Two Market Models for Demand Response in Power Networks

Transcription

1 Two Market Models for Demand Response n Power Networks Ljun Chen, Na L, Steven H. Low and John C. Doyle Engneerng & Appled Scence Dvson, Calforna Insttute of Technology, USA Abstract In ths paper, we consder two abstract market models for desgnng demand response to match power supply and shape power demand, respectvely. We characterze the resultng equlbra n compettve as well as olgopolstc markets, and propose dstrbuted demand response algorthms to acheve the equlbra. The models serve as a startng pont to nclude the applance-level detals and constrants for desgnng practcal demand response schemes for smart power grds. I. INTRODUCTION The usual practce n power networks s to match supply to demand. Ths s challengng because demand s hghly tmevaryng. The utlty company or generator needs to provson enough generaton, transmsson and dstrbuton capactes for peak demand rather than the average. As a result, the power network has a low load factor and s underutlzed most of the tme, whch s very costly. For example, the US natonal load factor s about 55%, and 1% of generaton and 25% of dstrbuton facltes are used less than 4 hours per year,.e., 5% of the tme [1]. Shapng the demand to reduce the peak and smooth the varaton can greatly mprove power system effcency and yeld huge savngs. An alternatve strategy mprovng effcency and reducng cost s to match the supply. As the proporton of renewable sources such as solar and wnd power steadly rses, power supply wll also become hghly tme-varyng. Matchng the supply wll become a more effectve and common way to mprove power system effcency and reduce cost [2]. In ths paper, we consder two abstract market models for desgnng demand response to match the supply and shape the demand, respectvely. Specfcally, n secton III, we consder a stuaton where there s an nelastc supply defct (or surplus) on electrcty, and study a supply functon bddng scheme for allocatng load sheddng (or load ncreasng) among dfferent customers/users to match the supply. Each customer submts a parameterzed supply functon to the utlty company, whch wll decde on a market-clearng prce based on the bds of customers, and s commtted to shed (or ncrease) ts load accordng to ts bd and the market-clearng prce [3]. We show that n a compettve market where customers are prce takng, the system acheves an effcent equlbrum that mzes the socal welfare. In an olgopolstc market where customers are prce antcpatng and strategc, the system acheves a unque Nash equlbrum that mzes another addtve, global objectve functon. Based on these optmzaton problem characterzatons of the market equlbra, we propose teratve, dstrbuted supply functon bddng schemes for the demand response to acheve the equlbra. In secton IV, we consder a stuaton where power supply s elastc, but customers are subjected to realtme spot prces and wll shft (and sometmes reduce) ther demands accordngly. Each customer has a lower bound as well as an upper bound on the total electrcty demand over a day. The customer wll allocate ts power usage for dfferent tmes, so as to mze ts aggregate net utltes over a day. We show that when customers are prce takng, the system acheves an effcent equlbrum that mzes socal welfare. When customers are prce antcpatng and strategc, the system acheves a unque Nash equlbrum that mzes another global objectve functon. Agan, based on the optmzaton problem characterzatons of the market equlbra, we propose dstrbuted algorthms for demand shapng to acheve the equlbra. The aforementoned demand response schemes requre tmely two-way communcatons between the customers and the utlty company and even drect communcatons between the customers. They also requre certan computng capablty of the customers. These communcaton and computng capabltes wll become norm n future smart grds [1]. Wth the ntegraton of state-of-the-art communcaton and computng technologes, future power systems wll become more ntellgent, more open, more autonomous, and wth much greater user partcpaton. Our demand response schemes are ntended to apply n exactly such smart power grds. II. RELATED WORK There exsts a huge amount of work on market models for varous aspects of power networks. We brefly dscuss those works that are drectly relevant to ths paper. The supply functon equlbrum model has been wdely used n the analyss of markets n many ndustres. It assumes that each suppler submts a supply functon to an auctoneer, who wll set a unform market clearng prce. In a semnal paper [3], Klemperer and Meyer study the supply functon equlbrum and gve condtons for the exstence and the unqueness of the Nash equlbrum n supply functons under uncertan demand, and show that the equlbra are contaned n a range of prces and allocatons between the Cournot and the Bertrand equlbra. The most notable applcaton of the supply functon equlbrum model s to the wholesale electrcty markets, see, e.g., [4], [5], [6], [7], [8]. In ths paper, nstead of applyng t to the electrcty supply sde, we apply the supply functon equlbrum concept to prcng and allocaton on the demand sde to match the electrcty supply, wth a specal form of parameterzed supply functons that can enable a smple mplementaton of the teratve supply functon bddng as an effectve demand response scheme n power networks. The model studed n secton IV-B s a straghtforward extenson of the compettve equlbrum models for the power /1/$ IEEE 397

2 network, see, e.g., [9]. In addton to tradng off the costs and utltes among dfferent customers, we also consder tradng off the costs and utltes over tme, whch ncentvzes customers to shft ther electrcty usage. III. DEMAND RESPONSE: MATCHING THE SUPPLY In ths secton we consder a stuaton where there s a supply defct or surplus on electrcty. The defct can be due to a decrease n power generaton from, e.g., a wnd or solar farm because of a change to worse weather condton, or an ncrease n power demand because of, e.g., a hot weather. The surplus can be due to an ncrease n power generaton from, e.g., a wnd or solar farm because of a change to better weather condton, or a decrease n power demand at, e.g., the late nght tme. We assume that t s very costly to ncrease the power supply n the case of a defct or decrease the supply n the case of a surplus,.e., the power supply s nelastc. If we have good estmaton of electrcty defct or surplus (e.g., an hour ahead or a day ahead), we can match the supply by customers/users sheddng or ncreasng ther loads. In the followng we focus on the case wth a supply defct and consder a bddng scheme for the demand response. The case wth a supply surplus can be handled n the same way. A. System Model Consder a power network wth a set N of customers/users 1 that are served by one utlty company (or generator). Assocated wth each customer N s a load q that t s wllng to shed n a demand response system. We assume that the total load shed needs to meet a specfc amount d> of electrcty supply defct,.e., q = d. (1) Assume that customer ncurs a cost (or dsutlty) C (q ) when t sheds a load of q. We assume that cost functon C ( ) s contnuous, ncreasng, strctly convex, and wth C () =. We consder a market mechansm for the load sheddng allocaton, based on supply functon bddng [3]. For smplcty of mplementaton of the demand response scheme, we assume that each customer s supply functon (for load sheddng) s parameterzed by a sngle parameter b, N, and takes the form of q (b,p)=b p, N. (2) The supply functon q (b,p) gves the amount of load customer s commtted to shed when the prce s p. The utlty company wll choose a prce p that clears the market,.e., q (b,p)= b p = d, (3) from whch we get p(b) = d. (4) b Here b =(b 1,b 2,,b N ), the supply functon profle. 1 Here a customer/user can be a sngle resdental or commercal customer, or represent a group of customers that acts as a sngle demand response entty. Remark: Supply functon as a strategc varable allows to adapt better to changng market condtons (such as uncertan demand) than does a smple commtment to a fxed prce or quantty [3]. Ths s one reason we use supply functon bddng, as we wll further study demand response under uncertan power network condtons. The other motvaton to use supply functon s to respect practcal nformatonal constrants n the power network. A customer mght not want to reveal ts cost functon because of ncentve or securty concerns, or the cost functon may requre a hgh descrpton complexty, whch means more communcaton/computaton. A properlychosen parameterzed supply functon controls nformaton revelaton whle demands less communcaton/computaton. B. Optmal demand response In ths subsecton, we consder a compettve market where customers are prce takng. Gven prce p, each customer mzes ts net revenue b pq (b,p) C (q (b,p)), (5) where the frst term s the customer s revenue when t sheds a load of q (b,p) at a prce of p and the second term s the cost ncurred. 1) Compettve equlbrum: We now analyze the equlbrum of the demand response system. A compettve equlbrum for the demand response system s defned as a tuple {(b ) N,p}, such that (C (q (b,q)) p)(ˆb b ), ˆb, (6) q (b,q) = d. (7) Theorem 1: There exsts a unque compettve equlbrum for the demand response system. Moreover, the equlbrum s effcent,.e., t mzes the socal welfare: q s.t. Proof: From equatons (6)-(7), we have C (q ) (8) q = d. (9) (C (q ) p)(ˆq q ), ˆq, q = d. Ths s just the optmalty condton of optmzaton problem (8)-(9) [1]. The unqueness of the equlbrum follows from the fact that problem (8)-(9) and ts dual are strctly convex. 2) Iteratve supply functon bddng: The socal welfare problem (8)-(9) can be easly solved by the dual gradent algorthm [1]. Ths suggests an teratve, dstrbuted supply functon bddng scheme for demand response that acheves the market equlbrum. At k-th teraton: Upon recevng prce p(k) announced by the utlty company over a communcaton network, each customer updates ts supply functon,.e., b (k), accordng to b (k) =[ (C ) 1 (p(k)) ] +, (1) p(k) 398

3 and submts t to the utlty company over the communcaton network. Here + denotes the projecton onto R +, the set of nonnegatve real numbers. Upon gatherng bds b (k) from customers, the utlty company updates the prce accordng to p(k +1)=[p(k) γ( b (k)p(k) d)] +, (11) and announces the prce p(k +1) to the customers over the communcaton network. Here γ > s a constant stepsze. When γ s small enough, the above algorthm converges [1]. The scheme requres only lght communcaton and computaton, and wll converge n short tme wth modern communcaton and computng technologes even for a very large network. The utlty company and customers jontly run the market (.e., the teratve bddng scheme) to fnd equlbrum prce and allocaton before the actual acton of load sheddng. The equlbrum prce wll be a market-clearng prce, and the actual load sheddng s suppled accordng to ths prce. C. Strategc demand response In ths subsecton, we consder an olgopoly market where customers know that prce p s set accordng to (4) and are strategc. Denote the supply functon for all customers but by b =(b 1,b 2,,b 1,b +1,,b N ) and wrte (b,b ) for the supply functon profle b. Each customer chooses b that mzes u (b,b ) = p(b)q (p(b),b ) C (q (p(b),b )) d 2 b = ( db C( ). (12) j bj)2 j bj Ths defnes a demand response game among customers. 1) Game-theoretc equlbrum: We now analyze the equlbrum of the demand response game. The soluton concept we use s the Nash equlbrum [11]. A supply functon profle b s a Nash equlbrum f, for all customers N, u (b,b ) u (b,b ) for all b. We see that the Nash equlbrum s a set of strateges for whch no player has an ncentve to change unlaterally. Lemma 2: If b s a Nash equlbrum of the demand response game, then j b j > for any N. Proof: We prove the result by contradcton. Suppose that t does not hold, and wthout loss of generalty, assume that j b j =for a customer. Then, the payoff for the customer s u (b,b )=f b =, and u (b,b )= d 2 /b C (d) f b >. We see that when b =,the customer has an ncentve to ncrease t, and when b > the customer has an ncentve to decrease t. Hence, there s no Nash equlbrum wth j b j =. The above Lemma also mples that at the Nash equlbrum at least two customers have b >. Let B = j b j.wehave u (b,b ) = d2 (B b ) b (B + b db db ) 3 (B + b ) 2 C ( ) B + b d 2 B b = [ B db (B + b ) 2 B + b d C ( )].(13) B + b The frst term n the square bracket s strctly decreasng n b and the second term s strctly ncreasng n b. So, f B d C () 1, b u (b,b ) for all b, and b = mzes the customer payoff u (b,b ) for the gven B b. If d C () < 1, b u (b,b ) = only at one pont b >. Furthermore, note that b u (,b ) > and b u (B,b ) <. So, ths pont b mzes the customer payoff u (b,b ) for the gven b. Thus, at the Nash equlbrum for the demand response game, b satsfes f B d C () 1; and otherwse, B b B + b B b =, (14) db d C ( B + )=. (15) b Lemma 3: If b s a Nash equlbrum of the demand response game, then b <B = j b j for any N,.e., each customer wll shed a load of less than d/2 at the equlbrum. Proof: The result holds when b =. Note that the second term on the left hand sde of equaton (15) s postve. So the frst term must be postve as well, whch requres B >b. The followng result follows drectly from Lemma 3. Corollary 4: No Nash equlbrum exsts when N =2. Theorem 5: Assume N > 2. The demand response game has a unque Nash equlbrum. Moreover, the equlbrum solves the followng convex optmzaton problem: D (q ) (16) q <d/2 s.t. q = d, (17) wth q D (q )=(1+ )C (q ) d 2q Proof: Frst, note that D (q )=(1+ q d C(x)dx. (18) (d 2x ) 2 q d 2q )C (q ), (19) whch s a postve, strctly ncreasng functon n b [,d/2). So, D (q ) s a strctly convex functon n [,d/2). Thus, the optmzaton problem (16)-(17) s a strctly convex problem and has a unque soluton. Based on the optmalty condton [1] and after a bt mathematcal manpulaton, the unque soluton q s determned by (p (1 + q )C (q d 2q ))(q q ), q, (2) q = d, (21) p >. (22) Second, note that the Nash equlbrum condton (14)-(15) can be wrtten compactly as d ( B + B b B C db ( b B + ))(b b b ), b. (23) 399

4 Recall that the (Nash) equlbrum prce p = d/ b and (Nash) equlbrum allocaton q = b p. We can wrte equaton (23) as (p (1 + q d 2q )C (q ))(b p q ). (24) Note that at the Nash equlbrum, p > snce b > by Lemma 2, and b s arbtrary. So, the above equaton s equvalent to equaton (2). Thus, the Nash equlbrum of the demand response game satsfes the optmalty condton (2)-(22), and solves the optmzaton problem (16)-(17). The exstence and unqueness of the Nash equlbrum follows from the fact that problem (16)-(17) admts a unque optmum. Suppose that there are n customers wth q > at the Nash equlbrum. By Lemma 3, n 3. There exsts at least one customer j such that D j (q j )=p and qj d/n. Thus, p (1 + 1 n 2 )C j (d/n) M, where M = D (d/n) D (d/3). Leth = (D ) 1 (M), wehaveq h for all N. Quanttes h d/ N and h d/n can be seen as measures of the heterogenety n the system. For a homogeneous system where customers have the same dsutlty functon, both measures equal zero. We can show that the Nash equlbrum prce p (1+h/(d 2h)) p, where p s the prce at compettve equlbrum dscussed n last subsecton. Remark: Theorem 5 can be seen as reverse-engneerng from the game-theoretc equlbrum nto a global optmzaton problem. 2) Iteratve supply functon bddng: By Theorem 5, we can solve the Nash equlbrum of the demand response game by solvng convex optmzaton problem (16)-(17). Ths suggests the followng teratve supply functon bddng scheme to acheve the market equlbrum. At k-th teraton: Upon recevng prce p(k) announced by the utlty company over the communcaton network, each customer updates ts supply functon,.e., b (k), accordng to b (k) =[ (D ) 1 (p(k)) ] +, (25) p(k) and submts t to the utlty company over the communcaton network. Upon gatherng bds b (k) from customers, the utlty company updates the prce accordng to p(k +1)=[p(k) γ( b (k)p(k) d)] +, (26) and announces prce p(k +1) to customers over the communcaton network. Note that the dstrbuted convergence to the Nash equlbrum s a dffcult problem n general, because of nformatonal constrants n the system. Here we nvolve the utlty company n medatng strategc nteracton among customers, see equaton (26), n order to acheve the equlbrum n a dstrbuted manner. The strategc acton of the customer s also partally encapsulated n equaton (25). IV. DEMAND RESPONSE: SHAPING THE DEMAND In ths secton, we consder demand shapng by subjectng customers to realtme spot prces and ncentvzng them to shft or even reduce ther loads. In the followng we study a utlty optmzaton model, based on whch propose dstrbuted algorthms for demand shapng. A. System Model Consder a power network wth a set N of customers/users that are served by one utlty company (or generator). Assocated wth each customer N s ts power load q (t) at tme t. 2 We assume that each customer has a mnmum total power requrement n a day 3 T q (t) Q, N, (27) correspondng to, e.g., basc daly routnes; and a mum total power requrement n a day T q (t) Q, N, (28) correspondng to, e.g., the total energy usage for a comfortable lfe style. Assume that each customer attans a utlty U (q,t) when ts power draw s q at tme t. The tme-dependent utlty models a general stuaton where the customer may have dfferent power requrements at dfferent tmes. We assume that U (q,t) as a functon of q s contnuously dfferentable, strctly concave, ncreasng, wth the curvatures bounded away from zero. On the supply sde, we assume that the utlty company has a tme-dependent cost of C(Q, t) when t supples power Q at tme t. The tme-dependent cost functon models a stuaton where energy generaton cost mght be dfferent at dfferent tmes. For example, when renewable energy such as solar s presented, the cost may depend on weather condtons, and a sunny hour may reduce the demand on power from a tradtonal power plant and result n a lower cost. The modelng of the cost functon s an actve research ssue. Here we assume that the cost functon C(Q, t) as a functon of Q s strctly convex, wth a postve, ncreasng margnal cost. We assume that the objectve of the utlty company s to mze ts net revenue. Gven prce p(t), t plans on ts supply so as to solve the followng mzaton problem: 4 Q(t) Q(t)p(t) C(Q (t),t), (29) whose soluton takes a smple form C (Q(t),t)=p(t), t T. (3) Note that the supply must equal the demand n the power network. So, the prce (p(t)) should settle down at a pont that clears the market 2 Note that we redefne the notaton. In ths secton q denotes the load of customer, whle n secton III q denotes the amount of load that customer s wllng to shed. 3 Each day s dvded nto T tmeslots of equal duraton, ndexed by t T = {1, 2,, T }. 4 Our focus n ths paper s on the demand sde. We thus do not consder possble strategc behavors of the utlty company or generator. 4

5 q (t) =Q(t), t T. (31) N B. Optmal demand response In ths subsecton, we consder a compettve market where customers are prce takng. Gven realtme spot prce p(t), customer allocates ts energy usage to mze ts aggregate net utlty subject to constrants (27)-(28): q (t) s.t. U (q (t),t) q (t)p(t) (32) q (t) Q, N (33) q (t) Q, N. (34) The above model captures two of the essental elements of demand response: realtme prcng and demand shftng. Demand shftng s acheved through optmzng over a certan perod of tme. 1) Compettve equlbrum: By ntroducng Lagrange multpler λ and λ for constrants (33) and (34) respectvely, the optmal q(t) of the problem (32)-(34) s determned by the followng condtons U (q (t),t)=p(t)+ λ λ, N, t T, (35) T λ (Q q (t)) =, N, (36) T λ ( q (t) Q )=, N. (37) When λ >,.e, constrant (33) s tght, the customer would equvalently pay a hgher prce than t should, n order to meet the mnmum demand on power. When λ >,.e, constrant (34) s tght, the customer would equvalently pay a lower prce than t should, whch can happen when the utlty company subsdzes the customer to encourage electrcty consumpton. A compettve equlbrum for the demand response system s defned as a trple {(q (t)) N,, (Q(t)), (p(t)) } that satsfes (35)-(37) and (3)-(31). Theorem 6: There exsts a unque compettve equlbrum for the demand response system. Moreover, the equlbrum s effcent,.e., t mzes the socal welfare: q (t) s.t. { U (q (t),t) C( q (t),t)} (38) N N T q (t) Q, N (39) T q (t) Q, N. (4) Proof: Pluggng equatons (3) and (31) nto equatons (35)-(37), we get U (q (t),t) C ( q (t),t)= λ λ, N, t T, T λ (Q q (t)) =, N, T λ ( q (t) Q )=, N, whch s just the optmalty condtons for the socal welfare problem. The unqueness of equlbrum comes from the fact that the socal welfare problem and ts dual are strctly convex. 2) Dstrbuted algorthm: The socal welfare problem (38)- (4) suggests a dstrbuted algorthm to compute the market equlbrum, based on the gradent algorthm [1]. At k-th teraton: The utlty company collects demands (q k(t)) from each customer over the communcaton network, calculates the total demand (Q k (t)) and the assocated margnal cost p k (t) =C (Q k (t),t), t T, (41) and announces (p k (t)) to customers over the communcaton network. Each customer updates ts demand q k (t) after recevng the update on prce p k (t), accordng to q k+1 (t) =[q k (t)+γ(u (q k (t),t) p k (t))] s, (42) where where γ> s a constant stepsze, and s denotes projecton onto the set S specfed by constrants (27)- (28). The projecton operaton s easy to do, as constrants (27)-(28) are local to customers. When γ s small enough, the above algorthm converges [1]. The utlty company and customers jontly run the market (.e., the above dstrbuted algorthm) to decde on power loads and supply for each tme t. C. Strategc demand response In ths subsecton, we consder an olgopoly market where customers know margnal cost (or supply curve) of the utlty company and are strategc. We can model demand response problem as a game among customers: Gven other customer power loads (q (t)) = {(q j (t)),j N/{}}, each customer chooses q (t) that mzes u (q (t),q (t)) = U (q (t),t) q (t)c ( q (t),t), (43) N subject to constrants (27)-(28). 1) Game-theoretc equlbrum: We now analyze the equlbrum of the demand response game. Note that the margnal cost C ( ) s postve and ncreasng by assumpton. Thus, customer s payoff u (q (t),q (t)) s concave n (q (t)). So, at the Nash equlbrum, (q (t)) satsfes u (q (t),q (t)) q (t) (q (t) q (t)), (q (t)) S. (44) Theorem 7: The demand response game has a unque Nash equlbrum. Moreover, t solves the convex problem: { U (q (t),t) q (t)c ( q (t),t)} (45) q (t) N N T s.t. q (t) Q, N (46) T q (t) Q, N. (47) 41

6 Proof: It s straghtforward to check that the objectve (45) s strctly concave, and the Nash equlbrum condton (44) s the optmalty condton (varatonal nequalty) for the convex problem (45)-(47). The theorem follows. 2) Dstrbuted algorthm: The above optmzaton problem characterzaton of the Nash equlbrum suggests a dstrbuted algorthm to compute the equlbrum. At k-th teraton: Customers exchange nformaton on ther demands (q k(t)) over the communcaton network. Each customer then calculates the total demand (Q k (t)) and updates ts demand q k (t), accordng to q k+1 (t) = [q k (t)+γ(u (q k (t),t) C (Q k (t),t) p k (t)c (Q k (t),t))] s, (48) where γ> s a constant stepsze. Customers need to communcate wth each other to jontly run the above algorthm to decde on ther power usage at each tme t. Note that we could also nvolve the utlty company n medatng strategc nteracton among customers, as n subsecton III-C.2. V. NUMERICAL EXAMPLES In ths secton, we provde numercal examples to complement the analyss n prevous sectons. We consder a smple power network wth 1 customers that jon n the demand response system. Due to the page lmt, we wll only report results on teratve supply functon bddng proposed n secton III. We assume that each customer has a cost functon C (q )=a q + h q 2 wth a and h >. The electrcty supply defct s normalzed to be 1, and the values for parameters a and h used to obtan numercal results are randomly drawn from [1, 2] and [2, 6], respectvely. Fgure 1 shows the evoluton of the prce and 5 customers supply functons wth stepsze γ =.2 for optmal supply functon bddng and for strategc supply functon bddng, respectvely. We see that the prce and supply functons approach the market equlbrum quckly. In order to study the mpact of dfferent choces of the stepsze on the convergence of the algorthms, we have run smulatons wth dfferent stepszes. We found that the smaller the stepsze, the slower the convergence, and the larger the stepsze, the faster the convergence but the system may only approach to wthn a certan neghborhood of the equlbrum, whch s a general characterstc of any gradent based method. In practce, the utlty company can frst choose large stepszes to ensure fast convergence, and subsequently reduce the stepszes once the prce starts oscllatng around some mean value. VI. CONCLUSION We have studed two market models for demand response n power networks. We characterze the resultng equlbra n compettve as well as olgopolstc markets, and propose dstrbuted demand response schemes and algorthms to match electrcty supply and to shape electrcty demand accordngly. As further research steps, we are characterzng effcency loss of the game-theoretc equlbra. We wll also search Prce Prce Number of Iteratons Number of Iteratons Supply functon b Supply functon b customer 2 customer 4.1 customer 6 customer 8 customer Number of Iteratons customer 2 customer 4.1 customer 6 customer 8 customer Number of Iteratons Fg. 1. Prce and supply functon evoluton of optmal supply functon bddng (upper panels) and strategc supply functon bddng (lower panels) for demand response. for other forms of parameterzed supply functons that are more expressve whle admt tractable analyss. As there are varous uncertantes n power networks, e.g., t may be dffcult to estmate or predct the power generaton from the solar or wnd farm precsely, we wll study demand response under uncertan power network condtons. Ths paper serves as a startng pont for desgnng practcal demand response schemes and algorthms for smart power grds. We wll further brng n the detaled dynamcs and realstc constrants of demand response applances. We expect that these new constrants wll not change the general structure of our models (n terms of, e.g, equlbrum characterzaton, and dstrbuted decomposton structure, etc), but they wll lead to hgher communcaton overhead and computng complexty as we come to the schedulng of ndvdual electronc applances. REFERENCES [1] The Smart Grd: An Introducton. The US Department of Energy, 28. [2] B. Krby and E. Hrst, Load as a resource n provdng ancllary servces, Techncal report, Oak Rdge Natonal Laboratory, [3] P. D. Klemper and M. A. Meyer, Supply functon equlbra n olgopoly under uncertanty, Econometrca, 57(6): , [4] R. Green and D. Newbery Competton n the brtsh electrcty spot market, Journal of Poltcal Economy, 1(5): , [5] A. Rudkevch, M. Duckworth and R. Rosen, Modelng electrcty prcng n a deregulated generaton ndustry: The potental for olgopoly prcng n a Poolco, Energy Journal, [6] R. Baldck and W. 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