Demand Response for Data Centers in Deregulated Markets: A Matching Game Approach

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1 Deman Response for Data Centers n Deregulate Markets: A Matchng Game Approach Shahab Bahram, Vncent W.S. Wong, an Janwe Huang? Department of Electrcal an Computer Engneerng, The Unversty of Brtsh Columba, Vancouver, Canaa? Department of Informaton Engneerng, The Chnese Unversty of Hong Kong, Hong Kong emal: {bahrams, vncentw}@ece.ubc.ca, jwhuang@e.cuhk.eu.hk Abstract Wth the fast evelopment of eregulate electrcty markets, a user can enter a contract wth a utlty company that offers the best rates among multple competng utlty companes. Meanwhle, a utlty company s motvate to ncrease ts market share by offerng eman response programs wth real-tme prcng (RT), whch can help ts customers to manage ther energy usage an save money. In ths paper, we focus on the eman response program n eregulate markets for ata centers, whch are often flexble n scheulng ther workloas. We capture the stochastc workloa process n a ata center as a multclass queung system. We moel the couple ecsons of utlty company choces an workloa scheulng of ata centers as a many-to-one matchng game wth externaltes. Analyzng such a game s challengng, as there oes not exst a general algorthm that guarantees to fn a stable outcome, where no player has an ncentve to unlaterally change ts strategy. We show that the ata center matchng game amts an exact potental functon, whose local mnma correspon to the stable outcomes of the game. We evelop an algorthm that can guarantee to converge to a stable outcome. Compare wth the scenaro wthout utlty company choces an eman response, smulatons show that our propose algorthm can reuce the cost of ata centers by.4% an ncrease the revenue of those utlty companes wth lower tarffs by up to 82%. The peak-to-average rato (AR) of the customers loa eman s also reuce by 7.2%. I. INTRODUCTION Recent avances n small-scale power plants an the ntegraton of communcaton technologes nto the power networks have enable utlty companes to enter the eregulate electrcty markets []. In such a market, a customer s free to purchase electrcty from one of several competng utlty companes. Meanwhle, the utlty companes can take avantage of such flexblty an choose ther retal prces. Ths motvates the utlty companes to evate from toay s common practce of flat-rate prcng an mplement real-tme prcng (RT) [2], [3]. By mplementng a eman response program wth RT, the utlty companes can beneft from a smoother energy eman profle, an acheve a hgher revenue by attractng more customers. The customers, on the other han, can take avantage of the lower prces. In ths paper, we focus on the choces of utlty company as well as eman response of a specal type of customers ata centers. Data center owners often closely montor an control the eman of ther nformaton technology (IT) equpment (e.g., servers, routers) an coolng facltes. Many typcal workloas n ata centers (e.g., hgh complexty scentfc computatons, an ata analytcs) are elay-tolerant, an hence may be rescheule to off-peak hours [4]. Ths motvates a recent rch boy of lterature on the eman response algorthm esgn for ata centers (e.g., [] [9]). We can classfy the relate lterature nto two man threas. The frst threa s concerne wth the soluton approaches for the workloa management problem n ata centers. Dfferent technques such as stochastc optmzaton [] an convex optmzaton [6], [7] are use to tackle the workloa management problem. In these works, the energy prce of the utlty company s fxe an the focus s to solve a cost mnmzng problem for the ata centers. The secon threa s concerne wth moelng the actve prcng ecsons of the utlty companes for ata centers. Wang et al. n [8] apple a two-stage optmzaton metho to moel the nteractons of a utlty company s prcng optmzaton an the ata centers energy eman optmzaton. The approach n [8] may not be rectly apple to the case wth multple ata centers. Tran et al. aresse a relate problem n [9], where the utlty companes nee to obtan the close-form soluton to the ata centers cost mnmzaton problem. Ths may not be always feasble n practce. In ths paper, we stuy the emergng eregulate markets, where multple utlty companes compete to supply electrcty to the same group of geographcally sperse ata centers. Each ata center can choose whch utlty company to sgn the contract an scheule ts workloas to mnmze ts payment. If the utlty companes aopt the RT scheme, the ata centers payments wll epen on the amount an the tme of ther electrcty consumptons. Thus, the ecsons of ata centers are couple among each other an wth the prcng ecsons of the utlty companes. We capture the ata centers couple ecsons of utlty company choces an workloa scheulng as a many-to-one matchng game. The unerlyng mechansm s a matchng wth externaltes [0], as the payments of the ata centers choosng the same utlty company epen on the workloa scheulng of each other. We characterze the stable outcome of the game, where no ata center has an ncentve to change ts matche utlty company an workloa scheule unlaterally. Such characterzaton s qute challengng, as there oes not exst a general algorthm that can guarantee to fn a stable outcome n matchng wth externaltes. The contrbutons of ths paper are as follows: Data Center Workloa Moel: We approxmate the workloas arrvals an executons n a ata center over the contract pero by a multclass queung system. Such a moel enables us to scheule the number of

2 operatng servers, meanwhle satsfyng the qualty-ofservce (QoS) requrements n executng fferent servce requests. Soluton Metho an Algorthm Desgn: We characterze an exact potental functon of the matchng game, an show that the stable outcomes of the game correspon to the local mnma of the potental functon. We evelop an algorthm that can be execute by the ata centers an utlty companes n a strbute fashon. We prove that the algorthm converges to a stable outcome of the game. erformance Evaluaton: We perform smulatons on a market wth 0 ata centers an 0 utlty companes. The results show that the propose algorthm reuces the cost of ata centers an the peak-to-average rato (AR) n the aggregate eman of ata centers connecte to the same utlty company by.4% an 7.2%, respectvely. Furthermore, the utlty companes that offer lower energy prces can ncrease ther revenue by up to 82%, as they can attract more ata centers as customers. The rest of ths paper s organze as follows. Secton II ntrouces the system moel. In Secton III, we propose a matchng game moel for the ata centers nteracton. We also evelop a strbute algorthm to obtan a stable outcome. In Secton IV, we evaluate the performance of the propose algorthm. Secton V conclues the paper. II. S YSTEM M ODEL Conser a system wth D ata centers an U utlty companes. Let D = {,..., D} an U = {,..., U } enote the set of ata centers an the set of utlty companes, respectvely. Data center 2 D can purchase electrcty from a utlty company n a preetermne set U U. Utlty company u 2 U s able to serve a preetermne subset of ata centers enote by Du D. Sets U, 2 D an Du, u 2 U are etermne base on the the topology of the network an the geographc locatons of the utlty companes an ata centers. Fg. (a) shows a system wth fve ata centers an three utlty companes. Fg. (b) shows the corresponng bpartte graph representaton. Each ata center possesses an energy management system (EMS), whch s connecte to the utlty companes n set U va a two-way communcaton network. The EMS enables exchangng nformaton such as the energy consumpton of the corresponng ata center an the energy prce for enterng a blateral contract. In eregulate markets, a ata center can enter a blteral contract wth one utlty company to purchase electrcty. Meanwhle, a utlty company can supply electrcty to multple ata centers. We can capture the contracts between ata centers an utlty companes as a many-to-one matchng [], whch s efne as follows. Defnton : A many-to-one matchng among the ata centers an utlty companes s a functon m : D [ U! (D [ U ), where m(u) Du represents the set of ata centers serve by utlty company u 2 U, an m() U wth m() = represents the utlty company choce of ata center 2 D. Here, enotes the carnalty an s the power set of a set. Fg.. (a) A system compose of fve ata centers equppe wth EMS an three utlty companes; (b) the corresponng bpartte graph representaton; (c) a feasble many-to-one matchng. Fg. (c) shows a feasble many-to-one matchng. Although short-term contracts are not common for resental customers n toay s eregulate markets, large loas such as ata centers can enter a contract wth utlty companes for a pero from a few hours to several ays [4], [2]. We assume that a ata center can enter a short-term contract (e.g., one ay) wth a utlty company. Wthout loss of generalty, we conser the same contract pero for all ata centers. We ve the ntene contract pero nto a set T = {,..., T } of T tme slots wth an equal length, e.g., one tme slot s mnutes. In matchng m, utlty company u sets ts retal prce pru (t), t 2 T, for the contracts wth the ata centers n set m(u). Data center specfes ts eman profle e (t), t 2 T, to be supple by ts utlty company choce m(). ) Contract rcng Moel: In general, a utlty company purchases electrcty from the wholesale market wth a prce p(t), t 2 T, etermne from the eman-supply balance n the wholesale market. The utlty companes may offer ynamc electrcty rates to the flexble large loas such as ata centers. In the ynamc prcng scheme, the retal prce of utlty company u epens on the tme of energy consumpton, as well as the total energy eman from ts customers. In partcular, the retal prce of utlty company u 2 U n tme slot t 2 T an matchng m s an ncreasng functon of the total energy eman eu (t) = eother (t) + u 2m(u) e (t), where eother (t) enotes the eman n tme slot t for the customers u other than the ata centers serve by utlty company u. The retal prce s greater than the wholesale prce, n orer to guarantee a postve proft for the utlty company. We conser the lnear approxmaton of the retal prce (aroun the wholesale prce) as follows [3]: pru (eu (t), m) = p(t)+ u (t) eu (t), u 2 U, t 2 T, () where u (t), u 2 U, t 2 T, are nonnegatve coeffcents wth the unt of $ / MWh2. The utlty companes can etermne u (t) accorng to the cost of supplyng electrcty. The ynamc prcng scheme n () motvates ata center towars scheulng ts energy eman e (t), t 2 T to beneft from the retal prce fluctuatons. Next, we escrbe how a ata center can manage ts energy eman. 2) Data Center s Operaton Moel: A ata center offers fferent servce classes (e.g., veo streamng, ata analytcs) to ts customers. Let C = {,..., C } enote the set of servce classes that are offere n ata center 2 D, where C = C. To meet the QoS requrements, the elay n

3 executng a workloa s lmte wthn a certan range. Let c, enote the elay that the executon of a workloa of servce class c 2C can tolerate. A small c, correspons to the nteractve servces that are nflexble ue to strngent elay requrements, such as web search, onlne gamng, an veo streamng. A large c, correspons to the elay-tolerant servces, such as scentfc applcatons, ata analytcs, an fle processng []. We now scuss how a ata center can scheule the number of operatng servers to meet the QoS requrements. We assume that both the workloas nter-arrval tme an executon tme follow the exponental strbuton [8], [9]. For ata center, a workloa requestng servce class c 2C arrves wth an average rate of c,(t), t2t, workloas per tme slot. Let c, enote the average tme t takes for a server n ata center to execute a workloa requestng servce class c. Let n (t) enote the average number of operatng servers of ata center n tme slot t. If all the servers n ata center execute the workloas of servce class c, the corresponng average executon rate n tme slot t s obtane as µ c, (t) = n (t). However, c, the servers n ata center execute C servce classes. Let c,(t) (t) = c2c µ c, (t) enote the server utlzaton of ata center n tme slot t. The proporton of tme that the servers are busy to execute the workloas of servce classes other than c s (t) c, (t), where c, (t) = c, µ c, (t). Hence, the proporton of tme that the servers are busy to execute the workloas of servce class c s ( (t) c, (t)). Thus, we can moel a ata center by a multclass M/M/ queung system, where the executon rate of the workloas of servce class c s µ c, (t) =( ( (t) c, (t)))µ c, (t). We can show that a workloa of servce class c experences the maxmum expecte watng tme ether at the begnnng or at the en of each tme slot t [4]. The watng tme at the begnnng of tme slot t epens on the number of workloas whose jobs have not been complete yet. For smplcty, we wll conser the steay state approxmaton of the average number of workloas at the en of tme slot t. Hence, an ncomng workloa of servce class c experences the c,(t ) average watng tme of + µ c, (t ) c,(t ) /µ c,(t) at the begnnng of tme slot t. To satsfy the elay requrement, t shoul be less than or equal to c,. We can rewrte µ c, (t ) an µ c, (t) n terms of n (t ) an n (t), respectvely. We can approxmate (t ) c, (t ) (t) c, (t) ue to the small changes n the proporton of tme that the servers are busy to execute the workloas of servce classes other than c over two consecutve tme slots. By performng some algebrac manpulatons, for tme slots t, t2t, we obtan ( c, / c, ) c2c + c, c,(t ) apple, c2c,2d. (2) n (t) n (t ) Inequalty (2) mples that f the number of servers n tme slot t s small, then the number of workloas wth ncomplete jobs ncreases. Hence, the number of servers n tme slot t shoul be suffcently large to meet the maxmum elay constrant at the begnnng of tme slot t. We use the steay state conton to approxmate the watng tme of an ncomng workloa of servce class c at the en of tme slot t. Thus, we nee to satsfy µ c, (t) c,(t) apple c,. By performng some algebrac manpulatons, we can rewrte the elay requrement as c, c, + X c2c c, c,(t) apple n (t), c 2C,2D,t2T. (3) In ata center, the number of operatng servers s upper boune by n max. That s n (t) apple n max, 2D,t2T. (4) Let E le an E peak enote the average le energy consumpton an the peak energy consumpton per tme slot of a server n ata center, respectvely. The average energy eman of ata center 2Dn tme slot t 2T can be obtane by e (t) = (t) n (t) E le +(E peak E le (t), () where (t) s the power usage effectveness (UE) of ata center n tme slot t. The typcal value of (t) for a ata center s between. an 2 []. III. ROBLEM FORMULATION AND ALGORITHM DESIGN Let a =(n (t), t2t) enote the scheulng ecson vector of ata center. Base on the prcng scheme n (), the contract payment of ata center to utlty company u = m() epens on the matchng m an the jont ecson vector a =(a,2d) of all ata centers. Hence, we have c (a,m)= t2t e (t) p r u(e u (t),m). (6) The ecson makng of ata centers are nterepenent. We capture the nteractons among the ata centers as a many-toone matchng game, whch s efne as follows []: Game Data Center Many-to-One Matchng Game: layers: The set of all ata centers D. Strateges: For ata center, the utlty company choce m() 2U an scheulng ecson a satsfy constrants (2) (). We enote the strategy of ata center by the tuple s =(a,m()). Let S enote the feasble strategy space for ata center efne by (2) () an constrant m() 2U. Let s =(s,2d) enote the jont strategy profle of ata centers. Let s enote the strategy profle of all ata centers except ata center. Costs: Data center ncurs a cost c (s, s ) as n (6), whch s a functon of strategy profle s of ata center an the strategy s of other ata centers. Notce that the cost of a ata center epens on the eman scheules of other ata centers that are matche to the same utlty company as. Hence, our game s a matchng game wth externaltes [0], []. The outcome of the game s a matchng m an the jont scheulng ecson profle a of the ata centers. The outcome s stable when no ata center wll ncur a lower cost from changng ether ts matche utlty company or ts acton profle unlaterally [].

4 Defnton 2: A stable outcome of the matchng game s the feasble strategy profle s? =(s?,2d) such that for 2D c (s?, s? ) apple c (s, s? ), s 2S. (7) A ata center s best response strategy s the choce that mnmzes ts own cost, assumng that the strateges of other ata centers are fxe. That s s best (s ) 2 arg mn c (s, s ), 2D. (8) s 2S A stable outcome s a fxe pont of the best responses of all ata centers. That s, s best (s? )=s? for all 2D. roblem (8) for ata center nvolves choosng a utlty company, an t s a nonconvex optmzaton problem wth screte varables. However, uner the gven matchng m, the objectve functon (6) an constrants (2) () can be expresse as posynomals. Hence, problem (8) s a geometrc program [6], whch can be transforme nto a convex optmzaton problem wth varables a. There are two steps nvolve n solvng problem (8) for ata center uner a gven strategy profle s : (a) solvng a convex optmzaton problem for a fxe matchng m, an (b) comparng the objectve value for all utlty company choces for ata center. In general, a stable outcome may not exst n a matchng game wth externaltes [0]. We prove the exstence of a stable outcome for Game by constructng an exact potental functon [7]. Such a functon s efne as follows: Defnton 3: A functon (s) s an exact potental for Game, f for any feasble strategy profles s =(s, s ) an es = (es, s ), we have c (s, s ) c (es, s )= (s, s ) (es, s ). (9) A potental functon (s) tracks the changes n the ata center s cost when ts strategy changes. In the followng theorem, we characterze an exact potental functon for Game. There s no generc metho of constructng a potental functon, an t requres explorng the structure of the problem. Theorem Game amts an exact potental functon (s) = u2u t2t 2m(u) p(t)+apple u (t)e other u (t) e (t) + apple u (t) e 2 (t) + apple u (t) < 0 2m(u) e (t) e 0(t). (0) The proof can be foun n Appenx A. Uner a gven matchng m, the potental functon (0) s a convex functon of a. Let a m enote the global mnmum of (s) uner a gven matchng m. Let M enote the set of tuples (a m,m) for all matchngs m. In Theorem 2, we show that the stable outcomes of the matchng game are n set M. Theorem 2 Game has at least one stable outcome. All stable outcomes are n set M. The proof can be foun n Appenx B. One can use the exstng algorthms base on the best response upate to etermne a stable outcome [8]. These algorthms, however, often suffer from a low convergence rate, as only one sngle ata center upates ts strategy per teraton. We propose Algorthm that can be execute by the ata centers an utlty Algorthm The Data Center Matchng Game Algorthm. : Set := an := : Ranomly assgn each ata center 2D to a utlty company m () 2U, an ntalze acton profle a. 3: Sen parameters apple u(t),t2t, to the ata centers n set D u. 4: Repeat : Each ata center sens e (t),t2t to utlty company m (). 6: Each utlty company u upates retal prces p r, u (e u(t),m ) for t 2T usng () an sens to the ata centers n set D u. 7: Each ata center chooses a utlty company n set U by computng ts best response strategy n (8). 8: Each ata center sens termnaton request to ts current utlty company f t s fferent from the chosen one. 9: Each utlty company u accepts at most one termnaton request. 0: Each ata center sens connecton request to ts chosen utlty company f ts termnaton request has been accepte. : Each utlty company u accepts at most one connecton request ranomly. 2: Each ata center wth an accepte connecton request upates m + () wth the chosen utlty. Otherwse, m + ():=m (). 3: Each ata center, that changes ts utlty company, upates ts acton profle wth ts best response,.e., a + := a best,. 4: Each utlty company u communcates the retal prce for the upate matchng m + to the ata centers n D u. : Each ata center, that oes not change ts utlty company, upates a + accorng to (). 6: := +. 7: Untl No ata center wants to change ts strategy,.e., m = m an a a <. companes n a strbute an parallel fashon to converge to a stable outcome. Let enote the teraton nex. The EMS of the ata centers are responsble for the computatons an message exchange. Fg. 2 shows the schematc of matchng upate n teraton for fve ata centers an three utlty companes n Fg. (b). Our algorthm nvolves the ntaton phase an matchng phase. Intaton phase: Lnes to 3 escrbe the ntalzaton for the ata centers an utlty companes. Matchng phase: The loop from Lnes 4 to 7 escrbes the matchng phase. It nclues the followng parts: a) Informaton exchange: Lnes an 6 escrbe the nformaton exchange between the ata centers an utlty companes about the energy emans an retal prces. Ths step s shown n Fg. 2 (a). b) Utlty company choce: Lnes 7 to escrbe how ata center chooses a utlty company an how utlty company u responses to the requests of the ata centers. For example, Fg. 2 (b) shows that ata centers 2, 3, an 4 sen termnaton request to utlty company 2. Data center sens termnaton request to utlty company 3. We allow a utlty company to accept at most one termnaton request an at most one connecton request n each teraton. A utlty company accepts the requests at ranom, snce t s nfferent between ata centers. Fg. 2 (b) shows that utlty company 2 accepts the termnaton request from ata center 2. Utlty company 3 accepts the termnaton request from ata center. Fg. 2 (c) shows that ata centers 2 an sen connecton requests to utlty companes an 2, respectvely. Fg. 2 () shows the upate matchng structure.

5 Number of Servers 2 0 Wthout Deman Resposne Wth Deman Rresponse 2 am 6 am 6 pm 2 pm 2 am Tme (hour) Total Energy Deman (MWh) (a) eak Loa Reucton Wthout Deman Response Wth Deman Response 2 am 6 am 2 pm 6 pm 2 am Tme (hour) Contract ayment ($) (b) Fg. 2. Matchng upate proceure n Algorthm. (a) Informaton exchange among ata centers an utlty companes n matchng m ; (b) termnaton requests from ata centers; (c) connecton requests from ata centers; () upate matchng m+. c) Strategy upate: Lnes 2 to escrbe how ata center upates ts strategy s = (a, m ()). If ata center changes ts matchng, then t upates ts scheulng ecson accorng to ts best response. By recevng the upate prce for the new matchng, each ata center that has not change ts utlty company wll upate ts ecson vector as follows. + a+ = a, () ra c a, a, m } where > 0 s a mnshng step sze wth =0 = 2 an =0 ( ) <, an [ ]} s the projecton onto the feasble space efne by (2) (). In Algorthm, ata centers use ther best response strateges an upate equaton () for ther utlty company choce an workloa scheulng. Each utlty company accepts at most one termnaton request an at most one connecton request n each teraton. We have Theorem 3 Algorthms globally converges to a stable outcome of the ata center matchng game. The proof can be foun n Appenx C. IV. ERFORMANCE E VALUATION In ths secton, we evaluate the performance of the stable outcome of the matchng game. We set the contract pero to be one ay. We ve a ay nto T = 96 tme slots, where each tme slot s mnutes. We conser the electrcty market wth 0 utlty companes servng 0 ata centers, an each ata center can choose a utlty company from a ranom subset of seven utlty companes. We use the wholesale market prce on Oct. 0, 206 of the Ontaro s wholesale market [9]. The hgh prce pero s from 2 pm to 6 pm. arameters u (t) for utlty companes u =, 2,..., 0 are set to 0.224, 0.208,..., 0.08 $/(MWh)2 for t 2 T, respectvely. To smulate the arrval rate of the workloas n a ata center, we use the ataset from [20]. Each ata center offers fve 0 Wthout Deman Response Wth Deman Response Data Center Number (c) Fg. 3. (a) Total number of servers; (b) total energy eman of ata center over one ay wth an wthout eman response; (c) contract payment of ata centers to 0 wth an wthout eman response. servce classes, an the workloas requestng servce class c =,..., can be elaye by at most c, = 0.0, 2, 2,, 20 tme slots, respectvely. For servce classes c =,...,, we set c, to 0., 0, 30, 0, 00 tme slots, respectvely. We conser nmax = 20,000 homogeneous servers wth power ratngs Ele = 0 W an Epeak = 300 W per tme slot n each ata center. arameters UE (t), t 2 T are chosen at ranom from nterval [., 2] for each ata center. The step sze n teraton s set to be = /( ). We scuss how Algorthm enables a ata center to manage ts energy eman. For the sake of comparson, we conser the scenaro wthout eman response, where each ata center ranomly chooses a utlty company an etermnes the number of servers base on constrants (2) (4) wthout conserng the prce values. Ths s a nontrval scenaro, as a ata center can elay the executon of a workloa. Let us conser ata center as an example. Fg. 3 (a) shows that wth eman response, the number of operatng servers n ata center ecreases urng the peak hours, e.g., t s reuce from 7,000 to 4,000 aroun 4 pm. Fg. 3 (b) shows that the energy eman of ata center s reuce by.% (from 7 MWh to. MWh urng peak hours). Fg. 3 (c) shows that the contract payment of ata centers s reuce by.4% on average as a result of server scheulng. We scuss how Algorthm affects the AR of the aggregate eman an the revenue of the utlty companes. We compare the AR of the utlty companes n the scenaros wth an wthout ata centers eman response. Fg. 4 (a) shows that, wth ata centers eman response, the AR of the utlty companes s reuce by 7.2% on average. A lower AR mproves the performance of the utlty companes urng peak hours. The revenue of utlty companes epens on the matchng structure. For the sake of comparson, we conser a

6 AR Revenue ($)..4.3 Wthout Deman Response Wth Deman Response Utlty Company's Number (a) 0 3 Wthout Utlty Company Choce Wth Utlty Company Choce Utlty Company's Number (b) Fg. 4. (a) AR n the energy eman; (b) revenue of the utlty companes n the scenaros wth an wthout utlty company choce. otental Functon Number of Iteratons Fg.. The convergence of the potental functon n Algorthm. scenaro where ata centers ranomly enter contracts wth utlty companes. In the stable outcome obtane from Algorthm, the number of ata centers connecte to utlty companes to 0 are 2, 2, 3, 4, 4, 4, 6, 6, 8, an, respectvely. A utlty company wth a lower apple u (t) can attract more ata centers as customers. When compare to the scenaro wthout utlty company choce, Fg. 4 (b) shows that the revenue of the utlty companes wth a hgher apple u (t) ecreases an the revenue of the utlty companes wth a lower apple u (t) ncreases (up to 82%). Thus, the results of Algorthm s consstent wth the utlty companes competton n eregulate markets. Fnally, we evaluate the convergence of Algorthm. Fg. epcts the convergence of the potental functon n one of our smulatons wth a ranom ntal conton. The potental functon ecreases n each teraton an converges to a stable outcome n 30 teratons. The runnng tme untl convergence s 47 secons. Regarng the computatonal complexty, n Lne 7 of Algorthm, ata center solves U optmzaton problems to etermne ts best response strategy. Hence, the per-teraton complexty of Algorthm for ata center s nepenent of the number of ata centers an epens only on the number of utlty companes n set U,.e., O( U ). V. CONCLUSION In ths paper, we stue the ata centers problem of choosng utlty companes an scheulng workloa n a eregulate electrcty market. We moele the nteracton among ata centers as a many-to-one matchng game wth externaltes. We constructe an exact potental functon, whose local mnma correspon to the stable outcomes of the game. We evelope an algorthm to etermne a stable outcome. Smulaton results showe that the ata centers can ecrease ther cost by.4% wth the propose algorthm, as they can purchase electrcty from ther preferre utlty companes an reuce ther emans urng peak hours. Meanwhle, the utlty companes can acheve 7.2% reucton n the AR. Those utlty companes that offer lower tarffs can ncrease ther revenue by up to 82%. For future work, we plan to exten the moel by conserng the competton among utlty companes through prce optmzatons. ACKNOWLEDGEMENT The work of S. Bahram an V.W.S. Wong was supporte by the Natural Scences an Engneerng Research Councl of Canaa. The work of J. Huang was supporte by the Themebase Research Scheme (roject No. T23-407/3-N) from the Research Grants Councl of the Hong Kong Specal Amnstratve Regon, Chna, an a grant from the Vce-Chancellors One-off Dscretonary Fun of The Chnese Unversty of Hong Kong (roject No. VCF20406). A. roof of Theorem AENDIX To prove Theorem, we substtute (0) nto the rght-han se of (9) an substtute (6) nto the left-han se of (9) for strateges s an es, an show that the results are the same. Data center changes ts utlty company choce from u to eu, an ts ecson profle from a to ea. Thus, the energy eman of ata center s change from e (t) to e (t) n tme slot t 2T. By substtutng (0) nto the rght-han se of (9) for s =(s, s ) an es =(es, s ), we obtan (s, s ) (es, s )= t2t p(t)+apple u (t) e other u (t) e (t)+apple u (t) e 2 (t) + apple u (t) 0 2m(u)\ e (t) e 0(t) p(t)+apple eu (t) e other eu (t) e (t) apple eu (t) e 2 (t) apple eu (t) 0 2 em(eu)\ e (t) e 0(t). (2) In (2), the terms relate to the ata centers other than ata center cancel each other. Substtutng (6) nto the left-han se of (9) for s =(s, s ) an es =(es, s ), we have c (s, s ) c (es, s )= t2t e (t) p r u(e u (t),m) e (t) p r eu (e eu(t), em). (3) By substtutng the retal prce () nto (3), the cost change for ata center wll be equal to the potental functon change n (2). Ths completes the proof. B. roof of Theorem 2 We frst show that the global mnmum of the potental functon (0) s a stable outcome. Let s? =(a m?,m? ) be the global mnmum of (s). Thus, f ata center changes ts acton profle to a or ts utlty company to m() unlaterally, then the value of the potental functon ncreases. The change n the exact potental functon s equal to the change n the cost of the evatng ata center. Hence, the cost of ata center ncreases as well. Consequently, no unlateral evaton from

7 s? can reuce the cost of any ata center, an hence s? s a stable outcome of Game. As the exact potental functon (0) has at least one global mnmum, we know that the matchng game has at least one stable outcome. Next we show that an arbtrary stable outcome (a,m) s n set M. We prove ths by contracton. Suppose that a stable outcome (a,m) s not n set M. Hence, we have a 6= a m. By efnton, a m s the global mnmum of the potental functon uner matchng m. We also know that (a,m) s a convex functon of a. Thus, a unlateral change of a for any ata center n the opposte recton of the graent r a (a,m) wll reuce the potental functon, an thus the cost of that ata center. It contracts the supposton that (a,m) s a stable outcome. Hence, (a,m) s n set M. C. roof of Theorem 3 Snce the potental functon (0) s lower-boune by zero, t s suffcent to show that the potental functon ecreases n each teraton of Algorthm. Lne 7 of Algorthm guarantees that f the algorthm converges, the result s a stable outcome. Next we prove the sketch of the proof. Step a) Conser teraton of Algorthm. We prove by nucton that the potental functon ecreases when k ata centers upate ther utlty company choces smultaneously, where k s an arbtrary number. The base case (.e., k =) correspons to the unlateral change n the strategy of one ata center. From (9), the potental functon ecreases, when the cost of a ata center ecreases. In the nucton step, we conser k = an suppose that the potental functon ecreases when ata centers change ther utlty company choces. We prove that the potental functon ecreases when k = + ata centers change ther utlty company choces. We ve the set of + ata centers nto two sets wth ata centers an one ata center, respectvely. We use the nucton supposton for k = to show that when ata centers change ther utlty company choces, the potental functon ecreases. Now, assume that ata center + eces to leave utlty company m (+) n teraton to connect to utlty company m + ( + ). In Algorthm, a utlty company accepts at most one termnaton request from ata centers per teraton. Thus, ata center + s the only one leavng utlty company m ( + ). Utlty company m ( + ) may accept a new connecton request from other ata centers, whch ncreases ts total eman. Thus, the payment of ata center + to utlty company m ( + ) wll ncrease after upatng the matchng of other ata centers. On the other han, n Algorthm, a utlty company accepts at most one connecton request from ata centers per teraton. Hence, ata center + s the only one that connects to utlty company m + ( + ). Utlty company m + (+) may accept a termnaton request from other ata centers, whch further ecreases ts total eman. Thus, the payment of ata center to utlty company m + () wll further ecrease after upatng the matchng of other ata centers. Consequently, the cost of ata center + ecreases even other ata centers change ther utlty company choces. The exact potental functon ecreases when the cost of ata center + ecreases. 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