Accepted Manuscript. Real-time pricing scheme based on Stackelberg game in smart grid with multiple power retailers

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1 Accepted Manuscrpt Real-tme prcng scheme based on Stacelberg game n smart grd wth multple power retalers Yemng Da Yan Gao Hongwe Gao Hongbo Zhu PII: S (17) DOI: /.neucom Reference: NEUCOM To appear n: Neurocomputng Receved date: 7 June 2016 Revsed date: 7 Aprl 2017 Accepted date: 9 Aprl 2017 Please cte ths artcle as: Yemng Da Yan Gao Hongwe Gao Hongbo Zhu Real-tme prcng scheme based on Stacelberg game n smart grd wth multple power retalers Neurocomputng (2017) do: /.neucom Ths s a PDF fle of an unedted manuscrpt that has been accepted for publcaton. As a servce to our customers we are provdng ths early verson of the manuscrpt. The manuscrpt wll undergo copyedtng typesettng and revew of the resultng proof before t s publshed n ts fnal form. Please note that durng the producton process errors may be dscovered whch could affect the content and all legal dsclamers that apply to the ournal pertan.

2 Real-tme prcng scheme based on Stacelberg game n smart grd wth multple power retalers Yemng Da a b * Yan Gao c Hongwe Gao a b Hongbo Zhu c a. School of Mathematcs and Statstcs Qngdao Unversty Qngdao Chna; b. Insttute of Appled Mathematcs of Shandong Qngdao Chna; c. School of Management Unversty of Shangha for Scence and Technology Shangha Chna; Abstract As an essental characterstc of smart grd demand response may reduce the power consumpton of users and the operatng expense of power supplers. Real-tme prcng s the ey component of demand response whch encourages power utlzaton n an effcent and economcal way. In ths paper we study the real-tme prcng scheme n smart grd wth multple retalers and multple resdental users usng Stacelberg game. Addtonally the prce competton among power retalers s formulated as a non-cooperatve game whle the coordnaton among the resdental users s formulated as an evolutonary game consderng the prvate nformaton of power retalers and resdental users. The exstence of Stacelberg equlbrum s proved. Moreover two specal algorthms are developed to solve the equlbrum. Numercal results show the convergence of algorthms and also confrm the effcency and effectveness of proposed real-tme prcng scheme. Keywords smart grd demand response real-tme prcng Stacelberg game 1. Introducton Advanced communcaton and nformaton technologes [1] have made energy management more flexble n smart grd [2-4]. As a ey component of smart grd technology demand response (DR) may mantan the balance of power supply and demand by pea load shavng. Among demand response schemes real-tme prcng (RTP) s regarded as an effcent way to manage prce-responsve loads [5-10]. In recent years RTP has drawn more attenton from polcy maers power companes and many academc researchers. Many methods and technologes such as optmzaton theory and game theory [11] have been appled to study RTP. A lot of wors on RTP usng game theory manly concentrate on the game relatonshp among the power generators or on the nteracton among users [12-14]. In addton the Stacelberg game approach s appled n [15-19] to study RTP problems. It s worth notng that the above wors merely consder one power suppler for reducng the computatonal complexty. Multple energy sources and the competton among them stll receve very lttle attenton [8]. In fact consderng the further openng of power maret and development of new renewable energy sources users especally the users who lve n resdental dstrct would fnd t easer to obtan power from dfferent power supplers than ever whch complcates the nteracton behavors between power supplers and resdental users [20]. Thus some sophstcated herarchcal games have been leveraged to shed lght on the mult-seller-mult-user RTP problems n the complex power maret see for [21-22]. Dfferent from [21] the smart grd system consdered by us n ths paper s a power system n smart resdental dstrct le [22] where all the resdental users have homogenety n the power consumpton process. The smart resdental dstrct grd system the latest hot research area n smart grd s also what our research focuses on. The power demand of users s accurately descrbed by solvng an optmzaton problem n [21] however snce the users are specal resdental users n ths paper solvng an optmzaton problem le [21] wll meet some obstacles: the power consumpton behavors of each resdental user are emboded n choosng a power retaler to purchase power but each resdental user doesn t now the choce of other resdental users whch s seen as a prvacy ssue. Then 1

3 the strategy of selectng power retaler to purchase power of each resdental user s a mxed strategy process. At last the evolutonary game s generated to descrbe the evoluton process of power consumpton behavors among resdental users. The above soluton s what maes our paper nnovatve and dfferent from [21]. The authors of [22] propose a two-level game where power utltes play a non-cooperatve game and resdental users play an evolutonary game whch s smlar to our paper. But n real smart grd system the retalers set the unt power prces based on avalable power and announce them to the resdental users thereby resdental users respond to the prces by an optmal power amount. Snce the retalers act frst and then the resdental users mae ther decson based on the prces t s a sequental acton for the two partcpants whch s gnored n [22]. Then though Nash equlbrum among power retalers and evolutonary equlbrum among resdental users are reached t does not guarantee that the strategy nteractons between power retalers and resdental users reman stable. That s to say the equlbrum of Stacelberg game between power retalers and resdental users does not necessarly eep exstence n [22]. Based on the above reasons both the power consumpton characterstcs of resdental users and the sequental competton between power retalers and resdental users are consdered at the same tme when demand response mechansm s desgned n ths paper. So the Stacelberg game model s adopted to study RTP n the power retalng maret wth multple retalers and multple resdental users. The retalers procure power from the power wholesale maret and set the real-tme electrcty prces. Thus the retalers play the role of the leaders and the resdental users have to be the followers. The optmzaton problems are consdered for each retaler and each resdental user respectvely. The users who lve n a neghborhood area are treated as a populaton. The evoluton process whch adusts power consumpton from the retalers s an optmal response to the real-tme power prces. Therefore we formulate an evoluton game for the resdental users. After the optmal power consumpton of resdental users s obtaned by the evoluton equlbrum the power demand of users s transmtted to power retalers and then the prce competton among the power retalers s formulated as a non-cooperatve game. Fnally each retaler sets the optmal real-tme prce accordng to the power demand of users. When the resdental users and the retalers reach ther equlbrums and the sequental competton could not change ther equlbrums the Stacelberg equlbrum (SE) s also acheved. The contrbutons of our paper are summarzed as follows. We formulate the RTP between multple power retalers and multple resdental users as a Stacelberg game. At the same tme an evolutonary game s generated for the resdental users whle a non-cooperatve game s proposed for the power retalers. We desgn an algorthm to acheve the equlbrum of generated evolutonary game. The exstence of Nash equlbrum (NE) s proved for the non-cooperatve game among the power retalers. Therefore after the evolutonary equlbrum s acheved we also desgn a dstrbuted algorthm for the power retalers to obtan NE and then the SE s also reached. The rest of ths paper s organzed as follows. We gve the system model n Secton 2. In Secton 3 we formulate the evolutonary game among the resdental users. An teratve algorthm s proposed to acheve evoluton equlbrum. In Secton 4 a non-cooperatve game s proposed for the prce competton behavors among the retalers. Secton 5 gves the Stacelberg game and exstence of ts equlbrum s proved. We provde numercal results and dscuss the performance of the proposed prcng scheme n Secton 6. The last Secton concludes ths paper. 2. System model Now we consder a smart power system wth multple retalers and multple resdental users consstng of a 2

4 set M 12...m of power retalers and a set N 12...n of resdental users. The smart meters are equpped for the resdental users to mae the resdental users schedule energy consumpton. A dstrbuton power staton supples power for the resdental users n a certan area. We tae one day as a perod and the perod s dvded nto K tme slots. Κ denotes the set of tme slots and denotes each tme slot where Κ. The power retalers accept electrcty demand of all users and send real-tme unt prces to the users n each tme slot. Let p be the prce of retaler n the th tme slot and p p 1 p pm ( ) be the prce strategy vector. All retalers purchase power and set prce to maxmze ther profts accordng to the real-tme power demand of all users. The retalers compete wth each other to maxmze ther profts by adustng the real-tme electrcty unt prces Utlty Functon of Resdental User User selects a power retaler to serve hmself n tme slot ts real-tme power demand s denoted by x x mn x x max where x mn and xmax represent mnmum and maxmum of electrcty consumpton of user respectvely. x ( x 1... x... xn) s the real-tme power demand vector of the resdental users n tme slot. The power demand of each resdental user vares from tme to tme. The dfferent behavors of users are depcted by dfferent utlty functons. In recent studes about RTP ([5] [11] [13]) the power demand s predcted by explorng the hstory dates of power consumpton. In ths paper we stll adopt the quadratc utlty functon u 2 x ( x ) 0 x 2 ( x ) 2 ( ) x 2 where and are user-specfc parameters. Same to [22] we tae 2 u( x ) x ( x ) mn x x xmax. (2) 2 After the m retalers announce ther prce vector p ( p 1... p... pm) at the th tme slot user pays px when consumng x amount of power f he selects retaler as the power suppler. Therefore the welfare functon of user s descrbed as follows U x u x p x x x p x 2 x x x Cost and Revenue Functon of Power Retaler 2 ( ) ( ) ( ) mn max The cost functonc of power retaler s defned as the cost for procurng the power amount of real-tme (1) (3) demand from the power wholesale maret at tme slot and t s denoted as C pl (4) where p s the electrcty prce procured by retaler n power wholesale maret and s set as a constant parameter L denotes the amount of power procured by retaler from power wholesale maret at slot tme. Hence the 3

5 revenue functon of retaler s gven by R ( p s ) p s pl (5) where s denotes the amount of power sold by retaler n retal power maret at slot tme s mn( L D ) D s the power demand of the users from retaler to be defned later n (9) Interacton between Electrcty Retalers and Resdental Users The retalers provde ther power to the resdental users n order to obtan larger revenues at lower cost whereas the users decde the power consumpton to maxmze ther satsfacton and welfare wth a lower payment. Accordng to the behavors of the power retalers and the resdental users we desgn a new study scheme whch s proposed to assure the proft maxmzaton of the two sdes. Therefore we develop approprate strateges to mantan the supply-demand balance of power between power retalers and resdental users. In vew of the sequence of actons of the retalers and the users the RTP problem s formulated as a Stacelberg game whch s dvded nto two sdes: the retaler sde and the user sde. Meanwhle besdes the game between the retalers and the users there s also other competton n each sde. One s the competton among the resdental users so as to reach welfare maxmzaton and the other one s the evolutonary process among the resdental users. Each resdental user selects one retaler to buy power at each tme slot n ths paper. 3. Evolutonary game among resdental users Evolutonary game [23] has general applcatons n the engneerng feld wth multple buyers and multple sellers [24-25]. An evolutonary game wth lmted ratonalty s formulated n our RTP problem where the player s resdental user the populaton s set of the resdental users and the strategy s the selecton of the retalers. Therefore the evolutonary equlbrum s acheved by desgnng a sutable replcator. Smlar to [22] we desgn a replcator to ensure that the evolutonary equlbrum among the resdental users s acheved an teratve algorthm s developed to carry out the replcator dynamcs Formulaton of Evolutonary Game We consder one populaton scenaro through usng b-drectonal communcaton structure n ths model. The strategy of each resdental user n the populaton s dentcal. Each user selects a retaler to buy power when resdental users receve the power prces announced by power retalers then each user gradually adusts ts behavors and acts ndependently n the selecton process. If the probablty of a user choosng retaler s expressed by 1 2 m m 1 y ( y y... y... y ) Replcator Dynamcs y (0 y 1 y 1) at tme slot the populaton state s denoted as Accordng to (3) when purchasng x amount of power from retaler the welfare functon of resdental user s denoted as 2 U ( x ) u ( x ) p x x ( x ) p x 2 x x x. mn max (6) where x mn and x max represent mnmum and maxmum of power consumpton of user from retaler 4

6 respectvely. Thus f x mn x x max the optmal power demand of user s acheved at p (7). x We next redefne the optmal power consumpton of user when purchasng electrcty from retaler at tme slot as follows p xmn x mn * p p ( x ) xmn xmax (8) p xmax xmax. So the total demand for electrcty comng from retaler at tme slot s n * ( ). 1 D y x (9) It s worth notcng that p and L eep unchanged n the evolutonary process of resdental users once they are * announced by retaler. ( x ) n (8) s also a constant. We stll use a so-called net utlty [22] to descrbe the accumulated welfare of users obtaned from retaler. Therefore the net utlty becomes where n L L D *2 ( x ) L D 2 ( ) n D *2 ( x ) L D 2 1 n *2 ( x ) s regarded as a constant n the evoluton process. 1 In the followng the replcator dynamcs s desgned to depct the selecton dynamcs of the populaton y t y ( ) (10) (11) where m y 1 s the average value of net utlty Evolutonary Equlbrum 5

7 The evolutonary equlbrum s acheved when the populaton eeps ts selecton unchanged. We determne the selecton accordng to the dfference between the net utlty and ts average value [22] thus the evolutonary equlbrum s acheved when the dfference s tny enough.e. Hence y t 0. (12) m y m m 1 y y t 1 1 (13) ( ) 0. Smlar to [22] Lyapunov method [26] s adopted to acheve the convergence of the evolutonary equlbrum (12) wth the replcator dynamcs (11). The evolutonary equlbrum s denoted by 3.4. Iteratve Algorthm y * ( y * y *... y *... y *). 1 2 m For descrbng the replcator dynamcs we use the dscrete replcator to gve an teratve algorthm as follows. y s 1 y s 1 y s( s s ) (14) where s s the teraton number and 1 denotes the step sze. The termnal crteron s expressed as (15) s where s an arbtrary small postve constant. The detaled algorthm s lsted as follows Table 1 Algorthm 1 1: User arbtrarly chooses one retaler to buy power 2: s 1; 3: Repeat. 4: Compute by (10); s 5: Compute the average value of net utlty after obtanng all ; s s N M; 6: Replace retaler to purchase power wth the probablty y s by (14); 7: ss 1; 8: End when (15) s satsfed. 4. Non-cooperatve game among retalers Each retaler ams at maxmzng ts own revenue through sellng the power to the users snce t s selfsh and ratonal then the non-cooperatve game s proposed to model the prce competton among the retalers Analyss of the Non-Cooperatve Game Defnton 1. ([27]) A non-cooperatve game s a trple G N ( S ) ( U ( l) where N mn max the set of players partcpatng n the game. S l l l l N N N s s the set of possble strateges that player 6

8 taes and U () l s the payoff functon. Defnton 2. ([27]) For a non-cooperatve game G N ( S ) ( U ( l) l l1 l2 l N * (... ) s sad to be a Nash equlbrum f and only f U l l N N ' ( ) U( l l ) a vector of strateges for all N and any other l ' S where l ( l1 l2... l 1 l 1... ln ) s the set of strateges selected by all the consumers except for consumer ( l l ) ( l1 l2... l 1 l l 1... ln ) s the strategy profle and U( l l ) s player 's resultng payoff gven the strateges of other players. In the proposed non-cooperatve game among the retalers the player s retaler the prce p s strategy of retaler at tme slot and retaler s revenue functon gven n (5) s wrtten as Lemma 1. ([27]) Nash equlbrum exsts n the game f 1) The player set s fnte; 2) The strategy sets are closed convex and bounded; p D pl L D R ( p s ) p L pl L D. 3) The utlty functons are contnuous and quas-concave n the strategy space. Based on lemma 1 we obtan the followng theorem. Theorem 1. Nash equlbrum exsts n the non-cooperatve game among the retalers. Proof. There are tghter lmts on such that p p mn p max n the non-cooperatve game among retalers. The lower lmt s the generaton costs and assocated operatng expenses. The retalers eep ther prce above p. The upper bound p max (16) mn s fxed by the polces from government. The retalers must consder these prce lmts. Therefore for the m retalers the strategy sets are nonempty closed bounded compact and convex subset of Substtutng (8) nto (9) we have Thus (16) s rewrtten as It s obvous that p p D y x x. n mn max 1 n p p y pl L D R ( p s ) 1 p L pl L D. m. R s contnuous wth respect to p n (18). We next consder the quas-concave of R. Snce the (17) (18) revenue R of retaler s ncreasng about the amount of power for a gven p each retaler ams to announce ts power prce and sell out all the procured power when the avalable power L procured from the power whole maret s gven..e. L D. Actually the non-cooperatve game ends when L D whch ensures the power supply-demand balance. 7

9 We consder two cases. 1) L D. we have 2 d R 0 2 ( ) d p (19) 2) L D. The second order dervatve of Combng (19) wth (20) leads to 2 d R 0 2 ( ) d p R wth respect to 2 d R 2 ( ) d p n 1 p s 1 2 y. (20). In summary R s always quas-concave n vrtue of Lemma 1 we conclude that Nash equlbrum exsts n the non-cooperatve game. 5. Stacelberg game between retalers and resdental users 5.1. Stacelberg Game p for the two cases. By From Secton 4 we now that the power retalers compete wth each other to set the power real-tme prces. After the power prces are announced to the resdental users by the retalers the resdental users are nvolved n the evolutonary game and fnally ther optmal power demand reachs the evolutonary equlbrum. After that the power demand of resdental users s transferred nto the retalers then the nteracton behavors between the retalers and the users are formulated as a Stacelberg game consderng the sequental competton acton of the retalers and the users where the retalers nvolved n the prce competton are the multple leaders and the resdental users are the multple followers. Ther obectves are to obtan the Stacelberg equlbrum. The equlbrum strategy for the users n the Stacelberg game s to consttute an optmal response for the announced Defnton 3. ([27] ) Let R and U p by the leaders. be the strategy sets for retaler and resdental user respectvely. The strategy sets of the retalers and the resdental users are R R1 R2... R m and U U 1 U 2... U N. Then p * s a Stacelberg equlbrum strategy of retaler f t satsfes R where p* p * : 1 2 n R ( p *y( p *)) R ( p p *y( p p *)) M (21) y y ;y ;...;y s the strategy of the users y( p *) s the optmal response of the users whch s obtaned by Algorthm 1. The above process cycle untl p *and y * reman unchanged and then vector ( p *y *) s a Stacelberg equlbrum Exstence of Stacelberg Equlbrum In the Stacelberg game when the retalers announce the power prce vector all resdental users receve the 8

10 announced prce nformaton and partcpate n the evolutonary game. Fnally the evolutonary equlbrum s acheved. As a result f the retalers adust ther power prce to converge to a NE n the non-cooperatve game among the retalers the Stacelberg game owns a Stacelberg equlbrum. Theorem 2. The Stacelberg equlbrum exsts between the power retalers and the resdental users. Proof. The exstence of NE s proved n Theorem 1. Based on the equlbrum prce of retalers the convergence to the evolutonary equlbrum wth the replcator dynamcs (11) s guaranteed.e. the optmal response of the users s obtaned. Then the equlbrum of Stacelberg game also exsts Dstrbuted algorthm for SE In the game model the resdental users acheve optmal power demand based on the power prces and procured power amount offered by the retalers. A dstrbuted algorthm s developed for the retalers to obtan the NE when each retaler does not now the nformaton of other retalers thus the Stacelberg equlbrum s reached. The prce updatng strategy of retaler s desgned by usng p s 1 p s 2 ( D s L ) (22) where 2 denotes a speed adustment parameter and s s the teraton number. The termnal crteron of the dstrbuted algorthm s p 1 s p D s L (23) where 0. After the power prce s adusted the resdental users evolve to obtan a new equlbrum. Then the prces of retalers are adusted agan. Ths process s operated by the followng algorthm. Table 2 Algorthm 2 Dstrbuted algorthm 1: For K do. 2: For s 1 arbtrarly choose 3: Repeat for ss 1; 4: User n; 5: Operatng Algorthm 1; 6: Compute power demand D 7: Transmt 8: Compute s p ; s 1 D to each power retaler ; p s 1 9: Untl (23) s satsfed. usng (22); 6. Numercal results of the resdental users accordng to (9); In ths secton we provde some numercal results to dscuss the performance of proposed algorthm and valdate the above model analyss then examne how the resdental users buy ther optmal electrcty based on the unt prce vector of the retalers and how the retalers optmze ther unt prce vector based on avalable power constrants. In the followng smulaton results we consder the scenaro consstng of two retalers and fve resdental users. The operatng tme s dvded nto 24 tme slots. s randomly selected from [410] and 0.5. In the retaler sde and the power wholesale prce p 0.3 the avalable power constrants of all retalers are L 1 22 L2 11n a tme slot Evolutonary Game To evaluate the convergence of Algorthm 1 we smulate for verfyng the users to reach the equlbrum by conductng Algorthm 1. We see the resdental users converge to the equlbrum qucly wth a vew to the probablty of buyng power from two retalers n Fg.1. Fg.2 shows the dynamc change process of the average net 9

11 utlty. Clearly the resdental users have better welfare. Fg1. Convergence process of resdental users Fg2. Convergence process of the average net utlty The scalablty of Algorthm 1 s well reflected n Fg. 3. Algorthm 1 has good scalablty wth the ncreasng user number. It s seen that the teraton number always eep n low range. Fg3. Iteratons of Algorthm 1 when the number of resdental users ncreases 6.2. Non-cooperatve Game among Retalers and Stacelberg Equlbrum For evaluatng the performance of Algorthm 2 and obtanng the Stacelberg Equlbrum between the retalers and the resdental users we consder the competton and convergence of Nash equlbrum among the retalers n the non-cooperatve game. Fg.4(a) shows that the welfare of the retalers s sgnfcantly mproved. 10

12 Fnally Nash equlbrum s reached and the welfare functons of the retalers are maxmzed. Fg.4 (b c) show the convergence process of the retalers accordng to the amount of power demand and power prce. It s obvous that Fg4 (a). Convergence process of retalers welfare Fg4 (b). Convergence process of retalers power demand Fg4 (c). Convergence process of retalers power prce 11

13 both the amount of power demand and the power prce converge to a contant whch ensures power supply and demand balance. From the above results we obtan the NE of the non-cooperatve game.e. equlbrum prce of the retalers. We next acheve the evolutonary equlbrum by vrtue of secton 6.1 then the Stacelberg equlbrum s reached between the power retalers and the resdental users Comparson wth Fxed Prcng Scheme To evaluate the performance of proposed real-tme prcng scheme we consder the fxed prcng scheme proposed n [5] for mang a comparson wth the new prcng scheme n ths paper. In ths fxed prcng scheme the retaler eeps a fxed value n the entre process of each tme slot. Therefore the resdental users have not taen demand response because of the lac of enthusasm. Fg.5 shows the power equlbrum prce of retaler 1 under two dfferent schemes n 24 hours. It s obvous that the proposed real-tme prcng scheme s very remarable n cuttng down the real-tme power prce. Therefore the resdental users are able to beneft from the proposed prcng scheme n savng payment. Fg5. Power equlbrum prce of retaler 1 under dfferent prcng schemes 7. Concluson In ths paper we have proposed a real-tme prcng scheme wth multple retalers and multple resdental users. We model the decson process of RTP as a Stacelberg game framewor. The prce competton among the power retalers are formulated as a non-cooperatve game whle the coordnaton among resdental users s modeled as an evolutonary game. We show that all the games converge to the correspondng equlbrums on the bass of proposed scheme. In addton we desgn two teratve algorthms to acheve the equlbrum strateges. Smulaton results confrm that the power prce s reduced sgnfcantly wth the proposed real-tme prcng scheme whle the power consumpton payment of resdental users s decreased. Correspondngly the resdental users are able to beneft from the proposed prcng scheme. However snce smart grd s a complex power system many uncertanty factors and nonlnear stochastc characterstcs exst n such system [28 29]. It s nterestng for us to extend and explore the uncertanty about the power loads of resdental users and consder the effect of prce predcton [30 31]. Acnowledgments 12

14 Ths wor s supported by the Natonal Natural Scence Foundaton of Chna (Proect No ) Proects of Internatonal (Regonal) Cooperaton and Exchanges of NSFC ( ) Natural Scence Foundaton of Shandong Provnce Chna (ZR2015GZ007) Chna Postdoctoral Scence Foundaton(2016M602104) Qngdao Postdoctoral Applcaton Research Funded Proect( ). References [1] H. Farhang The path of the smart grd IEEE Power and Energy Magazne 8(1) (2010) [2] X. Fang S. Msra G. Xue et al. Smart grd The new and mproved power grd: A survey IEEE Communcatons Surveys & Tutorals 14(4) (2012) [3] S. M. Amn B. F. Wollenberg Toward a smart grd: power delvery for the 21st century IEEE Power and Energy Magazne 3(5) (2005) [4] L. Wang G. We H. Shu State estmaton for complex networs wth randomly occurrng couplng delays Neurocomputng 122 (2013) [5] P. Samad A. H. Mohsenan-Rad R. Schober V. W. Wong J. Jatsevch Optmal real-tme prcng algorthm based on utlty maxmzaton for smart grd n: Proceedngs of the Frst IEEE Internatonal Conference on Smart Grd Communcatons 2010 pp [6] X. He T. Huang C. L et al. A recurrent neural networ for optmal real-tme prce n smart grd Neurocomputng 149 (2015) [7] G. L. Stort M. Paschero A. Rzz et al. Comparson between tme-constraned and tme-unconstraned optmzaton for power losses mnmzaton n Smart Grds usng genetc algorthms Neurocomputng 170 (2015) [8] R. Deng Z. Yang F. Hou et al. Dstrbuted real-tme demand response n multseller multbuyer smart dstrbuton grd IEEE Transactons on Power Systems 30(5) (2015) [9] Y. Wang S. Mao R. M. Nelms Dstrbuted onlne algorthm for optmal real-tme energy dstrbuton n the smart grd IEEE Internet of Thngs Journal 1(1) ( 2014) [10] K. Muraltharan R. Sathvel Y. Sh Multobectve optmzaton technque for demand sde management wth load balancng approach n smart grd Neurocomputng 177 (2015) [11] W. Saad Z. Han H. V. Poor et al. Game-theoretc methods for the smart grd: An overvew of mcrogrd systems demand-sde management and smart grd communcatons IEEE Sgnal Processng Magazne 29(5) (2012) [12] Y. Da Y. Gao Real-tme prcng decson mang for retaler-wholesaler n smart grd based on game theory Abstract and Appled Analyss do: /2014/ ( 2014) 1-8. [13] A. H. Mohsenan-Rad V. W. S. Wong J. Jatsevch et al. Autonomous demand-sde management based on game-theoretc energy consumpton schedulng for the future smart grd IEEE Transactons on Smart Grd 1(3) (2010) [14] C. Ibars M. Navarro L. Guppon Dstrbuted demand management n smart grd wth a congeston game n: Proceedngs of 2010 frst IEEE nternatonal conference on Smart grd communcatons 2010 pp [15] Y. Da Y. Gao Real-tme prcng decson based on leader-follower game n smart grd Journal of Systems Scence and Informaton 3(4) ( 2015) [16] Y. Da Y. Gao Real-tme prcng decson-mang n smart grd wth mult-type users and mult-type power sources Systems Engneerng-Theory and Practce 35(9) (2015) (n Chnese) [17] C. Chen S. Kshore L. V. Snyder An nnovatve RTP-based resdental power schedulng scheme for smart grds n: Proceedngs of 2011 IEEE Internatonal Conference on Acoustcs Speech and Sgnal Processng 2011 pp [18] Y. Da Y. Gao Real-tme prcng strategy wth mult-retalers based on demand-sde management for the smart grd Proceedngs of the Chnese Socety for Electrcal Engneerng 34(25) (2014) (n Chnese) 13

15 [19] J. Chen B. Yang X. Guan Optmal demand response schedulng wth stacelberg game approach under load uncertanty for smart grd n: Proceedngs of 2012 IEEE Thrd Internatonal Conference on Smart Grd Communcatons 2012 pp [20] Z. Fan P. Kularn S. Gormus et al. Smart grd communcatons: Overvew of research challenges solutons and standardzaton actvtes IEEE Communcatons Surveys & Tutorals 15(1) (2013) [21] S. Maharan Q. Zhu Y. Zhang et al. Dependable demand response management n the smart grd: A Stacelberg game approach IEEE Transactons on Smart Grd 4(1) (2013) [22] B. Cha J. Chen Z. Yang et al. Demand response management wth multple utlty companes: A two-level game approach IEEE Transactons on Smart Grd 5(2) (2014) [23] A. B. MacKenze L. A. DaSlva Game theory for wreless engneers Synthess Lectures on Communcatons 1(1) (2006) [24] D. Nyato E. Hossan Z. Han Dynamcs of multple-seller and multple-buyer spectrum tradng n cogntve rado networs: A game-theoretc modelng approach IEEE Transactons on Moble Computng 8(8) (2009) [25] D. Nyato E. Hossan Dynamcs of networ selecton n heterogeneous wreless networs: an evolutonary game approach IEEE Transactons on Vehcular Technology 58(4) (2009) [26] J. J. E. Slotne W. L Appled nonlnear control Englewood Clffs NJ: Prentce-Hall [27] R. B. Myerson Game theory: analyss of conflct Harvard Unversty Press [28] D. Dng Z. Wang G. We et al. Event-based securty control for dscrete-tme stochastc systems IET Control Theory & Applcatons 10(15) (2016) [29] S. Lu G. We Y. Song et al. Extended Kalman flterng for stochastc nonlnear systems wth randomly occurrng cyber attacs Neurocomputng 207 (2016) [30] Q. L B. Shen Y. Lu et al. Event-trggered H nfnty state estmaton for dscrete-tme stochastc genetc regulatory networs wth Marovan umpng parameters and tme-varyng delays Neurocomputng 174(2016) [31] E. Crsostom C. Gallccho A. Mchel et al. Predcton of the Italan electrcty prce for smart grd applcatons Neurocomputng 170 (2015)

16 1. Yemng DAI Bographes and pctures of all authors Yemng DAI receved her B.S. and M.S. degree n fundamental mathematcs from Henan normal Unversty Xnxang and Qngdao Unversty Qngdao Chna n 2006 and He obtaned hs Ph.D. degree from Unversty of Shangha for Scence and Technology Shangha Chna n Currently he s a assocate professor n Qngdao Unversty. Hs current research nterests nclude the area of smart grd game theory and ts applcaton. 2. Yan GAO gaoyao@usst.edu.cn Yan GAO receved hs Ph.D. degree from the Dalan Unversty of Technology n Snce 2001 he has been a char professor wth the Unversty of Shangha for Scence and Technology. Before 2000 he held teachng postons wth Yanshan Unversty and Chna Unversty of Mnng and Technology respectvely. Hs current research focus on dstrbuted generaton system demand-sde management ntegrated energy system real-tme prcng and nonsmooth optmzaton. 3. Hongwe GAO gaohongwe@qdu.edu.cn Hongwe GAO receved hs Ph.D. degree from the St.-Petersburg State Unversty n Snce 2004 he has been a full professor and specally-apponted professor of Qngdao Unversty wth the Unversty. He s secretary of Game Theory Branch of ORSC member of Executve Councl of SESC and Executve Board of ISDG respectvely. Hs current research focus on cooperatve game networ game. 15

17 4. Hongbo ZHU Hongbo ZHU receved her B.S. and M.S. degree n fundamental mathematcs both from Yanban Unversty Yan Chna n 2004 and She s currently a Ph. D. canddate n the School of Management Unversty of Shangha for Scence and Technology Shangha Chna. Her current research nterests nclude the area of smart grd demand sde management real-tme prcng and dstrbuted optmzaton. 16