17 storage size determination for grid-connected

68 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 4, NO. 1, JANUARY 2013

Storage Size Determination for Grid-ConnectedPhotovoltaic Systems

Yu Ru, Member, IEEE, Jan Kleissl, and Sonia Martinez, Senior Member, IEEE

Abstract—In this paper, we study the problem of determiningthe size of battery storage used in grid-connected photovoltaic (PV)systems. In our setting, electricity is generated from PV and is usedto supply the demand from loads. Excess electricity generated fromthe PV can be either sold back to the grid or stored in a battery, andelectricitymust be purchased from the electric grid if the PV gener-ation and battery discharging cannot meet the demand. Due to thetime-of-use electricity pricing and net metered PV systems, elec-tricity can also be purchased from the grid when the price is low,and be sold back to the grid when the price is high. The objective isto minimize the cost associated with net power purchase from theelectric grid and the battery capacity loss while at the same timesatisfying the load and reducing the peak electricity purchase fromthe grid. Essentially, the objective function depends on the chosenbattery size. We want to find a unique critical value (denoted as

) of the battery size such that the total cost remains the sameif the battery size is larger than or equal to , and the cost isstrictly larger if the battery size is smaller than . We obtain acriterion for evaluating the economic value of batteries comparedto purchasing electricity from the grid, propose lower and upperbounds on , and introduce an efficient algorithm for calcu-lating its value; these results are validated via simulations.

Index Terms—Battery, grid, optimization, photovoltaic (PV).

I. INTRODUCTION

T HE need to reduce greenhouse gas emissions due to fossilfuels and the liberalization of the electricity market have

led to large-scale development of renewable energy generatorsin electric grids [1]. Among renewable energy technologies suchas hydroelectric, photovoltaic (PV), wind, geothermal, biomass,and tidal systems, grid-connected solar PV continued to be thefastest growing power generation technology, with a 70% in-crease in existing capacity to 13 GW in 2008 [2]. However,solar energy generation tends to be variable due to the diurnalcycle of the solar geometry and clouds. Storage devices (suchas batteries, ultracapacitors, compressed air, and pumped hydrostorage [3]) can be used to 1) smooth out the fluctuation of thePV output fed into electric grids (“capacity firming”) [2], [4],2) discharge and augment the PV output during times of peakenergy usage (“peak shaving”) [5], 3) store energy for night-time use, for example in zero-energy buildings and residen-tial homes, or 4) power arbitrage under time-of-use electricity

Manuscript received June 18, 2011; revised April 05, 2012; acceptedMay 04,2012. Date of publication June 12, 2012; date of current version December 12,2012.The authors are with the Mechanical and Aerospace Engineering Depart-

ment, University of California, San Diego, La Jolla, CA 92093 USA (e-mail:[email protected]; [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSTE.2012.2199339

pricing, i.e., storing surplus PV when the time-of-use price islow and selling when the time-of-use price is high.Depending on the specific application (whether it is off-grid

or grid-connected), battery storage size is determined based onthe battery specifications such as the battery storage capacity(and the minimum battery charging/discharging time). For off-grid applications, batteries have to fulfill the following require-ments: 1) the maximum discharging rate has to be larger than orequal to the peak load capacity; 2) the battery storage capacityhas to be large enough to supply the largest nighttime energyuse and to be able to supply energy during the longest cloudyperiod (autonomy). The IEEE standard [6] provides sizing rec-ommendations for lead–acid batteries in standalone PV systems.In [7], the solar panel size and the battery size have been se-lected via simulations to optimize the operation of a standalonePV system, which considers reliability measures in terms of lossof load hours, the energy loss, and the total cost. In contrast,if the PV system is grid-connected, autonomy is a secondarygoal; instead, batteries can reduce the fluctuation of PV outputor provide economic benefits such as demand charge reduction,and power arbitrage. The work in [8] analyzes the relation be-tween available battery capacity and output smoothing, and es-timates the required battery capacity using simulations. In addi-tion, the battery sizing problem has been studied for wind powerapplications [9]–[11] and hybrid wind/solar power applications[12]–[14]. In [9], design of a battery energy storage system isexamined for the purpose of attenuating the effects of unsteadyinput power from wind farms, and solution to the problem via acomputational procedure results in the determination of the bat-tery energy storage system’s capacity. Similarly, in [11], basedon the statistics of long-term wind speed data captured at thefarm, a dispatch strategy is proposed which allows the batterycapacity to be determined so as to maximize a defined servicelifetime/unit cost index of the energy storage system; then a nu-merical approach is used due to the lack of an explicit mathe-matical expression to describe the lifetime as a function of thebattery capacity. In [10], sizing and control methodologies for azinc–bromine flow battery-based energy storage system are pro-posed to minimize the cost of the energy storage system. How-ever, the sizing of the battery is significantly impacted by spe-cific control strategies. In [12], a methodology for calculatingthe optimum size of a battery bank and the PV array for a stand-alone hybrid wind/PV system is developed, and a simulationmodel is used to examine different combinations of the numberof PVmodules and the number of batteries. In [13], an approachis proposed to help designers determine the optimal design ofa hybrid wind–solar power system; the proposed analysis em-ploys linear programming techniques to minimize the averageproduction cost of electricity while meeting the load require-

1949-3029/$31.00 © 2012 IEEE

RU et al.: STORAGE SIZE DETERMINATION FOR GRID-CONNECTED PV SYSTEMS 69

ments in a reliable manner. In [14], genetic algorithms are usedto optimally size the hybrid system components, i.e., to selectthe optimal wind turbine and PV rated power, battery energystorage system nominal capacity, and inverter rating. The pri-mary objective is the minimization of the levelized cost of en-ergy of the island system.In this paper, we study the problem of determining the

battery size for grid-connected PV systems for the purpose ofpower arbitrage and peak shaving. The targeted applicationsare primarily electricity customers with PV arrays “behind themeter,” such as most residential and commercial buildings withrooftop PVs. In such applications, the objective is to minimizethe investment on battery storage (or equivalently, the batterysize if the battery technology is chosen) while minimizing thenet power purchase from the grid subject to the load, batteryaging, and peak power reduction (instead of smoothing outthe fluctuation of the PV output fed into electric grids). Oursetting1 is shown in Fig. 1. We assume that the grid-connectedPV system utilizes time-of-use pricing (i.e., electricity priceschange according to a fixed daily schedule) and that the pricesfor sales and purchases of electricity at any time are identical.In other words, the grid-connected PV system is net metered,a policy which has been widely adopted in most states in theU.S. [15] and some other countries in Europe, Australia, andNorth America. Electricity is generated from PV panels, and isused to supply different types of loads. Battery storage is usedto either store excess electricity generated from PV systemswhen the time-of-use price is lower for selling back to the gridwhen the time-of-use price is higher (this is explained in detailin Section V-B), or purchase electricity from the grid whenthe time-of-use pricing is lower and sell back to the grid whenthe time-of-use pricing is higher (power arbitrage). Without abattery, if the load was too large to be supplied by PV generatedelectricity, electricity would have to be purchased from the gridto meet the demand. Naturally, given the high cost of batterystorage, the size of the battery storage should be chosen suchthat the cost of electricity purchase from the grid and the lossdue to battery aging are minimized. Intuitively, if the batteryis too large, the electricity purchase cost could be the sameas the case with a relatively smaller battery. In this paper, weshow that there is a unique critical value (denoted as ,refer to Problem 1) of the battery capacity such that the cost ofelectricity purchase and the loss due to battery aging remain thesame if the battery size is larger than or equal to , and thecost and the loss are strictly larger if the battery size is smallerthan . We obtain a criterion for evaluating the economicvalue of batteries compared to purchasing electricity from thegrid, propose lower and upper bounds on given the PVgeneration, loads, and the time period for minimizing the costs,and introduce an efficient algorithm for calculating the criticalbattery capacity based on the bounds; these results are validatedvia simulations.The contributions of this work are the following: 1) To the

best of our knowledge, this is the first attempt on determining

1Note that solar panels and batteries both operate on dc, while the grid andloads operate on ac. Therefore, dc-to-ac and ac-to-dc power conversion is nec-essary when connecting solar panels and batteries with the grid and loads.

Fig. 1. Grid-connected PV system with battery storage and loads.

lower and upper bounds of the battery size for grid-connected,net metered PV systems based on a theoretical analysis; incontrast, most previous works were based on trial and errorapproaches, e.g., the work in [8]–[12] (one exception is thetheoretical work in [16], in which the goal is to minimize thelong-term average electricity costs in the presence of dynamicpricing as well as investment in storage in a stochastic setting;however, no battery aging is taken into account and specialpricing structure is assumed). 2) A criterion for evaluating theeconomic value of batteries compared to purchasing electricityfrom the grid is derived (refer to Proposition 4 and Assumption2), which can be easily calculated and could be potentiallyused for choosing appropriate battery technologies for practicalapplications. 3) Lower and upper bounds on the battery size areproposed, and an efficient algorithm is introduced to calculateits value for the given PV generation and dynamic loads; theseresults are then validated using simulations. Simulation resultsillustrate the benefits of employing batteries in grid-connectedPV systems via peak shaving and cost reductions compared withthe case without batteries (this is discussed in Section V-B).The paper is organized as follows. In Section II, we lay

out our setting and formulate the storage size determinationproblem. Lower and upper bounds on are proposed inSection III. Algorithms are introduced in Section IV to calcu-late the value of the critical battery capacity. In Section V, wevalidate the results via simulations. Finally, conclusions andfuture directions are given in Section VI.

II. PROBLEM FORMULATION

In this section, we formulate the problem of determining thestorage size for a grid-connected PV system, as shown in Fig. 1.We first introduce different components in our setting.

A. PV Generation

We use the following equation to calculate the electricity gen-erated from solar panels:

(1)

where• GHI Wm is the global horizontal irradiation at the lo-cation of solar panels;

• m is the total area of solar panels; and


• is the solar conversion efficiency of the PV cells.The PV generation model is a simplified version of the one usedin [17] and does not account for PV panel temperature effects.2

B. Electric Grid

Electricity can be purchased from or sold back to the grid.For simplicity, we assume that the prices for sales and pur-chases at time are identical and are denoted as Wh .Time-of-use pricing is used ( depends on ) becausecommercial buildings with PV systems that would consider abattery system usually pay time-of-use electricity rates. In ad-dition, with increased deployment of smart meters and electricvehicles, some utility companies are moving towards residen-tial time-of-use pricing as well; for example, San Diego Gas& Electric (SDG&E) has the on-peak, semi-peak, and off-peakprices for a day in the summer season [18], as shown in Fig. 3.We use W to denote the electric power exchanged

with the grid with the interpretation that1) if electric power is purchased from the grid;2) if electric power is sold back to the grid.In this way, positive costs are associated with the electricity pur-chased from the grid, and negative costs with the electricity soldback to the grid. In this paper, peak shaving is enforced throughthe constraint that

where is a positive constant.

C. Battery

A battery has the following dynamic:

(2)

where Wh is the amount of electricity stored in thebattery at time , and W is the charging/dischargingrate; more specifically,• if the battery is charging;• if the battery is discharging.

For certain types of batteries, higher order models exist (e.g., athird-order model is proposed in [19] and [20]).To take into account battery aging, we use Wh to de-

note the usable battery capacity at time . At the initial time ,the usable battery capacity is , i.e., . Thecumulative capacity loss at time is denoted as Wh ,and . Therefore,

The battery aging satisfies the following dynamic equation:

ifotherwise

(3)

2Note that our analysis on determining battery capacity only relies oninstead of detailed models of the PV generation. Therefore, more complicatedPV generation models can also be incorporated into the cost minimizationproblem as discussed in Section II-F.

where is a constant depending on battery technolo-gies. This aging model is derived from the aging model in [5]under certain reasonable assumptions; the detailed derivationis provided in the Appendix. Note that there is a capacity lossonly when electricity is discharged from the battery. Therefore,

is a nonnegative and nondecreasing function of .We consider the following constraints on the battery:1) At any time, the battery charge should satisfy

2) The battery charging/discharging rate should satisfy

where , is the maximum battery dis-charging rate, and is the maximum batterycharging rate. For simplicity, we assume that

where constant is the minimum time required tocharge the battery from 0 to or discharge the batteryfrom to 0.

D. Load

W denotes the load at time . We do not make ex-plicit assumptions on the load considered in Section III exceptthat is a (piecewise) continuous function. In residentialhome settings, loads could have a fixed schedule such as lightsand TVs, or a relatively flexible schedule such as refrigeratorsand air conditioners. For example, air conditioners can be turnedON and OFF with different schedules as long as the room tem-perature is within a comfortable range.

E. Converters for PV and Battery

Note that the PV, battery, grid, and loads are all connectedto an ac bus. Since PV generation is operated on dc, a dc-to-acconverter is necessary, and its efficiency is assumed to be a con-stant satisfying

Since the battery is also operated on dc, an ac-to-dc converteris necessary when charging the battery, and a dc-to-ac converteris necessary when discharging, as shown in Fig. 1. For sim-plicity, we assume that both converters have the same constantconversion efficiency satisfying

We define

ifotherwise.


In other words, is the power exchanged with the ac buswhen the converters and the battery are treated as an entity. Sim-ilarly, we can derive

ifotherwise.

Note that is the round trip efficiency of the battery con-verters.

F. Cost Minimization

With all the components introduced earlier, now we can for-mulate the following problem of minimizing the sum of the netpower purchase cost3 and the cost associated with the battery ca-pacity loss while guaranteeing that the demand from loads andthe peak shaving requirement are satisfied:

(4)

ifotherwise

ifotherwise

(5)

where is the initial time, is the time period considered forthe cost minimization, Wh is the unit cost for thebattery capacity loss. Note the following:1) No cost is associated with PV generation. In other words,PV generated electricity is assumed free.

2) is the loss of the battery purchase investmentduring the time period from to due to the use ofthe battery to reduce the net power purchase cost.

3) Equation (4) is the power balance requirement for any time.

4) The constraint captures the peak shavingrequirement.

Given a battery of initial capacity , on the one hand,if the battery is rarely used, then the cost due to the capacityloss is low while the net power purchasecost is high; on the other hand, if thebattery is used very often, then the net power purchase cost

is low while the cost due to the capacityloss is high. Therefore, there is a trade-off onthe use of the battery, which is characterized by calculating

3Note that the net power purchase cost includes the positive cost to purchaseelectricity from the grid, and the negative cost to sell electricity back to the grid.

an optimal control policy on to the optimizationproblem in (5).Remark 1: Besides the constraint , peak shaving

is also accomplished indirectly through dynamic pricing.Time-of-use price margins and schedules are motivated by thepeak load magnitude and timing. Minimizing the net powerpurchase cost results in battery discharge and reduction in gridpurchase during peak times. If, however, the peak load for acustomer falls into the off-peak time period, then the constrainton limits the amount of electricity that can be purchased.Peak shaving capabilities of the constraint anddynamic pricing will be illustrated in Section V-B.

G. Storage Size Determination

Based on (4), we obtain

Let , then the optimization problem in (5) can berewritten as

ifotherwise

ifotherwise

(6)

Now it is clear that only is an independent variable. Wedefine the set of feasible controls as controls that guarantee allthe constraints in the optimization problem in (6).Let denote the objective function

If we fix the parameters , and , is a functionof , which is denoted as . If we increase , in-tuitively will decrease though may not strictly decrease (thisis formally proved in Proposition 2) because the battery can beutilized to decrease the cost by1) storing extra electricity generated from PV or purchasingelectricity from the grid when the time-of-use pricing islow; and

2) supplying the load or selling back when the time-of-usepricing is high.

Now we formulate the following storage size determinationproblem.


Problem 1 (Storage Size Determination): Given the opti-mization problem in (6) with fixed , and , de-termine a critical value such that• , , and• , .One approach to calculate the critical value is that we

first obtain an explicit expression for the function bysolving the optimization problem in (6) and then solve forbased on the function . However, the optimization problem in(6) is difficult to solve due to the nonlinear constraints onand , and the fact that it is hard to obtain analytical expres-sions for and in reality. Even though it mightbe possible to find the optimal control using the minimum prin-ciple [21], it is still hard to get an explicit expression for the costfunction . Instead, in Section III, we identify conditions underwhich the storage size determination problem results in non-trivial solutions (namely, is positive and finite), and thenpropose lower and upper bounds on the critical battery capacity

.

III. BOUNDS ON

Now we examine the cost minimization problem in (6). Since, or equivalently,

, is a constraint that has to be satisfiedfor any , there are scenarios in which either thereis no feasible control or for . Given

, and , we define

(7)

(8)

(9)

Note that4 . Intuitively, is the setof time instants at which the battery can only be discharged,is the set of time instants at which the battery can be dischargedor is not used (i.e., ), and is the set of time instantsat which the battery can be charged, discharged, or is not used.Proposition 1: Given the optimization problem in (6), if1) , or2) is empty, or3) (or equivalently, ), is nonempty,

is nonempty, and , , ,then either there is no feasible control or for

.Proof: Now we prove that, under these three cases, either

there is no feasible control or for .1) If , then .However, since , the battery cannot be dis-charged at time . Therefore, there is no feasible control.

2) If is empty, it means thatfor any , which implies that

the battery can never be charged. If is nonempty, thenthere exists some time instant when the battery has to bedischarged. Since and the battery can neverbe charged, there is no feasible control. If is empty, then

4 means and .

for any is the only feasible controlbecause .

3) In this case, the battery has to be discharged at time , butthe charging can only happen at time instant . If

, such that , then the batteryis always discharged before possibly being charged. Since

, then there is no feasible control.Note that if the only feasible control is for

, then the battery is not used. Therefore, we imposethe following assumption.Assumption 1: In the optimization problem in (6),, is nonempty, and either• is empty, or• is nonempty, but , , , where

are defined in (7)–(9).Given Assumption 1, there exists at least one feasible control.

Now we examine how changes when increases.Proposition 2: Consider the optimization problem in (6) with

fixed , and . If , then.

Proof: Given , suppose control achieves theminimum cost and the corresponding states for thebattery charge and capacity loss are and . Since

is also a feasible control for problem (6) with , andresults in the cost . Since is the minimum costover the set of all feasible controls which include , we musthave .In other words, is nonincreasing with respect to the pa-

rameter , i.e., is monotonically decreasing (though maynot be strictly monotonically decreasing). If , then

, which implies that . Inthis case, has the largest value

(10)Proposition 2 also justifies the storage size determination

problem. Note that the critical value (as defined in Problem1) is unique as shown below.Proposition 3: Given the optimization problem in (6) with

fixed , and , is unique.Proof: We prove it via contradiction. Suppose is not

unique. In other words, there are two different critical valuesand . Without loss of generality, suppose .

By definition, because is a criticalvalue, while because is a critical value.A contradiction. Therefore, we must have .Intuitively, if the unit cost for the battery capacity loss

is higher (compared with purchasing electricity from the grid),then it might be preferable that the battery is not used at all,which results in , as shown below.


Proposition 4: Consider the optimization problem in (6) withfixed , and under Assumption 1.1) If

then , which implies that ;2) if

then

where and are calculated for .Proof: The cost function can be rewritten as

where

and

Note that because the integrand isalways nonnegative.i) For , there aretwo possibilities:• , or equivalently,

. Thus, for any ,, which implies that .

Therefore, .•

. Let ,then . Let , then .Since ,. Now we have ,

and . Therefore,

Sinceand , we have

(11)

Therefore, we have .In summary, if ,we have , which implies that

. Since is a nonincreasingfunction of , we also have . Thus,we must have for , and

by definition.ii) For , we have

. Then

Since as argued in the proof to i),

. Now

we try to lower bound . Note that

[as shown in (11)] implies that

Given Assumption 1, is nonempty, which implies that. Since

,

, which implies that

(12)

In summary,, which proves the result.


Given the result in Proposition 4, we impose the followingadditional assumption on the unit cost of the battery capacityloss to guarantee that is positive.Assumption 2: In the optimization problem in (6)

In other words, the cost of the battery capacity loss duringoperations is less than the potential gain expressed as themargin between peak and off-peak prices modified by theconversion efficiency of the battery converters and the batteryaging coefficient.Remark 2: There are two implications of Assumption 2:1) If the pricing signal is given, then the condition in Assump-tion 2 provides a criterion for evaluating the economicvalue of batteries compared to purchasing electricity fromthe grid. In other words, only if the unit cost of the batterycapacity loss satisfies Assumption 2, it is desirable to usebattery storages. For example, if the price is constant, thenthe condition reduces to , which cannot be satisfiedby any battery. In other words, the use of batteries cannotreduce the total cost.

2) If the battery and its converter are chosen, i.e.,are all fixed, then the condition in Assumption 2 imposesa constraint on the pricing signal so that the battery can beused to lower the total cost.

Since is a nonincreasing function of and lowerbounded by a finite value givenAssumptions 1 and 2, the storagesize determination problem is well defined. Nowwe show lowerand upper bounds on the critical battery capacity in the fol-lowing proposition.Proposition 5: Consider the optimization problem in (6) with

fixed , and under Assumptions 1 and 2. Then, where

and

Proof: We first show the lower bound via con-tradiction. Without loss of generality, we assume that

(be-cause we require ). Suppose

or equivalently

Therefore, there exists such that. Since

, wehave

The implication is that the control does not satisfy the dis-charging constraint at . Therefore, .To show , it is sufficient to show that if, the electricity that can be stored never exceeds .Note that the battery charging is limited by , i.e.,

. Now we try to lower bound

in which the second inequality holds because is a nonde-creasing function of . By applying the aging model in (3), wehave

Using (12), we have

Since , we have

Therefore,, which

implies that . Thus, theonly constraint on related to the battery charging is

. In turn, duringthe time interval , the maximum amount of elec-tricity that can be charged is

which is less than or equal to . In summary, if ,the amount of electricity that can be stored never exceeds ,and therefore, .Remark 3: As discussed in Proposition 1, if is empty,

or equivalently, , then, which implies that ; this is consistent with

the result in Proposition 1 when is empty.


IV. ALGORITHMS FOR CALCULATING

In this section, we study algorithms for calculating the crit-ical battery capacity . Given the storage size determinationproblem, one approach to calculate the critical battery capacityis by calculating the function and then choosing thesuch that the conditions in Problem 1 are satisfied. Thoughis a continuous variable, we can only pick a finite number of

’s and then approximate the function . One way topick these values is that we choose from to withthe step size5 . In other words

where6 and . Supposethe picked value is , and then we solve the optimizationproblem in (6) with . Since the battery dynamics and agingmodel are nonlinear functions of , we introduce a binaryindicator variable , in which if and

if . In addition, we use as the samplinginterval, and discretize (2) and (3) as

ifotherwise.

With the indicator variable and the discretization of con-tinuous dynamics, the optimization problem in (6) can be con-verted to a mixed integer programming problem with indicatorconstraints (such constraints are introduced in CPlex [22]), andcan be solved using the CPlex solver [22] to obtain .In the storage size determination problem, we need to check if

, or equivalently, ; this isbecause due to and the defini-tion of . Due to numerical issues in checking the equality,we introduce a small constant so that we treatthe same as if . Similarly,we treat if .The detailed algorithm is given in Algorithm 1. At Step 4, if

, or equivalently,, we have . Because of the monotonicity

property in Proposition 2, we knowfor any . Therefore, the

for loop can be terminated, and the approximated critical bat-tery capacity is .

5Note that one way to implement the critical battery capacity in practice is toconnect multiple identical batteries of fixed capacity in parallel. In thiscase, can be chosen to be .6The ceiling function is the smallest integer which is larger than or equal

to .

Algorithm 1 Simple Algorithm for Calculating

Input: The optimization problem in (6) with fixed , , ,, , under Assumptions 1 and 2, the calculated bounds

, and parameters

Output: An approximation of

1: Initialize , whereand ;

2: for do

3: Solve the optimization problem in (6) with , andobtain ;

4: if and then

5: Set , and exit the for loop;

6: end if

7: end for

8: Output .

It can be verified that Algorithm 1 stops after at moststeps, or equivalently, after solving at most

optimization problems in (6). The accuracy of the critical batterycapacity is controlled by the parameters . Fixing

, the output is within

Therefore, by decreasing , the critical battery capacity canbe approximated with an arbitrarily prescribed precision.Since the function is a nonincreasing function of, we propose Algorithm 2 based on the idea of bisection

algorithms. More specifically, we maintain three variables

in which (or ) is initialized as (or ). We setto be . Due to Proposition 2, we have

If (or equivalently, ; thisis examined in Step 7), then we know that ;therefore, we update with but do not update7 the value

. On the other hand, if , then weknow that ; therefore, we updatewith and set with . In this case, we alsocheck if : if it is, then output sincewe know the critical battery capacity is between and ;otherwise, the while loop is repeated. Since every execution

7Note that if we update with , then the difference betweenand can be amplified when is updated again later on.


of the while loop halves the interval starting from, the maximum number of executions of the while

loop is , and the algorithm requiressolving at most

optimization problems in (6). This is in contrast to solvingoptimization problems in (6) using

Algorithm 1.

Algorithm 2 Efficient Algorithm for Calculating

Input: The optimization problem in (6) with fixedunder Assumptions 1 and 2, the

calculated bounds , and parameters

Output: An approximation of

1: Let and ;


3: Let sign ;

4: while sign do

5: Let ;


7: if then

8: Set , and ;

9: else

10: Set and set with ;

11: if then

12: Set sign ;

13: end if

14: end if

15: end while

16: Output .

Remark 4: Note that the critical battery capacity can be im-plemented by connecting batteries with fixed capacity in par-allel because we only assume that theminimum battery chargingtime is fixed.

V. SIMULATIONS

In this section, we calculate the critical battery capacity usingAlgorithm 2 , and verify the results in Section III via simula-tions. The parameters used in Section II are chosen based ontypical residential home settings and commercial buildings.

Fig. 2. PV output based onGHImeasurement at La Jolla, CA, where tickmarksindicate noon local standard time for each day. (a) Starting from July 8, 2010.(b) Starting from July 13, 2010.

A. Setting

The GHI data is the measured GHI in July 2010 at La Jolla,CA. In our simulations, we use , and m . Thus

GHI W . We have two choices for :1) is 0000 h local standard time (LST) on July 8, 2010,and the PV output is given in Fig. 2(a) for the followingfour days starting from (the interval is 30 min; this cor-responds to the scenario in which there are relatively largevariations in the PV output);

2) is 0000 h LST on July 13, 2010, and the PV outputis given in Fig. 2(b) for the following four days startingfrom (this corresponds to the scenario in which thereare relatively small variations in the PV output).

The time-of-use electricity purchase rate is1) 16.5¢/kWh from 11 A.M. to 6 P.M. (on-peak);2) 7.8¢/kWh from 6 A.M. to 11 A.M. and 6 P.M. to 10 P.M.(semi-peak);

3) 6.1¢/kWh for all other hours (off-peak).This rate is for the summer season proposed by SDG&E [18],and is plotted in Fig. 3.Since the battery dynamics and aging are characterized by

continuous ordinary differential equations, we useas the sampling interval, and discretize (2) and (3) as


Fig. 3. Time-of-use pricing for the summer season at San Diego, CA [18].

In simulations, we assume that lead–acid batteries are used;therefore, the aging coefficient is [5], the unitcost for capacity loss is Wh based on the cost of

kWh [23], and the minimum charging time is (h)[24].For the PV dc-to-ac converter and the battery dc-to-ac/

ac-to-dc converters, we use . It can be verifiedthat Assumption 2 holds because the threshold value for is8

For the load, we consider two typical load profiles: the res-idential load profile as given in Fig. 4(a), and the commercialload profile as given in Fig. 4(b). Both profiles resemble the cor-responding load profiles in [17, Fig. 8].9 Note that in the resi-dential load profile, one load peak appears in the early morning,and the other in the late evening; in contrast, in the commercialload profile, the two load peaks appear during the daytime andoccur close to each other. For multiple day simulations, the loadis periodic based on the load profiles in Fig. 4.For the parameter , we use W . Since the max-

imum of the loads in Fig. 4 is around W ,wewill illustratethe peak shaving capability of battery storage. It can be verifiedthat Assumption 1 holds.

B. Results

We first examine the storage size determination problemusing Algorithm 2 in the following setting (called the basicsetting):1) the cost minimization duration is h ;2) is 0000 h LST on July 13, 2010; and3) the load is the residential load as shown in Fig. 4(a).The lower and upper bounds in Proposition 5 are calculated as

Wh and Wh . When applying Algorithm 2 , we

8Suppose the battery is a Li–ion battery with the same aging coefficient as alead–acid battery. Since the unit cost Wh based on the cost of

kWh [23] (and Wh based on the 10-year projected costof kWh [23]), the use of such a battery is not as competitive as directlypurchasing electricity from the grid.9However, simulations in [17] start at 7 A.M. so Fig. 4 is a shifted version of

the load profile in [17, Fig. 8].

Fig. 4. Typical residential and commercial load profiles. (a) Residential loadaveraged at 536.8 (W). (b) Commercial load averaged at 485.6 (W).

choose Wh and . When running thealgorithm, 13 optimization problems in (6) have been solved.The maximum cost is while the minimum costis , which is larger than the lower bound ascalculated based on Proposition 4. The critical battery capacityis calculated to be Wh .Now we examine the solution to the optimization problem

in (6) in the basic setting with three different battery capacitiesWh , Wh (namely, a battery ca-

pacity larger than the critical capacity), and Wh(namely, a battery capacity smaller than the critical capacity).For the value , we solve the mixed integer programming

problem using the CPlex solver [22], and the objective functionis . , , and are plotted inFig. 5(a). The plot of is consistent with the fact that thereis capacity loss (i.e., the battery ages) only when the battery isdischarged. The capacity loss is around Wh , and

which justifies the assumption we make when linearizing thenonlinear battery aging model in the Appendix. The dynamicpricing signals , , , and are plotted inFig. 5(b). Now we examine the effects of power arbitrage. From8 A.M. to 11 A.M., there is surplus PV generation, as shown inFig. 5(c). However, the surplus generation is not sold back tothe grid [as during this period from the third plotof Fig. 5(b)] because currently the electricity price is not high


Fig. 5. Solution to a typical setting in which is on July 13, 2010, h ,the load is shown in Fig. 4(a), and Wh .

enough. Instead it is stored in the battery [which can be ob-served from the second plot of Fig. 5(b)], and is then sold whenthe price is high after 11 A.M. From the third plot in Fig. 5(b),it can be verified that, to minimize the cost, electricity is pur-chased from the grid when the time-of-use pricing is low, and


is sold back to the grid when the time-of-use pricing is high;in addition, the peak demand in the late evening (that exceeds

W ) is shaved via battery discharging, as shown indetail in Fig. 5(d).For the larger battery capacity Wh , we

solve the mixed integer programming problem, and the objec-tive function is , which is the same as thecost with as expected. , , and are plottedin Fig. 6(a). The dynamic pricing signals , , ,and are plotted in Fig. 6(b). It can be observed that in thiscase and are different from the ones with the batterycapacity even though the costs are the same. The implica-tion is that optimal control to the problem in (6) is not neces-sarily unique. Similarly, for the smaller battery capacity

Wh , we solve themixed integer programming problem,and the objective function is , which islarger than the cost with as expected. , , and

are plotted in Fig. 7(a). The dynamic pricing signals ,, , and are plotted in Fig. 7(b). Based on

the plots of in Figs. 5(b) and 7(b), it can be observed thatin this case, the amount of electricity sold back to the grid issmaller than the case with , which results in a larger cost.Those observations of the case with the battery capacity

for h also hold for h . In this case, we



change to be h in the basic setting, and solve the bat-tery sizing problem. The critical battery capacity is calculatedto be . Now we examine the solution to theoptimization problem in (6) with the critical battery capacity

Wh , and obtain . , ,and are plotted in Fig. 8(a), and the dynamic pricing signals

, , , and are plotted in Fig. 8(b). Notethat the battery is gradually charged in the first half of each day,and then gradually discharged in the second half to be empty atthe end of each day, as shown in the second plot of Fig. 8(a).To illustrate the peak shaving capability of the dynamic

pricing signal as discussed in Remark 1, we change the loadto the commercial load as shown in Fig. 4(b) in the basicsetting, and solve the battery sizing problem. The criticalbattery capacity is calculated to be , and

. The dynamic pricing signals ,, , and are plotted in Fig. 9. For the com-

mercial load, the duration of the peak loads coincides with thatof the high price. To minimize the total cost, during peak timesthe battery is discharged, and the surplus electricity from PVafter supplying the peak loads is sold back to the grid resultingin a negative net power purchase from the grid, as shown in thethird plot in Fig. 9. Therefore, unlike the residential case, thehigh price indirectly forces the shaving of the peak loads.


Now we consider settings in which the load could be eitherresidential loads or commercial loads, the starting time couldbe on either July 8 or July 13, 2010, and the cost optimizationduration can be h , h , or h . The results are shown inTables I and II. In Table I, is on July 8, 2010, while in Table II,is on July 13, 2010. In the pair , 24 refers to the cost

optimization duration, and stands for residential loads; in thepair , stands for commercial loads.We first focus on the effects of load types. From Tables I and

II, the commercial load tends to result in a higher cost (eventhough the average of the commercial load is smaller than thatof the residential load) because the peaks of the commercial loadcoincide with the high price. Commercial loads tend to result inlarger optimum battery capacity as shown in Tables I andII, presumably because the peak load occurs during the peakpricing period and reductions in surplus PV production have tobe balanced by additional battery capacity. If is on July 8,2010, the PV generation is relatively lower than the scenarioin which is on July 13, 2010, and as a result, the battery ca-pacity is smaller and the cost is higher. This is because it is moreprofitable to store PV generated electricity than grid purchasedelectricity. From Tables I and II, it can be observed that the costoptimization duration has relatively larger impact on the bat-tery capacity for commercial loads, and relatively less impactfor residential loads.


Fig. 9. Solution to a typical setting in which is on July 13, 2010, h ,the load is shown in Fig. 4(b), and Wh .

TABLE ISIMULATION RESULTS FOR ON JULY 8, 2010

TABLE IISIMULATION RESULTS FOR ON JULY 13, 2010

In Tables I and II, the row corresponds to the cost inthe scenario without batteries, the row corresponds tothe cost in the scenario with batteries of capacity , and therow Savings10 corresponds to . For Table I, wecan also calculate the relative percentage of savings using theformula , and get the row Percentage.One observation is that the relative savings by using batteries in-crease as the cost optimization duration increases. For example,when and the load type is residential, costcan be saved when a battery of capacity Wh is used;when and the load type is commercial, costcan be saved when a battery of capacity Wh is used.This clearly shows the benefits of utilizing batteries in grid-con-nected PV systems. In Table II, only the absolute savings areshown since negative costs are involved.

VI. CONCLUSION

In this paper, we studied the problem of determining the sizeof battery storage for grid-connected PV systems. We proposedlower and upper bounds on the storage size, and introduced anefficient algorithm for calculating the storage size. Batteries are

10Note that in Table II, part of the costs are negative. Therefore, the word“Earnings” might be more appropriate than “Savings.”

used for power arbitrage and peak shaving. Note that, when PVgeneration costs reach grid parity, abundant PV generation willcause demand peaks and high electricity price at night, and lowprices will occur during the day. Under this scenario, batteriescould also be used for energy shifting, i.e., saving energy whenPV generation is larger than load during the day for a future timewhen load exceeds PV generation at night.In our analysis, the conversion efficiency of the PV dc-to-ac

converter and the battery dc-to-ac and ac-to-dc converters is as-sumed to be a constant. We acknowledge that this is not the casein the current practice, in which the efficiency of converters de-pends on the input power in a nonlinear fashion [5]. This will bepart of our future work. Another implicit assumption wemade isthat there is no energy loss due to battery self-discharging. Cur-rent investigation on battery self-discharging (e.g., the work in[25] and [26]) shows that the self-discharging rate is very smallon the order of 0.2% per day. In addition, such work is usuallybased on electric circuit equivalent models and assumes that abattery is not used for a long time, both of which do not holdin our setting. We leave this issue as part of our future work. Inaddition, we would like to extend the results to distributed re-newable energy storage systems, and generalize our setting bytaking into account stochastic PV generation.

APPENDIXDERIVATION OF THE SIMPLIFIED BATTERY AGING MODEL

Equations (11) and (12) in [5] are used to model the batterycapacity loss, and are combined and rewritten as follows usingthe notation in this work:

(13)

If is very small, then . Therefore,. If

we plug in the approximation, divide on both sides of (13),and let go to 0, then we have

(14)

This holds only if as in [5], i.e., there could becapacity loss only when discharging the battery.Since (14) is a nonlinear equation, it is difficult to solve. Let

, then (14) can be rewritten as

(15)

If is much shorter than the life time of the battery, then thepercentage of the battery capacity loss is very closeto 0. Therefore, (15) can be simplified to the following linearODE:

when . If , there is no capacity loss, i.e.,


ACKNOWLEDGMENT

The authors would like to thank I. Altintas, M. Tatineni,J. Greenberg, R. Hawkins, and J. Hayes at the San DiegoSupercomputer Center for computing supports, as well as theanonymous reviewers for very insightful comments/sugges-tions that helped improve this work tremendously.

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Yu Ru (S’06–M’10) received the B.E. degreein industrial automation and the M.S. degree incontrol theory and engineering, both from ZhejiangUniversity, Hangzhou, China, in 2002 and 2005,respectively, and the Ph.D. degree in electrical andcomputer engineering from the University of Illinoisat Urbana-Champaign, Urbana, in 2010.He is a postdoctoral researcher at the Mechanical

and Aerospace Engineering Department, Universityof California, San Diego. His research focuses on op-timization of grid-connected renewable energy sys-

tems, coordination control of multiagent systems, estimation, diagnosis, andcontrol of discrete event systems, and Petri net theory.Dr. Ru was the recipient of the prestigious Robert T. Chien Memorial Award

and Sundaram Seshu International Student Fellowship from ECE at the Univer-sity of Illinois at Urbana-Champaign.

Jan Kleissl received the Ph.D. degree in environ-mental engineering from Johns Hopkins University,in 2004.He joined the University of California, San

Diego (UCSD) in 2006. He is an assistant professorat the Department of Mechanical and AerospaceEngineering, UCSD, and Associate Director, UCSDCenter for Energy Research. He supervises 12 Ph.D.students who work on solar power forecasting, solarresource model validation, and solar grid integrationwork funded by DOE, CPUC, NREL, and CEC.

He teaches classes in renewable energy meteorology, fluid mechanics, andlaboratory techniques at UCSD.Dr. Kleissl received the 2009 NSF CAREER Award, the 2008 Hellman

Fellowship for tenure-track faculty of great promise, and the 2008 UCSDSustainability Award.

Sonia Martinez (M’01–SM’07) received the Ph.D.degree in engineering mathematics from the Univer-sidad Carlos III de Madrid, Spain, in May 2002.She is an associate professor with the Mechanical

and Aerospace Engineering Department, at Univer-sity of California, San Diego. Following a year as aVisiting Assistant Professor of Applied Mathematicsat the Technical University of Catalonia, Spain, sheobtained a Postdoctoral Fulbright fellowship and heldpositions as a visiting researcher at UIUC and UCSB.Her main research interests include nonlinear control

theory, cooperative control, and networked control systems. In particular, herwork has focused on the modeling and control of robotic sensor networks, thedevelopment of distributed coordination algorithms for groups of autonomousvehicles, and the geometric control of mechanical systems. She is also interestedin control applications in energy systems.For her work on the control of underactuated mechanical systems, Dr. Mar-

tinez received the Best Student Paper award at the 2002 IEEE Conference onDecision and Control. She was the recipient of an NSFCAREERAward in 2007.For the paper “Motion Coordination with Distributed Information,” coauthoredwith Francesco Bullo and Jorge Cortes, she received the 2008 Control SystemsMagazine Outstanding Paper Award.

17 storage size determination for grid-connected

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