Optimization Decomposition of Resistive PowerNetworks with Energy Storage

Xin Lou, Student Member, IEEEand Chee Wei Tan,Senior Member, IEEE

Abstract—A fundamental challenge of a smart grid is: to whatextent can moving energy through space and time be optimizedto benefit the power network with large-scale energy storageintegration? With energy storage, there is a possibility togeneratemore energy when the demand is low and store it for lateruse. In this paper, we study a dynamic optimal power flowproblem with energy storage dynamics in purely resistive powernetworks. By exploiting a recently-discovered zero duality gapproperty in the optimal power flow (OPF) problem, we applyoptimization decomposition techniques to decouple the couplingenergy storage constraints and obtain the global optimal solutionusing distributed message passing algorithms. The decompositionmethods offer new interesting insights on the equilibrium loadprofile smoothing feature over space and time through therelationship between the optimal dual solution in the OPF andthe energy storage dynamics. We evaluate the performance ofthe distributed algorithms in several IEEE benchmark systemsand show that they converge fast to the global optimal solutionby numerical simulations.

Index Terms— Optimal power flow, energy storage, decomposi-tion method, distributed optimization, smart grid, message passingalgorithm.

I. INTRODUCTION

A key challenge in a smart grid design is the integrationof power flow control in the power grid network togetherwith the distributed renewable energy sources and energystorage at the endpoints of the network. Renewable energyis intermittent and difficult to predict and harness, and this inturn makes power flow in the power grid network harder tocontrol and optimize. An interconnected system of hundredsofmillions of distributed energy resources introduces rapid, largeand random fluctuations in power supply and demand. Suchdeployments create stronger demands and highly variable (intime and space) operating conditions than those experienced incurrent networks. The integration of a two-way communicationinfrastructure to connect the power grid network and differenttypes of demands and supply generation (see Figure 1), whilepromising in a smart grid, motivates the important question:to what extent can moving energy through space and timebe optimized to benefit the power network with large-scalestorage integration?

There are clearly benefits in a joint optimization of energystorage and the power flow [1], [2]. Energy production and

Manuscript received Oct 16, 2013; revised Apr 14, 2014. The work in thispaper was partially supported by grants from the Research Grants Council ofHong Kong Project No. RGC CityU 122013.

X. Lou and C. W. Tan are with the College of Science and Engineer-ing, City University of Hong Kong, Tat Chee Ave., Hong Kong (email:louxin86@gmail.com and cheewtan@cityu.edu.hk).

Copyright (c) 2012 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

Fig. 1. A general outline of the interconnection in a power grid network witha two-way communication infrastructure for message passing to coordinatethe distributed optimization of supply, energy storage anddifferent kinds ofworkloads with possibly time-varying demand, e.g., plug-in electric vehicles.

consumption can dynamically lead to new equilibrium oper-ating points over time. Energy storage at demand buses canabsorb the transients while the power loads are rebalanced.In particular, when the power price is low (due to a lowerdemand at off-peak hours), energy can be stored in batteries.On the other hand, when the power price is high (due to ahigher demand at peak hours), users can first draw energylocally from the batteries before consuming the energy directlyfrom the power grid network. Energy storage can also help inload smoothing [2]. For example, the energy storage batteryin plug-in electric vehicles1 plays an interestingly dual role:they represent a new type of demand load to be satisfied andalso as new energy storage resources (when power flows fromthe vehicle to grid) for load smoothing [3], [4]. With energystorage, how should power flow be optimized to maximizeefficiency and minimize power consumption cost?

The optimal power flow (OPF) problem is a classicalnonlinear optimization problem, whose solution optimizesanetwork-wide objective, e.g., generation cost or transmissionloss, subject to physical constraints on the power flow, thedemand specification and the network connectivity constraints[5], [6]. It is well-known that the OPF is generally hard tosolve due to its nonconvexity. In the vast literature on OPF,approximation methodologies, e.g., the direct current (DC)OPF linearization that assumes a constant voltage and smallphase angle, have been widely used [7]–[11]. However, recent

1A trend is expected in the growing use of plug-in electric vehicles thatuse chemical energy stored in rechargeable battery packs and installation ofelectric vehicle charging infrastructures, and this is oneof the driving forcesbehind a smart power grid network.

developments2 have shown that semidefinite programming(SDP) can be used to convexify the OPF and, under certainassumptions, solving this SDP convex relaxation is even exact[14], [15].3 For example, the relaxation is exact for distributionnetworks that have tree-like network topology [16], [17]. Thisis important because SDP is a convex optimization problemthat can be efficiently solved [18].

These developments have attracted a surge of interests indeveloping computational algorithms to solve the OPF exactlyor to obtain relaxations for the OPF and its special cases [19]–[23]. However, most efforts have been devoted to centralizedcomputation issues (which of course is still useful in thecontext of a tightly-regulated system) without focusing onenergy storage. Indeed, the dynamic OPF with energy storageis a more complex time-dependent power flow optimizationproblem (also known as the multi-period OPF in the powersystem literature [24], [25]). The energy storage dynamicshave unique time-dependent characteristics that couple withpower flow (over time) and the network topology (over space).In addition, the demand profile (over time) and the energystorage dynamics introduce a new degree of freedom tooptimize for load smoothing over time that is beneficial topower systems operation.

The authors in [26] first studied such a dynamic OPFwith energy storage using the DC OPF linearization andfinite horizon optimal control for a simple network topology(a single generator with a single battery connected to asingle load). The authors demonstrated a unique feature inthe optimal policy in which generation first exceeds demandin order to charge up the battery and subsequently demandexceeds generation with additional energy supplement comingfrom the battery till the battery is fully depleted by the endof the horizon. The authors in [25] applied the SDP convexrelaxation in [15] to a dynamic OPF but did not considerenergy storage. Gayme et al [20] solved this dynamic OPFproblem with energy storage constraints using the SDP convexrelaxation technique in a centralized manner. This howeverrequires a high computational complexity due to solvingan SDP of higher dimension (whose complexity increaseswith the number of periods in the horizon). The authors in[3] proposed valley-filling algorithms (for load smoothing)to tackle the problem in [20] for electric vehicle charging.The authors in [21] proposed message passing algorithms fordistributed optimization that can be combined with recedinghorizon control.

In this paper, we study a dynamic OPF problem for apurely resistive power network (i.e., no phase angle, reac-tive power variables and reactance parameters) with energystorage dynamics. This allows energy storage dynamics to beoptimized jointly with the demand and supply across multipletime periods. Our focus is to study optimization decompositiontechniques to enable distributed computation. Decomposition

2We refer the readers to [12], [13] for an overview on this developmentthat clarifies the relationships between various models of OPF and its convexrelaxation.

3The convex relaxation is exact if the optimal solution of a convexrelaxation is feasible for the original OPF problem and hence globally optimalas well. The relaxation gap (as well as the Lagrange duality gap in the OPFconsidered in this paper) is zero in this case.

methods (predominantly Lagrange dual decomposition) havebeen previously used in the literature to solve the OPF [7], [8],[24], [27], [28], but these prior work use approaches (usingthe DC OPF linearization to approximate the OPF) differentfrom that in this paper. Decomposition enables fundamentalunderstanding of architectural possibilities, especially in thedistributed coordination of functional modules in a largenetwork [29], [30]. We also study the interplay betweenoptimal generation schedule and the time-varying load profile,i.e., how to generate more energy when the demand is lowand store it for later use by charging and discharging theenergy storage. A resistive power network OPF with energystorage can be practically useful in HVDC (high-voltagedirect current) network or microgrid clusters integrated withrenewable energy sources, e.g., solar cells and photovoltaicgenerator, that produce real power flow.

Overall, the contributions of the paper are as follows:

1) We formulate a time-dependent dynamic OPF problemfor resistive power networks with energy storage dynam-ics. We leverage a zero duality gap property in the staticOPF to solve this dynamic OPF problem.

2) We propose various decomposition methodologies (anindirect Lagrange dual-dual and a primal-dual hierar-chical decomposition) that lead to different kinds ofefficient and distributed algorithms to solve the dynamicOPF.

3) We give a physical interpretation to the connection be-tween the dual solution corresponding to the power flowand energy storage as time-varying marginal price andstorage price respectively that explains the phenomenonof load profile smoothing over time.

4) Through extensive numerical experiments, we show thatthese algorithms converge fast to the global optimalsolution of the dynamic OPF. We also demonstrate theequilibrium load profile smoothing feature of the dy-namic OPF with energy storage under various problemsettings.

The rest of this paper is organized as follows. In Section II,we introduce the system model and formulate the dynamicOPF with energy storage problem formulation. In Section III,we review some preliminary results on the static OPF prob-lem and its convex relaxation. In Section IV, we proposedistributed algorithms to solve the dynamic OPF problem byan indirect dual-dual decomposition technique. In SectionV,we propose an alternative primal-dual decomposition. In Sec-tion VI, we explain the connection between the Lagrangedual variables and the energy storage dynamics. Numericalevaluations are presented in Section VII. In Section VIII, weconclude the paper.

We refer the readers to Figure 2 for an overview of theoverall relationship between the problem formulation, theoptimization decomposition, the proposed algorithms and theinterpretation of the dynamic OPF with energy storage.

II. OPTIMAL POWER FLOW WITH ENERGY STORAGE

Let us consider a purely resistive power network with busesN = f1; 2; : : : ; Ng andE � N � N transmission lines. Let

Optimal Power Flowwith Energy Storage

(8)

IndirectDual-Dual

Decomposition

AlternativePrimal-Dual

Decomposition

Energy StorageCoupling over

Time Interpretation

Methodology

Insight

Sec.II

Sec.IV Sec.V

Sec.VI

Algorithms 1-2 Algorithm 3

Lemmas 1-2

OptimizationDecompositionStructures

Algorithm 4

Fig. 2. An overview of the connection on the different optimization de-composition, algorithms and the interpretation of the dynamic OPF withenergy storage. The upper half of the line corresponds to thetwo optimizationdecomposition techniques, while the lower half corresponds to the insight onhow the dual solution of the two decomposition can be interpreted.

us denote the set of generation buses and demand buses byG andD respectively. We assume that the busi 2 D has anenergy storage battery attached to it. Let us denote the one-hop neighbors (connected by transmission lines) of busi bythe seti (jij � 1). The transmission lines have admittanceand this is modeled by the admittance matrixY. In particular,Y is symmetric, i.e.,Yij = Yji 2 R++ , if (i; j) 2 E andYij = Yji = 0 otherwise [14], [15].

We consider a multi-period power flow system indexed byt = 1; : : : ; T , whereT is the terminating time period. We useV(t) and I(t) to denote the column vectors for the voltages(Vi(t))i2N and the current(Ii(t))i2N for t = 1; : : : ; T ,respectively. By Kirchhoff’s Current Law and Ohm’s Law, wehaveI(t) = YV(t) for t = 1; : : : ; T , i.e.,0BB� I1(t)I2(t)

...IN (t)1CCA = 0BBBBB� Pj21 Y1j � � � �Y1N�Y21 � � � �Y2N...

. . ....�YN1 � � � Pj2N YNj1CCCCCA0BB� V1(t)V2(t)

...VN(t)1CCA : (1)

Using I(t) in (1), the nodal power and voltage constraintsare given respectively as follows:V(t)>YiV(t) � pi(t)� ri(t); t = 1; : : : ; T; (2)V � V(t) � V; t = 1; : : : ; T; (3)

where (�)> denotes the transpose,Yi = 12 (EiY + YEi),Ei = eie>i 2 Rn�n with ei being the standard basis vector inRn (i.e.,Ei is a matrix with a single one in the(i; i)th entryand zero everywhere else),ri(t) is the amount of power thatthe energy storage of busi charges (positive) or discharges(negative) at timet and fV;Vg are given voltage boundvalues. For a generation busi 2 G, �pi(t) is strictly positiveand models the generator capacity. On the other hand, for ademand busi 2 D, �pi(t) is strictly negative, and the magnitudej�pij represents the minimum power that has to be provided tosatisfy the load at busi. Similarly to [22], we make the load

over-satisfaction assumption, i.e., a demand bus can absorbthe powerj�pij to satisfy the basic demand requirement and tocharge the attached energy storage battery.

In addition to the basic OPF constraints, each battery at busi 2 N has to satisfy [20], [26]:4bi(t+ 1) = �ibi(t) + ri(t); t = 1; : : : ; T; (4)

where storage leakage in a general battery can be modeled byappropriately choosing0 < �i � 1 for all i 2 N in (4). Theinitial condition of the battery ati 2 N is given by:bi(1) = B0i ; (5)

whereB0i is a positive constant. At eachi 2 N , bi(t) andri(t) are constrained respectively by:0 � bi(t) � Bi; t = 1; : : : ; T + 1; (6)ri � ri(t) � �ri; t = 1; : : : ; T; (7)

whereBi is the battery capacity at busi. Our objective is tominimize the total transmission loss in the network:TXt=1V(t)>YV(t);and the energy storage cost:hi(bi(t); ri(t)) = h1i(bi(t)) + h2i(ri(t)); 8i 2 N ; 8t;whose first componenth1i(bi(t)) = �i(Bi � bi(t)) for all i 2 N ; t = 1; : : : ; T + 1;with �i � 0 for all i 2 N captures the instantaneous amount ofthe energy stored in the battery, i.e., the penalty is proportionalto the deviation from the storage capacity [26]. The secondcomponenth2i(ri(t)) captures the battery dynamics, i.e., thecharging/discharging amount of power in the battery. In addi-tion, we assume thath2i(ri(t)) is convex inri(t) for all i 2 Nandt = 1; : : : ; T [31], i.e., the battery cost increases with thecharge/discharge rate. In this paper, we leth2i(ri(t)) = ~�ir2i (t) for all i 2 N ; t = 1; : : : ; T;with ~�i � 0 for all i 2 N .

Consider the following time-dependent dynamic OPF prob-lem:

minimizeTXt=1V(t)>YV(t) +Xi2N T+1Xt=1 h1i(bi(t)) + TXt=1 h2i(ri(t))

subject toV(t)>YiV(t) � pi(t)� ri(t); 8i 2 N ; t = 1; : : : ; T;V � V(t) � V; t = 1; : : : ; T;bi(t+ 1) = �ibi(t) + ri(t); i 2 N ; t = 1; : : : ; T;b(1) = B0;0 � b(t) � B; t = 1; : : : ; T + 1;r � r(t) � �r; t = 1; : : : ; T;variables:V(t);b(t); r(t);

(8)

4We let the power quantities be in the unit of energy per unit time so thatenergy and power are used interchangeably [26].

whereb(t) andr(t) denote the column vectors for(bi(t))i2Nand (ri(t))i2N , respectively. The problem formulation in (8)is a nonconvex quadratic constrained quadratic programming(QCQP) problem, which is generally hard to solve. In thefollowing, we first review some results in [14], [15], [22] fora static OPF, and then leverage optimization decompositiontechniques in Sections IV and V to solve (8).

III. PRELIMINARIES ON CONVEX RELAXATION OF STATIC

OPF

Consider a fixed time period in (8) and suppose that thereis no battery, then (8) reduces to (omitting the time indext):

minimizeV>YVsubject toV>YiV � pi; 8i 2 N ;V � V � V;variables:V: (9)

Note that (9) is a static OPF problem that minimizes the totalpower losses in a resistive power network subject to voltageand power constraints. Even though (9) is nonconvex, it hasrecently been shown in [15] that a convex relaxation usingSDP can yield the global optimal solution of (9) under somemild conditions [15]. This remarkable result is a consequenceof a special type of quadratic programming problem withnonnegative problem parameters and solution [14], [32]. Inparticular, the SDP convex relaxation of (9) is given by:

minimize trace(YW)subject to trace(YiW) � pi; 8i 2 N ;V i2 � trace(EiW) � V i2; 8i 2 N ;W � 0; Wij � 0; 8i; j 2 N ;variables:W; (10)

whereW � 0 means thatW is a positive semidefinite matrixand trace(�) denotes the matrix trace operator. In fact, (10) isequivalent to (9) if the constraint rank(W) = 1 is includedin (10) sinceW = VV>. However, by observing thatY;Y1; : : : ;YN are matrices with nonpositive off-diagonalelements, the authors in [14], [15] show that (9) can be solvedexactly by (10). Furthermore, it has been shown that this tightrelaxation result also implies that the Lagrange duality gap of(9) is zero, and (9) can also be solved exactly by a secondorder cone programming relaxation [12], [13], [22]. A moregeneral result that applies to the AC (alternating current)OPFproblem can be found in [12], [13], [15].

In our recent work [22], we leverage this zero duality gapproperty to design low-complexity distributed algorithmstosolve (9) directly instead of solving the SDP convex relaxationin (10) with a centralized SDP solver (e.g., interior-pointmethod). In fact, the optimal solution of (9) can be unique(for different network topologies, e.g., line, radial and meshnetworks). In addition, this uniqueness characterizationof (9)has implications on how efficient distributed algorithms canbe designed to solve (9). The key idea is to exploit thezero duality gap result in [14], [15] and the Poincare-HopfIndex Theorem in [33], [34] to show uniqueness, and thendesign message passing algorithms to solve (9) using theLagrange dual decomposition. In another recent work [35], we

Subproblem:Problem (11)

First-levelDecomposition

Second-levelDecomposition· · ·

Second-level Master Problem:Problem (20)

Master Problem:Problem (12)

Subproblem:Problem (17)

at bus 1

Subproblem:Problem (17)at bus N

Dual Prices λi(t)

Dual Prices µi(t)

Fig. 3. Indirect dual-dual optimization decomposition to solve the OPF withenergy storage problem in (8). The partial dual decomposition is applied atthe two levels.

have studied a dynamic OPF problem similar to (8) but thework has some limitations: A simplified inequality constraintbi(t + 1) � bi(t) + ri(t) for t = 1; : : : ; T has been usedinstead of the equality constraint given in (4) that can modelthe storage dynamics more accurately, and only the batteryamount (i.e.,h1i(t)) is used to model the battery cost.

IV. I NDIRECT DUAL-DUAL DECOMPOSITION AND

ALGORITHMS

In this section, we study new optimization decompositionmethodologies that can offer novel architectural transformationinsights, and enable scalable and distributed algorithm designto solve (8). We study an indirect decomposition techniqueto decompose the dynamic OPF in (8) into simpler subprob-lems to unravel the coupling over space and time. Figure 3illustrates the overview of this decomposition. In the firstlevel, Lagrange dual decomposition is first applied. The dualvariable corresponding to (2) plays the role of power flowprice to control the decomposed subproblems. In the secondlevel, Lagrange dual decomposition is again applied. The dualvariable corresponding to (4) plays the role of energy storageprice to control the second-level subproblems.

In the following, we explain each of these two levels indetails. Let us apply a partial dual decomposition to de-compose (8) over time. Then, for the subproblems, they areseparable fort = 1; : : : ; T and each of them corresponds toone static OPF problem at time periodt:

minimizeV(t)>YV(t) +Xi2N �i(t)(V(t)>YiV(t))subject toV � V(t) � V;variables:V(t); (11)

where�i(t) for all i andt are the nonnegative Lagrange dualvariables corresponding to the constraintV(t)>YiV(t) �pi(t) � ri(t) in (8). Note that�i(t) can be interpreted as thepower price (by the shadow price interpretation of the optimaldual solution, cf. Chapter 5 in [18]). Due to the zero dualitygap, each decomposed subproblem (11) for a fixedt can besolved with the following message passing algorithm (for afixed �i(t) for all i andt, and we use to index the iterationin the algorithm) in [22].

Algorithm 1: Distributed computation of voltages

Compute voltageV(t):V `+1i (t) = max8<:V i;min8<:V i;Xj2i Aij(t)Vj (t)9=;9=;8i 2 N , whereAij(t) = 2Yij + �i(t)Yij + �j(t)Yij2(1 + �i(t)) Pj2i Yij ; 8(i; j) 2 E :Remark 1:Algorithm 1 converges to the unique optimal

solution of (11) for eacht = 1; : : : ; T [22]. The spectralproperty of the nonnegative matrixA(t) in Algorithm 1 canbe used to characterize its convergence performance [22]. Inaddition, the dependency on the network topology for messagepassing is captured by the matrixA(t).

The optimal power prices are obtained by solving thefollowing primary master dual problem:

maximizeTXt=1 g(�(t)) + �(t)>(r(t) � �p(t))

subject to�(t) � 0; t = 1; : : : ; T;variables:�(t); (12)

whereg(�(t))=V�(t)>YV�(t)+ Pi2N �i(t)(V�(t)>YiV�(t)):The dual function is differentiable at eacht and can be solvediteratively by the following gradient update:�k+1i (t)=[�ki (t)+�(V�(t)>YiV�(t)�pi(t)+ri(t))℄+; 8i 2 N ;

(13)where � is an appropriate stepsize, and[�℄+ denotes theprojection onto the nonnegative orthant. This update will inturn drive Algorithm 1.

Next, let us consider the decomposed problem correspond-ing to the energy storage:

minimizeT+1Xt=1 �>(B� b(t)) + TXt=1 �(t)>r(t) +Xi2N ~�ir2i (t)

subject tobi(t+ 1) = �ibi(t) + ri(t); i 2 N ; t = 1; : : : ; T;b(1) = B0;0 � b(t) � B; t = 1; : : : ; T + 1;r � r(t) � �r; t = 1; : : : ; T;variables:b(t); r(t):

(14)

Since bi(t + 1) = �ibi(t) + ri(t) for t = 1; : : : ; T; andbi(1) = B0i , then for eachi 2 N we have:bi(t+ 1) = �tibi(1) + tX�=1 �t��i ri(�); t = 1; : : : ; T: (15)

We thus rewrite (14) as:

minimizeT+1Xt=1 �>(B� �tb(1)� tX�=1�t��r(�))+ TPt=1�(t)>r(t) + Pi2N ~�ir2i (t)

subject tobi(t+ 1) = �tibi(1) +Pt�=1 �t��i ri(�); 8i 2 N ;r � r(t) � �r;variables:r(t);b(t); t = 1; : : : ; T:

(16)Now, for a fixedb(t), if we apply the partial dual decom-

position to (16), we get a second-level subproblem at eachi 2 N :

minimizeT+1Xt=1 �i(Bi � �tibi(1)� tX�=1 �t��i ri(�))+ TPt=1�i(t)ri(t)� (Pt�=1 �t��i ri(�)�i(t))+�i(t)(bi(t+ 1)� �tibi(1)) + ~�ir2i (t)

subject tori � ri(t) � �ri; i 2 N ; t = 1; : : : ; T;variables:ri(t);

(17)where �i(t) is the Lagrange dual variable correspondingto (4). Then, we have the following update on the charg-ing/discharging amount of power in the energy storage fort = 1; : : : ; T and i 2 N :rk+1i (t) = [rki (t)� �(��i (t)� TX�=t ���ti (�i + ��i (�))+2~�irki (t))℄riri ; (18)bki (t+ 1) = [�tibi(1) + tX�=1 �t��i rki (�)℄Bi0 ; (19)

where � is an appropriate stepsize, and[x℄ab denotes theprojectionmaxfb;minfa; xgg whereb < a.

Furthermore, the second-level master dual problem is givenby

maximizeTXt=1Xi2N gi(�(t)) + ~�ir�2i (t) + �(t)>r�(t)+�>(B� b�(T + 1))

variables:�(t); (20)

wheregi(�(t)) = �i(Bi� b�i (t))+�i(t)(b�i (t+1)��ib�i (t)�r�i (t))at each subproblem of (17). In particular, a subgradient updateof the dual variable�i(t) for t = 1; : : : ; T to solve (20) isgiven by�k+1i (t) = �ki (t) + (b�i (t+ 1)� �ib�i (t)� r�i (t)); 8i 2 N ;

(21)where is an appropriate stepsize. We summarize the varioussolution update of the above dual-dual decomposition in thefollowing (usingk to index algorithm iteration).

Algorithm 2:Dual-Dual Decomposition

1) Set the stepsizes�; �; 2 (0; 1).2) Compute the charging/discharging amount of power in

storage:rk+1i (t) = [rki (t)��(�ki (t)� TX�=t ���ti (�i+�ki (�))+2~�irki (t))℄riri8i 2 N and t = 1; : : : ; T .

3) Compute the battery amount:bki (t+ 1) = [�tibi(1) + tX�=1 �t��i rki (�)℄Bi08i 2 N and t = 1; : : : ; T .

4) Compute the storage dual variable:�k+1i (t) = �ki (t) + (bki (t+ 1)� �ibki (t)� rki (t))8i 2 N and t = 1; : : : ; T .

5) Run Algorithm 1 to compute the voltage and setAlgorithm 1 output asVk(t) for t = 1; : : : ; T .

6) Compute the power price:�k+1i (t) = [�ki (t)+�(Vk(t)>YiVk(t)�pi(t)+rki (t))℄+8i 2 N and t = 1; : : : ; T .

Update�, � and until convergence.

Remark 2: Steps 5 and 6 are equivalent to Algorithm2 in [22] for a fixed ri(t) at eacht, and its convergenceproof is given in [22]. Moreover, the subgradient methodof the second-level (i.e., Steps 2 and 4) can converge to aclosed neighborhood of the optimal solution by appropriatelychoosing the stepsize (using a diminishing stepsize rule orsufficiently small constant stepsizes) [30], [36].

Remark 3:Algorithm 2 is distributed, because each bus onlycommunicates with its local one-hop neighbors, i.e., at Steps 5and 6, each bus exchanges nodal voltage and nodal power priceinformation with its one-hop neighbors by message passing.An example is illustrated in Figure 4 for a 5-bus system ([6], Chapter 6, pp.327). Each nodei 2 N randomly choosesa feasibleV 0i (t) for t = 1; : : : ; T and broadcasts to its one-hop neighbors. Once a nodei 2 N receives allVj(t) wherej 2 i, it calculates�i(t). Next, the updated�i(t) is thenpassed to its neighbors. After collecting all�j(t) (j 2 i),Vi(t) is calculated by Algorithm 1 at busi. The other variablesri(t); bi(t) and�i(t) for j 2 i andt = 1; : : : ; T only requireinformation locally available at busi. This message passing isrepeated until convergence.

V. A LTERNATIVE PRIMAL -DUAL DECOMPOSITION AND

ALGORITHMS

We now present an alternative decomposition method (aprimal-dual decomposition) that leads to a distributed algo-rithm (without using Algorithm 1) to solve (8). The difference

G G

Bus 1

Bus 2 Bus 3

Bus 4 Bus 5

V ℓ

1(t), λk

1(t){ }

rk1(t), bk

1(t), µk

1(t){ }

rk2(t), bk

2(t), µk

2(t){ }

rk4(t), bk

4(t), µk

4(t){ } rk

5(t), bk

5(t), µk

5(t){ }

rk3(t), bk

3(t), µk

3(t){ }

V ℓ

2(t), λk

2(t){ }

V ℓ

3(t), λk

3(t){ }

V ℓ

5(t), λk

5(t){ }V ℓ

4(t), λk

4(t){ }

Fig. 4. Message passing in a 5-bus system ([6], Chapter 6, pp.327).

Subproblem:Problem (22)

First-levelDecomposition

Second-levelDecomposition· · ·

Second-level Master Problem:Problem (24)

Master Problem:Problem (25)

Subproblem:Problem (23)

at bus 1

Subproblem:Problem (23)at bus N

Primal Resources Vi(t)

Dual Prices {λi(t), µi(t)}

Fig. 5. Alternative primal-dual decomposition to solve theOPF with energystorage problem in (8). The primal and dual decomposition are applied at thefirst level and the second level respectively.

between these two decompositions is in how power pricesdrive the voltage computation. Consider a primal decompo-sition to (8) by fixingV(t) for t = 1; : : : ; T: Then, we obtainsubproblems of (8) given by:

minimizePi2N T+1Pt=1 h1i(bi(t)) + TPt=1h2i(ri(t))

subject toV(t)>YiV(t) � pi(t)� ri(t); 8i 2 N ; t = 1; : : : ; T;bi(t+ 1) = �ibi(t) + ri(t); i 2 N ; t = 1; : : : ; T;b(1) = B0;0 � b(t) � B; t = 1; : : : ; T + 1;r � r(t) � �r; t = 1; : : : ; T;variables:b(t); r(t):

(22)

We use Figure 5 to illustrate this alternative decompositionstructure from (22). In the first level, a primal decompositionis used. The primal variablevi(t) plays the role of resourcesto control the decomposed subproblems. In the second level,a partial Lagrange dual decomposition is applied to (22).The dual variables�i(t) and �i(t) play the role of powerprice and storage dual variable to control these second-levelsubproblems, respectively.

Now, by applying the partial dual decomposition to (22),

for eachi 2 N , we have the following subproblem:

minimizeT+1Xt=1 �i(Bi � bi(t)) + TXt=1 �i(t)ri(t) + ~�ir2i (t)+�i(t)(bi(t+ 1)� �ibi(t)� ri(t))

subject tobi(1) = B0i ;0 � bi(t) � Bi; t = 1; : : : ; T + 1;ri � ri(t) � �ri; t = 1; : : : ; T;variables:bi(t); ri(t):

(23)After using the relationbi(t + 1) = �ibi(t) + ri(t) fort = 1; : : : ; T; and fixingbi(t), we can rewrite the second-level

subproblem as (17) at eachi 2 N . Thus, we haveri(t) andbi(t) as given in (18) and (19) for alli 2 N andt = 1; : : : ; T .This second-level master dual problem is given by

maximizeTXt=1Xi2N gi(�(t);�(t)) + ~�ir�2i (t)+�>(B� b�(T + 1))

subject to�(t) � 0;variables:�(t);�(t); (24)

wheregi(�(t);�(t)) = �i(t)(V�(t)>YiV�(t)� �pi(t) + r�i (t))+�i(t)(b�i (t+ 1)� �ib�i (t)� r�i (t))+�i(Bi � b�i (t));in each subproblem of (17). The subgradient update of thedual variable�i(t) and�i(t) for t = 1; : : : ; T can be done by(13) and (21) respectively.

Going up one-level, the master primal problem can bewritten as

minimizef�(V(t))subject toV � V(t) � V;variables:V(t); (25)

wheref�(V(t)) is the optimal value of (8) for a givenV(t)for t = 1; : : : ; T: Recall that�(t) for t = 1; : : : ; T are theset of nodal power prices corresponding to (2). A subgradientmethod can solve (25) by iteratively updating the voltageVi(t):V `+1i (t) = �Vi (t)�Æ(2~��i (t) Xj2i YijVi (t)�Xj2i ~��ij(t)Vj (t))�V iV i ;where ~��i (t) = (1 + ��i (t));for all i 2 N and~��ij(t) = (2Yij + ��i (t)Yij + ��j (t)Yij);for all (i; j) 2 E andÆ is an appropriate stepsize.

We summarize the various solution update of the abovealternative decomposition in the following (usingk to indexalgorithm iteration).

Algorithm 3:Primal-Dual Decomposition

1) Set the stepsizesÆ; ; �; � 2 (0; 1).

2) Until convergence (of the index), compute the voltage:V `+1i (t) = [Vi (t)�Æ(2~�ki (t)Xj2i YijVi (t)�Xj2i ~�kij(t)Vj (t))℄V iV i ;where ~�ki (t) = (1 + �ki (t)); 8i 2 N and~�kij(t) = (2Yij + �ki (t)Yij + �kj (t)Yij); 8(i; j) 2 Eand t = 1; : : : ; T . Upon convergence, letVk(t) be thevoltage output.

3) Compute the storage dual variable:�k+1i (t) = �ki (t) + (bki (t+ 1)� �ibki (t)� rki (t))8i 2 N and t = 1; : : : ; T .

4) Compute the power price:�k+1i (t) = [�ki (t)+�(Vk(t)>YiVk(t)�pi(t)+rki (t))℄+:5) Compute the charging/discharging amount of power in

storage:rk+1i (t) = [rki (t)��(�ki (t)� TX�=t ���ti (�i+�ki (�))+2~�irki (t))℄riri8i 2 N and t = 1; : : : ; T .

6) Compute the battery amount:bki (t+ 1) = [�tibi(1) + tX�=1 �t��i rki (�)℄Bi08i 2 N and t = 1; : : : ; T .

UpdateÆ; ; � and� until convergence.

Remark 4:Unlike Algorithm 2, Algorithm 3 requires moresubgradient updates to converge to a closed neighborhood ofthe optimal solution by appropriately choosing the stepsizes(using a diminishing stepsize rule or sufficiently small constantstepsizes) [30], [36] and also synchronizing the updates ateach level of the decomposition [29]. The message passingin Algorithm 3 is similar to that illustrated in Figure 4.

VI. ENERGY STORAGE COUPLING OVER TIME

INTERPRETATION VIA LAGRANGE DUALITY

In this section, we discuss the physical interpretation ofthe optimization structures obtained through the two differentdecomposition methods in the previous two sections.

A. Physical interpretation of dual solution from indirect dual-dual decomposition

In the indirect dual-dual decomposition, for the subproblemcorresponding to the energy storage, i.e., (14) , the Lagrangian

at eachi 2 N can be expressed as:Li = T+1Xt=1 �i(Bi � bi(t)) + �i(t)(bi(t)�Bi)� �i(t)bi(t)+ TXt=1 �i(t)ri(t) + �i(t)(bi(t+ 1)� �ibi(t)� ri(t))+~�ir2i (t) + �i(t)(ri(t)� ri)� �i(t)(ri(t)� ri)+�i(1)(bi(1)�B0i ); (26)

where the dual variable�i(t) corresponds to (4),�i(t) and�i(t) respectively correspond to the lower and upper bounds of(6), �i(1) corresponds to (5), and�i(t) and�i(t) respectivelycorrespond to the discharge and charge bounds in (7). Inter-estingly, the difference between two successive dual variables�i�i(t)� �i(t� 1) plays a decoupling role over time:

Lemma 1:For a batteryi, if �i(t � 1) � �i�i(t) < �i,wheret = 2; : : : ; T or �i(T ) < �i, then it is saturated att orT + 1; If the inequality is reversed, then batteryi is drainedfor t = 2; : : : ; T or T +1. If the battery bound of busi in (8)is inactive fort = 1; : : : ; T , then�i(t) =PT�=t �T��i �i.

In addition, if ri(t) < 12~�i (�i(t) � �i(t)), where t =1; 2; : : : ; T , then the charging power amount upper bound is hitat t; If the inequality is reversed, then batteryi reaches its dis-charging power amount upper bound fort = 1; 2; : : : ; T . If thecharging/discharging power amount bounds of busi in (8) areinactive fort = 1; 2; : : : ; T , thenri(t) = 12~�i (�i(t)� �i(t)).

Proof: As the dynamic OPF has a zero duality gap, weuse the Karush-Kuhn-Tucker (KKT) conditions (cf. [18]) todeduce the following:2~�iri(t) = �i(t)� �i(t) + �i(t)� �i(t); t = 1; 2; : : : ; T;

(27)�i�i(t)� �i(t� 1) = ��i + �i(t)� �i(t); t = 2; 3; : : : ; T;(28)�i(T ) = �i � �i(T + 1) + �i(T + 1):

Suppose that batteryi never drains or saturates, i.e.,0 <bi(t) < Bi, then�i�i(t) � �i(t � 1) = ��i for t = 2; : : : ; Tand �i(T ) = �i. Thus, �i(t) = PT�=t �T��i �i for t =1; : : : ; T . If a batteryi drains, then�i�i(t)��i(t�1) � ��ifor t = 2; 3; : : : ; T (drains att) and �i(T ) � �i (drains atT +1). If a batteryi saturates, then�i�i(t)��i(t�1) � ��ifor t = 2; 3; : : : ; T (saturates att) and�i(T ) � �i (saturatesat T + 1).

In addition, suppose the charge/discharge bounds are nottight at the storage of busi, i.e., ri < ri(t) < �ri fort = 1; 2; : : : ; T , then the charging/discharging power amountis completely determined by the difference between two dualvariables (�i(t) and �i(t)), i.e., ri(t) = 12~�i (�i(t) � �i(t)).If a battery hits its charging power amount upper bound fort = 1; 2; : : : ; T , then ri(t) = 12~�i (�i(t) � �i(t) � �i(t)). Ifa battery hits its discharging power amount upper bound fort = 1; 2; : : : ; T , thenri(t) = 12~�i (�i(t)� �i(t) + �i(t)). Thisproves Lemma 1.

We now derive the battery amount. Rewriting the batteryamount (4) with dual variables for eachi 2 N and t =

1; 2; : : : ; T , we have:bi(t+ 1)= �ibi(t) + 12~�i (�i(t)� �i(t)� �i(t) + �i(t));= �ibi(t)+ 12~�i (��i(t)��i(t)+�i(t)+�i(1)��(t�1)i� t�1X�=1 ���i �i+ tX�=2(�i(�)��i(�))��(t+1��)i );(29)

where the second equality is due to (28). In (29), these twoentries corresponding to the lower and upper bound dualvariables�i(t) � �i(t) and

Pt�=2(�i(�) � �i(�)) can beobtained as follows:�i(t)� �i(t) = 2~�iri(t)� �i(t) + �i(t); (30)tX�=2�i(�)� �i(�) = tX�=2(�i�i(�) � �i(� � 1) + �i): (31)

Therefore, if we rewrite (29) by the iteration updates fort =1; 2; : : : ; T and i 2 N as:bki (t+ 1)= [�ibki (t) + 12~�i (��i(t) + 2~�iri(t)� �i(t) + �i(t)+ tX�=2(�i�i(�) � �i(� � 1) + �i)��(t+1��)i� t�1X�=1���i �i + �i(1)��(t�1)i )℄Bi0 ; (32)

then we have an extension of Algorithm 2 that modifiesthe battery amount update using the dual price iterates.

Algorithm 4:Extension of Algorithm 2

Replace Step 3 of Algorithm 2 by the update:bki (t+ 1)= [�ibki (t) + 12~�i (��ki (t) + 2~�irki (t)+ tX�=2(�i�ki (�) � �ki (� � 1) + �i)��(t+1��)i� t�1X�=1 ���i �i + �ki (1)��(t�1)i )℄Bi08i 2 N and t = 1; : : : ; T .

Remark 5: Similarly to Algorithm 2, by an appropriatechoice of the stepsizes, Algorithm 4 can converge to theglobal optimal solution, and we compare these modificationsin Section VII.

B. Physical interpretation of dual solution from alternativeprimal-dual decomposition

In the alternative primal-dual decomposition, the followingspecial case result quantifies the effect of load smoothingthrough the property of the dual solution.

Lemma 2:Suppose we ignore the battery costhi(t) (let� = 0 and ~� = 0), the battery storage leakage (let� = 1)and select the battery bounds to be large enough such that the

battery never hits the bounds beforeT+1, then the load profileis strictly smoothed, i.e.,V(t)>YiV(t) = V(t�1)>YiV(t�1) for t = 2; : : : ; T .

Proof: The Lagrangian that corresponds to the energystorage, i.e., (22) (V(t) is fixed), at eachi 2 N is given by:~Li = Li+PTt=1 �i(t)�V>(t)YiV>(t)�: We analyze the KKTconditions of~Li similarly to Lemma 1. Since the battery boundof bus i in (8) is inactive for alli 2 N and t = 1; : : : ; T �1, �i(t) = PT�1�=t �T���1i �i + �T�ti �i(T ). Thus, if �i = 0and �i = 1 for all i 2 N , we have�i(1) = �i(2) = : : : =�i(T ) for all i 2 N . Furthermore, by the stationarity of theLagrangian in the KKT conditions corresponding tori(t), if~�i = 0 and batteryi does not reach its charge/discharge boundfor all i 2 N andt = 1; : : : ; T , we have�i(t) = �i(t), whichthen implies that�i(1) = �i(2) = : : : = �i(T ) for all i 2 N .The problem in (22) is then no longer coupled over time andhence can be solved individually for eacht. Moreover, as thenodal power prices satisfy�i(1) = �i(2) = : : : ;= �i(T )for all i 2 N , this means that the nodal power in (22) at nodei for t = 1; : : : ; T must be the same, i.e.,V(t)>YiV(t) =V(t� 1)>YiV(t� 1) for t = 2; : : : ; T .

In summary, different decomposition methods can lead todifferent load profile smoothing features. Partial load profilesmoothing can still be achieved with a partial decomposition.As the energy storage batteries charge and discharge at differ-ent time periods, the dual solution of each OPF in the temporaldimension are coupled to achieve the load profile smoothing(see evaluations in Section VII-A).

VII. N UMERICAL EVALUATIONS

In this section, we evaluate the performance of our proposedmessage passing algorithms and demonstrate the load profilesmoothing features of the dynamic OPF problem in (8) fordifferent problem settings. In the following, unless otherwiseindicated, all the power values are normalized to per unitvalues (pu) like in [20].

A. Illustration of load profile smoothing by energy storage

To illustrate load profile smoothing, we use a 5-bus example([6], Chapter 6, pp.327) as shown in Figure 4, where a batteryis attached to each bus. We divide the time period of a dayinto six intervals (every four hours in an interval and thust = 1; 2; : : :6) and the demand profile starts at 6:00PM (peakhour). After solving this dynamic OPF under the demandvariation, we can observe a similar load profile smoothingfeature akin to “valley-filling” [3], [4] as shown in Figure 6.

Figure 6(a) shows the ideally smoothed load profile asdescribed in Lemma 2. The peak has been flattened while thevalley has been filled to the same level in the whole period. InFigure 6(b), we increase the demand at Bus 2 while keepingother parameters the same as before. Since Bus 2 reaches thebattery lower bound at timet = 3 (discharge the battery tosatisfy the high demand requirement), a strictly smoothed loadprofile is not shown according to Lemma 1 and Lemma 2.However, for the two time periodst = 1 � 2 and t = 3 � 6,as the relationship�i(t) = �i(t � 1) still holds at each timeperiod, the load profile is also strictly smoothed in each of

6:00PM 10:00PM 2:00AM 6:00AM 10:00AM 2:00PM0

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Fig. 6. Illustration of the actual and required demand underdifferent problemsettings. (a) Ideal load smoothing (b) Approximately idealload smoothing (c)A load smoothing case by choosing�i = ~�i = 0:1 (d) A load smoothingcase by choosing�i = 0:2; ~�i = 0:3.

these two periods. In Figures 6(c) and 6(d), we consider thebattery dynamic costh(t) and set the battery storage leakageparameter�i = 0:98. Incidentally, by an appropriate choiceof the battery cost parameter�i = ~�i = 0:1, the load canalso be smoothed. Specifically, at the demand peak hour, e.g.,6:00PM, the demand bus discharges the battery and less poweris absorbed from the grid. Conversely, the battery is charged atthe demand off-peak hour, e.g., 2:00AM. Thus, the high loadusage during the peak hours is shifted to the off-peak hours(valley bottom). However, when�i increases, the smoothingeffect is limited in order to minimize the total battery cost, andthus smoothing holds partially, e.g., the load is not balancedat Bus 3 when�i = 0:2; ~�i = 0:3.

Next, in connection to Figure 6(a), we plot the variationof the energy storage at Bus 2 in Figure 7 to illustrate theresult in Lemma 2. When the demand is high, the batteryis discharged to provide power to the attached bus. Duringthe off-peak hours, the battery absorbs power from the grid tocharge the battery. When the next peak hour arrives, the powerstored earlier is used to supply the demand for the attachedbus (realizing the load profile smoothing over the whole timeperiod).

B. Convergence performance

We evaluate the convergence performance of Algorithm2 and Algorithm 3 in the 5-bus system (Figure 4) usingthe time-varying demand of a day (six time intervals) asin Section VII-A. The parameter of the line admittance is[y12; y13; y23; y24; y35℄> = [4; 3; 6; 6; 7℄>pu. The initial step-sizes in Algorithm 2 are� = 0:3; � = 0:04, = 0:04. Theinitial stepsizes in Algorithm 3 are� = 0:35, � = 0:25, = 0:25, Æ = 0:1. We use a diminishing stepsize rule (cf.[30], [36]) to update the stepsizes. In Figure 8 and Figure 9,we

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Charge DischargeDischarge

Fig. 7. Illustration of Lemma 2 in the 5-bus system. This figure shows howthe actual demand, required demand and the battery storage amount vary inthe ideal load smoothing setting of a day as in Figure 6(a) at Bus 2.

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Fig. 8. Illustration of convergence of Algorithm 2 in the 5-bus system withthe initial point set close to the optimal solution.

show the convergence of Algorithm 2 for four buses at threetime intervals (t = 1; 3 and 6) using different initial points. InFigure 8, the initial point is set close to the optimal solution,i.e., in the neighborhood of the optimal solution. Algorithm2 converges by around 115 iterations. In Figure 9, we let theinitial point be set further away from the optimal solution,Algorithm 2 converges in less than 160 iterations. In Figure10,we choose the initial point the same as that in Figure 9 toshow the convergence behavior of Algorithm 3. By comparingFigure 10 and Figure 9, we see that Algorithm 3 converges tothe optimal solution slower than that of Algorithm 2.

Next, we evaluate the algorithm performance on energystorage dynamics, i.e., the charging/discharging power amountusing the same 5-bus system example. We choose the initialpoints randomly and letr0(t) = 0. In Figure 11 and Figure 12,we illustrate the convergence of Algorithm 2 and Algorithm4 onr(t) for two buses at all the six intervals with differently

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Fig. 9. Illustration of convergence of Algorithm 2 in the 5-bus system withthe initial point set further away from the optimal solution.

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Fig. 10. Illustration of convergence of Algorithm 3 in the 5-bus system withthe initial point set further away from the optimal solution.

random initial points, respectively. In Figure 11, the batteryamount is computed only by the primal variable updates (Step3 in Algorithm 2). We see that it converges in around 150iterations. In Figure 12, the battery amount is computed byusing the dual variable update (Step 3 in Algorithm 4), whichalso converges fast to the optimal solution like in Algorithm2.C. Algorithm performance in IEEE 14-bus and 30-bus systems

In this section, we evaluate the convergence performance ofAlgorithm 2 and Algorithm 3 for medium-sized systems usingthe IEEE 14-bus system and 30-bus system that have richerconnectivity. The IEEE 14-bus system and the IEEE 30-bussystem correspond to a portion of the Midwestern U.S. ElectricPower System as of February 1962 and December 1961respectively [37]. As originally there is no energy storage, we

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(c)Fig. 13. The IEEE test systems and the time-varying demand profiles used for numerical evaluations. (a) The topology of the IEEE 30-bus system. (b) Thetopology of the IEEE 14-bus system. (c) The total demand profiles created by using typical hourly demands averaged over 14and 30 different days in January2009 of the domestic customers in USA [38] for the IEEE 14-busand 30-bus system respectively.

have used the network topology and select appropriate valuesfor the other system parameters (e.g., line admittance) andbattery parameters to model energy storage. In both systems,Buses 1 and 2 are the generation buses. Moreover, similarlyto [20], the demand profiles for each bus are created by usingtypical hourly demands averaged over 14 and 30 differentdays in January 2009 of the domestic customers in USA forthe IEEE 14-bus and 30-bus systems respectively (availablein [38]). Figure 13(a) and Figure 13(b) show the topologyof the IEEE 30-bus system and the IEEE 14-bus systemrespectively. Figure 13(c) plots the scaled aggregated demandprofiles used in these two test systems, and the time period ofa day is divided into seven intervals (i.e.,t = 1; 2; : : :7) andthe demand profile starts at 1:00AM.

By appropriately choosing the stepsizes, we plot the conver-

gence behavior of Algorithms 2 and 3. Figure 14 and Figure 15show the convergence of both algorithms in the IEEE 14-bus system and the IEEE 30-bus system respectively. In thetop two sub-figures of Figure 14, we show the convergenceof Algorithm 2 at three buses (Buses 1, 4 and 14) for twotime intervals (t = 1 and4). In the bottom two sub-figures ofFigure 14, we show the convergence of Algorithm 3 at threebuses (Buses 2, 6 and 12) for two time intervals (t = 3 and6).Similarly, in the top two sub-figures of Figure 15, we show theconvergence of Algorithm 2 at three buses (Buses 1, 15 and30) for the time intervalst = 1 and4. In the bottom two sub-figures of Figure 15, we show the convergence of Algorithm3 at three buses (Buses 9, 14 and 23) for the time intervalst = 1 and4). From Figures 14 and Figure 15, we see that thealgorithms can converge to the optimal solution typically in

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Fig. 11. Illustration of convergence for the storage dynamics in Algorithm 2in the 5-bus system.

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t=1(Algo4)

t=2(Algo4)

t=3(Algo4)

t=4(Algo4)

t=5(Algo4)

t=6(Algo4)

Fig. 12. Illustration of convergence for the storage dynamics in Algorithm 4in the 5-bus system.

hundreds of iterations.

VIII. CONCLUSION

In this paper, we studied a dynamic OPF problem in a purelyresistive power network with energy storage. By exploitingarecently-discovered zero duality gap result for the staticOPF,we decomposed the dynamic OPF into simpler subproblemsusing an indirect dual-dual decomposition method and analternative primal-dual decomposition method for distributedoptimization by message passing algorithms. The optimizationdecomposition technique also revealed an interesting couplingover space and time of the Lagrange dual solution (to be inter-preted as power prices) and the energy storage dynamics. Weconducted extensive numerical evaluations on the performanceof the proposed algorithms and to demonstrate the equilibriumload profile smoothing feature in the dynamic OPF problem.

An important direction for future research is the extensionfor the general alternating current (AC) network. Furtherstudies that unify the various approximation methodologiesfor solving the AC OPF (e.g., the DC OPF linearization)and special cases (purely resistive network as given here or

100 200 300 4000.9

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Bus 1(opt)

Bus 4(opt)

Bus 14(opt)

Bus 1(Algo2)

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Bus 14(Algo2)

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Bus 2(opt)

Bus 6(opt)

Bus 12(opt)

Bus 2(Algo3)

Bus 6(Algo3)

Bus 12(Algo3)

Fig. 14. Illustration of convergence of Algorithms 2 and 3 inthe IEEE 14-bus system. The top two subfigures show the convergence of Algorithms 2for buses 1, 4, and 14 over two time intervals. The bottom two subfiguresshow the convergence of Algorithms 3 for buses 2, 6, and 12 over two timeintervals.

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Bus 15(opt)

Bus 30(opt)

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Bus 9(Algo3)

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Bus 23(Algo3)

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Fig. 15. Illustration of convergence of Algorithms 2 and 3 inthe IEEE 30-bus system. The top two subfigures show the convergence of Algorithms 2for buses 1, 15, and 30 over two time intervals. The bottom twosubfiguresshow the convergence of Algorithms 3 for buses 9, 14, and 23 over two timeintervals.

AC with special network topology) can allow us to betterunderstand which configuration of data, parameters and designvariables are important to load profile smoothing, and to shedfurther insights on the interaction between the physics ofpower flow and energy storage dynamics.

Another promising direction for future research is to exam-ine other important constraints in power system such as se-curity constraints, stability constraints and chance constraints.These coupling constraints can lead to new decomposabilitystructures and algorithms for a smart grid. Incorporating thecausality constraint (whenever the demand cannot be predictedand resources can only be optimized using only currently avail-able knowledge and not those in the future time) is even harderto analyze, but would be more practically useful. A possibilityis to study the extent to which online algorithm design can beused for distributed computation and to benchmark optimality

and performance with the offline setting.

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