A Simple Optimal Power Flow Model with Energy Storage

Transcription

1 49th IEEE Conference on Decision and Contro December 15-17, 2010 Hiton Atanta Hote, Atanta, GA, USA A Simpe Optima Power Fow Mode with Energy Storage K. Mani Chandy, Steven H. Low, Ufuk opcu and Huan Xu Abstract he integration of renewabe energy generation, such as wind power, into the eectric grid is difficut because of the source intermittency and the arge distance between generation sites and users. his difficuty can be overcome through a transmission network with arge-scae storage that not ony transports power, but aso mitigates against fuctuations in generation and suppy. We formuate an optima power fow probem with storage as a finite-horizon optima contro probem. We prove, for the specia case with a singe generator and a singe oad, that the optima generation schedue wi cross the time-varying demand profie at most once, from above. his means that the optima poicy wi generate more than demand initiay in order to charge up the battery, and then generate ess than the demand and use the battery to suppement generation in fina stages. his is a consequence of the fact that the margina storage cost-to-go decreases in time. I. INRODUCION he optima power fow (OPF probem is to optimize a certain objective over power network variabes under certain constraints. he variabes may incude rea and reactive power outputs, bus votages and anges; the objective may be the minimization of generation cost or maximization of user utiities; and the constraints may be bounds on votages or power eves, or that the ine oading not exceeding therma or stabiity imits. he OPF has been studied for over haf a century since the pioneering work of Carpentier 1]; see surveys for exampe in 2], 3], 4], 5], 6] and 7]. Its history is roughy a continuous appication of more and more sophisticated optimization techniques 7]. Most of the modes invove static optimization as, without arge scae storage, power suppy and demand must be matched exacty at a times, and therefore OPF can be soved in isoation from one period to the next. In this paper, we formuate a simpe OPF mode with storage and study how storage aows optimization of power generation across mutipe time periods. he mode is motivated by the intensifying trend to depoy renewabe energy such as wind or soar power. In the state of Caifornia, peak demand for power in 2003 reached 52 GW, with projections for the year 2030 exceeding 80 GW 8], 9]. With fossi fue and nucear pants retiring in the next few decades, and a required 15% reserve margin, an additiona 60 GW of new generation capacity wi be needed by ]. In 2006, Southern Caifornia Edison, the primary eectricity he authors are with Caifornia Institute of echnoogy, Pasadena, CA, 91125, USA. K. M. Chandy is with the Department of Computer Science; S. H. Low is with the Department of Computer Science and the Department of Eectrica Engineering; U. opcu is with the Department of Contro and Dynamica Systems; and H. Xu is with the Department of Mechanica Engineering. E- mai addresses: mani@catech.edu, sow@catech.edu, utopcu@cds.catech.edu, mumu@catech.edu. utiity company for southern Caifornia, signed a contract to provide 1.5 GW of power to its customers from wind projects in the ehachapi area in Caifornia 10]. Not ony is renewabe energy more environmentay friendy, but its potentia is high. It has been estimated that if 20% of the energy that coud be harvested from wind farms across the gobe were used, a of the word s eectricity demands coud be met severa times over 11]. here are however two major chaenges in integrating wind or soar power into the current system. First, their output fuctuates widey, rapidy, and randomy, a great operationa hurde. Second, the geographica ocations of wind or soar farms are inevitaby far from the oad centers. hese two probems cas for argescae storage that can absorb short-term fuctuations and transmission capacity that not ony transports power from generation to oad, but can aso provide spatia diversity in generation to mitigate intermittency of renewabe sources. See, for instance, 12] for an eary simuation study on how battery can hep with reguation and peak-shaving in economic dispatch. References 13], 14], 15] investigate the utiity of energy storage in the integration of renewabe energy resources and the concept of microgrids. he mode studied in this paper does not capture the fu spectrum of issues invoved in the integration of renewabe sources, but ony focuses on the effect of storage to optima power fow. For instance, it assumes time-varying but deterministic, as opposed to stochastic, generation and demand. Moreover, the cacuation of the optima power fow assumes the fu knowedge of the demand schedue. In practice, demands are typicay estimated 24 hours in advance to an accuracy of a few percentage points. he cassica OPF probem without storage is a static optimization as the need to baance suppy and demand at a times decoupes the optimization in different periods. he addition of storage introduces correation, and an opportunity to optimize, across time, e.g., charge when (and where the cost of generation is ow and discharge when it is high. Our mode characterizes the structure of optima generation and charge/discharge schedue. In Section II, we formuate the OPF mode with storage as a finite-horizon optima contro probem. In Section III, we consider the specia case with a singe generator and a singe oad (SGSL case. Without battery, the generation must be exacty equa to the oad in each period. With battery, we prove that the optima generation schedue wi cross the time-varying demand profie at most once, from above. his means that the optima poicy wi generate more than demand initiay in order to charge up the battery, and then generate ess than the demand and use the battery to suppement generation in fina stages. his is a /10/$ IEEE 1051

2 consequence of the fact that the margina storage cost-to-go decreases in time. In Section IV, we discuss the optimaity conditions for the genera network case and present some numerica exampes that suggest the intuition in the SGSL case might generaize to the network case. II. MODEL AND PROBLEM FORMULAION We now present a simpe OPF mode with energy storage and time-varying generation costs and demands. he mode ignores reactive power and makes other simpifying assumptions. Our goa is to understand the impact of storage on optima generation schedue. Consider a set G of generation nodes/buses connected to a set D of demand nodes/buses by a transmission network. Let N = G D be the set of a nodes. he transmission network is modeed by the admittance matrix Y, where Y i j = Y ji is the admittance between nodes i and j. If nodes i and j are not directy connected, Y i j = 0. he (rea power fow from node i to node j at time t is V i V j Y i j sin(θ i (t θ j (t for i j N. In this paper, we wi assume θ i (t θ j (t is sma and approximate sin(θ i (t θ j (t by θ i (t θ j (t 16]. he amount of power deivered over ink (i, j is imited by therma effects and network stabiity 17, chapter 4]. Current fows produce power osses aongside heat generation, reducing transmission efficiency and creating ine sag from temperature rises. We capture these constraints as V i V j Y i j (θ i (t θ j (t q i j (t, i j N (1 where q i j (t represents the ine capacity from nodes i to j. From Kirchoff s aws, the net power export from node i at time t is given by q i (t = V i V j Y i j (θ i (t θ j (t, i N. (2 j N If q i (t is positive, node i suppies power at time t. Otherwise, it consumes power at time t. Each node i D demands a fixed amount d i (t of power at time t that must be met by suppies from the generation nodes q i (t = d i (t, i D. (3 Each generation node i G has both a generator that produces g i (t amount of power at time t and a battery that can charge or discharge r i (t amount of power at time t. he net power export q i (t from generator node i at time t consists of the power from the node s generator and battery: q i (t = g i (t + r i (t, i G (4 g i (t 0, i G (5 Note that r i (t can either be negative (battery is charging or positive (battery is discharging. he battery energy eve b i (t at node i G evoves according to 1 b i (t = b i (t 1 r i (t, i G (6 1 In this paper, a power quantities such as g i (t,q i (t,r i (t,d i (t are in the unit of energy per unit time, so the energy produced/consumed in time period t are g i (t,q i (t,r i (t,d i (t, respectivey. with given initia energy eve b i (0 0. Battery storage is bounded by a minimum and a maximum capacity 0 b i (t B i i G. (7 We assume the cost c i (g i,t of generation at generator i G is a function of the amount of generated power g i and time t. here is a battery cost h i (b i,r i as a time-invariant function of energy eve b i and the power draw r i. Finay, there is a termina cost h i (b i( on the fina battery energy eve b i (. he cassica OPF probem without storage determines the votage anges θ i (t, i N, and power generations q i (t, i G, so as to minimize the tota generation cost. Without storage, there is no correation across time, and therefore the optimization probem is static and can be soved in isoation from one period to the next. he transmission network Y aows optimization across space. Storage aows optimization across time as we, i.e., charge when (and where the cost of generation is ow and discharge when it is high. he optimization probems in each period thus become couped, yieding an optima contro probem. Given admittance matrix Y, initia battery eves b i (0 0 and storage capacities B i 0 for i G, ink capacities q i j for i, j N, and demand profies d i (t for i D, t = 1,...,, the OPF probem with energy storage is OPF-S min i G (c i (g i (t,t + h i (b i (t,r i (t (8 + h i (b i ( i G over θ,q,r,g,b (9 s.t. (1,(2,(3,(4,(5,(6,(7, (10 where t = 1,..., in constraints (1-(7. Note that in this mode ony the generation costs c i (g i,t and demand profies d i (t are time-varying. In this paper, we restrict ourseves to the quadratic costs c i (g i (t,t := 1 2 γ i(tg 2 i (t, (11 where γ i (t modes the time-varying nature of the generation costs. he convexity of c i in g i refects the decreased efficiency of the generator when producing very high amounts of power 17, chapter 11]. We aso assume a battery cost that is independent of the power draw r i, but dependent ony on the energy eve b i, i.e., h i (b i,r i = h i (b i. (12 For exampe, if it is desirabe to maintain a fu battery eve at the end of each period, then a cost function h i (b i = α i (B i b i for some α i > 0, imposes a penaty proportiona to the deviation from its capacity. Notations. We wi use h (b(t and dh db (b(t interchangeaby. For both prima variabes, such as g i (t,b i (t, and their dua variabes, such as b i (t,b i (t,b i (t, a denotes their vaue at optimaity, e.g., g i (t,b i (t, b i (t, b i (t, b i (t. For x R, x] + := max(x,

3 III. SINGLE GENERAOR SINGLE LOAD (SGSL CASE In this section we sove the simpest case of a singe generator (with a battery connected to a singe oad. Remova of the network structure aows us to competey characterize the soution and make transparent the effect of storage on the optima generation schedue and the optima chargedischarge schedue of the battery. We expect the basic insight to generaize to the network case, the discussion of which is in the next section. A. SGSL mode From (2, (3, and (4 we have g(t + r(t = d(t (13 his simpifies the probem OPF-S into the foowing min over (c(g(t,t + h(b(t + h (b( g(t, b(t s. t. b(t = b(t 1 d(t + g(t (14 for g(t 0 (15 b(t 0 (16 B b(t 0 (17 t = 1,...,, where b(0 0 is given, and the cost functions c and h are given by (11 and (12 respectivey. he capacity constraint (1 degenerates into d(t q where q is the capacity of the ink connecting the generator to the oad. For feasibiity, we assume that this constraint is satisfied. For t = 1,...,, et b(t be the dua variabes associated with constraints (14, ˆλ(t with constraints (15, b(t with constraints (16, b(t with constraints (17. he probem OPF-S is a convex program and therefore the Karush-Kuhn-ucker (KK condition is both necessary and sufficient for optimaity. Differentiating with respect to b(t, the KK condition impies dh db (b (t + b (t b (t + 11(t < = b (t b (t where 1 denotes the indicator function. his defines a recursion on b (t whose soution, for t = 1,...,, is where H (t := B (t := b (t = H (t + B (t (18 dh db (b (τ dh db (b ( (b (τ b (τ. Here, b (tb (t = 0 and (B b (tb (t = 0 by compementary sackness. H (t is the margina storage cost-to-go at time t. By assumption (see A0 beow h (b < 0,(h (b < 0, and hence H (t > 0. For exampe, if h(b = α(b b and h (b = α (B b for some constants α,α > 0, then H (t = α( + 1 t + α. Differentiating the objective function with respect to g(t, the KK condition impies ( b(t = λ(t g (t = b (t + ˆλ (t, where ˆλ (tg (t = 0 by compementary sackness. Combining with (18, the optima soution is characterized by ] g (t = g(t + B (t + (19 ] b (t = b (t 1 d(t + g(t + B (t + (20 where g(t, the nomina generation at time t, is defined as g(t := H (t. (21 We now derive the structure of the optima generation schedue g and the optima battery energy eve b from (19 (21. B. Optima soution It is instructive to first ook at the case where the battery constraint is inactive, i.e., b(t (0,B. In this case, B (t 0, and the nomina generation schedue in (21 is optima, i.e., g = g. his satisfies the condition g (t = H (t = dh db (b (τ dh db (b (. he eft-hand side is the margina generation cost at time t and the right-hand side is the margina storage cost-to-go from t to. At optimaity, one cannot reduce the cost further by increasing (decreasing generation by a unit and raising (reducing battery eve by the same amount. Moreover, if h (b < 0, the margina storage cost-to-go, and therefore the margina generation cost, stricty decreases in time under an optima poicy. hese intuitions wi generaize to the genera case when the battery constraints may be active, as we now show. Our first main resut characterizes the optima generation schedue g (t under the assumptions A0: For t = 1,...,, d(t > 0, > 0. For b 0, h (b < 0 and (h (b < 0; A1: For t = 1,...,, d(t γ(t + 1d(t + 1 < h (b(t. Assumption A1 restricts how fast the product d(t can decrease. If the battery constraint is not active, then the optima generation g wi baance the margina generation cost and the margina storage cost-to-go. Hence, the optima margina generation cost g (t wi decrease at the same rate at which the margina storage cost-to-go H (t decreases, which is the margina storage cost h (b (t at time t. he case where the battery constraint is active is more compicated because the optima generation must anticipate future starvation and saturation. As we wi see beow, if the product d(t decreases more sowy than margina storage cost h (b(t, then the optima generation g retains 1053

4 a simpe goba structure where g crosses demand d(t at most once, from above, and the optima battery eve b is unimoda. Indeed, under assumption A1, the optima generation schedue generay has three phases. In the first phase, g (t > d(t and the battery charges. In the second phase, g (t = d(t and the battery remains saturated at b (t = B. In the third phase, g (t < d(t and the battery discharges. Degenerate cases where one or more of the phases are missing are possibe, depending on probem parameters, b(0,, and d(t. If the battery never saturates, the optima schedue degenerates into (at most two phases. If the battery does saturate, the optima schedue has three phases, defined by the beginning time + 1 and the end time m of the second phase, with 0 < m, where b (t = B for t =,...,m. o describe the optima generation g precisey, define the foowing time averages over the entire horizon and over the initia and fina phases d := 1 d 1 := 1 d 3 := 1 m t=m+1 ( d(t, γ :=, ( d(t, γ 1 :=, ( d(t, γ 3 := m. t=m+1 Our main resut impies that it is optima to charge the battery initiay and discharge it ater. heorem 1: Suppose A0 and A1 hod and 0 < b(0 < B. he optima generation g crosses the demand curve d(t at most once, from above, and therefore the optima battery eve b is unimoda. Moreover, they take one of the foowing two forms. 1 he battery never saturates, and the optima generation schedue is with g (t = g(t + γ d σ]+ σ := 1 g(t + b(0. Moreover, b ( = σ d] +. 2 here exist times and m, with 0 < m, for which b (t = B for t =,...,m. Before time +1, the battery charges and its stored energy stricty increases over time towards B. After time m, it discharges and its stored energy stricty decreases over time from B. he optima generation g has three phases: g(t γ 1 g (σ 1 d 1, t = 1,..., (t = d(t, t = + 1,...,m g(t + γ 3 d 3 σ 3 ] +, t = m + 1,..., b(0 where σ 3 := σ 1 := 1 1 m t=m+1 g(t B b(0 g(t + B m. Moreover, b ( = ( mσ 3 d 3 ] +. Case 1 of heorem 1 is iustrated in Figure 1. Here,!(! ( t g * ( t charge!( t d( t b * ( t discharge Fig. 1. Optima margina generation cost g (t, in comparison with d(t and battery eve b (t. his figure iustrates case 1 of heorem 1 with d > σ. the battery eve remains stricty beow B at a times. he threshod σ is the sum of average nomina generation and the initia battery energy per unit time. Under the nomina generation schedue, the tota avaiabe energy over the entire period t = 1,..., is σ. If the average demand d is ower than the threshod σ, then the nomina generation is optima, g = g, and the termina battery eve at the end of the contro horizon is b ( = (σ d. Otherwise, the nomina generation schedue wi depete the battery eve to zero before time, which is not optima, and hence the generation is shifted up by γ(d σ/ at each time t, so that the battery eve hits zero exacty (and ony at time. Case 2 of heorem 1 is iustrated in Figure 2. In this case, the battery eve rises from b(0 to B in phase 1, stays at B in phase 2, and drains in phase 3. he threshod σ 3 is the sum of the average nomina generation over the third phase and the per-period battery eve at the beginning of the third phase. If the nomina generation is used in the third phase, then the tota avaiabe energy over this phase is ( mσ 3. If the average demand d 3 over the third phase is ess than the threshod σ 3, then the nomina generation is optima in the third phase, g = g, and the termina battery eve is b ( = ( mσ 3 d 3 ] + (this is simiar to case 1 with initia battery eve B, but over ony the third phase. On the other hand, if d 3 > σ 3, then the nomina generation in the third phase is shifted up in the third phase so that the battery eve hits zero exacty at time. 1054

5 !( t g * ( t!( t d( t b * ( t b(0 charge saturated discharge Fig. 2. Optima margina generation cost g (t, in comparison with d(t, and battery eve b (t. his figure iustrates case 2 of heorem 1 with d 1 < σ 1 and d 3 < σ 3. Exampe 1: SGSL case without assumption A1. We present an SGSL exampe where assumption A1 does not hod. he horizon = 24 hours and each time step t represents 1 hour. he demand profie is ( 4π d(t = 10 sin (t GJ. 1 Battery capacity B is 25 GJ (biion joues, with an initia battery eve b(0 of 12.5 GJ. We set 1 in the generation cost c and use h(b = α(b b with α = 2. Figure 3 shows the numerica resut for one horizon representing a 24- hour day. he optima battery eve b increases to saturation, but because demand decreases faster than that required for Assumption A1, b discharges and recharges twice more before reaching zero at the fina time. he optima generation g remains inear (because h(b is affine and γ is timeinvariant when the battery charges and discharges, and foows demand when the battery is saturated, but does so severa times over the entire contro horizon. C. Proof idea In this subsection, we sketch the proof idea by describing some finer properties of the optima soution. he detaied proofs are omitted due to space constraint. he first emma states that, under an optima poicy, the battery is never exhausted unti possiby the ast period. his is a consequence of the convexity of the cost function c(g,t in g. Lemma 1: Suppose A0 and A1 hod. Under an optima poicy, b (t > 0 for t = 1,..., 1. he next resut describes the behavior in the third phase of heorem 1. Once an optima generation g (t drops beow the demand d(t, it wi remain stricty beow the (time-varying demand for subsequent periods. he battery therefore drains. Additionay, the optima margina cost g (t decreases at a rate equa to the margina battery cost h (b (t at time t. Fig. 3. Numerica resuts for a singe generator singe oad when assumption A1 does not hod. Optima generation g is inear when the battery charges and discharges, and matches demand when the battery is saturated. Lemma 2: Suppose A0 and A1 hod. If g (t 1 < d(t 1 for some t 1 {1,..., 1} under an optima poicy, then 1 0 < g (t < d(t, for t = t 1,...,. Moreover, for t = t 1,..., 1, γ(t + 1g (t + 1 = g (t + dh ] + db (b (t. 2 b (t < b (t 1 for t = t 1,...,. Moreover, for t = t 1,..., 1, b (t b (t + 1 > γ(t + 1 (b (t 1 b (t. he next resut describes the behavior in the first phase of the second case of heorem 1. If the optima generation exceeds the demand at a certain time, then it must have stayed stricty above the demand at a previous times. herefore the battery must have been in the charging mode throughout this period. Lemma 3: Suppose A0 and A1 hod. If b (t 0 < B and g (t > d(t for some t 0 {1,..., 1} under an optima poicy, then the foowing hod. 1 g (t > d(t for t = 1,...,t Moreover, for t = 1,...,t 0, γ(t + 1g (t + 1 = g (t + dh db (b (t. 2 b (t > b (t 1 for a t = 1,...,t Moreover, for t = 2,...,t 0 + 1, b (t b γ(t 1 (t 1 < (b (t 1 b (t 2. Lemmas 2 and 3 impy that the optima generation g (t can cross d(t at most once, from above. Moreover, the optima battery eve is unimoda, i.e., it can ony increase initiay, unti possiby reaching B. If the battery becomes 1055

6 saturated, it wi remain saturated unti the optima generation drops beow the demand, and can then ony decrease for the remaining times. Case (1 in heorem 1 describes the case where b is never saturated and Case (2 in heorem 1describes the case where b reaches saturation in phase two. IV. NEWORK CASE We now consider a genera network with mutipe generators and mutipe oads (i.e., the probem in (8-(10. o simpify, we eiminate the variabes q i (t by combining (2, (3, and (4 into g i (t + r i (t = Y i j (θ i (t θ j (t, i G, j N d i (t = Y i j (θ i (t θ j (t, i D. j N Since the probem is convex, the KK conditions are both necessary and sufficient. hey impy the foowing characterization of the optima generation where g (t = Γ 1 (th (t + B (t] + (22 H (t = diag B (t = diag ( dh i db (b i (τ ( (b i (τ b i (τ and Γ(t =diag(γ i (t. Here, b i (t is the optima battery eve at generator i, b i (t and b i (t are the Lagrange mutipiers associated with b i (t = 0 and b i (t = B i at generator i. he optima generation given in (22 is a direct extension of (19 and (21 from the SGSL case to the network case. he effect of network is iustrated most prominenty in the condition g (t + r ] (t = Yθ (t, (23 d(t where the network admittance matrix Y is given by { k N Y Y i j = ik, i = j Y i j, i j. We expect the macroscopic structure of the optima poicy to generaize from SGSL to the network case. For exampe, from (22, since the margina storage cost-to-go H (t decreases over time as ong as h i (b i < 0, the optima generation g (t tends to decrease over time as in the SGSL case. Battery reserves are used to meet demands toward the end of the horizon, reying more on power generation initiay. However, unike in the SGSL case where the generation and the demand are directy reated, g (t + r (t = d(t, in the network case, they are indirecty reated through the admittance matrix and the votage anges as shown in (23. his compicates the proof and the properties of optima poicy in the network case are under current study. Here, we present some numerica exampes. Exampe 2: Symmetric network with two generators. he network consists of two generators and 20 demand Fig. 4. Numerica resuts for the case with 2 generator nodes and 20 demand nodes. he configuration is symmetric, i.e. both generators are connected to every demand node, with equa admittance Y i j for every i G and j D. nodes. Every generator is directy connected to each of the demand nodes, but there is no direct connection between the two generators nor among the demand nodes. he admittance on every ink is Y i j = 1. Link capacities q i j = 1000 GJ are the same for a i, j. = 24 hours, B i = 16 GJ and b i (0 = 8 GJ. he oad profie is d i (t = 3sin(πt/ + 1 GJ. he storage cost is h i (b i = α(b i b i with α = 1; γ i (t = 1 in the generation cost function c i. he resuts are shown in Fig. 4. he optima generation decreases when the battery initiay charges and then tracks the net power output q i (t as the battery remains saturated. he optima generation then becomes stricty ess than the net power output and the battery discharges unti the battery drains competey and the generation again tracks net power output. Exampe 3: Cost savings. In this exampe, we consider both a SGSL case and a network case and iustrate how the time-varying nature of γ i (t in the generation cost can affect the cost saving. he network that is used is simiar to that in Exampe 2 except that the ink capacities are not symmetric: some of the ink capacities between generator 1 and the demand nodes are smaer. he behavior is shown in Fig. 5. Interestingy, the optima battery energy b (t of generator 2 is not unimoda, i.e., it charges and discharges mutipe times. Without battery, the tota generation must equa to tota demand at each time. In particuar, for SGSL, g (t = d(t for a t and the tota cost is t c(d(t,t without battery. With battery, the generation cost can be reduced through optimization over time. abe I compares the optima cost in (8 as a percentage of tota cost without battery, for both singe-generator-singe-oad and muti-generator-mutioad (MGML cases and for time-invariant and time-varying 1056

7 both the demand profie and the power generation are deterministic in our current formuation, which is unreaistic. We wi extend the current deterministic mode to incude randomness in demand and generation. Finay, we pan to incorporate a more reaistic battery mode that captures the dynamics of capacitance, as we as constraining the rate at which the battery can be charged or discharged. VI. ACKNOWLEDGMENS his work is partiay supported by Southern Caifornia Edison (SCE, the Boeing Corporation, and the Nationa Science Foundation. he authors wish to thank Christopher Cark, Jeff Gooding, and Percy Harason of SCE for hepfu discussions. Fig. 5. Numerica resuts for the case with 2 generator nodes and 20 demand nodes with unequa ink capacities. Demands vary sinusoiday. ABLE I OPIMAL COSS WIH BAERIES AS % OF OAL COS WIHOU BAERIES SGSL MGML time-invariant γ i 98% 97% time-varying γ i 83% 85% γ i. In the case with time-invariant γ i, the cost saving is sma. With time-varying γ i, the cost savings in both the SGSL and the network cases are significanty higher. his confirms the intuition that battery is more vauabe in the presence of fuctuations. he savings are greater in cases where the battery charges and discharges severa times over the entire contro period, which is a case that vioates Assumption A1. hen with arger or more frequent fuctuations, an optima poicy has more opportunities to charge the batteries when generation costs are ow and discharge when they are high. V. CONCLUSION In this paper, we have formuated a simpe optima power fow mode with storage. We have characterized the optima soution for the case with a singe generator and a singe oad when the generation cost c(g,t is quadratic in g and the battery cost h(b is stricty decreasing. In this case, under the assumption (A1 that the demand does not decrease too rapidy, the optima generation schedue wi cross the demand profie at most once, from above. he optima battery eve is unimoda where the battery is charged initiay, possiby reaching saturation, and then discharges ti the end of the contro horizon. We have presented some numerica exampes iustrating the behavior in the network case and when assumption A1 is not satisfied. We pan to extend the anaysis here in severa ways. First, we wi fuy characterize the optima soution in the network case by extending our resuts for the SGSL case. Second, REFERENCES 1] J. Carpentier, Contribution to the economic dispatch probem, Bu. Soc. Franc. Eectr., vo. 3, no. 8, pp , ] B. Chowdhury and S. Rahman, A review of recent advances in economic dispatch, IEEE ransactions on Power Systems, vo. 5, pp , ] M. Huneaut and F. D. Gaiana, A survey of the optima power fow iterature, IEEE ransactions on Power Systems, vo. 6, no. 2, pp , May ] J. A. Momoh, M. E. E-Hawary, and R. Adapa, A review of seected optima power fow iterature to 1993, part i: Noninear and quadratic programming approaches, IEEE ransactions on Power Systems, vo. 14, no. 1, pp , ], A review of seected optima power fow iterature to 1993, part ii: Newton, inear programming and interior point methods, IEEE ransactions on Power Systems, vo. 14, no. 1, pp , ] K. S. Pandya and S. K. Joshi, A survey of optima power fow methods, Journa of heoretica and Appied Information echnoogy, vo. 4, no. 5, pp , ] J. A. Momoh, Eectric power system appications of optimization. Marke Dekker, ] 2007 Summer Loads and Resources Operations Assessment, Caifornia ISO, CA, ech. Rep., ] Caifornias eectricity generation and transmission interconnection needs under aternative scenarios, Consutant Report, ] Southern Caifornia Edison Signs Largest Wind Energy Contract in U.S. Renewabe Industry History, Southern Caifornia Edison Press Reease, Onine]. Avaiabe: 11] C. Archer and M. Jacobson, Evauation of goba wind power, Journa of Geophysica Research, vo. 110, no. D12110, ]. Yau, L. Waker, H. Graham, and A. Gupta, Effects of battery storage devices on power system dispatch, IEEE ransactions on Power Apparatus and Systems, vo. PAS-100, no. 1, pp , ] N. Aguaci and A. J. Conejo, Mutiperiod optima power fow using benders decomposition, IEEE ransactions on Power Systems, vo. 15, no. 1, pp , ] E. Sortomme and M. A. E-Sharkawi, Optima power fow for a system of microgrids with controabe oads and battery storage, in Power Systems Conference and Exposition, ] M. Geid and G. Andersson, A modeing and optimization approach for mutipe energy carrier power fow, in In Proc. of IEEE PES Powerech, ] F. Wu, P. Varaiya, P. Spier, and S. Oren, Fok theorems on transmission access: Proofs and counterexampes, Journa of Reguatory Economics, vo. 10, no. 1, pp. 5 23, ] A. R. Bergen and V. Vitta, Power Systems Anaysis, 2nd ed. Prentice Ha,