A field investigation of the intermediate light switching by users

A field investigation of the intermediate light switching by users

David Lindelof *, Nicolas Morel

Solar Energy and Building Physics Laboratory (LESO-PB), EPFL, CH-1015 Lausanne, Switzerland

Abstract

This paper describes how data collected during a continuously running data acquisition program on the LESO building in Lausanne,

Switzerland, was used to measure the intermediate light switch probability by users as a function of current illuminance levels, i.e. the probability

for a given timestep that the user will switch on or off the electric lighting, excluding such actions that happen upon user entry to or exit from the

office. We assume such a probability to be independent of the user’s history and further derive some theoretical consequences of this postulate. In

particular, we show how a history-less user leads naturally to patterns of behaviour already observed in real buildings.

# 2006 Elsevier B.V. All rights reserved.

Keywords: Intermediate light switching; Visual comfort; Poisson process

www.elsevier.com/locate/enbuild

Energy and Buildings 38 (2006) 790–801

1. Introduction

Understanding the way users interact with building services

(blinds, electric lighting, cooling, ventilation, window opening,

etc.), and the impact of their use on the building’s total energy

consumption, helps us attain two goals. First, we may elaborate

better models of the user’s behaviour for simulation software

that will help building planners to predict and optimize the

energy use or the comfort provided to the user. Secondly,

advanced control algorithms may use this information to

increase their acceptance by users and help achieve energy

savings.

Several building simulation software packages that need a

good simulation of user’s behaviour are available [1,2] or will

shortly be [3,4]. The software packages just mentioned are all

based on the Lightswitch-2002 algorithm described in [2]. An

underlying assumption behind this algorithm is that users use

the manual controls at their disposal in a conscious and

consistent way, which allows us to predictively model their

behaviour.

The algorithm seeks also to model the intermediate light

switch-on probability, i.e. the probability that a user switches on

the artificial lighting without leaving or arriving in the office. It

uses a probability function that depends on the workplane

illuminance, derived from previous work by the author of the

algorithm [5]. For 5-min timesteps, it finds that the intermediate

* Corresponding author. Tel.: +41 21 693 55 56; fax: +41 21 693 27 22.

E-mail address: [email protected] (D. Lindelof).

0378-7788/$ – see front matter # 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.enbuild.2006.03.003

switch-on probability is about 2% between 0 and 200 lux

workplane illuminance, and sharply drops to about 0.002 for

higher illuminances. One purpose of this paper is to verify this

model.

More specifically, we will focus on the following themes

concerning lighting actions (daylight or electric lighting):

� The lapse of time between the entry of the user into the room

and the use of controls, or between use and subsequent exit

from the room;

� T
he probability of a user switching on or off the electric
lighting as a function of ambient illuminance levels;

� T
he lighting conditions immediately preceding and imme-
diately following a user’s action with the electric lighting;

� T
he correlation between the delay before the user action and
the illuminance level.

We will also discuss some theoretical consequences of the

modelling of users’ actions and their relationship to experi-

mentally observed data.

2. User simulation

Some models of user behaviour assume that the time

between user actions, given constant environmental conditions,

is a random variable distributed according to an exponential

distribution with sole parameter l satisfying l ¼ 1=T where Tis the average time before the action. In other words, its

probability density function is given by

flðtÞ ¼ lexp ð�ltÞ

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http://dx.doi.org/10.1016/j.enbuild.2006.03.003

D. Lindelof, N. Morel / Energy and Buildings 38 (2006) 790–801 791

This distribution is believed to hold, with different l parameters

of course, for most of the user’s actions, such as use of artificial

lighting controls, window opening or closing, and exit or

arrival.

The exponential distribution function is used for modelling

the occurrence of events ranging from earthquakes to phone

calls. Similarly, the time remaining until, for instance, the

user’s next opening of windows can be modelled in much the

same way as the time remaining until the user’s next phone

call.

This postulate is justified by strong evidence that the number

of user actions of a given kind for constant or near-constant

environmental conditions in a given time frame follows a

Poisson distribution. From that fact follows that the intervals

between events are distributed according to an exponential

distribution. In the case of user entry and exit, Wang et al. [6],

for example, has recently verified experimentally that the

duration of user absences from a room indeed follows an

exponential distribution.

The problem is that environmental conditions are seldom

constant in an office. Temperatures, air quality and illuminance

levels change over time. How are we to compute the probability

density function for user events under varying conditions,

assuming we know how to do so for constant conditions?

Consider the example of the use of artificial lighting

controls. If we could find a relationship between the time a user

tolerates given visual conditions and the variables describing

these visual conditions (e.g., the illuminance levels), then we

would be in an advantageous position to simulate the behaviour

of the user in a computer simulation of the building. This

section will describe how.

Let us assume that such a relationship exists between the

average time T before user action and the illuminance E, and

that we have found it. We do not specify this relationship; we

are in no position to do so yet. But we assume that the

probability density function of the time the user spends

before switching on the lights follows the exponential

distribution, and that its l parameter is given by some

function of E.

We begin at time t ¼ 0. What is now the probability PðTÞthat the user has not turned on the lights yet at a time t ¼ T? Let

us discretize the time between t ¼ 0 and t ¼ T into n equal

timesteps Dt ¼ T=n. Let us assume that these timesteps are

sufficiently small that the workplane illuminance can be taken

as constant during each timestep, noted Ei, with i running from

0 to n� 1. The corresponding l parameters are noted li.

The probability that the user did not switch on his lights

during the first timestep is high because Dt is small, but not

quite equal to unity. It is given by

PðDtÞ ¼ 1�Z Dt

0

fl0ðtÞ dt ¼ exp ð�Dtl0Þ

By the exponential distribution’s lack of memory, the prob-

ability that the user did not switch on the lights between Dt and

2Dt knowing that he did not do so between 0 and Dt is similarly

equal to exp ð�Dtl1Þ, and so on.

The probability that the user has not switched on his lights by

the time T is thus the product of all these probabilities. We obtain:

PðTÞ ¼Yn�1

i¼0

exp ð�liDtÞ ¼ exp

�� Dt

Xn�1

i¼0

li

�¼ exp ð�T lÞ;

where l ¼Pn�1

i¼0 li=n is the average of all li.

But if pðtÞ denotes the probability distribution function of the

time at which the user switches on the lights for non-constant

environmental conditions, then the following must hold:Z 1T

pðtÞ dt ¼ PðTÞ;

since the left-hand term is the probability that the user switches

on the lights at a time between T and1 and the right-hand term

is the probability that the user did not switch the lights on

between 0 and T. They are, of course, the same thing.

Replacing with the value for PðTÞ found previously, and

deriving with respect to T on both sides, we obtain:

pðtÞ ¼ lexp ð�ltÞ

Note that along the way we lost any reference to Dt, so this

formula holds for Dt vanishingly small. In fact, it does not hold

if Dt is too large for the environmental conditions to be

considered as constant.

We thus have an expression for the distribution function of

the time of user action for varying environmental conditions. It

is however difficult to use this expression in practice since it

cannot readily be integrated over time, due to the non-constant

l parameter.

A computer running a building simulation should therefore

rather compute the evolution of environmental conditions in

steps of Dt, and compute at each timestep PðDtÞwith parameter

l ¼ li where i indexes the timestep.

The question remains, how does one measure this l

parameter? One cannot put a user in his office, bid the sun and

the clouds not to move too much for the next hour and wait until

the user gets up and switches on his lights, repeat N times, and

derive the average waiting time. Rather, since a process obeying

an exponential distribution implicitly assumes that the

probability that the user should use his controls within the

next few minutes is independent of the user’s history, one

should rather measure for a given illuminance level what is the

probability p that the user should use his controls in a given

time window Dt. Then the average waiting time between two

user actions, if Dt is small enough, is 1=l ¼ Dt=p.

This probability p can be obtained over the course of a

measuring campaign by counting how many times the user

finds himself in a given illuminance, and how many times that

illuminance leads to a user action.

3. Methodology

The data representing the base material for the study

discussed in this paper is part of a continuous recording

program by LESO-PB (Solar Energy and Building Physics

Laboratory, EPFL, Lausanne, Switzerland), on the LESO

D. Lindelof, N. Morel / Energy and Buildings 38 (2006) 790–801792

building. The LESO building is a small office building (20

office rooms of around 20 m2 floor area each, about half of

them with a single user and the other half with two), hosting the

activities of LESO-PB (for a detailed description of the

building, see [7]). Additionally, this inhabited building has been

used for the experimentation of new passive solar systems and

advanced control algorithms for building services (heating,

blinds and electric lighting). It is equipped with a commercial

EIB building bus system, with the following sensors and

actuators for each room:

� Sensors: inside and outside air temperatures, diffuse and

m

ca

ac

ill

global solar radiation, solar illuminance, wind speed and

direction, interior illuminance,1 occupancy, window opening;

� A
Fig. 1. Boxplots of the times between user entry and his use of manual controls,
ctuators: blinds, electric lighting (continuous dimming),

heating;
per office. � U ser interface: blinds, electric lighting, heating temperature
setpoint.

Finally, the building is also equipped with a legacy central

data logger used for recording miscellaneous data such as room

temperatures at different positions (measuring, in effect, the

stratification of the temperature) or the electricity consumption

for each room (counting separately the heating and the other

appliances).

The EIB was installed in 2000. Since then, two research

projects focused on control systems have been carried out at

LESO-PB that made use of measurements on the LESO

building, AdControl [8–10] and Ecco-Build, for which no

publication is available yet. The data acquisition has been

running independently of any research project and been stored

to a MySQL database [11], representing slightly more than 2

years of continuous monitoring. For data due to user actions

alone, we have accumulated at the time of writing about three

million datapoints.

The data we consider covers the period from mid-November

2002 to mid-January 2005. Any time one of the physical

variables changes by more than an adjustable threshold, that

change is logged to the database together with the time of the

event. It is trivial to use this data to reconstruct a time series of

any variable for any given constant timestep.

The data was analyzed with the open-source R data analysis

environment [12].

4. Data analysis

4.1. Definition of a user action

In the following a user action is defined as a set of

interventions on individual controls available to the user not

more than 1 min apart from each other. For example, a user

might come in the morning, switch on the lights, and open the

1 We use Siemens brightness sensors GE 252, which are actually ceiling-

ounted luminance sensors shielded from the window’s luminance. They were

librated by Guillemin [8] with LMT reference luxmeters with an estimated

curacy of �1:9%. The conversion from the workspace’s luminance to its

uminance is a programmable feature of the sensor.

blinds. During the day, he or she might decide that the sunlight

is enough to illuminate the office so he or she switches off the

light but lowers the blinds sufficiently to prevent direct sunlight

from hitting the computer screen. These would be considered

two user actions. As long as the individual interventions are not

spaced more than 1 min apart, they are considered the same

user action.

The data recorded on the building bus is used each night to

rebuild a table with all user actions since data acquisition began.

Each entry of that table records the time of the beginning of the

action, the time of its end, as well as the time when the user

came into the office before the action occurred and the time at

which the user left the office afterwards. We also record which

control(s), among blinds, artificial lighting, or windows, were

affected.

4.2. Time between user action and user entry/exit

In this study, we would like to concentrate on those user

actions where the user was known to have been present in his

office for a certain time before using the controls. We also

naturally require that the user still be present in the office for

some time after using the controls, in order to filter out those

events where the user, for instance, switches off the lights

before going home in the evening.

Box plots of the times in seconds between user entry and

user action are given for each office in Fig. 1. The middle-bar in

each box is the median time. The box’s edges (noted t25 and t75)

are placed at the 25 and the 75% quantiles.2 The boxes are then

extended with so-called ‘‘whiskers’’ that extend to the most

extreme data point not further away from the box than 1.5 times

the interquartile range t75– t25. Any datapoints beyond the

whiskers (‘‘outliers’’) are plotted as small circles.

For instance, office 102 has 953 actions recorded. 244 of

these, or approximately 25%, happened within 3 s after the user

had entered the room, so the 25% quantile edge of the box is

2 I.e., the values below which we have respectively 25 and 75% of the total

number of events.


Fig. 2. Boxplots of the time between user’s use of manual controls and his

departure, per office.

placed at 3 s. 479 actions, practically half of all actions,

occurred within 31 s after user entry, so the median bar in the

box is placed at 31 s. Similarly, the upper edge of the box is

placed at 160 s.

The interquartile range is 160� 3 ¼ 157 s, so the upper

whisker is placed at the most extreme data point not exceeding

1:5� 157þ 160 ¼ 395:5 s. The highest such data point is at

393 s, so the upper whisker lands there. The lower whisker ends

Fig. 3. Distribution of illuminance levels before user actions, per office. The histogram

after the user action. On top of this the events where the illuminance decreased o

on the minimum of the data points, at 0 s. The remaining data

points (115 of them in total) are considered outliers.

It is apparent that for most offices, except offices 001, 104

and 201, three quarters of all user actions occurred less than

300 s, or 5 min, after the user entered the office. In other words,

users usually use the controls available to them while they are

‘‘on the move’’. Users do not leave their seats to adjust their

settings unless the situation is clearly uncomfortable.

Fig. 2 is also a box plot but of the times between the action

and the departure of the user. The distribution seems similar,

but note the shift upwards of the lower 25% quantile box

edge. This is due to the intrinsic 30-s timeout on the occu-

pancy sensors. Again, about 75% of all user actions happen

about 5 min before the user’s departure. In other words,

again, users use their controls mostly when coincidentally

passing by.

This has important consequences for the choice of

placement of the user’s controls in an office, which should

be as close and as convenient as possible to the user, who will

otherwise simply not use them. This observation also

highlights again the obvious need for smart building control

systems since users, unless particularly energy-conscious,

usually will not adjust their controls if the only benefit is the

saving of energy. A clear discomfort is required for the user to

take action.

s are first filled in black with events where the workplane illuminance increased

r stayed constant are added in white.


4.3. Intermediate light switching

By ‘‘intermediate light switching’’ we mean the act of using

the artificial lighting controls in circumstances other than upon

arrival to or before departure from the office.

We therefore select from the database those user actions that

concerned artificial lights only (i.e., no blinds action) and where

the user was present at least 5 min before, and at least 1 min

after the action.

For each such action we can query the database for the

values of physical variables 1 min before the beginning of the

action and 1 min after the end. For each office considered, a

histogram of horizontal workplane illuminance before and after

the action are given in Figs. 3 and 4, respectively. The sensors’

accuracy is estimated to be 15%. Their non-linear behaviour for

illuminances above 500 lux has been corrected by the EIB

monitoring software.

That the artificial lighting should increase the amount of

available workplane illuminance is hardly surprising. Neither

should one ascribe too much importance to the differences in

the distribution of illuminance after the action. Past certain

hours, in particular those hours where lighting is needed

most, the lighting can only provide so many lux and we doubt

that the users in office 101 have deliberately and system-

Fig. 4. Distribution of illuminance levels after user actions, per offi

atically fiddled with the dimming controls to get to the shown

average of about 300 lux. That value represents more likely a

rough estimate of the maximum workplane illuminance the

lighting can provide.

More interesting are the disparities seen in the distribution of

illuminance right before the action. Some users (e.g. 104) never

allow the illuminance to go below about 200 lux before turning

the lights on. Others (such as the people in office 001) seem less

bothered and tolerate even very low light levels before turning

the lights on.

Only office 004 shows odd results, but the measurement of

this office’s illuminance values is known to be faulty. A new

user moved in during 2003, as a result of which the main

luminaire (a lamp projecting its light on the ceiling) has been

moved right under the luminance sensor doubling as an

illuminance sensor. This mistake has now have been corrected,

but the data taken on this office will be excluded from further

analysis in this paper and will not be included in plots obtained

by pooling together all data. The histogram obtained by

lumping together all offices except office 004 are given in

Fig. 5.

Fig. 6 shows a scatterplot for each office of the illumina-

nce level after vs. before user action. Data points beneath

the diagonal represent events when the user found the

ce. See Fig. 3 for the explanation of the black and white bars.


Fig. 5. Distribution of illuminance, all data. See Fig. 3 for the explanation of the

black and white bars. (a) Before user action and (b) after user action.

light too strong and decreased or turned it off, while points

over the diagonal represent events where the illuminance

level was deemed insufficient. Fig. 7 groups together all

the events for all offices, with a small jitter applied to each

point, in order to prevent the discrete illuminance values

provided by the measurement from hiding the real data point

density.

There is a marked tendency for all offices to prefer switching

on the lights to switching them off, something coherent with our

personal experience. People are, in general, more concerned

with their visual comfort than with unnecessary energy

expenditures.3 Of all offices, only the occupant of office 104

seems to switch off somewhat regularly the lights when not

necessary.

Note also that for most offices, but most notably in offices

001, 002, 101, 104, 106, 203 and 204, there is a clear clustering

of the points along two diagonals running parallel to the main

diagonal, and equally distanced from that diagonal. This

reflects the fact that users usually don’t bother fiddling with the

light dimming commands and content themselves with

switching the lights on or off, thus resulting in a constant

increase or decrease of available illuminance of roughly

300 lux, the maximum the current light installation can

provide. This shows that even when dimming commands are

available, few users will make use of them in an energetically

3 Or even forget they have left the lights on.

optimal way when they are placed close to the office’s entrance

rather than close to the user.4

The occupants of offices 001, 104 and 201 obviously use

their electric lighting controls much more than the other

users. Since the placement of the electric lighting controls is

similar in all offices (close to the office’s entrance), one

can only conclude that these users are much more concerned

with a rational use of the artificial lighting than the others.

This leads credence to the notion of active vs. passive users

found in the Lightswitch-2002 model, which distinguishes

users based on their willingness to use the controls at their

disposal.

From now on we shall assume that all user actions on the

artificial lights are switch on/off events and neglect the

extremely rare dimming events.

4.4. Intermediate switch probability

We now turn to the determination of the intermediate switch

probability, i.e. the probability that the user will switch on or off

the lights in a given time window as a function of the

illuminance.

We choose a time window of 5 min, and slice up the periods

of user presence into periods of 5 min each, always beginning

5 min after the initial user entry into the room. Two periods of

presence separated by no more than 2 min absence are

considered as an uninterrupted presence.

We query the database for the value of the illuminance at

each such timestep, and check two things: whether an

interaction with the artificial lighting (alone) occurred within

the next 5 min, and whether any interaction with blinds or

artificial lighting occurred. Remember that all interactions with

the artificial lighting are assumed to be switching events.

If an interaction with the artificial lighting alone occurred,

we count it as an intermediate switch event. If no interaction

with the artificial lighting nor with the blinds occurred, we

count it as a situation where the user was satisfied with his

visual environment. If only an interaction with the blinds

occurred we exclude the timestep.

For a given range of illuminance values we can thus compute

the ratio between the number of times the user acted on the

artificial lighting at that illuminance, and the total number of

times the user spent at that illuminance without altering the

visual environment by means of the blinds.

We obtain thus respectively the switch-on and switch-off

probabilities for a time window of 5 min for different ranges of

illuminances. These probabilities are given for each individual

office in Figs. 8 and 9 respectively, and again for the combined

data from all offices in Fig. 10.

4.4.1. Switch-on probability

The behaviour of users with respect to switch-on probability

shows remarkable consistency. Most users seem to have an

4 A user even told one of the authors that she actually did not even know she

could dim her light.


Fig. 6. Illuminances recorded right after (Y-axis) vs. right before (X-axis), per office.

illuminance threshold under which the switch-on probability

sharply rises to a level of between 1% and 10%. As long as the

illuminance is above that threshold, the switch-on probability is

negligible. That threshold varies from user to user but lies

between 100 and 200 lux.

Fig. 7. Illuminances recorded right after vs. right before, all offices.

Fig. 10, obtained by lumping together the data from all office

rooms except rooms 003 and 004 (which both seem to display

an abnormal behaviour) shows that our average user has a

switch-on probability of about 3.3% between 0 and 100 lux,

which drops to about 1.4% between 100 and 200 lux, then to

about 0.6% between 200 and 300 lux, and which then becomes

more or less negligible.

Should the switch-on process be considered as a Poisson

process, the above figures would then correspond to an average

switch-on time of about 150, 360 and 830 min, respectively. In

terms of ‘‘half-lives’’, it means that if left in constant

conditions, half the users will have switched their lights on

after 104, 250 and 575 min, respectively (assuming anyone is

still left in the office).

Fig. 11 shows the combined switch-on probability for low

illuminance values, detailing what happens below 100 lux. We

see that below 50 lux the intermediate switch-on probability

continues to rise up to about 4%. It is difficult, however, to see

whether the probability should rise up to one for vanishing

illuminance.

The figures obtained are very comparable to the ones

proposed in the Lightswitch-2002 model [2], where the

intermediate switch-off probability almost constant and

equal to 2% between 0 and 200 lux, and drops to 0.002 for

higher illuminances. This model, and the Suntool program

based on it [3], further set this probability equal to 1 for zero


Fig. 8. Intermediate switch-on probability, per office.

illuminance. This makes arguably sense but cannot be

confirmed nor ruled out from our data alone, for a practical

reason: it is simply extremely unlikely that a user would allow

the workplane illuminance to drop to zero without switching

on the lights before, and hence, we do not have in our data

these events.

4.4.2. Switch-off probability

The switch-off probability poses more problems. The

statistics are poor; for instance there are roughly 10 times

less intermediate switch-off events in office 001 than switch-on

events, which explains the huge error bars on the graphics. This

indicates that the users switch off their lights mostly on the way

out of the office, rather than as an intermediate switch-off event.

Fig. 10 shows that the intermediate switch-off probability for

the data gathered from all users (except offices 003 and 004) is

rather flat and lies at roughly 0.1%, rather independently of the

illuminance value.

4.5. Correlation between action delay and illuminance

Some models of user behaviour postulate that the time

users tolerate a visual discomfort before deciding to use the

manual controls should be correlated to the level of their

discomfort. In other words, a user in a very dark or very

bright room will act on the controls earlier than a user in a

room whose visual environment is just at the discomfort

threshold.

The exact relationship between this delay and the level of

discomfort is a question left unanswered for the moment.

Furthermore, it is difficult to measure such a delay until

the user acts on the controls since, usually, visual conditions

in the room vary over time and the user acts only when

some discomfort threshold has been crossed. The best we

can do, since we have non-constant environmental condi-

tions, is to see if there is at least a correlation between the

illuminance at the time of the user’s action and the time since

the user’s entry in the room, in the hope that environmental

conditions remain more or less constant during the user’s

presence.

Unfortunately, as can be seen on Fig. 12 on a per-office basis

or on Fig. 13 for all data grouped together, there is no such

readily discernible pattern. However, this lack of a pattern could

be entirely due to two statistical reasons. First, users are

unlikely to allow lighting conditions at the end of the day to

degrade far beyond the discomfort limit and will thus deny us


Fig. 9. Intermediate switch-off probability, per office.

data points for low illuminance levels. In other words, data

points for high discomfort levels will not exist simply because

the users will have adjusted their controls before. Secondly,

users can tolerate lighting conditions just at the lower limit of

the discomfort zone (roughly 200–300 lux) indefinitely and

will switch the lights on only when moving close to the

Fig. 10. Intermediate switch probability, all data. (a) S

controls, again depriving us of data points for higher

illuminance levels closer to the comfort zone.

Finding a correlation between the time a user spends before

deciding to use manual controls (placed within arm’s length) at

given environmental conditions is probably a project best suited

for laboratory conditions, not a real-life building.

witch-on probability and (b) switch-off probability.


Fig. 11. Intermediate switch-on probability for low illuminances, all data. Fig. 13. Time between user entry and use of controls vs. illuminance levels, all

data.

Fig. 12. Time between user entry and use of controls vs. illuminance levels, per office.

5. Conclusion

Three-quarters of the LESO users’ use of manual controls

occur less than 5 min after their arrival in their office or before

their departure, most likely a direct consequence of the

traditional placement of their controls close to the door rather

than close to their desk.

We have measured intermediate switch-on probabilities for

the users of the LESO building as a function of horizontal

workplane illuminance for 5-min time intervals. The increase in

probability for lower illuminances is consistent between users

and can be taken as being equal to about 3.3% between 0 and

100 lux, to about 1.4% between 100 and 200 lux, to about 0.6%

between 200 and 300 lux, and negligible for higher illuminances.


5 We obtain exactly the same result if we consider the state of the window

over time as a Markov process with two states. The transition probability from

closed to open is Tco ¼ 1� exp ð�loDtÞ, and the one from open to closed is

Toc ¼ 1� exp ð�lcDtÞ. The Markov process asymptotically tends to a state in

which the probability of having the window closed is TocTcoþToc

, which is none

other than the equation above.

When it comes to actual use of electric lighting controls, we

have observed that the users (about 30 people) behave quite

differently from each other. It is remarkable that some users

give very frequent commands, while others seem more passive

and do not bother using the system. This lends credence to the

classification in the Lightswitch-2002 algorithm of users into

different types according to their dynamic or static use of

manual controls.

We have also realized that in the LESO building, where

lighting controls are placed close to the offices’ entrances, the

users very seldomly use the dimmable feature of their electric

lighting. They almost always switch it completely on or

completely off.

Acknowledgements

We wish to thank Jessen Page (LESO-PB/EPFL) for kindly

reviewing this paper and providing helpful and insightful

comments on the more mathematical aspects.

We thank Lee-Ann Nicol (LESO-PB/EPFL) for going

through this paper and leaving no page without comments for

improvements of style or english grammar.

We thank Denis Bourgeois (Ecole d’Architecture, Uni-

versite Laval, Canada) for having been the catalyst behind some

of the more theoretical developments in this work.

We thank Christoph Reinhart (National Research Council,

Canada) for going through this paper and offering his helpful

comments for improvement.

Appendix A. Link with Nicol’s probit function

For this final section we will try to establish a link between

user actions described by an exponential distribution and the

results given by Nicol in his 2001 paper [13].

It will be recalled that Nicol found that the fraction of offices

within a given building exhibiting a certain user behaviour

(such as having the windows open, or having the fans running)

was a function of an external stimulus; in his case, he considers

only the external temperature, a hypothesis born out by the

study by Fritsch et al. [14]. That function was given by pðxÞ ¼exp ðaþ bxÞ=ð1þ exp ðaþ bxÞÞ where x was the external

temperature and a and b were parameters to be fitted from

experimental data. We will see if we can derive this result from

our assumption about an exponential distribution of delays

between user actions.

Let us consider the case of window opening by the user. We

choose this example because, although the theory is similar, an

open window will usually be closed after a short while and

reopened again during the day, whereas a lamp that has been

turned on might remain on until the end of the day. Therefore,

we expect the average time between window openings and

closings to be shorter than the average time between the turning

on or off of artificial lighting, and it should be easier to verify

this theory with windows during field measurements. For this

reason, we are going to derive an expression for the fraction of

windows open after a long enough time, but which will hold

only for ‘‘reversible’’ user actions, i.e. actions that the user

effectively undoes during a day. The user will usually close a

window he or she has opened, but will seldom switch off a light

he or she has turned on before leaving the office.

Let us assume that the mean time before a user opens a

window is To, and once the window is open, the mean time

before the user closes it again is Tc. The distribution function of

the time remaining until the next user action is therefore an

exponential function with parameter lo ¼ 1=To or lc ¼ 1=Tc,

respectively.

For simplicity’s sake, let us assume all windows in the

building start closed. We choose a timestep Dt sufficiently small

so that the probability of the user both opening and closing the

window during that timestep is vanishingly small. If the

window begins a timestep closed, the probability that it should

be open at the next timestep is exp ð�lcDtÞ. Similarly, if the

window begins the timestep open, the probability that it should

be closed at the next timestep is exp ð�loDtÞ.At time t ¼ 0, the probability that the window is closed is

Pcð0Þ ¼ 1. At t ¼ Dt we have:

PcðDtÞ ¼ exp ð�loDtÞ

At t ¼ 2Dt the probability that the window is closed is given by

the probability that the window was closed at t ¼ Dt and that it

remained so at t ¼ 2Dt, plus the probability that it was open at

t ¼ Dt but that it closed again at t ¼ 2Dt:

Pcð2DtÞ ¼ PcðDtÞexp ð�loDtÞ þ ð1� PcðDtÞÞ

� ð1� exp ð�lcDtÞÞ

By recursion, we see that for arbitrary n,

PcðnDtÞ ¼ Pcððn� 1ÞDtÞexp ð�loDtÞ þ ð1� Pcððn� 1ÞDtÞÞ

� ð1� exp ð�lcDtÞÞ

Expanding the right-hand side all the way down to Pcð0Þ, we

finally obtain:

PcðnDtÞ ¼ ðexp ð�loDtÞ þ exp ð�lcDtÞ � 1Þn

þ ð1� exp ð�lcDtÞÞXn�1

i¼0

ðexp ð�loDtÞ

þ exp ð�lcDtÞ � 1Þi

For large n, and writing T ¼ nDt!1, the first term in that sum

vanishes and the second one is the sum of a geometric series.

We obtain thus:5

PcðT ¼ 1Þ ¼1� exp ð�lcDtÞ

1� exp ð�loDtÞ þ 1� exp ð�lcDtÞ


Taking the limit Dt! 0, we obtain:

limDt! 0

PcðT ¼ 1Þ ¼lc

lo þ lc

In his paper, Nicol observed that the fraction of open windows

obeyed a relationship with the outside temperature. Reordering

a little bit, he found essentially that the fraction of closed

windows could be written in the form:

PcðT ¼ 1Þ ¼1

1þ exp ðaþ btÞ

where t is the outside temperature (not the time). Identifying his

finding with ours, we see immediately that,

lo

lc

¼ exp ðaþ btÞ

We have just found a theoretical relationship between Nicol’s a

and b probit parameters and the average times between user’s

opening and closing of windows.

Notice also that the preceding equation can be rewritten by

taking lc ¼ 1=Tc, where Tc is the average time before window

closure, and similarly for average time before window opening,

and taking the logarithm on both sides:

log Tc � log To ¼ aþ bt

But since a was an arbitrary constant, and assuming indepen-

dence between To and t if the user’s need to open the window is

taken as independent of the outside temperature, we can

redefine it as a ¼ log To þ a, and we see thus that

log Tc ¼ aþ bt

In other words, we find a theoretical affine relationship between

the logarithm of the mean time before window closure, and the

outside temperature.

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A field investigation of the intermediate light switching by users

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