《博弈论》（英文版）STRATEGIC BIDDING IN ELECT

Dept. of Electrical Computer Engineering OOCL Transportation Co Dept, of Economics University of Ilinois at Urbana-Champaign 300 Central Ave Urbana IL 6180 University of Illinois at Urbana-Champaign Mountain view, CA 94043 Champaign, IL 61820 Abstract We provide a brief review of the EWPP operation [1].The Pool dispatcher is charged with determining on a daily basis We formulate a general framework of a competitive electric- the schedule for the so-called availability declaration period ity generation supply market(CEM, embodying the salient (ADP),a 39-bour

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STRATEGIC BIDDING IN ELECTRICITY GENERATION SUPPLY MARKETS George Gross David J. Finlay George Deltas Dept. of Electrical Computer Engineering OOCL Transportation Co Dept, of Economics University of Ilinois at Urbana-Champaign 300 Central Ave Urbana IL 6180 University of Illinois at Urbana-Champaign Mountain view, CA 94043 Champaign, IL 61820 Abstract We provide a brief review of the EWPP operation [1].The Pool dispatcher is charged with determining on a daily basis We formulate a general framework of a competitive electric- the schedule for the so-called availability declaration period ity generation supply market(CEM, embodying the salient (ADP),a 39-bour period running from 9: 00 p. m. on day 1, attributes of the poolco concept. This framework serves two the bid submittal day, to 12: 00 noon on day &. The gener principal purposes: to solve the selection by the CEM opera- tion schedule for the period known as the schedule day, running tor of the winners in a sealed bid auction for the right to serve from 5: 00 a. m. on day 1 to 5: 00 a.m. on day z, is then accepted load in each period of the auction horizon; and, to determine as the actual schedule for the next day. By 10: 00 a m. each the profit-maximising strategic bids of a generation supplier. day, the dispatcher produces a forecast of national demand for The formulation represents the physical and operating con- every half hour of the ADP. Also by 10: 00 a m, each bidder siderations of the electric generation system, the multi-period must submit an ofer file for each of his gensets. a genset is a nature of the auction as well as the market economics. The re- unit or a group of units which are considered together for the sulting large-scale nonlinear programming model has a struc- purposes of the dispatch. The offer file contains informatio are that is effectively exploited for solution by Lagrangian on the availability[maximum capacity] of the genset for each relaxation, Under conditions of a perfectly competitive mar- of the 78 ADP half hours; the offer price of the genset;the ket, the strategic bids of a player can be derived analytically. genset start-up prices; and, the genset operational characteris- Numerical results illustrate the effectiveness of the strategic tics. The prices charged from the Pool for operation and start bids. Directions for future research are discussed gation on a bidder to reveal its genset's true costs. The genset Introduction ffer price is specified as piece-wise linear function known as the Willans line[2]. A maximum of three segments can be The Poolco concept is based on the England and Wales submitted per genset Figure l shows an example. Power Pool(EWPP)[1]. The Pool is ontrols the scheduling and dispatch of generation to meet load around the clock and operates the electricity spot market. vir ofer Price tually all power is transacted through the pool and the multi ple buyers and sellers have set up what has become the largest Proe io ch itive mark in to such a structure. In particular, we consider the task such take best advantage of the sealed bid auction given the g ating resources, costs and constraints. We refer to this as the optimal bidding strategy problem. To formulate and attack this problem, we develop a mathematical framework of the op eration of a competitive electricity generation supply market (CEM). Our work explicitly considers the competitive bidding Output Power mechanism in electricity and takes into account the unique fea tures and problems associated with the generation of electrical Figure 1: Example of a willans line The Willans line is completely specified by at most 8 parame- two elbow points and the minimum and maximum output of the genset pi and prse Patcher determines the schedule of generation to meet the fore asted demand at minimum cost to the pool. This problem is ntially the unit commitment. Basically, the dispatcher ar- 0-7803-4403-098s10.00@1998IEEE 309

ranges the bidding genset blocks in order of increasing price to The bid variable price b((: describes the V< form a merit order list for each ADP half hour to the ceM as a function of mw provided. The price of the most expensive genset dispatched in any half that the bid variable price is a p hour t is designated as the System Marginal Price(SMP.)[3], e into R, the set of real for that half hour as is shown in Figure 2. Each genset that here pin(pi)is the minimum(maximum)output of ates during half hour t receives a payment that includes bidder is unit MPt for each MWh of energy generated during that time The bid start-up price b(: describes the cost incurred ( Hence every genset is paid more than or equal to the price by the CPP whenever unit i is started up. We assume specified in the offer file. Generators also receive the submitted that the start-u is a function of the down time start-up price each time the unit is started up the unit with 62:[0,∞)→R The bid ofered capacity gi=[ai, 1, ai, 2. by bidder i to the CPP dispatcher for use We assume that the bidder submi correct operational data for the unit lest a schedule be infeasible. These consist of pin, spinning reserve capability of unit Ti and d the unit minimum up and down times, respectively. The generator not obliged to reveal any information concerning true costs. The bid variable price, bid start-up price and bid offered ca- pacity are strategic decision variables that the bidde er selec maximize profits. he define a bid of bidder i to be the triple Bi=( 4(), Figure 2: Determination of the system marginal pric 620, gi). A bid p: is admissible if b! ()ccp?im, p7 a],b EC[0, oo)and ga 20cR, where C[a, bl(C(a, b1)denotes the set of continuous(piece-wise linear)functions on the interval [a, b] We construct a framework that embodies the salient charac- We state the CEM operator problem using the nota teristics of the EWPP. The bidder, thus, submits a bid for the tion of Table 1 and the definition of the T-dimensione right to serve load Under competitive conditions, the bidder vectors D= [D1, D2,, Dr], R=[Ri, Ra,Rr ui= [ui lect the unit to be included in the commitment list. Since the ],g2=四p,i,…;r]and al vectors bidder receives a payment which is greater or equal to its bi der to maximise profits. Given the large-scale and nonlinear li,r,IMl The cem operator problem determine nature of the problem, the auction theory literature (4). (5). (6 most economic dispatch that satisfies the forecasted demands and exploits well the structural characteristics of the analyti- constraints. This is denoted by cal based on decision analysis CEM framework. The strength of the results lies in the explicit representation of the variou constraints and considerations under which power systems op ) crate. The analytical development not only allows the optimal bidding strategy formulation but also is useful in providing es- d analytical 区+ ing the performance of generating units. Extensive numerical (1) results illustrate the robustness and superiority of the analyti- ally developed optimal bidding strategies subject to D4-∑A1Pi;tui;=0 The Framework Rt-∑1r;tw;t≤0 We develop a general competitive electricity (CEM) framework which we use to formulate and anal pn≤p,t≤p The commitment and dispatch of units in the CEM are based 0≤pi;≤ai;t on a competitive auction procedure. The market sellers, typi- 0≤rt≤min{r,pa-p; =1,2,…,M cally generators, submit a sealed bid stating the price at which they are willing to sell power. The CEM operator, the en tity responsible for coordinating all energy transactions with Tie satisfies the t and i constraints the Cem, selects the set of least expensive units to meet the We formulate roblems by considering he bids received from the set of M bidders. Each bid B: has we refer to equations( 1)-(3)as the primal form of the CEm ator problem(CEMP). The triple 2=uip r], is 310

T is the number of time periods in the scheduling horizon System Parameters Rt is the system reserve requirement in time period t Bidder da M is the number of bidders participating in the CPP i=1,2,... M is the bidder inde Unit variables if the unit if the unit is shut do is the status of unit i in time period t is the real Ti, t is the reserve provided by unit i in time period t Ti, t is the downtime of unit i at the end of time period t an operating schedule for unit i and 2=u, p,r)is sche The CEMP is the determination of the opt system schedule >=yp, pop rop) that minimizes 如m +)A2D CEM cost. We denote by subject to n)4{∑∑=色里 satisfiesequation()4) ,Er}(a)=1,2,…M the set of feasible operating schedules for unit We can rewrite the Lagrangian relaxation as re function in equation (1)is the sum of th variable and the start-up prices. For each unit i, the CEM ost when unit i demand Pi, t in period t is given by bA (pi, t ) ui, t. CEM start-up costs arise if unit i is shut down in 1>∑()+(x)1--) time period t-1 and is operating in period t, i.e., if ui, t-1=0 and ui, t= 1. The downtime when started up is the downtime AePi t -uri, Jui +2d+FR as the length of the time period we express r, t recursively in subject to erms of ui, t, t= l,,T and Ti,o ;2r;eg)v=1,2,…M Ti, t=(Ti,t-1+Ar)(1-ui,t)Ti,o is given. (5) Here we removed the constant terms aD and AR from the The objective function is nonconvex. The state space minimand. The Lagrangian function complex minimum up and down time constraints and is 叭(AED,B)min ∑!(p:) is 78 half hour periods and the number of units can exceed 200, this is a large scale and complex nonlinear optimization +b(r-1)(1-wt-1)w+2D+出2 problem. in the solution of the CEMP may be effectively exploited. This approach leads ,2r}(g) the decomposition of the problem in terms of each bid- is separable in terms of bidders as there is no inter-unit er and results in the economic interpretation of the Lagrange pling in the constraints. This allows us to decompose the multipliers as prices. The Lagrangian relaxation technique in- problem into M subproblems. The subproblem for bidder i volves the construction and solution of modified problem in hich the system-wide constraints on demand and reserve con straints, which couple all bidders, are used to augment primal objective function with their associated Lagrange mul- p(,E) tipliers. The new problem does not enforce the demand and s:1>p+ reserve constraints and is therefore "relaxed ". All bidder con- traints,however, are enforced AePi, t -Her Jui,t:u,P,rJeni (a )J We define the T-dimensional vectors A=(A1, A2,,Ar]T r. Here A: and 420 are the La For given A and A, the M subproblems can be independ (o), for the demand and reserve constraints in time period t, of the CPP dispatcher problem can be solved efficiently for multipliers, which are non-negative for inequality constraint Ived in an efficient manner. Hence the Lagr particular values of A and u, giving the value (,E D, an be shown that the Lagrangian function中(Δ丛; B ∑∑)+(+---) P(DB)≥虮A;DR) 311

In particular, if (A, u)is the optimal Lagrange multipliers the profits I (Bi, A'(i), u(Bi))of bidder i are equal to the revenues less the costs incurred with 中(2’D,B)=max{4(△;D,B):AH≥0}(1) I(G;'G),()=∑()+(, then, L(DP)全中(A,D,R) c(p:)-c(r:H)(1-t:-)u (15) The optimal bidding strategy calls for the maximization of provides a tighter lower bound on the optimal cost P(D, R)of I(Bi, A(Bi), u(B ) )over the set of admissible bids, i.e the primal problem P(D,R)≥L(D,R) b(),b(),g (c(p)+c(rt-1)(1 As a by-product of the process of maximizing P(A,A; D,R), we obtain the optimal Lagrange multipliers 2 and A:(B)p; t-i(o)ri t ]ui, t system schedule >=fy, p,r) resulting from the solution to the Lagrangian relaxation for A= a.and u= u.The bEClp?m,P1,6;ec[, oo),:20(16) given in Equation(3),i.e, >i=ui,P,ti]en (ai )for all i 1, M. In certain cases, does satisfy the demand and re- serve constraints making it feasible for the primal problem. If in addition,> satisfies the complementary slackness condi- 卫;r ∑()+(4-)(1-;-)-A(B) ∑=∑ the optimal schedule to the primal probler 9]. Practical approaches for computing a near-optimal sched- (1)r;wt:{u;;ren()} (17) ule have been developed [7].[8]. For all practical purposes the with a (gi )as defined in equation(4) difference between 2 and the near optimal schedule >"is We next introduce the assumption of perfect competition in assumed to be negligible. Moreover we also assume that the the CEM. Under such a condition, no single bidder may affect optimal Lagrange multiplier A'(u*), associated with demand prices and is consequently a price taker. In other words, any (reserve)in time period t, differs negligibly from the marginal change in the bid submitted by bidder i will have a small effect price(reserve price)in the same period on the prices determined by the CEMP. Formally, we state the Strategic Bid Formulation ny bidder has a negligible effect on the system marginal and reserve prices We use the CEM framework to solve the bidder's problem: This assumption holds when ne bidder controls a signifi. formulation of a bidding strategy to earn maximum profit. We cant portion of the total CEM tion and capacity. The nsider the problem of bidder i who submits bid Bi. For market price is determined by ds of the set of competing dder i, the bids Pi,j=1, 2. 1,i+1,.,M of the other bidders. From the viewpoint of bidder i, the market clearing bidders are fired but unknown. The CEMP determines the prices are independent of Bi so that optimal system price pair('A), the optimal system schedule X'(日)=2°and'()= (18) The prices depend on the bid of all generators in the CEM, B,, j=1,.M. However, bidder i It is convenient to define the loss function A2-II and re- place equation(16)by the 4.(Bi). The dependence of 2 on the bid B: is of Ai. We restate the problem relaxation of the CEMP b(),b(),g ∑c!()+(1-n-)- b!(p;)+b(r,t)( 1)-A:(B -r:bCpm,n],eC,∞),a;≥0}(19) Li,P: ri where ui,p*r i minimizes the proble where the set ni (a )is a defined in equation(4).The bidder generation costs incurred in c!()u;+c(r-)(1 u;t-1)u,. We use cf((c1(]to denote the fuel and variable operations and maintenance costs(start-up costs]of unit i. We Given the structural similarity between the minimizations in sume both functions to be continuous. The total costs of unit equations(19)and(20)we make use of the [e(p2)+c(r-1)(1 paid to generato time period is Ai(Pi)per MWh of 1 In effect collaboration among bidders, i. e,gener- tors behave noncooperatively and there is ergy and ui(P:)per MW of no cartel of generators re served. It follows that who act together to set prices

Theorem [13]: A global optimal solution to the problem in(19)and(20)is the bid pope=1bf, 64, a: 1, b!()=啭(p) Pepa,P“" b:()=c(r)r≥0 ai t = Pi eopt=c,c, P as] is a globally optimal bidding strategy No other bidding strategy can result in a greater profit to the bidder. This does not preclude some other from also achieving the same profit. We also not that the global optimality is independent of the system price pair(A, u). Regardless of the bid is optimal if it equals pi. Numerical results presented in the next section demonstrate these analytic results. A salient feature of this optimal bidding strategy is that it reveals the true cost of operation of the unit to the CeM oper Figure 3: System marginal price data tor. This highly desirable outcome is due to the construction of this auction for the right to serve load in the Cem. We can refer to the optimal bid as bidding at cost or the truth-revealing the generator's bid and the assumed system marginal price bid. The result can be extended to cases in which a generator data, we can determine the unit's schedule for the week and owns multiple units consequently its profits. The variable costs of the bidding unit The expression for the profit realised by bidder i under the are represented by a piecewise linear function of the unit output power, with three segments. The parameters that describe this :(B)生m(,A)=(4()()2:e)1出mm plot of the variable cost function for the unit is shown in Figure AePi, f+ATi t-df(pi, t) -c(rt-1)1-;t-1);:{u;,p; ri]en(g)}(22) This expression for the optimal profit IQ,F allows the iden of several key properties including convexity and non negativity(11]. These are useful in developing sensitivity info tion and in quantifying the effects of volatility in the prices Eon the optimal profit of bidder i. Moreover, these prop- erties can be used in evaluating the impacts on profits of a change in the costs of the bidder to assess the retur sible invest med at improving the performance of the unit [11] Simulation Studies 600c7o We illustrate the formulation of optimal bidding strategies in the CEM under the assumption that the system prices are dependent of the bid of any one generator. Given the syst Figure 4: Variable cost in C/h for the bidding unit ical studies reported here, corresponiding to the bid pi the bid variable price bf is submitted as piecewise linear function The start-up costs of the unit are assumed to be an expo f unit power(willans line)as is the case in the EwPP. We nential function of cooling time xamine the variation of the profit of bids with respect to var us parameter values and compare that to the optimal strategy (T)=c(0)+c(∞)-c(0)[1-ezp(-T/r bid. We ignore the reserve price in these numerical studies so as not to detract from the focus on the system marginal price. T' is the cooling time constant for the unit. The selection of m marginal price data derived from the EWPP c(o), c(oo)and r completely specifies this function. We f.unit.a set of half hourly values of xf for a week was c(oo), r". c (0), ct(oo),T), which specify the variable and astructed using data in [10]. The time plot of the in start-up cost functions as the cost parameters of the unit. The of h is shown in Figure 3. Hour 0 ponds to cost parameters and the minimum up and down times of the Sunday bidding unit are presented in Table 2 bid, we calculate the profit corresponding to We consider for the bidding unit the effect on profits of the the bid for the week of system marginal price data. Given submission of bids different from bidding at cost. To this end 313

Table 2: Cost and operational parameters for the bidding unit eter mwtMwtiaejbo t 300500 MWA/Mwh[∠/Mwh/MwhA we examine the change in profits as the bid is changed varying the bid parameters. It is assumed that the unit made fully available for every hour of the week. We restrict the bid variations to one parameter at a time while each of the other bid parameters is kept constant at the values of the corresponding cost parameters. The parameters that describe the bid variable price function are the bid no-load price, 60 the bid n in the same way the variable cost function of the unit is specified by the parameters c(o), e, ea, mi,m2 and m The bid start-up 46o. aeters b(O)“"(∞).nd+, We refer to the set of values d,2η,η2,n,b(0),b'(∞)r2} as the bid parameters f the unit. The profits made by the submission of the bid B are denoted by I(p, A ). In Figure 5, plot of I(e, A)against variations in e is given. 10 Figure 6: Plot of I(B, A )versus n2 under conditions of perfect comp re noteworthy for the explicit inclusion and detailed represen tion of the yar with the generation for electrical power. We have developed bidding to supply ration at cost and at marimum ca- pacity. The adoption of the POOLCO concept [12 ]makes th a result of practical interest in California. has focused only on one aspect of CPP- the duced a problem which can be effectively solved using the Cem framework. There are several facets of the CEM that require Figure 5: Plot of I(e, A)versus e additional work. For ex the optimal bidding strategy problem for buyers from the CEM may be formulated In Figure 6 plot is given of I(p, A)against variations in n. to optimize profits given resources, constraints and costs. The Once again, it is seen that maximum profit occurs when the bid oligopoly situation in EWPP generation markets[13]is an area trameter sponding cost parameter. The that may be explored ame theoretic notions writhin the aatness'of the profit with respect to variations in n' deserves CEM framework. A game theoretic formulation will allow the comment. Units that submit a piecewise linear bid variable analysis of the interaction between competing bidders. Such ice function are dispatched at the elbow points e or a interaction is not represented in the CEM the maximum power point, framework of the present n do not result in the cPP dispatcher redispatching the unit siderable work is the to other elbow points; hence, the profit remains unchanged. and pricing [14] into the CEM framework ults are observed for changes in n Conclusions References This paper has reported the development of a work which incorporates the salient features of the Companies of England and Wales, " James Capel and Co. sequently the England and Wales (EWPP). We have applied the framework to form [2]S. C. Littlechild, Report on Pool Price Inquiry. United termine the optimal bidding strategies of s bidder y Reg

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