An Optimal Real-time Pricing Algorithm for the Smart Grid: A Bi-level Programming Approach

Transcription

1 An Optimal Real-time Pricing Algorithm for the Smart Grid: A Bi-level Programming Approach Fan-Lin Meng and Xiao-Jun Zeng School of Computer Science, Univerity of Mancheter Mancheter, United Kingdom mengf@c.man.ac.uk, x.zeng@mancheter.ac.uk Abtract Thi paper propoe an improved approach to our previou work [11]. [11] ue Stackelberg game to model the interaction between electricity retailer and it cutomer and genetic algorithm are ued to obtain the Stackelberg Equilibrium (SE). In thi paper, we propoe a bi-level programming model by conidering benefit of the electricity retailer (utility company) and it cutomer. In the upper level model, the electricity retailer determine the real-time retail price with the aim to maximize it profit. The cutomer react to the price announced by the retailer aiming to minimize their electricity bill in the lower level model. In order to make it more tractable, we convert the hierarchical bi-level programming problem into one ingle level problem by replacing the lower lever problem with hi Karuh Kuhn Tucker (KKT) condition. A branch and bound algorithm i choen to olve the reulting ingle level problem. Experimental reult how that both the bi-level programming model and the olution method are feaible. Compared with the genetic algorithm approach propoed in work [11], the branch and bound algorithm in thi paper i more efficient in finding the optimal olution ACM Subject Claification G.1.6 Optimization Keyword and phrae Real-time Pricing, Demand Repone, Smart Gird, Bi-level Programming, Branch and Bound Algorithm Digital Object Identifier /OASIc.ICCSW Introduction The traditional electricity grid i facing many exiting and potential problem with the increaed demand from cutomer in recent year, and the reliability of the grid ha been put in danger. In addition, the average houehold electricity load ha the potential to double with the deployment of plug-in hybrid electric vehicle (PHEV), which will further endanger the exiting grid. Intead of building more power plant to meet the peak demand of cutomer, demand repone i a better choice for olving the above problem, epecially with the development of the mart grid. Real-time pricing (RTP) i one of the mot important DR trategie, where the price announced by retailer change typically hourly to reflect variation of the price in the wholeale market over time. Generally, cutomer are notified of RTP price the day before or a few hour before the delivery time. One of the mot typical type of RTP i day-ahead RTP, in which cutomer receive the price for the next 24 hour [6]. Part of thi paper appeared in the proceeding of UKCI 2012 [11]. Fan-Lin Meng and Xiao-Jun Zeng; licened under Creative Common Licene CC-BY 2013 Imperial College Computing Student Workhop (ICCSW 13). Editor: Andew V. Jone, Nichola Ng; pp OpenAcce Serie in Informatic Schlo Dagtuhl Leibniz-Zentrum für Informatik, Dagtuhl Publihing, Germany

2 82 An Optimal Real-time Pricing Algorithm for the Smart Grid There exit much literature on RTP. However, the reult and analyi in thi paper differ from the related work in everal apect: In the work of [16], they analytically model the cutomer preference and cutomer electricity conumption pattern in form of utility function and it how that the propoed algorithm can benefit both cutomer and energy provider. However, no explicit form of the cutomer utility function i given. [8] and [18] further develop the work of [16]. Both work ue the ame concept of utility function to model the atifaction of cutomer a [16], but imilarly no explicit form of the utility function are given. A a reult, the approach i unable to help the cutomer to find the bet cheme to minimize their bill. To overcome thi weakne, the approach given in thi paper aim to provide the bet olution for cutomer to achieve the minimal bill. Since the RTP deign need the participation of electricity retailer and it cutomer and the deciion making are equential, i.e., the electricity retailer announce the price firt, then it cutomer react to the price by hifting the energy ue. The interaction between electricity retailer and it cutomer can be repreented a a leader-follower Stackelberg game, and thu can be modelled uing a bi-level programming model. In fact, the bi-level programming problem i a tatic Stackelberg game where two player try to maximize their individual objective function [1]. Due to the hierarchical tructure of the Stackelberg game or bi-level programming model, many real-world problem with two deciion level can be modelled uing the bi-level programming approach. Price etting problem are two deciion level problem and have been tudied for everal year uing bi-level programming approach [14, 15, 7]. [2] propoe a deciion-making cheme for electricity retailer baed on Stackelberg game. They model the cutomer preference and atifaction a utility function. [4] preent an optimal demand repone cheduling with Stackelberg game approach. Similar to [2], they model the cutomer behaviour pattern a utility function. However, no explicit form of utility function are given. The difference between our work and [2] and [4] lie in that we model the follower level problem (lower level problem) with appliance-level detail, which i more practical and thu more difficult to olve. The main focu of thi paper i to propoe a deciion making cheme baed on bi-level programming model for the electricity retailer and it cutomer by conidering the benefit of both participant and give efficient olution to the propoed model. The ret of thi paper i organized a follow. The background of bi-level programming model i introduced in Section 2. In Section 3, the ytem model and the olution method are given. Experimental reult are preented in Section 4. The paper i concluded in Section 5. 2 Background of Bi-level Programming Model Deciion making problem in decentralized organization are often modelled a Stackelberg game, and they are formulated a bi-level mathematical programming problem. The major feature of a bi-level programming problem i that it include two optimization problem within a ingle intance. The lower level execute it own optimal policy after deciion are made at the upper level [10]. The general formulation of a bi-level programming problem can be repreented a follow: (U pper Level) min x F (x, y).t. G(x, y) 0

3 F. Meng and X. Zeng 83 where y = y(x) i implicitly defined by: (Lower Level) min y f(x, y).t. g(x, y) 0 where F i the objective function of the upper level problem; f i the objective function of the lower level problem; G i the contraint et of the upper level deciion vector; g i the contraint et of the lower level deciion vector; x i the deciion vector of the upper level; and y i the deciion vector of the lower level. y = y(x) i called reaction function. To olve the bi-level programming model, one need to obtain the reaction function by olving the lower-level problem and replace the variable y in the upper level problem with the reaction function [17]. However, for many real application, the reaction function y = y(x) i not able to be explicitly repreented. Thu, the problem can not be handled in the above way and become more difficult to olve. Even though the imple linear bi-level programming problem are proven to be NP-hard, there are many method ariing in the lat thirty year for olving the bi-level programming problem, uch a the extreme-point approach for the linear bi-level programming, branch and bound method, decent method, penalty function method and trut region method [5]. In thi paper, the branch and bound method i choen to olve our propoed bi-level programming problem a thi algorithm wa proved to be able to obtain the global optimal olution [1]. 3 Sytem Model and Solution Method In thi ection, we provide a mathematical repreentation of the conidered deciion making problem. Firtly, our focu i to formulate the electricity conumption cheduling problem in repone to the real-time pricing in each houehold a an optimization problem that aim to minimize the payment bill. Secondly, we model the profit optimization problem for the retailer who will offer the 24 hour real-time price to the cutomer. We define P = [p 1, p 2,..., p h,...p H ] a the leader trategy pace, where p h repreent the electricity price at hour h and H repreent the cheduling time window. We aume that p min p h p max, where p min repreent the minimum price that the retailer (utility company) can offer to the cutomer and p max repreent the maximum price that the retailer can offer. It i alo reaonable to aume that the price that the retailer can offer i greater than the wholeale price of each hour. The price of p min and p max are uually deigned baed on hitory data and condition of the wholeale price. However, ince mot of the retail market now are regulated, there exit a price cap for the retail price and p max hould be le than the price cap. For the following part of thi paper, we et H {1, 2,..., H}. Uually, H = 24. We define the et of appliance in the cutomer houehold S. In thi paper, we only conider the one-leader, one-follower cae, i.e., in our model, only one electricity retailer and one cutomer are conidered. 3.1 Lower-level Model Problem Thi model improve that of [13]. In their work, a upper limit for hourly electricity uage i et for each houehold, but we do not have uch contraint for the optimization problem at cutomer ide a there i no uch uage limit in practice. Intead, we conider the total I C C S W 1 3

4 84 An Optimal Real-time Pricing Algorithm for the Smart Grid upper limit of hourly uage in the optimization problem at retailer ide. Thi i to repreent the maximum load capacity of power network. Therefore, we can actually control the hourly ue of electricity of each houehold by properly determining the retail price, which i more practical from an application point of view. For each appliance S, we define an electricity conumption cheduling vector: e = [e 1,..., e h,..., e H ] (1) where H i the cheduling window. For each hour h H {1, 2,..., H}, e h 0 repreent the cutomer electricity conumption of appliance at time h. It i reaonable to aume that the energy conumption of each appliance during a typical day i maintained at the ame level and the total electricity conumed by appliance in a typical day i defined a E. Moreover, the cutomer need to et a valid cheduling window H {α,..., β } by pecifying the beginning operation time α H and the end operation time β H of appliance. Baed on the above analyi, we have β h=α e h = E (2) and e h = 0, h H\H (3) After defining the minimum power level γ min each appliance S, we have and the maximum power level γ max for γ min e h γ max, h H. (4) Then, the payment bill optimization problem for the cutomer can be modelled a follow: H min e h h=1 ph ( S eh ).t. β h=α e h = E, e h = 0, h H\H, γ min e h γ max, h H. (5) 3.2 Upper-level Model Problem In thi ection, we model the profit of the retailer by uing the revenue ubtracting the energy cot impoed on the retailer. We will dicu about the energy cot model firt, and then a profit maximization model will be propoed. In the practical application cenario, to determine the retail price, we need to conider many factor uch a running cot of the retailer including the payment incurred in the wholeale market and o on. For implicity, we define a cot function C h (L h ) indicating the cot of providing electricity by the retailer at each hour h H, where L h repreent the amount of power provided to the cutomer at each hour of the day. We aume that the cot function C h (L h ) i increaing in L h for each h [12, 8, 3]. In view of thi, we deign the cot function a follow [12]. C h (L h ) = a h L h + b h (6)

5 F. Meng and X. Zeng 85 where a h > 0 and b h 0 at each hour h H. For each hour h H, by defining the minimum price that the retailer (utility company) can offer p min and the maximum price p max, we have p min p h p max. Note that there i uually a maximum load capacity, denoted a Eh max, of power network at each hour. Thu, we have following contraint: S e h E max h, h H (7) Then the profit maximization problem can be modelled a (8). max{ p h e h p h C h ( e h )} h H S h H S.t. p min p h p max e h Eh max, h H S (8) 3.3 Solution Method Intead to olve the bi-level problem in it hierarchical form (Eq.(8) and (5)), we convert it into a tandard mathematical program by replacing the follower problem (lower level problem, Eq.(5)) with hi Karuh Kuhn Tucker (KKT) condition. Then a branch and bound algorithm i choen to olve the reulting non-linear programming problem [1]. We adopt the YALMIP olver, which i baed on the above mentioned algorithm and implemented in Matlab, to olve our bi-level programming mode [9]. 4 Experimental Reult We imulate a imple one energy retailer (utility company), one cutomer cae. It i aumed that the cutomer ha 4 appliance: dih waher, wahing machine, clothe dryer and PHEV. Note that the cheduling horizon i from 8AM to 8AM (the next day). For the cot of the energy provided to the cutomer by utility company, we model thi a a cot function. We chooe a imple linear cot function: C h (L h ) = a h L h + b h, where L h repreent the amount of power provided to the cutomer at each hour of the day. For implicity we aume that b h = 0 for all h H. Alo, we have a h = 5.5 cent during the day, i.e., from 8AM to 12AM and a h = 4.0 cent at night hour, i.e., from 13AM to 8AM (the next day). Finally, the parameter etting of thee home appliance can be found in Table 1. Getting idea from the time-of-ue pricing (ToU), we divide the 24 hour price into three level, i.e., peak hour( 5PM-12AM), mid-peak hour(8am-5pm) and off-peak hour(12am- 8AM). For peak hour, the price range from 12 cent to 14 cent. Similarly, the price range Table 1 Home Appliance Parameter Setting. Appliance Name E H γ min γ max Dih waher 1.8kwh 8PM-6AM 0.1kwh 1.0kwh Wahing machine 1.94kwh 8AM-8PM 0.1kwh 1.0kwh Clothe dryer 3.4kwh 7PM-7AM 0.25kwh 3.0kwh PHEV 9.9kwh 8PM-7AM 0.3kwh 2.0kwh I C C S W 1 3

6 86 An Optimal Real-time Pricing Algorithm for the Smart Grid Table 2 24 Hour Optimal Price Offered by the Retailer. Time 8AM 9AM 10AM 11AM 12PM 1PM 2PM 3PM Price(cent) Time 4PM 5PM 6PM 7PM 8PM 9PM 10PM 11PM Price (cent) Time 12AM 1AM 2AM 3AM 4AM 5AM 6AM 7AM Price(cent) Electricity Price (cent) AM 11AM 2PM 5PM 8PM 11PM 2AM 5AM Hour Figure 1 Optimal Real-time Price Offered by the Retailer. from 8 cent to 12 cent for mid-peak hour while the price float between 6 cent to 10 cent for off-peak hour. The aim of our propoed optimal RTP cheme i to find the optimal 24 hour price by maximizing the retailer profit (upper level problem). Beide thi, with thi identified price information, the cutomer can achieve hi bet benefit, i.e., minimize hi payment bill (lower level problem). Applying the open ource olver YALMIP to our propoed bi-level programming problem, we can get the optimal 24 hour price hown a Table 2 and Figure 1. With the purpoe to deign a benchmark, we aume, without our propoed optimal appliance cheduling cheme, the appliance tart working right at the beginning of the time interval H and at it typical power level. The energy conumption comparion of appliance with and without cheduling can be een from Figure 2. We can eaily find from Figure 2 that the cutomer hift the energy ue from high price period to low price period. A a reult, with our propoed cheduling cheme, the electricity bill of the cutomer for one day i reduced from 2.35 $ to 1.94 $. Baed on the above analyi, we can ee that with thi bi-level programming model not only the electricity retailer can maximize hi benefit, but the cutomer can alo benefit from the reduced electricity bill. Lat but not leat, the propoed branch and bound algorithm i more efficient compared

7 F. Meng and X. Zeng with cheduling without cheduling electricity price kwh 2 12 cent AM 11AM 2PM 5PM 8PM 11PM 2AM 5AM Hour 10 Figure 2 Energy Conumption Comparion with Scheduling and without Scheduling under Obtained Optimal Real-time Pricing. with genetic algorithm ued in [11]. Ten eparate experiment for each approach have been done for the computation time comparion. The average time cot of the genetic algorithm approach in obtaining the optimal olution to our propoed bi-level programming model i around 120 econd while the branch and bound algorithm take only around 8 econd. 5 Concluion and Future Work We propoe a bi-level programming approach to model the interaction between the retailer and it cutomer. Firt, a electricity bill minimization model (lower level model) ha been propoed for the cutomer to incentive him to change hi electricity ue pattern. Second, a profit maximization model (upper level model) for the retailer ha been modelled. A branch and bound algorithm i choen to olve thi propoe bi-level programming problem. A the imulation reult how that both the retailer and the cutomer can benefit from the propoed framework, it ha great potential to improve the implementation of current energy pricing program, help cutomer to reduce the increaing energy bill, and change their energy uage pattern. Thi work can be extended in everal direction. Firt, we will enrich the lower level problem by conidering the trade-off between minimizing bill and atifying cutomer comfort. Second, we will extend the current one-leader one follower cae to one-leader multiple-follower bi-level programming model in our future work. Reference 1 Jonathan F Bard and Jame T Moore. A branch and bound algorithm for the bilevel programming problem. SIAM Journal on Scientific and Statitical Computing, 11(2): , S. Bu, F. Richard Yu, and Peter X. Liu. A game-theoretical deciion-making cheme for electricity retailer in the mart grid with demand-ide management. In 2011 IEEE I C C S W 1 3

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