Two Market Models for Demand Response in Power Networks

Transcription

1 Two Market Moels for Deman Response n Power Networks Ljun Chen, Na L, Steven H. Low an John C. Doyle Engneerng & Apple Scence Dvson, Calforna Insttute of Technology, USA Abstract In ths paper, we conser two abstract market moels for esgnng eman response to match power supply an shape power eman, respectvely. We characterze the resultng equlbra n compettve as well as olgopolstc markets, an propose strbute eman response algorthms to acheve the equlbra. The moels serve as a startng pont to nclue the applance-level etals an constrants for esgnng practcal eman response schemes for smart power grs. I. INTRODUCTION The usual practce n power networks s to match supply to eman. Ths s challengng because eman s hghly tmevaryng. The utlty company or generator nees to provson enough generaton, transmsson an strbuton capactes for peak eman rather than the average. As a result, the power network has a low loa factor an s unerutlze most of the tme, whch s very costly. For example, the US natonal loa factor s about 55%, an 1% of generaton an 25% of strbuton facltes are use less than 4 hours per year,.e., 5% of the tme [1]. Shapng the eman to reuce the peak an smooth the varaton can greatly mprove power system effcency an yel huge savngs. An alternatve strategy mprovng effcency an reucng cost s to match the supply. As the proporton of renewable sources such as solar an wn power stealy rses, power supply wll also become hghly tme-varyng. Matchng the supply wll become a more effectve an common way to mprove power system effcency an reuce cost [2]. In ths paper, we conser two abstract market moels for esgnng eman response to match the supply an shape the eman, respectvely. Specfcally, n secton III, we conser a stuaton where there s an nelastc supply efct (or surplus) on electrcty, an stuy a supply functon bng scheme for allocatng loa sheng (or loa ncreasng) among fferent customers/users to match the supply. Each customer submts a parameterze supply functon to the utlty company, whch wll ece on a market-clearng prce base on the bs of customers, an s commtte to she (or ncrease) ts loa accorng to ts b an 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 an strategc, the system acheves a unque Nash equlbrum that mzes another atve, global objectve functon. Base on these optmzaton problem characterzatons of the market equlbra, we propose teratve, strbute supply functon bng schemes for the eman response to acheve the equlbra. In secton IV, we conser a stuaton where power supply s elastc, but customers are subjecte to realtme spot prces an wll shft (an sometmes reuce) ther emans accorngly. Each customer has a lower boun as well as an upper boun on the total electrcty eman over a ay. The customer wll allocate ts power usage for fferent tmes, so as to mze ts aggregate net utltes over a ay. We show that when customers are prce takng, the system acheves an effcent equlbrum that mzes socal welfare. When customers are prce antcpatng an strategc, the system acheves a unque Nash equlbrum that mzes another global objectve functon. Agan, base on the optmzaton problem characterzatons of the market equlbra, we propose strbute algorthms for eman shapng to acheve the equlbra. The aforementone eman response schemes requre tmely two-way communcatons between the customers an the utlty company an even rect communcatons between the customers. They also requre certan computng capablty of the customers. These communcaton an computng capabltes wll become norm n future smart grs [1]. Wth the ntegraton of state-of-the-art communcaton an computng technologes, future power systems wll become more ntellgent, more open, more autonomous, an wth much greater user partcpaton. Our eman response schemes are ntene to apply n exactly such smart power grs. II. RELATED WORK There exsts a huge amount of work on market moels for varous aspects of power networks. We brefly scuss those works that are rectly relevant to ths paper. The supply functon equlbrum moel has been wely use n the analyss of markets n many nustres. 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 an Meyer stuy the supply functon equlbrum an gve contons for the exstence an the unqueness of the Nash equlbrum n supply functons uner uncertan eman, an show that the equlbra are contane n a range of prces an allocatons between the Cournot an the Bertran equlbra. The most notable applcaton of the supply functon equlbrum moel s to the wholesale electrcty markets, see, e.g., [4], [5], [6], [7], [8]. In ths paper, nstea of applyng t to the electrcty supply se, we apply the supply functon equlbrum concept to prcng an allocaton on the eman se to match the electrcty supply, wth a specal form of parameterze supply functons that can enable a smple mplementaton of the teratve supply functon bng as an effectve eman response scheme n power networks. The moel stue n secton IV-B s a straghtforwar extenson of the compettve equlbrum moels for the power

2 network, see, e.g., [9]. In aton to trang off the costs an utltes among fferent customers, we also conser trang off the costs an utltes over tme, whch ncentvzes customers to shft ther electrcty usage. III. DEMAND RESPONSE: MATCHING THE SUPPLY In ths secton we conser a stuaton where there s a supply efct or surplus on electrcty. The efct can be ue to a ecrease n power generaton from, e.g., a wn or solar farm because of a change to worse weather conton, or an ncrease n power eman because of, e.g., a hot weather. The surplus can be ue to an ncrease n power generaton from, e.g., a wn or solar farm because of a change to better weather conton, or a ecrease n power eman at, e.g., the late nght tme. We assume that t s very costly to ncrease the power supply n the case of a efct or ecrease the supply n the case of a surplus,.e., the power supply s nelastc. If we have goo estmaton of electrcty efct or surplus (e.g., an hour ahea or a ay ahea), we can match the supply by customers/users sheng or ncreasng ther loas. In the followng we focus on the case wth a supply efct an conser a bng scheme for the eman response. The case wth a supply surplus can be hanle n the same way. A. System Moel Conser a power network wth a set N of customers/users 1 that are serve by one utlty company (or generator). Assocate wth each customer N s a loa q that t s wllng to she n a eman response system. We assume that the total loa she nees to meet a specfc amount > of electrcty supply efct,.e., q =. (1) Assume that customer ncurs a cost (or sutlty) C (q ) when t shes a loa of q. We assume that cost functon C ( ) s contnuous, ncreasng, strctly convex, an wth C () =. We conser a market mechansm for the loa sheng allocaton, base on supply functon bng [3]. For smplcty of mplementaton of the eman response scheme, we assume that each customer s supply functon (for loa sheng) s parameterze by a sngle parameter b, N, an takes the form of q (b, p) = b p, N. (2) The supply functon q (b, p) gves the amount of loa customer s commtte to she when the prce s p. The utlty company wll choose a prce p that clears the market,.e., q (b, p) = b p =, (3) from whch we get p(b) = b. (4) Here b = (b 1, b 2,, b N ), the supply functon profle. 1 Here a customer/user can be a sngle resental or commercal customer, or represent a group of customers that acts as a sngle eman response entty. Remark: Supply functon as a strategc varable allows to aapt better to changng market contons (such as uncertan eman) than oes a smple commtment to a fxe prce or quantty [3]. Ths s one reason we use supply functon bng, as we wll further stuy eman response uner uncertan power network contons. 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 escrpton complexty, whch means more communcaton/computaton. A properlychosen parameterze supply functon controls nformaton revelaton whle emans less communcaton/computaton. B. Optmal eman response In ths subsecton, we conser 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 shes a loa of q (b, p) at a prce of p an the secon term s the cost ncurre. 1) Compettve equlbrum: We now analyze the equlbrum of the eman response system. A compettve equlbrum for the eman response system s efne as a tuple {(b ) N, p}, such that (C (q (b, q)) p)(ˆb b ), ˆb, (6) q (b, q) =. (7) Theorem 1: There exsts a unque compettve equlbrum for the eman response system. Moreover, the equlbrum s effcent,.e., t mzes the socal welfare: q Proof: From equatons (6)-(7), we have C (q ) (8) q =. (9) (C (q ) p)(ˆq q ), ˆq, q =. Ths s just the optmalty conton of optmzaton problem (8)-(9) [1]. The unqueness of the equlbrum follows from the fact that problem (8)-(9) an ts ual are strctly convex. 2) Iteratve supply functon bng: The socal welfare problem (8)-(9) can be easly solve by the ual graent algorthm [1]. Ths suggests an teratve, strbute supply functon bng scheme for eman response that acheves the market equlbrum. At k-th teraton: Upon recevng prce p(k) announce by the utlty company over a communcaton network, each customer upates ts supply functon,.e., b (k), accorng to b (k) = [ (C ) 1 (p(k)) ] +, (1) p(k)

3 an submts t to the utlty company over the communcaton network. Here + enotes the projecton onto R +, the set of nonnegatve real numbers. Upon gatherng bs b (k) from customers, the utlty company upates the prce accorng to p(k + 1) = [p(k) γ( b (k)p(k) )] +, (11) an 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 an computaton, an wll converge n short tme wth moern communcaton an computng technologes even for a very large network. The utlty company an customers jontly run the market (.e., the teratve bng scheme) to fn equlbrum prce an allocaton before the actual acton of loa sheng. The equlbrum prce wll be a market-clearng prce, an the actual loa sheng s supple accorng to ths prce. C. Strategc eman response In ths subsecton, we conser an olgopoly market where customers know that prce p s set accorng to (4) an are strategc. Denote the supply functon for all customers but by b = (b 1, b 2,, b 1, b +1,, b N ) an 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 )) 2 b = ( j b j) C ( b 2 j b ). (12) j Ths efnes a eman response game among customers. 1) Game-theoretc equlbrum: We now analyze the equlbrum of the eman 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 eman response game, then j b j > for any N. Proof: We prove the result by contracton. Suppose that t oes not hol, an wthout loss of generalty, assume that j b j = for a customer. Then, the payoff for the customer s u (b, b ) = f b =, an u (b, b ) = 2 /b C () f b >. We see that when b =, the customer has an ncentve to ncrease t, an when b > the customer has an ncentve to ecrease 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. We have u (b, b ) = 2 (B b ) b (B + b ) B b 3 (B + b ) 2 C ( ) B + b 2 = (B + b ) [ B b B b 2 B + b C ( )].(13) B + b The frst term n the square bracket s strctly ecreasng n b an the secon term s strctly ncreasng n b. So, f B C () 1, b u (b, b ) for all b, an b = mzes the customer payoff u (b, b ) for the gven B b. If C () < 1, b u (b, b ) = only at one pont b >. Furthermore, note that b u (, b ) > an 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 eman response game, b satsfes f B C () 1; an otherwse, B b B + b B b =, (14) b C ( B + ) =. (15) b Lemma 3: If b s a Nash equlbrum of the eman response game, then b < B = j b j for any N,.e., each customer wll she a loa of less than /2 at the equlbrum. Proof: The result hols when b =. Note that the secon term on the left han se of equaton (15) s postve. So the frst term must be postve as well, whch requres B > b. The followng result follows rectly from Lemma 3. Corollary 4: No Nash equlbrum exsts when N = 2. Theorem 5: Assume N > 2. The eman response game has a unque Nash equlbrum. Moreover, the equlbrum solves the followng convex optmzaton problem: D (q ) (16) wth q </2 q D (q ) = (1 + )C (q ) 2q Proof: Frst, note that D (q ) = (1 + q =, (17) q C(x)x. (18) ( 2x ) 2 q 2q )C (q ), (19) whch s a postve, strctly ncreasng functon n b [, /2). So, D (q ) s a strctly convex functon n [, /2). Thus, the optmzaton problem (16)-(17) s a strctly convex problem an has a unque soluton. Base on the optmalty conton [1] an after a bt mathematcal manpulaton, the unque soluton q s etermne by (p (1 + q 2q )C (q ))(q q ), q, (2) q =, (21) p >. (22) Secon, note that the Nash equlbrum conton (14)-(15) can be wrtten compactly as b ( B + B b B C ( b B + ))(b b b ), b. (23)

4 Recall that the (Nash) equlbrum prce p = / b an (Nash) equlbrum allocaton q = b p. We can wrte equaton (23) as (p (1 + q )C (q 2q ))(b p q ). (24) Note that at the Nash equlbrum, p > snce b > by Lemma 2, an b s arbtrary. So, the above equaton s equvalent to equaton (2). Thus, the Nash equlbrum of the eman response game satsfes the optmalty conton (2)-(22), an solves the optmzaton problem (16)-(17). The exstence an unqueness of the Nash equlbrum follows from the fact that problem (16)-(17) amts 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 an qj /n. Thus, p (1 + 1 n 2 )C j (/n) M, where M = D (/n) D (/3). Let h = (D ) 1 (M), we have q h for all N. Quanttes h / N an h /n can be seen as measures of the heterogenety n the system. For a homogeneous system where customers have the same sutlty functon, both measures equal zero. We can show that the Nash equlbrum prce p (1+h/( 2h)) p, where p s the prce at compettve equlbrum scusse 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 bng: By Theorem 5, we can solve the Nash equlbrum of the eman response game by solvng convex optmzaton problem (16)-(17). Ths suggests the followng teratve supply functon bng scheme to acheve the market equlbrum. At k-th teraton: Upon recevng prce p(k) announce by the utlty company over the communcaton network, each customer upates ts supply functon,.e., b (k), accorng to b (k) = [ (D ) 1 (p(k)) ] +, (25) p(k) an submts t to the utlty company over the communcaton network. Upon gatherng bs b (k) from customers, the utlty company upates the prce accorng to p(k + 1) = [p(k) γ( b (k)p(k) )] +, (26) an announces prce p(k + 1) to customers over the communcaton network. Note that the strbute convergence to the Nash equlbrum s a ffcult problem n general, because of nformatonal constrants n the system. Here we nvolve the utlty company n meatng strategc nteracton among customers, see equaton (26), n orer to acheve the equlbrum n a strbute manner. The strategc acton of the customer s also partally encapsulate n equaton (25). IV. DEMAND RESPONSE: SHAPING THE DEMAND In ths secton, we conser eman shapng by subjectng customers to realtme spot prces an ncentvzng them to shft or even reuce ther loas. In the followng we stuy a utlty optmzaton moel, base on whch propose strbute algorthms for eman shapng. A. System Moel Conser a power network wth a set N of customers/users that are serve by one utlty company (or generator). Assocate wth each customer N s ts power loa q (t) at tme t. 2 We assume that each customer has a mnmum total power requrement n a ay 3 T q (t) Q, N, (27) corresponng to, e.g., basc aly routnes; an a mum total power requrement n a ay T q (t) Q, N, (28) corresponng 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 raw s q at tme t. The tme-epenent utlty moels a general stuaton where the customer may have fferent power requrements at fferent tmes. We assume that U (q, t) as a functon of q s contnuously fferentable, strctly concave, ncreasng, wth the curvatures boune away from zero. On the supply se, we assume that the utlty company has a tme-epenent cost of C(Q, t) when t supples power Q at tme t. The tme-epenent cost functon moels a stuaton where energy generaton cost mght be fferent at fferent tmes. For example, when renewable energy such as solar s presente, the cost may epen on weather contons, an a sunny hour may reuce the eman on power from a tratonal power plant an result n a lower cost. The moelng 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 eman n the power network. So, the prce (p(t)) shoul settle own at a pont that clears the market 2 Note that we reefne the notaton. In ths secton q enotes the loa of customer, whle n secton III q enotes the amount of loa that customer s wllng to she. 3 Each ay s ve nto T tmeslots of equal uraton, nexe by t T = {1, 2,, T }. 4 Our focus n ths paper s on the eman se. We thus o not conser possble strategc behavors of the utlty company or generator.

5 q (t) = Q(t), t T. (31) N B. Optmal eman response In ths subsecton, we conser 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) U (q (t), t) q (t)p(t) (32) q (t) Q, N (33) q (t) Q, N. (34) The above moel captures two of the essental elements of eman response: realtme prcng an eman shftng. Deman shftng s acheve through optmzng over a certan pero of tme. 1) Compettve equlbrum: By ntroucng Lagrange multpler λ an λ for constrants (33) an (34) respectvely, the optmal q(t) of the problem (32)-(34) s etermne by the followng contons 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 woul equvalently pay a hgher prce than t shoul, n orer to meet the mnmum eman on power. When λ >,.e, constrant (34) s tght, the customer woul equvalently pay a lower prce than t shoul, whch can happen when the utlty company subszes the customer to encourage electrcty consumpton. A compettve equlbrum for the eman response system s efne as a trple {(q (t)) N,, (Q(t)), (p(t)) } that satsfes (35)-(37) an (3)-(31). Theorem 6: There exsts a unque compettve equlbrum for the eman response system. Moreover, the equlbrum s effcent,.e., t mzes the socal welfare: q (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) an (31) nto equatons (35)-(37), we get U (q (t), t) C ( q (t), t) = λ λ, N, t T, λ (Q T q (t)) =, N, T λ ( q (t) Q ) =, N, whch s just the optmalty contons for the socal welfare problem. The unqueness of equlbrum comes from the fact that the socal welfare problem an ts ual are strctly convex. 2) Dstrbute algorthm: The socal welfare problem (38)- (4) suggests a strbute algorthm to compute the market equlbrum, base on the graent algorthm [1]. At k-th teraton: The utlty company collects emans (q k(t)) from each customer over the communcaton network, calculates the total eman (Q k (t)) an the assocate margnal cost p k (t) = C (Q k (t), t), t T, (41) an announces (p k (t)) to customers over the communcaton network. Each customer upates ts eman q k (t) after recevng the upate on prce p k (t), accorng 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, an s enotes projecton onto the set S specfe by constrants (27)- (28). The projecton operaton s easy to o, as constrants (27)-(28) are local to customers. When γ s small enough, the above algorthm converges [1]. The utlty company an customers jontly run the market (.e., the above strbute algorthm) to ece on power loas an supply for each tme t. C. Strategc eman response In ths subsecton, we conser an olgopoly market where customers know margnal cost (or supply curve) of the utlty company an are strategc. We can moel eman response problem as a game among customers: Gven other customer power loas (q (t)) = {(q j (t)), j N/{}}, each customer chooses q (t) that mzes q (t), t), (43) u (q (t), q (t)) = U (q (t), t) q (t)c ( N subject to constrants (27)-(28). 1) Game-theoretc equlbrum: We now analyze the equlbrum of the eman response game. Note that the margnal cost C ( ) s postve an 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 eman 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 q (t) Q, N (46) T q (t) Q, N. (47)

6 Proof: It s straghtforwar to check that the objectve (45) s strctly concave, an the Nash equlbrum conton (44) s the optmalty conton (varatonal nequalty) for the convex problem (45)-(47). The theorem follows. 2) Dstrbute algorthm: The above optmzaton problem characterzaton of the Nash equlbrum suggests a strbute algorthm to compute the equlbrum. At k-th teraton: Customers exchange nformaton on ther emans (q k(t)) over the communcaton network. Each customer then calculates the total eman (Q k (t)) an upates ts eman q k (t), accorng 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 nee to communcate wth each other to jontly run the above algorthm to ece on ther power usage at each tme t. Note that we coul also nvolve the utlty company n meatng strategc nteracton among customers, as n subsecton III-C.2. V. NUMERICAL EXAMPLES In ths secton, we prove numercal examples to complement the analyss n prevous sectons. We conser a smple power network wth 1 customers that jon n the eman response system. Due to the page lmt, we wll only report results on teratve supply functon bng propose n secton III. We assume that each customer has a cost functon C (q ) = a q + h q 2 wth a an h >. The electrcty supply efct s normalze to be 1, an the values for parameters a an h use to obtan numercal results are ranomly rawn from [1, 2] an [2, 6], respectvely. Fgure 1 shows the evoluton of the prce an 5 customers supply functons wth stepsze γ =.2 for optmal supply functon bng an for strategc supply functon bng, respectvely. We see that the prce an supply functons approach the market equlbrum quckly. In orer to stuy the mpact of fferent choces of the stepsze on the convergence of the algorthms, we have run smulatons wth fferent stepszes. We foun that the smaller the stepsze, the slower the convergence, an the larger the stepsze, the faster the convergence but the system may only approach to wthn a certan neghborhoo of the equlbrum, whch s a general characterstc of any graent base metho. In practce, the utlty company can frst choose large stepszes to ensure fast convergence, an subsequently reuce the stepszes once the prce starts oscllatng aroun some mean value. VI. CONCLUSION We have stue two market moels for eman response n power networks. We characterze the resultng equlbra n compettve as well as olgopolstc markets, an propose strbute eman response schemes an algorthms to match electrcty supply an to shape electrcty eman accorngly. 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 customer 6 customer 8 customer Number of Iteratons customer 2 customer 4 customer 6 customer 8 customer Number of Iteratons Fg. 1. Prce an supply functon evoluton of optmal supply functon bng (upper panels) an strategc supply functon bng (lower panels) for eman response. for other forms of parameterze supply functons that are more expressve whle amt tractable analyss. As there are varous uncertantes n power networks, e.g., t may be ffcult to estmate or prect the power generaton from the solar or wn farm precsely, we wll stuy eman response uner uncertan power network contons. Ths paper serves as a startng pont for esgnng practcal eman response schemes an algorthms for smart power grs. We wll further brng n the etale ynamcs an realstc constrants of eman response applances. 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