Joint Optimal Pricing and Electrical Efficiency Enforcement for Rational Agents in Micro Grids
Riccardo Bonetto, Michele Rossi, Stefano Tomasin, Carlo Fischione
aa r X i v : . [ c s . S Y ] J un Joint Optimal Pricing and Electrical EfficiencyEnforcement for Rational Agents in Micro Grids
Riccardo Bonetto ⋆ † , Michele Rossi † , Stefano Tomasin † , Carlo Fischione ‡ Abstract —In electrical distribution grids, the constantly in-creasing number of power generation devices based on renew-ables demands a transition from a centralized to a distributedgeneration paradigm. In fact, power injection from DistributedEnergy Resources (DERs) can be selectively controlled to achieveother objectives beyond supporting loads, such as the minimiza-tion of the power losses along the distribution lines and thesubsequent increase of the grid hosting capacity. However, thesetechnical achievements are only possible if alongside electricaloptimization schemes, a suitable market model is set up to pro-mote cooperation from the end users. In contrast with the existingliterature, where energy trading and electrical optimization ofthe grid are often treated separately or the trading strategyis tailored to a specific electrical optimization objective, in thiswork we consider their joint optimization. We also allow for amodular approach, where the market model can support anysmart grid optimization goal. Specifically, we present a multi-objective optimization problem accounting for energy trading,where: 1) DERs try to maximize their profit, resulting fromselling their surplus energy, 2) the loads try to minimize theirexpense, and 3) the main power supplier aims at maximizingthe electrical grid efficiency through a suitable discount policy.This optimization problem is proved to be non convex, and anequivalent convex formulation is derived. Centralized solutionsare discussed first, and are subsequently distributed transformingthe optimization problem into an equivalent one that can beefficiently solved through the alternating direction method ofmultipliers. Numerical results to demonstrate the effectivenessof the so obtained optimal policies are then presented, showingthe proposed model results in economic benefits for all the users(generators and loads) and in an increased electrical efficiencyfor the grid.
I. I
NTRODUCTION
In traditional power grids, two main challenges are emerg-ing: increasing power demand, and uncoordinated injectionof electrical power from distributed generators. On the onehand, the constantly increasing power demand calls for radicalchanges on how the energy is generated and delivered to thefinal users. On the other hand, the uncoordinated injection ofelectrical power from distributed generators based on renew-ables [1], [2] tends to destabilize the power network, possiblyleading to outages.To address these problems, recent work has shown thatDistributed Energy Resources (DERs) can be effectively usedto boost the grid efficiency [3]–[7]. This research work hasresulted in the proposal of several grid optimization tech-niques [8]–[10], each exploiting some existing communication ⋆ Corresponding author. † Department of information Engineering, Univer-sity of Padova, via Gradenigo 6/b, 35131, Padova, Italy. ‡ Electrical Engi-neering and ACCESS Linnaeus Center, KTH Royal Institute of Technology,Osquldas V¨ag 10, 10044, Stockholm, Sweden. This work has been supportedby the University of Padova through the Junior Research Grant ENGINE. infrastructure and relying on online smart metering proce-dures [11]. A common trait of these techniques is that acoordinated and intelligent control of the distributed generationcapabilities (from renewables) holds the potential of enhancingthe electrical grid performance, ameliorating the aforemen-tioned problems and, at the same time, increasing the gridhosting capacity.In this paper, we target residential micro grids where someof the end-users behave as DERs, through the exploitation ofrenewable energy such as solar, wind, biomass, geothermal,etc. In these micro grids, each DER is normally equipped withan energy storage device (i.e., a battery) and it is assumed tofulfill its own power demand. In addition, during each networkcycle, DERs can independently decide to either sell part of thestored energy to the main power supplier, which is addressedas the Point of Common Coupling (PCC), or directly to someselected end-users (loads). Without any further regulation,DERs would sell all their energy to the agents ensuring thehighest revenue. However, this could lead to inefficient operat-ing points for the grid (e.g., high distribution power losses orinstability problems). Control techniques for DERs [12]–[16]can prevent this, by significantly reducing distribution powerlosses, relieving the PCC from some of the power load andgranting stability. According to these strategies, after local loadsatisfaction, end-users fine tune their energy injection into theelectricity grid so as to reduce the distribution power lossesand the total power demand from the mains.Nevertheless, previous control studies for electrical powergrids ignore that, in real-world scenarios, DERs energy in-jection’s attitude depends on economic advantages . Thus,previous approaches may not be viable in practice if notpaired with suitable market rules. New market models forthe smart grid have been studied so far in terms of demand-response control and dynamic pricing strategies. Some workaddressed the case where a single energy provider determinesthe best real time pricing policy, maximizing its own economicbenefit [17], [18] or a specific quality of service functionaccounting for the main supplier revenue and the aggregatedend-users experience [19], [20]. Other papers exploit dynamicpricing policies to control the power demand from end-users,thus reducing the chance of instability events as, for example,power outages [21], [22]. Moreover, effort has been devoted tothe definition of pricing models that enforce the efficiency ofspecific electrical optimization techniques, see, for example,[23]–[25]. Here, we recognize that real users are expectedto change their behavior and positively contribute to the gridoptimization if this leads to economic benefits (i.e., a monetaryincome). Moreover, we note that several electrical optimizationtechniques may already be deployed in the same micro grid, such as, peak shaving [26] or power loss minimization [27].Thus, we propose an optimization framework that jointlyaccounts for economic rewards (i.e., lowering the energy con-sumers expenses and guaranteeing higher profit to the DERs)and for the execution of a selected electrical optimizationtechnique (so to increase the energy efficiency of the powergrid and assure its stability). The proposed framework definesa market model that can be optimized to support any combi-nation of electrical optimization techniques, as long as theydetermine the amount of power that each DER has to injectat any given time. Also, as we discuss below, the proposedmodel does not require that preexisting contract terms andconditions for the electric supply are renegotiated. Hence, theapproach can be readily deployed with minimal modificationsto existing grids/regulations. Although we are aware that incurrent markets DERs sell their surplus energy to the PCC(i.e., direct selling to distributed users is not permitted), weassume that energy trading among end-users will be allowedin the future and we disregard regulatory restrictions. In fact,current research trends are promoting distributed architectureswhere end users trade energy in a peer-to-peer fashion [28],as we do here.The proposed market scenario is naturally formulated asa multi-objective optimization problem. Each grid user (i.e.,loads, DERs, and the PCC) is assumed to act as a rationalagent and, in turn, it always tries to maximize its own benefit.Hence, each DER maximizes its own profit, each load mini-mizes its own expense and the PCC aims at assuring the grid’selectrical efficiency. Energy can be traded directly among end-users (i.e., DERs and loads) or between end-users and thePCC. In this paper, after formally defining the optimizationproblem, its centralized solution is presented. In this case, thePCC acts as a central controller and regulator for the grid. Thissolution provides a complete description of the Pareto-optimal(P-optimal) trading strategy (i.e., energy prices, and energyallocation matrix) for each grid agent. In this centralizedcase, the PCC solves the problem and then distributes the P-optimal parameters to the network agents. Since the solutionis guaranteed to provide economic benefit to all the DERs andloads, it is in their best interest to adopt the P-optimal tradingstrategy. In particular, each DER receives from the PCC theenergy prices that such a DER should apply and the amountof energy it should sell to each load in the same grid. Theloads can then decide whether to buy energy from the DERs(according to the proposed prices) or from the PCC (accordingto a fixed common price). The PCC enforces the grid electricalefficiency by applying a discount policy to the price paid bythe loads when buying from DERs. After characterizing thecentralized solution, we present a distributed formulation ofthe problem. This is achieved through a transformation of theoriginal problem into an equivalent one, which is shaped asa general form consensus with regularization [29]. This newproblem can be efficiently solved in a decentralized mannerusing the alternating direction method of multipliers [29].We seek for a P-optimal policy that provides the best tradingstrategy (in terms of economic benefit) for each end-user,while also driving the system toward the best electrical condi-tion (according to a selected grid optimization technique). Weremark that the proposed model is transparent to the chosen G L L G PCC B B B B Fig. 1: Electrical network example, where B i , i = 1 , . . . , are the electrical distribution lines, L j , j = 1 , are the loads, G k , k = 1 , are the DERs, and the PCC is the Point ofCommon Coupling.grid optimization strategy. In fact, any electrical optimizationtechnique can be plugged into our framework as long as itprovides an optimal power allocation for the nodes. Hencethe applicability of our model does not reduce to a singlescenario and it does not exclude future improvements in termsof electrical optimization. Moreover, we allow the PCC (i.e.,the electrical utility) to set the importance of each performanceobjective, i.e., end-user revenue vs grid electrical efficiency.This is achieved by means of a maximum discount factor thatlimits the individual discount that can be applied in the energytrading between DERs and loads.The rest of this paper is structured as follows. Section IIintroduces the considered electrical / market scenarios and twouse cases for the proposed optimization framework. Section IIIpresents the mathematical notation for the market model, theassociated multi-objective optimization problem, its discussionand solution. In this section, we first show that the consideredoptimization problem is non-convex. Thus, a bijective trans-formation yielding a convex version of the original problemis found and the solution of the new convex problem isassessed. Finally, a decentralized approach based on ADMMis proposed. In Section IV, the electrical grid topology and theparameters used to obtain the numerical results are given. InSection V, the numerical results obtained through the setupof Section IV are shown. Finally, in Section VI, we drawthe conclusions of our work and discuss the validity of theproposed model.II. S YSTEM M ODEL AND U SE C ASES
In this section, the electrical, communication and marketscenarios are introduced. Some relevant use cases are de-scribed, showing how two electrical optimization techniquesfrom the literature can be plugged into the proposed optimiza-tion framework. In particular, we define the communicationrequirements and infrastructure needed to support the proposedmodel and discuss the interactions among the involved (ratio-nal) agents.
A. Electrical Scenario
We consider a steady-state low voltage power micro grid.For ease of computation, and without loss of generality, theconsidered grid is modeled as a directed tree. The root ofthe tree represents the Point of Common Coupling (PCC)and the other nodes represent loads and Distributed Energy In steady-state, the network has reached equilibrium and transient phe-nomena are no longer relevant.
Resources (DERs). Loads are either modeled by constantcomplex impedances or by constant current sources, the PCCis modeled as a voltage generator setting the voltage andphase references for the entire grid, while DERs are modeledeither as power or current generators. Here, we assume thatthe PCC is always able to supply the grid with the neededpower. Hence, no power outages or voltage instabilities (i.e.,overvoltages and voltage sags due to the DERs operations)can occur in the considered electrical setup. This model hasbeen widely considered in the literature, and in particular forpower loss minimization algorithms [15], [16], [30]–[32].Fig. 1 shows an example power grid is shown. DER i = 1 , and load j = 1 , are respectively denoted by G i and L j .Distribution lines are assumed to have a constant section [15],[16], and hence each line has a constant impedance per unitlength. The length of the z -th distribution line is denotedby B z . Each DER is equipped with a finite-size energystorage device (e.g., rechargeable battery). The size of theenergy storage devices determines the total amount of availablepower. Moreover, each DER is assumed to be feeding anassociated load and to have the capability of injecting partof its energy surplus into the grid. The surplus power thatDER G i can inject into the grid is denoted by E i . For thesake of terminology, the quantity E i will be referred to as G i ’s surplus energy. The amount of energy E i that each DERwishes to inject into the grid is not regulated by a centralauthority, but it depends on his local decision. This decision,in turn, depends on the specific energy storage policy that eachDER implements, as, for example, the minimization of theprobability of not being able to feed its associated load withina given time horizon. In this paper, it is assumed, withoutloss of generality, that E i ≥ ∀ i . Each load L j is assumedto have a non-negative power demand, which is denoted by D j ≥ . These assumptions are common in the literature, seefor example [14], [16], [26], [30]. B. Communication Scenario
Each node (i.e., loads, DERs and the PCC) in the gridis equipped with a transceiver, whose communication perfor-mance depends on the requirements of the selected electricaloptimization technique. These details are however neither con-sidered here nor fundamental to the solution of the presentedoptimization problem. In fact, our optimization framework isindependent of the specific communication technology andinfrastructure, as long as these allow a timely bi-directionalcommunication between each pair of nodes.
C. Market Scenario
We propose a market scenario where each DER can eithersell its surplus power to the PCC or directly to the loads.The monetary revenue that each DER obtains by selling (partof) its energy to the PCC is determined by a PCC-imposedunitary buying price . The monetary revenue that each DERobtains by selling its energy directly to a specific load isdetermined by a DER-imposed unitary selling price . EachDER can independently set a different selling price for eachload, i.e., this price is not controlled by the PCC. Also, eachload can fulfill its power demand by buying the needed power
Micro Grid ...
PCC buying price: γ PCC i PCC selling price: π PCC j Discount: s j,i Energy price: p i,j PCC G i DER i Load jL j Fig. 2: Market scenario example. Here, the PCC can buyenergy from DER G i paying unitary price γ PCC i , and it cansell energy to load L j for a unitary price π PCC j . Moreover, L j can buy energy from G i paying a discounted unitary price p i,j − s j,i , where the discount s j,i is imposed by the PCC.from the PCC or directly from the DERs. DERs and loadsare assumed to behave as rational agents . That is, each DERwill sell its power to the agents (PCC and loads) ensuring thehighest revenue, while each load will buy the power it needsfrom the agents (PCC and DERs) ensuring the lowest expense.Note that this trading model is consistent with the expectedevolution of the smart grid market [28].Fig. 2 shows an example for the considered market scenario.On the one hand, for each DER G i , the PCC determines theunitary price γ PCC i . This is the unitary price that the PCC payswhen buying power from DER G i . On the other hand, for eachload L j the PCC determines the unitary price π PCC j . This isthe unitary price that load L j pays when buying power fromthe PCC. The prices γ PCC i and π PCC j do not depend on theoptimization process. They are imposed by the PCC accordingto existing energy trading contracts. Each network agent mayhave different contract terms and conditions for buying andselling energy from and to the PCC. These conditions set thebaseline for the proposed optimization to determine the newtrading strategy. With respect to the current practice, whereeach DER sells E i energy to the PCC for γ PCC i revenue andeach load buys D j energy from the PCC spending π PCC j D j ,the new trading strategy results in economic benefits for thenetwork agents, while also enforcing the electric optimizationpolicy already in place. In the proposed market model, eachDER G i proposes a unitary price p i,j to each load L j . Theunitary price p i,j determines the monetary revenue that G i obtains when selling power to L j . In order to move the gridelectrical state toward the optimal solution (dictated by theselected electrical grid optimization technique), the PCC canapply a discount to the unitary prices p i,j proposed by theDERs to the loads. The discount proposed by the PCC to L j when buying from G i is denoted by s j,i and, in turn,the unitary price that L j pays to G i is p i,j − s j,i . Similarinteraction models have been previously used in the scientificliterature see, e.g., [33]. We remark that, since DERs andloads are assumed to be rational agents, any energy trading solution resulting in higher revenues for the DERs and in lowerexpenses for the loads with respect to the current practicewill be embraced. Here, we propose an optimization problemwhose solution guarantees better trading conditions for DERsand loads with respect to the current energy trading paradigmand, at the same time, enforces the operation of an electricaloptimization technique that is possibly already in place. Withrespect to the current energy market, the only regulatory act toenable the proposed optimized market is to allow direct energytrading between the grid agents. As noted before, this scenariois expected to become a solid reality in the near future.To optimize the electrical status of the power network, weallow the PCC to influence the amount of energy that the DERsinject into the grid. In particular, PCC can prevent DERs frominjecting energy by buying it. In this case, the energy boughtby the PCC is for example stored by the DERs into theirbatteries, and could be made available for future use. D. Use Cases i) In the first use case, we consider the current based sur-round control (CBSC) proposed in [15]. When this techniqueis implemented, the electrical network is divided into clusters.Each cluster is made of a pair of DERs connected such ason the path between them there are only loads. Within eachcluster, the DERs cooperate to feed the loads on the pathconnecting them. The framework proposed in this paper canbe utilized to enforce CBSC by defining market rules thatguarantee economic advantages for the loads and the DERs.First, the DERs in each cluster have to solve an electricaloptimization problem. To do that, the DERs collect the amountof power needed by the loads. Then, they determine theoptimal amount of power that they should inject into the grid tominimize the distribution losses as described in [15]. Once thisprocess is complete, the DERs send to the PCC the informationthey obtained (i.e., the optimal amount of power that theyshould inject and the power needed by the loads) together withthe maximum amount of power they are willing to sell (i.e.,to inject). The PCC collects this information and, by solvingthe optimization problem defined in this paper determines theprices p i,j and the discounts s j,i of Fig. 2 which guarantee that,when acting as rational agents, DERs and loads experienceeconomic benefits while minimizing the grid power losses.This process can be carried out according to different timescales (i.e., real time, on a minute basis, or on an hourlybasis), moreover the electrical optimization time scale andthe market one can be decoupled. For example, the electricaloptimization can operate in real time, while the market onecan operate on an hourly basis. In this case, DERs and loadsmust send to the PCC hourly generation and consumptiondata. This can be done in a day ahead fashion by forecastingthe power generation and consumption of the next hour ineach cluster. This information is then sent to the PCC, whosolves the electrical optimization problem assuming a constantpower consumption and generation during the next hour anddetermines the prices p i,j and the discounts s j,i to be appliedaccordingly. Once this is done, the electrical optimization isperformed in real time, but the energy trading strategy remainthe same until the next update of market quantities (prices). ii) The second use case deals with the peak shavingtechnique in [26]. This optimization requires that the DERssend to the PCC a day ahead forecast of the generated power.Once the PCC receives this information, it performs a dayahead forecast of the grid power consumption and, based onthese two predictions, it computes two parameters that arebroadcast to the DERs. Once they receive these parameters,the DERs determine the amount of active and reactive powerthat they will inject into the grid. Differently from CBSC, inthis case only the aggregated power consumption from theloads is required. Nevertheless, the computation of a suitabletrading strategy, through the optimization that we proposein this paper, requires one to know the power consumptionof each load. According to the growing diffusion of homedeployed smart meters [34], this fine-grained information iseasily gathered and is expected to be available in nearlyall future power grids. With this information, it is possibleto perform the market optimization proposed here. To dothat, the PCC determines the optimal amount of power thatthe DERs have to generate to prevent power consumptionpeaks. Moreover, it determines the amount of power that theDERs have to sell to the loads to fulfill their power demand.For example, consider Fig. 1 and assume that the electricaloptimization process dictates that DERs G and G inject E and E power, respectively. Moreover, assume that loads L and L will need D and D power, respectively. Then, apossible power allocation scheme could be: • min( E , D ) power is sold by G to D ; • min( E , D ) power is sold by G to D ; • the remaining available (needed) power is sold to (boughtfrom) the PCC.This information is then sent to the DERs and loads, respec-tively. After that, the PCC solves the optimization problempresented here, obtaining the prices p i,j and discounts s j,i of Fig. 2. As in the previous case, different time scales can beused for the electrical and market optimization processes.III. N OTATION AND M ULTI -O BJECTIVE O PTIMIZATION P ROBLEM
In this section, we present the first core contribution of thispaper, which consists in an original optimization modeling ofthe multi-agent system. Our goal is to propose an optimizedmarket model aiming at increasing the DERs monetary rev-enue and reducing the loads expenses while enforcing the gridelectrical efficiency. Given that these are differing optimizationobjectives, a multi-objective optimization problem is proposed.First, we introduce the mathematical notation that is usedthroughout the paper. Next, this problem is posed, character-ized and transformed into an equivalent convex formulation,which allows for a convenient computation of the optimalPareto frontier.
A. Notation
We now introduce the mathematical notation that is used forthe market model throughout the paper, and the multi-objectiveoptimization problem.Let G be the set of active DERs in the grid, |G| = G, G ∈ N ,where G is the cardinality of G . Let L be the set of active loadsin the grid, |L| = L, L ∈ N , where L is the cardinality of L . Domains:
Let P = { P ∈ R G × L ++ : p i,j ≤ P i , ∀ i ∈ G} (1)be the set of matrices P whose elements are the unitary prices p i,j that the DERs propose to the loads. The i, j element,for i = 1 , . . . , G and j = 1 , . . . , L , of matrix P is denotedby p i,j and represents the unitary price that G i proposes to L j , ∀ i ∈ G , ∀ j ∈ L . Let p i, · and p ] · ,j denote the i -throw and the j -th column of P , respectively. Moreover, let P i , ∀ i ∈ G be the PCC imposed maximum unitary price that G i can propose to the loads.Let H = { H ∈ R G × ( L +1)+ : L X j =0 h i,j = E i , ∀ i ∈ G} (2)be the set of matrices H representing the amount of powerthat the DERs can sell to each buyer (the loads or the PCC).The i, j element, for i = 1 , . . . , G and j = 1 , . . . , L , of matrix H is denoted by h i,j and represents the amount of power that G i sells to the buyer j (where j = 0 denotes the PCC and j = 1 , . . . , L denotes load L j ). With h i, · and h · ,j , we denotethe i -th row and the j -th column of H , respectively.Let D = { D ∈ R L × ( G +1)+ : G X j =0 d i,j = D i , ∀ i ∈ L} (3)be the set of matrices D representing the amount of power thatthe loads can buy from each seller (the DERs or the PCC).The i, j element, for i = 1 , . . . , L and j = 1 , . . . , G , of matrix D is denoted by d i,j and represents the amount of power thatload L i buys from the seller j (where j = 0 denotes the PCCand j = 1 , . . . , G denotes G j ). With d i, · and d · ,j , we denotethe i -th row and the j -th column of D , respectively.We aim at bounding the maximum expense from the PCCand, to this end, we introduce a further parameter α . Let S = { S ∈ R L × G + : s i,j ≤ αp j,i , ∀ i ∈ L , ∀ j ∈ G} (4)be the set of matrices S representing the discounts that thePCC applies to the unitary prices that the DERs proposeto the loads. We recall that the discount policy is meant todrive the electrical grid state toward the optimal one, whichis determined by a selected electrical optimization technique .The i, j element, for i = 1 , . . . , L and j = 1 , . . . , G , of matrix S is denoted by s i,j and represents the discount that the PCCis willing to apply to the unitary price p i,j that G j proposesto L i . Moreover, let ≤ α ≤ be the PCC defined maximumdiscount factor, i.e., the PCC is willing to discount at most (100 α )% for each proposed unitary price.Let H ∈ H , we define the index sets: ˜ H i, · = { k ∈{ , . . . , L } , k : h i,k = 0 } and ˜ H · ,j = { k ∈ { , . . . , G } , k : h k,j = 0 } . These two sets determine the row and columnindices, respectively, of the non zero elements of H .Similarly, for the demand we define ˜ D i, · = { k ∈ { , . . . , G } : d i,k = 0 } ˜ D · ,j = { k ∈ { , . . . , L } : d k,j = 0 } . (5)As above, these two sets respectively determine the row andcolumn indices of the non zero elements of D . B. Objective Functions
Each DER will support the proposed market model, asdescribed in Section II-C, only if it guarantees a highermonetary revenue with respect to solely selling its energy tothe PCC. We represent the monetary revenue of the DER i when selling its E i amount of energy to the loads, as specifiedby the vector h i, · and using the unitary prices defined by p i, · ,by the following equation U G i ( p i, · , h i, · ) = L X j =1 p i,j h i,j + ( E i − L X j =1 h i,j )) γ PCC i ( ∀ i ∈ G ) . (6)Note that with the second addend we model the fact that allthe excess energy from DER i that is not sold to the loads,will be bought by the PCC. This is, in fact, what occurs incurrent markets and what we also consider here.In contrast to the DERs behavior, each load will endorse theproposed market model only if it guarantees lower expenseswith respect to solely buying energy from the PCC. Usingthe demand vector d i, · and the discounted unitary prices p ] · ,i − s i, · , we represent the expense incurred by L i whenbuying energy from the DERs as U L i ( p ] · ,i , d i, · , s i, · ) = G X j =1 ( p j,i − s i,j ) d i,j + ( D i − G X j =1 d i,j ) π PCC i , ( ∀ i ∈ L ) . (7)We assume that an electrical grid optimization strategy isavailable and, through it, we obtain the optimal demands d ⋄ i, · , ∀ i ∈ L for the loads from an electrical standpoint.These are optimal in the sense that they will drive the gridtoward a certain desirable electrical state. Any optimizationscheme from the state of the art can be used to obtain theseoptimal demands, a possible technique is discussed shortly inSection IV.Within our market scenario, the PCC drives the power gridas close as possible to the optimal working point and, to thisend, it enforces a discount s i,j to each unitary price p j,i that G j proposes to L i . The effect of the discounts is determined, foreach load L i , by computing the squared distance between thechosen demand vector d i, · and the most electrically efficientone d ⋄ i, · . Such distance is computed through the followingequation: U PCC i ( d i, · ) = || d i, · − d ⋄ i, · || ( ∀ i ∈ L ) . (8)The goal of the PCC is then to determine a discount matrix S that minimizes the distance in Eq. 8. We recall that eachindividual discount s i,j is upper bounded by s i,j ≤ αp j,i . C. Constraints
The electrical state of the system induces a set of constraintsto account for the physical consistency of the grid. Moreover,an additional set of constraints limits the maximum pricesthat the DERs can propose to the loads, and the maximumdiscounts that can be applied to these prices. Next, theseconstraints are presented and discussed.
We impose that each DER G i ∈ G sells no more than itssurplus energy E i , by the following equation L X j =1 h i,j ≤ E i ∀ i ∈ G . (9)With this constraint we make sure that G i can not sell moreenergy than the amount remaining after fulfilling its ownneeds.We model the fact that the loads are not equipped withenergy storage devices, and hence each load must buy theexact amount of energy needed to fulfill its current powerdemand. This is implied by the following equation G X j =1 d i,j ≤ D i ∀ i ∈ L , (10)and by the second addend of Eq. (7). The reason for theinequality in Eq. (10) is because the loads are not requiredto buy all the energy they need from the DERs, but they canalso buy part of it from the PCC, which is referred to as d i, , ∀ i ∈ L . It follows that P Gj =0 d i,j = D i .Furthermore, the amount of energy that DER G i ∈ G isselling to load L j ∈ L must be equal to the amount of energythat L j is buying from G i , i.e., h i,j = d j,i ∀ i ∈ G , ∀ j ∈ L . (11)The limits imposed by PCC to the prices that the DERspropose to the loads are modeled by the following constraints p i,j ≤ P i ∀ i ∈ G , ∀ j ∈ L , (12)where the maximum prices act as market regulators, preventingthe prices from growing unboundedly and also determine themaximum unitary discount that the PCC is willing to apply.The maximum fraction of the unitary prices proposed bythe DERs that can be discounted by the PCC is set by theconstraint s i,j ≤ αp j,i ∀ i ∈ L , ∀ j ∈ G . (13) D. Optimization Problem
With the objective functions defined in Eqs. (6)-(8) andthe constraints of Eqs. (9)-(13), the following multi-objectiveoptimization problem can be formulated:min P , H , D , S U G i ( p i, · , h i, · ) − ∀ i ∈ G U L i ( p ] · ,i , d i, · , s i, · ) ∀ i ∈ L U PCC i ( d i, · ) ∀ i ∈ L (14a)s.t. L X j =1 h i,j ≤ E i ∀ i ∈ G , ∀ j ∈ L G X j =1 d i,j ≤ D i ∀ i ∈ L p i,j ≤ P i ∀ i ∈ G , ∀ j ∈ L s j,i ≤ αp i,j ∀ i ∈ G , ∀ j ∈ L h i,j = d j,i ∀ i ∈ G , ∀ j ∈ L . (14b)Solving the multi-objective optimization problem (14) doesnot lead to a unique solution, because the objective functions inEqs. (6)-(8) are contrasting. Hence, simultaneously minimizingthese objective functions leads to a set of solutions called Pareto Frontier (PF) . In this paper, we adopt the Pareto multi-objective optimality definition [35]. Next, we investigate thePareto-optimal (P-optimal) solution, with particular emphasison its domain and on the non-convexity of its objectivefunctions. With D ∗ , H ∗ , s ∗ i,j and p ∗ i,j we mean any solutionof (14) resting on the PF. Proposition 1:
Consider the optimization problem Eq. (14)and let p j,i (1 − α ) > π PCC i for some j ∈ { , . . . , G } and let d ∗ i, · be the i -th row of the P-optimal demand matrix D ∗ ∈ D .Then d ∗ i,j = 0 , ∀ s i,j ∈ ]0 , αp j,i ] and the P-optimal discountvalue s ∗ i,j admits infinite solutions. Proof:
Let p j,i (1 − α ) > π PCC i and let d ∗ i, · be the P-optimal demand vector for load i . If d ∗ i,j = 0 , then a newvector ¯ d i, · such that ¯ d i,j = 0 and ¯ d i, = d ∗ i, + d ∗ i,j can bedefined. It holds by construction that U L i ( p ] · ,i , ¯ d i, · , s i, · )
Consider optimization problem (14) and let p i,j < γ PCC i for some j ∈ { , . . . , L } , and let h ∗ i, · be the i -th row of the P-optimal allocation matrix H ∗ ∈ H . Then h ∗ i,j = 0 and the P-optimal discount value s ∗ j,i admits infinitesolutions. Proof:
Let p i,j < γ PCC i and let h ∗ i, · be the P-optimalallocation vector for DG i . If h ∗ i,j = 0 , then a new allocationvector ¯ h i, · such that ¯ h i,j = 0 and ¯ h i, = h ∗ i, + h ∗ i,j canbe defined. It is true, by construction, that U G i ( p i, · , ¯ h i, · ) >U G i ( p i, · , h ∗ i, · ) , but this is not possible because h ∗ i, · is P-optimaland hence h ∗ i,j = 0 .Proposition 2 states that if the revenue that G i obtains byselling its power to the PCC is greater than the maximumrevenue that can be obtained by selling it to L j , then againthere is no way for the PCC to enforce the electrical gridefficiency. To summarize, Propositions 1 and 2 imply that,in order for the PCC to enforce the grid electrical efficiencythrough a discount policy, the following conditions must hold ∀ i ∈ G , j ∈ L : p i,j ≥ γ PCC i and p i,j (1 − α ) ≤ π PCC j (15)We recall that, as discussed in Section II-C, the PCC is notallowed to act on γ PCC i and π PCC i as they depend on preexistingcontract terms and conditions. Hence, the only way it can actto promote the electrical efficiency of the grid is to allowthe DERs to have a higher revenue through selling energyto a specific load, dictated by the joint optimization that isproposed here, as opposed to selling it to any other loads orto the PCC itself. To do this, the PCC discounts the pricesthat DERs propose. However, to limit its expenses the PCCimposes a maximum discount factor (i.e., α ) that dictates thelimit to within the DERs proposed prices can grow before theloads (being rational agents) stop buying energy from them,i.e., when the price paid to buy energy from a DER is higherthan buying it from the PCC ( p i,j (1 − α ) > π PCC j ). Proposition 3:
Consider optimization problem (14) and let h ∗ i,j , d ∗ j,i respectively be the P-optimal i, j allocation and j, i demand values (according to the respective indexing). Then,either one of the following holds:1) h ∗ i,j = d ∗ j,i = 0 if p i,j (1 − α ) > π PCC j or p i,j < γ PCC i ;2) h ∗ i,j = d ∗ j,i = 0 otherwise . Proof:
By considering Propositions 1 and 2, and recallingthat both the DERs and the loads are rational agents, we seethat the only case where it is economically convenient for G i to sell power to L j is when it can get a higher revenue thanthe one it would obtain selling the same amount of power tothe PCC. This concludes the proof.Proposition 3 follows from Propositions 1 and 2. It states thatthe PCC can enforce the grid efficiency only if the conditionsof Eq. (15) are met. Moreover, if these conditions are met, therational behavior for DERs and loads will be to adhere to thediscount policy proposed by the PCC and trading energy withthe agents guaranteeing higher revenues and smaller expensesfor the DERs and loads, respectively. Proposition 4:
Consider the optimization problem Eq. (14)and let p ∗ i,j be the P-optimal i, j price value for the optimiza-tion problem. Let h ∗ i,j = d ∗ j,i = 0 . Then, it must hold that1) π PCC j < p ∗ i,j − s j,i for at least one value of s j,i ;2) p ∗ i,j > γ PCC i . Proof:
By considering Propositions 1-3 and recalling thatloads are rational agents, we see that the only case where L j will buy power from G i is when the discounted price proposedby G i is lower than the price it would pay to the PCC.Moreover, by recalling that DERs are also rational agents, wesee that the only case in which G i will sell power to L j is theone where its revenue is higher than the one it can get fromthe PCC. This concludes the proof.Proposition 4 descends from Proposition 3. It states that, inorder for the DERs and loads to adhere to the proposed model,the discounts (limited to a fraction α of the proposed prices)must meet the rational behavior of the trading agents.Propositions 1-4 characterize the PF of problem (14). Fromthese propositions, it follows that, by construction, no solu-tion on the PF can lead to situations where 1) some DERsexperience smaller revenues with respect to the case where allthe energy is sold to the PCC or 2) some loads experiencehigher expenses with respect to the case where all the energyis bought from the PCC. Hence, network agents deciding toadopt the proposed optimized market model have no reasonnot to accept the trading strategy resulting from the solutionof the optimization problem (14).In the following proposition, we show that the domainsdefined through Eqs. (1)-(4) are convex. Proposition 5:
The sets defined in Eqs. (1)-(4) are convexwith respect to the matrix sum operation.
Proof:
Let P , P ∈ P and ≤ θ ≤ . Let P = θ P + (1 − θ ) P , then L X j =1 p i,j = θ L X j =1 p i,j + (1 − θ ) L X j =1 p i,j . (16)Since P , P ∈ P , it holds true that ∀ i ∈ G , L X j =1 p i,j ≤ P i and L X j =1 p i,j ≤ P i (17) hence L X j =1 p i,j ≤ θP i + (1 − θ ) P i = P i , ∀ i ∈ G , (18)thus P ∈ P .The convexity of sets H , D and S can be shown by similararguments.Next, we show that although the domains are convex themulti-objective optimization problem of Eq. (14) is not. Proposition 6:
Optimization problem (14) is not convex.
Proof:
In order to prove the non convexity of Eq. (14)it is sufficient to show that one of its objective functions isnot convex. Considering U G i ( p i, · , h i, · ) , since U G i ( p i, · , h i, · ) is twice differentiable in its domain the Hessian matrix Φ U G i ( p i, · , h i, · ) can be computed: Φ U G i ( p i, · , h i, · ) = (cid:20) A BB A (cid:21) , (19)where A ∈ { } L × L and B is the L × L identity matrix. Φ U G i ( p i, · , h i, · ) is a permutation matrix. Let z ∈ R L and let z , z ∈ R L : z T = [ z T z T ] , then z T Φ U G i ( p i, · , h i, · ) z = [ z T z T ] z , (20)and hence Φ U G i ( p i, · , h i, · ) is not positive semidefinite nor it isnegative semidefinite.According to Proposition 6, solving problem (14) with stan-dard convex multi-objective solution methods could not leadto the actual PF. In the following subsection, a transformationof problem (14) is proposed, which establishes an equivalentconvex optimization problem whose solutions are the sameas those of problem (14). The convexity of the new problemallows the application of standard solution methods. E. Geometric Programming Formulation
Since the DERs’ and loads’ objective functions can beexpressed in posynomial form, part of the non-convex multi-objective minimization problem (14) can be formulated as ageometric programming problem [36]–[38]. Next, the stepsleading to this transformation will be presented and discussed.First, we can express Eq. (6) in the form U G i ( p i, · , h i, · ) = L X j =1 p i,j h i,j + h i, γ PCC i ∀ i ∈ G . (21)Now, we consider the following definitions: α G j = ( if j = 01 otherwise , c G ij = ( γ PCC i if j = 01 otherwise (22)and let p i, ∈ R ∀ i ∈ G . Then, Eq. (21) can be rewritten as aposynomial function: U G i ( p i, · , h i, · ) = L X j =0 c G ij p i,jα G j h i,j , ∀ i ∈ G . (23)Similarly, Eq. (7) can be re-formulated as U L i ( p ] · ,i , d i, · , s i, · ) = G X j =1 ( p j,i − s i,j ) d i,j + d i, π PCC i , (24) ∀ i ∈ L . If we define α L j = ( if j = 01 otherwise and c L ij = ( π PCC i if j = 01 otherwise , (25)Eq. (24) can also be formulated as a posynomial function: U L i ( p ′ · ,i , d i, · , s i, · ) = G X j =0 c L ij p ′ j,iα L j d i,j , ∀ i ∈ L . (26)If we apply the geometric programming transformationdetailed in [37], Eq. (23) and Eq. (26) can be transformedinto convex functions: U ′ i G ( p i, · , h i, · ) = X j ∈ ˜ H i, · e α G j log p i,j +log h i,j +log c G ij , (27) U ′ i L ( p ′ · ,i , d i, · , s i, · ) = X j ∈ ˜ D i, · e α L j log p ′ j,i +log d i,j +log c L ij . (28)Based on Eqs. (27) and (28), the non-convex multi-objectiveoptimization problem (14) can be transformed into a convexmulti-objective optimization problem: min P , H , D , S − log (cid:16) U ′ i G ( p i, · , h i, · ) (cid:17) ∀ i ∈ G log (cid:16) U ′ i L ( p ′ · ,i , d i, · , s i, · ) (cid:17) ∀ i ∈ L U PCC i ( d i, · ) ∀ i ∈ L (29a)s.t. constraints in Eq. (14b) p ′ j,i = p j,i − s i,j ∀ i ∈ L , ∀ j ∈ G . (29b)Note that this optimization problem is equivalent to (14) in thesense that the P-optimal solutions of problem (29) are identicalto those of problem (14). F. Solution
Given the convexity of optimization problem (29), whosesolutions are identical to (14), its solution can be obtainedthrough standard convex solvers. Since problem (29) is convex,the duality gap is zero and the Karush-Kuhn-Tucker (KKT)optimality conditions can be applied to the scalarized formof problem (26). To do so, let λ = [ λ , λ , . . . , λ G +2 L ] T ∈ [0 , G +2 L : P G +2 Li =1 λ i = 1 , then the scalarized objectivefunction is U ( P , H , D , S ) = − G X i =1 λ i log (cid:16) U ′ i G ( p i, · , h i, · ) (cid:17) + L X i =1 λ i + G log (cid:16) U ′ i L ( p ′ · ,i , d i, · , s i, · ) (cid:17) + L X i =1 λ i + G + L U PCC i ( d i, · ) . (30) Then, the scalarized convex minimization problem can bedefined min P , H , D , S U ( P , H , D , S ) (31a)s.t. L X j =1 h i,j ≤ E i ∀ i ∈ G , ∀ j ∈ L G X j =1 d i,j ≤ D i ∀ i ∈ L p i,j ≤ P i ∀ i ∈ G , ∀ j ∈ L s j,i ≤ αp i,j ∀ i ∈ G , ∀ j ∈ L h i,j = d ji ∀ i ∈ G , ∀ j ∈ L p ′ j,i = p j,i − s i,j ∀ i ∈ L , ∀ j ∈ G . (31b)Eq. (31) is a standard convex minimization problem, hence,if the problem is feasible, a P-optimal solution is guaranteedto exist ∀ λ ∈ [0 , G +2 L : P G +2 Li =1 λ i = 1 . The Paretofrontier is the set of all the P-optimal solutions to problem(31) obtained for every possible weight vector λ . We recall thatall the points in the Pareto frontier are equally P-optimal (in thesense that all these solutions yield the same value of the scalar-ized objective function), it is up to the decision maker (i.e.,the PCC) to determine the particular weight vector satisfyingher/his own needs. In the following results, λ was heuristicallychosen to guarantee that the best electrical working point isreached for low values of the discount factor α . The rationale isto drive the grid toward the wanted electrical operating pointby maintaining the expenses incurred by the PCC low. Asdiscussed in Section III-D, all the solutions of problem (31)lying on the PF guarantee economic benefits to all the DERsand the loads participating in the proposed market. Hence,regardless of the specific λ that is selected, all the networkagents will benefit from embracing the corresponding P-optimal solution. The choice of λ determines the entity ofthe benefit that each individual agent will achieve thanks tothe optimization process. In the proposed model, it is thePCC that selects, during each optimization phase, the mostsuitable λ with respect to each specific scenario. For example,in certain scenarios it might be appropriate to specificallyenforce the electrical optimization in some regions of thegrid. In such a case, the components of λ associated withagents connected to these particular regions will be higherthan the others. By doing so, minimizing these functions willhave a greater impact on the overall minimization with respectto the functions of agents in other portions of the grid. Weremark that the choice of the vector λ determines the particularsolution on the PF, but it is not part of the optimizationprocess. G. Distributed Solution
Here, we present a decentralized solution of problem(31) exploiting the alternating direction method of multipli-ers (ADMM) [36]. Consider the function U ′ i G ( p i, · , h i, · ) ofEq. (27), and, for the ease of notation, define y i,j = e α G j log p i,j +log h i,j +log c G ij ∀ j ∈ ˜ H i, · . (32)Then, Eq. (27) can be rewritten as U ′ i G ( p i, · , h i, · ) = X j ∈ ˜ H i, · y i,j = U ′G i ( y i, · ) , (33) and, for the Jensen’s inequality: log | ˜ H i, · | U ′G i ( y i, · ) ! ≥ | ˜ H i, · | X j ∈ ˜ H i, · log ( y i,j ) , (34) ∀ i ∈ G . Proposition 7:
The feasible vector y ∗ i, · = argmax y i, · X j ∈ ˜ H i, · log ( y i,j ) , (35)subject to the constraints of problem (29), is unique andmaximizes U ′G i ( y i, · ) . Proof:
Since the logarithm is a strictly concave function,the sum of logarithms is strictly concave. Hence, y ∗ i, · is unique.Since y ∗ i, · is unique, it follows that ∀ feasible y i, · = y ∗ i, · y i,j ≤ y ∗ i,j ∀ j ∈ ˜ H i, · . Then, ∀ feasible y i, · = y ∗ i, · X j ∈ ˜ H i, · y i,j ≤ X j ∈ ˜ H i, · y ∗ i,j . (36)This concludes the proof.As a consequence of Proposition 7, the feasible solution thatmaximizes the right-hand side of Eq. (34), also maximizes theoriginal objective function log (cid:16) U ′G i ( y i, · ) (cid:17) . Define ˙ U G i ( y i, · ) = X j ∈ ˜ H i, · log ( y i,j ) , ∀ i ∈ G . (37)Similarly, define ˙ U L i ( z i, · ) = X j ∈ ˜ D i, · log ( z i,j ) , ∀ i ∈ L , (38)where z i,j = e α L j log p ′ j,i +log d i,j +log c L ij . For the same argumentof Proposition 7, the vector that minimizes ˙ U L i ( z i, · ) is uniqueand also minimizes log (cid:16) U ′L i ( y i, · ) (cid:17) . By substituting U ′G i ( y i, · ) and U ′L i ( z i, · ) with ˙ U G i ( y i, · ) and ˙ U L i ( z i, · ) , respectively, inEq. (30), the objective function of problem (31) becomesseparable with respect to the interactions between any pairof network agents. In the following, we show how to exploitthis separability to express problem (31) as a general formconsensus with regularization [29]. Consider a smart grid with K > branches departing from the PCC. Define region R k as the set of grid agents (i.e., loads and DERs) connected tothe PCC through the k -th branch. Then, the DER set G canbe partitioned into K subsets G R k , k = 1 , . . . , K such that G R ∪ . . . ∪ G R K = G , and G R ∩ . . . ∩ G R K = ∅ where the set G R k is the set of DERs belonging to region k . Likewise, the setof loads L can be partitioned into K subsets L R , . . . , L R K .According to these partitions, and assuming that g ∈ G R k and l ∈ L R k , ˙ U G g ( y g, · ) and ˙ U L l ( z l, · ) can be rewritten as: ˙ U G g ( y g, · ) = X j ∈ ˜ H g, · ∩G Rk log ( y g,j ) + X j ∈ ˜ H g, · ∩ ( G\G Rk ) log ( y g,j ) , (39)and ˙ U L l ( z l, · ) = X j ∈ ˜ D l, · ∩L Rk log ( z l,j ) + X j ∈ ˜ D l, · ∩ ( L\L Rk ) log ( z l,j ) , (40)respectively. For the ease of notation, Eq. (39) can be rewrittenas ˙ U G g ( y g, · ) = ˙ U G Rk g ( y kg, · ) + ˙ U G\G Rk g ( y ˘ kg, · ) , (41) where the first term in the RHS corresponds to the first sumin the RHS of Eq. (39), and the second term corresponds tothe second sum in the RHS of Eq. (39), where y kg, · modelsinteractions between g ∈ G R k and agents in the same region R k , while y ˘ kg, · models interactions across regions. The samedecomposition can be applied to loads ( L ) and the PCC,leading to: ˙ U L l ( z l, · ) = ˙ U L k l ( z kl, · ) + ˙ U L\L k l ( z ˘ kl, · ) ,U PCC l ( d l, · ) = U PCC l ( d kl, · ) + U PCC l ( d ˘ kl, · ) . (42)The separability of all the objective functions allows defininga scalarized optimization problem whose solution is equivalentto the one of problem (31). This can be done by defining thenew scalarized objective function as ˙ U ( P , H , D , S ) = K X k =1 X i ∈G Rk λ i ˙ U G Rk i ( y ki, · ) + X i ∈L Rk (cid:16) λ i + G ˙ U L k l ( z ki, · ) + λ i +2 G U PCC l ( d ki, · ) + K X k =1 X i ∈G λ i ˙ U G\G Rk i ( y ˘ ki, · ) + X i ∈L Rk (cid:16) λ i + G ˙ U L\L k i ( z ˘ ki, · ) + λ i +2 G U PCC i ( d ˘ ki, · ) , (43)which can be rewritten in compact form as: ˙ U ( P , H , D , S ) = K X i =1 ˙ U R k ( V k ) + ˙ U cross ( W ) , (44)where V k is a vector containing the portions of P , H , D , S pertaining only agents in region R k , ˙ U cross ( W ) contains thecross-terms modeling the interactions across regions, while W = ( P , H , D , S ) . For construction, V k is independent ofany other V r : r = k . Replacing Eq. (30) with Eq. (44) intoproblem (31) and adding the constraints that guarantee thateach V k is consistent with W , generates a new problem thatis equivalent to the previous one but is posed as a general formconsensus with regularization. This problem can be efficientlysolved in a decentralized fashion using ADMM, as shownin [29], [39]. We recall that, since the new considered problemis convex, ADMM is guaranteed to converge to the uniqueP-optimal solution (for each given λ ). Hence, the tradingstrategies that are obtained through this decentralized approachare the same as those obtained through the centralized one.In the following section, we present numerical results tocharacterize the P-optimal solutions of our original multi-objective optimization problem (14) based on the establishedequivalent scalarized problem (31).IV. S IMULATION S ETUP
In this section, we present the electrical grid topology andthe electrical scenarios, in terms of power demand at the loadsand surplus energy at the DERs, that will be considered forour subsequent performance analysis of Section V.We consider the electrical grid of Fig. 1 as a case study. Todetermine the optimal power demand matrix, i.e., d ⋄ i, · , ∀ i ∈ L , we selected the Current Based Surround Control algorithm(CBSC) [15]. The reason why the CBSC algorithm has beenchosen is twofold. On the one hand, it drives the grid towardits theoretically optimal working point, and hence it allowsthe assessment of the optimization process ability to drivethe power grid toward its maximum electrical efficiency. Onthe other hand, the communication infrastructure requirementsneeded to implement CBSC are the same needed to implementthe proposed optimization strategy. Both techniques, indeed,require that each node is equipped with a smart metering de-vice (to determine the exact power availability, power demandand line impedance) and a transceiver (to communicate themeasured data and implement control actions). Lastly, CBSCwas proven to be very efficient and to lead to optimal results interms of power loss minimization along the distribution lines.Still, we also recall that other optimization techniques can beused in combination with our optimization framework.CBSC groups the nodes into clusters . Clusters are definedby checking, for any pair of DERs, whether their connectingpath includes any other DER or the PCC. If this is not thecase, a cluster is defined as the set containing the two DERs,the associated nodes, and all the nodes between them in theelectrical network topology. For each cluster, the DER thatis closest to the PCC is elected as the cluster head (CH).In the case when one of the two DERs in the cluster isthe PCC, this is elected as the CH (i.e., we assume that thePCC has better communication and computational resourceswith respect to the other nodes). The current injected foroptimization purposes is scaled by a real factor ≤ ξ ≤ .If we refer to I C as the total current demand in the cluster,the currents injected by the two DESs therein are ξI C and (1 − ξ ) I C . The parameter ξ is determined for each clusteraccording to the instantaneous power demand from its loadsand their branch impedances. Hence, this technique requiresthat every node is a smart node (i.e., equipped with metering,communication and control capabilities).According to CBSC, the optimal power allocation matrix,for the considered grid topology, is D ⋄ = B D B + B D , (45)where with B i , i = 1 , . . . , we indicate the length of thedistribution lines, see table Tab. I, whereas D and D arethe power demands associated with the two loads L and L ,which have been set to D = D = 100 kW. According toEq. (45) and Tab. I, the optimal power demand matrix is: D ⋄ = (cid:20) (cid:21) . (46)Given the optimal power demand matrix of Eq. (45), threeelectrical scenarios have been considered: 1) the first scenariois referred to as tight power offer as it addresses the case wherethe individual surplus energy for each DER equals the totalenergy that it should inject according to CBSC. 2) The secondscenario is referred to as unbalanced tight power offer . Here,the total surplus energy equals the optimal one that shouldbe injected, but the individual surplus energy does not matchthat dictated by CBSC. In this case, the optimal electrical gridconditions can not be reached. 3) The third is referred to as TABLE I: Distribution lines length in meters B B B B m m m m TABLE II: DERs surplus energy ( E i ) Scenario 1 Scenario 2 Scenario 3 G G loose power offer . This scenario considers the case where thetotal surplus energy exceeds the total energy demand.The DERs surplus energy for each considered scenario isshown in Tab. II. V. R ESULTS
In this section, we discuss the P-optimal solutions obtainedthrough the proposed optimization approach to the case studyof Section IV.For each scenario, the performance of the optimizationprocess has been assessed using the following metrics: • the DERs gain, obtained asgain = revenue opt − revenue no revenue no × , where “revenue opt ” is the aggregated revenue of DERswhen our joint optimization is used, whereas “revenue no ”corresponds to the DERs aggregated revenues in thenon-optimized case, i.e., where the surplus power isentirely sold to the PCC; • the loads gain, obtained asgain = expense no − expense opt expense no × , where “expense opt ” is the aggregated expense of loadswhen our joint optimization is used, whereas “expense no ”corresponds to the loads aggregated expense in thenon-optimized case, i.e., where the needed power isentirely bought from the PCC; • the achieved electrical efficiency with respect to thetheoretical optimal working point, achieved by CBSC.Moreover, we set γ PCC i = 20 , ∀ i ∈ G and π PCC i = 50 , ∀ i ∈L . This models the real-world scenario where the PCC buysenergy for less than what it sells it for. Given these values,the prices p i,j can range from to α ) . To select oneparticular solution on the PF we chose a specific vector λ .This vector has the following properties: • functions of the same class have the same weight, i.e., λ = λ (for the DERs), λ = λ (for the loads), and λ = λ (for the PCC); • λ = 0 . λ ; • λ = 0 . λ .By doing so, we give the higher weight to the functionsenforcing the electrical efficiency, while the revenue of theDERs becomes less important in the global optimizationprocess. Next, we discuss the performance for the threescenarios identified in Section IV. G a i n [ % ] α DERsLoads
Fig. 3: Aggregated gains for revenues (DERs) and expenses(loads). Tight power offer case.
A. Scenario 1: Tight Power Offer
In the tight power offer scenario, G and G sell the exactamount of power dictated by the PCC.Fig. 3 shows the DERs and the loads aggregate gains (withrespect to the case in which no optimization is performed)obtained through the proposed optimization method, i.e., solv-ing (14) for a discount factor α ranging from to .We recall that α is used as a free parameter to bound themaximum expense from the PCC, according to the proposeddiscount strategy. A first noticeable result is that, for everyvalue of α , the optimized aggregate revenue is always largerthan that in the non-optimized case. Moreover, the aggregateexpense is always smaller than in the non-optimized one.These facts are highly desirable, since they guarantee thatendorsing the proposed market model leads to a substantialeconomic convenience for all the agents involved in the energytrading process.When computing the distance from the electrically efficientcondition, the norm of of the difference d i, · − d ⋄ i, · is computedfor each load i ∈ L . The plotted distance is thus the sumof the L individual distances. Fig. 4 shows this distance for α = 10% , . . . , . We emphasize that, as the maximumdiscount factor reaches , the electrical efficiency obtainedthrough the proposed optimization equals the theoretical opti-mal electrical efficiency obtained through CBSC.Remarkably, Fig. 3 and Fig. 4 show that for a maximumdiscount factor of , the P-optimal solution reaches themaximum achievable electrical efficiency, while doubling theaggregate revenue of DERs with respect to the non-optimizedcase. At the same time, the consumers will incur sensiblysmaller expenses. B. Scenario 2: Unbalanced Tight Power Offer
In the unbalanced tight power offer scenario, G is willingto sell more power than the amount dictated by CBSC, while G sells less power than what dictated by CBSC.Fig. 5 shows the DERs and the loads aggregate gainsobtained through the proposed optimization when the maxi-mum discount factor α varies from to . As in the D i s t an c e α Optimized Distance
Fig. 4: Electrical efficiency in terms of distance from optimalelectrical working point. Tight power offer case. G a i n [ % ] α DERsLoads
Fig. 5: Aggregated gains for revenues (DERs) and expenses(loads). Unbalanced tight power offer case.previous case, endorsing the proposed optimization will leadto economical benefits for both the DERs and the loads.Fig. 6 shows the distance between the power demand matrixobtained through the proposed optimization and the optimalone obtained through CBSC. The considered scenario does notallow to reach the theoretical optimal electrical efficiency. Asa matter of fact, even though the total available power equalsthat required by CBSC, G has more available power thanwhat is needed, while G has less. Hence, no configurationexists for which the power allocation matrix obtained throughthe proposed optimization approach can match the optimalpower demand matrix. It can nevertheless be noted that, fora maximum discount factor of α = 20% , the optimizationprocess reaches the minimum achievable distance from thetheoretical optimal working point. In contrast with the previouscase, in this scenario there exists, for the selected weightvector λ , a single maximum discount factor that allows tomaximize the electrical grid efficiency (i.e., the one leadingto the minimum in Fig. 6). In fact, configurations exist whereDERs and loads individual interests drive the grid toward anon-optimal power allocation condition, i.e., G , instead of
10 11 12 13 14 15 160.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 D i s t an c e α Optimized Distance
Fig. 6: Electrical efficiency in terms of distance from optimalelectrical condition. Unbalanced tight power offer case. G a i n [ % ] α DERsLoads
Fig. 7: Aggregated gains for revenues (DERs) and expenses(loads). Loose power offer case.selling kW to the PCC, it starts trading with L leading toa sub-optimal electrical efficiency.As for the previous case, Fig. 5 and Fig. 6 show thatthe proposed optimization always ensures economical benefitsfor DERs and loads while, at the same time, leading to anincreased electrical grid efficiency. C. Scenario 3: Loose Power Offer
In the loose power offer scenario, G sells more power thanwhat dictated by CBSC, while G sells the exact amount ofpower dictated by the CBSC algorithm.Fig. 7 shows the performance of the proposed optimizationin terms of the aggregated gains obtained by the DERs andthe loads. As for the previous cases, we see that the proposedoptimization always guarantees higher revenues and smallerexpenses with respect to the case where the PCC is the onlyagent trading electrical power, i.e., all power has to be uniquelysold to or bough from the PCC.Fig. 8 shows that for α = 20% the optimal electricalworking point is reached. In this case, G sells kW to D i s t an c e α Optimized Distance
Fig. 8: Electrical efficiency in terms of distance from optimalelectrical condition. Loose money case. L and the remaining available power is sold to the PCC. As α grows, G starts selling more power to L and hence thedistance from the optimal electrical condition starts increasing.As for the previous case, for the selected weight vector λ , asingle value of α exists for which the electrical efficiency ismaximized, i.e., the distribution power losses are minimized.The presented results show that, for every considered powerconfiguration, the proposed optimization approach results insubstantial economical benefits (even though the objectivefunctions of the DERs were given the smallest weight inthe global optimization process) and is likely to drive thepower grid toward its maximum electrical efficiency. It isworth noting that, in the considered examples, the discountfactor that is required to reach the electrical grid efficiency isnever higher than . This is appealing as it shows that themaximum discount remains rather small, irrespective of thenetwork configuration. This may be especially convenient forthe grid operator in practical scenarios.VI. C ONCLUSIONS
In this paper, an original market model for smart gridswas presented. The proposed framework jointly accounts forend users economical benefits and electrical grid efficiencymaximization. This model was formally described as a nonconvex multi-objective optimization problem, which was thentransformed into a convex one through a bijective transforma-tion based on geometric programming. Pareto-optimal tradingand discount policies were devised through the solution ofthe equivalent convex formulation. Both a centralized anddecentralized solution have been devised. The performanceof the proposed market model was then assessed in termsof electrical efficiency for the power grid and achievableeconomical benefit for all involved actors, i.e., profit madeby DERs and expense incurred by the loads. Several networkconfigurations were considered so as to systematically test theefficacy of the proposed market model. Numerical results showthat considerable economical benefits can be reached for allagents and that the micro grid can be concurrently driven toward an optimal working point through the use of smalldiscount factors from the regulating authority.R EFERENCES[1] EIA, “Annual Energy Outlook 2013 with Projections to 2040,” Office ofCommunications, EI-40 Forrestal Building, Independence Avenue, S.W.Washington, DC 20585, 2013.[2] IEA, “Key World Energy STATISTICS,” 9 rue de la F´ed´eration, 75739Paris Cedex 15, France, 2013.[3] S. Goel, S. F. Bush, and D. Bakken, Eds.,
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