Data Center Demand Response in Deregulated Electricity Markets

Transcription

1 This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/TSG , IEEE Transactions on Smart Gri Data Center Deman Response in Deregulate Electricity Markets Shahab Bahrami, Stuent Member, IEEE, Vincent W.S. Wong, Fellow, IEEE, an Jianwei Huang, Fellow, IEEE Abstract With the evelopment of eregulate electricity markets, a customer can enter a contract with one of several competing utility companies. Meanwhile, a utility company is motivate to increase its market share by helping its customers manage their energy usage an save money through eman response programs. In this paper, we stuy the eman response program in eregulate electricity markets for ata centers that often have significant flexibility in workloa scheuling. We consier the real-time pricing RTP) an moel the ata centers couple ecisions of utility company choices an workloa scheuling as a many-to-one matching game with externalities. To solve such a game, we show that it amits an exact potential function, whose local minima correspon to the stable outcomes of the game. We further evelop a istribute algorithm that guarantees to converge to a stable outcome. Compare with the scenario without ata centers eman response, we show through simulation that the propose algorithm can reuce the average contract payment of ata centers by 8.7% an increase the revenue of the utility companies that offer lower electricity tariffs up to 80% by attracting more ata centers as customers. Keywors: eregulate electricity market, many-to-one matching game, stable outcome, istribute algorithm. I. INTRODUCTION The avances in small-scale power plants an the integration of communication technologies into the power gris have enable eregulate electricity markets for many parts of the worl [], such as the states of Texas an Pennsylvania in the Unite States [2], Alberta in Canaa [3], an Noric countries in Europe [4]. In a eregulate market, a customer is free to purchase electricity from one of several competing utility companies. Meanwhile, the utility companies can take avantage of such flexibility an ynamically set their retail price to attract more customers an gain a larger market share [5]. This motivates the utility companies to eviate from toay s common practice of flat-rate pricing an implement real-time pricing RTP) [6]. When the customers respon to such a pricing scheme through eman response, the utility companies benefit from a smoother energy eman profile an sometimes a higher revenue. The customers, on the other han, can take avantage of the lower prices. Manuscript was receive on Sept. 6, 207, revise on Jan. 0, 208, an accepte on Feb. 2, 208. This work is supporte by the Natural Sciences an Engineering Research Council of Canaa NSERC), the Theme-base Research Scheme Project No. T23-407/3-N) from the Research Grants Council of the Hong Kong Special Aministrative Region, China, an a grant from the Vice-Chancellor s One-off Discretionary Fun of the Chinese University of Hong Kong Project No. VCF20406). S. Bahrami an V.W.S. Wong are with the Department of Electrical an Computer Engineering, The University of British Columbia, Vancouver, BC, Canaa, V6T Z4. J. Huang is with the Department of Information Engineering, The Chinese University of Hong Kong, Hong Kong, {bahramis, vincentw}@ece.ubc.ca, jwhuang@ie.cuhk.eu.hk In this paper, we focus on the ata centers utility company choices an price-base eman response. Data centers often monitor an control the emans of their information technology IT) equipment e.g., servers, routers). Many typical workloas e.g., scientific computations, ata analytics) in ata centers are often elay-tolerant, hence may be rescheule to off-peak hours [7], [8]. This motivates a recent rich boy of literature on the esign of the eman response algorithm for ata centers, which can be ivie into two main threas. The first threa of literature is concerne with the ata centers workloa management problem. Different techniques such as stochastic optimization [9], convex optimization [0], [], an mixe-integer linear programming [2] have been use to tackle the workloa management problem. These works often assume fixe energy price of the utility company an aresse the cost minimizing problem from the ata centers point of view. The secon threa of literature is concerne with the utility companies pricing ecisions for ata centers. Wang et al. in [3] applie a two-stage optimization metho to optimize the energy pricing rates for a utility company, in aition to the optimization of the energy eman profiles for ata centers. Tran et al. in [4] stuie the interactions among utility companies an ata centers by a Stackelberg game moel, where the utility companies obtain a close-form solution to the cost minimization problems of ata centers. In this paper, we stuy the emerging eregulate markets, where multiple utility companies compete to supply electricity to the same group of geographically isperse ata centers. Each ata center can choose a utility company to sign the contract an scheule its workloa to minimize its contract payment. We emphasize that signing contracts with multiple utility companies may not be practical for a ata center in a eregulate electricity market. In particular, market eregulation promotes competition among the utility companies to offer better proucts, rates, an plans to retain an attract customers. Hence, in practice, a ata center can choose the utility company with the best offer that fits its requirements. If a ata center is not satisfie with its current utility company, it can switch to another utility company. A viable example for our propose market mechanism is the eregulate electricity market in Houston in the Unite States, where there are 42 ata centers [5] that can sign bilateral contracts with eight competing utility companies [6]. We consier the RTP from the utility companies, where the payments from the ata centers epen on the amount an time of their electricity usage. With the RTP, the ata centers ecisions are couple together through their choices of utility companies. We capture the ata centers couple ecisions of utility company choices c) 208 IEEE. 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2 This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/TSG , IEEE Transactions on Smart Gri an workloa scheuling as a many-to-one matching game, where each utility company can supply electricity to multiple ata centers, an a ata center can match with one utility company. The unerlying mechanism is a matching with externalities [7], ue to the coupling ecision of ata centers. We characterize the stable outcome of the game, where no ata center has an incentive to change its matche utility company an workloa scheule unilaterally. Such characterization is challenging as there oes not exist a general methoology for solving a matching with externalities. This paper is an extension of our previous work [8] by consiering the energy eman uncertainties, as well as the preference of ata centers an utility companies in their matching choice. The contributions of this paper are as follows: Data Center Workloa Moel: We moel the workloas arrivals an executions in a ata center by a timeepenent multiclass queuing system. Such a moel provies a framework to compute the optimal number of active servers an the execution time of elay-tolerant services, subject to the constraints on the waiting time of the interactive real-time services over the contract perio. Risk-Aware Contract: Data centers have uncertainty about their workloa eman an local renewable generation. We introuce a risk measure calle the conitional valueat-risk CVaR), which enables the ata centers to limit the risk of eviation in the energy eman from the contracte amount. To the best of our knowlege, this is the first work that incorporates risk management in the ata center eman response problem. Solution Metho an Algorithm Design: We characterize an exact potential function an show that the stable outcomes correspon to the local minima of the potential function. One can etermine a local optimal solution to the potential function by solving a mixe-integer nonlinear optimization problem, which is NP har. Instea, we evelop an algorithm that can be execute by the ata centers an utility companies in a istribute fashion an converges to a stable outcome of the game. Performance Evaluation: We perform simulations on a market with 50 ata centers an 0 utility companies. When compare with the scenario without the eman response, our propose algorithm reuces the cost of ata centers an the peak-to-average ratio PAR) of the aggregate eman of ata centers connecte to the same utility company by 8.7% an 8%, respectively. The propose algorithm also enables the utility companies to attract more ata centers as customers by setting lower energy tariffs, an as a result increase their revenue up to 80%. The computational complexity of the propose algorithm is linear with the number of utility companies an inepenent of the number of ata centers. The rest of this paper is organize as follows. Section II introuces the system moel. In Section III, we propose a matching game moel for the ata centers interaction. We also evelop a istribute algorithm to obtain a stable outcome. In Section IV, we evaluate the performance of the propose algorithm. Section V conclues the paper. II. SYSTEM MODEL Consier a system with D ata centers an U utility companies. Let D = {,..., D} an U = {,..., U} enote the set of ata centers an the set of utility companies, respectively. Data center D can purchase electricity from a utility company in a preetermine set U U. Utility company u U is able to serve a preetermine set D u D of ata centers. Sets U, D, an D u, u U, are etermine base on the geographic locations of the utility companies an ata centers as well as the topology of the power network. Fig. a) shows a system with five ata centers an three utility companies. Fig. b) shows the corresponing bipartite graph representation. For example, utility company can sell electricity to ata centers in set D = {, 2, 3}. Data center 4 can choose a utility company from set U 4 = {2, 3}. Each ata center possesses an energy management system EMS) connecte to the utility companies in set U via a twoway communication network. It enables exchanging information such as the energy emans of the corresponing ata center an the energy price for entering a bilateral contract. A. Bilateral Contract an Contract Pricing Moels In eregulate markets, a ata center can enter a bilateral contract with one utility company to purchase electricity. We can capture the contracts between ata centers an utility companies as a many-to-one matching function [9]. Definition : A many-to-one matching among the ata centers an utility companies is a function m : D U PD U), where mu) D u represents the set of ata centers serve by utility company u U, an m) U with m) = represents the utility company choice of ata center D. Here, enotes the carinality an P is the power set of a set. Fig. c) shows a feasible many-to-one matching, where a ata center enters a contract with one utility company, while a utility company can enter contracts with multiple ata centers. We assume that a ata center can enter a short-term contract e.g., one ay) with a utility company [7], [20], [2]. Without loss of generality, we assume that the contract perio of all ata centers starts an finishes at the same time. We ivie the intene contract perio into a set T = {,..., T } of T time slots with an equal length, e.g., 5 minutes per time slot. In matching m, ata center D specifies its energy eman profile e t), t T, to be satisfie by its utility company choice. Each utility company purchases electricity from the wholesale market with a price pt), t T. The utility companies may offer RTP rates to the flexible large loas such as ata centers. In an RTP scheme, the retail price of utility company u epens on both the volume an time of the energy consumption of all its customers. The unit) retail price of utility company u U, in time slot t T, an matching m is an increasing convex function of the total energy eman e u t) = e other u t) + mu) e t), where t) enotes the eman in time slot t for the customers e other u serve by utility company u excluing the ata centers. The retail price is greater than the wholesale price, in orer to guarantee a positive profit for the utility company. We use the c) 208 IEEE. Personal use is permitte, but republication/reistribution requires IEEE permission. See for more information.

3 This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/TSG , IEEE Transactions on Smart Gri 3 Utility companies 2 3 Utility companies Data centers Data centers ) a b) ) c Communication link Electricity line Utility company Data center Energy management system Fig.. a) A system compose of five ata centers equippe with EMS an three utility companies; b) the corresponing bipartite graph representation; c) a feasible many-to-one matching. t 2 [t, t + c, ] with a time-shift probability p c, t, t 2 ). For ata center, let p c, t ) = p c, t, t 2 ), t 2 T ) enote the time-shift probability mass function for the arriving workloas of service class c in time slot t. The EMS of ata center ecies on the probability mass functions p c, = p c, t), t T ) for service classes c C. We have p c, t, t 2 ) = 0, if t 2 [t, t + c, ], 2a) p c, t, t 2 ) [0, ], if t 2 [t, t + c, ], 2b) p c, t, t 2 ) =, t T. 2c) t 2 T first-orer Taylor approximation of the retail price near the wholesale price as follows: p r ue u t), m) = pt)+ κ u t) e u t), u U, t T, ) where κ u t), u U, t T, are nonnegative coefficients with the unit of $ / MWh) 2. The utility companies can etermine κ u t) accoring to the cost of supplying electricity to the ata centers. The linear retail price moel in ) has been use in ifferent problem settings such as the Cournot an Bertran competitive markets [22] an retail power markets [23]. This linear price moel can be extene to a piecewise linear moel, which can be use to approximate nonlinear ynamic pricing schemes. The pricing scheme in ) epens on the energy eman across all ata centers. It motivates ata center towars scheuling its energy eman e t), t T to take avantage of the retail price fluctuations an lower its contract payment t T e t) p r ue u t), m) in matching m, where m) = {u}. B. Data Center s Operation Moel The energy eman of a ata center inclues the eman for the workloas execution [9], [4]. A ata center offers ifferent service classes e.g., vieo streaming, ata analytics) to its customers. Consier ata center D. Let C = {,..., C } enote the set of service classes, where C = C. We assume that both the workloas inter-arrival time an execution time follow the exponential istribution [3], [4]. We can moel the workloas arrivals an executions by a time-epenent multiclass M/M/ queuing system [24]. To manage the energy eman for executing the workloas in a ata center, we consier the possibility of eferring the execution of an incoming workloa to future time slots. To meet the quality-of-service requirements, the elay in executing an incoming workloa nees to be controlle within a certain range, which epens on the type of service request. Let c, enote the maximum number of time slots that the execution of a workloa of service class c C, D, can be elaye. If c, = 0, then the service cannot be elaye ue to its interactive nature. Examples of such interactive services inclue web search, online gaming, an vieo streaming. For the elay-tolerant flexible services, such as scientific applications, ata analytics, an file processing, we have c, [25]. We propose a probabilistic workloa scheuling framework. The EMS in ata center may efer the execution of a workloa of service class c from time slot t T to Let λ c, t) enote the average arrival rate of workloas of service class c C in ata center an time slot t. By workloa scheuling, the workloas in time slot t inclue the newly arriving workloas, which will not be eferre to future time slots. The newly arriving workloas of service class c have an average arrival rate λ c, t) = p c, t, t)λ c, t). The workloas in time slot t also inclue the workloas that are initially in the system at the beginning of time slot t. These workloas inclue the eferre workloas from the previous time slots, an the workloas with incomplete job from time slot t. The average number of workloas of service class c in ata center that are eferre from the previous time slots to time slot t is I c, p c,, t) = t =t t = p c, t, t) λ c, t ). 3) We now compute the average number of workloas from time slot t with incomplete job. Let n t) enote the number of operating active) servers of ata center in time slot t. We consier homogeneous servers in ata center an use ς c, to enote the average time that it takes for a server in ata center to execute a workloa of service class c. In the case with heterogeneous servers, we can use the average processing time over ifferent server types. If there exists only one service class c in ata center, then all the operating servers execute the workloas of service class c in time slot t with the average execution rate µ c, t) = n t)/ς c,. Nevertheless, the operating servers execute multiple service classes simultaneously. In Appenix A, we show that the process of the workloas of service class c can be moele by an equivalent single-class M/M/ queuing system with the workloas average arrival rate λ c, t) an execution rate µ c, t) = µ c, t) ρ t) λ c,t) ) ), 4) µ c, t) where ρ t) = c C λ c, t)/µ c, t) is the average server utilization in time slot t in ata center. We use the steay state approximation to compute the average number of backlog workloas at the en of time slot t as follows [24]: J c, p c,, t) = λ c, t ) µ c, t ) λ c, t ). 5) By characterizing the workloas, we etermine the bouns on the number of operating servers in ata center. The scheule workloas in each time slot shoul be execute in a c) 208 IEEE. 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4 This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/TSG , IEEE Transactions on Smart Gri short perio of time. Let δ c, enote the maximum execution time i.e., the waiting time in the queue plus the servers service time) for the workloas of service class c in ata center. Due to the large number of operating servers in a ata center, the value of δ c, is generally much smaller than the length of one time slot, e.g., δ c, often correspons to a few secons. Our approach, however, is applicable to any value of δ c,. In Appenix A, we show that a workloa of service class c in ata center experience the maximum expecte execution time either at the beginning or at the en of time slot t [26]. We use the steay state approximation of the expecte execution time of an incoming workloa of service class c at the en of time slot t. Thus, we have / µ c, t) λ c, t) ) δ c,. By performing some algebraic manipulations, we obtain ς c, δ c, + c C ς c, λ c,t) n t), c C, D, t T. 6) Moreover, inequality / µ c, t ) λ c, t ) ) δ c, implies that J c, p c,, t) in 5) is upper boune by δ c, λ c, t ). Hence, we can approximate the number of initial workloas of service class c at the beginning of time slot t as I c, p c,, t)+δ c, λ c, t ). The expecte execution time at the beginning of time slot t for a workloa of service class c is +I c, p c,, t)+δ c, λ c, t ) )/ µ c, t). It shoul be less than or equal to δ c,. By performing some algebraic manipulations, we obtain ς c, + ς c, λ c,t) + ς c, λc, t ) λ c, t) ) δ c, c C + ς c, I c, p c,, t) δ c, n t), c C, D, t T. 7) In ata center, the number of operating servers is also upper boune by n max. We have n t) n max, D, t T. 8) Next, we approximate energy eman of ata center for workloas execution by the energy consumption of the operating servers, which have an average utilization ρ t), over the steay state perio in time slot t. Let E ile an E peak enote the average ile energy consumption an the peak energy consumption per time slot of a single server in ata center, respectively. The average energy eman of ata center D in time slot t T can be obtaine by e w t) = η t) n t) E ile + E peak E ile )ρ t) ), 9) where η t) > is the power usage effectiveness of ata center in time slot t. The typical value of η t) for most ata centers is between.5 an 2 [27]. We assume that a ata center possesses a small-scale renewable generator e.g., photovoltaic PV) panel) to partially supply its eman. We also assume that a ata center possesses an energy storage system. The ata center can charge an ischarge the energy storage system to smooth out the fluctuations in the energy eman an renewable generation. Data center scheules the energy storage s charging an ischarging profile e b = eb t), t T ), where eb t) enotes the amount of energy being charge e b t) > 0) to or ischarge e b t) < 0) from the battery energy storage in time slot t. The charging/ischarging rate of the energy storage in ata center has limits e b,min e b,min < 0 an e b,max > 0. That is, e b t) e b,max, D, t T. 0) Let E b,init enote the initial energy level of the energy storage in ata center. The store energy in the storage of ata center until time T T is nonnegative an upper boune by the limit E b,max. Thus, we have 0 E b,init + T t= eb t) Eb,max, D, T T. ) Finally, the total energy eman of ata center in time slot t is obtaine by e t) = e w t) + e b t) e r t), D, t T. 2) C. Risk-Aware Energy Deman Scheuling The accurate preiction of renewable generation is a challenge. The preicte renewable generation e r t) often oes not exactly match with the actual generation level ê r t) in time slot t T. The preicte arrival rate λ c, t) of the workloas of service class c in ata center an time slot t often oes not exactly match with the actual arrival rate λ c, t) either. Let vectors ê r = ê rt), t T ) an er = er t), t T ) enote the actual an preicte generation profiles of the renewable generator in ata center, respectively. Let vectors λ c, = λ c, t), t T ) an λ c, = λ c, t), t T ) enote the profiles of actual an preicte arrival rate of the workloas of service class c in ata center, respectively. We efine vectors λ = λ c,, c C ) an λ = λ c,, c C ) for ata center. The uncertainty in the renewable generation an workloas arrival causes the actual energy eman ê t) to eviate from the preicte energy eman e t). We use 9) an 2) to express ê t) e t) as ê t) e t) = ϕ c, t) λc, t) λ c, t) ) c C where an t + ψ c, t, t) λc, t ) λ c, t ) )) t = ê r t) e r t) ), 3) ϕ c, t) = ς c, η t) p c, t, t) E peak E ile ), t T, ψ c, t, t) = ς c, η t) p c, t, t) E peak, t, t T. The excess energy eman of ata center in time slot t is equal to [ê t) e t)] +, where [ ] + = max{, 0}. Utility company u can set penalties p + u t), t T with the unit of $/MWh to prevent the ata centers from uner-estimating their eman. We efine the cost of risk associate with the excess energy eman of ata center in matching m as R λ, λ, ê r, e r ) = t T p + u t) [ê t) e t)] +, 4) c) 208 IEEE. Personal use is permitte, but republication/reistribution requires IEEE permission. See for more information.

5 This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/TSG , IEEE Transactions on Smart Gri where m) = {u}. Utility company u can etermine the penalties p + u t), t T, by taking into account the real-time prices in the spot electricity market. In this paper, we consier the scenario that each ata center purchases electricity from a utility company to meet its excess energy eman. In a scenario that ata center purchases electricity irectly from the spot market to meet its excess energy eman, the cost moel in 4) without subscript u in p + u t), t T ) can be interprete as the payment of ata center to the spot market. We consier CVaR to etermine appropriate vectors λ an e r [28]. Optimizing the CVaR enables a ata center to use the historical ata recor about its workloas an renewable energy generation to limit the risk of high penalty for the excess energy eman within a confience level. The CVaR for ata center is efine for a confience level β 0, ), an vectors λ an e r as CVaR,β λ, e r ) = E [ R λ, λ, ê r, e r ) R λ, λ, ê r, e r ) α β ], 5) where E[ ] is the expectation over the ranom variables λ an ê r, an we have α β = min { α Pr ) } R ) α β. Minimizing the CVaR in 5) results in the appropriate vectors λ an e r that minimize the expecte value of the penalty R ) when it is higher than α β. In general, the explicit characterization of the probability istributions of the ranom variables λ an ê r are not available. However, it is possible to estimate the CVaR in 5) by aopting the sample average approximation SAA) technique [29]. We use the set J {,..., J} of J samples of ranom variables λ an ê r to estimate Prλj, er,j ), the probability of the scenario with jth sample. The CVaR function in 5) can be approximate by [29] where CVaR,β λ, e r ) min α R Γ,β α, λ, e r ), 6) Γ,β α, λ, e r ) = α + Pr λ j, ) er,j [ ] +. R λ j β, λ, e r,j, er ) α 7) j J D. Preference of the Data Center an Utility Company The total cost of ata center in matching m inclues the bill payment an the CVaR function Γ,β ). Let a = pc,, c C ), e b, µ, λ, e r, α ) enote the scheuling ecision vector of ata center. The contract payment of ata center epens on its utility company choice m) = {u} in matching m an the joint ecision vector a = a, D) of all ata centers through the pricing scheme in ). We have c a, m) = ω cvar ) e t) p r ue u t), m) t T + ω cvar Γ,β α, λ, e r ), 8) where ω cvar is a weight coefficient in the interval [0, ]. For ata center D, we efine preference relation over the pairs a, m) an a, m ) as a, m) a, m ) c a, m) c a, m ). 9) Utility company u prefers a contract with a higher revenue. It also prefers a lower PAR of the aggregate energy eman to improve the performance of the energy network uring peak hours. Reucing the PAR helps the utility company lower its retail price, hence can attract more customers. We consier the following objective function for utility company u U: f u a, m) = ω u )fu rev a, m) ω u fu PAR a, m), 20) where ω u is a weight coefficient in the interval [0, ]. The revenue of utility company u U in matching m is fu rev a, m) = e t) p r ue u t), m), mu) t T an the PAR of the energy eman of the ata centers in set mu) is { max e other fu PAR u t) + mu) e t) } t T a, m) = T t T e other u t) + mu) e t) ). The total energy supply from utility company u is upperboune by e max u, ue to the limite energy buget an transmission capacity in the network. We efine the preference relation u for utility company u U over the pairs a, m) an a, m ) as a, m) u a, m ) { f u a, m) f u a, m ), e u t), e ut) e max u, t T. III. PROBLEM FORMULATION AND ALGORITHM DESIGN 2) Data center aims to minimize its cost in 8) an achieve the most preferre scheuling ecision an utility company choice base on the preference relation in 9). The ecision making of ata centers are interepenent, since the cost of a ata center in 8) is a function the joint ecision vector a of all ata centers as well as the matching structure m. We capture the interactions among the ata centers as a many-toone matching game [9], which is efine as follows: Game Data Center Many-to-One Matching Game): Players: The set of all ata centers D. Strategies: The strategy of ata center is the tuple s = a, m)). Let S enote the feasible strategy space for ata center efine by 2) 3), 7), an constraint m) U. Let s = s, D) enote the strategy profile of all ata centers. Let s enote the strategy profile of all ata centers except ata center. Costs: Data center aims to minimize the cost c s, s ) as in 8), which is a function of strategy profiles of ata center an other ata centers. We emphasize that in the above-mentione matching game, the utility companies are not players an the function forms of their prices are fixe base on ), i.e., parameters κ u t), t T are fixe for each utility company u. That is, Game consiers the perspective of the ata centers. However, utility company u signs a contract with the most preferre ata centers base on the preference relation in 2) c) 208 IEEE. Personal use is permitte, but republication/reistribution requires IEEE permission. See for more information.

6 This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/TSG , IEEE Transactions on Smart Gri Here, the cost of ata center epens on the eman scheules of other ata centers matche to the same utility company as. Hence, our game is a matching game with externalities among the ata centers [7], [9]. For our problem setting, the outcome of the game is a matching m an the joint scheuling ecision profile a of the ata centers. The outcome is stable if there exists no ata center that incurs a lower cost from changing either its matche utility company or its action profile unilaterally [7], [9]. Definition 2 Stable Outcome): A stable outcome of the matching game is the feasible strategy profile s = s, D) such that for D c s, s ) c s, s ), s S. 22) In general, a stable outcome may not exist in a matching game with externalities [7]. Furthermore, there oes not exist a general algorithm that guarantees to compute a stable outcome in a matching game with externalities. To stuy the stable outcomes in Game, we consier the concept of best response strategy of ata center, which is efine as s best s ) arg min c s, s ), D. 23) s S A stable outcome is a fixe point of the best responses of all ata centers. That is, s best s ) = s for all D. Problem 23) for ata center involves choosing a utility company. Hence, it is a non-convex optimization problem with iscrete variables. However, uner the given strategy profile s an matching m, problem 23) can be transforme into a convex optimization problem with quaratic objective function an linear constraints with variables a. There are two steps involve in solving problem 23) for ata center uner a given strategy profile s : a) solving a convex optimization problem for a fixe matching m, an b) comparing the objective value for all utility company choices for a ata center. We prove the existence of a stable outcome for Game by constructing an exact potential function [30]. Such a function is efine as follows: Definition 3 Exact Potential Function): A function P s) is an exact potential for Game, if for any feasible strategy profiles s = s, s ) an s = s, s ), we have c s, s ) c s, s ) = P s, s ) P s, s ). 24) A potential function P s) tracks the changes in the ata center s cost when its strategy changes. In the following theorem, we characterize an exact potential function for Game. Theorem : Game amits an exact potential function P s) = ω cvar ) κ u t) e 2 t) u U t T mu) + e other u t)e t) + e t) e t) ) ) ) + pt)e t) + D < mu) ω cvar Γ,β α, λ, e r ). 25) The proof can be foun in Appenix B. Uner a given matching m, the potential function 25) is a convex function of a. Let a m enote the global minimum of P s) uner a given matching m. Let M enote the set of tuples a m, m) for all matchings m. In Theorem 2, we show that the stable outcomes of the matching game are in set M. Theorem 2: Game has at least one stable outcome, which is in set M. The proof can be foun in Appenix C. We now propose Algorithm execute by the ata centers an utility companies in a istribute fashion to converge to a stable outcome. The propose algorithm is base on the graient ecent metho an best response upate of multiple ata centers. Let i enote the iteration inex. In Algorithm, Lines to 3 escribe the initialization for the ata centers an utility companies. Lines 5 an 6 escribe the information exchange between the ata centers an utility companies about the energy emans an retail prices. Lines 7 to escribe how a ata center chooses a utility company an how a utility company u respons to the requests of the ata centers. To compute the best response strategy s best,i s i ), ata center solves problem 23) for ifferent possible utility company choices m) U, as it is a convex optimization problem uner the given strategy profile s i an utility company choice m). If its current utility company is ifferent from the utility company choice in its best response strategy, ata center sens a termination request to its current utility company. A utility company u uses the preference relation u in 2) to accept the most preferre termination request. Next, each ata center sens a connection request to the utility company in its best response strategy if its termination request has been accepte. A utility company u uses the preference relation u in 2) to accept the most preferre connection request. Note that each utility company can accept at most one termination request an at most one connection request per iteration. Lines 2 to 5 escribe how a ata center upates its strategy s i. If ata center changes its matching, then it upates its scheuling ecision profile with its best response, i.e., a i+ := a best,i. By receiving the upate retail price for the new matching m i+, each ata center that has not change its utility company) computes its upate action profile a i+ base on the following graient upating process: a i+ = [ a i γ i a i c a i, a i, m i+) ], 26) where γ i > 0 is a iminishing step size with i=0 γi = an i=0 γi )2 <, an [ ] is the projection onto the feasible space efine by 2) 3), an 7). The effect of externalities among the ecision making of the ata centers in Game can be observe in Lines 6, 7, 9, an of Algorithm. In Line 6, utility company u upates its retail price, which epens on the scheuling ecisions of all ata centers connecte to that utility company. In Line 7, ata center solves the optimization problem 23), the solution of which epens on the retail prices of the utility companies. In Lines 9 an, the preference relation 2) of utility company u epens on the joint scheuling ecisions of all ata centers connecte to that utility company. Remark : In Lines 9 an of Algorithm, each utility company is allowe to accept at most one termination request an at most one connection request per iteration. As we show c) 208 IEEE. Personal use is permitte, but republication/reistribution requires IEEE permission. See for more information.

7 This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/TSG , IEEE Transactions on Smart Gri Algorithm The Data Center Matching Game Algorithm. Initiation phase : Set i := an ξ := : Ranomly assign each ata center D to a utility company m ) U, an initialize action profile a. 3: Sen parameters κ ut), t T to the ata centers in set D u. Matching phase 4: Repeat Information exchange 5: Each ata center sens e i t), t T to utility company m i ). 6: Each utility company u upates retail prices p r,i u e i ut), m i ) for t T using ) an sens to the ata centers in set D u. Utility company choice 7: Each ata center chooses a utility company in set U by computing its best response strategy in 23). 8: Each ata center sens termination request to its current utility company if it is ifferent from the chosen one. 9: Each utility company u accepts the most preferre termination request base on the preference relation u in 2). 0: Each ata center sens connection request to its chosen utility company if its termination request has been accepte. : Each utility company u accepts the most preferre connection request base on the preference relation u in 2). Strategy upate 2: Each ata center with an accepte connection request upates m i+ ) with the chosen utility. Otherwise, m i+ ):=m i ). 3: Each ata center, that changes its utility company, upates its action profile with its best response, i.e., a i+ := a best,i. 4: Each utility company u communicates the retail price for the upate matching m i+ to the ata centers in D u. 5: Each ata center, that oes not change its utility company, upates a i+ accoring to 26). 6: i := i +. 7: Until No ata center wants to change its strategy, i.e., m i = m i an a i a i < ξ. in Theorem 3, this assumption enables us to prove the convergence of Algorithm to a stable outcome. The intuition behin this assumption is as follows. In Line 9, a utility company accepts the termination requests one by one per iteration, an thereby reucing its retail price graually in orer to retain those ata centers who has submitte termination requests. In Line, a utility company accepts the connection requests one by one per iteration to avoi a suen price increase an losing many ata centers as customers. In the following theorem, we show that Algorithm converges to a stable outcome. Theorem 3: Algorithm globally converges to a stable outcome of the ata center matching game. The proof can be foun in Appenix D. In the following, we evaluate the per-iteration complexity of Algorithm for each ata center an the average number of iterations for Algorithm to converge to a stable outcome. Remark 2: In Line 7 of Algorithm, each ata center solves U convex optimization problems to etermine its best response strategy. Furthermore, the ata centers upate their utility company choice an workloa scheule in parallel. Hence, the per-iteration complexity of Algorithm for ata center is inepenent of the number of ata centers in the system an epens only on the number of utility companies in set U, i.e., O U ). Remark 3: Depening on the number of termination requests an connection requests in Lines 8 an 0 of Algorithm, multiple ata centers between 0 an U ) upate their strategies using their best responses. Hence, Algorithm can be interprete as a best response algorithm with multiple upates per iteration. In [3, Theorem 2], it is shown that in a potential game, the average number of iterations to converge for a best response algorithm with one upate per iteration) is linear with the number of players. The strategy upate of multiple ata centers in one iteration of Algorithm can be roughly interprete as multiple consecutive iterations with one upate. Hence, we can use [3, Theorem 2] to conclue that uner the given number of utility companies, the average number of iterations for Algorithm to converge to a stable outcome is linear with the number of ata centers, i.e., OD). IV. PERFORMANCE EVALUATION In this section, we evaluate the performance of the stable outcome of the matching game. We set the contract perio to be one ay. We ivie a ay into T = 96 time slots, where each time slot is 5 minutes. We consier the electricity market with 0 utility companies serving 50 ata centers, which are free to choose a utility company from a ranom preetermine subset of seven utility companies. Fig. 2a) shows the wholesale market price on Oct. 0, 206 of the Ontario s wholesale market [32]. Each ata center is equippe with a PV plant. We use the historical generation ata for Ontario, Canaa power gri atabase from June, 206 to Oct., 206 [32] to obtain the samples for PV generation. Fig. 2b) shows the average output power of a PV unit over 24 hours obtaine from the historical generation ata. For each ata center, we scale the available historical ata such that the average generation level of the PV plants becomes 0.5 MW per time slot. Each ata center is equippe with an energy storage system. The maximum charging/ischarging rate an capacity of an energy storage is set to 0.05 MW an 0.5 MWh, respectively. The initial energy level of the storage system in each ata center is set to 50% of the maximum capacity. To simulate the arrival rate of the workloas in a ata center, we use the Worl Cup 98 web hits ata [33], which consists of all the requests mae to the 998 Worl Cup website between Apr. 30, 998 an Jul. 26, 998 i.e., 89 ays). This ataset inclues several service classes e.g., HTML, JPEG) for ifferent locations. For simulations, we ranomly select 5 service classes for each ata center an consier the average workloa arrival rates of those service classes in the ataset. We also assume some aitional specifications for the service classes. In ata center, the workloas of service classes c =,..., 5 can be elaye by at most c, = 0, 4, 8, 6, 20 time slots, respectively. We set δ c, at ranom from the interval of [0. sec, 3 sec] for service classes in each ata center. The time of the requests are available in the Worl Cup 98 web hits ataset. We ivie the collection perio into 2 time intervals of seven consecutive ays. Next, for each time interval, we etermine the average number of workloas of service class c in time slot t. We obtain 2 samples of the c) 208 IEEE. Personal use is permitte, but republication/reistribution requires IEEE permission. See for more information.

8 This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/TSG , IEEE Transactions on Smart Gri Wholesale market price cents/kwh) Average PV generation MWh) am 6 am 2 pm 6 pm 2 am Time hour) a) am 6 am 2 pm 6 pm 2 am Time hour) b) Fig. 2. a) Wholesale ay-ahea market prices over 24 hours; b) Average PV generation over 24 hours. average arrival rate for the workloas of service class c in each time slot. We consier n max = 4,000 homogeneous servers with power ratings E ile = 00 kw an E peak = 200 kw per time slot in each ata center. Parameters ς c,, c C for the servers are chosen at ranom from interval [0. sec, 0 sec]. The coefficient ω cvar is set to 0.5, an parameters η t), t T are set to.5 for all ata centers. The confience level β for ata center is chosen at ranom from interval [0.75, 0.85]. Parameters κ u t) for utility companies u =, 2,..., 0 are set to 0.224, 0.208,..., 0.08 $/MWh) 2 for t T, respectively. We set p + u t) = 20 $/MWh, t T, an ω u = 0.8 for all utility companies. We perform simulations using Matlab in a PC with processor IntelR) CoreTM) i7-3770k CPU@3.5 GHz. We first emonstrate how Algorithm enables a ata center to manage its energy eman. The step size in iteration i is set to γ i = / i). For the sake of comparison, we consier the benchmark scenario, where each ata center ranomly chooses a utility company an oes not scheule its workloas. Consier ata center as an example. Fig. 3a) shows the preicte arrival rate of the elay-tolerant workloas in ata center requesting service class 5. With workloa scheuling, the number of workloas uring peak hours is reuce. Fig. 3b) shows that with workloa scheuling, the total number of operating servers in ata center ecreases over the ay, e.g., it is reuce from 4,000 to 2,000 aroun 5 pm. Moreover, Fig. 3c) shows that the energy eman of ata center is reuce by 7.3% from MWh to 9. MWh on average) uring the perio from 2 pm to 6 pm, as a result of workloa scheuling an reucing the number of servers. We then stuy the changes in the cost of ata centers when they implement Algorithm. Fig. 4 shows the aily cost of ata centers to 0 in three scenarios: i) the benchmark scenario without eman response an utility company choice, ii) the scenario without eman response an with utility company choice, an iii) the scenario with both eman response an utility company choice. When comparing the secon scenario with the first scenario, the total cost of the ata centers ecreases by 8.8% on average as a result of choosing their preferre utility company. When comparing the thir scenario with the first scenario, the total cost of the ata centers ecreases by 8.7% as a result of both workloa scheuling Arrival rate workloa per time slot) Average number of operating servers Total energy eman MWh) Without workloa scheuling With workloa scheuling am 6 am 2 pm 6 pm 2 am Time hour) a) am 6 am 2 pm 6 pm 2 am Time hour) b) 8 6 Peak loa reuction 4 2 am 6 am 2 pm 6 pm 2 am Time hour) c) Fig. 3. a) Arrival rates of the workloas of service class 5; b) Total number of operating servers; c) Total energy eman over 24 hours in ata center with an without workloa scheuling. Cost $) Benchmark With utility company choice, 0 3 With utility company choice, scenario without loa scheuling with loa scheuling Fig. 4. Total aily cost of ata center by consiering the opportunities of utility company choice or workloa scheuling. an choosing their preferre utility company. Note that in the secon an thir scenarios, there may exist some ata centers e.g., ata centers 6 an 9) with a higher cost compare with the first scenario. This is ue to the effect of externalities, i.e., the workloa scheuling an utility company choices of other ata centers on the price values. Nevertheless, in the stable outcome, no ata center has an incentive to change its utility company choice an workloa scheuling. Next, we stuy the impact of the confience level β on the energy eman preiction an risk of high excess energy eman for the ata centers. It enables us to stuy the strategy of a ata center in reucing the risk of penalty for excess energy eman in the contract. Fig. 5a) shows the values of of Γ,β ) in 7) for ata center with β in the range between 0.05 to When β increases, the ata center becomes more risk averse an will try to ecrease the risk of high excess energy eman by assigning a higher preicte workloa energy eman an a lower preicte PV generation. Hence, the value of Γ,β ) ecreases when β increases. Fig. 5b) shows the contracte energy eman profiles of ata center with confience levels β = 0.95 an β = 0.8. The contracte energy eman of the ata center is larger in most time slots with the confience level β = That is, the ata center is risk-averse an prefers to sign a contract for a larger amount c) 208 IEEE. Personal use is permitte, but republication/reistribution requires IEEE permission. See for more information.

9 This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/TSG , IEEE Transactions on Smart Gri Γ,β ) Paramater β a) = 0.95 = am 6 am 2 pm 6 pm 2 am Revenue $) a) b) b) Expecte excess eman to the contracte eman %) Parameter β c) Fig. 5. a) The value of Γ,β α, λ, e r ) in 7) for ata center with β within the range between 0.05 an 0.95; b) Total energy eman profile of ata center with β = 0.95 an β = 0.8; c) Total energy eman of ata center with β within the range between 0.05 an of its energy eman, in orer to reuce the risk of penalty payment for the excess energy eman. Fig. 5c) shows the ratio of the expecte excess energy eman to the contracte energy eman for ata center with the confience level β. When β increases from 0.05 an 0.95, the expecte excess energy eman which shoul be supplie either by the utility company or irectly from the spot market) ecreases from 7.8% to 0.07% of the contracte energy eman. It also shows that in general this ata center prefers to sign the bilateral contract with a utility company for most of its energy eman. We now compare the PAR an revenue of the utility companies in three scenarios: i) the benchmark scenario without ata centers eman response an utility company choice, an with the utility companies ranomly selection of a ata center in responing to the connection or termination requests from ata centers, ii) the scenario with ata centers eman response an utility company choice, an with parameter ω u = for the preference relation 2) for the utility companies, an iii) the scenario with ata centers eman response an utility company choice, an with parameter ω u = for the preference relation 2). Recall that the value of ω u inicates the importance of reucing the PAR compare with increasing the revenue in responing to the connection or termination requests from ata centers. Fig. 6a) shows that, compare with the first scenario, the PAR is reuce by 4.6% an 8% on average in the secon an thir scenarios, respectively. The PAR is the lowest in the thir scenario, as for ω u =, reucing the PAR is the most ominant factor for the utility companies in responing to the connection or termination requests. Fig. 6. a) PAR in the generate power; b) Revenue of the utility companies in the benchmark scenario, an the scenarios with ω u = 0 an ω u =. Fig. 6b) shows the revenue of the utility companies in the above-mentione three scenarios. Utility company has the highest parameter κ u t) an utility company 0 has the lowest parameter κ u t). Fig. 6b) shows that the utility companies with a higher parameter κ u t) have a higher average revenue in the benchmark scenario, since ata centers ranomly choose their matche utility company, an utility companies ranomly accept the connection an termination requests from ata centers. However, if ata centers choose their matche utility company accoring to preference relation 9) an utility companies have the preference relation in 2) with ω u = 0, then the revenue of those utility companies with a higher κ u t) ecreases up to 70%) an the revenue of those utility companies with a lower κ u t) increases up to 80%). In fact, in the stable outcome for the secon scenario, the number of ata centers connecte to utility companies to 0 are 2, 3, 3, 3, 4, 5, 5, 7, 8, an 0, respectively. It illustrates that parameter κ u t) affects the market share, an hence the revenue for a utility company. When comparing the secon scenario with the thir scenario, the revenue of the utility companies is 9.8% higher on average, since increasing the revenue is the ominant criterion in the secon scenario. Finally, we evaluate the average number of iterations of Algorithm. Fig. 7a) epicts the convergence of the potential function in one of our simulations for 50 ata centers. The potential function ecreases in each iteration an converges to a stable outcome in 40 iterations. Fig. 7b) shows the average number of iterations versus the number of ata centers for an error tolerance ξ = 0 3. We perform simulations for 20 ranom initial conitions for the matching structure an ata centers energy emans. The number of utility companies is set to 0 in all scenarios. For each scenario, we increase the step size graually an report the smallest number of iterations for convergence. The number of iterations increases in the number of ata centers. However, our algorithm always converges in a reasonable number of iterations. V. CONCLUSION In this paper, we aresse the ata centers problem of choosing utility companies an scheuling workloa in c) 208 IEEE. Personal use is permitte, but republication/reistribution requires IEEE permission. See for more information.

10 This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/TSG , IEEE Transactions on Smart Gri Average number of iterations Number of ata centers b) Fig. 7. a) The value of potential function P s i) per iteration for 50 ata centers; b) The require number of iterations for convergence versus the number of ata centers. a eregulate electricity market. We showe that with the RTP, the ata centers ecisions of utility company choices an workloa scheuling become couple with each other an with the pricing ecisions of the utility companies. We moele the interaction among ata centers as a many-toone matching game with externalities. It was a challenge to prove the existence of the stable outcome of the unerlying matching game. We aresse this challenge by constructing an exact potential function, whose local minima correspon to the stable outcomes of the game. Constructing an exact potential function also enable us to evelop a istribute algorithm to reach a stable outcome. Simulation results showe that the ata centers can ecrease their costs by 8.7% with the propose algorithm, as they can purchase electricity from their preferre utility company an shift their eman to off-peak hours. Meanwhile, the utility companies can achieve an 8% reuction in the PAR. The utility companies with lower tariffs can increase their revenue by 80% through attracting more ata centers as customers. For future work, we plan to exten the propose ata center s operation moel by consiering a time-epenent multiclass G/G/ queuing system, an exten our propose matching algorithm by taking into account the transmission lines congestion an power flow moels. APPENDIX A. Multiclass M/M/ Queuing System Moel In the first step, we etermine an equivalent M/M/ queuing moel for the arrival an execution of the workloas of service class c in ata center an time slot t. In the secon step, we use the transient behaviour of the unerlying queuing system to show that a workloa of service class c in ata center experience the maximum expecte execution time either at the beginning or at the en of each time slot t. a) Consier ata center D in time slot t T. There are multiple service classes in ata center. Hence, ρ t) λ c, t)/µ c, t) = c C, c c λ c, t)/µ c,t) is the proportion of time slot t that the servers are busy with executing the workloas of the service classes other than c. Therefore, ρ t) λ c, t)/µ c, t)) is the proportion of time slot t that the servers in ata center are not busy with executing the workloas of service classes other than c. Note that the parameter µ c, t) = n t)/σ c, is the average execution rate of the workloas in time slot t if all the operating servers only execute the workloas of service class c. Hence, in the unerlying multiclass queuing system, by allocating ρ t) λ c, t)/µ c, t)) of time slot t to execute the workloas of service class c, the average workloa execution rate becomes µ c, t) = µ c, t) ρ t) λ c, t)/µ c, t)) ). Consequently, we can capture the process of the workloas of service class c in ata center an time slot t by an equivalent M/M/ queuing system with an average workloa arrival rate λ c, t) an an execution rate µ c, t). b) We use the transient behaviour of the M/M/ queuing system corresponing to the workloas of service class c. We consier time slot t an use the continuous parameter τ to represent the time elapse since the beginning of time slot t. Hence, τ varies from 0 to one time slot. We rop time inex t from the parameters in the rest of the proof to avoi the potential of confusion. Consier the M/M/ queuing system corresponing to the workloas of service class c in ata center an time slot t. The state of the queue represents the number of workloas in the system in time τ. Let q c, k, k 2, τ) enote the probability of being in state k 2 in time τ when the initial state in time τ = 0 is k. For ata center, let W c, k, τ) enote the expecte execution time of a workloa of service class c in time τ when the initial state is k. We have ) k + W c, k, τ) = q c, k, k, τ). 27) We can rewrite 27) as W c, k, τ) = k=0 µ c, k k=0 + µ c, k=0 µ c, ) q c, k, k, τ) ) q c, k, k, τ). 28) The value of k=0 k q c,k, k, τ) in the first summation of 28) is equal to the expecte number of workloas of service class c in time τ when the queue s initial state is k. We enote this summation by Q c, k, τ). For the summation in the secon term, we have k=0 q c,k, k, τ) =. We can rewrite 28) as W c, k, τ) = Q c,k, τ) + µ c,. 29) We now show that a workloa of service class c in ata center experience the maximum expecte execution time either at the beginning or at the en of each time slot t. It is sufficient to show that function W c, k, τ) in 29) is maximize when either τ = 0 or τ is equal to one time slot. From 29), it is sufficient to etermine the time τ that maximizes function Q c, k, τ). We can compute the erivative of function Q c, k, τ) with respect to τ as [26] Q c,k, τ) = λ c, µ c, q c, k, 0, τ)). 30) c) 208 IEEE. Personal use is permitte, but republication/reistribution requires IEEE permission. See for more information.

11 This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/TSG , IEEE Transactions on Smart Gri We now consier the following three scenarios for k, the initial number of workloas of service class c. ) k = 0: In this scenario, the queuing system corresponing to the workloas of service class c is initially empty. Thus, with probability of one, there is no workloa in the queue in time τ = 0, i.e., we have q c, k, 0, 0) =. From 30), we have Q c, k, 0) = λ c,, which is nonnegative. When time τ elapses from 0 to one time slot, then the value of q c, k, 0, τ) ecreases from one to its steay state value. Meanwhile, from 30), the value of Q c, k, τ) ecreases from λ c, an converges to zero graually. Therefore, the value of Q c, k, τ) increases graually an converges to its steay state value, an hence functions Q c, k, τ) an W c, k, τ) are maximize when τ is equal to one time slot. In this scenario, the maximum value of W c, k, τ) is µ c, λ c,. 2) 0 < k λ c, : Initially there exist some workloas µ c, λ c, of service class c in the queue, but the number of initial workloas is at most λ c, µ c, λ c,. Thus, we have q c, k, 0, 0) = 0. From 30), we have Q k, 0) = λ c, µ c,, which is negative. When time τ elapses from 0 to one time slot, then the value of q c, k, 0, τ) increases from zero to its steay state value. Meanwhile, Q c, k, τ) increases graually from its initial value, i.e., λ c, µ c,, becomes positive, an then ecreases to converges to zero. Therefore, the value of Q c, k, τ) ecreases at the beginning, an then increases graually an converges to its steay state value. Hence, Q c, k, τ) an W c, k, τ) are maximize when τ is equal to one time slot. 3) k > λ c, : We have q µ c, λ c, k, 0, 0) = 0. From 30), c, we obtain Q k, 0) = λ c, µ c,, which is negative. When time τ elapses from 0 to one time slot, then q c, k, 0, τ) increases from zero to its steay state value, an hence the value of Q c, k, τ) increases from its initial negative value, i.e., λ c, µ c, an converges to zero. Thus, Q c, k, τ) ecreases graually an converges to its steay state value. In this scenario, Q c, k, τ) an W c, k, τ) are maximize when τ = 0. The proof is complete B. Proof of Theorem To prove Theorem, we substitute 25) into the right-han sie of 24) an substitute 8) into the left-han sie of 24) for strategies s an s, an show that the results are the same. By changing the strategy of ata center from s to s, its utility company choice is change from u to ũ, an its ecision profile is change from a to ã. Hence, the matching structure is change from m to m an the energy eman of ata center is change from e t) to ẽ t) in time slot t T. By substituting 25) into the right-han sie of 24) for s = s, s ) an s = s, s ), we obtain P s, s ) P s, s ) = ω cvar ) κ u t) e 2 t) + e other u t) e t) t T + < mu) e t) e t) ) + pt) e t) + ω cvar κũt) ẽ 2 t) + e other ũ t) ẽ t) + ẽ t) e t) ) ) ) pt) ẽ t) < mũ) Γ,β α, λ, e r ) Γ,β α, λ ), ẽ) r. 3) In 3), the terms relate to the ata centers other than ata center cancel each other, since the strategy of other ata centers are unchange in s = s, s ). By substituting 8) into the left-han sie of 24) for s = s, s ) an s = s, s ), we have c s, s ) c s, s ) = ) ) ω cvar ) e t) p r ue u t), m) ẽ t) p r ũ e ũt), m) t T + ω cvar Γ,β α, λ, e r ) Γ,β α, λ ), ẽ) r. 32) By substituting the retail price ) into 32), the cost change for ata center will be equal to the potential function change in 3). This completes the proof. C. Proof of Theorem 2 We first show that the global minimum of the potential function 25) is a stable outcome. Let s = a m, m ) be the global minimum of P s). Thus, if ata center changes its action profile to a or its utility company to m) unilaterally, then the potential function increases. The change in the exact potential function is equal to the change in the cost of the eviating ata center. Hence, the cost of ata center increases as well. Thus, no unilateral eviation from s can reuce the cost of any ata center, an hence s is a stable outcome of Game. As the exact potential function 25) has at least one global minimum, the matching game has at least one stable outcome. Next we show that an arbitrary stable outcome a, m) is in set M. We prove this by contraiction. Suppose that a stable outcome a, m) is not in set M. Hence, we have a a m. By efinition, a m is the global minimum of the potential function uner matching m. We also know that P a, m) is a convex function of a. Thus, a unilateral change of a for any ata center in the opposite irection of the graient a P a, m) will reuce the potential function, an thus the cost of that ata center. It contraicts the supposition that a, m) is a stable outcome. Hence, a, m) is in set M. D. Proof of Theorem 3 Since the potential function 25) is lower boune it is always positive), it is sufficient to show that the potential function ecreases in each iteration of Algorithm. Line 7 of Algorithm guarantees that if the algorithm converges, the result is a stable outcome. Our proof involves two steps: Step a) We show that the potential function ecreases when the ata centers upate their utility company choices in Line 2 of Algorithm. Each ata center upates its utility company choice to reuce its cost. Nevertheless, we cannot irectly use 24) in Definition 3 to conclue that the potential c) 208 IEEE. Personal use is permitte, but republication/reistribution requires IEEE permission. See for more information.

12 This article has been accepte for publication in a future issue of this journal, but has not been fully eite. Content may change prior to final publication. Citation information: DOI 0.09/TSG , IEEE Transactions on Smart Gri function ecreases accoringly. In fact, equality 24) hols if a ata center changes its utility company choice unilaterally. However, in Line 2 of Algorithm, several ata centers may upate their utility company choices simultaneously. Consier iteration i of Algorithm. We prove by inuction that the potential function ecreases when k ata centers upate their utility company choices simultaneously, where k is an arbitrary number. Base case: Base case: Consier k =. If one ata center upates its utility company choice, then it correspons to a unilateral change in the strategy of the ata center to reuce its cost. Hence, from 24) in Definition 3, the potential function P s i ) ecreases, when the cost of a ata center ecreases. Inuction step: Consier k =. Suppose that the potential function P s i ) ecreases, when ata centers change their utility company choices in Line 2 of Algorithm. We prove that the potential function also ecreases when k = + ata centers change their utility company choices simultaneously. Without loss of generality, we enote the inex set of ata centers that change their utility company choices as {,..., +}, which is partitione into two sets: {,..., } an {+}. The simultaneous changes in the utility company choices of ata centers {,..., +} correspon to the simultaneous changes in the utility company choices of ata centers {,..., }, an the change of utility company choice for ata center +. From our inuction assumption, when ata centers {,..., } change their utility company choices, the potential function ecreases. In the following, we show that the potential function further ecreases when ata center + changes its matche utility company. Suppose that ata center + ecies to leave utility company m i + ) = {u i + } in iteration i an connect to utility company m i+ +) = {u i+ + }. Accoring to Line 7 of Algorithm, such a ecision is the best response of ata center +. Hence, the cost of ata center + shoul ecrease from c + i i+ i+ to c +, i.e., c + < c + i if other ata centers o not change their utility company choices. However, ata centers {,..., } change their utility company choices at the same time. Therefore, the cost of ata center + changes from c i + to ci+ +. We show that ci + c + i an c i+ i+ i+ + c +. Then, inequality c + < c + i will lea to inequality c i+ + < ci +. We first show that c i + c + i. A utility company accepts at most one termination request from ata centers per iteration. Thus, ata center + is the only one that leaves utility company u i +. However, utility company ui + may accept a new connection request from a ata center in set {,..., }, which increases the total eman from this utility company. Therefore, the payment of ata center + to utility company u i + will increase after upating the matching of ata centers {,..., }. That is, we have c i + c + i. Next we show that c i+ i+ + c +. A utility company accepts at most one connection request from ata centers per iteration. Hence, ata center + is the only one that connects to utility company u i+ +. Nevertheless, utility company ui+ + may accept a termination request from a ata center in set {,..., }, which ecreases the total eman from this utility company. Hence, the payment of ata center + to utility company u i+ + will ecrease after upating the matching of ata centers {,..., }. That is, we have c i+ i+ + c +. We can conclue that after ata centers in set {,..., } change their utility company choices, we have c i+ + < ci +. From 24) in Definition 3, the exact potential function ecreases when the cost of ata center + ecreases. We have shown that the potential function will ecrease when k = + ata centers change their utility company choices simultaneously. By the principle of inuction, the potential function ecreases when any number of ata centers change their utility company choices simultaneously. Step b) Uner a given matching m i+, 24) implies that a i c a i, a i, mi+) = a i P a i, ai, mi+). If all ata centers use 26) for upate, the potential function varies in the opposite irection of its graient, a i P a i, ai, mi+). Uner a given matching m i+, P ) is a convex function of a i an has Lipschitz continuous erivative. Thus, for sufficiently small step size, the opposite graient irection is a ecreasing irection. We note that the step sizes γ i are not require to be equal for all ata centers D. Unequal but sufficiently small step sizes lea to upating the ecision vector a i in the opposite subgraient irection of the potential function. 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Wong S 94, M 00, SM 07, F 6) receive the B.Sc. egree from the University of Manitoba, Winnipeg, MB, Canaa, in 994, the M.A.Sc. egree from the University of Waterloo, Waterloo, ON, Canaa, in 996, an the Ph.D. egree from the University of British Columbia UBC), Vancouver, BC, Canaa, in From 2000 to 200, he worke as a systems engineer at PMC-Sierra Inc. now Microsemi). He joine the Department of Electrical an Computer Engineering at UBC in 2002 an is currently a Professor. His research areas inclue protocol esign, optimization, an resource management of communication networks, with applications to wireless networks, smart gri, mobile clou computing, an Internet of Things. Dr. Wong is an Eitor of the IEEE Transactions on Communications. He has serve as a Guest Eitor of IEEE Journal on Selecte Areas in Communications an IEEE Wireless Communications. He has also serve on the eitorial boars of IEEE Transactions on Vehicular Technology an Journal of Communications an Networks. He was a Technical Program Co-chair of IEEE SmartGriComm 4, as well as a Symposium Co-chair of IEEE ICC 8, IEEE SmartGriComm 3, 7) an IEEE Globecom 3. He is the Chair of the IEEE Vancouver Joint Communications Chapter an has serve as a Chair of the IEEE Communications Society Emerging Technical Sub-Committee on Smart Gri Communications. He receive the 204 UBC Killam Faculty Research Fellowship. Jianwei Huang F 6) is a Professor in the Department of Information Engineering at the Chinese University of Hong Kong. He is the co-author of 9 Best Paper Awars, incluing IEEE Marconi Prize Paper Awar in Wireless Communications 20. He has co-authore six books, incluing the textbook on Wireless Network Pricing. He has serve as the Chair of IEEE TCCN an MMTC. He is an IEEE ComSoc Distinguishe Lecturer an a Clarivate Analytics Highly Cite Researcher. Shahab Bahrami S 2) receive the B.Sc. an M.A.Sc. egrees both from Sharif University of Technology, Tehran, Iran, in 200 an 202, respectively. He receive the Ph.D. egree from the University of British Columbia UBC), Vancouver, BC, Canaa in 207. Dr. Bahrami continue to work as a post-octoral research fellow at UBC until Jan His research interests inclue optimal power flow analysis, game theory, an eman sie management, with applications to smart gri c) 208 IEEE. Personal use is permitte, but republication/reistribution requires IEEE permission. See for more information.