A comparison of residual demand models for oligopolistic markets

A COMPARISON OF RESIDUAL DEMAND MODELS FOR OLIGOPOLISTIC MARKETS Agust´ın R. Marulanda Guerra Jose L. Mart´ınez Ramos Antonio G´omez Exp´osito Universidad del Zulia Universidad de Sevilla Universidad de Sevilla Maracaibo, Venezuela Sevilla, Espa˜na Sevilla, Espa˜na [email protected] [email protected] [email protected] Abstract - In many pool-based electricity markets two types of generating companies coexist, namely price-taking and leader companies. In this context, the optimal bidding of a leader company, and consequently the market clearing price, is mainly determined by its residual demand curve, which can be modeled in different ways. In this work, three different residual demand curves have been adopted to simulate the optimal strategy of an oligopolistic company. The influence of these models on the gap between the expected and actual market prices, as well as on the resulting leader company profit is analyzed. Keywords - competitive electricity markets, nonlinear programming, residual demand. 1 INTRODUCTION IN oligopolistic markets one or very few generating companies have the ability to manipulate the marketclearing prices. For this purpose, they resort to the so-called residual demand curves, relating the marketclearing price, for a given hour, with their own power production [1]. A generating company exercises market power when it reduces its output or raises the price at which it is willing to sell its energy in order to change the market-clearing price. Different models have been adopted to materialize this kind of curves, such as: Taylor series expansion [2], piecewise-linear approximation [3, 4] and multi-step curves [1]. However, as those curves are necessarily built from incomplete and aggregated information, where relevant individual constraints affecting market agents are ignored, there will always exist a gap between the optimal prices expected by the leader company and actual market-clearing prices determined by the market operator on the basis of all bids submitted to the pool. The motivation for this work is to show how the model adopted to represent the residual demand curves influences the market-clearing prices, when the leader company is willing to exercise its market power. The reminder of the paper is organized as follows, Section 2 summarizes the mathematical model used to simulated the leader generating company, presented as a non-linear optimization problem, and the Market Operator problem, presented as a complex auction optimization problem based on net social profit. Section 3 shows the methodology used to compare the optimal prices and the market-clearing prices. In Section 4 a numerical test and the results obtained are presented. Finally, Section 5 presents several conclusion derived from this work. 2 OPTIMIZATION MODELS In this section, the optimization problems respectively faced by the leader generating company and market operator are briefly summarized. 2.1 Leader Generating Company Model The profit function is defined as the total company income minus the total costs. The goal is then to determine the production that maximizes the total benefit max 24 X h=1 λh(φh)·φh − nj X j=1 CP h j+CUj·ah j+CSj·sh j   (1) whereλh(φh)representsthe residualdemandfunction,φh is the total energy produced at hour h,njrepresents the number of generating units belonging to the leader company, CP h jis the variable production cost of unit jat hour h,CUjand CSjare the start-up and shut-down costs of unit j, respectively. Finally, ah jand sh jare binary variables which are null if unit jis not starting up or shutting down at the beginning of hour h, respectively. Apart from the expressions characterizing the residual demand curves, the above maximization problem is subject to a set of constraints, which can be defined in terms of piecewise-linear equations [5][1]: • Total generation, φh, is the sum of individual productions, φh= nj X j=1 Ph j(2) where Ph jis the average output of unit jat hour h. • Upper and lower generation limits for each unit, Pm j·xh j≤Ph j≤PM j·xh j(3) where Pm jand PM jare the minimum and maximum power output of unit j, and xh jis binary variable which is null if unit jis not committed at hour h. • Logical relationships among status changes, xh j−xh−1 j=ah j−sh j(4) ah j+sh j≤1(5) 15th PSCC, Liege, 22-26 August 2005 Session 3, Paper 1, Page 1 • Maximum up and down ramps, −RDj≤Pj,h −Pj,h−1≤RUj(6) where RUjand RDjare the ramp-up and rampdown limit of unit j, respectively. • Variable production cost, Cph j(Ph j) = fcj·xh j+ nc X k=1 mj,k ·σh j,k (7) Ph j=Pm j·xh j+ nc X k=1 σh j,k (8) 0≤σh j,k ≤pmax j,k (9) where Cph j(Ph j)is the piecewise linear cost function of unit jat hour h,fcjis the fixed cost of unit j,ncrepresents the number of segments of the piecewise linear cost function, σh j,k is a positive variable that represents the power of segment kat hour h,mj,k represents the slope of the segment k, and pmax j,k is the upper limit of segment kof the cost function of unit j. • Hydraulic power output versus water discharge, Ph j(Qh j) = Pm jxh j+ nl X l=1 mqj,l ·αh j,l (10) Qh j=qm j·xh j+ nl X l=1 αh j,l (11) 0≤αh j,l ≤qmax j,l (12) where Ph j(Qh j)is the piecewise linear output function of hydro unit jat hour h,Qh jis the water discharge, qm jis the minimum water discharge, nlrepresents the numberof segments ofthe piecewise linear output function, αh j,l is a positive variable that represents the water discharge of segment lat hour h,mcj,l is the slope of the segment lof the piecewise linear function, and qmax j,l is the upper limit of segment lof the piecewise linear output function of hydro plant j. • Reservoir spill-out as a function of the reservoir volume, Sh j(Vh j) = nm X m=1 msj,m ·ψh j,m (13) Vh j= nm X m=1 ψh j,m (14) 0≤ψh j,m ≤vmax j,m (15) where Sh j(Vh j)is the piecewise linear spill function of reservoir jat hour h,Vh jis the water volume of reservoir jat the end of hour h,msj,m is the slope of the segment mof the piecewise linear spill function, ψh j,m is a positive variable that represents the spillage rate of segment m, and vmax j,m represents the upperlimit of segment vof the piecewise linear spill function of reservoir j. • Water dynamic balance with travel delay, Vh j=Vh−1 j+rh j+Qh−τj j−1+ +Sh−τj j−1−Qh j−Sh j(16) Vm j≤Vh j≤VM j(17) where rh jin the net inflow to the reservoir jduring hour h,τjis the water delay time in hours between reservoir jand the next reservoir downstream, Vm j and VM jare the minimum and maximum reservoir volumen limits, respectively. The above nonlinear maximization problem is solved using the commercial package DICOPT [7] under GAMS [8]. 2.2 Market Operator Model In a market where network losses and constraints can be neglected, the market price is determined by maximizing the total surplus of generators and consumers. The objective function is defined as max ϕ,δ 24 X h=1 (X i∈IX d∈D ϕh i,d ·P Ch i,d −X j∈JX b∈B δh j,b ·P Gh j,b   (18) where Jis the set of generating units, Iis the set of consumers, Dis the set of energy blocks offered by consumers, Bis the set of energy blocks offered by producers, ϕh i,d is the amount of energy corresponding to block dof consumer iat hour h,P Ch i,d is the associated energy price, δh j,b is the amount of energy corresponding to block bof producer jat hour hand P Gh j,b is the corresponding energy price. The objective function is subject to the following constraints (h= 1,...,24) [6]: • At any time, the total energy sold has to be equal to the total energy bought. Also, each block of cleared energy must not exceed the amount of energy offered by each producer or consumer for that block. X i∈IX d∈D ϕh i,d =X j∈JX b∈B δh j,b (19) ϕh i,d ≤Qch i,d (20) δh i,d ≤Qgh i,d (21) where Qci,h,d is the energy that consumer iis willing to buy at hour hof block d, and Qgj,h,b is the energy that unit jis willing to produce at hour hof block b. 15th PSCC, Liege, 22-26 August 2005 Session 3, Paper 1, Page 2 • Upper and lower limits for each unit and consumer, Pm j·xh j≤Ph j= B X b=1 δh j,b ≤PM j·xh j(22) Cm i≤Ch i= D X d=1 ϕh i,b ≤CM i(23) where Pj,h is the energyof unit jat hourh, and Ci,h is the energy served to consumer jat hour h. • Logicalrelationships among status changesof units, xh j−xh−1 j=ah j−sh j(24) ah j+sh j≤1(25) where ah j, and sh jare binary variables which are equal to one if unit jis started-up or shut-down at the beginning of hour h, respectively. • Maximum up and down ramps of units, −RDj≤Pj,h −Pj,h−1≤RUj(26) where RUjand RDjare the ramp-up and rampdown limit of unit j, respectively. The above optimization problem has been implemented and solved by the linear programming solver CPLEX [9], under GAMS commercial software [8]. 3 ASSESSMENT METHODOLOGY In this section, the methodology adopted to assess the differences between the optimal price and the marketclearing price is described. The optimal price is defined as the price determined by the leader companywhen it solves its own optimization problem, assuming the remaining agents are not able to exercise market power. The marketclearing price, on the other hand, represents the pool price obtained through the auction mechanism, when the leader company has submitted its strategic bid to exercise market power. It is assumed, like in the Spanish pool, that all agents pay/get the same price for the cleared energy they demand/offer,irrespective of their own bids for each block energy. Figure 1 shows a flow diagram representing the major steps involved in the comparison process. The information about the rivals’ bidding curves and the aggregated demand are assumed to be known. Consequently, it is possible to obtain the residualdemand curves of the leader companyfor each hourof the biddingperiod. The residual demand is obtained as the difference between the aggregated demand and the rivals’ bidding curves for each level of prices. Figure 1: Methodology adopted to compare optimal and market-clearing prices. Obtaining in practice the residual demand model for a given hour is an involved stochastic process which, based on historical data, leads to a set of sample points. In order to use this informationwithin a deterministic optimization problem, the set of points is replaced or approximatedby a certain analyticalfunction. Usually,in simulation environments, we come upwith a step-wise curve, whosestep size is related to the energy block sizes, but this does not mean that the step-wise model is the true or best substitute in real life for the original cloud of points. The oligopolistic company will be interested in using the model for which the resulting market prices better match its expectations. In order to find the optimal price and the optimal amount of energy that the leader company has to offer at each period, the daily benefit function described in Section 2.1 is maximized, where residual demand curves play a critical role. To exercise its potential market power the leader company must establish the prices for each of its energy blocks, in accordance with the total quantity of energy determined previously in the optimization process. Then, knowing all generators’ bids, including that of the leader company,and the aggregateddemand, the complex auction algorithm described in Section 2.2 is used to obtain the energy blocks assigned to each generating unit. Afterwards, the market-clearing price, determined by the most expensive block of accepted energy, is compared with the optimal price, in order to assess the gap between both prices. Although both prices should be rather similar, it is evident that they will not be necessarily equal, as a consequence of the different data handled by each optimization model. While the market operator knows all of the details associated with each individual bid, the leader company must rely to a set of residual demand curves in which the aggregated competitors’ information is somehow embedded. In this regard,note that the residual demand curve for a given houris not coupled with those of previousor future hours, whereasthe detailed bidshandledbythe marketoperator are usually coupled along the 24-hour period. 15th PSCC, Liege, 22-26 August 2005 Session 3, Paper 1, Page 3 4 CASE STUDY The oligopolistic generation company simulated in this work is composed of eight thermal units plus two hydro plants, coupled by the same water stream, whose power-water characteristics, water dynamics and spill out behavior have been modeled as piece-wise linear functions. 0 5 10 15 20 25 1600 1800 2000 2200 2400 2600 2800 3000 Time [h] Energy [MWh] Figure 2: Hourly demand for the simulated market. A 24-hour period for a day-ahead auction market has been considered. Figure 2 shows the total hourly demand corresponding to the scenario simulated. For simplicity, the demand is considered inelastic, which is equivalent to replacing the first term in (18) by a constant value Dh. In this study, the followingresidual demandcurves havebeen modeled: 1) second-degree polynomial approximation; 2) piecewise linear approximation; 3) step-wise approximation. Such models are represented in Figure 3 for hour 21. 500 1000 1500 2000 2500 26 28 30 32 34 36 38 Polynomial Model Energy [Mwh] Market−Clearing Price [Euro/MWh] 500 1000 1500 2000 2500 26 28 30 32 34 36 38 Piecewise Model Energy [Mwh] Market−Clearing Price [Euro/MWh] 500 1000 1500 2000 2500 26 28 30 32 34 36 38 Energy [Mwh] Market−Clearing Price [Euro/MWh] Step−wise Model Figure 3: Residual demand approximations for the same hour. 4.1 Polynomial approximation In this case, the residual demand curve is expressed as λh(φ) = αh 1+αh 2φh+αh 3(φh)2 where the respective coefficients are obtained by secondorder polynomial regression. Figure 4 shows the optimal and market prices, along with the hourly gap between both prices. The average error is 1.13%, the largest errors taking place at hours 4, 5 and 6, when the energy cleared is smaller. One of the sources which can justify in part the resulting error lies in the fact that the market-clearing algorithm takes into accountthe ramp limits of rival generatingunits, which are ignored by the leader company. In order to analyze the influence of this factor,the experimentis repeated after removing such constraints from the auction-based algorithm. Figure 5 shows that, as expected, the error decreases during hours 4, 5, and 6, the average error for the 24-hour period being just 0.84%. This result suggests that energy ramps, which do not explicitly appear in the aggregated bidding curves of rival generating units, constitute a crucial decision variable when formulating the optimal bidding problem of a leader company. Ramp limits will be considered in the sequel. 19 20 21 22 23 24 25 26 27 28 29 30 31 1 3 5 7 9 11 13 15 17 19 21 23 Time [h] Price [  /MWh] 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% Error Error Optimal Price Market-Clearing Price Figure 4: Prices and error for the polynomial model with ramp constraints. 19 20 21 22 23 24 25 26 27 28 29 30 31 1 3 5 7 9 11 13 15 17 19 21 23 Time [h] Price [  /MWh] 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% Error Error Optimal Price Market-Clearing Price Figure 5: Prices and error for the polynomial model without ramp constraints. 4.2 Piecewise linear approximation In this case, the residual demand curves have been approximated by a piecewise linear function comprising eight segments. The comparison between the optimal and marketclearing prices is shown in Figure 6. Note that the market prices are smaller than the expected optimal prices, except for hours 4, 5, 15 and 19. The average error is this time 1.58%, the largest and smallest differences being 3.75% and 0.059% at hours 10 and 4 respectively. 4.3 Step-wise model Figure 7 shows the differencebetween the optimal and market clearing prices when the residual demand curves are represented by a multi-step model. Compared to previous models, it is apparent that the error is much larger in this case at peak hours; in fact, it is null for the first 8 hours. The average error for the 24-hour period is 0.92%, compared to 1.13% for the polynomial regression and 1.93% for the piecewise linear approximation. Therefore, it can be concluded that the multi-step model for residual demand curves provides a closer agree15th PSCC, Liege, 22-26 August 2005 Session 3, Paper 1, Page 4 ment between expected and resulting prices when the leader company is interested in exercising its potential market power. Accuracy of the piecewise linear approximation could be improved at the expense of increasing the number of linear intervals. 19 20 21 22 23 24 25 26 27 28 29 30 31 1 3 5 7 9 11 13 15 17 19 21 23 Time [h] Price [  /MWh] 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% Error Error Optimal Price Market-Clearing Price Figure 6: Prices and error for the piecewise linear model. 19 20 21 22 23 24 25 26 27 28 29 30 31 1 3 5 7 9 11 13 15 17 19 21 23 Time [h] Price [  /MWh] 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% Error Error Optimal Price Market-Clearing Price Figure 7: Prices and error for the step-wise model. 4.4 Oligopolistic company profit and demand share Even more important than the gap between the market expected prices and the resulting prices are the noticeable differences among the hourly prices of Figures 4, 6 and 7, which may significantly affect the net income of all market agents, in particular that of the leader company. This is somewhat contrary to intuition, considering the minor quantitative differences among the three models adopted to represent the residual demand curves. Table 1 compares the average market price, the total amount paid by consumers, the fraction of the total energy which is actually supplied by the leader company, as well as its profit, for the three residual demand models adopted. Note that the multi-step model yields both the largest daily profit and the largest percentage of market share for the leader company, whereas the piecewise linear model leads to opposite results. Such differences can be partly attributed to the complexity and nonconvexity of the resulting optimization problems, comprising many integer variables and nonlinear constraints, which are hence prone to converge to different local optimum points. In this kind of applications, the optimization tools employed behave like black boxes, whose output is rather sensitive to minor model changes. For comparison, the perfect market case, in which the leader company does not exercise market power, is included in the last row. It is worth noting that, in spite of the leader company serving a larger portion of the total load, its total profit is, as expected, much smaller in this case as a consequence of the lower clearing prices. Model Mean Demand Market Daily price cost share profit [Euros/MWh] [Euros] [Euros] [Euros] Polynomial 27.93 1,678,390 57.65 420,964 Piecewise 28.13 1,690,913 56.66 419,242 Multi-Step 27.89 1,677,260 59.46 427,054 Perf. Market 25.39 1,531,484 71.64 336,538 Table 1: Average market-clearing price, cost to consumers, leader company’s market share and daily profit. 5 CONCLUSIONS This paperanalyzes the influenceof the model adopted to represent the residual demand curves of an oligopolistic company, which is willing to exercise market power, on the clearing prices of a day-ahead auction-based energy pool. First, the nonlinear mixed-integer optimization problems respectively faced by the market operator and the leader company are briefly reviewed. Then, three residual demand curve models have been simulated to compare the gap between optimal prices predicted by the leader company and actual prices resulting from a complex auction process. It is shown that energy ramps of rival companies are partly responsible for these differences, which are smaller when the multi-step model is adopted. Finally, it is analyzed how such models affectthe net profit and market share of the leader company. ACKNOWLEDGMENTS The authors would like to acknowledge the financial support provided by the Spanish MCYT and Junta de Andaluc´ıa, under grants DPI2001-2612 and ACC-1021-TIC2002 respectively. REFERENCES [1] A. J. Conejo, J. Contreras, J. M. Arroyo and S. 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