72e7e52aaa41a76fc6

8/13/2019 72e7e52aaa41a76fc6

http://slidepdf.com/reader/full/72e7e52aaa41a76fc6 1/14

Exact sizing of battery capacity for photovoltaic systemsYu Ru a , Jan Kleissl b , Sonia Martinez b ,n

a GE Global Research at Shanghai, Chinab Mechanical and Aerospace Engineering Department, University of California, San Diego, United States

a r t i c l e i n f o

Article history:Received 22 July 2013Accepted 16 August 2013

Recommended by Astol Alessandro

Keywords:PV GridBatteryOptimization

a b s t r a c t

In this paper, we study battery sizing for grid-connected photovoltaic (PV) systems. In our setting, PV generated electricity is used to supply the demand from loads: on one hand, if there is surplus PV generation, it is stored in a battery (as long as the battery is not fully charged), which has a xedmaximum charging/discharging rate; on the other hand, if the PV generation and battery dischargingcannot meet the demand, electricity is purchased from the grid. Our objective is to choose an appropriatebattery size while minimizing the electricity purchase cost from the grid. More speci cally, we want to

nd a unique critical value (denoted as E cmax ) of the battery size such that the cost of electricity purchaseremains the same if the battery size is larger than or equal to E cmax , and the cost is strictly largerotherwise. We propose an upper bound on E cmax , and show that the upper bound is achievable for certainscenarios. For the case with ideal PV generation and constant loads, we characterize the exact value of E cmax , and also show how the storage size changes as the constant load changes; these results arevalidated via simulations.

& 2013 European Control Association. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Installations of solar photovoltaic (PV) systems have been growingat a rapid pace in recent years due to the advantages of PV such asmodest environmental impacts (clean energy), avoidance of fuelprice risks, coincidence with peak electrical demand, and the abilityto deploy PV at the point of use. In 2010, approximately 17,500 mega-watts (MW) of PV were installed globally, up from approximately7500 MW in 2009, consisting primarily of grid-connected applica-tions [2] . Since PV generation tends to uctuate due to cloud coverand the daily solar cycle, energy storage devices, e.g., batteries,ultracapacitors, and compressed air, can be used to smooth out the

uctuation of the PV output fed into electric grids (capacity rming)[14] , discharge and augment the PV output during times of peakenergy usage (peak shaving) [16] , or store energy for nighttime use,

for example in zero-energy buildings.In this paper, we study battery sizing for grid-connected PV systems to store energy for nighttime use. Our setting is shown inFig. 1. PV generated electricity is used to supply loads: on onehand, if there is surplus PV generation, it is stored in a battery forlater use or dumped (if the battery is fully charged); on the otherhand, if the PV generation and battery discharging cannot meetthe demand, electricity is purchased from the grid. The battery has

a xed maximum charging/discharging rate. Our objective is to

choose an appropriate battery size while minimizing the electri-city purchase cost from the grid. We show that there is a uniquecritical value (denoted as E cmax , refer to Problem 1 ) of the batterycapacity (under xed maximum charging and discharging rates)such that the cost of electricity purchase remains the same if thebattery size is larger than or equal to E cmax , and the cost is strictlylarger otherwise. We rst propose an upper bound on E cmax giventhe PV generation, loads, and the time period for minimizing thecosts, and show that the upper bound becomes exact for certainscenarios. For the case of idealized PV generation (roughly, it refersto PV output on clear days) and constant loads, we analyticallycharacterize the exact value of E cmax , which is consistent with thecritical value obtained via simulations.

The storage sizing problem has been studied for both off-grid

and grid-connected applications. For example, the IEEE standard[11] provides sizing recommendations for lead-acid batteries instand-alone PV systems. In [18] , the solar panel size and thebattery size have been selected via simulations to optimize theoperation of a stand-alone PV system. If the PV system is grid-connected, batteries can reduce the uctuation of PV output orprovide economic bene ts such as demand charge reduction,capacity rming, and power arbitrage. The work in [1] analyzesthe relation between available battery capacity and outputsmoothing, and estimates the required battery capacity usingsimulations. In addition, the battery sizing problem has beenstudied for wind power applications [21,5,12] and hybrid wind/solar power applications [4 ,8,20] . Most previous work completely

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/ejcon

European Journal of Control

0947-3580/$ - see front matter & 2013 European Control Association. Published by Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.ejcon.2013.08.002

n Corresponding author. Tel.: þ 1 858 822 4243.E-mail addresses: [email protected] (Y. Ru) , [email protected] (J. Kleissl) ,

[email protected] (S. Martinez) .

Please cite this article as: Y. Ru, et al., Exact sizing of battery capacity for photovoltaic systems, European Journal of Control (2013), http://dx.doi.org/10.1016/j.ejcon.2013.08.002 i

European Journal of Control ∎ (∎∎∎∎) ∎∎∎– ∎∎∎

http://www.sciencedirect.com/science/journal/09473580

http://www.elsevier.com/locate/ejcon


mailto:[email protected]

























http://www.elsevier.com/locate/ejcon

http://www.sciencedirect.com/science/journal/09473580

8/13/2019 72e7e52aaa41a76fc6


relies on trial and error approaches to calculate the storage size.Only very limited work has contributed to the theoretical analysisof storage sizing, such as [13 ,9,17] . In [13] , discrete Fourier trans-forms are used to decompose the required balancing power intodifferent time-varying periodic components, each of which can beused to quantify the physical maximum energy storage require-ment. In [9], the storage sizing problem is cast as an in nitehorizon stochastic optimization problem to minimize the long-term average cost of electric bills in the presence of dynamicpricing as well as investment in storage. In [17] , we cast thestorage sizing problem as a nite horizon deterministic optimiza-tion problem to minimize the cost associated with the net powerpurchase from the electric grid and the battery capacity loss due toaging while satisfying the load and reducing peak loads. Lower andupper bounds on the battery size are proposed that facilitate theef cient calculation of its value. The contribution of this work isthe following: exact values of battery size for the special case of ideal PV generation and constant loads are characterized; incontrast, in [17] , only lower and upper bounds are obtained. Inaddition, the setting in this work is different from that of [9] inthat a nite horizon deterministic optimization is formulated here.These results can be generalized to more practical PV generationand dynamic loads (as discussed in Remark 10 ).

We acknowledge that our analysis does not apply to the typicalscenario of “ net-metered ” systems, 1 where feed-in of energy tothe grid is remunerated at the same rate as purchase of energyfrom the grid. Consequently, the grid itself acts as a storage systemfor the PV system (and E cmax becomes 0). However, from a gridoperator standpoint it would be most desirable if the PV systemcould just serve the local load and not export to the grid. Thismotivates our choice of no revenue for dumping power to the grid.Our scenario also has analogues at the level of a balancing area byavoiding curtailment or intra-hour energy export. For load balan-cing, in a balancing area (typically a utility grid) steady-stateconditions are set every hour. This means that the power importsare constant over the hour. The balancing authority then has tobalance local generation with demand such that the steady statewill be preserved. This also corresponds to avoiding “ out ow ” of energy from the balancing area. In a grid with very high renewablepenetration, there may be more renewable production than load.In that case, the energy would be dumped or “ curtailed ” . However,with demand response (e.g., loads with relatively exible sche-dules) or battery storage, curtailment could be avoided.

The paper is organized as follows. In the next section, weintroduce our setting, and formulate the battery sizing problem.An upper bound on E cmax is proposed in Section 3 , and the exactvalue of E cmax is obtained for ideal PV generation and constantloads in Section 4 . In Section 5 , we validate the results viasimulations. Finally, conclusions and future directions are givenin Section 6 .

2. Problem formulation

In this section, we formulate the problem of determining thestorage size for grid-connected PV system, as shown in Fig. 1. Solarpanels are used to generate electricity, which can be used tosupply loads, e.g., lights, air conditioners, microwaves in a resi-dential setting. On one hand, if there is surplus electricity, it can bestored in a battery, or dumped to the grid if the battery is fullycharged. On the other hand, if there is not enough electricity topower the loads, electricity can be drawn from the electric grid.Before formalizing the battery sizing problem, we rst introducedifferent components in our setting.

2.1. Photovoltaic generation

We use the following equation to calculate the electricitygenerated from solar panels:

P pv ðt Þ ¼GHIðt Þ S η ; ð1Þ

where GHI (W m 2 ) is the global horizontal irradiation at thelocation of solar panels, S (m 2 ) is the total area of solar panels, andη is the solar conversion ef ciency of the PV cells. The PV generation model is a simpli ed version of the one used in [15]and does not account for PV panel temperature effects.

2.2. Electric grid

Electricity can be drawn from (or dumped to) the grid. Weassociate costs only with the electricity purchase from the grid,and assume that there is no bene t by dumping electricity tothe grid. The motivation is that, from a grid operator standpoint, itwould be most desirable if the PV system could just serve thelocal load and not export to the grid. In a grid with very highrenewable penetration, there may be more renewable productionthan load. In that case, the energy would have to be dumped (orcurtailed).

We use C gp ðt Þ (¢/kW h) to denote the electricity purchase rate,P gp ðt Þ ðW Þ to denote the electricity purchased from the grid at timet , and P gd ðt Þ ðW Þ to denote the surplus electricity dumped to thegrid or curtailed at time t . For simplicity, we assume that C gp ðt Þ is

time independent and has the value C gp . In other words, there is nodifference between the electricity purchase rates at different timeinstants.

2.3. Battery

A battery has the following dynamic:

dE Bðt Þdt

¼ P Bðt Þ; ð2Þ

where E Bðt Þ ðW h Þ is the amount of electricity stored in the batteryat time t , and P Bðt Þ ðW Þ is the charging/discharging rate (more

speci cally, P Bðt Þ4 0 if the battery is charging, and P Bðt Þo 0 if the

Solar Panel

Electric Grid

Load

Battery

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

1 Note that in [17] , we study battery sizing for “ net-metered ” systems under

more relaxed assumptions compared with this work.

Y. Ru et al. / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎2









8/13/2019 72e7e52aaa41a76fc6


battery is discharging). We impose the following constraints onthe battery:

(i) At any time, the battery charge E B(t ) should satisfy E Bmin r

E Bðt Þr E Bmax , where E Bmin is the minimum battery charge, E Bmax

is the maximum battery charge, and 2 0 o E Bmin r E Bmax .(ii) The battery charging/discharging rate should satisfy P Bmin r

P Bðt Þr P Bmax , where P Bmin o 0, P Bmin is the maximum battery

discharging rate, and P Bmax 4 0 is the maximum batterycharging rate.

For lead-acid batteries, more complicated models exist (e.g., athird order model is proposed in [7 ,3]).

2.4. Load

P load ðt Þ ðW Þ denotes the load at time t . We do not make explicitassumptions on the load considered in Section 3 except that P load ðt Þis a (piecewise) continuous function. Loads could have a xedschedule such as lights and TVs, or a relatively exible schedulesuch as refrigerators and air conditioners. For example, air condi-tioners can be turned on and off with different schedules as long as

the room temperature is within a comfortable range. In Section 4 , weconsider constant loads, i.e., P load ðt Þ is independent of time t .

2.5. Minimization of electricity purchase cost

With all the components introduced earlier, now we canformulate the following problem of minimizing the electricitypurchase cost from the electric grid while guaranteeing that thedemand from loads are satis ed:

minP B;P gp ;P gd Z

t 0 þ T

t 0C gp P gp ðτ Þ dτ

s:t : P pv ðt Þþ P gp ðt Þ ¼P gd ðt Þþ P Bðt Þþ P load ðt Þ; ð3Þ

dE Bðt Þ

dt ¼ P Bðt Þ; E Bðt 0 Þ ¼E Bmin ;E Bmin r E Bðt Þr E Bmax ;P Bmin r P Bðt Þr P Bmax ;P gp ðt ÞZ 0 ; P gd ðt ÞZ 0; ð4Þ

where t 0 is the initial time, T is the time period considered for thecost minimization. Eq. (3) is the power balance requirement forany time t A ½t 0 ; t 0 þ T ; in other words, the supply of electricity(either from PV generation, grid purchase, or battery discharging)must meet the demand.

2.6. Battery sizing

Based on Eq. (3) , we obtain

P gp ðt Þ ¼P load ðt Þ P pv ðt Þþ P Bðt Þþ P gd ðt Þ:Then the optimization problem in Eq. (4) can be rewritten as

minP B;P gd Z

t 0 þ T

t 0C gp ðP load ðτ Þ P pv ðτ Þþ P Bðτ Þþ P gd ðτ ÞÞ dτ

s:t : dE Bðt Þ

dt ¼ P Bðt Þ; E Bðt 0 Þ ¼E Bmin ;

E Bmin r E Bðt Þr E Bmax ;P Bmin r P Bðt Þr P Bmax ;P gd ðt ÞZ 0 :

Now there are two independent variables P B(t ) and P gd ðt Þ. To

minimize the total electricity purchase cost, we have the followingkey observations:

(A) If the battery is charging, i.e., P Bðt Þ4 0 and E Bðt Þo E Bmax , then thecharged electricity should only come from surplus PV generation.

(B) If the battery is discharging, i.e., P Bðt Þo 0 and E Bðt Þ4 E Bmin ,then the discharged electricity should only be used to supplyloads. In other words, the dumped electric power P gd ðt Þ should

only come from surplus PV generation.

In observation (A), the battery can be charged by purchasingelectricity from the grid at the current time and be used later onwhen the PV generated electricity is insuf cient to meet demands.However, this incurs a cost at the current time, and the saving of costs by discharging later on is the same as the cost of charging(or could be less than the cost of charging if the dischargedelectricity gets dumped to the grid) because the electricity purchaserate is time independent. That is to say, there is no gain in terms of costs by operating the battery charging in this way. Therefore, wecan restrict the battery charging to using only PV generated electricpower. In observation (B), if the battery charge is dumped to thegrid, potentially it could increase the total cost since extra electricitymight need to be purchased from the grid to meet the demand. Insummary, we have the following rule to operate the battery anddump electricity to the grid (i.e., restricting the set of possiblecontrol actions) without increasing the total cost.

Rule 1. The battery gets charged from the PV generation only whenthere is surplus PV generated electric power and the battery can stillbe charged, and gets discharged to supply the load only when theload cannot be met by PV generated electric power and the batterycan still be discharged. PV generated electric power gets dumped tothe grid only when there is surplus PV generated electric powerother than supplying both the load and the battery charging.

With this operating rule, we can further eliminate the variableP gd ðt Þ and obtain another equivalent optimization problem. On one

hand, if P load ðt Þ P pv ðt Þþ P Bðt Þo

0, i.e., the electricity generated fromPV is more than the electricity consumed by the load and charging thebattery, we need to choose P gd ðt Þ4 0 to make P gp ðt Þ ¼0 so that thecost is minimized; on the other hand, if P load ðt Þ P pv ðt Þþ P Bðt Þ4 0, i.e.,the electricity generated from PV and battery discharging is less thanthe electricity consumed by the load, we need to choose P gd ðt Þ ¼0 tominimize the electricity purchase costs, and we have P gp ðt Þ ¼P load ðt Þ

P pv ðt Þþ P Bðt Þ. Therefore, P gp ðt Þ can be written as

P gp ðt Þ ¼max ð0; P load ðt Þ P v ðt Þþ P Bðt ÞÞ;

so that the integrand is minimized at each time.Let

xðt Þ ¼E Bðt ÞE Bmax þ E Bmin

2 ;

uðt Þ ¼P Bðt Þ, and

E max ¼E Bmax E Bmin

2 :

Note that 2 E max ¼ E Bmax E Bmin is the net (usable) battery capacity,which is the maximum amount of electricity that can be stored inthe battery. Then the optimization problem can be rewritten as

J ¼ minu Z

t 0 þ T

t 0C gp max ð0; P load ðτ Þ P pv ðτ Þþ uðτ ÞÞ dτ

s:t : dxðt Þ

dt ¼ uðt Þ; xðt 0 Þ ¼ E max ;

j xðt Þjr E max ; P Bmin r uðt Þr P Bmax : ð5Þ

Now it is clear that only u(t ) (or equivalently, P B(t )) is an

independent variable. As argued previously, we can restrict u(t )

2 Usually, E Bmin is chosen to be larger than 0 to prevent fast battery aging. For

detailed modeling of the aging process, refer to [16] .

Y. Ru et al. / European Journal of Control ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3









8/13/2019 72e7e52aaa41a76fc6


to satisfying Rule 1 without increasing the minimum cost J . We de nethe set of feasible controls (denoted as U feasible ) as controls that satisfythe constraint P Bmin r uðt Þr P Bmax and do not violate Rule 1 .

If we x the parameters t 0 ; T ; P Bmin , and P Bmax , J is a function of E max , which is denoted as J ðE max Þ. If E max ¼ 0, then uðt Þ ¼0, and J reaches the largest value

J max ¼

Z t 0 þ T

t 0C gp max ð0; P load ðτ Þ P pv ðτ ÞÞ dτ : ð6Þ

If we increase E max , intuitively J will decrease (though may notstrictly decrease) because the battery can be utilized to store extraelectricity generated from PV to be potentially used later on whenthe load exceeds the PV generation. This is justi ed in thefollowing proposition.

Proposition 1. Given the optimization problem in Eq . (5), if E 1max o E 2max , then J ðE 1max ÞZ J ðE 2max Þ.

Proof. Refer to Appendix A .

In other words, J is monotonically decreasing with respect tothe parameter E max , and is lower bounded by 0. We are interestedin nding the smallest value of E max (denoted as E cmax ) such that J remains the same for any E max Z E c

max, and call it the battery sizing

problem.

Problem 1 (Battery sizing ). Given the optimization problem inEq. (5) with xed t 0 ; T ; P Bmin and P Bmax , determine a critical valueE cmax Z 0 such that 8 E max o E cmax , J ðE max Þ4 J ðE cmax Þ, and 8 E max Z

E cmax , J ðE max Þ ¼ J ðE cmax Þ.

Remark 1. In the battery sizing problem, we x the charging anddischarging rate of the battery while varying the battery capacity.This is reasonable if the battery is charged with a xed charger,which uses a constant charging voltage but can change thecharging current within a certain limit. In practice, the chargingand discharging rates could scale with E max , which results inchallenging problems to solve and requires further study. □

Note that the critical value E cmax is unique as shown in thefollowing proposition, which can be proved via contradiction.

Proposition 2. Given the optimization problem in Eq . (5) with xedt 0 ; T ; P Bmin and P Bmax , E cmax is unique .

Remark 2. One idea to calculate the critical value E cmax is that werst obtain an explicit expression for the function J ðE max Þ by

solving the optimization problem in Eq. (5) and then solve forE cmax based on the function J . However, the optimization problemin Eq. (5) is dif cult to solve due to the state constraint j xðt Þjr E max

and the fact that it is hard to obtain analytical expressions forP load ðt Þ and P pv ðt Þ in reality. Even though it might be possible to

nd the optimal control using the minimum principle [6] , it is stillhard to get an explicit expression for the cost function J . Instead, inthe next section, we rst focus on bounding the critical value E cmaxin general, and then study the problem for speci c scenarios inSection 4 . □

3. Upper bound on E cmax

In this section, we rst identify necessary assumptions to ensure anonzero E cmax , then propose an upper bound on E cmax , and nallyshow that the upper bound is tight for certain scenarios.

Proposition 3. Given the optimization problem in Eq . (5) with xedt 0 ; T ; P Bmin and P Bmax , E cmax ¼ 0 if any of the following conditions holds :

(i) 8 t A ½t 0 ; t 0 þ T , P pv ðt Þ P load ðt Þr 0,

(ii) 8 t A ½t 0 ; t 0 þ T , P pv ðt Þ P load ðt ÞZ 0,

(iii) 8 t 1 A S 1 , 8 t 2 A S 2 , t 2 o t 1 , whereS 1≔ft A ½t 0 ; t 0 þ T jP pv ðt Þ P load ðt Þ4 0g; ð7Þ

S 2≔ft A ½t 0 ; t 0 þ T jP load ðt Þ P pv ðt Þ4 0g: ð8Þ

Proof. Refer to Appendix B .

Remark 3. The intuition of condition (i) in Proposition 3 is that if

8 t A ½t 0 ; t 0 þ T , P pv ðt Þ P load ðt Þr 0, no extra electricity is generatedfrom PV and can be stored in the battery to strictly reduce thecost. The intuition of condition (ii) in Proposition 3 is that if 8 t A ½t 0 ; t 0 þ T , P pv ðt Þ P load ðt ÞZ 0, the electricity generated fromPV alone is enough to satisfy the load all the time, and extraelectricity can be simply dumped to the grid. Note that J max ¼ 0 forthis case. As de ned in condition (iii), S 1 \ S 2 ¼ ∅ because it isimpossible to have both P pv ðt Þ P load ðt Þ4 0 and P load ðt Þ P pv ðt Þ4 0at the same time for any time t . □

Based on the result in Proposition 3 , we impose the followingassumption on Problem 1 to make use of the battery.

Assumption 1. There exists t 1 and t 2 for t 1 ; t 2 A ½t 0 ; t 0 þ T , suchthat t 1 o t 2 , P pv ðt 1 Þ P load ðt 1 Þ4 0 and P pv ðt 2 Þ P load ðt 2 Þo 0.

Remark 4. P pv ðt 1 Þ P load ðt 1 Þ4 0 implies that at time t 1 there issurplus electric power available from PV. P pv ðt 2 Þ P load ðt 2 Þo 0implies that at time t 2 the electric power from PV is not suf cientfor the load. If t 1 o t 2 , the electricity stored in the battery at time t 1can be discharged to supply the load at time t 2 to strictly reducethe cost. □

Proposition 4. Given the optimization problem in Eq . (5) with xedt 0 ; T ; P Bmin and P Bmax under Assumption 1, 0 o E cmax r min ð A; BÞ=2,where

A¼ Z t 0 þ T

t 0min ðP Bmax ; max ð0; P pv ðt Þ P load ðt ÞÞÞ dt ; ð9Þ

and

B ¼ Z t 0 þ T

t 0min ð P Bmin ; max ð0; P load ðt Þ P pv ðt ÞÞÞ dt : ð10 Þ

Proof. Refer to Appendix C .

Remark 5. Note that if 8 t A ½t 0 ; t 0 þ T we have P pv ðt Þ P load ðt Þr 0,then A¼ 0 following Eq. (9) ; therefore, the upper bound for E cmax inProposition 4 becomes 0, which implies that E cmax ¼ 0. If 8 t A ½t 0 ; t 0 þ T we have P pv ðt Þ P load ðt ÞZ 0, then B¼ 0 followingEq. (10) ; therefore, the upper bound for E cmax in Proposition 4becomes 0, which implies that E cmax ¼ 0. Both results are consistentwith the results in Proposition 3 . □

Proposition 5. Given the optimization problem in Eq . (5) with xedt 0 ; T ; P Bmin and P Bmax under Assumption 1, if 8 t 1 A S 1 , 8 t 2 A S 2 ,t 1 o t 2 , then E cmax ¼ min ð A; BÞ=2, where S 1 (or S 2 , A, B, respectively )is de ned in Eq . (7) (or (8) – (10) , respectively ). In addition , if E max ischosen to be min ð A; BÞ=2, xðt 0 þ T Þ ¼ E max (i.e., no battery chargeleft at time t 0 þ T ).

Proof. Refer to Appendix D .

4. Ideal PV generation and constant load

In this section, we study how to obtain the critical value for thescenario in which the PV generation is ideal and the load is constant.Ideal PV generation occurs on clear days; for a typical south-facing PV array on a clear day, the PV output is zero before about sunrise, risescontinuously and monotonically to its maximum around solar noon,

then decreases continuously and monotonically to zero around










8/13/2019 72e7e52aaa41a76fc6


sunset, as shown in Fig. 2(a). In other words, there is essentially noshort time uctuation (at the scale of seconds to minutes) due toatmospheric effects such as clouds or precipitation. By constant load,we mean P load ðt Þ is a constant for t A ½t 0 ; t 0 þ T . A typical constantload is plotted in Fig. 2(b). To further simplify the problem, weassume that t 0 is 0000 h Local Standard Time (LST), and T ¼ t 0 þ k 24 ðhÞ where k is a nonnegative integer, i.e., T is a duration of multiple days. Fig. 2 plots the ideal PV generation and the constantload for T ¼ 24 ðhÞ. Now we summarize these conditions in thefollowing assumption.

Assumption 2. The initial time t 0 is 0000 h LST, T ¼ k 24 ðhÞwhere k is a positive integer, P pv ðt Þ is periodic on a timescale of 24 h, and satis es the following property for t A ½0; 24 ðhÞ : thereexist three time instants 0 o t sunrise o t max o t sunset o 24 ðhÞ suchthat

1. P pv ðt Þ ¼0 for t A ½0; t sunrise [ ½t sunset ; 24 ðhÞ ;2. P pv ðt Þ is continuous and strictly increasing for t A ½t sunrise ; t max ;3. P pv ðt Þ achieves its maximum P max

pv at t max ;4. P pv ðt Þ is continuous and strictly decreasing for t A ½t max ; t sunset ,

and P load ðt Þ ¼P load for t A ½t 0 ; t 0 þ T , where P load is a constantsatisfying 0 o P load o P max

pv .

It can be veri ed that Assumption 2 implies Assumption 1 .

Proposition 6. Given the optimization problem in Eq . (5) with xedt 0 ; T ; P Bmin and P Bmax under Assumption 2 and T ¼ 24 ðhÞ, E cmax ¼min ð A1 ; B1 Þ=2, where

A1 ¼ Z t 2

t 1min ðP Bmax ; max ð0; P pv ðt Þ P load ÞÞ dt ;

and

B1 ¼ Z 24

t 2min ð P Bmin ; max ð0; P load P pv ðt ÞÞÞ dt ;

in which t 1 o t 2 and P pv ðt 1 Þ ¼P pv ðt 2 Þ ¼P load .

Proof. Refer to Appendix E .

Remark 6. A1 and B1 in Proposition 6 are shown in Fig. 3. Inwords, A1 is the amount of extra PV generated electricity that canbe stored in a battery, and B1 is the amount of electricity that isnecessary to supply the load and can be provided by batterydischarging. Note that t 1 and t 2 depend on the value of P load . Toeliminate this dependency, we can rewrite A1 as

A1 ¼ Z 24

0min ðP Bmax ; max ð0; P pv ðt Þ P load ÞÞ dt ; ð11 Þ

and rewrite B1 as

B1 ¼ Z 24

t max

min ð P Bmin ; max ð0; P load P pv ðt ÞÞÞ dt ; ð12 Þ

where t max is de ned in Assumption 2 . □

Remark 7. If the PV generation is not ideal, i.e., there areuctuations due to clouds or precipitation, the E cmax value in

Proposition 6 based on ideal PV generation naturally serves asan upper bound on E cmax for the case with the non-ideal PV generation. Similarly, if the load varies with time but is boundedby a constant P load , the E cmax values in Proposition 6 based on theconstant load P load naturally serves as an upper bound on E cmax forthe case with the time varying load. □

Now we examine how E cmax changes as P load varies from 0 toP max

pv .

Proposition 7. Given the optimization problem in Eq . (5) with xedt 0 ; T ; P Bmin and P Bmax under Assumption 2 and T ¼ 24 ðhÞ, then

(a) there exists a unique critical value of P load A ð0; P maxpv Þ (denoted as

P c load ) such that E c

max achieves its maximum ;(b) if P load increases from 0 to P c

load , E cmax increases continuously andmonotonically from 0 to its maximum ;

(c) if P load increases from P c load to P max

pv , E cmax decreases continuouslyand monotonically from its maximum to 0.

Proof. Refer to Appendix F .

Remark 8. A typical plot of E cmax as a function of P load is given inFig. 4. Note that the slopes at 0 and P max

pv are both 0, which can bederived from the expressions of dA1 =dP load and dB1=dP load . Theresult has the implication that there is a ( nite) unique batterycapacity that minimizes the grid electricity purchase cost for anyP load 4 0. Fig. 10(a) veri es the plot via simulations. □

Now we focus on the case with multiple days.

Proposition 8. Given the optimization problem in Eq . (5) with xedt 0 ; T ; P Bmin and P Bmax under Assumption 2 and T ¼ k 24 ðhÞ wherek 4 1 is a positive integer , E cmax ¼ min ð A2 ; B2 Þ=2, where

A2 ¼

Z t 2

t 1

min ðP Bmax ; max ð0; P pv ðt Þ P load ÞÞ dt ;

0 t sunrise t sunset 24

P pv (W )

t (h )

0 24

P load (W )

t (h )

Fig. 2. Ideal PV generation and constant load. (a) Ideal PV generation for a clear day. (b) Constant load.

0 t 1 t 2 24

P pv (W )

t (h )

P Bmax

P Bmin

A 1

B 1

Fig. 3. PV generation and load, where P Bmax (or P Bmin ) is the maximum charging(or discharging) rate, and A1 ; B1 ; t 1 ; t 2 are de ned in Proposition 6 .










8/13/2019 72e7e52aaa41a76fc6


and

B2 ¼ Z t 3


in which t i A T crossing ≔ft A ½0; k 24 jP pv ðt Þ ¼P load g, t 1 ; t 2 ; t 3 are thesmallest three time instants in T crossing and satisfy t 1 o t 2 o t 3 .

Proof. Refer to Appendix G .

Remark 9. A2 and B2 in Proposition 8 are shown in Fig. 5. Inwords, A2 is the amount of extra PV generated electricity that can

be stored in a battery in the time interval ½t 1 ; t 3 , and B2 is theamount of electricity that is necessary to supply the load and canbe provided by battery discharging in the time interval ½t 1 ; t 3 . Notethat t 1 ; t 2 ; and t 3 depend on the value of P load . To eliminate thisdependency, we can rewrite A2 as

A2 ¼ Z 24

0min ðP Bmax ; max ð0; P pv ðt Þ P load ÞÞ dt ; ð13 Þ

and rewrite B2 as

B2 ¼ Z t max þ 24

t max

min ð P Bmin ; max ð0; P load P pv ðt ÞÞÞ dt ; ð14 Þ

where t max is de ned in Assumption 2 . □

It can be veri ed that the following result on how E cmax changesholds based on an analysis similar to the one in Proposition 7using Eqs. (13) and (14) , and Fig.10(b) and (c) veri es the trend viasimulations.

Proposition 9. Given the optimization problem in Eq . (5) with xedt 0 ; T ; P Bmin and P Bmax under Assumption 2 and T ¼ k 24 ðhÞ wherek 4 1 is a positive integer , then

(a) there exists a unique critical value of P load A ð0; P maxpv Þ (denoted as

P c load ) such that E c

max achieves its maximum ;(b) if P load increases from 0 to P cload , E cmax increases continuously and

monotonically from 0 to its maximum ;(c) if P load increases from P c

load to P maxpv , E cmax decreases continuously

and monotonically from its maximum to 0.

Remark 10. Note that Assumption 2 can be relaxed. Given T ¼ 24,if P load ðt Þ and P pv ðt Þ are piecewise continuous functions, andintersect at two time instants t 1 ; t 2 , in addition S 1 (as de ned inEq. (7) ) is the same as the open interval ðt 1 ; t 2 Þ, then the result inProposition 6 also holds, which can be proved similarly based onthe argument in Proposition 6 . Besides these conditions, if P load ðt Þand P pv ðt Þ are periodic with period 24 h, then the result inProposition 8 also holds. However, with these relaxed conditions,

the results in Propositions 7 and 9 do not hold any more since theload might not be constant. □

5. Simulations

In this section, we verify the results in Sections 3 and 4 viasimulations. The parameters used in Section 2 are chosen based ona typical residential home setting.

The GHI data is the measured GHI between July 13 and July 16,2010 at La Jolla, California. In our simulations, we use η ¼ 0:15 andS ¼ 10 m 2 . Thus P pv ðt Þ ¼1:5 GHIðt Þ ðW Þ. We choose t 0 as 0000 hLST on July 13, 2010, and the hourly PV output is given in Fig. 6 for thefollowing four days starting from t 0 . Except the small variation on July15, 2010 and being not exactly periodic for every 24 h, the PV generation roughly satis es Assumption 2 , which implies thatAssumption 1 holds. Note that 0 r P pv ðt Þo 1500 W for t A ½t 0 ; t 0 þ 96 .

The electricity purchase rate C gp is chosen to be 7.8 ¢/kW h,which is the semipeak rate for the summer season proposed by

0 t 1 t 2 24

P pv (W )

t (h )

P Bmax

P Bmin

A 2 B 2

t 3 48

Fig. 5. PV generation and load (two days), where P Bmax (or P Bmin ) is the

maximum charging (or discharging) rate, and A2 ; B2 ; t 1 ; t 2 ; t 3 are de ned inProposition 8 .

0

500

1000

1500

Jul 13, 2010 Jul 14, 2010 Jul 15, 2010 Jul 16, 2010

P p v

( w )

PBmax

−PBmin

Fig. 6. PV output for July 13 – 16, 2010, at La Jolla, California. For reference, a constantload 200 W is also shown (solid line) along with P Bmax ¼ P Bmin ¼ 200 W. The shadedarea above the 200 W horizontal line corresponds to the amount of electricity that canbe potentially charged to a battery, while the shaded area below the 200 W linecorresponds to the amount of electricity that can potentially be provided bydischarging the battery.

0 P c

load P max

pv P load (W )

E cmax (Wh )

B 12

A 12

Fig. 4. E cmax as a function of P load for 0 r P load r P maxpv .

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25100

200

300

400

500

600

700

800

900

1000

Time (h)

L o a

d ( w )

Fig. 7. A typical residential load pro le.










8/13/2019 72e7e52aaa41a76fc6


SDG&E (San Diego Gas & Electric) [19] . For the battery, we chooseE Bmin ¼ 0:4 E Bmax , and then

E max ¼E Bmax E Bmin

2 ¼ 0:3 E Bmax :

The maximum charging rate is chosen to be P Bmax ¼ 200 W, andP Bmin ¼ P Bmax . Note that the battery dynamic is characterized bya continuous ordinary differential equation. To run simulations, weuse 1 h as the sampling interval, and discretize Eq. (2) as

E Bðk þ 1Þ ¼E BðkÞþ P BðkÞ:

5.1. Dynamic loads

We rst examine the upper bound in Proposition 4 usingdynamic loads. The load pro le for one day is given in Fig. 7,which resembles the residential load pro le in Fig. 8(b) in [15] .3

Note that one load peak appears in the early morning, and theother in the evening. For multiple day simulations, the load isperiodic based on the load pro le in Fig. 7. We study how the costfunction J of the optimization problem in Eq. (5) changes as afunction of E max by increasing the battery capacity E max from 0 to1500 W h with the step size 10 W h. We solve the optimizationproblem in Eq. (5) via linear programming using the CPlex solver[10] . If T ¼ 24 ðhÞ (or T ¼ 48 ðhÞ, T ¼ 96 ðhÞ, respectively), the plotof the minimum costs versus E max is given in Fig. 8(a) (or (b), (c),

respectively). The plots con rm the result in Proposition 1 , i.e., theminimum cost is a decreasing function of E max , and also show theexistence of the unique E cmax . If T ¼ 24 ðhÞ, E cmax is 700 W h, whichcan be identi ed from Fig. 8(a). The upper bound in Proposition 4is calculated to be 900 W h. Similarly, if T ¼ 48 ðhÞ, E cmax is 900 W hwhile the upper bound in Proposition 4 is calculated to be1800 W h; if T ¼ 96 ðhÞ, E cmax is 900 W h while the upper boundin Proposition 4 is calculated to be 3402 W h. The upper boundholds for these three cases though the difference between theupper bound and E cmax increases when T increases. This is due tothe fact that during multiple days battery can be repeatedlycharged and discharged; however, this fact is not taken intoaccount in the upper bound in Proposition 4 . Since the load pro leroughly satis es the conditions imposed in Remark 10 , we cancalculate the theoretical value for E cmax based on the results inPropositions 7 and 9 even for this dynamic load. If T ¼ 24 ðhÞ, thetheoretical value is 690 W h which is obtained from Proposition 6by evaluating the integral in A1 and B1 using the sum of theintegrand for every hour from t 0 to t 0 þ T . Similarly, if T ¼ 48 ðhÞ orT ¼ 96 ðhÞ, then the theoretical value is 900 W h which is obtainedfrom Proposition 8 . Due to the step size 10 W h used in the choiceof E max , these theoretical values are quite consistent with resultsobtained via simulations.

5.2. Constant loads

We now study how the cost function J of the optimization

problem in Eq. (5) changes as a function of E max with a constant

0 500 1000 15001.8

1.9

2

2.1

2.2

2.3

2.4

2.5

Emax (wh)

M i n i m u m

C o s

t ( $ )

0 500 1000 15000.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

Emax (wh)

M i n i m u m

C o s

t ( $ )

0 500 1000 15000.85

0.9

0.95

1

1.05

1.1

1.15

1.2

Emax (wh)

M i n i m u m

C o s

t ( $ )

Fig. 8. Plots of minimum costs versus E max obtained via simulations given the load pro le in Fig. 7 and P Bmax ¼ 200 W. (a) One day. (b) Two days. (c) Four days.

3 However, simulations in [15] start at 7 AM so Fig. 7 is a shifted version of the

load pro le in Fig. 8(b) in [15] .










8/13/2019 72e7e52aaa41a76fc6


load, and the load is used from t 0 to t 0 þ T to satisfy Assumption 2 .We x the load to be P load ¼ 200 W, and increase the batterycapacity E max from 0 to 1500 W h with the step size 10 W h. Wesolve the optimization problem in Eq. (5) via linear programmingusing the CPlex solver [10] . If T ¼ 24 ðhÞ (or T ¼ 48 ðhÞ, T ¼ 96 ðhÞ,respectively), the plot of the minimum costs versus E max is given inFig. 9(a) (or (b), (c), respectively). The plots con rm the result inProposition 1 , i.e., the minimum cost is a decreasing function of E max , and also show the existence of the unique E cmax .

Now we validate the results in Propositions 6 – 9. We vary theload from 0 to 1500 W with the step size of 100 W, and for eachP load we calculate the E cmax and the minimum cost correspondingto the E cmax . In Fig.10(a), the left gure shows how E cmax changes asa function of the load for T ¼ 24 ðhÞ, and the right gure shows thecorresponding minimum costs. The plot in the left gure isconsistent with the result in Proposition 7 except that the max-imum of E cmax is not unique. This is due to the fact that the load ischosen to be discrete with step size 100 W. The right gure isconsistent with the intuition that when the load is increasing,more electricity needs to be purchased from the grid (resulting ina higher cost). Note that the blue solid curve corresponds to thecosts with E cmax , while the red dotted curve corresponds to J max , i.e., the costs without battery. The plots for E cmax and the minimumcost for T ¼ 48 ðhÞ and T ¼ 96 ðhÞ are shown in Fig. 10(b) and (c).The plots in the left gures of Fig. 10(b) and (c) are consistent withthe result in Proposition 9 . Note that as T increases, the criticalload P cload decreases as shown in the left gures of Fig. 10. Oneobservation on the left gures of Fig. 10 is that E cmax increases

roughly linearly with respect to the load when the load is small.

The justi cation is that when the load is small, E cmax is determinedby B1 in Fig. 3 (or B2 in Fig. 5 for multiple days) and B1 (or B2 )increases roughly linearly with respect to the load as can be seenfrom Fig. 3 (or Fig. 5).

Now we examine the results in Proposition 6 . For T ¼ 24 ðhÞ, weevaluate the integral in A1 and B1 using the sum of the integrandfor every hour from t 0 to t 0 þ T given a xed load, and then obtainE cmax ; this value is denoted as the theoretical value. The theoreticalvalue is plotted as the red curve (with the circle marker) in the leftplot of Fig. 11(a). The value E cmax calculated based on simulations isplotted as the blue curve (with the square marker) in the left plotof Fig. 11(a). In the right plot of Fig. 11(b), we plot the differencebetween the value obtained via simulations and the theoreticalvalue. Note that the value obtained via simulations is always largerthan or equal to the theoretical value because E max is chosen to bediscrete with step size 10 W h. The differences are always smallerthan 9 W h, 4 which con rms that the theoretical value is veryconsistent with the value obtained via simulations. The sameconclusion holds for T ¼ 48 ðhÞ, as shown in Fig. 11(b). ForT ¼ 96 ðhÞ, the largest difference is around 70 W h as shown inFig. 11 (c); this is more likely due to the slight variation in the PV generation for different days. Note that the differences for 10

16 ¼ 58 of

the load values (which range from 0 to 1500 W with the step size100 W) are within 10 W h.

0 500 1000 15000.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

Emax (wh)

M i n i m u m

C o s

t ( $ )

0 200 400 600 800 1000 1200 1400 16000.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Emax (wh)

M i n i m u m

C o s

t ( $ )

0 500 1000 15000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Emax (wh)

M i n i m u m

C o s t

( $ )

Fig. 9. Plots of minimum costs versus E max obtained via simulations given P load ¼ 200 W and P Bmax ¼ 200 W. (a) One day. (b) Two days. (c) Four days.

4 The step size for E max is 10 W h, so the difference between the value obtainedvia simulations and the theoretical value is expected to be within 10 assuming the

theoretical value is correct.










8/13/2019 72e7e52aaa41a76fc6


0 500 1000 15000

100

200

300

400

500

600

700

Load (w)

E m a x c

( w h )

0 500 1000 15000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Load (w)

C o s t

( $ )

Minimum costJ

max

0 500 1000 15000

200

400

600

800

1000

1200

Load (w)

E m a x c

( w h )

0 500 1000 15000

0.5

1

1.5

2

2.5

3

3.5

4

Load (w)

C o s

t ( $ )

Minimum costJ

max

0 500 1000 15000

200

400

600

800

1000

1200

Load (w)

E m

a x c

( w h )

0 500 1000 15000

1

2

3

4

5

6

7

8

Load (w)

C

o s

t ( $ )

Minimum costJ

max

Fig.10. E cmax (left), and costs (both J max and the cost corresponding to E cmax , right) versus the xed load obtained via simulations for P Bmax ¼ 200 W. (a) One day. (b) Two days.(c) Four days.










8/13/2019 72e7e52aaa41a76fc6


0 500 1000 15000

100

200

300

400

500

600

700

Load (w)

E m a x c

( w h )

TheoreticalSimulation

0 500 1000 15000

1

2

3

4

5

6

7

8

9

Load (w)

E r r o r ( w

h )

0 500 1000 15000

200

400

600

800

1000

1200

Load (w)

E m a x c

( w h )


0 500 1000 15000

1

2

3

4

5

6

7

8

9

Load (w)

E r r o r

( w h )

0 500 1000 15000

200

400

600

800

1000

1200

Load (w)

E m

a x c

( w h )


0 500 1000 15000

10

20

30

40

50

60

70

Load (w)

E r r o r

( w h )

Fig. 11. Plots of E cmax versus the xed load for the theoretical value (the curve with the circle marker) and the value obtained via simulations (the curve with the squaremarker). (a) One day. (b) Two days. (c) Four days.










8/13/2019 72e7e52aaa41a76fc6


6. Conclusions

In this paper, we studied the problem of determining the size of battery storage for grid-connected PV systems. We proposed anupper bound on the storage size, and showed that the upperbound is achievable for certain scenarios. For the case with idealPV generation and constant load, we characterized the exactstorage size, and also showed how the storage size changes as

the constant load changes; these results are consistent with theresults obtained via simulations.There are several directions for future research. First, the dynamic

time-of-use pricing of the electricity purchase from the grid could betaken into account. Large businesses usually pay time-of-use elec-tricity rates, but with increased deployment of smart meters andelectric vehicles some utility companies are moving towards differentprices for residential electricity purchase at different times of the day(for example, SDG&E has the peak, semipeak, offpeak prices for a dayin the summer season [19] ). New results (and probably newtechniques) are necessary to deal with dynamic pricing. Second, wewould like to study how batteries with a xed capacity can beutilized (e.g., via serial or parallel connections) to implement thecritical battery capacity for practical applications. Last, we would alsolike to extend the results to wind energy storage systems, andconsider battery parameters such as round-trip charging ef ciency,degradation, and costs.

Acknowledgments

This work was funded by NSF grant ECCS-1232271.

Appendix A. Proof to Proposition 1

Given E 1max , suppose a feasible control u1 ðt Þ achieves theminimum electricity purchase cost J ðE 1max Þ and the correspondingstate x is x1 ðt Þ. Since j x1 ðt Þjr E 1max o E 2max , u1 ðt Þ is also a feasiblecontrol for problem (5) with the state constraint E 2max and satisfy-ing Rule 1 , and results in the cost J ðE 1max Þ. Since J ðE 2max Þ is theminimal cost over the set of all feasible controls which includeu1 ðt Þ, we must have J ðE 1max ÞZ J ðE 2max Þ. □

Appendix B. Proof to Proposition 3

Condition (i) holds : Since 8 t A ½t 0 ; t 0 þ T , P pv ðt Þ P load ðt Þr 0, wehave P load ðt Þ P pv ðt ÞZ 0. Denote the integrand in J of Eq. (5) as α , i.e., α ðt Þ ¼C gp max ð0; P load ðt Þ P pv ðt Þþ uðt ÞÞ. If P load ðt Þ P pv ðt Þ ¼0,then we could choose uðt Þ ¼0 to make α to be 0. If P load ðt Þ P pv ðt Þ4 0, we could choose P pv ðt Þ P load ðt Þr uðt Þo 0 todecrease α , i.e., by discharging the battery. However, since xðt 0 Þ ¼ E max , there is no electricity stored in the battery at theinitial time. To be able to discharge the battery, it must have beencharged previously. Following Rule 1 , the electricity stored in thebattery should only come from surplus PV generation. However,there is no surplus PV generation at any time because8 t A ½t 0 ; t 0 þ T , P pv ðt Þ P load ðt Þr 0. Therefore, the cost is notreduced by choosing P pv ðt Þ P load ðt Þr uðt Þo 0. In other words, ucan be chosen to be 0. In both cases, u(t ) can be 0 for anyt A ½t 0 ; t 0 þ T without increasing the cost, and thus, no battery isnecessary. Therefore, E cmax ¼ 0.

Condition (ii) holds : Since 8 t A ½t 0 ; t 0 þ T , P pv ðt Þ P load ðt ÞZ 0, wehave P load ðt Þ P pv ðt Þr 0. Denote the integrand in J as α , i.e., α ðt Þ ¼C gp max ð0; P load ðt Þ P pv ðt Þþ uðt ÞÞ. If P load ðt Þ P pv ðt Þr 0, we couldchoose uðt Þ ¼0, and then α ¼ max ð0; P load ðt Þ P pv ðt Þþ uðt ÞÞ ¼

max ð0; P load ðt Þ P pv ðt ÞÞ ¼0. Since u(t ) can be 0 for any t A ½t 0 ; t 0 þ T

without increasing the cost, no battery is necessary. Therefore,E cmax ¼ 0.

Condition (iii) holds : S 1 is the set of time instants at which there isextra amount of electric power that is generated from PV aftersupplying the load, while S 2 is the set of time instants at which thePV generated power is insuf cient to supply the load. According toRule 1 , at time t , the battery could get charged only if t A S 1 , andcould get discharged only if t A S 2 . If 8 t 1 A S 1 , 8 t 2 A S 2 , t 2 o t 1 implies

that even if the extra amount of electricity generated from PV isstored in a battery, there is no way to use the stored electricity tosupply the load. This is because the electricity is stored after the timeinstants at which battery discharging can be used to strictly decreasethe cost and initially there is no electricity stored in the battery.Therefore, the costs are the same for the scenario with battery andthe scenario without battery, and E cmax ¼ 0. □

Appendix C. Proof to Proposition 4

It can be shown, via contradiction, that under Assumption 1 , A4 0 and B4 0, which implies that min ð A; BÞ=2 4 0.

We show E cmax 4 0 via contradiction. Since E cmax Z 0, we need to

exclude the case E cmax ¼ 0. Suppose E

cmax ¼ 0. If we choose

E max 4 E cmax ¼ 0, J ðE max Þo J ðE cmax Þ because under Assumption 1 abattery can store the extra PV generated electricity rst and thenuse it later on to strictly reduce the cost compared with the casewithout a battery (i.e., the case with E max ¼ 0). A contradiction tothe de nition of E cmax .

To show E cmax r min ð A; BÞ=2, it is suf cient to show that if E max Z min ð A; BÞ=2, then J ðE max Þ ¼ J ðmin ð A; BÞ=2Þ. There are twocases depending on if A r B or not:

1. Ar B. Then min ð A; BÞ ¼ A. At time t , max ð0; P pv ðt Þ P load ðt ÞÞ isthe extra amount of electric power that is generated from PV after supplying the load, and

min ðP Bmax ; max ð0; P pv ðt Þ P load ðt ÞÞÞ

is the extra amount of electric power that is generated from PV after supplying the load and can be stored in a battery subjectto the maximum charging rate. Then

A¼ Z t 0 þ T

t 0min ðP Bmax ; max ð0; P pv ðt Þ P load ðt ÞÞÞ dt

is the maximum total amount of extra electricity that can bestored in a battery while taking the battery charging rate intoaccount. Even if 2 E max Z A, i.e., E max Z A=2, the amount of electricity that can be stored in the battery cannot exceed A.Therefore, any control that is feasible with j xðt Þjr E max isalso feasible with j xðt Þjr A=2. Therefore, J ðE max Þ ¼ J ð A=2Þ ¼ J ðmin ð A; BÞ=2Þ.

2. A4 B. Then min ð A; BÞ ¼B. At time t , max ð0; P load ðt Þ P pv ðt ÞÞ isthe amount of electric power that is necessary to satisfy theload (and could be supplied by either battery power or gridpurchase), and

min ð P Bmin ; max ð0; P load ðt Þ P pv ðt ÞÞÞ

is the amount of electric power that can potentially be dischargedfrom a battery to supply the load subject to the maximumdischarging rate (in other words, if P load ðt Þ P pv ðt Þ4 P Bmin ,electricity must be purchased from the grid). Then

B ¼ Z t 0 þ T

t 0min ð P Bmin ; max ð0; P load ðt Þ P pv ðt ÞÞÞ dt

is the maximum total amount of electricity that is necessary to be

discharged from the battery to satisfy the load while taking the










8/13/2019 72e7e52aaa41a76fc6


battery discharging rate into account. When 2 E max Z B, i.e.,E max Z B=2, the amount of electricity that can be charged canexceed B because A4 B; however, the amount of electricity that isstrictly necessary to be (and, at the same time, can be) dischargeddoes not exceed B. In other words, if the stored electricity in thebattery exceeds this amount B, the extra electricity cannot helpreduce the cost because it either cannot be discharged or is notnecessary. Therefore, any control that minimizes the total cost

with the battery capacity being B also minimizes the total costwith the battery capacity being 2 E max . Therefore, J ðE max Þ ¼ J ðB=2Þ¼ J ðmin ð A; BÞ=2Þ. □

Appendix D. Proof to Proposition 5

From Proposition 4 , we have E cmax r min ð A; BÞ=2. To proveE cmax ¼ min ð A; BÞ=2, we show that E cmax o min ð A; BÞ=2 is impossiblevia contradiction. Suppose E cmax o min ð A; BÞ=2. If 8 t 1 A S 1 , 8 t 2 A S 2 ,t 1 o t 2 , then during the time interval ½t 0 ; t 0 þ T , the battery is rstcharged, and then discharged following Rule 1 . In other words,there is no charging after discharging. There are two casesdepending on A and B:

1. Ar B. In this case, E cmax o A=2, i.e., 2 E cmax o A. If the batterycapacity is 2 E cmax , then the amount of electricity A 2E cmax 4 0(which is generated from PV) cannot be stored in the battery. If we choose the battery capacity to be A, this extra amount canbe stored and used later on to strictly decrease the cost because Ar B. Therefore, J ð A=2Þo J ðE cmax Þ. A contradiction to the de ni-tion of E cmax . In this case, if E max is chosen to be A=2, then thebattery is rst charged with A amount of electricity, and thencompletely discharged before (or at) t 0 þ T because Ar B.Therefore, we have xðt 0 þ T Þ ¼ E max .

2. A4 B. In this case, E cmax o B=2, i.e., 2E cmax o B. If the batterycapacity is 2 E cmax , at most 2 E cmax o Bo A amount of PV gener-ated electricity can be stored in the battery. Therefore, theamount of electricity B 2E cmax 4 0 must be purchased from thegrid to supply the load. If we choose the battery capacity to beB, the amount of electricity B 2E cmax purchased from the gridcan be provided by the battery because the battery can becharged with the amount of electricity B (since A4 B), and thusthe cost can be strictly decreased. Therefore, J ðB=2Þo J ðE cmax Þ. Acontradiction to the de nition of E cmax . In this case, if E max ischosen to be B=2, then the battery is rst charged with Bamount of electricity (that is to say, not all extra electricitygenerated from PV is stored in the battery since A4 B), andthen completely discharged at time t 0 þ T . Therefore, we alsohave xðt 0 þ T Þ ¼ E max . □

Appendix E. Proof to Proposition 6

Due to Assumption 2 , P load ðt Þ intersects with P pv ðt Þ at two timeinstants for T ¼ 24 ðhÞ; the smaller time instant is denoted as t 1 ,and the larger is denoted as t 2 , as shown in Fig. 3. It can be veri edthat P pv ðt Þ4 P load for t A ðt 1 ; t 2 Þ and P pv ðt Þo P load for t A ½0; t 1 Þ [ðt 2 ; 24 following Assumption 2 . For t A ½0; t 1 Þ, a battery could onlyget discharged following Rule 1 ; however, it cannot be dischargedbecause xð0Þ ¼ E max . Therefore, u(t ) can be 0 while achieving thelowest cost for the time period ½0; t 1 Þ. Then the objective functionof the optimization problem in Eq. (5) can be rewritten as

J ¼ min

Z 24

0

C gp max ð0; P load P pv ðτ Þþ uðτ ÞÞ dτ ¼ J 0 þ J 1 ;

where J 0 ¼ R t 10 C gp ðP load P pv ðτ ÞÞ dτ is a constant which is indepen-

dent of the control u , and

J 1 ¼ min Z 24

t 1C gp max ð0; P load P pv ðτ Þþ uðτ ÞÞ dτ :

In other words, the optimization problem is essentially the sameas minimizing J 1 for t A ½t 1 ; 24 ; accordingly, the critical value E cmaxwill be the same since the battery is not used for the time interval½0; t 1 . For the optimization problem with the cost function J 1 underAssumption 2 , S 1 ¼ ðt 1 ; t 2 Þ and S 2 ¼ ðt 2 ; 24 Þ according to Eqs.(7) and (8) . Since 8 t ′1 A S 1 , 8 t ′2 A S 2 , t ′1 o t 2 o t ′2 , the conditions inProposition 5 are satis ed. Thus, we have E cmax ¼ min ð A; BÞ=2,where

A¼ Z 24

t 1min ðP Bmax ; max ð0; P pv ðt Þ P load ÞÞ dt

¼ Z t 2


which is essentially A1 , and

B ¼ Z 24

t 1min ð P Bmin ; max Þð0; P load P pv ðt ÞÞÞ dt

¼ Z 24


which is essentially B1 . Thus the result holds. □

Appendix F. Proof to Proposition 7

Let f ≔ A1 B1 , where A1 and B1 are de ned in Eqs. (11) and (12) .Note that f is a function of P load . If P load ¼ 0, then

A1 ¼ Z 24

0min ðP Bmax ; max ð0; P pv ðt ÞÞÞ dt 4 0;

according to Eq. (11) , and

B1 ¼ Z 24

t maxmin ð P Bmin ; max ð0; P pv ðt ÞÞÞ dt ¼ 0;

according to Eq. (12) . Therefore, f ð0Þ ¼ A1 ð0Þ B1 ð0Þ4 0. If P load ¼ P max

pv , then

A1 ¼ Z 24

0min ðP Bmax ; max ð0; P pv ðt Þ P max

pv ÞÞ dt ¼ 0;

and

B1 ¼ Z 24

t max

min ð P Bmin ; max ð0; P maxpv P pv ðt ÞÞÞ dt 4 0:

Therefore, f ðP maxpv Þ ¼ A1 ðP max

pv Þ B1 ðP maxpv Þo 0. In addition, since f is

an integral of a continuous function of P load , f is differentiable withrespect to P load , and the derivative is given as

df dP load

¼ dA1

dP load

dB1

dP load:

Since for P load A ð0; P maxpv Þ,

dA1

dP load¼ Z

24

0ð 1Þ I 0 o P pv ðt Þ P load r P Bmax dt

o 0;

and

dB1

dP load¼ Z

24

t max

1 I 0 o P load P pv ðt Þr P Bmin dt

4 0;

we have df =dP load o 0, where I f0 o P pv ðt Þ P load r P Bmax g is the

indicator function (i.e., if 0 o P pv ðt Þ P load r P Bmax , the function










8/13/2019 72e7e52aaa41a76fc6


has value 1, 0 otherwise). Therefore, f is continuous and strictlydecreasing for P load A ½0; P max

pv . Since f ð0Þ4 0 and f ðP maxpv Þo 0, there

is one and only one value of P load such that f is 0. We denote thisvalue as P c

load and have A1 ðP c load Þ ¼B1 ðP c

load Þ.If P load A ½0; P c

load Þ, f 4 0, i.e., A1 4 B1 . Therefore, E cmax ¼ B1 =2. SincedB1 =dP load 4 0, E cmax increases continuously (since B1 is differentiablewith respect to P load ) and monotonically from 0 to the valueB1 ðP c

load Þ=2. On the other hand, if P load A ðP c load ; P max

pv , f o 0, i.e., A1 o B1 . Therefore, E cmax ¼ A1 =2. Since dA1=dP load o 0, E cmax decreasescontinuously (since A1 is differentiable with respect to P load ) andmonotonically from the value A1 ðP c

load Þ=2 ¼ B1 ðP c load Þ=2 to 0. Therefore,

E cmax achieves its maximum at P c load . This completes the proof. □

Appendix G. Proof to Proposition 8

Due to Assumption 2 , P load ðt Þ intersects with P pv ðt Þ at 2 k timeinstants for T ¼ k 24 ðhÞ; we denote the set of these time instantsas T crossing ≔ft A ½0; k 24 jP pv ðt Þ ¼P load g. We sort the time instantsin an ascending order and denote them as t 1 ; t 2 ; t 3 ; … ;t 2i 1 ; t 2i; … ; t 2k 1 ; t 2k , where 2 r i r k. Following Rule 1 , at time t ,a battery could get charged only if t A ðt 1 ; t 2 Þ [ ðt 3 ; t 4 Þ

[ ⋯ðt 2k 1 ; t 2kÞ, and could get discharged only if t A ð0; t 1 Þ[ðt 2 ; t 3 Þ [ ðt 4 ; t 5 Þ [ ⋯ðt 2k; k 24 Þ. As shown in the proof toProposition 6 , u(t ) can be zero for t A ð0; t 1 Þ, and results in thelowest cost J 0 ¼ R

t 10 C gp ðP load P pv ðτ ÞÞ dτ , which is a constant. At

time t 1 , there is no charge in the battery. Then the battery isoperated repeatedly by charging rst if t A ðt 2i 1 ; t 2iÞ and thendischarging if t A ðt 2i; t 2i þ 1 Þ for i ¼ 1; 2; … ; k and t 2k þ 1 ¼ k 24.Naturally, we could group the charging interval ðt 2i 1 ; t 2iÞ withthe discharging interval t A ðt 2i; t 2i þ 1 Þ to form a complete batteryoperating cycle in the interval ðt 2i 1 ; t 2i þ 1 Þ.

Now the objective function of the optimization problem in Eq.(5) satis es

J ¼ minu

Z k 24

0C gp max ð0 ; P load P pv ðτ Þþ uðτ ÞÞ dτ

¼ J 0 þ minu

∑k 1

i ¼ 1Li þ Lk !;

where

Li ¼ Z t 2i þ 1

t 2i 1

C gp max ð0; P load P pv ðτ Þþ uðτ ÞÞ dτ ;

for i ¼ 1; 2; … ; k 1, and

Lk ¼ Z t 2k þ 1

t 2k 1

C gp max ð0; P load P pv ðτ Þþ uðτ ÞÞ dτ :

Note that given a certain E max , if the battery charge at the endof the rst battery operating cycle is larger than 0 (i.e., xðt 3 Þ4 E max ), then E max 4 E cmax . This can be argued as follows. If the battery charge at the end of the rst cycle is larger than 0 (thisalso implies that the battery charge at the end of the ith cycle isalso larger than 0 due to periodic PV generations and loads), i.e.,there is more PV generation than demand in the time intervalðt 1 ; t 3 Þ, then E max can be strictly reduced to a smaller capacity sothat xðt 3 Þ ¼ E max without increasing the electricity purchase costin the interval ðt 1 ; t 3 Þ. Due to periodicity of the PV generation andthe load, the smaller E max can be used for the interval ðt 2i 1 ; t 2i þ 1 Þfor i ¼ 2; … ; k 1 without increasing the electricity purchase cost.Therefore, this E max must be larger than E cmax . In other words, if E cmax is used, then xðt 2i þ 1 Þ for i ¼ 1; 2; … ; k 1 has to be E cmax , i.e.,no charge left at the end of each operating cycle. Now we onlyconsider E max such that at the end of each operating cycle xðt 2i þ 1 Þ ¼ E max for i ¼ 1 ; 2; … ; k 1 (necessarily, E cmax is smaller

than or equal to any such E max ). For such E max , the control actions

for each operating cycle are completely decoupled. 5 Therefore, thetotal cost J can be rewritten as

J ¼ J 0 þ ∑k 1

i ¼ 1 J i þ J k;

where J i ¼ min u Li for i ¼ 1; 2; … ; k.Now we focus on J 1 . For the optimization problem with the cost

function J 1 under Assumption 2 , S 1 ¼ ðt 1 ; t 2 Þ and S 2 ¼ ðt 2 ; t 3 Þ

according to Eqs. (7) and (8). Since 8 t ′1 A S 1 , 8 t

′2 A S 2 , t

′1 o t 2 o t

′2 ,

the conditions in Proposition 5 are satis ed. Thus, we haveE cmax ð1Þ ¼min ð A; BÞ=2, where E cmax ð1Þ is the E cmax when we onlyconsider the cost function J 1 ,

A¼ Z t 3

t 1min ðP Bmax ; max ð0; P pv ðt Þ P load ÞÞ dt

¼ Z t 2


which is essentially A2 , and

B¼ Z t 3

t 1min ð P Bmin ; max ð0; P load P pv ðt ÞÞÞ dt

¼ Z t 3

t 2 min ð P Bmin ; max ð0; P load P pv ðt ÞÞÞ dt ;

which is essentially B2 . Thus we have E cmax ð1Þ ¼min ð A2 ; B2 Þ=2.Based on Proposition 5 , we also know that xðt 3 Þ ¼ E cmax ð1Þ. Thus,this E cmax ð1Þ satis es the requirement that at the end of theoperating cycle there is no charge left.

For the cost function J 2 , the optimization problem is essentiallythe same as the problem with the cost function J 1 because

1. P pv ðt Þ ¼P pv ðt 24 Þ for t A ½t 3 ; t 5 because P pv ðt Þ is periodic withperiod 24 h. Note that t 24 A ½t 1 ; t 3 ;

2. P load ðt Þ is a constant; and3. there is no charge left at t 3 .

In other words, there is no difference between the optimizationproblem with the cost function J 2 and the one with J 1 other thanthe shifting of time t by 24 h. Therefore, the E cmax ð2Þ will be thesame as E cmax ð1Þ. The same reasoning applies to the optimizationproblem with the cost function J i for i ¼ 3; … ; k 1. Therefore, wehave E cmax ðiÞ ¼E cmax ð1Þ for i ¼ 2; … ; k 1.

For the optimization problem J k, there is no charge left at time2k 1. This problem is essentially the same as the problem studiedin the proof to Proposition 6 with the cost function J 1 except theshifting of time t by ðk 1Þ 24 h. The solution E cmax ðkÞ is given asmin ð Ak ; BkÞ=2, where

Ak ¼ Z t 2k

t 2k 1

min ðP Bmax ; max ð0; P pv ðt Þ P load ÞÞ dt ;

and

Bk ¼ Z t 2k þ 1

t 2k

min ð P Bmin ; max ð0; P load P pv ðt ÞÞÞ dt :

Note that Ak ¼ A2 , and Bk o B2 . If we choose E max ¼ min ð A2 ; B2 Þ=2which is larger than or equal to E cmax ðkÞ, we have J kðE max Þ ¼ J kðE cmax ðkÞÞ.

5 Note that the control action for t A ðt 2i 1 ; t 2iÞ and the control action fort A ðt 2i ; t 2i þ 1 Þ for i ¼ 1; 2; … ; k are coupled in the sense that battery cannot bedischarged if at time t 2 i there is no charge in the battery. In general, the controlaction for t A ðt 2i ; t 2i þ 1 Þ and the control action for t A ðt 2i þ 1 ; t 2i þ 2 Þ for i ¼ 1; k 1 canalso be coupled if at time t 2i þ 1 there is extra charge left in the battery because theextra charge will affect the charging action in the interval t A ðt 2i þ 1 ; t 2i þ 2 Þ. Here,there is no such coupling for the latter case when we restrict E max so that at the end

of each operating cycle there is no charge left.










8/13/2019 72e7e52aaa41a76fc6


Now we claim that E cmax when considering the cost function J isexactly min ð A2 ; B2 Þ=2. If we choose E max o min ð A2 ; B2 Þ=2, then J ðE max Þ4 J ðmin ð A2 ; B2 Þ=2Þ by an argument similar to the one inProposition 5 . On the other hand, if we choose E max Z min ð A2 ; B2 Þ=2,then J ðE max Þ ¼ J ðmin ð A2 ; B2 Þ=2Þ. Therefore, E cmax to the optimizationproblem with the cost function J is min ð A2 ; B2 Þ=2. □

References

[1] M. Akatsuka, et al., Estimation of battery capacity for suppression of a PV power plant output uctuation, in: IEEE Photovoltaic Specialists Conference(PVSC), 2010, pp. 540 – 543.

[2] G. Barbose, et al., Tracking the Sun IV: An Historical Summary of the InstalledCost of Photovoltaics in the United States from 1998 to 2010, Technical Report,LBNL-5047E, 2011.

[3] S. Barsali, M. Ceraolo, Dynamical models of lead-acid batteries: implementa-tion issues, IEEE Transactions on Energy Conversion 17 (2002) 16 – 23 .

[4] B. Borowy, Z. Salameh, Methodology for optimally sizing the combination of abattery bank and PV array in a wind/PV hybrid system, IEEE Transactions onEnergy Conversion 11 (1996) 367 – 375 .

[5] T. Brekken, et al., Optimal energy storage sizing and control for wind powerapplications, IEEE Transactions on Sustainable Energy 2 (2011) 69 – 77 .

[6] A.E. Bryson, Y.-C. Ho, Applied Optimal Control: Optimization, Estimation andControl, Taylor & Francis, New York, USA, 1975 .

[7] M. Ceraolo, New dynamical models of lead-acid batteries, IEEE Transactions onPower Systems 15 (2000) 1184 – 1190 .

[8] R. Chedid, S. Rahman, Unit sizing and control of hybrid wind-solar powersystems, IEEE Transactions on Energy Conversion 12 (1997) 79 – 85 .

[9] P. Harsha, M. Dahleh, Optimal sizing of energy storage for ef cient integrationof renewable energy, in: Proceedings of 50th IEEE Conference on Decision andControl and European Control Conference, 2011, pp. 5813 – 5819.

[10] IBM, IBM ILOG CPLEX Optimizer, 2012. URL: http://www-01.ibm.com/software/integration/optimization/cplex-optimizer .

[11] IEEE, IEEE Recommended Practice for Sizing Lead-Acid Batteries for Stand-Alone Photovoltaic (PV) Systems, IEEE Std 1013-2007, 2007.

[12] Q. Li, et al., On the determination of battery energy storage capacity and short-term power dispatch of a wind farm, IEEE Transactions on Sustainable Energy2 (2011) 148 – 158 .

[13] Y. Makarov, et al., Sizing Energy Storage to Accommodate High Penetration of Variable Energy Resources, Technical Report, PNNL-SA-75846, 2010.

[14] W. Omran, et al., Investigation of methods for reduction of power uctuations

generated from large grid-connected photovoltaic systems, IEEE Transactionson Energy Conversion 26 (2011) 318 – 327 .[15] M.H. Rahman, S. Yamashiro, Novel distributed power generating system of PV-

ECaSS using solar energy estimation, IEEE Transactions on Energy Conversion22 (2007) 358 – 367 .

[16] Y. Riffonneau, et al., Optimal power ow management for grid connected PV systems with batteries, IEEE Transactions on Sustainable Energy 2 (2011)309 – 320 .

[17] Y. Ru, et al., Storage size determination for grid-connected photovoltaicsystems, IEEE Transactions on Sustainable Energy 4 (2013) 68 – 81 .

[18] G. Shrestha, L. Goel, A study on optimal sizing of stand-alone photovoltaicstations, IEEE Transactions on Energy Conversion 13 (1998) 373 – 378 .

[19] UCAN, SDG& E Proposal to Charge Electric Rates for Different Times of the Day:The Devil Lurks in the Details, 2012. URL: http://www.ucan.org/energy/electricity/sdge_proposal_charge_electric_rates_different_times_day_devil_lurks_details .

[20] E. Vrettos, S. Papathanassiou, Operating policy and optimal sizing of a highpenetration RES-BESS system for small isolated grids, IEEE Transactions on

Energy Conversion 26 (2011) 744–

756 .[21] X. Wang, et al., Determinant of battery energy storage capacity required in abuffer as applied to wind energy, IEEE Transactions on Energy Conversion 23(2008) 868 – 878 .


http://refhub.elsevier.com/S0947-3580(13)00155-6/othref0005










http://refhub.elsevier.com/S0947-3580(13)00155-6/sbref3





































http://www-01.ibm.com/software/integration/optimization/cplex-optimizer
















































http://www.ucan.org/energy/electricity/sdge_proposal_charge_electric_rates_different_times_day_devil_lurks_details













































































Documents

72e7e52aaa41a76fc6