probability den-sity function (pdf) 119891

119883(119909) is

119891119883(119909)

asymp (1

212058711987010158401015840119909( (119909))

)

12

exp 119870119883( (119909)) minus (119909) 119909

(34)

where (119909) is the solution to the saddlepoint equation

1198701015840

119883(119904) = 119909 (35)

Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865

119883(119904) Note that wemay

use the empirical mgf 119883(119904) instead so that a nonparametric

estimation of the pdf 119891119883(119909) is possible

The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901

119894119895 Given a transition from zone 119911

119894to

zone 119911119895 the occupation time in 119911

119894has distribution function

119865119894119895(119909) The matrix Q(119909) with elements 119876

119894119895(119909) = 119901

119894119895119865119894119895(119909)

is the so called semi-Markov kernel The occupation timedistribution in zone 119911

119894independent of the state to which a

transition is made is 119865119894(119909) = sum

119895119876119894119895(119909) Consider now the

case that at time 119905 occupants exist in zone 119911119894and the node of

zone 119911119899has tomake a decision onwhether to start heating the

zone or not For the decision process the distribution of timeuntil entrance to zone 119911

119899 119884119899(119905) is needed 119884

119899(119905) is the sum

of two terms The first term is the residual occupation timeat zone 119911

119894and the second term is the time to reach 119911

119899upon

departure from zone 119911119894 The distribution 119877

119894(119909) of the first

term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911

119894

Thus if zone 119911119894neighbors119873

119894zones namely 119911

119896 119896 = 1 119873

119894

then the time to reach zone 119911119899upon departure is distributed

according to

119866119894119899(119909) =

119873119894

sum

119896=1

119901119894119896119867119896119899(119909) (36)

where119867119896119899(119909) is the distribution of the first passage from zone

119911119896to zone 119911

119899 Pyke [12] andMason [13 14] provided solutions

for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867

119894119895(119909) is given by

119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]

minus1)119889minus1

(37)

where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865

119894119895(119909) Notation 119872

119889denotes the

matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ

119894119895= (minus1)

119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of

119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get

1198671119899(119904) =

Δ1198991(119904)

Δ119899119899(119904)

(38)

By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899

8 Journal of Energy

I

III

II

IV

V

IXVI

VIII

VII

To

To

To

To

Figure 4 Multizone model

In summary

119865119884119899(119905)

(119904) = 119877119894(119904) sdot 119866

119894119899(119904) = 119877

119894(119904) sdot (

119873119894

sum

119896=1

119901119894119896119867119896119899(119904)) (39)

with 119867119896119899(119904) computed using (38) Next the saddlepoint

approximation technique can be used to calculate the density119891119884119899(119905)

(119909)We have tomention that the process of estimating the first

passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865

119894119895(119904)

is obtained Having in our disposal the transforms 119865119894119895(119904) we

can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867

119894119895(119904) using (38) Finally numerical

methods may be used to invert these transforms in order toobtain the necessary probability density functions

4 Simulation Results

We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])

Table 1 Training and testing parameters for outer temperaturemodel and heat gain

Walterrsquos model Heat gainPhase 119879max 119879min 119879

119898119902(119905)

Training 15 2 7 600Testing 16 1 6 700

Table 2 Training and testing zonesrsquo target temperatures

Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21

5 10 15 20 250Time of day (h)

0

5

10

15

Tem

pera

ture

To

(∘C)

Figure 5 Empirical estimation of 119879119900

In Figure 4 119879119900denotes the outside temperature which

is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879

119898

need to be provided in order to estimate 119879119900 As an example if

119879max = 15∘C 119879min = 2

∘C and 119879119898= 7∘C the daily variations

of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm

41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879

119904=

1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892

to those of Table 2 (first row) with a margin equal to119879119898119892

= 1∘C That is for a specific zone the heater is on until

temperature119879119905119892+119879119898119892

is reached After this point the heater isturned off until temperature hits the lower threshold119879

119905119892minus119879119898119892

and the whole process is repeated

Journal of Energy 9

0 1 2 3 4 5 6 7 8 9 10 11Order

Zone VZone I

Zone II

10minus15

10minus10

10minus5

100

Sing

ular

val

ues

Figure 6 Singular values of zones I II and V

Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2

Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase

We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

02

4

6

8

10

12

14

16

18

20

Time (sec)

Zone I (real)

Zone I (predicted)

Zone IV (real)

Zone IV (predicted)

Tem

pera

ture

Figure 7 Real and predicted temperature of zones I and IV

zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model

Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is

119891119883(119909) =

120573120572

Γ (120572)119909120572minus1

119890minus120573119909

(40)

with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862

119894

The results are averages over 200 runs of the simulation Thecase of 119862

119894= 0 is the full reactive case where heating of

zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862

119894 the decision process

acts proactively by spending energy in order to reduce thediscomfort level

Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone

IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases

10 Journal of Energy

0

100

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 8 Total comfort cost versus119862119894 The occupation time of zone

I is Gamma distributed

15

2

25

3

Tota

l ene

rgy

cost

(kW

h)

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 9 Total energy cost versus 119862119894 The occupation time of zone

I is Gamma distributed

In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty

The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is

119891119883(119909) =

120572119909120572

119898

119909120572+1 119909 ge 119909

119898

0 119909 lt 119909119898

(41)

Zone I

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Zone IV(C = 0)

Zone IV(C = 10000)

Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0

1000 2000 3000 4000 5000 6000 70000Time (sec)

0

05

1

15

2

25

3

35Pa

reto

pdf

times10

minus3

xm = 300

xm = 900

xm = 1800

xm = 3600

Figure 11 Pareto pdf for various values of 119909119898

with expected value

119864 [119883] =

120572119909119898

120572 minus 1 120572 gt 1

infin 120572 le 1

(42)

Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=

300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases

Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862

119894 As it is observed a 50 improvement on the

comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909

119898= 300) which is due

to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909

119898

Journal of Energy 11

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

xm = 300xm = 900

xm = 3600

Figure 12 Total comfort cost for Pareto distributed occupationperiod

xm = 300

xm = 900

xm = 1800

xm = 3600

1

15

2

25

Tota

l ene

rgy

cost

(kW

h)

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

Figure 13 Total energy cost for Pareto distributed occupationperiod

and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13

Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution

Table 3 Parameters used in cascade movement scenario

Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20

Table 4 Energy and comfort costs for constant occupancy times

Comfort weight 119862 500 1000 1500 2000

119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020

119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025

I

IV

V

II

III VI

VII

VIII

IX

Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)

The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on

Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess

12 Journal of Energy

Fixed proactivecomfort cost

Energy cost

Comfort cost

1000

1500

2000

2500

3000

3500

4000

4500

Tota

l com

fort

cost

600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)

8

85

9

95

10

105

11

Tota

l ene

rgy

cost

(kW

h)

proactiveFixed

energy cost

Figure 15 Total energy and comfort costs (cascade movement sce-nario)

Zone I

Zone IX

Zone II

Zone VIII

Zone V

Zone IV

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Figure 16 Temperature profiles of zones (cascade movement sce-nario)

Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process

5 Conclusions and Future Work

Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone

to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique

Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862

119894may depend

on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov

[2] US Energy Information Administration June 2015 httpwwweiagov

[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013

[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013

[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011

[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013

[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014

[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014

[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015

Journal of Energy 13

[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996

[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954

[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961

[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953

[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956

[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005

[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015

[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967

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High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

6 Journal of Energy

the state (5) for a predetermined period of time using themeasurements of the surrounding zones and the zonersquos heaterstatus Note that the training and the convergence phaseneed to be executed only once prior to the normal operationof the nodes After the convergence phase the nodes enterthe ldquodecisionrdquo phase During this phase the nodes updatethe state of the model (as in the convergence phase) andperiodically they take decisions on whether to turn the zonersquosheater on or not Suppose that at the decision epoch 119905 thestate of the tagged unoccupied node is 119909

119905 The node bases its

decision on the estimates 119878on119894(119905) and 119878off

119894(119905) Recall that 119878on

119894(119905)

is the time needed to reach the comfort level 119879119888if the heater

is on That is

119878on119894(119905) = min ℓ 119910

119905+ℓge 119879119888 (21)

The node ldquofreezesrdquo the input-vector 119906119896to its current value 119906

119905

(more recent temperatures of the surrounding zones and thetagged zonersquos heater is on) and starting from state 119909

119905repeats

(5) (6) until 119910119905+119896

exceeds the value 119879119888 Note that

119910119905+119896

= 119862119860119896119909119905+ 119862119860119896minus1

119861119906119905+ 119862119860119896minus2

119861119906119905+1

+ sdot sdot sdot

+ 119862119860119861119906119905+119896minus2

+ 119862119861119906119905+119896minus1

+ 119863119906119905+119896

(22)

and for 119906119905= 119906119905+1

= sdot sdot sdot = 119906119905+119896

we obtain

119910119905+119896

= 119862119860119896119909119905+ 119862 (119868 + 119860 + sdot sdot sdot + 119860

119896minus1) 119861119906119905+ 119863119906119905

= 119862119860119896119909119905+ 119862 (119868 minus 119860)

minus1(119868 minus 119860

119896) 119861119906119905+ 119863119906119905

(23)

If we set 119910119905+119896

= 119879119888and

119908 = 119879119888minus 119862 (119868 minus 119860)

minus1119861119906119905minus 119863119906119905 (24)

then (23) takes the form

119908 = 119862119860119896119909119905minus 119862 (119868 minus 119860)

minus1119860119896119861119906119905

= 119862119881Λ119896119881minus1119909119905minus 119862 (119868 minus 119860)

minus1119881Λ119896119881minus1119861119906119905

(25)

where we have considered the diagonalization of the matrix119860 as 119860 = 119881Λ119881

minus1 with Λ being the diagonal matrix of theeigenvalues and 119881 being the matrix of the correspondingeigenvectors Numerical methods can be used to solve (25)for 119896 In a similar manner we compute 119878off

119894(119905) In this case

however we first repeat (5) 119863 times with input-vector 119906119896

reflecting the fact that the heater is off and then starting fromthe new state 119909

119905+119863we estimate the time needed to reach 119879

119888if

the heater is on

32 On the Distribution of 119884119899(119905) The random variable 119884

119899(119905)

expresses the remaining time from the current time instant 119905until entrance to zone 119911

119899 That is we assume that occupants

move around as a swarm visiting zones either in a prede-termined order or in a random fashion and finally they endup in zone 119911

119899 Each time the occupants visit a zone 119911

ℓthey

znz120001

Figure 2 Movement of occupants from zone 119911ℓto the adjacent

unoccupied zone 119911119899

stay in it for a nonzero time period which is itself a ran-dom variable with distribution 119865

ℓ(119909) We use the variable 119905

to designate calendar time the time elapsed since the processstarted and 120591 to designate the time elapsed since entry to azone The zone occupation time distribution 119865

ℓ(119909) may be

given parametrically either a priori or inferred from exam-ination of historic data by suitable fitting of the distributionrsquosparameters We may use the empirical distribution func-tion (edf) instead which is the nonparametric estimate of119865ℓ(119909) That is if a sample of119873 occupation periods of zone 119911

ℓ

1199091 1199092 119909

119873 is available then

ℓ(119909) =

1

119873

119873

sum

119898=1

119868 (119909119898isin [0 119909]) (26)

where 119868(sdot) is the indicator functionLet us consider first the simple case of two adjusting zones

119911ℓ 119911119899with zone 119911

ℓbeing occupied and 119911

119899unoccupied as it is

shown in Figure 2 We assume that the occupants remain tozone 119911

ℓfor a random period of time 119883

ℓwhich is distributed

according to 119865ℓ(119909) and then upon departure they enter zone

119911119899 In this case 119884

119899(119905) expresses the remaining waiting time

to zone 119911ℓand follows the residual life distribution which is

defined by

119877119884119899(119905)

(119910) = 1 minus 119877119884119899(119905)

(119910) = 119875 119883ℓgt 120591 + 119910 | 119883

ℓgt 120591

=119865ℓ(120591 + 119910)

119865ℓ(120591)

(27)

Next we consider the case that occupants are currently ina zone (call it 119911

0) and upon departure they visit in cascade

zones 1199111 1199112 119911

119899as it is shown in Figure 3 Then

119884119899(119905) = 119884

0+ 1198831+ 1198832+ sdot sdot sdot + 119883

119899minus1 (28)

where 1198840is the remaining time in zone 119911

0and 119883

119894is the

occupation period of zone 119911119894 It is well known that the

distribution of 119884119899(119905) is the convolution of the distributions of

the random variables 1198840 1198831 119883

119899minus1 That is

119865119884119899(119905)

(119910) = 119875 119884119899(119905) le 119910

= 1198770(119910) ⋆ 119865

1(119910) ⋆ 119865

2(119910) ⋆ sdot sdot sdot ⋆ 119865

119899minus1(119910)

(29)

with 1198770(119910) = 1 minus 119865

0(120591 + 119910)119865

0(120591) A more compact form is

obtained by considering a transform of the distributions that

Journal of Energy 7

zn

z0 z1

Figure 3 Movement of occupants in cascade from zone 1199110to zone

119911119899

is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865

119883(119909) of a nonnega-

tive random variable119883 is defined by

119865119883(119904) = int

infin

0

119890119904119909119889119865119883(119909) (30)

(29) takes the form

119865119884119899(119905)

(119904) = 1198770(119904) sdot 119865

1(119904) sdot 119865

2(119904) sdot sdot 119865

119899minus1(119904) (31)

In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865

119884119899(119905)(119910)

For example if the occupation periods 119883119894 119894 = 0 119899 minus 1

are exponentially distributed with corresponding parameters120582119894 then 119884

0is also exponentially distributed with parameter

1205820due to the memoryless property of the exponential

distribution and the probability density function of 119884119899(119905) is

(assuming that the parameters 120582119894are all distinct)

119891119884119899(119905)

(119909) =

119899minus1

sum

119894=0

1205820sdot sdot sdot 120582119899minus1

prod119899minus1

119895=0119895 =119894(120582119895minus 120582119894)

exp (minus119909120582119894) (32)

(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891

119884119899(119905)(119909) does not exist or it is difficult to be

obtained we resort to the saddlepoint approximation [11] toinvert 119865

119884119899(119905)(119904) To this end for a random variable 119883 with

cumulative distribution function (cdf) 119865119883(119909) we define the

cumulant generating function (cgf) 119870119883(119904) as the logarithm

of the corresponding mfg 119865119883(119904) that is

119870119883(119904) = log119865

119883(119904) (33)

Then the saddlepoint approximation of the probability den-sity function (pdf) 119891

119883(119909) is

119891119883(119909)

asymp (1

212058711987010158401015840119909( (119909))

)

12

exp 119870119883( (119909)) minus (119909) 119909

(34)

where (119909) is the solution to the saddlepoint equation

1198701015840

119883(119904) = 119909 (35)

Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865

119883(119904) Note that wemay

use the empirical mgf 119883(119904) instead so that a nonparametric

estimation of the pdf 119891119883(119909) is possible

The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901

119894119895 Given a transition from zone 119911

119894to

zone 119911119895 the occupation time in 119911

119894has distribution function

119865119894119895(119909) The matrix Q(119909) with elements 119876

119894119895(119909) = 119901

119894119895119865119894119895(119909)

is the so called semi-Markov kernel The occupation timedistribution in zone 119911

119894independent of the state to which a

transition is made is 119865119894(119909) = sum

119895119876119894119895(119909) Consider now the

case that at time 119905 occupants exist in zone 119911119894and the node of

zone 119911119899has tomake a decision onwhether to start heating the

zone or not For the decision process the distribution of timeuntil entrance to zone 119911

119899 119884119899(119905) is needed 119884

119899(119905) is the sum

of two terms The first term is the residual occupation timeat zone 119911

119894and the second term is the time to reach 119911

119899upon

departure from zone 119911119894 The distribution 119877

119894(119909) of the first

term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911

119894

Thus if zone 119911119894neighbors119873

119894zones namely 119911

119896 119896 = 1 119873

119894

then the time to reach zone 119911119899upon departure is distributed

according to

119866119894119899(119909) =

119873119894

sum

119896=1

119901119894119896119867119896119899(119909) (36)

where119867119896119899(119909) is the distribution of the first passage from zone

119911119896to zone 119911

119899 Pyke [12] andMason [13 14] provided solutions

for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867

119894119895(119909) is given by

119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]

minus1)119889minus1

(37)

where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865

119894119895(119909) Notation 119872

119889denotes the

matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ

119894119895= (minus1)

119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of

119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get

1198671119899(119904) =

Δ1198991(119904)

Δ119899119899(119904)

(38)

By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899

8 Journal of Energy

I

III

II

IV

V

IXVI

VIII

VII

To

To

To

To

Figure 4 Multizone model

In summary

119865119884119899(119905)

(119904) = 119877119894(119904) sdot 119866

119894119899(119904) = 119877

119894(119904) sdot (

119873119894

sum

119896=1

119901119894119896119867119896119899(119904)) (39)

with 119867119896119899(119904) computed using (38) Next the saddlepoint

approximation technique can be used to calculate the density119891119884119899(119905)

(119909)We have tomention that the process of estimating the first

passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865

119894119895(119904)

is obtained Having in our disposal the transforms 119865119894119895(119904) we

can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867

119894119895(119904) using (38) Finally numerical

methods may be used to invert these transforms in order toobtain the necessary probability density functions

4 Simulation Results

We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])

Table 1 Training and testing parameters for outer temperaturemodel and heat gain

Walterrsquos model Heat gainPhase 119879max 119879min 119879

119898119902(119905)

Training 15 2 7 600Testing 16 1 6 700

Table 2 Training and testing zonesrsquo target temperatures

Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21

5 10 15 20 250Time of day (h)

0

5

10

15

Tem

pera

ture

To

(∘C)

Figure 5 Empirical estimation of 119879119900

In Figure 4 119879119900denotes the outside temperature which

is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879

119898

need to be provided in order to estimate 119879119900 As an example if

119879max = 15∘C 119879min = 2

∘C and 119879119898= 7∘C the daily variations

of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm

41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879

119904=

1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892

to those of Table 2 (first row) with a margin equal to119879119898119892

= 1∘C That is for a specific zone the heater is on until

temperature119879119905119892+119879119898119892

is reached After this point the heater isturned off until temperature hits the lower threshold119879

119905119892minus119879119898119892

and the whole process is repeated

Journal of Energy 9

0 1 2 3 4 5 6 7 8 9 10 11Order

Zone VZone I

Zone II

10minus15

10minus10

10minus5

100

Sing

ular

val

ues

Figure 6 Singular values of zones I II and V

Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2

Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase

We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

02

4

6

8

10

12

14

16

18

20

Time (sec)

Zone I (real)

Zone I (predicted)

Zone IV (real)

Zone IV (predicted)

Tem

pera

ture

Figure 7 Real and predicted temperature of zones I and IV

zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model

Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is

119891119883(119909) =

120573120572

Γ (120572)119909120572minus1

119890minus120573119909

(40)

with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862

119894

The results are averages over 200 runs of the simulation Thecase of 119862

119894= 0 is the full reactive case where heating of

zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862

119894 the decision process

acts proactively by spending energy in order to reduce thediscomfort level

Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone

IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases

10 Journal of Energy

0

100

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 8 Total comfort cost versus119862119894 The occupation time of zone

I is Gamma distributed

15

2

25

3

Tota

l ene

rgy

cost

(kW

h)

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 9 Total energy cost versus 119862119894 The occupation time of zone

I is Gamma distributed

In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty

The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is

119891119883(119909) =

120572119909120572

119898

119909120572+1 119909 ge 119909

119898

0 119909 lt 119909119898

(41)

Zone I

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Zone IV(C = 0)

Zone IV(C = 10000)

Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0

1000 2000 3000 4000 5000 6000 70000Time (sec)

0

05

1

15

2

25

3

35Pa

reto

pdf

times10

minus3

xm = 300

xm = 900

xm = 1800

xm = 3600

Figure 11 Pareto pdf for various values of 119909119898

with expected value

119864 [119883] =

120572119909119898

120572 minus 1 120572 gt 1

infin 120572 le 1

(42)

Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=

300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases

Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862

119894 As it is observed a 50 improvement on the

comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909

119898= 300) which is due

to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909

119898

Journal of Energy 11

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

xm = 300xm = 900

xm = 3600

Figure 12 Total comfort cost for Pareto distributed occupationperiod

xm = 300

xm = 900

xm = 1800

xm = 3600

1

15

2

25

Tota

l ene

rgy

cost

(kW

h)

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

Figure 13 Total energy cost for Pareto distributed occupationperiod

and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13

Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution

Table 3 Parameters used in cascade movement scenario

Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20

Table 4 Energy and comfort costs for constant occupancy times

Comfort weight 119862 500 1000 1500 2000

119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020

119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025

I

IV

V

II

III VI

VII

VIII

IX

Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)

The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on

Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess

12 Journal of Energy

Fixed proactivecomfort cost

Energy cost

Comfort cost

1000

1500

2000

2500

3000

3500

4000

4500

Tota

l com

fort

cost

600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)

8

85

9

95

10

105

11

Tota

l ene

rgy

cost

(kW

h)

proactiveFixed

energy cost

Figure 15 Total energy and comfort costs (cascade movement sce-nario)

Zone I

Zone IX

Zone II

Zone VIII

Zone V

Zone IV

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Figure 16 Temperature profiles of zones (cascade movement sce-nario)

Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process

5 Conclusions and Future Work

Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone

to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique

Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862

119894may depend

on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov

[2] US Energy Information Administration June 2015 httpwwweiagov

[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013

[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013

[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011

[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013

[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014

[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014

[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015

Journal of Energy 13

[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996

[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954

[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961

[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953

[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956

[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005

[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015

[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967

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Renewable Energy

Submit your manuscripts athttpwwwhindawicom

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High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of Energy 7

zn

z0 z1

Figure 3 Movement of occupants in cascade from zone 1199110to zone

119911119899

is the Laplace transform the moment generating functionor the characteristic function Using the moment generatingfunction (mgf) which for a distribution 119865

119883(119909) of a nonnega-

tive random variable119883 is defined by

119865119883(119904) = int

infin

0

119890119904119909119889119865119883(119909) (30)

(29) takes the form

119865119884119899(119905)

(119904) = 1198770(119904) sdot 119865

1(119904) sdot 119865

2(119904) sdot sdot 119865

119899minus1(119904) (31)

In some cases (unfortunately few) (26) can be inversedanalytically in order to calculate the distribution 119865

119884119899(119905)(119910)

For example if the occupation periods 119883119894 119894 = 0 119899 minus 1

are exponentially distributed with corresponding parameters120582119894 then 119884

0is also exponentially distributed with parameter

1205820due to the memoryless property of the exponential

distribution and the probability density function of 119884119899(119905) is

(assuming that the parameters 120582119894are all distinct)

119891119884119899(119905)

(119909) =

119899minus1

sum

119894=0

1205820sdot sdot sdot 120582119899minus1

prod119899minus1

119895=0119895 =119894(120582119895minus 120582119894)

exp (minus119909120582119894) (32)

(The expression for the general case of nondistinct parame-ters is slightly more involved but still exists) If a closed formfor the density 119891

119884119899(119905)(119909) does not exist or it is difficult to be

obtained we resort to the saddlepoint approximation [11] toinvert 119865

119884119899(119905)(119904) To this end for a random variable 119883 with

cumulative distribution function (cdf) 119865119883(119909) we define the

cumulant generating function (cgf) 119870119883(119904) as the logarithm

of the corresponding mfg 119865119883(119904) that is

119870119883(119904) = log119865

119883(119904) (33)

Then the saddlepoint approximation of the probability den-sity function (pdf) 119891

119883(119909) is

119891119883(119909)

asymp (1

212058711987010158401015840119909( (119909))

)

12

exp 119870119883( (119909)) minus (119909) 119909

(34)

where (119909) is the solution to the saddlepoint equation

1198701015840

119883(119904) = 119909 (35)

Estimation (34) relies on numerical computations based oncgf119870119883(119878) which in turn depends on 119865

119883(119904) Note that wemay

use the empirical mgf 119883(119904) instead so that a nonparametric

estimation of the pdf 119891119883(119909) is possible

The more general case is to model the movement of theoccupants in themultizone system as a semi-Markov processThe states of the process are the zones of the system and at thetransition times they form an embedded Markov chain withtransition probabilities 119901

119894119895 Given a transition from zone 119911

119894to

zone 119911119895 the occupation time in 119911

119894has distribution function

119865119894119895(119909) The matrix Q(119909) with elements 119876

119894119895(119909) = 119901

119894119895119865119894119895(119909)

is the so called semi-Markov kernel The occupation timedistribution in zone 119911

119894independent of the state to which a

transition is made is 119865119894(119909) = sum

119895119876119894119895(119909) Consider now the

case that at time 119905 occupants exist in zone 119911119894and the node of

zone 119911119899has tomake a decision onwhether to start heating the

zone or not For the decision process the distribution of timeuntil entrance to zone 119911

119899 119884119899(119905) is needed 119884

119899(119905) is the sum

of two terms The first term is the residual occupation timeat zone 119911

119894and the second term is the time to reach 119911

119899upon

departure from zone 119911119894 The distribution 119877

119894(119909) of the first

term is given by (27) whereas for the second term we have toconsider the various paths takenupondeparture from state 119911

119894

Thus if zone 119911119894neighbors119873

119894zones namely 119911

119896 119896 = 1 119873

119894

then the time to reach zone 119911119899upon departure is distributed

according to

119866119894119899(119909) =

119873119894

sum

119896=1

119901119894119896119867119896119899(119909) (36)

where119867119896119899(119909) is the distribution of the first passage from zone

119911119896to zone 119911

119899 Pyke [12] andMason [13 14] provided solutions

for the first passage distribution in a semi-Markov processFor example Pykersquos formula states that119867(119904) the elementwisetransform of the matrix119867(119909) = 119867

119894119895(119909) is given by

119867(119904) = T (119904) [119868 minusT (119904)]minus1([119868 minusT (119904)]

minus1)119889minus1

(37)

where T(119904) is the transmittance matrix with the transformsT119894119895(119904) of the distributions 119865

119894119895(119909) Notation 119872

119889denotes the

matrix that is formed by the diagonal elements of 119872 Asimplified version of (37) is obtained if we let Δ(119904) = det(119868 minusT(119904)) and Δ

119894119895= (minus1)

119894+119895 det(119868 minus T(119904))119894119895 the 119894119895th cofactor of

119868 minusT(119904) Using this notation for a semi-Markov process of 119899states we get

1198671119899(119904) =

Δ1198991(119904)

Δ119899119899(119904)

(38)

By relabeling the states 1 119899 minus 1 of the process we canfind the transform of the first passage from any zone to zone119911119899

8 Journal of Energy

I

III

II

IV

V

IXVI

VIII

VII

To

To

To

To

Figure 4 Multizone model

In summary

119865119884119899(119905)

(119904) = 119877119894(119904) sdot 119866

119894119899(119904) = 119877

119894(119904) sdot (

119873119894

sum

119896=1

119901119894119896119867119896119899(119904)) (39)

with 119867119896119899(119904) computed using (38) Next the saddlepoint

approximation technique can be used to calculate the density119891119884119899(119905)

(119909)We have tomention that the process of estimating the first

passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865

119894119895(119904)

is obtained Having in our disposal the transforms 119865119894119895(119904) we

can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867

119894119895(119904) using (38) Finally numerical

methods may be used to invert these transforms in order toobtain the necessary probability density functions

4 Simulation Results

We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])

Table 1 Training and testing parameters for outer temperaturemodel and heat gain

Walterrsquos model Heat gainPhase 119879max 119879min 119879

119898119902(119905)

Training 15 2 7 600Testing 16 1 6 700

Table 2 Training and testing zonesrsquo target temperatures

Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21

5 10 15 20 250Time of day (h)

0

5

10

15

Tem

pera

ture

To

(∘C)

Figure 5 Empirical estimation of 119879119900

In Figure 4 119879119900denotes the outside temperature which

is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879

119898

need to be provided in order to estimate 119879119900 As an example if

119879max = 15∘C 119879min = 2

∘C and 119879119898= 7∘C the daily variations

of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm

41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879

119904=

1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892

to those of Table 2 (first row) with a margin equal to119879119898119892

= 1∘C That is for a specific zone the heater is on until

temperature119879119905119892+119879119898119892

is reached After this point the heater isturned off until temperature hits the lower threshold119879

119905119892minus119879119898119892

and the whole process is repeated

Journal of Energy 9

0 1 2 3 4 5 6 7 8 9 10 11Order

Zone VZone I

Zone II

10minus15

10minus10

10minus5

100

Sing

ular

val

ues

Figure 6 Singular values of zones I II and V

Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2

Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase

We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

02

4

6

8

10

12

14

16

18

20

Time (sec)

Zone I (real)

Zone I (predicted)

Zone IV (real)

Zone IV (predicted)

Tem

pera

ture

Figure 7 Real and predicted temperature of zones I and IV

zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model

Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is

119891119883(119909) =

120573120572

Γ (120572)119909120572minus1

119890minus120573119909

(40)

with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862

119894

The results are averages over 200 runs of the simulation Thecase of 119862

119894= 0 is the full reactive case where heating of

zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862

119894 the decision process

acts proactively by spending energy in order to reduce thediscomfort level

Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone

IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases

10 Journal of Energy

0

100

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 8 Total comfort cost versus119862119894 The occupation time of zone

I is Gamma distributed

15

2

25

3

Tota

l ene

rgy

cost

(kW

h)

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 9 Total energy cost versus 119862119894 The occupation time of zone

I is Gamma distributed

In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty

The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is

119891119883(119909) =

120572119909120572

119898

119909120572+1 119909 ge 119909

119898

0 119909 lt 119909119898

(41)

Zone I

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Zone IV(C = 0)

Zone IV(C = 10000)

Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0

1000 2000 3000 4000 5000 6000 70000Time (sec)

0

05

1

15

2

25

3

35Pa

reto

pdf

times10

minus3

xm = 300

xm = 900

xm = 1800

xm = 3600

Figure 11 Pareto pdf for various values of 119909119898

with expected value

119864 [119883] =

120572119909119898

120572 minus 1 120572 gt 1

infin 120572 le 1

(42)

Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=

300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases

Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862

119894 As it is observed a 50 improvement on the

comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909

119898= 300) which is due

to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909

119898

Journal of Energy 11

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

xm = 300xm = 900

xm = 3600

Figure 12 Total comfort cost for Pareto distributed occupationperiod

xm = 300

xm = 900

xm = 1800

xm = 3600

1

15

2

25

Tota

l ene

rgy

cost

(kW

h)

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

Figure 13 Total energy cost for Pareto distributed occupationperiod

and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13

Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution

Table 3 Parameters used in cascade movement scenario

Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20

Table 4 Energy and comfort costs for constant occupancy times

Comfort weight 119862 500 1000 1500 2000

119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020

119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025

I

IV

V

II

III VI

VII

VIII

IX

Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)

The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on

Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess

12 Journal of Energy

Fixed proactivecomfort cost

Energy cost

Comfort cost

1000

1500

2000

2500

3000

3500

4000

4500

Tota

l com

fort

cost

600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)

8

85

9

95

10

105

11

Tota

l ene

rgy

cost

(kW

h)

proactiveFixed

energy cost

Figure 15 Total energy and comfort costs (cascade movement sce-nario)

Zone I

Zone IX

Zone II

Zone VIII

Zone V

Zone IV

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Figure 16 Temperature profiles of zones (cascade movement sce-nario)

Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process

5 Conclusions and Future Work

Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone

to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique

Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862

119894may depend

on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov

[2] US Energy Information Administration June 2015 httpwwweiagov

[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013

[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013

[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011

[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013

[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014

[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014

[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015

Journal of Energy 13

[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996

[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954

[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961

[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953

[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956

[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005

[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015

[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967

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Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombustionJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

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Nuclear InstallationsScience and Technology of

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Solar EnergyJournal of

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Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

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High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

8 Journal of Energy

I

III

II

IV

V

IXVI

VIII

VII

To

To

To

To

Figure 4 Multizone model

In summary

119865119884119899(119905)

(119904) = 119877119894(119904) sdot 119866

119894119899(119904) = 119877

119894(119904) sdot (

119873119894

sum

119896=1

119901119894119896119867119896119899(119904)) (39)

with 119867119896119899(119904) computed using (38) Next the saddlepoint

approximation technique can be used to calculate the density119891119884119899(119905)

(119909)We have tomention that the process of estimating the first

passage distributions needs to be performed only prior to thedecision process Thus given the topology of the multizonesystem we posit a parametric family for the occupation timedistributions and we estimate the distributionsrsquo parametersfrom sample data Then the transition probabilities areestimated based on the data and the occupation mgf 119865

119894119895(119904)

is obtained Having in our disposal the transforms 119865119894119895(119904) we

can form the transmittance matrix T(119904) and solve for thefirst passage transform 119867

119894119895(119904) using (38) Finally numerical

methods may be used to invert these transforms in order toobtain the necessary probability density functions

4 Simulation Results

We simulated a multizone system consisting of a squaredarrangement of rooms (zones) where each room (zone) isequipped with a wireless sensor node as shown in Figure 4We assume that heat transfer takes place between roomrsquos airand the walls as well as between roomrsquos air and the roofFurthermore we neglect the effects of ground on the roomtemperature North and south walls have the same effect onthe room temperature and we assume the same for the eastand west walls According to these assumptions there is asymmetry in the dynamics of the zones For example zonesI III VII and IX exhibit the same behavior Note that forthe multizone system of Figure 4 several state variables suchas wall temperatures are common to the individual zonesystems In the simulations we treat the multizone system asa single system with 42 states and nine outputs (the zonesrsquotemperatures)The parameter values for this lumped capacityzone model were taken from [15] (see also [16])

Table 1 Training and testing parameters for outer temperaturemodel and heat gain

Walterrsquos model Heat gainPhase 119879max 119879min 119879

119898119902(119905)

Training 15 2 7 600Testing 16 1 6 700

Table 2 Training and testing zonesrsquo target temperatures

Target temperaturesZone I II III IV V VI VII VIII IXTraining 16 18 16 15 14 13 12 16 14Testing 17 20 17 17 17 14 18 19 21

5 10 15 20 250Time of day (h)

0

5

10

15

Tem

pera

ture

To

(∘C)

Figure 5 Empirical estimation of 119879119900

In Figure 4 119879119900denotes the outside temperature which

is assumed to be uniform with no loss of generality Forsimulation purposes daily outside temperature variations areobtained using Waltersrsquo model [17] For this model the max-imum temperature of day 119879max the minimum temperatureof day 119879min and the mean of the 24 hourly temperatures 119879

119898

need to be provided in order to estimate 119879119900 As an example if

119879max = 15∘C 119879min = 2

∘C and 119879119898= 7∘C the daily variations

of temperature in Figure 5 are obtained which shows thattemperature is higher at 200ndash300 pm

41 Training Phase For the deterministic subsystem identi-fication we used 86400 samples with sampling period 119879

119904=

1 sec For the outside temperature we used Walterrsquos modelwith parameters given in the first row of Table 1 and a heatergain equal to 600W We set the zonesrsquo target temperatures119879119905119892

to those of Table 2 (first row) with a margin equal to119879119898119892

= 1∘C That is for a specific zone the heater is on until

temperature119879119905119892+119879119898119892

is reached After this point the heater isturned off until temperature hits the lower threshold119879

119905119892minus119879119898119892

and the whole process is repeated

Journal of Energy 9

0 1 2 3 4 5 6 7 8 9 10 11Order

Zone VZone I

Zone II

10minus15

10minus10

10minus5

100

Sing

ular

val

ues

Figure 6 Singular values of zones I II and V

Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2

Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase

We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

02

4

6

8

10

12

14

16

18

20

Time (sec)

Zone I (real)

Zone I (predicted)

Zone IV (real)

Zone IV (predicted)

Tem

pera

ture

Figure 7 Real and predicted temperature of zones I and IV

zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model

Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is

119891119883(119909) =

120573120572

Γ (120572)119909120572minus1

119890minus120573119909

(40)

with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862

119894

The results are averages over 200 runs of the simulation Thecase of 119862

119894= 0 is the full reactive case where heating of

zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862

119894 the decision process

acts proactively by spending energy in order to reduce thediscomfort level

Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone

IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases

10 Journal of Energy

0

100

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 8 Total comfort cost versus119862119894 The occupation time of zone

I is Gamma distributed

15

2

25

3

Tota

l ene

rgy

cost

(kW

h)

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 9 Total energy cost versus 119862119894 The occupation time of zone

I is Gamma distributed

In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty

The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is

119891119883(119909) =

120572119909120572

119898

119909120572+1 119909 ge 119909

119898

0 119909 lt 119909119898

(41)

Zone I

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Zone IV(C = 0)

Zone IV(C = 10000)

Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0

1000 2000 3000 4000 5000 6000 70000Time (sec)

0

05

1

15

2

25

3

35Pa

reto

pdf

times10

minus3

xm = 300

xm = 900

xm = 1800

xm = 3600

Figure 11 Pareto pdf for various values of 119909119898

with expected value

119864 [119883] =

120572119909119898

120572 minus 1 120572 gt 1

infin 120572 le 1

(42)

Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=

300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases

Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862

119894 As it is observed a 50 improvement on the

comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909

119898= 300) which is due

to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909

119898

Journal of Energy 11

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

xm = 300xm = 900

xm = 3600

Figure 12 Total comfort cost for Pareto distributed occupationperiod

xm = 300

xm = 900

xm = 1800

xm = 3600

1

15

2

25

Tota

l ene

rgy

cost

(kW

h)

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

Figure 13 Total energy cost for Pareto distributed occupationperiod

and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13

Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution

Table 3 Parameters used in cascade movement scenario

Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20

Table 4 Energy and comfort costs for constant occupancy times

Comfort weight 119862 500 1000 1500 2000

119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020

119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025

I

IV

V

II

III VI

VII

VIII

IX

Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)

The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on

Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess

12 Journal of Energy

Fixed proactivecomfort cost

Energy cost

Comfort cost

1000

1500

2000

2500

3000

3500

4000

4500

Tota

l com

fort

cost

600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)

8

85

9

95

10

105

11

Tota

l ene

rgy

cost

(kW

h)

proactiveFixed

energy cost

Figure 15 Total energy and comfort costs (cascade movement sce-nario)

Zone I

Zone IX

Zone II

Zone VIII

Zone V

Zone IV

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Figure 16 Temperature profiles of zones (cascade movement sce-nario)

Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process

5 Conclusions and Future Work

Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone

to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique

Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862

119894may depend

on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov

[2] US Energy Information Administration June 2015 httpwwweiagov

[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013

[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013

[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011

[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013

[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014

[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014

[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015

Journal of Energy 13

[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996

[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954

[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961

[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953

[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956

[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005

[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015

[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967

TribologyAdvances in

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International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

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High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of Energy 9

0 1 2 3 4 5 6 7 8 9 10 11Order

Zone VZone I

Zone II

10minus15

10minus10

10minus5

100

Sing

ular

val

ues

Figure 6 Singular values of zones I II and V

Having collected the data over the period of 24 hoursthe measurements of the zonesrsquo temperature the outer tem-perature and the heat gains of each zone are organized intoinput-output data for the subsystem identification processFor example for zone I the outer temperature the heatinggain of zone I and the temperatures of zones II and IVare the input data to the identification process whereas thetemperature of zone I itself is the output data Note that thenumber of input signals differs fromzone to zone Zones I IIIVII and IX use 4 input signals whereas the rest of the zonesuse 5 input signals Based on the input-output datawe identifythe matrices 119860 119861 119862 and 119863 for each zone as described inSection 3 During this process we have to decide on the orderof the subsystems This is achieved by looking at the singularvalues of the SVD decomposition in (20) and deciding onthe number of dominant ones Figure 6 depicts the singularvalues for zones I II and V All subsystems exhibit similarbehavior regarding the profile of their singular values andtherefore we set the order of all subsystems equal to 2

Next we run a test on the obtained state-space modelsWe set the outer temperature parameters as in the secondrow of Table 1 and the target temperatures as in the secondrow of Table 2 The initial state of the subsystem model wasset to [minus10 0] whereas the elements of the 42-state vector ofthe multizone system were set equal to 3 (the initial outputtemperature) Figure 7 shows the evolution of the actual andpredicted temperatures for zones I and IV For zone I theheater was on (heat gain equal to 700) whereas the heaterfor zone IV was off As it is observed after 5000 samples(appoximately 80 minutes period) the state of the subsystemshas converged to one that produces almost the same outputas the original system After this point the WSN nodes canenter in the ldquodecisionrdquo phase

We should notice at this point that the ldquooriginalrdquo simu-lated system (42-state-space model) is also linear and thusthe use of linear subspace identification may be questionablefor more complex and possibly nonlinear systems As thesimulation results indicate although the dynamics of each

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

02

4

6

8

10

12

14

16

18

20

Time (sec)

Zone I (real)

Zone I (predicted)

Zone IV (real)

Zone IV (predicted)

Tem

pera

ture

Figure 7 Real and predicted temperature of zones I and IV

zone are more complex (including the roof temperatureeg)a low dimensionality subsystem of order 2 can capture itsbehavior For more complex systems we can choose the orderof the identified subsystem to be high enough For nonlinearsystems the identification of a time varying system is possibleby using a recursive update of the model

Next we simulate the scenario of Figure 2 We assumethat occupants enter zone I (at time instant 5000) and thenafter a random period of time they enter zone IV where theyremain for 3600 sec Zone I is heated since the beginning ofthe process (heat gain 900W) until the occupants leave thezone to enter zone IV Zone IV uses a heater with gain 1200Wwhich is turned on (or off) according to the decisions of itsnode The target temperatures for zones I and IV are 17∘Cand 16∘C respectively For the first set of experiments theoccupation period of zone I is Gamma distributed that is

119891119883(119909) =

120573120572

Γ (120572)119909120572minus1

119890minus120573119909

(40)

with the shape parameter 120572 = 5 and 120573 such that the expectedvalue of 119883 119864[119883] = 120572120573 is 3600 and 7200 Decisions weretaken every119863 = 67 seconds Figure 8 shows the total comfortcost achieved for various values of the comfort weight 119862

119894

The results are averages over 200 runs of the simulation Thecase of 119862

119894= 0 is the full reactive case where heating of

zone IV is postponed until occupants enter the zone On theother extreme for high values of 119862

119894 the decision process

acts proactively by spending energy in order to reduce thediscomfort level

Figure 9 shows the total energy cost in KWh for the twocases of Figure 8 As it is observed the higher the value of119862119894the more the energy consumed since then heating of zone

IV starts earlier The difference between the two curves ofFigure 9 is justified by the difference of the means of theoccupation period 119864[119883] = 3600 and 119864[119883] = 7200 for thetwo cases

10 Journal of Energy

0

100

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 8 Total comfort cost versus119862119894 The occupation time of zone

I is Gamma distributed

15

2

25

3

Tota

l ene

rgy

cost

(kW

h)

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 9 Total energy cost versus 119862119894 The occupation time of zone

I is Gamma distributed

In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty

The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is

119891119883(119909) =

120572119909120572

119898

119909120572+1 119909 ge 119909

119898

0 119909 lt 119909119898

(41)

Zone I

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Zone IV(C = 0)

Zone IV(C = 10000)

Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0

1000 2000 3000 4000 5000 6000 70000Time (sec)

0

05

1

15

2

25

3

35Pa

reto

pdf

times10

minus3

xm = 300

xm = 900

xm = 1800

xm = 3600

Figure 11 Pareto pdf for various values of 119909119898

with expected value

119864 [119883] =

120572119909119898

120572 minus 1 120572 gt 1

infin 120572 le 1

(42)

Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=

300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases

Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862

119894 As it is observed a 50 improvement on the

comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909

119898= 300) which is due

to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909

119898

Journal of Energy 11

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

xm = 300xm = 900

xm = 3600

Figure 12 Total comfort cost for Pareto distributed occupationperiod

xm = 300

xm = 900

xm = 1800

xm = 3600

1

15

2

25

Tota

l ene

rgy

cost

(kW

h)

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

Figure 13 Total energy cost for Pareto distributed occupationperiod

and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13

Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution

Table 3 Parameters used in cascade movement scenario

Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20

Table 4 Energy and comfort costs for constant occupancy times

Comfort weight 119862 500 1000 1500 2000

119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020

119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025

I

IV

V

II

III VI

VII

VIII

IX

Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)

The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on

Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess

12 Journal of Energy

Fixed proactivecomfort cost

Energy cost

Comfort cost

1000

1500

2000

2500

3000

3500

4000

4500

Tota

l com

fort

cost

600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)

8

85

9

95

10

105

11

Tota

l ene

rgy

cost

(kW

h)

proactiveFixed

energy cost

Figure 15 Total energy and comfort costs (cascade movement sce-nario)

Zone I

Zone IX

Zone II

Zone VIII

Zone V

Zone IV

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Figure 16 Temperature profiles of zones (cascade movement sce-nario)

Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process

5 Conclusions and Future Work

Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone

to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique

Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862

119894may depend

on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov

[2] US Energy Information Administration June 2015 httpwwweiagov

[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013

[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013

[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011

[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013

[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014

[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014

[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015

Journal of Energy 13

[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996

[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954

[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961

[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953

[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956

[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005

[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015

[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

10 Journal of Energy

0

100

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 8 Total comfort cost versus119862119894 The occupation time of zone

I is Gamma distributed

15

2

25

3

Tota

l ene

rgy

cost

(kW

h)

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

00

Comfort (C)

E[X] = 3600

E[X] = 7200

Figure 9 Total energy cost versus 119862119894 The occupation time of zone

I is Gamma distributed

In Figure 10 we plot the temperature profiles for zones IIV and for two different values of119862 As it is observed the valueof 119862 = 0 corresponds to the reactive case that is the heater ofzone IV remains off until occupants are detected in the zoneOn the other extreme a large value of 119862 (10000) will force theheater of zone IV to be turned on as soon as possible in aneffort to reduce the discomfort penalty

The scenario of Figure 2 was also simulated for Paretodistributed occupation times of zone I The pdf for Paretodistribution is

119891119883(119909) =

120572119909120572

119898

119909120572+1 119909 ge 119909

119898

0 119909 lt 119909119898

(41)

Zone I

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Zone IV(C = 0)

Zone IV(C = 10000)

Figure 10 Temperature versus time for 119862 = 10000 and 119862 = 0

1000 2000 3000 4000 5000 6000 70000Time (sec)

0

05

1

15

2

25

3

35Pa

reto

pdf

times10

minus3

xm = 300

xm = 900

xm = 1800

xm = 3600

Figure 11 Pareto pdf for various values of 119909119898

with expected value

119864 [119883] =

120572119909119898

120572 minus 1 120572 gt 1

infin 120572 le 1

(42)

Figure 11 shows the Pareto pdf for four values of 119909119898(119909119898=

300 900 1800 3600) and 120572 suitably chosen so that the meanvalue of the occupation period is 7200 in all cases

Figures 12 and 13 show the total comfort cost and thetotal energy consumed respectively for various values of thecomfort gain119862

119894 As it is observed a 50 improvement on the

comfort cost (compared to the reactive case) can be achievedfor a value of119862 equal to 1000However there is a performancefloor (this is evident for the case of 119909

119898= 300) which is due

to the nature of the Pareto distribution Most of the samplesfor the occupation time period are concentrated close to 119909

119898

Journal of Energy 11

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

xm = 300xm = 900

xm = 3600

Figure 12 Total comfort cost for Pareto distributed occupationperiod

xm = 300

xm = 900

xm = 1800

xm = 3600

1

15

2

25

Tota

l ene

rgy

cost

(kW

h)

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

Figure 13 Total energy cost for Pareto distributed occupationperiod

and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13

Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution

Table 3 Parameters used in cascade movement scenario

Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20

Table 4 Energy and comfort costs for constant occupancy times

Comfort weight 119862 500 1000 1500 2000

119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020

119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025

I

IV

V

II

III VI

VII

VIII

IX

Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)

The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on

Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess

12 Journal of Energy

Fixed proactivecomfort cost

Energy cost

Comfort cost

1000

1500

2000

2500

3000

3500

4000

4500

Tota

l com

fort

cost

600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)

8

85

9

95

10

105

11

Tota

l ene

rgy

cost

(kW

h)

proactiveFixed

energy cost

Figure 15 Total energy and comfort costs (cascade movement sce-nario)

Zone I

Zone IX

Zone II

Zone VIII

Zone V

Zone IV

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Figure 16 Temperature profiles of zones (cascade movement sce-nario)

Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process

5 Conclusions and Future Work

Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone

to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique

Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862

119894may depend

on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov

[2] US Energy Information Administration June 2015 httpwwweiagov

[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013

[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013

[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011

[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013

[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014

[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014

[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015

Journal of Energy 13

[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996

[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954

[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961

[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953

[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956

[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005

[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015

[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of Energy 11

200

300

400

500

600

700

800

900

Tota

l com

fort

cost

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

xm = 300xm = 900

xm = 3600

Figure 12 Total comfort cost for Pareto distributed occupationperiod

xm = 300

xm = 900

xm = 1800

xm = 3600

1

15

2

25

Tota

l ene

rgy

cost

(kW

h)

100 200 300 400 500 600 700 800 900 10000Comfort gain (C)

Figure 13 Total energy cost for Pareto distributed occupationperiod

and therefore there is not enough time to preheat the nextvisiting zone which in turn implies high discomfort costsMoreover the heavy tail of the Pareto distribution causessome extremely high values of the occupation period whichresult in an increase of the total consumed energy as it can beobserved clearly from Figure 13

Next we simulate the scenario of Figure 3The occupantsmove in cascade from zone to zone as it is depicted inFigure 14 The target temperatures for the zones are given inthe first row of Table 3 whereas the heat gain of each zone isgiven in the second row of Table 3 For the occupancy timeof each zone we considered an exponential random variablewith mean provided in the last row of Table 3 Note thatthis assumption is the least favorable for adjacent zones dueto the memoryless property of the exponential distribution

Table 3 Parameters used in cascade movement scenario

Zone I IV V VIII IX VII III IITarget ∘C 17 16 17 17 16 14 17 18Power (KW) 09 12 09 09 09 12 09 12Mean time times 102 36 27 18 40 54 45 30 20

Table 4 Energy and comfort costs for constant occupancy times

Comfort weight 119862 500 1000 1500 2000

119863 = 67Comfort cost 2615 72 67 67Energy cost 775 907 973 1020

119863 = 167Comfort cost 2856 72 71 64Energy cost 779 911 977 1025

I

IV

V

II

III VI

VII

VIII

IX

Figure 14 Movement of occupants in cascade (zones I IV V VIIIIX VI III and II)

The total comfort and energy costs for this scenario areplotted in Figure 15 for various values of the comfort weight119862 The results are averages over 100 runs The two curvesfor each cost correspond to 119863 = 67 and 119863 = 167 respec-tively As it is observed considerable comfort gains areachieved for a moderate increase of the consumed energy InFigure 15 we also show the costs for a ldquofixedrdquo proactive sce-nario in which the heating of a zone starts as soon as occu-pants are detected to its neighbor zone For example whenoccupants enter zone IV the heater of zone V is turned onand so on

Table 4 shows the comfort and energy costs obtainedwhen there is no randomness on the occupancy times forvarious values of the comfort weight 119862 and for two differentvalues of 119863 (119863 = 67 and 167) The occupancy times of thezones were set to the mean values provided in the last row ofTable 3 As it is observed extremely low comfort costs can beachieved in this case which implies that the right modellingof the occupancy times is of paramount importance to theprocess

12 Journal of Energy

Fixed proactivecomfort cost

Energy cost

Comfort cost

1000

1500

2000

2500

3000

3500

4000

4500

Tota

l com

fort

cost

600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)

8

85

9

95

10

105

11

Tota

l ene

rgy

cost

(kW

h)

proactiveFixed

energy cost

Figure 15 Total energy and comfort costs (cascade movement sce-nario)

Zone I

Zone IX

Zone II

Zone VIII

Zone V

Zone IV

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Figure 16 Temperature profiles of zones (cascade movement sce-nario)

Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process

5 Conclusions and Future Work

Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone

to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique

Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862

119894may depend

on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov

[2] US Energy Information Administration June 2015 httpwwweiagov

[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013

[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013

[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011

[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013

[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014

[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014

[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015

Journal of Energy 13

[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996

[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954

[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961

[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953

[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956

[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005

[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015

[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

12 Journal of Energy

Fixed proactivecomfort cost

Energy cost

Comfort cost

1000

1500

2000

2500

3000

3500

4000

4500

Tota

l com

fort

cost

600 800 1000 1200 1400 1600 1800 2000 2200400Comfort weight (C)

8

85

9

95

10

105

11

Tota

l ene

rgy

cost

(kW

h)

proactiveFixed

energy cost

Figure 15 Total energy and comfort costs (cascade movement sce-nario)

Zone I

Zone IX

Zone II

Zone VIII

Zone V

Zone IV

2

4

6

8

10

12

14

16

18

20

Tem

pera

ture

5000 10000 150000Time (sec)

Figure 16 Temperature profiles of zones (cascade movement sce-nario)

Finally Figure 16 shows a sample of the zonesrsquo tempera-ture profile Note that the temperature of zone II althoughzone II is the last zone visited starts increasing (at a slowerrate) much earlier This is because zone II neighbors zones Iand V which are heated in earlier stages of the process

5 Conclusions and Future Work

Amethod that fuses temperature and occupancy predictionsin an effort to balance the thermal comfort condition andthe consumed energy in a multizone HVAC system waspresented Temperature predictions are based on a subspaceidentification technique used to model the thermal dynamicsof each zone independently This technique is quite robustand accurate predictions are possible after a convergenceperiod which may last 1-2 hours With regard to the occu-pancy predictions the decision process makes use of thedistribution of the first passage time from an occupied zone

to an unoccupied one Simulation results for different firstpassage distributions demonstrated the effectiveness of theproposed technique

Several extensions and modifications of the proposedmethod are possible The comfort parameter 119862

119894may depend

on the zone the number of occupants and the currentstatus of the zones Moreover this parameter may be timevariable that is different values may be used for day andnight hours Regarding the decision process itself there isno need that it be executed at regular time epochs If theenergy consumed by the wireless sensor nodes is an issuewe may take decisions at irregular time epochs dependingon the occupancy predictions of the zones The computationof the risks used by the decision process may take intoconsideration additional parameters that affect energy con-sumption andor the thermal comfort of the occupants Forexample open windows situation may be easily detected andincorporated suitably in the decision process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] US Energy Information Administration ldquoAnnual Energy Out-look 2015rdquo DOEEIA-0383 April 2015 httpwwweiagov

[2] US Energy Information Administration June 2015 httpwwweiagov

[3] S Lee Y Chon Y Kim R Ha and H Cha ldquoOccupancy pre-diction algorithms for thermostat control systems using mobiledevicesrdquo IEEE Transactions on Smart Grid vol 4 no 3 pp1332ndash1340 2013

[4] V L Erickson S Achleitner and A E Cerpa ldquoPOEM power-efficient occupancy-based energy management systemrdquo in Pro-ceedings of the 12th International Conference on InformationProcessing in Sensor Networks (IPSN rsquo13) pp 203ndash216 ACMPhiladelphia Pa USA April 2013

[5] V L Erickson M A Carreira-Perpinan and A E CerpaldquoOBSERVE occupancy-based system for efficient reduction ofHVAC energyrdquo in Proceedings of the 10th ACMIEEE Interna-tional Conference on Information Processing in Sensor Networks(IPSN rsquo11) pp 258ndash269 Chicago Ill USA April 2011

[6] B Balaji J Xuy A Nwokafory R Guptay and Y AgarwalldquoSentinel occupancy based HVAC actuation using existingWiFi infrastructure within commercial buildingsrdquo in Proceed-ings of the 11th ACMConference on Embedded Networked SensorSystems (SenSys rsquo13) Rome Italy November 2013

[7] BDong andK P Lam ldquoA real-timemodel predictive control forbuilding heating and cooling systems based on the occupancybehavior pattern detection and local weather forecastingrdquoBuilding Simulation vol 7 no 1 pp 89ndash106 2014

[8] J R Dobbs and B M Hencey ldquoPredictive HVAC control usinga Markov occupancy modelrdquo in Proceedings of the AmericanControl Conference (ACC rsquo14) pp 1057ndash1062 Portland OreUSA June 2014

[9] J Brooks S Kumar S Goyal R Subramany and P BarooahldquoEnergy-efficient control of under-actuated HVAC zones incommercial buildingsrdquo Energy and Buildings vol 93 pp 160ndash168 2015

Journal of Energy 13

[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996

[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954

[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961

[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953

[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956

[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005

[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015

[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of Energy 13

[10] P Van Overschee and B De Moor Subspace Identification forLinear Systems Theory-Implementation-Applications KluwerAcademic New York NY USA 1996

[11] H EDaniels ldquoSaddlepoint approximations in statisticsrdquoAnnalsof Mathematical Statistics vol 25 pp 631ndash650 1954

[12] R Pyke ldquoMarkov renewal processes with finitely many statesrdquoAnnals of Mathematical Statistics vol 32 pp 1243ndash1259 1961

[13] S J Mason ldquoFeedback theory-some properties of signal flowgraphsrdquo Proceedings of Institute of Radio Engineers vol 41 no9 pp 1144ndash1156 1953

[14] S JMason ldquoFeedback theorymdashfurther properties of signal flowgraphsrdquo Proceedings of the IRE vol 44 no 7 pp 920ndash926 1956

[15] B Tashtoush MMolhim andM Al-Rousan ldquoDynamic modelof an HVAC system for control analysisrdquo Energy vol 30 no 10pp 1729ndash1745 2005

[16] D Sklavounos E Zervas O Tsakiridis and J Stonham ldquoA sub-space identificationmethod for detecting abnormal behavior inHVAC systemsrdquo Journal of Energy vol 2015 Article ID 69374912 pages 2015

[17] A Walter ldquoNotes on the utilization of records from third orderclimatological stations for agricultural purposesrdquo AgriculturalMeteorology vol 4 no 2 pp 137ndash143 1967

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Renewable Energy

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

StructuresJournal of

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear InstallationsScience and Technology of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solar EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Wind EnergyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nuclear EnergyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014