Distributed Energy Trading: The Multiple-Microgrid Case

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1 Dstrbuted Energy Tradng: 1 The Multple-Mcrogrd Case Davd Gregoratt, Member, IEEE, and Javer Matamoros Ths s an extended verson of a paper wth the same ttle that appeared n the IEEE Transactons on Industral Electroncs, vol. 62, no. 4, pp , Apr arxv: v5 [math.oc] 13 Apr 2015 Abstract In ths paper, a dstrbuted convex optmzaton framework s developed for energy tradng between slanded mcrogrds. More specfcally, the problem conssts of several slanded mcrogrds that exchange energy flows by means of an arbtrary topology. Due to scalablty ssues and n order to safeguard local nformaton on cost functons, a subgradent-based cost mnmzaton algorthm s proposed that converges to the optmal soluton n a practcal number of teratons and wth a lmted communcaton overhead. Furthermore, ths approach allows for a very ntutve economcs nterpretaton that explans the algorthm teratons n terms of supply demand model and market clearng. Numercal results are gven n terms of convergence rate of the algorthm and attaned costs for dfferent network topologes. Index Terms Energy tradng, smart grd, dstrbuted convex optmzaton I. INTRODUCTION Worldwde energy demand s expected to ncrease steadly over the ncomng years, drven by energy demands from humans, ndustres and electrcal vehcles: more precsely, t s expected that the growth wll be n the order of 40% by year Ths demand s fueled by an ncreasngly energy-dependent lfestyle of humans, the emergence of electrcal vehcles as the major source of transportaton, and further automaton of processes that wll be facltated by machnes. In today s power grd, energy s produced n centralzed and large energy plants macrogrd energy generaton; then, the energy s transported to the end clent, often over very large dstances and through complex energy transportaton meshes. Such a complex structure has a reduced flexblty and wll hardly adapt to the demand growth, thus ncreasng the probablty of grd nstabltes and outages. The mplcatons are enormous as demonstrated by recent outages n Europe and North Amerca that have caused losses of mllons of Euros [1]. Gven these problems at macro generaton, t s of no surprse that a lot of efforts have been put nto replacng or at least complementng macrogrd energy by means of local renewable energy sources. In ths Authors are wth the Centre Tecnològc de Telecomuncacons de Catalunya CTTC, Parc Medterran de la Tecnologa, Castelldefels, Barcelona Span, emals: frst.last@cttc.es Ths work was partally supported by the Catalan Government under grant 2014 SGR 1567 and by the Spansh Government under grants TEC C03-01 and TEC P. Aprl 14, 2015

2 2 context, mcrogrds are emergng as a promsng energy soluton n whch dstrbuted renewable sources are servng local demand [2]. When local producton cannot satsfy mcrogrd requests, energy s bought from the man utlty. Mcrogrds are envsaged to provde a number of benefts: relablty n power delvery e.g., by slandng, effcency and sustanablty by ncreasng the penetraton of renewable sources, scalablty and nvestment deferral, and the provson of ancllary servces. From ths lst, the capablty of slandng [3] [5] deserves specal attenton. Islandng s one of the hghlghted features of mcrogrds and refers to the ablty to dsconnect the mcrogrd loads from the man grd and energze them exclusvely va local energy resources. Intended slandng wll be executed n those stuatons where the man grd cannot support the aggregated demand and/or operators detect some major grd problem that may potentally degenerate nto an outage. In these cases, the mcrogrd can provde enough energy to guarantee, at least, a basc electrcal servce. The connecton to the man grd wll be restored as soon as the entre system stablzes agan. Clearly, these are nontrval functonaltes that may cause nstablty. In ths regard, the sequel of papers [6], [7] provde a recent survey on decentralzed control technques for mcrogrds. In order to mprove the capabltes of the Smart Grd, a typcal approach s to consder the case where several mcrogrds exchange energy to one another even when the mcrogrds are slanded, that s dsconnected from the man grd [8], [9]. In other words, there exst energy flows wthn a group of contguous mcrogrds but not between the mcrogrds and the man grd. In ths context, the optmal power flow problem has recently attracted consderable attenton. For nstance, n [10] the authors consder the power flow problem jontly wth coordnated voltage control. Alternatvely, the work n [11] focuses on unbalanced dstrbuton networks and proposes a methodology to solve three-phase power flow problems based on a Newton-lke descend method. Due to the fact that these centralzed solutons may suffer from scalablty [11], [12] and prvacy ssues, dstrbuted approaches based on optmzaton tools have been proposed n [13], [14] and more recently n [15] [17]. In general, the optmal power flow problem s nonconvex and, thus, an exact soluton may be too complex to compute. For ths reason, suboptmal approaches are often adopted. As an example, references [16], [18] show that semdefnte relaxaton see [19] for further detals on ths technque can, n some cases, help to approxmate the global optmal soluton wth hgh precson. Alternatvely, [17] resorts to the so-called alternatng drecton method of multplers see [20] for further nformaton to solve the power flow problem n a dstrbuted manner. In ths paper, conversely to the aforementoned works, we consder an abstract model that allows us to focus on the tradng process rather than on the electrcal operatons of the grd. In terms of tradng, the massve spread of dstrbuted energy resources s expected to drve the transton from today s olgopolstc market to a more open and flexble one [21]. Ths new pcture of the market has trggered the nterest on new energy tradng mechansms [8], [22] [29]. For nstance, the authors n [22] consder a scenaro where a set of geographcally dstrbuted energy storage unts trade ther stored energy wth other elements of the grd. The authors formulate the problem as a noncooperatve game that s shown to have at least one Nash equlbrum pont. In the context of demand response, the work n [24] proposes an effcent energy management polcy to control a cluster of demands. Interestngly, [23] consders demand-response capabltes as an asset to be offered wthn the market and, thus, they can be traded between the dfferent agents retalers, dstrbuton system operators, aggregators, etc.. From a more general perspectve, the problem of energy tradng between mcrogrds or, market agents has been consdered n [8], [25] [29]. Whereas these works manly focus on Aprl 14, 2015

3 3 smulaton studes and archtectural ssues, ths paper attempts to provde a comprehensve analytcal soluton for the energy tradng problem between mcrogrds that, besdes ts theoretcal appeal, can be dstrbutedly mplemented wthout the need of a central coordnator. More specfcally, our settng conssts of M mcrogrds n whch: each mcrogrd has an assocated energy generaton cost; there exsts a cost mposed by the dstrbuton network operator for transferrng energy between adjacent mcrogrds; and each mcrogrd has an assocated power demand that must be satsfed. Under these consderatons, we am to fnd the optmal amounts of energy to be exchanged by the mcrogrds n order to mnmze the total operatonal cost of the system energy producton and transportaton costs. Of course, a possble approach would be to solve the optmzaton problem by means of a central controller wth global nformaton of the system. However, such a centralzed soluton presents a number of drawbacks snce mcrogrds mght be operated by dfferent utltes and nformaton on producton costs cannot be dsclosed. Therefore, n order to safeguard crtcal nformaton on local cost functons and make the system more scalable, we propose an algorthm based on dual decomposton that teratvely solves the problem n a dstrbuted manner. Interestngly, each teraton of the resultng algorthm has a straghtforward nterpretaton n economcal terms, once the new operatonal varables we ntroduce are gven the meanng of energy prces. Frst, each mcrogrd locally computes the amounts of energy t must produce, buy and sell to mnmze ts local cost accordng to the current energy prces. Then, after exchangng the mcrogrd bds, a regulaton phase follows n whch, n a dstrbuted way, the energy prces are adjusted accordng to the law of demand. Ths two-step process terates untl a global agreement s reached about prces and transferred energes. The remander of ths paper s organzed as follows. In Secton II we present the system model. Next, Secton III shows how the dstrbuted optmzaton framework provdes a soluton for the local subproblems and gves an nterpretaton from an economcal pont of vew. Fnally, n Sectons IV and V, we present the numercal results and draw some conclusons. II. SYSTEM MODEL Consder a system composed of M nterconnected mcrogrds µgs operatng n slanded mode. Durng each schedulng nterval, each mcrogrd µg- generates E g unts of energy 1 and consumes E c unts of energy. Moreover, µg- may be allowed to sell energy E,j to µg-j, j, and to buy energy E k, from µg-k, k. Then, energy equlbrum wthn the µg requres E g + e T A T E b = E c + e T AE s, 1 where the two M-dmensonal column vectors E b = E 1,. and E s = E,1. 2 E M, E,M gather all the energes bought and sold, respectvely, by µg-. Also, we have ntroduced the adjacency matrx A = [a,j ] M M : element a,j s equal to one f there exsts a connecton from µg- to µg-j and zero otherwse. 1 For smplcty, we assume that all energes correspond to constant power generaton/absorpton/transfer over the schedulng nterval. Aprl 14, 2015

4 4 Note that, generally, A may be nonsymmetrc, meanng that at least two µgs are allowed to exchange energy n one drecton only. By conventon, we fx a, = 0. Also, a,j = 0 E,j = 0 MWh for all, j = 1,..., M. Next, let C E g and γe,j be the costs of producng E g unts of energy at µg- and transferrng E,j unts of energy on the µg- µg-j lnk, respectvely. Ths transferrng cost functon may model several factors. For example, the dstrbuton network operator, as the enabler of the energy transfer between µgs, may charge a tax for energy transactons. In addton, ths transfer cost functon may also account for lne congestons by ntroducng soft constrans on the maxmum capacty of the lne see Secton II-A for further nformaton. Even though the soluton below may be extended to the case where dfferent lnks have dfferent transfer cost functons, we assume that γ s common among all the lnks n order to avod further complexty. As t wll be clearer later, our approach s qute general and only requres that all cost functons both producton and transport satsfy some mld convexty constrants, summarzed n Secton II-A. Fnally, we further assume that all µgs agree to cooperate wth one another n order to mnmze the total cost of the system. In other words, the energy quanttes exchanged by nterconnected µgs form the equlbrum pont of the followng mnmzaton problem: C = mn M {E,j} =1 C E c s. to E,j 0,, j, E c + e T + e T AE s AE s A T E b A T E b 0,, + M =1 e T A T γe b where e s the -th column of the M M dentty matrx and, wth some abuse of notaton, we wrote [ ] T γe b g = γe 1, γe M,. Also, we used 1 to get rd of the varables {E }. Note that problem 3 consders the µgs as parts of a common system for nstance, they are controlled by the same operator and the am s at mnmzng the global cost, wthout focusng on the benefts/losses of each ndvdual µg. We wll see later on, however, that the proposed dstrbuted and teratve mnmzaton algorthm opens to a wder-sense nterpretaton where achevng the global objectve mples a cost reducton at every µg. 3 A. The cost functons As mentoned before, the algorthm proposed hereafter works wth any set of generaton/transfer cost functons, as long as they all satsfy some mld convexty constrants. Specfcally, t s requred that {C } and γ are postve valued, monotoncally ncreasng, convex and twce dfferentable. Even though these requrements may seem abstract and dstant from real systems, one should take nto account that the cost functon of a common electrcal generator as, e.g., ol, coal, nuclear,... s often modeled as a quadratc polynomal Cx = a+bx+cx 2, where the coeffcents a, b and c depend on the generator type, see [30] and, especally [31]. Such a generaton model clearly satsfes our assumptons and, wthout avalable counterexamples, we extended t to the transfer cost functon γ. It s worth commentng here that the assumptons on the cost functons also allow for a smple way to ntroduce upper bounds on the energy generated by the µgs or supported by the transfer connectons. Indeed, one can ntroduce soft constrants by desgnng the cost functon wth a steep rse at the nomnal maxmum value see also the frst paragraph of Secton IV, where we comment on the cost functons used for smulaton. Aprl 14, 2015

5 5 By dong so, the maxmum generated/transferred energes are controlled drectly by the cost functons, wthout the need for hard constrants.e., well specfed nequaltes such as E g E g,max that would ncrease the complexty of the mnmzaton problem n 3. Note that ths expedent translates, somehow, to a more flexble system: when needed, a µg can produce more energy than the nomnal maxmum f t s wllng to pay an sgnfcant extra cost. Indeed, ths stuaton arses n practcal systems when backup generators are actvated. A. Decentralzng the problem III. ITERATIVE DISTRIBUTED MINIMIZATION Problem 3 s known to have a unque mnmum pont snce both the objectve functon and the constrants are strctly convex. However, dealng wth MM 1 unknowns can be very nvolved. Moreover, a centralzed soluton would requre a control unt that s aware of all the system characterstcs. Ths fact mples a consderable amount of data traffc to gather all the nformaton and can mss some prvacy requrements, snce µgs may prefer to keep producton costs and quanttes prvate. To avod these ssues, we propose here a dstrbuted teratve approach that reaches the mnmum cost by decomposng the problem nto M local, reduced-complexty subproblems solved by the µgs wth lttle nformaton about the rest of the system. 1 Identfyng local subproblems: In order to decompose 3 nto M µg subproblems, let us rewrte t n the followng equvalent form C = {ε s mn M },{E,j} =1 C E c s. to E,j 0,, j, E c ε s + ε s + ε s e T A T E b = e T AE s,. e T A T E b The dea s that, for each µg, we frst use the new varable ε s 0,, + =1 e T A T γe b 4 to represent the energy sold by µg- and only later we force t to be equal to all the energy bought by other µgs from µg-, namely ε s couplng constrant. = e T AEs, the Due to the convexty propertes of the prmal problem 3 or, equvalently, 4, one can fnd the mnmum cost by relaxng the M couplng constrants and solvng the dual problem where Cλ = M =1 Cl λ wth terms C l λ = mn ε s,e b C = max Cλ 5 λ C ε s s. to ε s E c, E b, λ 0, E j, 0, j + ε s e T A T E b 0. In the last defnton, whch s a local mnmzaton subproblem gven the parameters λ, we ntroduced C ε s, E b c, λ = C E + ε s e T A T E b + e T A T γe b + e T A T dag{λ}e b λ ε s, 7 6 Aprl 14, 2015

6 6 ] T that s the contrbuton of µg- to the Lagrangan functon relatve to 4. The parameter vector λ = [λ 1 λ M gathers all the Lagrange multplers λ correspondng to the couplng constrants ε s and for all = 1,..., M. = e T AEs, respectvely 2 Iteratve dual problem soluton: To solve the dual problem 5, we resort to the teratve subgradent method [32, Chapter 8], whch bascally fnds a sequence {λ[k]} that converges to the optmal pont of the dual problem 5, namely λ = arg max λ Cλ. More specfcally, for each pont λ[k], each µg mnmzes ts contrbuton to the Lagrangan functon by solvng the local subproblem 6 and determnng the mnmum pont ε s [k], E b [k] = ε s λ[k], E b λ[k]. Then, the Lagrange multplers are updated accordng to e T 1 AE s 1 [k] εs 1 [k] λ[k + 1] = λ[k] + α[k]., 8 e T M AEs M [k] εs M [k] where α[k] s a postve step factor. Also, recall from 2 that the set of vectors {E s } can be readly derved knowng the set {E b }. Note that the vector ς = [e T AEs [k] [ε s [k]] M 1 s a subgradent of the dual concave functon Cλ n λ = λ[k],.e. Cλ Cλ[k] + ς T λ λ[k], λ. Fnally, 8 also says that λ can be updated at µg- once the vector E s [k] has been bult wth the nputs E,j [k] collected from the neghborng µgs. 3 Interpretaton Market clearng: Algorthm 1 summarzes the dstrbuted mnmzaton procedure. One can readly notce that all necessary data s computed at the µgs, wth no need for an external, centralzed control unt. The nformaton exchanged by the µgs s lmted to the Lagrange multplers {λ } and the demanded energes {E j, }, computed at µg- and communcated only to the correspondng µg-j. Both prvacy and traffc lmtatons are hence satsfed. Algorthm 1 Dstrbuted approach µg- ntalze λ [0] repeat µgs exchange {λ [k]} µg- computes ε s [k] and E b [k] by solvng 6 wth fxed λ[k] µg- nforms µg-j, j, about the energy t s wllng to buy, namely E j, [k], at the gven prce λ j [k] [ ] T wth energy requests E,j [k] from neghborng µgs, µg- bulds E s [k] E,1 [k] E,M [k] µg- computes λ [k + 1] λ [k] + α[k]e T AEs k k + 1 untl convergence condton s verfed [k] ε s [k] As commented before, ths algorthm allows for an nterestng nterpretaton: each Lagrange multpler λ may be understood as the prce per energy unt requested by µg- to sell energy to ts neghbors. Then, the Lagrangan functon 7 can be seen as the net expendture the opposte of the net ncome for µg-: each µg pays for producng energy, for buyng energy and for transportng the energy t buys. Conversely, the µg s payed for the energy t sells. By solvng problem 6, µg- s thus maxmzng ts beneft for some gven sellng Aprl 14, 2015

7 7 λ [k] and buyng λ j [k], j prces per energy unt. Accordng to ths vew, the updatng step 8 s clearng the market: prces should be modfed untl, globally, energy demand matches energy offer. Note that 8 s an example of the law of demand: f the energy offered by µg- ε s [k] s less than all the energy demanded by the neghborng µgs from µg-, that s e T AEs, then the sellng prce must ncrease and λ [k + 1] λ [k]. B. The µg subproblem In the prevous secton we have shown how the cost mnmzaton problem 3 can be solved by means of successve teratons between the soluton of local problems 6 and the update of the Lagrange multplers accordng to 8. We wll gve now a closed-form soluton to the local subproblem 6 to be solved by the generc µg-. In order to keep notaton as smple as possble, and wthout loss of generalty, we assume that the Lagrange multplers {λ j }, j are ordered n ncreasng order,.e. λ mn = λ 1 λ 2 λ 1 λ +1 λ M. Also, wth some abuse of notaton, we fx λ j = + when a j, = 0: as far as µg- s concerned, the fact that there s no connecton from µg-j to µg- s equvalent to assume that the prce of the energy sold by µg-j s too hgh to be worth buyng. Besdes, we wll make use of the functons C and γ the frst dervatves of the cost functons C and γ and of ther nverse functons, respectvely χ and Γ. It s nterestng to menton that, n economcs, the dervatve of a cost functon s called the margnal cost the cost of ncreasng nfntesmally the argument. To see ths, consder for nstance the generaton cost functon and assume that we ncrease producton from E g to E g generaton cost can be approxmated as C E g + ɛ, wth ɛ representng a small amount of energy. Then, the new + ɛ C E g + C E g ɛ, 9 showng that the cost vares proportonally wth ɛ and that the coeffcent s C Eg. Note that ɛ can be ether postve more energy s produced for, e.g., sellng purposes or negatve because, e.g., some energy s bought from outsde. Analogously, γ E j, s the margnal transportaton cost. The soluton to the mnmzaton subproblem 6 at µg- behaves accordng to sx dfferent cases. Each case s characterzed by a specfc relatonshp between the generaton/transportaton margnal costs C and γ and the untary sellng/buyng prces {λ }. We report next the mathematcal defnton of the sx cases whereas, n Secton III-C, some addtonal comments wll help n graspng ther electrcal/economcal meanng. Case 1 µg- nether sells nor buys: If λ C Ec and λ mn C Ec γ 0, then µg- wll decde to reman n a self-contaned state and generate all and only the energy t consumes. Namely, E g = E c, ε s = 0, E j, = 0 j. Case 2 µg- buys but nether generates nor sells: Let us assume that λ C 0 and λ mn λ γ E c. Moreover, we can dentfy a partton {S, S 0 } of {j = 1,..., M : j } that satsfes the followng assumptons: t exsts η > 0 such that η > λ j λ + γ 0 for all j S ; η λ j λ + γ 0 for all j S 0 ; Aprl 14, 2015

8 8 η s the unque postve soluton to Γη λ j + λ = E c ; ether j S : λ j λ γ 0 or j S, λ j < λ γ 0 and Γλ λ j E c ; for all j S, one has λ j C 0 γ 0 and C 0 λ j Γ E c. Then, µg- buys all and only the energy t consumes,.e. t nether generates nor sells any energy. More specfcally E j, = Γη + λ λ j j S, E j, = 0 j S 0, ε s = 0 E g = 0. Case 3 µg- generates and buys but does not sell: Let us assume that λ < C Ec whle λ mn > max{c 0, λ } γ E c and λ mn < C Ec γ 0. Moreover, we can dentfy a partton {S, S 0 } of {j = 1,..., M : j } that satsfes the followng assumptons: t exsts η > 0 such that η > λ j λ + γ 0 for all j S ; η λ j λ + γ 0 for all j S 0 ; η s the unque postve soluton to C E c Γη + λ + λ j = η + λ ether j S : λ j λ γ 0 or j S, λ j < λ γ 0 and λ C ether j S : λ j > C 0 γ 0 or j S, λ j C 0 γ 0 and C 0 λ j > E c. Γ E c Γλ λ j ; Then, µg- does not sell any energy. Furthermore, t buys some energy to supplement the local generator and feed all the loads. The exact amounts are as follows: E j, = Γη + λ λ j j S, E j, = 0 j S 0, ε s = 0 E g = χ η + λ. Case 4 µg- generates and sells but does not buy: If λ > C Ec and λ mn λ γ 0, then µg- does not buy any energy. Conversely, t generates all the energy t needs plus some extra energy for the market. More specfcally, E g = χ λ, ε s = E g E c, E j, = 0 j. Case 5 µg- sells and buys but does not generate: Assume that λ C 0 and λ mn < λ γ 0. Also, let S = {j : λ j < λ γ 0}. Note that S snce, at least, λ mn S. Then, µg- does not generate any Aprl 14, 2015

9 9 µg-2 µg-2 µg-1 µg-4 µg-1 µg-4 µg-3 µg-3 a Fully connected b Rng µg-1 µg-2 µg-3 µg-4 c Lne Fg. 1. Consdered connecton topologes. energy: t buys all the energy t consumes, plus some extra energy for the market, from all µgs n the set S. The exact amounts are as follows: E j, = Γλ λ j j S, E j, = 0 j / S, ε s = E j, E c, E g = 0. Case 6 µg- sells, buys and generates: Assume that λ > C 0, λ mn < λ γ 0 and λ > C E c Γλ λ j, where we ntroduced the set S = {j : λ j < λ γ 0}. Then, the local generator s actvated but µg- also buys energy from all µgs n S. After feedng all local loads wth E c, some extra energy s left for sellng n the market: E j, = Γλ λ j j S, E j, = 0 j / S, ε s = E g + E j, E c, E g = χ λ. Proof: The proof of these results s a cumbersome convex optmzaton exercse. From the Karush- Kuhn-Tucker condtons assocated to 6, one must suppose all the dfferent cases above and realze that the correspondng assumptons are necessary for each gven case. Furthermore, one can also derve the exact values of all energy flows. Once all cases have been consdered, a careful nspecton shows that the derved necessary condtons form a partton of the hyperplane {λ, {λ j } j }. Hence, the condton are also suffcent, along wth necessary, and the proof s concluded. All the detals are gven n the appendx. E c As a fnal remark, note that we are assumng a postve load at the µgs,.e. E c > 0,. The case where = 0 may be handled analogously and brngs to smlar results. More specfcally, Cases 4, 5 and 6 extend drectly because of contnuty. Conversely, Cases 2 and 3 dsappear and expand the doman of Case 1 to λ C 0 and λ mn λ γ 0. C. Interpretaton and summary It s nterestng to note that the optmal soluton provded n the prevous secton has a straghtforward nterpretaton n economcal terms. Recall that, at µg-, the Lagrange multplers can be nterpreted as the Aprl 14, 2015

10 10 untary sellng prce λ and the untary buyng prces from the other µgs {λ j }, j. Moreover, the dervatves C and γ are the margnal generaton cost and the margnal transportaton cost, respectvely, that s the lnear varaton on the cost due to an nfntesmal varaton of the generated or transported energy, respectvely. Bearng ths n mnd, let us focus on Case 1. By means of 7 and 9, one readly realzes that µg- s not nterested n sellng energy snce the sellng prce λ s lower than the margnal producton cost C Eg. Indeed, the ncome λ ε s wll be lower than the extra producton cost, namely C E c +ε s C E c > C Ec ε s the last nequalty s due to the convexty of C. Smlarly, buyng s not proftable ether snce the mnmum energy prce λ mn s larger than the margnal beneft 2 C Ec γ 0. The condtons for Case 1 are hence justfed. Analogous consderatons hold for the other cases. Another nterestng pont s that mcrogrds are always wllng to trade snce ther local cost wthout tradng,.e. C E c +γ0, wll always be hgher than ther net expendture C l λ n 7, where λ stands for the optmal pont of 5. For the sake of brevty, we prove ths result for Case 6 only, although the same reasonng holds true for the rest of cases. In Case 6, the net expendture reads where ε s C l λ c = C E + ε s E j, + γe j, + λ j E j, λ ε s, {E j, } = ε s λ, {E j, λ } s now the mnmum pont of 6 for λ = λ. Next, by means of the results of Case 6, one has λ above can be rewrtten as follows: wth E 0 ε s turns out that C E c leads to the desred result. = C Eg and λ j = λ γ E j, for all j S. Then, the equaton C l λ c = C E + E 0 C c E + E 0 E0 + γej, γ E j, E j,, E j,. Fnally, snce the cost functons are monotoncally ncreasng and convex, t + E 0 C Ec + E 0 E 0 < C E c and γe j, γ E j, E j, < γ0 = 0, whch IV. NUMERICAL RESULTS As for the numercal results, we have consdered a system composed of four mcrogrds and, for smplcty, we have assumed the same generaton cost functon at all mcrogrds. More specfcally, we have consdered the U12 generator n [31], whose quadratc cost functon Cx = a + bx + cx 2 has coeffcents a = $, b = $/MWh and c = $/MWh 2. Besdes, as motvated n Secton II-A, the orgnal cost functon has been multpled by the functon x/e g max 30 n order to nclude a soft constrant that accounts for the maxmum energy generaton E g max = 10 MWh. Regardng the transfer cost functon, we have modeled t as the cubc polynomal 3 γx = x + x 3, wth x n MWh and γx n US dollars. Numercal results are gven for the three topologes shown n Fg. 1: fully connected, rng and lne. 2 When buyng energy, the producton cost reduces but the transportaton cost ncreases. The margnal beneft wth respect to E g = E c the µg generates all and only the energy t consumes s thus C Ec γ 0. 3 Note that ths partcular choce s arbtrary and ths functon may depend on physcal parameters and the busness model. However, smlar results are expected wth other setups. Aprl 14, 2015

11 Dualty gap, left Total cost, rght [$] [$] Iteraton number a Dualty gap/total cost 0.0 Prce [$/MWh] λ 1 λ 2 λ 3 λ Iteraton number b Sellng prces λ Fg. 2. Algorthm convergence for E c = [8, 11, 11, 6] T MWh: a shows the evoluton of the dualty gap left y-axs and the evoluton of the total cost rght y-axs, whle b shows the evoluton of the sellng prces λ. Fg. 2 assesses the convergence speed of the dstrbuted mnmzaton algorthm. The curves refer to a fully connected system where the mcrogrd loads are E c = [8, 11, 11, 6] T MWh. Frst, Fg. 2a shows the dualty gap and the total cost of the system as a functon of the teraton number. As we can observe, the algorthm converges the dualty gap s almost null after a reasonable number of teratons. More nterestngly, Fg. 2b shows the evoluton of the sellng prces. Note that the relatonshp between prces after convergence reflects the one between local loads: the more energy the µg consumes locally, the hgher ts sellng prce s. Whle ths s a natural consequence of the generaton cost functon beng strctly ncreasng f the local load s low, the µg can generate extra energy for sellng purposes at a lower cost, we can also see t as a manfestaton of the law of demand. Indeed, the mcrogrd energy demand may be seen as composed by two terms, an nternal one correspondng to the local loads and an external one from the other mcrogrds. The sellng prce wll hence ncrease wth the resultng total demand. Some more nsghts about how and how fast the algorthm converges can be found n [33]. There, only two mcrogrds are consdered: such a smple case allows for a centralzed closed-form soluton and ts drect comparson wth the dstrbuted approach. Next, n order to get some more nsght nto the evoluton of prces and energy flows, we consder a scenaro where all local loads are held constant at 11 MWh just above E g max, except for µg-4, whose load vares from 1 to 11 MWh. In Fgures 3a, 4 and 5, for the four mcrogrds, we report the local cost after convergence C l λ, that s the mnmum net expendture 6, wth λ the maxmum pont of 5. For benchmarkng purposes, we have also depcted the costs at each mcrogrd n the dsconnected case.e. when no tradng s performed, the dashed lnes. As shown n Secton III-C, when usng 7 as the local cost, we can observe that Aprl 14, 2015

12 12 Dsconnected µg-4 Dsconnected µg-{1,2,3} µg-1 µg-2 µg-3 µg Cost [k$] E c 4 [MWh] a Local costs [MWh] 2 Sold energy, left Prce per unt, rght Income, rght 1.0 [k$] E c 4 [MWh] b Sold energy left y-axs, unt prce and revenues rght y-axs Fg. 3. Fully-connected topology: local costs a and µg-4 metrcs b for E c = [11, 11, 11, E c 4 ]T MWh. optmal tradng always brngs some beneft cost reducton to all mcrogrds. Let us now focus on the fully connected topology of Fg. 3. In Fg. 3a we see that the cost attaned by µg-4 after tradng ntally follows the cost of the dsconnected mcrogrd. It s only when the local load grows above 6 MWh that the gan becomes notceable, reaches ts maxmum for E c 4 9 MWh and then decreases agan untl t becomes null at E c 4 = 11 MWh. There, all µgs have the same nternal demand and, for symmetry reasons, there s no energy exchange. The gan, ndeed, s a result of the energy sold by µg-4 to the other mcrogrds, whose amount, unt prce and correspondng ncome s depcted n Fg. 3b. At frst sght, t may be dsconcertng to see that the neglgble gan obtaned by µg-4 for E c 4 < 6 MWh s the result of sellng a large amount of energy at a very low prce both almost constant for E c 4 < 6 MWh. For E c 4 > 6 MWh, Aprl 14, 2015

13 13 however, the µg sells less energy but the unt prce ncreases fast enough to mprove the gan the Income curve: the best trade-off between the amount of energy sold by the µg and ts unt prce s reached, as sad before, for E c 4 9 MWh. For larger values of E c 4, the hgh sellng prce cannot compensate for the decrease of sold energy and the ncome goes to zero. To understand why ths happens, let us consder µg-4. After convergence, µg-4 s defned by Case 4 generates and sells. Then, partcularzng 7, the local costs at µg-4 s C 4 = CE c 4 + ε s 4 λ 4ε s 4, 10 where we used the fact that we are assumng C = C for = 1, 2, 3, 4. The optmal λ 4 = λ 4 s gven by the margnal cost, that s λ 4 = C E c 4 + ε s. 11 Now, consder 10, 11 and recall that the cost functon CE g 4 s the one wth label Dsconnected µg-4 n Fg. 3a: for E g 4 < 9 MWh approxmately, the cost functon s almost lnear. Thus, the unt prce 11 s nearly constant and so s the cost 10 as a functon of ε s 4 for all E c 4 and ε s 4 such that E g 4 = E c 4 +εs 4 < 9 MWh. In other words, the µg cost C 4 only depends on the consumed energy and not on the sold one snce the ncome from sellng some extra energy s canceled out by the extra cost needed to generate t. All these consderatons are reflected by the curves n Fgures the total energy sold by µg-4 n ths regme s approxmately 3 MWh. 4 3a and 3b for E c 4 < 6 MWh or, equvalently, E g 4 < 9 MWh, snce Now, wthout loss of generalty, let us focus on µg-1. Note that, for symmetry reasons, µg-1, µg-2 and µg-3 are all buyng the same amount of energy from µg-4 namely, E 4, = ε s 4 /3 and they are not exchangng energy to one another. Intutvely, µg-1 falls wthn Case 3 generates and buys and ts local cost s C 1 = CE c 1 ε s 4 /3 + γεs 4 /3 + λ 4ε s 4 /3. The optmal prce λ 4 = λ 4 s gven by 11 but, from µg-1 perspectve, can also be rewrtten as λ 4 = C E c 1 ε s 4 /3 γ ε s 4 /3. 12 Let us neglect, for the moment, the transfer cost functon γ. Also, recall that C s agan the one depcted n Fg. 3a wth label Dsconnected µg-4. Snce E c 1 = 11 MWh, the producton cost CE c 1 ε s 4 /3 takes values above the curve elbow for all E 4,1 = ε s 4 /3 < 1 MWh, approxmately. In ths regme, the generaton cost s very hgh and µg-1 s certanly tryng to buy energy to reduce ts expendture. For E 4,1 = ε s 4 /3 > 1 MWh, however, the generaton cost takes values below the curve elbow and has an almost lnear behavor see also comments above. Then, t s not worth to buy more energy, snce ts prce wll cancel out the generaton savngs. The convex nature of the transfer cost functon γ accentuates ths trend. These consderatons explan why ε s 4 = 3 E 4,1 3 MWh for all E c 4 < 6 MWh. For E c 4 > 6 MWh, the four µgs work n the nonlnear part of the generaton cost functon. Matchng the energy prce at both the sellng sde 11 and the buyng sde 12 results n the trade-off of Fg. 3b, whch turns out to be qute frutful for µg-4. Ths fact compensates somehow for the lttle beneft wth respect to the benefts of the other µgs experenced by µg-4 at low values of E c 4. Aprl 14, 2015

14 14 Dsconnected µg-4 Dsconnected µg-{1,2,3} µg-1 µg-2 µg-3 µg Cost [k$] E c 4 [MWh] Fg. 4. Costs for the rng topology E c = [11, 11, 11, E c 4 ]T MWh. Dsconnected µg-4 Dsconnected µg-{1,2,3} µg-1 µg-2 µg-3 µg Cost [k$] E c 4 [MWh] Fg. 5. Costs for the lne topology E c = [11, 11, 11, E c 4 ]T MWh. Fnally, let us recall that the purpose of the orgnal problem 3 s to mnmze the total cost of the system and not to maxmze local benefts. Ths, together wth the fact that we do not allow µgs to cheat, explans why µg-4 always sells at a unt prce gven by the margnal cost and does not look for extra gans. Even though the general deas dscussed above about the cost behavors apply to all systems, connecton topologes see Fg. 1 other than the full connected one present some specfc characterstcs. For nstance, Aprl 14, 2015

15 15 Fg. 4 report the local cost for the four µgs n rng topology. We may see that µg-2 and µg-3 get some extra beneft from actng as ntermedares between µg-4 and µg-1. To be more precse, after the tradng process, the local solutons at µg-{2, 3} wll fall wthn Case 6: energy s bought not only to satsfy nternal needs but also to be resold to µg-1. By dong so, µg-{2, 3} can reduce ther local cost substantally. In Fg. 1c we depct another stuaton where topology heavly mpacts on the costs. In ths case, mcrogrds are connected by means of a lne topology wth µg-4 at the end of the lne. Therefore, µg-3 can be regarded as the bottleneck of the system, snce all the energy that goes from µg-4 to µg-{1,2} has to pass through t nevtably. As t can be observed n the fgure, ths stuaton benefts µg-3. V. CONCLUSIONS In ths paper, we have addressed a problem n whch several mcrogrds nteract by exchangng energy n order to mnmze the global operaton cost, whle stll satsfyng ther local demands. In ths context, we have proposed an teratve dstrbuted algorthm that s scalable n the number of mcrogrds and keeps local cost functons and local consumpton prvate. More specfcally, each algorthm teraton conssts of a local mnmzaton step followed by a market clearng process. Durng the frst step, each mcrogrd computes ts local energy bd and reveals t to ts potental sellers. Next, durng the market clearng process, energy prces are adjusted accordng to the law of demand. As for the local optmzaton problem, t has been shown to have a closed form expresson whch lends tself to an economcal nterpretaton. In partcular, we have shown that no matter the local demand that a mcrogrd wll be always wllng to start the tradng process snce, eventually, ts net expendture wll be lower than ts local cost when operatng on ts own. Fnally, numercal results have confrmed that the algorthm converges after a reasonable number of teratons and there certanly s a gan over nonconnected µgs whch strongly depends on the energy demands and network topology. APPENDIX A SOLUTION TO THE MICROGRID PROBLEM Ths appendx proves the soluton to the µg problem gven n Secton III-B. As explaned before, we wll suppose that the µg best opton s to operate n one of the sx dfferent states defned accordng to whether the µg s or s not sellng, buyng and generatng any energy, as descrbed by the sx cases of Secton III-B. By dong so, one can compute all the energy values of nterest and dentfy what constrants the prces {λ, {λ j } j } must satsfy for the consdered case to be feasble. After all cases have been consdered, the sx sets of necessary condtons should form a partton of the λ, {λ j } j hyperplane. Indeed, ths fact mples that each set of condtons s suffcent, along wth necessary, for the correspondng µg state and that the computed energy values are those mnmzng the local cost functon 7 for gven prces {λ, {λ j } j }. Before delvng nto the dfferent cases, some common prelmnares are needed. For the local problem 6 the Lagrangan functon s as follows: L = C E c + ε s e T A T E b + e T A T γe b + e T A T ΛE b λ ε s ηε s µ T E b ω E c + ε s e T A T E b, Aprl 14, 2015

16 where we have ntroduced the Lagrange multplers η, ω and µ = hence wrte L ε s L E b = C c E + ε s = C c E + ε s 16 [ µ 1 µ 2 µ M ] T. The KKT condtons e T A T E b λ η ω = 0 13a e T A T E b Ae + dag{ae }γ E b + ΛAe µ + ωa T e = 0 13b ε s 0, η 0, ηε s = 0, 13c E j, 0, µ j 0, µ j E j, = 0, j = 1,..., M, 13d E c + ε s e T A T E b 0, ω 0, ω E c + ε s e T A T E b = 0. 13e By recallng the defnton of A n Secton II, the elements of the gradent n 13b can be wrtten n a much smpler form, namely C c E + ε s e T A T E b γ E j, λ j + µ j ω = 0, j : a j, = 1, µ j = 0, E j, = 0, j : a j, = 0. The dervaton of the sx possble solutons gven n Secton III-B s based on the analyss of the KKT condtons above, as explaned hereafter. A. Proof of Case 1 Let us suppose that the soluton of the mnmzaton problem tells us that the µg nether sells nor buys any energy, that s ε s wrte = 0, E b C C = 0 and E g = E c. If ths was the case, then the KKT condtons 13 would c E λ η = 0, c E γ 0 λ j + µ j = 0, j, η 0, ω = 0 and µ j 0, j. The frst condton mples η = C E c λ 0 λ C c E, whle the second condton yelds µ j = γ 0 + λ j C E c whch are the two necessary condtons correspondng to Case 1. 0 λj C c E γ 0, B. Proof of Case 2 We now look for the necessary condtons for Case 2, that s µg- sells no energy.e. ε s = 0 and buys energy from at least another µg.e. j = 1,..., M, j : E j, > 0. Besdes, µg- generates no energy and Aprl 14, 2015

17 17 E g = E c e T AT E b = 0. The KKT condtons 13 smplfy to C 0 λ η ω = 0, 14a C 0 γ E j, λ j ω = 0 and µ j = 0, j S, 14b C 0 γ 0 λ j + µ j ω = 0 and µ j 0, j S 0, 14c η 0 and ω 0, 14d where we have ntroduced the sets S = {j = 1,..., M : j and E j, > 0}, S 0 = {j = 1,..., M : j and E j, = 0}. The frst necessary condton λ C 0 s a straghtforward consequence of 14a and 14d. Next, for all j S, 14a and 14b mply λ + η λ j γ E j, = 0, 15 whch means that λ j < λ + η γ 0, j S. 16 Smlarly, because of 14a and 14c, one has λ j λ + η γ 0, j S 0. By comparng the last two nequaltes, one sees that λ j < λ k for all j S, k S 0, meanng that λ mn = mn j λ j s certanly part of the set S when ths soluton s correct. From 15, and takng 16 nto account, we can nfer that E j, = Γη λ j + λ, where Γ s the nverse of γ, whch exsts because of the contnuty and convexty assumptons on the cost functon γ. Snce we are supposng that the optmal workng pont satsfes E c see that η shall satsfy the equalty e T AT E b = 0, we Γη λ j + λ = E c. 17 Gven that Γ s an ncreasng functon, and recallng 16, t can be easly shown that equaton 17 n the varable η has a unque soluton, whose value allows us to compute the energes E j, bought from neghbor µg-j, j S compare wth the statement of Case 2. To derve the other necessary condtons for ths case, let us focus on 17. Snce Γ s a non-negatve functon and λ j = λ mn j S, t follows: Γη λ mn + λ E c. Recallng that Γ s the nverse of γ, ths mples 0 η γ E c + λmn λ and, hence, λ mn λ γ E c s a necessary condton for Case 2. Aprl 14, 2015

18 18 C E c λ 1 λ + γ 0 λ 2 λ + γ 0 η λ 3 λ + γ 0 C 0 λ η C 0 η + λ C E c Γη λ j + λ Fg. 6. Graphcal representaton of nequalty 19 and of the soluton η = η to C 0 = C c E Γη λ j + λ. Wthout loss of generalty, we assume here that λ 1 = λ mn λ 2 λ 1 λ +1 λ M. Note that, as η ncreases, a new mode s actvated each tme a pont λ j λ + γ 0 s crossed. Now, let S = {j = 1,..., M : j and λ j < λ γ 0}, whch s a subset of S because of 16. Then, by comparson wth 17, E c Γη λ j + λ Γλ λ j, j S j S where the second nequalty s a consequence of Γ beng an ncreasng functon. Also, we are lettng η 0, whch s possble snce λ λ j > γ 0 for the consdered j. In partcular, f S = S then Γλ λ j E c, as reported n Case 2. We wll see later that ths condton s mportant to dentfy the boundary between the soluton regons of Case 2 and Case 5. Consder now 14a and 16, whch gve and, n turn, a new necessary condton: Moreover, 14a and 17 yeld 0 ω = C 0 λ η < C 0 λ j γ 0 C 0 = C λ j < C 0 γ 0, j S. 18 E c Γη λ j + λ η + λ. 19 Substtutng nto 17, we can wrte the last necessary condton of Case 2, namely Γ C c 0 λ j E. 20 Note that the left-hand sde has a meanng accordng to 18. The last nequalty may also be deduced from the graphcal representaton n Fg. 6. Aprl 14, 2015

19 19 C. Proof of Case 3 We suppose agan a soluton where ε s to the prevous case, however, we also suppose that E c = 0 and E j, > 0 for at least one j = 1,..., M, j. As opposed e T AT E b Wth ths soluton, the Lagrange multpler ω s zero and the KKT condtons become where, agan, C E c C E c C E c η 0, > 0,.e. µg- produces some energy. e T A T E b λ η = 0, 21a e T A T E b γ E j, λ j = 0 and µ j = 0, j S, 21b e T A T E b γ 0 λ j + µ j = 0 and µ j 0, j S 0, 21c S = {j = 1,..., M : j and E j, > 0}, S 0 = {j = 1,..., M : j and E j, = 0}. Condton 21a drectly gves the frst requrement for the case n hand, namely λ < C Ec. Next, by combnng 21a and 21c, one has λ λ j + η γ 0 + µ j = 0 and, hence, λ j λ + η γ 0, j S 0, 22 snce µ j 0. Smlarly, 21a and 21b yeld λ λ j + η γ E j, = 0, whch mples snce γ s an ncreasng functon, and λ j < λ + η γ 0, j S, 23 E j, = Γη + λ λ j, 24 where Γ s the nverse of γ. Inequalty 23 guarantees that E j, s postve. Followng, by njectng 24 nto 21a, t turns out that η must satsfy C E c Γη + λ λ j = η + λ. 25 It s straghtforward to show that the last equaton n η admts a unque soluton see also the graphcal representaton n Fg. 7 that allows us to compute the energes bought from neghbor mcrogrds accordng to 24, as stated by Case 3. Also, knowng η and recallng that χ s the nverse functon of C, 21a allows us to compute the generated energy E g = E c E j, = χ η + λ. Combnng 25 wth 23, we get λ j + γ 0 < C Ec for all λ j, j S and n partcular for λ mn = mn j λ j see the statement of Case 3. We are partcularly nterested n λ mn snce the correspondng mcrogrd wll certanly belong to the set S and possbly be ts only element f ths case s the soluton to the mnmzaton problem. Ths can be deduced by comparng 22 and 23. Another necessary condton for ths soluton can be derved by notng that E c Γη +λ λ j > 0 mples E c Γη + λ λ mn > 0 and, hence, η < γ E c λ + λ mn. Ths bound requres λ mn > λ γ E c and, together wth 25, λ mn > C 0 γ E c. Aprl 14, 2015

20 20 C E c λ 1 λ + γ 0 λ 2 λ + γ 0 C 0 λ η λ 3 λ + γ 0 η C 0 η + λ C E c Γη λ j + λ Fg. 7. Graphcal representaton of nequalty 26 and of the soluton η = η to η + λ = C c E Γη λ j + λ. Wthout loss of generalty, we assume here that λ 1 = λ mn λ 2 λ 1 λ +1 λ M. Note that, as η ncreases, a new mode s actvated each tme a pont λ j λ + γ 0 s crossed. From 25 one further has η > C 0 λ. Then, agan because of Γ beng ncreasng n η, C 0 < C E c Γη + λ λ j < C E c Γη + λ λ j j S < C E c Γ C 0 λ j, 26 j S and, equvalently, j S Γ C 0 λ c j < E, where we have ntroduced S = {j = 1,..., M : j and λ j < C 0 γ 0} S see also Fg. 7. In partcular, f S = S then Γ C 0 λ c j < E, whch represents the complementary set of 20 n Case 2. Fnally, let S = {j = 1,..., M : j and λ j < λ γ 0}. Then, λ η + λ = C E c Γη + λ λ j C E c Γη + λ λ j j S C E c Γλ λ j, whch s equvalent to χ λ + j S Γλ λ j E c. In partcular, note that χ λ + Γλ λ j E c when S = S and compare t wth Case 6 below. D. Proof of Case 4 The next potental soluton provdes E j, = 0 for all j µg- does not buy any energy, ε s > 0 and, obvously, E g = E c +ε s > 0 the µg sells and generates some energy. Accordng to the KKT condtons, Aprl 14, 2015

21 21 we also have η = ω = 0 and C c E C c E + ε s λ = 0, + ε s γ 0 λ j + µ j = 0, j. The frst KKT condton can be satsfed only f λ > C Ec the frst necessary condton for the current case. Moreover, combnng both condtons, we get µ j = λ j + γ 0 λ j, whch requres λ j λ γ 0 the second necessary condton for the current case n order to have µ j 0. The value taken by ε s Ths also means E g = χ λ. s obtaned by nvertng C n the frst condton above, namely εs = χ λ E c. E. Proof of Case 5 E c We now analyze potental solutons of the type ε s > 0, E j, > 0 for at least one ndex j and + ε s e T AT E b = 0. In other terms, the mcrogrd does not generate any energy but buys more than consumes and sells the surplus. In ths case, the KKT condtons wrte C 0 λ ω = 0, 27a C 0 γ E j, λ j ω = 0 and µ j = 0, j S, 27b C 0 γ 0 λ j + µ j ω = 0 and µ j 0, j S 0, 27c η = 0 and ω 0, where we have ntroduced the sets S = {j = 1,..., M : j and E j, > 0}, S 0 = {j = 1,..., M : j and E j, = 0}. Snce ω 0, condton 27a requres λ C 0, the frst necessary condton for the current case. Next, jonng 27a and 27c, we readly see that µ j = λ j λ + γ 0 and λ j λ γ 0 j S Smlarly, 27a and 27b mply λ λ j γ E j, = 0 for all j S. By usng the functon Γ, nverse of γ, the last equaton allows computng the value of the energy bought from µg-j, namely E j, = Γλ λ j, as we wanted to show. Note, however, that ths s a meanngful expresson only f λ j < λ γ 0 for all j S. By comparng ths last requrement wth 28, we see that j S λ j < λ γ 0, whch s the defnton of the set S as stated by Case 5. Fnally, the total sold energy s Snce we requre ε s Case 2. ε s = e T A T E b E c =. Γλ λ j E c > 0, t must be Γλ λ j > E c, our last necessary condton compare t wth Aprl 14, 2015

22 22 F. Proof of Case 6 The last type of soluton we have to deal wth s characterzed by ε s j and E g = E c > 0, E j, > 0 for at least one ndex + ε s e T AT E b > 0,.e. µg- generates, buys and sells certan postve amounts of energy. The KKT condtons become C E c + ε s e T A T E b λ = 0, 29a C E c + ε s e T A T E b γ E j, λ j = 0 and µ j = 0, j S, 29b C E c + ε s e T A T E b γ 0 λ j + µ j = 0 and µ j 0, j S 0, 29c η = 0 and ω = 0, where we have ntroduced the sets S = {j = 1,..., M : j and E j, > 0}, S 0 = {j = 1,..., M : j and E j, = 0}. Condton 29a drectly mples λ > C 0 the frst necessary condton for Case 6 because of the convexty assumptons on C. Also, for all j S 0, the Lagrangan multplers µ j are gven by 29a and 29c, namely µ j = λ j λ + γ 0. Ths value s non-negatve only f λ j λ γ 0 j S On the other hand, 29a and 29b lead to λ λ j γ E j, = 0 for all j S. Ths dentty allows us to express the quantty E j, = Γλ λ j the desred result, wth Γ the nverse of γ, provded that λ j < λ γ 0 for all j S. Ths requrement, together wth 30, results n the defnton of S for Case 6, namely j S λ j < λ γ 0. ε s To conclude, by means of the functon χ, nverse of C, 29a s equvalent to Eg = χ λ or, also, = χ λ E c + Γλ λ j, whch are the requred expressons of the total generated and sold energy, respectvely. Note that ths s a postve and meanngful quantty only f Γλ λ j +χ λ > E c, the last necessary condton for Case 6 compare also wth Case 3. G. Summary and remarks So far, we have analyzed all the cases representng possble operatng states for the mcrogrd. For each case we have found the correspondng values taken by the energy flows E g, ε s and E j,, j. Also, each case s characterzed by a set of condtons that are necessary for the feasblty of the case tself. A close nspecton of these condtons shows that they are mutually exclusve see also the representaton of Fg. 8. Thus, each set of condtons s also suffcent, together wth necessary, for the respectve soluton case, meanng that the local mnmzaton problem s unvocally solved. REFERENCES [1] G. Andersson, P. Donalek, R. Farmer, N. Hatzargyrou, I. Kamwa, P. Kundur, N. Martns, J. Paserba, P. Pourbek, J. Sanchez-Gasca, R. Schulz, A. Stankovc, C. Taylor, and V. Vttal, Causes of the 2003 major grd blackouts n North Amerca and Europe, and recommended means to mprove system dynamc performance, IEEE Trans. Power Syst., vol. 20, no. 4, pp , Aprl 14, 2015

23 23 ext3 ext2b ext2 ext1 ext4 ext5 λ mn = C Ec γ 0 3 λ mn = C 0 γ 0 1 λ = C Ec nt1 4 λ mn = λ γ E c 6 exte extd extc ext6 2 nt2 extb 5 λ ext7 ext7b ext8 ext9 exta λ mn = λ γ 0 λ = C 0 λ mn = C 0 γ E c λ mn Fg. 8. Example of parttonng of the λ, {λ j }-space accordng to the soluton regons of Cases 1 to 6 of Secton III-B. Note that the fgure only represents a slce at a gven λ, λ mn -plane: ths s enough to dentfy most of the boundares, whch only depend on these two parameters. Some boundares the dashed ones, however, depend on all {λ j } and cannot be represented properly. [2] N. Hatzargyrou, H. Asano, R. Iravan, and C. Marnay, Mcrogrds: An overvew of ongong research, development, and demonstraton projects, IEEE Power Energy Mag., vol. 5, no. 4, pp , Jul./Aug [3] I. Balaguer, Q. Le, S. Yang, U. Supatt, and F. Z. Peng, Control for grd-connected and ntentonal slandng operatons of dstrbuted power generaton, IEEE Trans. Ind. Electron., vol. 58, no. 1, pp , Jan [4] V. K. Sood, D. Fscher, J. M. Eklund, and T. Brown, Developng a communcaton nfrastructure for the smart grd, n IEEE Electrcal Power Energy Conference EPEC, Oct. 2009, pp [5] F. Katrae, R. Iravan, N. Hatzargyrou, and A. Dmeas, Mcrogrds management, IEEE Power Energy Mag., vol. 6, no. 3, pp , May/Jun [6] J. M. Guerrero, M. Chandorkar, T.-L. Lee, and P. C. Loh, Advanced control archtectures for ntellgent mcrogrds - part I: Decentralzed and herarchcal control, IEEE Trans. Ind. Electron., vol. 60, no. 4, pp , Apr [7] J. M. Guerrero, P. C. Loh, T.-L. Lee, and M. Chandorkar, Advanced control archtectures for ntellgent mcrogrds - part II: Power qualty, energy storage, and AC/DC mcrogrds, IEEE Trans. Ind. Electron., vol. 60, no. 4, pp , Apr [8] H. S. V. S. K. Nunna and S. Doolla, Multagent-based dstrbuted-energy-resource management for ntellgent mcrogrds, IEEE Trans. Ind. Electron., vol. 60, no. 4, pp , Apr [9] M. Fath and H. Bevran, Statstcal cooperatve power dspatchng n nterconnected mcrogrds, IEEE Trans. Softw. Eng., vol. 4, no. 3, pp , Jul [10] L. F. Ochoa and G. P. Harrson, Mnmzng energy losses: Optmal accommodaton and smart operaton of renewable dstrbuted generaton, IEEE Trans. Power Syst., vol. 26, no. 1, pp , Feb [11] S. Bruno, S. Lamonaca, G. Rotondo, U. Stecch, and M. L. Scala, Unbalanced three-phase optmal power flow for smart grds, IEEE Trans. Ind. Electron., vol. 58, no. 10, pp , Oct [12] S. Paudyal, C. A. Cañzares, and K. Bhattacharya, Three-phase dstrbuton OPF n smart grds: Optmalty versus computatonal burden, n Innovatve Smart Grd Technologes ISGT Europe, nd IEEE PES Internatonal Conference and Exhbton on, 2011, pp [13] B. H. Km and R. Baldck, Coarse-graned dstrbuted optmal power flow, IEEE Trans. Power Syst., vol. 12, no. 2, pp , [14] R. Baldck, B. H. Km, C. Chase, and Y. Luo, A fast dstrbuted mplementaton of optmal power flow, IEEE Trans. Power Syst., vol. 14, no. 3, pp , [15] M. Kranng, E. Chu, J. Lavae, and S. Boyd, Dynamc network energy management va proxmal message passng, Foundatons and Trends n Optmzaton, vol. 1, no. 2, pp. 1 54, Aprl 14, 2015