An ordinal optimization based bidding strategy for electric ... - CiteSeerX

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001

An Ordinal Optimization Based Bidding Strategy for Electric Power Suppliers in the Daily Energy Market Xiaohong Guan, Senior Member, IEEE, Yu-Chi (Larry) Ho, Life Fellow, IEEE, and Fei Lai

Abstract—The deregulation and reconstruction of the electric power industry worldwide raises many challenging issues related to the economic and reliable operation of electric power systems. Traditional unit commitment or hydrothermal scheduling problems have been integrated with generation resource bidding, but the development of optimization based bidding strategies is only at a very preliminary stage. This paper presents a bidding strategy based on the theory of ordinal optimization that the ordinal comparisons of performance measures are robust with respect to noise and modeling error, and the problems become much easier if the optimization goal is softened from asking for the “best” to “good enough” solution. The basic idea of our approach is to use a approximate model that describes the influence of bidding strategies on the market clearing prices (MCP). A nominal bid curve is obtained by solving optimal power generation for a given set of MCPs via bids are generated by perturbing Lagrangian relaxation. Then the nominal bid curve. The ordinal optimization method is applied to isolate a good enough set that contains some good bids with high probability by performing rough evaluation. The best bid is then selected by solving full hydrothermal scheduling or unit commitment problems for each of the bids in . Using ordinal optimization approach we are able to obtain a good enough bidding strategy with reasonable computational effort. Numerical results using historical MCPs from the California market and a generation company with 10 units show that the ordinal optimization based method is efficient and good bidding strategies are obtained. Index Terms—Bidding strategies, electric energy bidding, electric power market deregulation, hydrothermal scheduling.

I. INTRODUCTION

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HE ELECTRIC power industry worldwide is experiencing unprecedented restructuring. The core of the restructuring is deregulation of the industry and introduction of competition among power suppliers and demanders [1]–[6]. In California, the first state with a new deregulated market, there is a Power Exchange (PX) with “day-ahead” and “hour-ahead” energy markets, and a real-time market for energy balancing operated by the Independent System Operator (ISO). The PX and the ISO are independent, nonprofit Manuscript received May 8, 2000; revised March 29, 2001. This work was supported in part by EPRI Contract WO 8333-03, ONR N00014-98-1-0720, AFOSR F49620-98-1-0387. X. Guan and F. Lai were supported in part by the National Outstanding Young Investigator Grant 6970025 and a Key Project 59937150 of National Natural Science Foundation, China. X. Guan is with the Division of Engineering and Applied Science, Harvard University, Cambridge, MA 02138 USA and Xian Jiaotong University, Xian, China. Y.-C. Ho is with the Division of Engineering and Applied Science, Harvard University, Cambridge, MA 02138 USA. F. Lai is with the Systems Engineering Institute, Xian Jiaotong University, Xian, Shaanxi 710049, China. Publisher Item Identifier S 0885-8950(01)09436-6.

Fig. 1.

Market clearing price.

organizations. Electricity is primarily traded through bidding in the PX market. Independent from the PX, the ISO controls and operates the transmission grid, and facilitates transactions and transmission. Electric power suppliers can sell energy to the PX, and ancillary services (including automatic generation control (AGC), real-time load balance, spinning reserve, and generating capacity required for grid congestion management) to the ISO. Energy is eventually distributed to end-customers through distribution networks belonging to “utility energy service companies,” where ancillary services are used to support system operation. In the day-ahead market, an energy supplier submits to the PX piece-wise linear and monotonically increasing power-price “supply bid curves” for each generator or for a portfolio of generating units, one for each hour of the next day. These supply bid curves are aggregated by the PX to create a single “supply bid curve.” On the other hand, an energy service company submits to the PX an hourly power-price “demand bid curve” reflecting its forecasted demand. These demand bid curves are also aggregated by the PX to create a single “demand bid curve.” Based on the demand and supply bid curves, the PX determines a “Market Clearing Price” (MCP) for each hour as shown in Fig. 1. The power to be awarded to each bidder is then determined based on the individual bid curves and the MCP. All the power awards will be compensated at the MCP. After the auction closes, each bidder aggregates all its power awards as its system demand, and performs a traditional unit commitment or hydrothermal scheduling to meet its obligations at the minimum cost over the bidding horizon. As pointed out in [6], suppliers’ bidding decisions are coupled with generation scheduling since generator characteristics and how they will be used to meet the accepted bids

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GUAN et al.: AN ORDINAL OPTIMIZATION BASED BIDDING STRATEGY FOR ELECTRIC POWER SUPPLIERS IN THE DAILY ENERGY MARKET

in the future have to be considered before bids are submitted. Therefore bidding decision must consider the anticipated MCP, generation award and costs, and competitor’s decisions. It can be seen from the above that many challenging issues arise under the new competitive market structure. Instead of centralized decision-making when the power industry was a regulated monopoly, many market participants with different goals are now involved and competing in the market. The information available to a party may be limited, regulated, and received with time delay, and decisions made by one participant may influence the decisions of others. These difficulties are compounded by the underlying uncertainties inherent in the system such as the demand for electricity, fuel prices, outages of generators and transmission lines, tactics by other market participants, etc. Consequently, the market is full of uncertainty and risk. California’s one year experience with deregulation has already shown that MCPs can be quite volatile and unpredictable especially when the demand is high [7]. In fact, the price spikes in the summer of 1998 in California and other markets had ignited many debates suggesting changing market regulation including tighter price-caps. Many approaches have been presented in the literature to address the deregulated power markets. Prior to the deregulation in the US, the market structure model discussed the most was the “British Model,” which is different from the “California Model” but the two share some similar framework [2]. Ideally each participant in the energy market will select the bidding strategy that maximize its profit. Optimal bidding strategies to maximize a bidder’s profit based on the pool model of England and Wales were presented in [2] under the assumption of perfect market, that is, individual bids do not affect the MCP. For markets where bids consist of start-up prices, variable prices, and generator capacity, it was demonstrated that profit would be maximized by bidding each generator at its physical cost curve and maximum capacity. This is done by showing that such a strategy is a “Nash equilibrium” for the market, i.e., there is no incentive for a bidder to unilaterally deviate from this strategy if everyone else bids this way. The conditions assumed in [2], however, may not be practical since a bid may affect MCP. A bidder would want to increase or decrease bidding prices from physical marginal costs to maximize its profit. Various methods for modeling and solving the bid selection problem at different levels of the market have been discussed. In [8], a bid clearing system in New Zealand is presented. Detailed models are used, including network constraints, reserve constraints, and ramp-rate constraints, and linear programming is used to solve the problem. In [9], bids are selected to minimize the total system cost, and the market clearing price is determined as the highest accepted price for each hour. Other approaches addressing various aspects of generation and ancillary service bidding can be found in [10]–[12], where Lagrangian relaxation, decision trees, and expert systems were used to analyze and support the bidding process. For example, a bidding strategy considering revenue adequacy was presented in [13] based on Lagrangian relaxation and an iterative bid adjustment process. A bidding method considering the uncertainties of other bidders, the ISOs bid selection process and self-scheduling in New England’s power market is presented in [14]. The problem is

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solved within a simplified game theoretical framework, where the ISO has a closed form solution, and the utility’s problem is solved by using Lagrangian relaxation. The exact “gaming” phenomenon among bidders however is not captured. Bidding behaviors under a simple auction market are studied in [15]. The results show that power suppliers would tend to bid above their production costs to hedge against the possibility of winning on the margin. The factors affecting bidding strategies in Australia power market are analyzed in [16]. It can be seen from above that tools to support the bidding process are far from satisfactory in view of the inherent complexity (multiple participants with their own objectives in a dynamic and uncertain environment) and the sizes of practical problems (tens to hundreds of generators with various constraints). High quality systematic and computationally efficient approaches are critically needed to address the new challenges and to develop effective optimization based bidding strategies. Since bidding problems are generally associated with inherent computational difficulties, it is more desirable to ask which solution is better as opposed to looking for an optimal solution. In this paper, a systematic bid selection method based on ordinal optimization is developed to obtain “good enough bidding” strategies for generation suppliers. Ordinal optimization provides a way to obtain reasonable solution with much less effort. The ordinal optimization method has been developed to solve complicated optimization problems possibly with or without uncertainties [17]–[23]. Ordinal optimization is based on the following two tenets: 1) it is much easier to determine “order” than “value.” To determine whether A is better or worse than B is a simpler task than to determine how much better is A than B (i.e., the value of A-B) especially when uncertainties exist. 2) In stead of asking the “best for sure,” we seek the “good enough with high probability.” This goal softening makes the optimization problem much easier. The ordinal optimization method has been applied to solve the well-known Whitsenhausen problem [24] which had been unresolved for thirty years since its formulation. A numerical solution was finally obtained by using the ordinal optimization method sampling and selection in 1999 [22], [23]. To apply ordinal optimization framework to integrated generation scheduling and bidding, major efforts are needed to build power market simulation models and to devise a strategy to construct a small but good enough select set . The basic idea of our approach is to use a rough model that describes the influence of bidding strategies on the MCP. This model can be replaced by any model with more accuracy if needed. A nominal bid curve is obtained by solving optimal power generation for a given set of the MCPs with the Lagrangian relaxation framework. Then bids are generated by perturbing the nominal bid curve. The ordinal optimization method is applied to form a good enough bid set , which contains some good bids with high probability, by performing rough evaluation. The best bid is then searched and selected over by solving hydrothermal scheduling or unit commitment problems within reasonable computational time. Numerical testing results based on historical MCPs in California market and a generation company with 10 units show that the ordinal optimization based method is efficient and good bidding strategy is obtained.


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II. BIDDING PROBLEM FORMULATION The bidding problem of the daily energy market for energy providers is formulated in this section. The purpose is to establish a model describing the relationship of submitted bids and the associated profits, and therefore, establishing a basis for developing an ordinal optimization based bidding strategy in the next section. Assume the bidding strategy developed in this paper is for an energy or generation supply company . Suppose there are generating units in . Supply bids can be submitted for individual units or an aggregated bid can be submitted in the PX market. The bidding objective for is to select to maximize its profit over a its supply bid curves time horizon , i.e.,

with

(1) where aggregated energy (generation)price supply bid curve of for hour ; generation and demand bids of other bidders unknown to for hour ; generation cost for delivering gener; ation award power generation award for depending on the bid of and the bids of other bidders; bidding time horizon (24 hours for the day-ahead market); market clearing price (MCP) determined by the aggregated supply bid curve and the aggregated demand bid curve of all market participants as shown in Fig. 1. In this paper, the bid curves of other participants are assumed to be fixed and we are only interested in the influence of ’s own bids on the MCP, which will be modeled later. According to the PX rules, if a supplier bidder is awarded , it will be compensated by the dollar amount regardless of the original bid submitted by that supplier. Startup costs should be accounted for in bid curves since there is no direct startup compensation. The problem described in (1) is thus an optimization problem to to determine the optimal supply bid curves maximize the profit subject to relevant operating constraints such as the minimum down/up time, ramp-rate constraints, etc. Since the PX rules require bid curves to be piece-wise linear and monotonically increasing, searching the optimal bidding strategy is to determine the corner points of the bid curves. Note that MCPs are determined by the bids submitted by all

Fig. 2. Generation award with the MCP given.

the bidders, and when submitting the bids, a bidder does not know the bid curves submitted by others. is determined by the PX, it is trivial Once the MCP to determine the generation award for individual units from their from its bid curves or for the generation supply company aggregated bid curve as shown in Fig. 2. Although can submit bid for each individual unit, the PX only view total generation as obligation. Given total generation award, it is award to desirable for to schedule or to reallocate all generating units across the biding horizon to deliver its total award at the minimal cost. This can be formulated as a traditional unit commitment or hydrothermal scheduling problem with the total generation award as system demand as follows: with

(2)

subject to (3) and other individual operating constraints such as minimum down/up times described in [25], [26], where total generation cost over the entire bidding horizon; cost of unit for generating at hour ; startup cost associated with down-up transitions for unit at hour ; system demand for scheduling problem at hour , equal to total generation award to ; power generation of unit at hour ; generation award for unit at hour according to unit’s individual bid curve. If the market clearing price MCP is known, and consequently the generation award is known, the profit calculation of (1) can be re-written as (4) is from the scheduling result of (2), and MCP and the total award are assumed to be known. Note that (2) and (4) establish a

where



procedure to calculate the profit once the MCP is known. The next step is to model how bids influence the MCP. In the PX day-ahead market, two types of the factors influence the MCP: 1) system demand or load, the power import/ export to the market by long-term contracts, transmission network limits due to congestion and other “objective” factors; and 2) those related to the bidding strategies of the market participants. The two types of factors are largely independent of each other. Influence of the first type can usually be forecasted with reasonable accuracy by a regression model such as artificial neural network (ANN) using historical MCPs and other factors as inputs. The nominal MCP forecasting using an ANN model has been reported in our recent work [29]. However, a generation supplier is more interested in knowing the impact of its bidding strategy on the MCP in the PX auction and making use of its market power to optimize its bidding strategy. It is desirable to build a model that describes the relationship between bidding strategy and the MCP. Generally, the influence of bidding strategies can be modeled by simulation, regression, game theory, heuristics, etc. The biggest advantage of the ordinal optimization method is that any model can be used without affecting the effectiveness of the method whose essence is to allow crude model to be used to quickly determine what bidding strategies are highly likely to be better than others. Suppose the nominal and the deviated piece-wise linear bid curves are specified by corner point pairs and , respectively.1 Consider the following regression model for the MCP (5) is the nominal MCP forecasted by where , the forecasted system demand the historical MCPs and other forecasted factors such as fuel prices, etc. The influence of bidding strategies is described by influence func, where and tion are the difference between corner points of the nominal and the deviated bid curves. The influence funccan also be established by regression tion method based on the historical MCPs and records of one’s own bids plus bidders’ experience. Understandably the above model is only an approximation. as modeling error can be To reflect the error, i.i.d. noise added to (5) as

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Fig. 3. A profolio of MCPs.

III. ORDINAL OPTIMIZATION BASED BIDDING STRATEGY As mentioned earlier, it is recognized that the pursuit of optimal bidding strategy under the deregulated market structure is impractical because of problem complexity and the uncertainties involved. Instead, a near-optimal approach that seeks “good enough” bids with high probability are presented below. The basic idea of our approach includes the following steps: 1) generating a nominal bid curve by using Lagrangian relaxation method for hydrothermal scheduling; bid curves by perturbing the nominal bid 2) generating curve and obtaining the MCPs associated with these bids; generating a “good enough” select bid set by evaluating bids using a very crude model with little comthese putation effort, and ranking and selecting them based on ordinal optimization method; 3) evaluating the bids in the select bid set using accurate model and solving computationally time-consuming hydrothermal scheduling problem (2), and then selecting the best one. A. Generating the Nominal Bid Curve A nominal bid curve should serve two purposes: 1) it is a basis in (5) to define relationship between bid curves and the MCP; and 2) it is an optimal bid curve if bidding strategy has no influence on the MCP. That is, if is a “price-taker,” the nominal bid curve should maximize ’s profit for any MCP determined by the market. To achieve this goal, let a profolio of MCPs across hours be given as bid prices as shown in Fig. 3, and the optimal generation for an individual unit to maximize its profit can be solved based on (4): with (7)

(6) or It should be emphasized that the ordinal optimization method has no restriction on any model used to describe bidding strategy influence on the MCP. In fact (5) can be replaced by a Monte Carlo simulation procedure or other game-theoretic models. The purpose of the next section is to demonstrate the effectiveness of the approach and not necessarily the validity of (5).

1The variable b (t) means price rather than the bid curve b(1; t) defined in Section II.

with (8) is subject to the individual operating constraints, where the point index on a bid curve. For the given series of MCPs , (8) is a parameter optimization problem. It is in fact a subproblem within the unit commitment or hydrothermal scheduling context when solved using Lagrangian can be viewed as the Larelaxation technique, where grange multipliers given by the high level problem. Clearly this


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matches economical explanation of the Lagrange multipliers as shadow prices. The problem can thus be efficiently solved by dynamic programming as in our previous work [25], [26]. for all The results are the optimal generation levels units at each hour. The individual bid curve for unit and the aggregated nominal bid curve for are thus generated as

and

The bid curves for different hours obtained by solving (4) may be inter-temporally related. This is reasonable reflection of the power system that will deliver the generation award since there may be many inter-temporal individual constraints such as minimum up/down times, ramp-rate constraints imposed on the system. The nominal bid curve generated above is equivalent to the production cost curve bid in [2] where the British model is used. It is the best bidding strategy for a “price-taker” bidder who has no influence on the MCP. Based on the procedure where the nominal bid curve is created, it is seen that if the MCP de, the generation award termined by the market is equal to would maximize the profit of an individual unit as in (7). B. Perturbing the Nominal Bid Curve The nominal bid curve given above is perturbed to generate bid curves as

(9) is a perturbation function. A simple way to where is to keep the power generation of a bidimplement the same and sample equally ding generation point spaced in the neighborhood

Fig. 4.

Perturbing the nominal bid curve.

be delivered by that unit. This may not be true since generation award to a unit may not even satisfy the inter-temporal individual operating constraints. A generation company will schedule or re-allocate all its units to meet its total generation award with the minimal total generation cost for delivering its total obligation and satisfying every individual operating can be obtained by constraint of its units. The true profit solving (2)–(4) and there may be significant error due to the above approximation (12) is error. The advantage of the ordinal optimization where method is its capability to separate the good from bad even with very crude model, namely the performance “order” is relatively immune to large approximation error. Even if the rough estimation is used to rank bid curves, some good enough bids will be kept within the select set with high probability. See the Appendix for details. The major task in applying ordinal optimization is to construct the selected subset containing “good enough” bids with . high probability, including the determination of its size The quantitative measure of the “good enough” is the alignment probability defined as (13)

(10) as shown in Fig. 4, so that always specifies a monotonically increasing piece-wise linear bid curve as required by the PX. Based on a perturbed bid curve , the corresponding MCP estimated using (5) can be obtained. C. Selecting Good Bids The bid curves obtained in (9) can be evaluated and ranked by ordinal optimization. The estimated profit of each set of bid curves is calculated as

(11) Note (11) is just a rough profit evaluation with the MCP given but without solving scheduling problems. It is assumed that the generation award to a unit based on its bid curve will also

where is the “good enough” bid set and is called the alignment level. Intuitively the alignment probability is the probability of the event that there are at least elements in the good enough set matched in the select set . perturbed bids generated To select good ones from the by (9), the profits are estimated by (11) and ranked. The top bids are then selected as and its size is determined by a regressed nonlinear equation to satisfy certain confidence requirement [21]. The value of can be estimated by (14) is the size of and and are coefwhere ficients or parameters obtained by nonlinear regression. These , the value coefficients depend on the alignment probability and the shape of the ordered performance curve, where of the profits of bids are ordered and plotted (OPC, see the Apand the values of pendix). For are published in [21, Table I] through intensive simulation for



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TABLE I CASE DEFINITION

Fig. 6. Two sets of nominal MCPs.

Fig. 7. Fig. 5. Flowchart of the bidding strategy.

different OPCs. The results can be directly used to determine in this paper. bidding strategies using (11) is computationEvaluating ally efficient and the ordinal optimization method can guarantee that good enough bids will be among the selected strategies. The result of [21] tells us how large should be. More accurate but time-consuming evaluation which requires solving generation scheduling problems is applied to evaluate the selected bids. For each bid with the associated forecasted MCPs, a traditional generation scheduling or unit commitment problem described by (2) and (3) is solved to calculate ’s profit using the Lagrangian relaxation based algorithm in [25], [26]. The best bid is then selected by evaluating those bids in the subset based on the estimates of the MCP. Since is much smaller than , the ordinal optimization method is extremely efficient in comparison with brute and force method by solving scheduling problems. Fig. 5 gives a flowchart to summarize the ordinal optimization based bidding strategy developed in this paper. IV. NUMERICAL TESTING RESULTS The numerical testing results are based on a power system with 10 thermal units. The system and unit parameters including cost functions and operating constraints are presented in [27]. Four cases, defined in Table I, are tested based on two sets in (5) and (6). of nominal MCPs with and without noise For testing purpose, the published MCPs for May 1, 1998 and

OPC and normalized rough profit of Case 1.

January 4, 1999 on the California PX day-ahead energy market shown in Fig. 6 are used as the nominal MCPs in (5). The is simplified as bidding influence function a linear function just to demonstrate the effectiveness of the ordinal optimization method. The parameters for ordinal optimization are selected as follows. ; Search space size: ; Alignment probability: top 50 bids among bids, i.e., Good enough set: ; 5, i.e., . Alignment level: To use (13) to obtain , the knowledge on the shape of OPC is needed (see the Appendix and [21]). The profit defined in (4) is calculated for bids by solving the scheduling problem (2) times. The profits are ordered, normalized and plotted together with the normalized rough profits calculated using (11) shown in Figs. 7 and 8. It can be observed that the OPCs in the two cases are of bell shape as described in [21]. The average CPU times for calculating the accurate OPC is about 11 hours using the scheduling algorithm implemented in Matlab 5.1 on a Pentium II-350 PC computer. Therefore evaluating bids without applying ordinal optimization method can be computationally prohibitive in practical application since it is necessary to solve 1000 difficult generation scheduling problems. It is also interesting to note that the approximation errors of the rough profits are not related to bids. That is, the errors are evenly distributed along the OPC. Therefore the dependence bias discussed in the Appendix is not a major concern in the bidding problem.


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Fig. 8.


OPC and normalized rough profit of Case 2.

TABLE II TESTING RESULTS OF ORDINAL OPTIMIZATION METHOD

Fig. 10.

Error distribution.

Fig. 11.

Concept of ordinal optimization.

Fig. 9. Aggregated bid curve at hour 20 in Case 1.

The testing results using ordinal optimization method are summarized in Table II. The aggregated bid curve obtained at hour 20 of Case 1 is shown in Fig. 9 as an example. To use the results in [21] to calculate , the variance of error distribution is required. Fig. 10 shows the histogram of the in Case 1 distribution of normalized approximation error and Case 2, where the variances are 0.089 and 0.0343 respectively. These variances match those of uniform distributions with and . Therefore the in [21] can be used. It should be noted results for that error distributions in Fig. 11 have clear characteristics of normal distribution, which is not consistent with the uniform distribution assumption in [21]. Therefore, calculating , the is an approximation. Our testing results size of select set may be over-selected, i.e., is larger later show that the than required. This would not affect our result but may only unnecessarily increase computational time.

The ordinal optimization based bidding algorithm is also implemented in Matlab 5.1 and tested on a Pentium II-350 PC computer. Table II lists , the size of the select set by applying (12), which is much smaller than . In another word, only generation scheduling problems need to be solved by applying the ordinal optimization method and CPU time is about 40 minutes. Moreover, the current prototype biding strategy is implemented by Matlab, an interpretive language on a low end PC. Efforts are being made to incorporate the bidding into an integrated generation resource scheduling and bidding package implemented in C++. By considering the fact that the C++ implementation would be several times faster than Matlab, it should be possible to obtain a good enough bid or curve in about half hour for a generation company with practical size. Therefore ordinal optimization method is quite efficient.



TABLE III BIDS IN AND

G

S

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TABLE IV

jG \ Sj

To test the influence of MCP forecast errors on bid selecis added to tion, uniformly distributed noise the MCP as in (5).3 The testing results of Case 3 and Case 4 is reduced in in Tables III and IV show that the size of comparison with Case 1 and 2, i.e., less good bids are in because of the noise. However, it is still within tolerance: the bids are much more than 5. Therefore the selected bids still in contain enough good bids. For the 4 cases tested, the best bid is so that the results in Table II look the same. not missed in However this may not be true in other cases. V. CONCLUSION An ordinal optimization based bidding strategy for electric power suppliers is developed in this paper for applications in daily electric energy market. Since the bidding problem is integrated with hydrothermal generation problems and subject to uncertainties, it is extremely difficult to obtain the optimal bidding strategy. Ordinal optimization seeks “good enough” bids with high probabilities, and turns out to be an effective and systematic with much reduced computational efforts. The essence of this approach is to evaluate a large number of bids using a very crude model and to form a select set, where good enough bids are contained with high probability. Accurate evaluation is then applied to the select set by solving hydrothermal generation scheduling problems with much less computational efforts. Numerical testing results based on a 10-unit generation company and historical California PX market clearing prices as the nominal MCP show that a good enough bid can be obtained in reasonable computational time for day-ahead market bidding. The ordinal optimization based approach is being integrated with the MCP forecast method under development and to be tested in a practical power generation company on California PX market. APPENDIX ORDINAL OPTIMIZATION METHOD

To observe “order is relatively immune from error” claimed by the ordinal optimization method, the good enough bid set , 2 , i.e., top 50 bids, and , the selected bids by rough profit evaluation are listed in Table III. The ordinal optimization method tells us that at least 5 bids in should be also be in . It is and listed in observed that sizes of the intersections of Table IV are greater than 5 for all cases. Therefore the good enough bids will not “slip away” from select set because of very crude approximation and the ordinal optimization method is very effective.

G

2Obtaining usually requires huge computational efforts and it is generally not known in ordinal optimization applications. It is listed here for comparison as a result of computing the OPC.

Consider, for example, a search on a bidding strategy space . Define a “good enough” subset, with size , as the top bids in strategy space. Further define a selected subset, with size , is the space to be searched. The goal of ordinal optimization is to construct a selected subset such that the alignment probability

where is the alignment level. In another word, there should be good bids in . The detailed simulation or search can be per, a major speedup in search for formed over . Since a good bid can be achieved. The concept of ordinal optimization is shown in Fig. 11. 3Based on the experience of the California PX day-ahead market, in the MCP forecast is a reasonable assumption.


6$1 error

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where is the decision variable and is the model dependent error and is independent error. It is proposed in [28] that an adjusted performance function

(17) be used and the model dependent error by regression or other methods.

be estimated

ACKNOWLEDGMENT Fig. 12.

Ordered performance curve.

The major task in applying ordinal optimization is to construct the selected subset containing “good enough” bids with high probability, including the determination of its size . The quantitative measure of the “good enough” is the alignment probability defined in (13). The alignment probability is related to the ordered performance curves (OPC) of a particular problem defined in [21] where the performance indices (the profit in bidding problem) or objective function values of randomly sampled decisions are ordered and plotted against the order indices. The OPC is monotonic by definition and a typical OPC is shown in Fig. 12. Although the exact OPC is generally very difficult to know in a practical problem, its shape may help us estimate alignment probability. There are two ways to pick sets of bids from a space of sets of perturbed bid curves: Blind Pick (BP) and Horse Race (HR). In Blink pick, sets of bids are randomly selected, sets of bid curves are preliminarily whereas in Horse race evaluated and the best sets of strategies are selected. For the BP method, the size can be determined in closed form [19]. The HR method, however, is often preferable since it can make use of results based on a simplified problem or a crude model, and generally ends up with a smaller as compared to the BP method. The value of can be estimated by (14). If there is no noise added to the MCP in (5), the ordinal optimization discussed in this paper is to use a simple performance evaluation model to approximate a complex but deterministic problem. The approximation error

(15) is the accurate profit is considered as “random noise,” where obtained by solving (4). A problem may rise if the error deis correlated with the bid pends on the bid, i.e., the “noise” curve. However, the results of the ordinal optimization method are based on the assumption that the error or noise is identical and independently distributed (i.i.d.). This issue has been discussed in [28] where the objective or performance function is described as

(16)

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Xiaohong Guan received the B.S. and M.S. degrees in automatic control from Tsinghua University, Beijing, China in 1982 and 1985, and the Ph.D. degree in electrical engineering from the University of Connecticut, in 1993, respectively. He was with Xian Jiaotong University from 1985 to 1988. He was a consulting engineer with PG&E from 1993 to 1995. Since 1995, he has been a professor with the Systems Engineering Institute, and now the Director of the National Lab for Manufacturing Systems Engineering, Xian Jiaotong University, Xian, China. He is currently visiting Division of Engineering and Applied Science, Harvard University. His research interests include scheduling of power and manufacturing systems, bidding strategies for deregulated electric power markets and optimization of large-scale systems. Dr. Guan received the 1996 Li Heritage Prize for Excellence in Creative Activities, Li Foundation, San Francisco, USA and National Outstanding Young Investigator Award, National Natural Science Foundation of China.

Yu-Chi (Larry) Ho received the S.B. and S.M. degrees in electrical engineering from M.I.T. and the Ph.D. degree in applied mathematics from Harvard University. Except for three years of full time industrial work he has been on the Harvard Faculty where he is the T. Jefferson Coolidge Chair in Applied Mathematics and the Gordon McKay Professor of Systems Engineering. He has published over 140 articles and three books. He is on the editorial boards of several international journals and is the editor-in-chief of the international Journal on Discrete Event Dynamic Systems. He is the recipient of various fellowships and awards including the Guggenheim (1970) and the IEEE Field Award for Control Engineering and Science (1989), the Chiang Technology Achievement Prize (1993), the Bellman Control Heritage Award (1999) of the American Automatic Control Council, and the ASME Rufus Oldenburger Award (1999). He is a Life Fellow of IEEE, and was elected a member of the U.S. National Academy of Engineering (1987). His current research interests lie at the intersection of Control System Theory, Operations Research, and Computational Intelligence.

Fei Lai received the B.S. and M.S. degrees in applied mathematics from Xian Jiaotong University, Xian, China in 1993 and 1996, respectively. He is currently a Ph.D. candidate in Systems Engineering Institute, Xian Jiaotong University, working on optimization of large scale systems and integrated resource bidding and scheduling in the deregulated electric power market.
