Distributed Vehicle Grid Integration Over Communication and Physical Networks
aa r X i v : . [ ee ss . S Y ] A ug Distributed Vehicle Grid Integration OverCommunication and Physical Networks
Dimitra Apostolopoulou, Rahmat Poudineh, and Anupama Sen
Abstract —This paper proposes a distributed framework forvehicle grid integration (VGI) taking into account the communi-cation and physical networks. To this end, we model the electricvehicle (EV) behaviour that includes time of departure, time ofarrival, state of charge, required energy, and its objectives, e.g.,avoid battery degradation. Next, we formulate the centralisedday ahead distribution market (DADM) which explicitly rep-resents the physical system, supports unbalanced three phasenetworks with delta and wye connections, and incorporates thecharging needs of EVs. The solution of the centralised marketrequires knowledge of EV information in terms of desired energy,departure and arrival times that EV owners are reluctant inproviding. Moreover, the computational effort required to solvethe DADM in cases of numerous EVs is very intensive. Assuch, we propose a distributed solution of the DADM clearingmechanism over a time-varying communication network. Weillustrate the proposed VGI framework through the 13-bus, 33-bus, and 141-bus distribution feeders.
Index Terms —electric vehicle charging, unbalanced threephase network, distributed optimisation, uncertainty
I. I
NTRODUCTION
A rapid increase in the adoption of electric vehicles (EVs)has been recorded worldwide in the last years. This is partlydue to emissions reduction goals, e.g., the UK has committedto reduce its emissions by 80% by 2050; and the decreasingprices of lithium ion batteries. However, there are severalobstacles that need to be surpassed in order to promote a vastadaptation of EVs. In particular, the power consumption oftypical household appliances, e.g., washing machines, refrig-erators, is very different compared to that of an EV charger.It has been shown in [1] that an uncontrolled EV chargingscheme would increase the peak national demand in the UKby 20 GW assuming all vehicles were electrified. Moreover,the after diversity maximum demand in distribution systemsat a house level increases by 1 kW with the integration of anEV [2]. These effects and the associated costs with upgradingthe electric power network (e.g., [3]) may be mitigated ifinstead of uncontrolled charging, active charging techniquesare used. In unidirectional active charging, EVs can modulatethe charging power; and in the bidirectional case, EVs canalso inject power back to the grid. We refer to the first oneas Vehicle Grid Integration (VGI) and the latter is knownas Vehicle to Grid (V2G) [4]. VGI may be seen as anintermediate solution between uncontrolled charging and V2Gthat requires less communication infrastructure. An additionalbenefit of VGI is that it is more considerate towards the EV
D. Apostolopoulou is with the Department of Electrical and ElectronicEngineering at City, University of London, London, UK EC1V 0HB. E-mail:
R. Poudineh and A. Sen are with the Oxford Insti-tute for Energy Studies, Oxford, UK OX2 6FA. E-mail: { Rahmat.Poudineh,Anupama.Sen } @oxfordenergy.org battery compared to V2G. Under a VGI framework EVs mayoffer services to the transmission system operator, distributionsystem operator and facilitate the integration of renewableresources. For instance, EVs can offer peak shaving, powerlosses reduction, voltage regulation and frequency controlin distribution systems (see, e.g., [5]–[7]). The services thatmay be provided by an individual EV owner are small whencompared to the large scale complex power system. However,an EV aggregator, who is in charge of operating the chargingschedule of numerous EVs, may offer services to the gridranging from kWs to MWs.In this paper we present a framework for a network-awaredistributed VGI. More specifically, we propose a methodologyto coordinate the services and operational constraints of threeentities: the EV owner, the EV aggregator and the Distributionsystem operator (DSO). Each of these players has a differentobjective and privacy concerns. To this end, we formulate anoptimisation problem that represents the day ahead distributionmarket (DADM) where the EVs and the aggregator participate.We explicitly model the EV behavior in terms of time of arrivaland departure, required charging energy, and its objectives aswell as the network constraints of an unbalanced three-phasedistribution system. A centralised structure of the DADMby the DSO requires knowledge of EV information that EVowners are reluctant in providing. In this regard, we propose adistributed solution of the DADM clearing mechanism wherethe entities exchange limited insensitive information over atime-varying communication network, as is in reality, untilthey reach consensus. One challenge in this setting is thatthe aggregator would not be certain of the number of EVsavailable at a certain time instant, their state of charge andavailable energy, their arrival and departure times. As suchwe propose a methodology to provide the aggregators witha given confidence on the amount of capacity that theyhave available to participate in the market. In the numericalresults section, we use the proposed framework in large-scaledistribution feeders using realistic data for the EV behaviourand demonstrate its applicability and how it may be used byaggregators as well as DSOs for the system benefit.Next, we discuss some relevant works in the literature whichhave also looked at the smooth integration of EVs in powersystems. A thorough review of EV management schemes isgiven in [8]. Some studies have focused on modelling andcontrol problems of EVs for valley-filling (e.g., [9]), fre-quency regulation (e.g., [10]), and facilitating the integrationof renewable resources (e.g., [11]). Charging algorithms arecategorised into two broad classes: centralised and decen-tralised approaches. In [12] a framework for centralised real-time EV charging management from an EV aggregator thatparticipates in the energy and regulation markets is developed. A decentralised algorithm to optimally schedule EV chargingthat fills the valleys in electric load profiles is given in [9].The authors design an algorithm that only requires each EVsolving its local problem, hence its implementation requireslow computation capability. In [10] the authors propose adistributed EV charging coordination mechanism to meet thedaily mobility energy requirement of an EV fleet with respectto the day-ahead schedule of the EV aggregator and to meetthe regulation dispatch signals sent to the EV aggregator bythe system. However in the formulations above the networkeffects are neglected. In [13] a decentralised EV chargingcontrol scheme to achieve “valley-filling” while meeting het-erogeneous individual charging requirements and satisfyingdistribution network constraints is proposed. However, thenetwork formulation is based on a simplified representationthat is not sufficiently accurate as shown in [14]. In [5] theauthors propose a coordination methodology for the operationof EV owners, EV fleet operators and the DSO by only con-sidering the cost minimisation subject to aggregated capacityto approximate the network effects.The remainder of the paper is organised as follows. InSection II, we describe the EV modelling, the EV objectivesand constraints associated with charging decisions. In Sec-tion III, we introduce the network modelling and define theDADM clearing problem taking into account the EV chargingneeds and objectives. In Section IV, we formulate a distributedsolution over a time-varying communication network to theDADM that addresses privacy and computational issues of themarket participants. In Section V, we propose a methodologyto provide the EV aggregators with a given confidence onthe amount of capacity that they have available to participatein the DADM. In Section VI, we illustrate the proposedVGI framework through the 13-bus, 33-bus, and 141-busdistribution feeders. In Section VII, we summarize the resultsand make some concluding remarks.II. EV M
ODELLING
In this section, we describe the EV modelling, as well as theobjectives and constraints associated with EV charing decisionmaking. The principal sources of uncertainty for an EV are (i)the time intervals that an EV is connected to the grid; (ii)the distances traveled by an EV, i.e., the amount of energyconsumed from the battery due to driving; and (iii) the state ofcharge (SOC) of an EV at any point in time [15]. We considera collection of E EVs denoted by the set E = { , , . . . , E } and a study period of T = { , . . . , T } with T intervals ofsize ∆ t . We assume that the EVs are connected at a three-phase network with N bus nodes denoted by the sets N bus = { , . . . , N bus } and phases Φ = { a, b, c } . In order to determinethe location of the EVs, we need to determine the node andthe phase that they are connected to. To this end, we define foreach EV j ∈ E the triplet H j = { n j , φ j , ξ j } , where n j ∈ N bus is the node that the EV is connected to, φ j ∈ Φ the phase,and ξ j the type of connection which takes values delta orwye. For the entire set of vehicles E we define the collectionof triplets H = { H , . . . , H E } . We introduce the energyconsumed by EV j for commuting at period T by e j , j ∈ E .We denote by T dep j = { t dep j, , . . . , t dep j,D j } the set that indices thetimes EV j departed within period T from home, where D j is the number of times that car j departs from the house in T . T arr j = { t arr j, , . . . , t arr j,D j } is the set that defines the timesEV j arrived at period T at home. We denote by π j ( t ) theavailability of EV i at time t by: π j ( t ) = , ≤ t < t dep j, , , t dep j, ≤ t < t arr j, , , t arr j, ≤ t < t dep j, , ... , t dep j,D j ≤ t < t arr j,D j , , t arr j,D j ≤ t < T. (1)In this work, we only consider home charging however, theproposed framework can easily be expanded to include workand other public space charging. We denote by y j ( t ) thecharging power of vehicle j at time interval t .The charging constraints associated with the charging vari-ables are the following: X t ∈ T π j ( t ) y j ( t )∆ t = e j , ∀ j ∈ E , (2)which ensures that each vehicle has received the right amountof energy at the end of the time horizon. The initial and finalSOC are implicitly represented in (2) by appropriately defining e j , for j ∈ E . There are limits associated with each chargingpower which can be expressed as follows: ≤ y j ( t ) ≤ π j ( t ) y max j ( t ) , (3)where y max j ( t ) is the maximum value, e.g., . kW for slowcharging. Equation (3) ensures that at times when the EV j isnot available for charging y j ( t ) will be zero.The degradation cost of the EV battery is taken into accountby minimising the second order polynomial of the chargingrates [16]: X t ∈ T X j ∈ E y j ( t ) . (4)In this paper, we do not consider EVs as curtailable priceresponsive loads; thus we do not include a cost component forthe charging power y j ( t ) . As a result, the amount of energynecessary e i is pre-defined and stays constant. However, theEVs charge at the minimum possible cost of e j total energydue to the formulation of the day-ahead distribution market,which is formulated in Section III-B. In future work, we willaddress the willingness of EV owners to participate in themarket and modifying their desired energy e j based on pricesignals; thus making the energy e j a decision variable of theframework.III. C ENTRALISED D AY -A HEAD D ISTRIBUTION M ARKET (DADM) C
LEARING P ROBLEM
We follow the DADMs model as described in [17]. In thissection, we first introduce the network modelling and definethe DADM clearing problem. We solve the market clearing forthe period T and find the distribution location marginal prices(DLMPs); the real and reactive power quantities consumedor produced at each point in the network, so as to minimisethe distribution network operator’s cost minus the distributedparticipant benefits subject to linearised power flow relationsand voltage magnitude constraints. We also model distributedgeneration (DG), such as photovoltaic (PV) resources whose capacity can be used for reactive power compensation andvoltage control. A. Network Modelling
To reduce the computational complexity, a linear model isused for the modelling of three phase unbalanced networks,as described in [14]. The authors have validated its accuracycompared to a full AC power flow. Let us assume thatthe system has N bus three-phase buses denoted by the sets N bus = { , . . . , N bus } and the phases Φ = { a, b, c } ; and ℓ lines denoted by the set L = { , . . . , ℓ } . We denote by Y ∈ C N bus × N bus the admittance matrix; by s Y ∈ C N bus ( s ∆ ∈ C N bus ) the phase to line (phase to phase) complexpower injections at each bus and v ∈ R N bus the magnitudeof the bus complex voltages. We assume node 0 is theslack bus and partition the admittance matrix and the voltagemagnitude vector as following Y = (cid:20) Y Y L Y L Y LL (cid:21) , where Y ∈ C × , Y L ∈ C N bus − × , Y L ∈ C × N bus − , and Y LL ∈ C N bus − × N bus − ; and v = [ v , v L ] ⊤ where v ∈ R is the slack bus voltage magnitude and v L ∈ R N bus − thevoltage magnitudes at remaining buses. Let us assume that thereal (reactive) power phase to line injections are denoted by p Y ∈ C N bus − ( q Y ∈ C N bus − ) and the real (reactive)power phase to line load is denoted by p Yd ∈ C N bus − ( q Yd ∈ C N bus − ) for all buses than the slack bus, i.e., ∀ n ∈ N bus / { } . The real (reactive) power phase to phaseinjections are denoted by p ∆ ∈ C N bus − ( q ∆ ∈ C N bus − )and the real (reactive) power phase to phase load is denotedby p ∆ d ∈ C N bus − ( q ∆ d ∈ C N bus − ) for all buses than theslack bus, i.e., ∀ n ∈ N bus / { } .The fixed-point linearisation around a nominal point (ˆ s Y , ˆ s ∆ , ˆ v ) renders the following relationships for the networkrepresentation: v = K Y (cid:20) p Y − p Yd q Y − q Yd (cid:21) + K ∆ (cid:20) p ∆ − p ∆ d q ∆ − q ∆ d (cid:21) + b, (5)where K Y = diag ( h ) Re ( diag ( h ) − M Y ) , K ∆ = diag ( h ) Re ( diag ( h ) − M ∆ ) , b = | h | , with M Y = (cid:20) × N bus − × N bus − Y − LL diag (ˆ v L ) − − jY − LL diag (ˆ v L ) − (cid:21) ,M ∆ = (cid:20) × N bus − × N bus − Y − LL H T diag ( H ˆ v L ) − − jY − LL H T diag ( H ˆ v L ) − (cid:21) , and h = (cid:20) ˆ v − Y − LL Y L ˆ v (cid:21) , where Re ( · ) denotes the real part ofa complex number and ( · ) its conjugate. The complex powerat the substation denoted by s = p + jq ∈ C is given by: s = G Y (cid:20) p Y − p Yd q Y − q Yd (cid:21) + G ∆ (cid:20) p ∆ − p ∆ d q ∆ − q ∆ d (cid:21) + c, (6)where G Y = diag (ˆ v ) Y L M Y , G ∆ = diag (ˆ v ) Y L M ∆ and c = diag (ˆ v ) (cid:16) Y ˆ v − Y L Y − LL Y L ˆ v (cid:17) . B. Day-Ahead Distribution Market Clearing Formulation
The constraints associated with the DADM are the networkconstraints given by the linearised load flow relationships in (5)-(6), modified to include the charging variables y j ( t ) , ∀ j ∈ E , t ∈ T as loads. Thus we have: v ( t ) = K Y (cid:20) p Y ( t ) − p Yd ( t ) − ˜ y Y ( t ) q Y ( t ) − q Yd ( t ) (cid:21) + K ∆ (cid:20) p ∆ ( t ) − p ∆ d ( t ) − ˜ y ∆ ( t ) q ∆ ( t ) − q ∆ d ( t ) (cid:21) + b, ∀ t ∈ T , (7)and s ( t ) = G Y (cid:20) p Y ( t ) − p Yd ( t ) − ˜ y Y ( t ) q Y ( t ) − q Yd ( t ) (cid:21) + G ∆ (cid:20) p ∆ ( t ) − p ∆ d ( t ) − ˜ y ∆ ( t ) q ∆ ( t ) − q ∆ d ( t ) (cid:21) + c, ∀ t ∈ T , (8)where ˜ y Y ( t ) ∈ R N bus is vector that has zero entries for busesand phases that do not have an EV, and is y j ( t ) for bus n j and phase φ j with a wye connection as determined by thetriplet H j = { n j , φ j , ξ j } as defined in Section II. Similarly wemay define ˜ y ∆ ( t ) ∈ R N bus for delta connection. Equation (8)represents two equations, one for the real and one for reactivecomponent. The voltage magnitude constraints are denoted by v φ, min n ≤ v φn ( t ) ≤ v φ, max n , ∀ n ∈ N bus , φ ∈ Φ , ∀ t ∈ T . (9)For both wye and delta connections the real and reactive powerinjections by DG are formulated as: p φ, min n ≤ p φn ( t ) ≤ p φ, max n , ∀ n ∈ N gen , φ ∈ Φ , ∀ t ∈ T , (10) q φ, min n ≤ q φn ( t ) ≤ q φ, max n , ∀ n ∈ N gen , φ ∈ Φ , ∀ t ∈ T , (11)where N gen ⊆ N bus is the set of nodes that contain DG. For n ∈ N bus / N gen we have p φn = q φn = 0 for all φ ∈ Φ . For n ∈ N gen , if DG is connected to only one phase p φn = q φn = 0 for the remaining phases. The EV charging related constraintsgiven in (2), (3) that describe the intertemporal state of chargedynamics, non-negativity and charging rate constraints are alsoincluded.The objectives of the DADM refer to the minimisation ofthe cost of real power procured at the substation: X t ∈ T X φ ∈ Φ λ ( t ) p φ ( t )∆ t, (12)where λ ( t ) is the locational marginal price (LMP) at thesubstation at time t and p φ ( t ) is the injection at phase φ attime t at the substation and ∆ t is the time interval that theDADM is cleared, e.g., 5 minutes. A byproduct of (12) is thateach EV j ∈ E procures the desired energy e j at minimumcost, as stated in Section II. The objective also includes aterm that ensures that voltage levels throughout the networkare operating close to the reference voltage: X t ∈ T X n ∈ N bus X φ ∈ Φ ( v φn ( t ) − v ref ) , (13)where v ref is the reference voltage. The cost of distributedgeneration is also taken into account with X t ∈ T X n ∈ N gen X φ ∈ Φ c φn ( t ) p φn ( t )∆ t, (14)where c φn ( t ) is the cost of DG generation connected to node n and phase φ at time t . The degradation cost of EV batteriesgiven in (4) is also included in the formulation. The decision variables for each n ∈ N bus and φ ∈ Φ is the real powerinjection p φn ( t ) ; the reactive power injection q φn ( t ) ; the voltagemagnitude v φn ( t ) ; and for each EV j ∈ E is the chargingschedule y j ( t ) , for all t ∈ T . The DADM is formulated asfollows: min { p φn ( t ) ,q φn ( t ) , v φn ( t ) ,y j ( t ) } t ∈ T ,n ∈ N bus ,φ ∈ Φ ,j ∈ E (4) + (12) + (13) + (14) subject to (2) , (3) , (7) − (11) (15)In the formulation above we only consider one-directionalcharging under the VGI framework. This can be easily ex-tended to bi-directional charging. In this work we focus onone-directional charging as an intermediate step between theuncontrolled charging and V2G which requires a more intensecommunication network.IV. P ROPOSED D ISTRIBUTED
DADMIn this section, we formulate the distributed solution to theDADM that addresses privacy and computational issues of themarket participants. The solution of (15) by the DSO requiresknowledge of EV information in terms of desired energy,departure and arrival times, and SOC. However, EV ownersare reluctant in providing such information. Moreover, if thenumber of EVs is very large it can be very computationallyintensive for the DSO to solve the DADM. In this regard,there is a need to propose a distributed solution of the DADMclearing mechanism. We divide the DADM participants into E + 1 agents, i.e., the EV owners ( E ) and the DSO. Theproposed framework could be extended to any number ofagents, e.g., PV owners could also be separate agents or eveneach network bus; however, since the focus of this paper ison EV charging we limit the number of agents to E + 1 .We assume that the communication network that these agentsuse to exchange information is time-varying as is in reality.The DADM clearing mechanism given in (15) may be seenas an optimisation problem where each agent optimises alocal objective subject to local constraints, but needs to agreewith the other agents in the network on the value of somedecision variables that refer to the usage of shared resources,i.e., the power at the substation and the network usage, whichare represented by coupling constraints. More specifically,each agent i has its own vector x i ∈ R n i of n i decisionvariables, e.g., the voltage magnitude, the charging schedule;its local linear constraint set A i x i = b i and D i x i ≤ ,these include constraints such as (2),(3), (9)-(11); and itsobjective f i ( x i ) : R n i → R , e.g., (4), (12)-(14). The couplingconstraints refer to (7) and (8); (7) has N bus T constraints and(8) has T (since (8) refers to two equality constraints per timestep) thus in total the coupling constraints are T ( N bus + 2) .We denote the coupling constraints as P E +1 i =1 Z i x i = ζ ,where Z i ∈ R T ( N bus +2) × n i and ζ ∈ R T ( N bus +2) . Each agentcontributes to the coupling constraints with Z i . Now we may rewrite (15) in compact form as min { x i } E +1 i =1 E +1 X i =1 f i ( x i )subject to A i x i = b i , i = 2 , . . . , E + 1 ,D i x i ≤ , i = 1 , . . . , E + 1 , E +1 X i =1 Z i x i = ζ. (16)The construction and definitions of all variables and param-eters, e.g., x i or A i may be found in the Appendix to facilitatethe readability of the paper. A. Proposed distributed algorithm
A distributed strategy that addresses both privacy and com-putational issues of the DADM given in (16) is:
Algorithm
Distributed DADM Initialization k = 0 . Consider ˆ x i (0) such that A i ˆ x i (0) = b i , D i ˆ x i (0) ≤ ,for all i = 1 , . . . , E + 1 . Consider κ i (0) ∈ R r , for all i = 1 , . . . , E + 1 . For i = 1 , . . . , E + 1 repeat until convergence ℓ i ( k ) = P E +1 j =1 a ij ( k ) κ j ( k ) . X i = { x i : A i x i = b i , D i x i ≤ } x i ( k + 1) ∈ arg min x i ∈ X i f i ( x i ) + ℓ i ( k ) ⊤ Z i x i . κ i ( k + 1) = ℓ i ( k ) + c ( k )( Z i x i ( k + 1) − ζE +1 ) ˆ x i ( k + 1) = ˆ x i ( k ) + c ( k ) P kr =0 c ( r ) ( x i ( k + 1) − ˆ x i ( k )) . ˜ x i ( k + 1) = ˆ x i ( k + 1) , k < k i,s P kr = ki,s c ( r ) x i ( k +1) P kr = ki,s c ( r ) , k ≥ k i,s . k ← k + 1 .In the Algorithm above r is the row dimension of the Z i matrices, i.e., the number of coupling constraints which are r = 3 T ( N bus + 2) , c ( k ) is the subgradient step-size usuallyset to c ( k ) = βk +1 for some β > , k s,i ∈ N + is the iterationindex related to a specific event, namely, the convergence ofthe Lagrange multipliers, as detected by agent i .The steps of the algorithm may be explained as follows.Each agent i , i = 1 , . . . , E + 1 initialises the estimate of itslocal decision vector with ˆ x i (0) that needs to satisfy its localconstraints, i.e., ˆ x i (0) such that A i ˆ x i (0) = b i , D i ˆ x i (0) ≤ (step 3 of algorithm), and the estimate of the common dualvariables vector of the equality constraints with a κ i (0) ∈ R r ,e.g., κ i (0) = 0 r × , i = 1 , . . . , E + 1 . At every iteration k eachagent i computes a weighted average ℓ i ( k ) of dual variablesvector based on the estimates κ j ( k ) , j = 1 , . . . , E + 1 , ofthe other agents and its own estimate. The weight a ij ( k ) thatagent i attributes to the estimate of agent j at iteration k is setequal to zero if agent i does not communicate with agent j at iteration k . The conditions that the communication networkweights must satisfy are the following: a ij ( k ) ∈ [0 , , for all k ≥ , P E +1 j =1 a ij ( k ) = 1 , ∀ i = 1 , . . . , E +1 , P E +1 i =1 a ij ( k ) = 1 , ∀ j = 1 , . . . , E + 1 . Agent i updates its local variable ˜ x i untilconvergence. Some important characteristics of the algorithm are that nolocal information related to the primal problem is exchangedbetween the agents. In particular, only the estimates of the dualvector are communicated; thus addressing privacy concernsof the DSO and the EV owners. Furthermore, the algorithmreduces computational complexity by distributing the burdenbetween the agents. The communication network of the DSOand the EV owners may be time-varying and has to satisfythe constraints mentioned above for a ij ( k ) . Step 9 of thealgorithm is a running average of the primal iterates which areconstructed as they are shown to exhibit superior convergenceproperties with respect to x i ( k ) while step 10 performs a resetof this average at a certain iteration index as this has beenshown to speed up practical convergence [18]. It has beenshown that the dual iterates κ i ( k ) generated by the algorithmconverge to an optimal dual vector which ˆ x i ( k ) achieveasymptotically the optimal objective value. More details aboutthe algorithm may be found in [18].V. U NCERTAINTY MODELLING
In this section we propose a methodology to provide the EVaggregators with a given confidence on the amount of capacitythat they have available to participate in the DADM. Theproposed framework may be used by aggregators to provideservices to the grid. Given that the EV owners do not share anyprivate information they would be willing to let an aggregatorbe responsible to charge their vehicle subject to their desires,e.g., final state of charge, departure times, etc. for receivingmonetary benefits. The aggregator would communicate withthe DSO and other aggregators to determine the clearing ofthe DADM. One challenge in this setting is that the aggregatorwould not be certain of the number of EVs available at acertain time instant, their state of charge and available energy,their arrival and departure times.In order for the aggregators to have a given confidence onthe amount of capacity that they have available to participatein the market we use a simulation approach that contains inde-pendent Monte Carlo simulations and requires the constructionof multiple independent and identically distributed (i.i.d.)sample paths for each output random variable to evaluate theperformance metrics [19, p. 10]. More specifically, we carryout simulation runs to determine the probability distributionof the amount of available capacity by the aggregator forevery time interval, i.e., γ ( t ) = P j ∈ E y j ( t ) . The performancemetrics we select is the expected value. Let M be the numberof Monte Carlo simulations, the estimated average availablecapacity by the aggregator for every time interval, i.e., γ ( t ) = 1 M M X m =1 γ ( m ) ( t ) , (17)where γ ( m ) ( t ) is the realization of the random variable γ ( t ) in simulation run m . The number of simulation runs M depends on the statistical reliability requirements specified forthe estimation of the desired expected values. We define thestatistical reliability of the hourly sample mean estimator γ ( t ) to be the length of the − ν )% confidence interval with < ν < for the true mean of γ ( t ) . According to theCentral Limit Theorem, the sample mean estimator γ ( t ) is Fig. 1: LMP at the substation over a 24 hour period.approximately normally distributed for large M [20]. Thuswe can establish that the true mean of γ ( t ) lies in the interval (cid:20) γ ( t ) − z − ν/ σ γ ( t ) √ M , γ ( t ) + z − ν/ σ γ ( t ) √ M (cid:21) , (18)with a − ν )% probability, where σ γ ( t ) is the standarddeviation of γ ( t ) and z − ν/ = Φ − (1 − ν ) , with Φ − theinverse of the cumulative distribution of the standard normaldistribution. The length of the confidence interval is a functionof √ M − with decays slowly for large M . Thus beyond acertain value of M , the improvement in statistical reliabilityis generally too small to warrant the extra computing timeneeded to perform additional simulation runs.The confidence interval may be further tuned if more refinedhistorical data are used to construct an empirical pdf, sothat the aggregator may make a more informed decision. Forinstance, if the aggregator holds data for the location of thefeeder and it is part of a rural or urban area, then the predicteddaily energy consumption of an EV may be tuned accordingly(e.g., [21]). VI. N UMERICAL R ESULTS
In this section, we present several numerical examples todemonstrate the capabilities of the proposed VGI framework.We use small systems, the unbalanced 13-bus and 33-bus dis-tribution feeders to provide insights into the results presented.We demonstrate the scalability of the proposed distributed al-gorithm in Section IV with the 141-bus distribution feeder [22]with 11 agents communicating with each other. Additionally,we demonstrate how the amount of EVs affects the confidencelevel that the EV aggregator has when participating in theDADM.
A. 13-bus distribution feeder
A lot of work has conducted into obtaining realistic data forEV behaviour (see, e.g., [21], [23], [24]). In this work, we useFig. 2: Charging schedule of 10 EVs over a 24 hour period.
Fig. 3: Voltage magnitude at the EVs connections of the 13-bus feeder over a 24 hour period.data from [21] to model realistically the charging behaviourof an EV. In the numerical results we assume that charging istaking place at home.In the first case study we consider the 13-bus distributionfeeder [25] with no renewable resources. We consider acollection of 10 EVs denoted by the set E = { , , ..., } that are wye connected in various phases of the 13-bus feederand a study period of T = { , ..., } with intervals of size ∆ t = 1 h. For instance, EV 1 is at node 632 in phace cwith times of arrival and departure specified by t dep , = 11 and t arr , = 12 and required energy e = 3 . kWh. The maximumcharging value is set to y maxj = 10 kW for all j = 1 , . . . , .The LMP at the substation λ ( t ) for t = 1 , . . . , is depictedin Fig 1. The minimum (maximum) allowed voltage level is0.95 pu (1.06 pu). The outcome of the charging schedule ofthe EVs is depicted in Fig. 2. As it may be seen the EVs thatare available select to charge at the hours when the LMP atthe substation is lower, i.e., 16:00-22:00. At the same timeinterval we may notice in Fig. 3 that the voltage levels arenear the minimum value for the nodes and phases where theEVs are connected. B. 33-bus distribution feeder
The solution of (15) by the DSO requires knowledge ofEV information in terms of desired energy, departure andarrival times, and SOC. However, EV owners are reluctantin providing such information. Moreover, if the number ofEVs is very large it can be very computationally intensive forthe DSO to solve the DADM. In this regard, we formulated adistributed solution of the DADM clearing mechanism givenin Section IV. We validate the proposed methodology in a 33-bus with study period of T = { , ..., } with intervals of size ∆ t = 1 h. We divide the DADM participants into agents, i.e.,the EV owners ( ) and the DSO. The optimisation problemFig. 4: Evolution of the agents estimates { κ i ( k ) } i =1 where allagents communicate with each other for the 33-bus system. Fig. 5: Evolution of objective function until it reaches theoptimal value (red line) where all agents communicate witheach other for the 33-bus system.of the DSO has 2016 decision variables and local constraintsset defined by 4032 inequalities. The optimisation problemof each EV has 96 decision variables and local constraintsset defined by 1 equality and 192 inequalities. There are840 coupling equality constraints, and therefore we have 840Lagrange multipliers associated with them. It is assumed thatall agents communicate with each other and the c ( k ) = k +1 , k i,s = 2000 for i = 1 , . . . , . We ran the proposed algorithmfor 2500 iterations with κ i (0) = 0 , i = 1 , . . . , and theevolution of the Lagrange multipliers is depicted in Fig. 4.As we may see they converge to the optimal value fromaround 2000 iterations. In Fig. 5 the evolution of the objectivevalue is depicted. We may see a jump at iteration number k i,s = 2000 for i = 1 , . . . , since the Lagrangian multipliershave converged and we only use estimates for ˆ x i ( k ) basedon values after iteration k i,s (see step 10 of the algorithm).In order to test how the communication network affects therate of convergence we modify the communication networkso that only half the agents talk with the other half at anytime-step. The communication network is depicted in Fig. 6and corresponds to a connected graph, whose edges are dividedinto two groups: the blue and the red ones, which are activatedalternatively. Here, we have k i,s = 3000 for i = 1 , . . . , .According to step 10 of the proposed distributed algorithmthe “jump” at iteration k i,s speeds up practical convergence agent 1 agent 2agent 3agent 4agent 5agent 6 Fig. 6: Time-varying communication network where half theagents communicate with each other for the 33-bus system.Fig. 7: Evolution of objective function until it reaches theoptimal value (red line) where half the agents communicatewith each other for the 33-bus system.
Fig. 8: Evolution of the agents estimates { κ i ( k ) } i =1 where allagents communicate with each other for the 141-bus system.by “resetting” the running average estimate. We may noticein Fig. 7 that the objective function now converges to theoptimal value at a greater number of iterations compared tothe previous case where all agents communicated to each otheras seen in Fig. 5. C. 141-bus distribution feeder
To demonstrate the scalability of the proposed methodologywe implement the distributed algorithm in the 141-bus distri-bution feeder. In this case the DADM has 11 agents, i.e., theEV owners (10) and the DSO. The optimisation problem ofthe DSO has 9408 decision variables and local constraints setdefined by 18816 inequalities. The optimisation problem ofeach EV has 96 decision variables and local constraints setdefined by 1 equality and 192 inequalities. There are 3408coupling equality constraints, and therefore we have 3408Lagrange multipliers associated with them. It is assumed thatall agents communicate with each other and the c ( k ) = k +1 , k i,s = 1000 for i = 1 , . . . , . We ran the proposed algorithmfor 2000 iterations with κ i (0) = 0 , i = 1 , . . . , and theevolution of the Lagrange multipliers is depicted in Fig. 8.As we may see they converge to the optimal value fromaround 1000 iterations. In Fig. 9 the evolution of the objectivevalue is depicted. We may see a jump at iteration number k i,s = 1000 for i = 1 , . . . , since the Lagrangian multipliershave converged and we only use estimates for ˆ x i ( k ) based onvalues after iteration k i,s .In order to perform an analysis on the number of vehiclesthat are necessary for an aggregator to participate in theDADM with a confidence interval we perform Monte Carlosimulations in the 141-bus feeder. We modify the level of EVdeployment from: i) 30, ii) 80, and iii) 120 thousand EVs andrun 500 Monte Carlo simulations for each integration level. InFig. 10, we depict the mean value of the available capacity forFig. 9: Evolution of objective function until it reaches theoptimal value (red line) for the 141-bus system. Fig. 10: Hourly mean values of available capacity for theEV aggregator with 95% confidence for various levels of EVintegration for the 141-bus system. TABLE I: EV aggregator confidence level as a function ofEVs for hour t = 14 .every hour of the day; we may notice that as the number ofEVs increases capacity is also available at more hours of theday. In Fig. 11, we depict the intervals with 95% confidencefor these mean values, the values are normalised with themean value for every level of integration so that the graphis more readable. We may notice that as the number of EVsincreases the level of confidence that the EV aggregator hasalso increases. In Table I we show the available capacity forhour t = 14 as a function of various confidence intervals andlevels of EV integration.VII. C ONCLUSIONS
In this paper, we developed a distributed VGI frame-work that enables the smooth integration of EVs. Throughthe numerical examples, we demonstrated that the proposedframework is scalable and performs well in a variety ofcircumstances. We also demonstrated that this framework isuseful for EV aggregators. More specifically, we provideda detailed model of EVs, i.e., representing their times ofarrival and departure, SOC, required energy and objectives.Next, we formulated the centralised DADM that incorporatesthe charging needs of EVs and has a detailed representationof the underlying three phase power network. We proposeda distributed solution to the DADM that converges to theoptimal solution under a time-varying communication networkFig. 11: Normalised mean value intervals with 95% confidencefor the 141-bus system. with no exchange of sensitive information. We provided theEV aggregator with a methodology to quantify the level ofconfidence for the available capacity that can participate inthe DADM based on the number of EVs available.There are natural extensions of the work presented here. Forinstance, we will investigate the incentives for an EV ownerto participate in this market setup. In our future studies, weplan on incorporating uncertainty in the formulation of theDADM due to load variations and the intermittent nature ofrenewable resources and propose a distributed algorithm thatconverges under this uncertain environment. We will report onthese developments in future papers.A
PPENDIX
The three-phase network studied has E EVs connected toit and a total of N bus connection points. Thus N bus − E arenodes/phases that do not contain an EV and are part of theDSO agent. The DSO agent, which without loss of generality,is indexed by , has a decision variable x ∈ R n with n =3(3 N bus − E ) T . More specifically: x = [ p φn ( t ) , q φn ( t ) , v φn ( t ) : { n, φ, ξ } / ∈ H , n ∈ N bus , φ ∈ Φ , t ∈ T ] ⊤ . Its objective is defined as f = P { n,φ,ξ } / ∈ H n ∈ N bus ,φ ∈ Φ t ∈ T f φn ( t ) , where f φn ( t ) = ( v φn ( t ) − v ref ) + c φn ( t ) p φn ( t ) , n ∈ N gen ( v φn ( t ) − v ref ) , n ∈ N bus / N gen ∪ { } λ ( t ) p φ ( t ) + ( v φ ( t ) − v ref ) , n = 0 (19)The limiting constraints for agent 1 given in (9)-(11) maybe represented as the matrix D = [ C φn ( t ) : { n, φ, ξ } / ∈ H , n ∈ N bus , φ ∈ Φ , t ∈ T ] ⊤ , (20)where C φn ∈ R n × n , i.e., one row for the minimum andanother for the maximum limit associated with each of thethree variables.For agent i : i = 2 , .. . . . , E + 1 , i.e., an EV owner j connected to node n j and phase φ j determined by the duplet { n j , φ j , ξ j } ∈ H j , we define the vector x i ∈ R n i with n i = 4 T by x i = [ p φ j n j ( t ) , q φ j n j ( t ) , v φ j n j ( t ) , y j ( t ) : t ∈ T ] ⊤ , (21)and the objective function f i = X t ∈ T (cid:16) ( v φ j n j ( t ) − v ref ) + y j ( t ) (cid:17) . (22)We rewrite (2) as A i x i = b i , where A i ∈ R × n i and b i ∈ R and the limiting constraints given in (3) and (9)-(11) as D i =[ C φ j n j ( t ) : t ∈ T ] ⊤ , where C φ j n j ∈ R × n i , i.e., one row for theminimum and another for the maximum limit associated witheach of the four variables.R EFERENCES[1] C. Crozier, D. Apostolopoulou, and M. McCulloch, “Mitigating theimpact of personal vehicle electrification: A power generation perspec-tive,”
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