Battery Sizing for Grid Connected PV Systems with Fixed Minimum Charging/Discharging Time

Transcription

1 Battery Sizing for Grid Conneted PV Systems with Fixed Minimum Charging/Disharging Time Yu Ru, Jan Kleissl, and Sonia Martinez Abstrat In this paper, we study a battery sizing problem for grid-onneted photovoltai (PV) systems assuming that the battery harging/disharging limit sales linearly with its apaity. The objetive is to seek a value of the battery size suh that the eletriity purhase ost from the grid is minimized while satisfying the loads. We propose an upper bound on the storage size, and an algorithm to alulate the exat storage size for the ase with ideal PV generation and onstant loads; we verify that these are onsistent with the results obtained via simulations. I. INTRODUCTION Distributed renewable energy systems have seen signifiant deployment on eletri distribution systems to redue apaity needs and fossil fuel emissions. Among the available renewable energy tehnologies suh as hydroeletri, photovoltai (PV), wind, geothermal, biomass, and tidal systems, grid-onneted solar PV has ontinued to be the fastest growing distributed power generation tehnology [1]. Meanwhile, solar energy generation tends to be variable due to the diurnal yle of the solar geometry and louds. To omplement the PV output during times of peak energy usage and store surplus PV energy for nighttime use, storage devies (suh as batteries, ultraapaitors, and pumped hydro storage) an be applied. In this paper, we study the problem of determining the battery size for grid-onneted PV systems. Our setting is shown in Fig. 1. Eletrial energy is generated from solar panels, and is used to supply loads. If the PV generation is larger than the loads, extra eletrial energy an be stored in batteries to redue energy imports later on; if the PV generation is smaller than loads, eletriity has to be either disharged from the battery or purhased from the grid to meet the loads. Given the high ost of battery storage systems, the size of the battery storage should be small yet minimize eletriity purhases from the grid. We thus formulate a battery sizing problem in whih the maximum battery harging/disharging rate sales linearly with the battery apaity, and show that there is a unique ritial value (denoted as E max, refer to Problem 1) suh that the ost of eletriity purhase remains the same if the net usable battery apaity is larger than or equal to E max and the ost is stritly larger otherwise. We propose an upper bound on E max, and an algorithm to alulate the exat value for the ase of typial maximum PV generation on lear days and Yu Ru, Jan Kleissl, and Sonia Martinez are with the Mehanial and Aerospae Engineering Department, University of California, San Diego ( yuru@usd.edu, jkleissl@usd.edu, soniamd@usd.edu). Solar Panel Eletri Grid Battery Fig. 1. Grid-onneted PV system with battery storage (we assume that the onversion effiieny of DC-to-AC or AC-to-DC onverters is 1; therefore, onverters are not drawn here). onstant loads; the results are very onsistent with the ritial value obtained via simulations. Storage sizing problems for grid-onneted systems have been studied extensively, e.g., [] [6]. In these studies, batteries are used to redue the flutuation of PV output [], maximize a defined servie lifetime/unit ost index of the energy storage system [4], or minimize the ost of the energy storage system [3]. Almost all of these previous studies on storage sizing are based on trial and error approahes exept [5], [6]. In [5], the goal is to minimize the longterm average eletriity osts in the presene of dynami priing as well as investment in storage in a stohasti setting. In [6], we study a battery sizing problem in a setting similar to this work. The key differene is that the maximum harging/disharging rate is fixed in [6], whih is relaxed in this work to allow the maximum rate to sale linearly with the battery apaity; this is more realisti beause battery ells an be onneted in parallel to implement the ritial battery apaity. A. Photovoltai Generation Load II. PROBLEM FORMULATION To alulate the eletrial power generated from solar panels, we use P pv (t) = GHI(t) S η, where GHI (W/m ) is the global horizontal irradiation at the loation of solar panels, S (m ) is the total area of solar panels, and η is the onversion effiieny of the solar ells. The PV generation model is a simplified version of the one used in [7]. B. Eletri Grid Eletriity an be drawn from (or dumped to) the grid. We assoiate osts only with the eletriity purhase from the grid, and assume that there is no benefit by dumping eletriity into the grid. The motivation is that, from a grid operator standpoint, it would be most desirable if PV system served only the loal load and did not export to the grid. We use C gp (t) ( /kwh) to denote the eletriity purhase rate, P gp (t) (W ) to denote the eletrial power purhased

2 from the grid at time t, and P gd (t) (W ) to denote the exess eletrial power dumped to the grid or urtailed at time t. For simpliity, we assume that C gp (t) is time independent and has the value C gp. C. Battery A battery has the dynami deb(t) dt = P B (t), where E B (t) (Wh) is the amount of eletrial energy stored in the battery at time t, and P B (t) (W ) is the harging/disharging rate satisfying P B (t) > when harging and P B (t) < when disharging. We impose the following onstraints on the battery: i) E Bmin E B (t) E Bmax, where E Bmin (or E Bmax ) is the minimum (or maximum) battery eletrial energy (usually E Bmin is hosen to be larger than to prevent fast battery aging), and < E Bmin E Bmax, and ii) P Bmin P B (t) P Bmax, where P Bmin > (or P Bmax > ) is the maximum battery disharging (or harging) rate. We assume that P Bmax = P Bmin = EBmax EBmin, where is the minimum time neessary to harge (or disharge) the battery from E Bmin (or E Bmax ) to E Bmax (or E Bmin ). In other words, the maximum harging/disharging rate sales linearly with the usable apaity of the battery. This assumption is reasonable for senarios in whih the battery is omposed of multiple battery ells onneted in parallel. D. Minimization of Eletriity Purhase Cost With all the omponents introdued earlier, now we an formulate the following problem: min P B,P gp,p gd t+t t C gp P gp (τ)dτ s.t. P pv (t)+p gp (t) = P gd (t)+p B (t)+p load (t), (1) de B(t) dt = P B (t), E Bmin E B (t) E Bmax, E B (t ) = E Bmin, P Bmin P B (t) P Bmax, P Bmax = P Bmin = EBmax EBmin, P gp (t),p gd (t), where t is the initial time, T is the time period onsidered for the ost minimization, and P load (t) (W ) denotes the load at timet. Note that the objetive is to minimize the eletriity purhase ost from the eletri grid during the time interval [t,t + T] while guaranteeing that the eletriity supply an meet the eletriity demand (i.e., the power balane equation (1)) and battery onstraints are not violated. E. Battery Sizing Problem Formulation Before we formulate the battery sizing problem, we first simplify the ost minimization problem in Setion II-D by eliminating variables P gd (t) and P gp (t). Using Eq. (1), we obtain P gp (t) = P load (t) P pv (t) + P B (t)+p gd (t). If P load (t) P pv (t)+p B (t) <, we need to hoose P gd (t) > to make P gp (t) = so that the ost is minimized at the urrent time t; if P load (t) P pv (t) + P B (t) >, we need to hoose P gd (t) = to minimize the eletriity purhase osts, and we have P gp (t) = P load (t) P pv (t)+p B (t). Therefore, P gp (t) an be written as P gp (t) = max(,p load (t) P pv (t) + P B (t)) so that the integrand in the objetive funtion is minimized at the urrent time t. We an plug in the expression of P gp (t) in the objetive funtion, and obtain an optimization problem whih has only one independent variable P B (t). We now hange variables to make the state variable and the ontrol variable expliit. Let x(t) = E B (t) EBmax+EBmin, u(t) = P B (t), and E max = EBmax EBmin (note that the division by is to obtain a simple onstraint on x(t): x(t) E max, and E max is the net (usable) battery apaity; thus, we an equivalently use E max as the battery apaity). Then we have P Bmax = P Bmin = E max, () and an rewrite the optimization problem as t+t J =min C gp max(,p load (τ) P pv (τ)+u(τ))dτ u t s.t. dx(t) = u(t), x(t) E max, x(t ) = E max, dt P Bmin u(t) P Bmax, P Bmax = P Bmin = Emax. (3) We define the set of feasible ontrols (denoted as U feasible ) as ontrols that satisfy the onstraints Emax u(t) Emax. If we fix the parameters t, T, and, J is a funtion of E max, whih is denoted as J(E max ). If E max =, then u(t) =, and J reahes the largest value J max = t+t t C gp max(,p load (τ) P pv (τ))dτ. If we inrease E max, intuitively J will derease (though may not stritly derease) beause the battery an be utilized to store extra eletrial energy generated from PV to be potentially used later on when the load exeeds the PV generation. This is formally justified by the following lemma. Lemma 1 Given the optimization problem in Eq. (3) with fixed t, T, and, if Emax 1 < Emax, then J(Emax) 1 J(E max). Proof: Given Emax, 1 suppose a feasible ontrol u 1 (t) ahieves the minimum eletriity purhase ost J(Emax), 1 and the orresponding state x is x 1 (t). Sine x 1 (t) Emax 1 < Emax and u 1 (t) E1 max < E max, u 1 (t) is also a feasible ontrol for problem (3) with the state onstraint Emax, and results in the ost J(Emax). 1 Sine J(Emax) is the minimal ost over the set of all feasible ontrols whih inlude u 1 (t), we must have J(Emax) 1 J(Emax). In other words, J is dereasing with respet to the parameter E max, and is lower bounded by. We are interested in finding the smallest value of E max (denoted as E max) suh that J remains the same for any E max E max. Problem 1 (Storage Size Determination) Given the optimization problem in Eq. (3) with fixed t,t, and, determine a ritial value E max suh that i) E max < E max, J(E max ) > J(E max), and ii) E max E max, J(E max ) = J(E max). Note that the ritial value Emax is bounded (refer to Proposition 1), and unique as stated below (whih an be

3 proved via ontradition based on the definition of E max). Lemma Given the optimization problem in Eq. (3) with fixed t,t, and, E max is unique. One may try to alulate E max by first obtaining an expliit expression for J(E max ) and then solve for E max. However, the optimization problem in Eq. (3) is diffiult to solve due to the state and ontrol onstraints. Instead, we first fous on bounding the ritial value E max in the next setion, and then study the problem for speifi senarios in Setion IV. III. UPPER BOUND ON E max To make use of the battery, we impose the following assumption on Problem 1, whih guarantees that the battery an be harged at t 1 and the battery disharging at t > t 1 an stritly redue the ost. Assumption 1 There exist t 1,t [t,t + T], suh that t 1 < t, P pv (t 1 ) P load (t 1 ) > and P pv (t ) P load (t ) <. Proposition 1 Given the optimization problem in Eq. (3) with fixed t,t, and under Assumption 1, where < E max max( min(a,b) A = t+t, max(c,d) ), t max(,p pv (t) P load (t))dt, (4) t+t B = max(,p load (t) P pv (t))dt. (5) t C = max (P load(t) P pv (t)), (6) t [t,t +T] D = max t [t,t +T] (P pv(t) P load (t)). (7) Proof: We first show Emax > via ontradition. Sine Emax, we need exlude the ase Emax =. Suppose =. If we hoose E max > Emax =, J(E max ) < J(Emax) beause under Assumption 1 a battery an store the extra PV generated eletrial energy first and then use it later on to stritly redue the ost ompared with the ase without a battery (i.e., the ase with E max = ). A ontradition to the definition of Emax. To show Emax max( min(a,b), max(c,d)t ), it is suffiient to show that if E max max( min(a,b), max(c,d)t ), then J(E max ) = J(max( min(a,b), max(c,d)t )). Before we prove the result, we briefly disuss the meanings of A,B,C,D. In Eq. (4), max(,p pv (t) P load (t)) is the extra PV generated eletrial power after supplying the load, and therefore, A is the maximum total amount of eletrial energy that an be stored in the battery during the interval [t,t + T] without onsidering the maximum harging rate P Bmax. In Eq. (5), max(,p load (t) P pv (t)) is the eletrial power stritly neessary to supply the load at time t via either battery disharging or grid purhase. Therefore, B is the total amount of eletrial energy that needs to be disharged from the battery or purhased from the grid to meet the load. Sine the objetive is to minimize the eletriity purhase from the grid, we would like to fulfill B using the eletrial energy disharged from the battery (whenever possible). The atual disharged eletrial energy an be smaller than B if the maximum disharging rate P Bmin is smaller than max(,p load (t) P pv (t)) for some t. In Eq. (6), C is the minimum of the battery disharging rate so that battery disharging and PV generation an meet the load. In Eq. (7), D is the minimum of the battery harging rate so that the battery an be harged whenever there is extra PV generated power after supplying the load. We first onsider min(a,b) max(c,d)t. In this ase, the upper bound beomes min(a,b), and E max min(a,b) max(c,d). Note that E max max(c,d)t implies that P Bmax = P Bmin = Emax max(c,d). Therefore, the maximum battery harging/disharging rate is large enough to ensure that PV generated extra power an be stored (as long as the battery apaity is large enough) and the battery disharging an meet the demand at any time (as long as there is enough eletrial energy stored). Thus, in this ase, the only fator that matters is the eletrial energy stored and the eletrial energy that an be disharged. If E max min(a,b), the battery is large enough to store all extra PV generated eletrial energy (when A B) or supply the load all the time (when A > B). Then it is not possible to lower the ost with an E max min(a,b), whih implies that J(E max ) = J( min(a,b) ). min(a,b) We now onsider < max(c,d)t. In this ase, the upper bound beomes max(c,d)t, and E max max(c,d) > min(a,b), i.e., the battery apaity is large enough to hold all extra PV generated eletrial energy (when A B) or eletrial energy neessary to power the load (when A > B). Thus, in this ase, the only fator that matters is the maximum harging/disharging rate. If E max max(c,d)t, then P Bmax = P Bmin max(c,d). In other words, the maximum battery harging/disharging rate is large enough to ensure that PV generated extra eletrial power an be stored and the battery disharging an meet the demand at any time. Therefore, it is not possible to lower the ost, whih implies that J(E max ) = J( max(c,d)t ). This ompletes the proof. IV. IDEAL PV GENERATION AND CONSTANT LOAD In this setion, we study how to obtain the ritial value for the senario in whih the PV generation is ideal and the load is onstant as speified below. Assumption The initial time t is h LST, T = k 4(h) where k is a nonnegative integer, P pv (t) is periodi on a timesale of 4 hours, and satisfies the following property fort [,4(h)]: there exist three time instants < t sunrise < t max < t sunset < 4(h) suh that P pv (t) = for t [,t sunrise ] [t sunset,4(h)]; P pv (t) is ontinuous and stritly inreasing (or dereasing) for t [t sunrise,t max ] (or t [t max,t sunset ]); P pv (t) ahieves its maximum P max pv at t max, and P load (t) = P load for t [t,t + T], where P load is a onstant satisfying < P load < P max pv.

4 It an be verified that Assumption implies Assumption 1. A. E max under Fixed Maximum Charging/Disharging Rate If P Bmin and P Bmax are fixed, we have the following results regarding E max. Proposition [6] Given the optimization problem in Eq. (3) with fixed t,t,p Bmin and P Bmax under Assumption. I) If T = 4(h), Emax = min(a1,b1), where A 1 = B 1 = 4 4 min(p Bmax,max(,P pv (t) P load ))dt, (8) t max min( P Bmin,max(,P load P pv (t)))dt ; (9) II) If T = k 4(h) where k > 1 is a positive integer, E max = min(a,b), where A = A 1 and B = t max+4 t max min( P Bmin,max(,P load P pv (t)))dt. (1) Remark 1 Note that if T = 4(h), A 1 is the amount of extra PV generated eletrial energy that an be stored in a battery, and B 1 is the amount of eletrial energy that is neessary to supply the load and an be provided by battery disharging. Instead, if we fix the harging time, then P Bmin and P Bmax hange as E max hanges due to Eq. (). Based on Proposition, one might try to alulate Emax by replaing P Bmax and P Bmin in Proposition with Eq. () and then solving Emax = min(a1,b1) as an impliit equation of Emax. Numerially we an solve the impliit equation; however, the obtained Emax may not be onsistent with the value obtained via simulations for ertain, whih is illustrated in Setion V-B. Therefore, new tehniques are neessary to handle the senario with fixed. B. E max under Fixed Minimum Charging/Disharging Time Now we study how to alulate Emax when is fixed. This alulation is based on the analysis of Proposition, its appliation in the urrent setting, and insights provided by simulations. We first onsider the senario in whih P load < Ppv max P load and T = 4(h). As illustrated in the simulation Setion V-B, the value of Emax given by Proposition is orret for ertain values of. Therefore, we begin our analysis based on the result in Proposition. For the quantity A 1 defined in Eq. (8), the maximum value of max(,p pv (t) P load ) is Ppv max P load. Therefore, if P Bmax Ppv max P load, A 1 beomes A 1 = 4 max(,p pv (t) P load )dt, (11) whih does not diretly depend on E max (reall P Bmax = E max ). Similarly, If P Bmin P load, B 1 beomes B 1 = 4 t max max(,p load P pv (t))dt, (1) whih does not diretly depend on E max either (reall P Bmin = Emax ). Now we study different ranges for P Bmax under the onstraint P load < Ppv max P load. There are three possibilities: i) P Bmax < P load, or equivalently, E max < Emax, L where Emax L = PloadT. In this ase, it means that P Bmin = P Bmax is not large enough to supply the demand from the load for t [t sunset,4], whih results in grid purhase. By inreasing the battery apaity (whih inreases P Bmax ), the eletriity purhased from the grid will be lowered (thus, also lowering the ost) beause more extra PV generation an be stored due to P Bmax < P load < Ppv max P load. Therefore, Emax annot be smaller than Emax. L ii) P load P Bmax < Ppv max P load, or equivalently, Emax L E max < Emax, H where Emax H max (Ppv P load) =. In this ase, the quantity B 1 is given in Eq. (1) but the quantity A 1 is still given in Eq. (8). Based on the meanings of A 1 andb 1 in Remark 1, to minimize the grid purhase ost, we need to make sure A 1 B 1 and E max B 1 ; in other words, the supply is at least as large as the demand and the battery an store the needed eletrial energy B 1. By inreasing E max starting from Emax, L A 1 will stritly monotonially inrease based on Eq. (8), and therefore, A 1 B 1 will eventually be satisfied (though E max may inrease even beyond Emax). H Therefore, the smallest value of E max satisfying Emax L E max < Emax, H A 1 B 1, and E max B 1, will be the ritial value Emax, whih an be proved via ontradition. If no suh value an be found, we need to onsider E max Emax. H P load P Bmax, or equivalently, E max Emax. H In this ase, the quantity A 1 is given in Eq. (11) and B 1 is given in Eq. (1). Based on the meanings of A 1,B 1, and the result in Proposition, the battery apaity an be hosen to be min(a1,b1). However, we iii) P max pv also need to guarantee that E max Emax. H Therefore, max(emax, H min(a1,b1) ) is the ritial value, whih an be proved via ontradition. IfP load P max (P max pv P load) pv P load, we an redefine PloadT ase H max, and as Emax. L Similar analysis an be applied to this senario and the differene lies in the ase Emax L E max < Emax. H In this ase, it might not be possible that A 1 B 1 due to larger loads; when A 1 < B 1, we require E max A 1, i.e., when the supply annot meet the demand, the battery should store all extra PV generated eletrial energy to lower the grid purhase. When T = k 4 for k > 1, we an do the same analysis based on the result in part II of Proposition. Now an algorithm to alulate the value Emax is given in Algorithm 1. The inputs are t,t,,p load,p pv (t), and the output is Emax. The struture of the algorithm follows the previous disussion: Steps 1- set Emax L and Emax; H Steps 3-7 deal with the ase in whih Emax L E max < Emax H while Steps 1-1 handle the ase in whih E max Emax. H Based on the expression for Emax at Step 11, it an be verified that the upper bound in Proposition 1 also holds in the urrent

5 P pv (w) Load (w) Jul 13, 1 Jul 14, 1 Jul 15, 1 Jul 16, Time (h) Fig.. setting. PV output for July 13-16, 1, at La Jolla, California. Algorithm 1 Calulation of Emax Input: Fixed t,t,,p load, and P pv (t) for t [t,t +T] satisfying Assumption Output: Emax 1: Set E L max = : Set Emax H = 3: if P load < Ppv max max min(pload, Ppv P load) ; max max(pload, Ppv P load) ; P load then 4: Find the smallest value of E max satisfying Emax L E max < Emax H and: i) A 1 B 1 and E max B 1 for T = 4; ii) A B and E max B for T = k 4; 5: else 6: Find the smallest value of E max satisfying Emax L E max < Emax H and: i) either A 1 B 1 and E max B 1, or A 1 < B 1 and E max A 1 for T = 4; ii) either A B and E max B, or A < B and E max A for T = k 4; 7: end if 8: if a solution is found then 9: Set E max as the smallest value of E max ; 1: else 11: i) If T = 4, alulate A 1,B 1, and set E max = max(e H max, min(a1,b1) ); ii) If T = k 4, alulate A,B, and set E max = max(e H max, min(a,b) ); 1: end if 13: Output E max. V. SIMULATIONS In this setion, we verify the results in Setions III and IV via simulations. The parameters used in Setion II are hosen based on a typial residential home setting. The GHI data is the measured GHI between July 13 and July 16, 1 at La Jolla, California. In our simulations, we use η =.15, and S = 1(m ). Thus P pv (t) = 1.5 GHI(t)(W). We hoose t as h LST on Jul 13, 1, and the hourly PV output is given in Fig. for the following four days starting from t. The PV generation roughly satisfies Assumption. Note that P pv (t) < 15(W) for t [t,t +96]. We set C gp = 7.8 /kwh, whih is the semipeak rate for the summer season proposed by SDG&E (San Diego Gas & Fig. 3. A typial residential load profile. Eletri) [8]. For the battery, we hoosee Bmin =.4 E Bmax, and then E max =.3 E Bmax. is hosen to be between (h) and 14(h) with the step size 1(h). We hoose P load to be between and 15(W), and the load is used from t to t + T to satisfy Assumption. To run simulations, we disretize the battery dynami equation in Setion II-C as E B (k + 1) = E B (k) + P B (k) using one hour as the sampling interval. A. Dynami Loads We first examine the upper bound in Proposition 1 using dynami loads. The load profile for one day is given in Fig. 3, whih resembles the residential load profile in Fig. 8(b) in [7]. For multiple day simulations, the same load profile is used for all days. We study how the ost funtion J hanges as E max varying from to 6(Wh) with the step size 1(W h), by solving the optimization problem in Eq. (3) via linear programming using the CPlex sover [9]. If T = 4(h) and = (h) (or = 7(h), respetively), the plot of the minimum osts versus E max is given in Fig. 4(a) (or (b), respetively). The plots onfirm the result in Lemma 1, and also show the existene of the unique Emax. The orresponding Emax values are shown in the olumn Sim with T = 4(h) in Table I, whih an be identified from Fig. 4. The upper bounds in Proposition 1 are shown in the olumn UB with T = 4(h) in Table I. Similarly, for T = 48(h) and T = 96(h), we an alulate the values of Emax given = (h) or = 7(h), and also the orresponding upper bounds. The results are listed in Table I. The upper bound holds for all ases though the differene between the upper bound andemax inreases when T inreases. This is due to the fat that for multiple days battery an be repeatedly harged and disharged; however, this fat is not taken into aount in the upper bound in Proposition 1. TABLE I VALUES AND UPPER BOUNDS ON E max(wh) T = 4(h) T = 48(h) T = 96(h) (h) Sim UB Sim UB Sim UB B. Constant Loads In this subsetion, we first set P load = (W), and inrease the battery apaity E max from to 1(Wh) with the step size 1(Wh). Then for = (h) (or = 7(h), respetively), the value for Emax is alulated to be 499(Wh) (or 7(Wh), respetively). If we try to use the result in Proposition by replaing P Bmax and P Bmin in Proposition

6 Minimum Cost ($) Emax (Wh) Ex Th 4 Ex 4 Th 6 Ex 6 Th 8 Ex 8 Th 1 Ex 1 Th 1 Ex 1 Th E max (wh) (a) = (h) T (h) (a) One day Minimum Cost ($) Emax (Wh) Ex Th 4 Ex 4 Th 6 Ex 6 Th 8 Ex 8 Th 1 Ex 1 Th 1 Ex 1 Th E max (wh) (b) = 7(h) Fig. 4. Plots of minimum osts versus E max obtained via simulations with the load profile in Fig. 3 and T = 4(h). with Eq. () and then solving Emax = min(a1,b1) as an impliit equation of Emax, we obtain Emax = 499(Wh) for = (h), and Emax = 1(Wh) for = 7(h). Even though the theoretial value for = (h) is very onsistent with the value obtained via simulations, the theoretial value for = 7(h) is far from orret. However, if we apply Algorithm 1 to both ases, the theoretial values are 499(Wh) and 7(W h), whih are onsistent with the values obtained via simulations. Now we run more simulations to examine the results obtained using Algorithm 1. We hoose P load from (W) to 1(W) with the step size (W), from (h) to 14(h) with the step size 1(h), and T to be 4(h), 48(h), and 96(h), and then alulate Emax. The results are plotted in Fig. 5. For example, in Fig. 5(a), we fix T = 4(h), and plot the theoretial values (i.e., the urves with the marker) and the values obtained via simulations (i.e., the urves with the + marker) of Emax as varies from to 14 hours. The theoretial values are very lose to the values obtained via simulations for T = 4(h), whih is also true for T = 48(h) as shown in Fig. 5(b). However, for T = 96(h), there are slightly larger variations in the differene between the theoretial values and the values obtained via simulations for P load = (W) and P load = 4(W), as shown in Fig. 5(). By examining the eletriity purhase history (namely, P gp (t)), the disrepany is due to the fat that the PV generation is not perfetly periodi, whih an be verified based on the PV output plot in Fig.. VI. CONCLUSIONS In this paper, we studied optimal sizing of battery storages used in grid-onneted PV systems assuming that the maximum battery harging/disharging rate sales linearly with its apaity. In future work, we would like to take into aount the dynami time-of-use priing of the eletriity purhase from the grid, and extend the results to wind energy storage systems. Emax (Wh) (h) Ex Th 4 Ex 4 Th 6 Ex 6 Th 8 Ex 8 Th 1 Ex 1 Th 1 Ex 1 Th (b) Two days (h) () Four days Fig. 5. Plots of the values obtained via simulations (namely, Ex ) and theoretial values (namely, Th ) of Emax for P load from (W) to 1(W) with the step size (W) and T = 4,48,96(h). REFERENCES [1] S. Teleke, M. E. Baran, S. Bhattaharya, and A. Q. Huang, Rule-based ontrol of battery energy storage for dispathing intermittent renewable soures, IEEE Transations on Sustainable Energy, vol. 1, pp , 1. [] M. Akatsuka, R. Hara, H. Kita, T. Ito, Y. Ueda, and Y. Saito, Estimation of battery apaity for suppression of a PV power plant output flutuation, in IEEE Photovoltai Speialists Conferene (PVSC), Jun. 1, pp [3] T. Brekken, A. Yokohi, A. von Jouanne, Z. Yen, H. Hapke, and D. Halamay, Optimal energy storage sizing and ontrol for wind power appliations, IEEE Transations on Sustainable Energy, vol., pp , Jan. 11. [4] Q. Li, S. S. Choi, Y. Yuan, and D. L. Yao, On the determination of battery energy storage apaity and short-term power dispath of a wind farm, IEEE Transations on Sustainable Energy, vol., pp , Apr. 11. [5] P. Harsha and M. Dahleh, Optimal sizing of energy storage for effiient integration of renewable energy, in Pro. of 5th IEEE Conferene on Deision and Control and European Control Conferene, De. 11, pp [6] Y. Ru, J. Kleissl, and S. Martinez, Storage size determination for grid-onneted photovoltai systems, submitted to IEEE Transations on Sustainable Energy, 11. [Online]. Available: V1 [7] M. H. Rahman and S. Yamashiro, Novel distributed power generating system of PV-ECaSS using solar energy estimation, IEEE Transations on Energy Conversion, vol., pp , Jun. 7. [8] SDG&E proposal to harge eletri rates for different times of the day: The devil lurks in the details. [Online]. Available: proposal harge eletri rates different times day devil lurks details [9] IBM ILOG CPLEX Optimizer. [Online]. Available: ibm.om/software/integration/optimization/plex-optimizer