Voltage rise is an undesirable side-effect of solar photovoltaic (PV) generation, arising from the flow of surplus electrical power back into the grid when PV generation exceeds local demand. Customers deploying residential-scale battery storage are likely to further exacerbate voltage rise problems for electrical utilities unless the charge/discharge schedules of batteries are appropriately coordinated. In this paper, we present a real-time pricing mechanism for use in a network of distributed residential energy systems (RESs), each employing solar PV generation and battery storage. The pricing mechanism proposed in this paper is based on a Market Maker algorithm in which predicted power profiles and real-time pricing information is iteratively exchanged between a central entity and each of the RESs. The Market Maker formulation presented in this paper is shown via simulation studies to converge to a fixed price vector, thereby reducing the price volatility observed in an earlier formulation, while achieving the same reduction in power usage variability as a centralised model predictive control (MPC) scheme presented previously

Braun, Philipp

Grüne, Lars

Kellett, Christopher M.

Weller, Steven R.

Worthmann, Karl

Gruene, Lars

University of Newcastle's Digital Repository

A real-time pricing scheme for residential energy systems using a market maker

EPub Bayreuth

A Real-Time Pricing Scheme for Residential Energy Systems Using aMarket Maker*Philipp Braun1, Lars Gru¨ne1, Christopher M. Kellett2, Steven R. Weller2, and Karl Worthmann3Abstract— Voltage rise is an undesirable side-effect of solarphotovoltaic (PV) generation, arising from the flow of surpluselectrical power back into the grid when PV generation exceedslocal demand. Customers deploying residential-scale batterystorage are likely to further exacerbate voltage rise problemsfor electrical utilities unless the charge/discharge schedulesof batteries is appropriately coordinated. In this paper, wepresent a real-time pricing mechanism for use in a network ofdistributed residential energy systems (RESs), each employingsolar PV generation and battery storage. The pricing mech-anism proposed in this paper is based on a Market Makeralgorithm in which predicted power profiles and real-timepricing information is iteratively exchanged between a centralentity and each of the RESs. The Market Maker formulationpresented in this paper is shown via simulation studies toconverge to a fixed price vector, thereby reducing the pricevolatility observed in an earlier formulation, while achievingthe same reduction in power usage variability as a centralisedmodel predictive control (MPC) scheme presented previously.I. INTRODUCTIONWidespread deployment of solar photovoltaics (PV) atthe residential level can cause network difficulties in theform of voltage rise caused by households generating morepower than they consume; e.g., in the middle of a sunnyday while the residents are not at home. If in addition solarPV is augmented with residential battery storage, voltagerise problems may well be exacerbated by multiple batteriesdischarging to the grid at the same time as local generationalready exceeds the local load. Hence, an important questionis: how to schedule battery storage?A natural performance metric in this context is the reduc-tion in variation of the grid usage profile or, in other words,the achieved reduction in power demand variability relativeto the average demand over some time window. In [1],[2], we presented centralised, decentralised, and distributedmodel predictive control (MPC) schemes aimed at reducingthis deviation. Here, we differentiate between decentralisedcontrol, where any individual residence chooses how toschedule its battery without any external information, and*C. M. Kellett is supported by ARC Future Fellowship FT1101000746.L. Gru¨ne is supported by the Deutsche Forschungsgemeinschaft, Grant GR1569/13-1.1P. Braun and L. Gru¨ne are with the Mathematical Institute, Uni-versita¨t Bayreuth, 95440 Bayreuth, Germany, e-mail: {philipp.braun,lars.gruene}@uni-bayreuth.de.2C. M. Kellett and S. R. Weller are with the School of Electri-cal Engineering and Computer Science at the University of Newcas-tle, Callaghan, New South Wales 2308, Australia, e-mail: {chris.kellett,steven.weller}@newcastle.edu.au.3K. Worthmann is with the Institute for Mathematics, Technische Uni-versita¨t Ilmenau, 99693 Ilmenau, Germany, e-mail: karl.worthmann@tu-ilmenau.de.distributed control where some level of communication be-tween residences is permitted. Not surprisingly, in [2], it isobserved that the centralised MPC scheme performs betterthan the distributed MPC scheme which, in turn, performsbetter than the decentralised MPC scheme.As a result of the curse of dimensionality, the centralisedMPC scheme presented in [2] does not scale up to alarge number of residences. Hence, in [3], we developed ahierarchical distributed optimisation algorithm that recoversthe performance of the centralised MPC scheme. However,the algorithm presented in [3], similar to the centralised MPCscheme of [2], is essentially a cooperative scheme wherebyall residences cooperate to achieve the goal of reducingnetwork deviation from the average.By contrast, the distributed MPC scheme of [2] is based ona real-time pricing mechanism referred to as a Market Maker[4], [5]. In this scheme, an iterative process is employedwhereby residences communicate predicted power profiles toa central entity that computes prices that are then broadcast toall residences. This process is repeated a number of times.A drawback of the Market Maker proposed in [2] is thatthis iterative process may not converge to a fixed pricevector, with the possibility of marked price volatility fromone iteration to the next.In this paper, we present an alternate Market Makerformulation that, at least in simulation, appears to convergeto a fixed price vector. Furthermore, using this new MarketMaker, we provide simulation results which recover theperformance of the centralised MPC scheme.The paper is organised as follows. In Section II weintroduce the mathematical model of the Residential EnergySystem (RES) and define the desired performance metrics.The centralised MPC approach is presented in Section III andour new Market Maker algorithm is described in Section IV.A simulation study using data from an Australian electricitydistribution company, Ausgrid, is undertaken in Section V.Concluding remarks are provided in Section VI.II. THE RESIDENTIAL ENERGY SYSTEMWe consider Residential Energy Systems (RESs) com-prised of residential load, generation, battery storage, anda connection to the electricity network. Let I ∈ N be thenumber of RESs in the local area under consideration. Asimple model of RES i, i ∈ {1, . . . , I}, is given byxi(k + 1) = xi(k) + Tui(k), (1)zi(k) = wi(k) + ui(k) (2)where xi is the state of charge of the battery in kilowatt-hours(kWh), ui is the battery charge/discharge rate in kilowatts(kW), wi is the static load minus the local generation inkilowatts, and zi is the power supplied by/to the grid in kilo-watts. Here, T represents the length of the sampling intervalin hours. While the system dynamics (1) is autonomous,the performance output (2) depends on the time varyingquantity wi(·).The RES network is then defined by the followingdiscrete-time systemx(k + 1) = f(x(k), u(k)),z(k) = h(u(k), w(k))where x, u, w, z ∈ RI , and the definitions of f and h aregiven componentwise by (1) and (2), respectively. For eachRES i ∈ {1, . . . , I}, the constraints on the battery capacityand charge/discharge rates are described by the constantsCi, ui ∈ R>0 and ui ∈ R<0, i.e.,0 ≤ xi(k) ≤ Ci and ui ≤ ui(k) ≤ ui ∀k ∈ N0. (3)Our aim is to design a pricing mechanism so as to avoidpeaks in supply and demand or, put another way, to flattenthe performance output z. To this end, letΠ(k) :=1II∑i=1zi(k)denote the average power demand at time k and let N denotethe number of samples in a simulation. The performance met-ric of peak-to-peak (PTP) variation of the average demandof all RESs is given by(maxk∈{0,...,N−1}Π(k))−(mink∈{0,...,N−1}Π(k)). (PTP)The second performance metric of the root-mean-square(RMS) deviation about the average is defined as√√√√ 1NN−1∑k=0(Π(k)− Υ )2 (RMS)with the average demand Υ := 1NI∑N−1k=0∑Ii=1 wi(k).III. CENTRALISED MODEL PREDICTIVE CONTROLWe recall the model predictive control (MPC) algorithmfor the control of a network of RESs introduced in [2]and [1], respectively. This approach is a centralised MPC(CMPC) scheme, in which full communication of all relevantvariables for the entire network as well as a known modelof the network are required.MPC iteratively minimises an optimisation criterion withrespect to predicted trajectories and implements the firstpart of the resulting optimal control sequence until the nextoptimisation is performed (see, e.g., [6] or [7]). To this end,we assume that we have predictions of the residential loadand generation some time into the future that is coincidentwith the horizon of the predictive controller. In other words,given a prediction horizon N ∈ N, we assume knowledgeof wi(j) for j ∈ {k, . . . , k + N − 1}, where k ∈ N0 is thecurrent time.To implement the CMPC algorithm, we compute thenetwork-wide average demand at every time step k over theprediction horizon byζ¯(k) :=1II∑i=11Nk+N−1∑j=kwi(j)− xi(k) . (4)After the average demand is computed, the joint cost functionV (x(k); k) := minuˆ(·)k+N−1∑j=k(ζ¯(k)− 1II∑i=1(wi(j) + uˆi(j))︸ ︷︷ ︸zˆi(j))2is minimised with respect to the predicted control in-puts uˆ(·) = (uˆ1(·), uˆ2(·), . . . , uˆI(·))T with uˆi(·) =(uˆi(j))k+N−1j=k , i ∈ {1, 2, . . . , I}, subject to the systemdynamics (1), the current state x(k) = (x1(k), . . . , xI(k))T ,terminal constraints xˆ(k+N) = xend, and the constraints (3)for i ∈ {1, . . . , I}. The vector of the predicted performanceoutput zˆ(·) is defined in the same way as the predictedcontrol uˆ(·). Explicitly, we use the hat notation to denotepredicted values, with the absence of a hat denoting theactual value. Additionally we use the notation u(j) =(u1(j), . . . , uI(j))T for a fixed time j ∈ N. The same holdsfor the other variables x, w, and z.IV. AN IMPROVED MARKET MAKER SCHEMEThe CMPC approach contains an implicit assumptionthat individual RESs willingly contribute to the goal of thegrid operator, i.e., reducing variance in electricity supply.However, the benefit to an individual RES of such a schemeis separate to any price mechanism. Here, we will design areal-time pricing mechanism that recovers the performanceof the CMPC scheme and which corresponds to the naturalbehaviour of an RES, namely minimisation of cost.We assume that the price for energy at a fixed time isgiven by a quadratic function l : I ⊂ R→ R of the forml(z; b) ={T · p (z + a1(b− z)2 + cb) , z ≤ bT · p (z + a2(z − b)2 + cb) , z ≥ bwhere cb ∈ R is given bycb ={ −b2 · a1, if b > 0−b2 · a2, if b ≤ 0and depends on the parameter b while p, a1, a2 ∈ R+ arepositive constants. Note that cb is defined so that l(0; b) = 0holds. The function l needs to be monotonically increasingto capture the fact that increasing energy usage incursincreasing costs. Hence, we restrict the domain to I =[b − a−11 /2,∞). In this case, l(z; b) > 0 for all z > 0 andl(z; b) < 0 for all z < 0; i.e., energy demand produces costsand energy production leads to a profit.The parameter b defines a variable threshold. An energydemand smaller than the threshold is penalised with addi-tional costs a1 ·(z−b)2 while an energy demand bigger thanthe threshold is penalised by a2 · (z − b)2. In other words,both excessive usage and excessive generation are penalised,dependent on the threshold b. The constant p is not necessaryfor the analysis of the cost function but is used to scale thecosts to realistic energy prices.We assume that the parameter b is time dependent and canbe chosen by the Market Maker. Minimising the energy costsfor an individual system in an MPC context can be achievedby minimising the cost functionv(zi; b) :=k+N−1∑j=kl(zi(j); b(j)). (5)The question which has to be addressed by the Market Makeris how to choose the parameters b(j) to achieve the samevalue of PTP and RMS as obtained by CMPC.Assuming that zi(j) ≥ 0 implies costs and zi(j) < 0yields a profit, it is clear that each RES wants to havea fully discharged battery at the end of the predictionhorizon, i.e., xˆi(k + N) = 0 in order to minimise costor maximise profit. Hence, the average energy consumption1IN∑Ii=1∑k+N−1j=k zˆi(j) during the prediction window cor-responds to ζ¯(k) given by (4), and thus, does not dependon the individual strategies of the RESs as long as eachRES follows the paradigm of profit maximisation. In order tomake a fair comparison between the Market Maker algorithmand CMPC, we set the terminal constraint in the CMPCscheme to be xˆ(k +N) = xend = 0.The idea behind the choice of b(j), j ∈ {0, . . . , N − 1},is to penalise the deviation from the average ζ¯(k). Here,we assume that the Market Maker knows the desired powerprofile zˆi = (zˆi(j))k+N−1j=k . Ifζ¯(k)− 1II∑i=1zˆi(j) > 0(ζ¯(k)− 1II∑i=1zˆi(j) < 0)holds, the RESs should be motivated to use more (less)energy, i.e., b(j) should be increased (decreased).Note that no information on the battery capacity, thecharging rate, the load data w(k), w(k+1), . . . , w(k+N−1),or the initial state of the batteries is transmitted to the MarketMaker, and the energy demand of an RES is the outcome ofits own optimisation. The scaling factor κ, computed in (7)below, guarantees boundedness of the threshold values.Algorithm 4.1: Set the iteration index ` = 0, the maximaliteration number `max ∈ N, and a bound bmax ∈ R+.Initialisation: The Market Maker collects the desired powerprofile zˆ0i = (zˆ0i (j))k+N−1j=k , i ∈ {1, 2, . . . , I}, from eachRES i, computes ζ¯(k), and sets b0 = (ζ¯(k), . . . , ζ¯(k))T .(1) Increment the iteration index ` by one.(2) The Market Maker computes the average predictedenergy demand at time j, j ∈ {k, 1, . . . , k +N − 1}:Π`(j) =1II∑i=1zˆ`−1i (j).(3) The Market Maker setsb`(j) = b`−1(j) + (ζ¯(k)−Π`(j)), (6)computes the scaling factorκ = bmaxmaxj=0,...,N−1 |(ζ¯(k)−Π`(j))|maxj=0,...,N−1 |b`(j)| (7)for j = k, k + 1, . . . , k +N − 1, and broadcasts κ · b`to the RESs.(4) Each RES determines its energy usage profile bysolving an optimal control problem, i.e.zˆ`i := argmin v(zˆi;κ · b`)s.t. xˆi(k) = xi,0s.t. xˆi(j + 1) = xˆi(j) + T uˆi(j)s.t. zˆi(j) = wi(j) + uˆi(j)s.t. 0 ≤ xˆi(j) ≤ Ci, ui ≤ uˆi(j) ≤ uis.t. ∀ j = k, . . . , k +N − 1and sends zˆ`i to the Market Maker.(5) If ` < `max go to step (1). Otherwise, return zˆ`.V. NUMERICAL RESULTSIn this section we compare the performance of CMPC withthe performance of Algorithm 4.1. Additionally we show thebenefits of Algorithm 4.1 in terms of a price interpretation.All simulations use a setting of 30 RESs, initial conditionsxi(0) = 0[kWh], constraints ui = −ui = 0.3[kW] and Ci =2[kWh], a discretisation of T = 0.5[h], and a simulationlength of N = 387. (N is chosen such that xi(387) = 0 forall i ∈ {1, . . . , I} holds.)A. Performance of the improved Market Maker scheme0 5 10 15 200.20.40.60.8z [kW]0 5 10 15 20−0.0200.02∆z [kW]time in [h]Fig. 1. Open loop solution of the aggregated power demand1I∑Ii=1 zˆi(j) (j = 0, . . . , 47) of CMPC (red) and Algorithm 4.1(top) and the difference between the open loop CMPC solution andAlgorithm 4.1 (bottom). The blue line is with bmax = 50 and theblack line with bmax = 100.Figure 1 compares the (open loop) solution at a fixed timestep of the aggregated power demand 1I∑Ii=1 zˆi(j) (j =0, . . . , 47) solved with CMPC and Algorithm 4.1 with twovalues of bmax; bmax = 50 and bmax = 100. We observe thatthe solution of Algorithm 4.1 is close to the CMPC solutionand the difference appears to decrease with increasing bmax.Additionally, the parameters p = 0.3, a1 = 5 · 10−3 anda2 = 2 · 10−2 for bmax = 50, and p = 0.3, a1 = 5 · 10−4and a2 = 2 · 10−3 for bmax = 100 are used in Algorithm4.1 to ensure that the cost function l(z; b) is monotonicallyincreasing on the given data set. Note that according to ournumerical experience the parameters p, a1, and a2 are notimportant for the performance of the algorithm as long as themonotonicity assumption holds. The specific values chosenhere provide reasonable prices when `(z; b) is assumed to begiven in cents per kWh. The differences with respect to ourperformance metrics for the three settings are negligible (cf.Table I).Method PTP RMSUncontrolled 1.3362 0.2281CMPC 0.7362 0.0711Alg. 4.1, bmax = 50 0.7362 0.0712Alg. 4.1, bmax = 100 0.7362 0.0711TABLE IPERFORMANCE OF THE DIFFERENT MPC SCHEMES.B. Benefits for individual RES in terms of energy pricesAlgorithm 4.1 recovers the performance of CMPC bysolving the problem in a distributed fashion where each RESsolves its own optimal control problem. Additionally the costfunction can be interpreted as actual energy prices. Figure2 shows the time varying prices for 1[kWh] of continuoususage in the corresponding time interval of 30 minutes andthe parameters connected to bmax = 50.0 50 100 15000.511.5Costs for 1[kWh]time in [h]Fig. 2. Half hourly time varying prices (in cents) for continuoususage of 1[kWh].Not only does our proposed scheme meet the goal of thegrid operator, i.e., flattening the aggregated power demand,but also the individual RESs can benefit from increasingtheir battery capacity to lower their electricity costs. Figure 3shows the reduction of the electricity costs of RES 1 andthe impact on the other RESs, if the battery size of RES1 is given by C1 = 2 · c and charging/discharging rateu1 = −u1 = 0.3·c for c = 0, . . . , 5. The energy consumptionof RES 1 during the simulation is 126[kWh]. The averageenergy consumption of the other systems is 92[kWh]. InTable II the costs for energy with changing battery sizesare given.We observe that the installation of additional capacityprovides a benefit to RES 1, albeit with a diminishing return.0 2 4 6 8 100123Battery sizeCost ReductionFig. 3. Cost reduction of the RESs if RES 1 (blue line) increasesthe battery capacity from 0 to 10 in steps of 2.The impact on the other systems is mixed. On the onehand, the installation of additional capacity results in lowerprices since the overall network deviations are reduced. Onthe other hand, RESs who previously benefited from sellingpower at high prices see a reduction in their profit.VI. CONCLUSIONIn this paper, we have presented a novel real-time pric-ing mechanism for networks of residential energy systems(RESs) with the aim of reducing variation in network us-age. This pricing mechanism is based on a central MarketMaker entity that sets prices based on individual predictedusage. Simulations indicate that, when each RES acts so asto minimise its cost, this pricing mechanism recovers theperformance of a centralised optimal control algorithm.REFERENCES[1] K. Worthmann, C. M. Kellett, L. Gru¨ne, and S. R. Weller. Distributedcontrol of residential energy systems using a market maker. In 19thIFAC World Congress, South Africa, pages 11641–11646, 2014.[2] K. Worthmann, C. M. Kellett, P. Braun, L. Gru¨ne, and S. R. Weller.Distributed and decentralized control of residential energy systemsincorporating battery storage. IEEE Transactions on Smart Grid, 2015.Doi: 10.1109/TSG.2015.2392081.[3] P. Braun, L. Gru¨ne, C. M. Kellett, S. R. Weller, and K. Worthmann. Pre-dictive control of a smart grid: A distributed optimization algorithm withcentralized performance properties. Preprint, University of Bayreuth,2015.[4] A. Beja and M. B. Goldman. On the dynamic behavior of prices indisequilibrium. The Journal of Finance, 35(2):235–248, 1980.[5] M. B. Garman. Market microstructure. Journal of Financial Economics,3:257–275, 1976.[6] J. B. Rawlings and D. Q. Mayne. Model Predictive Control: Theoryand Design. Nob Hill Publishing, 2009.[7] L. Gru¨ne and J. Pannek. Nonlinear Model Predictive Control. Theoryand Algorithms. Springer London, 2011.Battery Capacity RES1 av. RES2−30C1 [kWh] ($) ($)0 42.5720 30.87572 41.4419 30.86174 40.5828 30.84746 39.9673 30.82838 39.5356 30.804810 39.2373 30.7848TABLE IIENERGY COSTS FOR RES1 AND THE AVERAGE OF THE COSTS FOR THEOTHER RESS DEPENDING ON THE BATTERY CAPACITY OF RES1 .

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