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STOCHASTIC OPTIMIZATION, INCLUDING CONDITIONAL VALUE AT RISK CONSTRAINTS, OF LONG TERM ELECTRICITY TRADING JESÚS M. VELÁSQUEZ BERMÚDEZ DECISIONWARE LTDA. jvelasquez@decisionware-ltd.com presented in SIMMAC XIII INTERNATIONAL SYMPOSIUM ON MATHEMATICAL METHODS APPLIED TO THE SCIENCES SUBMITTED TO THE SPECIAL VOLUME ON "OR MODELS FOR ENERGY POLICY, PLANNING AND MANAGEMENT" OF ANNALS OF OPERATIONS RESEARCH ABSTRACT The necessary modeling to optimize electricity trading through standardized financial instruments and long term bilateral contracts is analyzed based on the existing practical and theoretical developments in the area of investment portfolios optimization. As a result, a linear stochastic optimization model is formulated. Experimental results for a "simple" electricity market, integrated by a spot market and a non standardized long term contracts market, are presented. The experiments compare different utility functions, including Conditional Value-at-Risk constraints. Keywords: Electricity Trading, Stochastic Optimization, Conditional Value-at-Risk Constraints INTRODUCTION The decisions associated with marketing electricity in the long run, through bilateral contracts or through standardized financial instruments, as options and forwards, are equivalent to those which are taken when the distribution portfolio is optimized. In the case of the electricity market exists, at least, an option of purchase, or of sale, by each hour of the planning horizon. The decision maker, buyer or seller, must face at least two risks: the price risk and the volumetric risk. The combination of these two risks causes that the traditional tools oriented to the solution of simple cases will not be effective upon facing this type of problems; more yet if we have in the mind that the relationship of the previous risks with the hydro-meteorological processes, in the supply and in the demand side, establishes strong and complex correlation structures. The decision process is complicated upon considering that the negotiation modalities imply permanent commitments of purchase/sale electricity by long periods, what increases the magnitude of the extreme risks. The use of stochastic optimization models to support financial decisions is wide. Rachev and Tokat (2001) present a summary of the different operations research methodologies that have been used to solve real problems. Other paper that includes an overview is presented by Mulvey et al. (1997). In electricity trading, the decisions are prices and quantities. The stochastic optimization models combined with the appropriate measurement of the risks, as Value at Risk -VaR-, are an appropriate way for this, since the primal variables are associated with the quantities, and the dual variables with the prices. 1. MATHEMATICAL MODELING In the following, it is considered a decision process in which the vector X represents the long term market decisions, the vector Y the parameters associated with random scenario, the function f(X|Y) the income, or costs, associated with a decision X since occurs the random condition Y, and p(Y) the probability distribution function of Y. 1.1. REFERENCE FRAMEWORK A model to optimize the decisions of electricity trading is described. The modeling process must consider that the regional regulatory entities have different conceptions about what "must be" a competitive electricity market. Therefore, aspects of the models that depend on each specific market exist. In this document it is assumed a "simple" market in which the agents can accomplish long term transactions under different modalities and short term transactions in the spot market in which the agents should sell the surpluses, or buy the shortages, of electricity that result as consequence of the transactions consolidation in the long term market consider their supply and their demand. In a long term commodity market the main factor of the decisions is the expectations about the future price of the commodity in the spot market. If the probabilistic characteristics of the stochastic process that determines the future spot price are known, it is possible to generate synthetic series of spot prices to use them in a stochastic optimization model and thus to represent the random environment of the decision process. There are many alternatives to generate spot prices in an electricity market. One of them it is to consider that in a hydrothermal system the spot price depends on the hydro-climatic conditions, the existing industrial infrastructure, the demand level and the "market regulation" and to obtain synthetic series of spot prices as functions of the marginal costs of the demand equation in a minimum cost hydrothermal dispatch model (Velásquez and Nieto 1999, Pereira and Campodómico 1995). Another possibility is to use equilibrium models more oriented to competitive markets. Also it is possible to generate synthetic series using statistic models and/or artificial intelligence methodologies taking as reference the historical series. This document does not has as objective to analyze the mathematical models that can be used to determine long run projections of the spot price. Obviously, the generation of this type of series makes part of a deep formal study since that the "optimal" decisions depend on the knowledge of the random environment; if this knowledge is weak, we can not wait "good" decisions. In the following sections is assumed that exist synthetic series of spot prices that represent the random environment of the decision making process. Each synthetic series is associated to a possible future scenario in a stochastic optimization model called as OPTMER (in Spanish "OPTmización del MERcadeo"). OPTMER is a stochastic linear optimization model that supports the decisions of an electricity generator, of a distributor, of a trader and/or of a vertical integrated company. This document only consider the math formulation for a trader that accomplishes purchase/sale transactions with multiple agents or clients. In the spot market the trader purchase/sells the deficits/surpluses not covered in the long term market. The agent attends a demand that it can be composed by regulated clients of mandatory attention and/or by bilateral contracts subscribed with unregulated clients or with other agents. For simplicity, the math formulation does not consider commercial differentiation factors between the clients, the contracts and/or the agents. According to the typical characteristics of the prices structure in the electricity markets the modeling is based on the following concepts: Planning Periods: one, or several months Day Types: ordinary, Saturday and holidays. Load Blocks: a hour or a group of hours Contracts: Take or Pay and Options Negotiation Modalities: free blocks: irregular electricity sales for any load block monthly blocks: electricity blocks for all hours of a month, or a period. annual blocks: electricity blocks for all hours of the year, or the planning horizon. 1 monthly blocks modulated by a load curve: electricity blocks for all hours of a month as a percentage of the buyer load curve annual blocks modulated by a load curve: electricity blocks for all the hours of a year as a percentage of the buyer load curve monthly options: options of electricity blocks for all hours of a month annual options: options of electricity blocks for all hours of a year The previous negotiation modalities are some examples that can exist in the long term market. Each modality implies a set of variables and restrictions especially oriented to describe the associated financial flows. To include new modalities implies to include new equations and variables. OPTMER can be used from two points of view: Physical: to determine the quantities to buy or to sell of a set of offers Economic: to determine prices and quantities to include in an offer that an agent will present to another agent. In the first case the decisions are the quantities to contract in each hour under each negotiation modality, related with the primal variables. In the second, the decisions are limits for the prices, called equilibrium prices, for those which are convenient to accomplish the transactions, related with the dual variables. Based on a sensibility analysis is possible to build agent's supply-demand curves for the long term market. 1.2. MATH FORMULATION To be short, the detailed math formulation is limited to present the parameters, the equations and the variables related to purchase offers and ignores the possibility of sale offers, those which can be included using a similar procedure to the described in the present document. 1.2.1. DEFINITIONS The index used in the model are t for period (month), d for type of day, b for load block, g for agent that sells electricity and h for random scenario. The parameters used in the model are (in cursive letter): Deterministic parameters Electricity prices ($/MWh) PCBAg Agent g price for annual blocks PCBMt,g Agent g price for monthly blocks in month t PCLIt,d,b,g Agent g price for free blocks in month t type day d block b PCMAg Agent g price for modulate annual blocks PCMMt,g Agent g price for modulate monthly blocks in month t PCOAg Agent g price for annual options PCSAg Agent g strike price for annual options PCOMt,g Agent g price for monthly options in month t PCSMt,g Agent g strike price for monthly options in month t Electricity quantities (MWh) DVBAg Agent g availability for annual blocks DVMMt,g Agent g availability for modulate monthly blocks in month t DVBMt,g Agent g availability for monthly blocks in month t DVMAg Agent g availability for modulate annual blocks DVLIt,d,b,g Agent g availability for free blocks in month t type day d block b DVOAg Agent g availability for annual options DVOMt,g Agent g availability for monthly options Random Parameters (components of the vector Yh) DEMt,d,b,h Demand (regulated plus contracts) in the month t type day d block b under random condition h PSPt,d,b,h Spot price in month t type day d block b under random condition h 2 t,h t,h Coefficient associated with the payment capacity of the spot market in month t random condition h Coefficient associated with payment time of the spot market in month t random condition h The variables used in the model are (in normal letter) : Long term market decisions (deterministic variables, components of X) CLPt,d,b,g Total electricity bought to the agent g in month t type day d block b CBAg Electricity bought in annual blocks to the agent g CBMt,g Electricity bought in monthly blocks to the agent g in month t CLIt,d,b,g Electricity bought in free blocks to the agent g in month t type day d block b CMAg Fraction of electricity demand bought in annual modulate blocks to the agent g CMMt,g Fraction of electricity demand bought in monthly modulate blocks to the agent g in month t COAg Annual options of electricity blocks bought to the agent g COMt,g Monthly options of electricity blocks bought to the agent g in month t Simulated variables (random variables) VMSt,d,b,h Sales in the spot market in month t type day d block b under random condition h CMSt,d,b,h Purchases in the spot market in month t type day d block b under random condition h INCOME FUNCTION 1.2.2. The income are split into deterministic and stochastic. The deterministic part, d(X), does not dependent of the random conditions and corresponds to commitments derived from the decisions in the long term market: the costs of blocks of electricity and the costs of the options. d(X) can be expressed as d(X) = g[ CBAg PCBA + COAg PCOAg + t CBMt,c PCBMt + COMt,g PCOMt,g + db CLIt,d,b,g PCLIt,d,b,g + CMMt,g PCMMt,g DEMt,d,b + CMAg PCMAg DEMt,d,b ] (1) where X represents the decision vector related with long term market transactions. The stochastic income, r(X|Yh), correspond to the purchases or sales in the spot market and to the exercise, or not, of the options. They depend on the random condition h, and can be expressed as COP(X|Yh): expenditures by exercises the options COP(X|Yh)g t db Minimum (PCSAg ,PSPt,d,b,h) COAg + Minimum (PCSMt,g ,PSPt,d,b,h) COMt,g (2) IMS(X|Yh): incomes by sales in the spot market ISM(X|Yh) = t db Maximum(0, g CLPt,d,b,g - DEMt,d,b) PSPt,d,b,h t,h t,h EMS(X|Yh): expenditures by purchases in the spot market ESM(X|Yh) = t db Maximum(0, DEMt,d,b - g CLPt,d,b,g) PSPt,d,b,h (3) (4) where Yh represents the random parameters vector associated with the condition h. The income and the expenditures in the spot market are considered independently due to the fact that would exist asymmetry in the spot market payment conditions (t,h and t,h). The previous financial movements are caused in the future and represent the risk of the decision. Their net value is r(X|Yh) r(X|Yh) = ISM(X|Yh) - COP(X|Yh) -ESM(X|Yh) The total income, deterministic plus stochastic, is f(X|Yh) = d(X) + r(X|Yh) (6) (5) 3 If a stochastic linear programming model is formulated, the variables contained within the Maximum function must be represented by a set of linear equations using the process that is described below. The following expression is considered Maximum [ 0 , z ] P where the variable z is not restricted and it is represented as the difference of two positive variables z = z + - zbased on the previous change of variables we have Maximum [ 0 , z ] P = z+ P (9) (8) (7) For a correct representation, it should be to guarantee that one of the two z-variables will be equal to zero, what is procured in linear programming due to the colineality between z+ and z-. Based on the foregoing, the following definitions can be considered zt,d,b,h = VMSt,d,b,h - CMSt,d,b,h = g CLPt,d,b,g - DEMt,d,b (10) where CLPt,d,b,g represents the total purchases of electricity to the agent g in month t type day d block b under random condition h and it is defined by the sum of negotiation modalities CLPt,d,b,g = CBAg + CBMt,g + COAg + COMt,g + CLIt,d,b,g + CMMt,g DEMt,d,b + CMAg DEMt,d,b Then ISM(X|Yh)t db VMSt,d,b,h PSPt,d,b,h t,h t,h ESM(X|Yh) = t db CMSt,d,b,h PSPt,d,b,h (11) (12) (13) The equations set {1, 2, 5, 6, 10, 11, 12, 13} constitute a linear system that describes the income/expenditures that will have the agent and should make part of the optimization model; it will be called "the marketing process constraints". 1.2.3. OTHER CONSTRAINTS 1.2.3.1. LOAD MODULATION The variables related to blocks modulated by a load curve are demand fractions and they should be ranged between 0 and 1 CMMt,g (14a) CMAg (14b) 1.2.3.2. AVAILABILITY TO SALE Normally, the agents receive electricity offers in those which a price is associated to a quantity that the seller is prepared to committing. This implies bounds for the quantities to buy CBAg DVBAg CBMt,g DVBMt,g CMAg DVMAg CMMt,g DVMMt,g COAg DVOAg COMt,g DVOMt,g CLIt,d,b,g DVLIt,d,b,g (15a) (15b) (15c) (15d) (15e) (15f) (15g) 4 1.2.3.3. FINANCIAL CONSTRAINTS Having in mind that the decisions of long term electricity marketing have as purpose the financial risks hedging, it can be necessary, and/or convenient, to include in the model constraints associated with these flows. This topic is not studied in this document and the reader is referred to other specialized articles, for example Cariño and Ziemba (1998). 1.3. UTILITY FUNCTIONS In a stochastic optimization model exists multiple possibilities to determine the decisions utility function. At least two functions can be considered to measure the kindness of the decision (yield measure): the revenue expected value and the regret due to the decision. 1.3.1. MAXIMIZE THE EXPECTED INCOME In this case the objective of the optimization will be maximize the expected income. This is Maximize d(X) + h r(X|Yh) /NE (16) where NE represents the number of random conditions. This approach does not imply the rationalization of the of the risk management, and in many cases is equivalent to a deterministic model based on the expected value of the random parameters (Yh). 1.3.2. MAXIMIZE THE MINIMAL INCOME To maximize the minimal income can be expressed as Maximize { d(X) + Minimumh [ r(X|Yh) ] } The previous objective function can be represented by the following formulation { Maximize d(X) + Rmin | Rmin r(X|Yh) ; h=1,NE } (18) (17) where Rmin is a not restricted variable that it should be substitute by the difference of two positive variables { Maximize d(X) + Rmin+ - Rmin- | Rmin+ - Rmin- r(X|Yh) ; h=1,NE } (19) The previous problem join with the marketing process constraints is a linear programming problem. 1.3.3. MAXIMUM REGRET The regret is the difference between the revenue of the decision X since occurred the random condition Yh with the revenue associated with the optimum decision, X*(Yh), that must be taken if we a priori known that Yh was going to occur. It can be expressed as (X/Yh) = d(X*(Yh)) + r(X*(Yh)|Yh) - d(X) - r(X|Yh) (20) where d(X*(Yh)) + r(X*(Yh)|Yh) represents the maximum revenue under random condition h. X*(Yh) is the solution to the deterministic problem of maximizing the revenue given the random condition h. This case considers the minimization of the maximum regret, also known as Savage criterion (Raiffa 1968), Minimize Maximumh [ (X/Yh) ] The previous objective function must be represented based on a set of constraints (21) 5 { Minimize Rmax | Rmax d(X*(Yh)) + r(X*(Yh)|Yh) - d(X) - r(X|Yh) ; h=1,NE } (22) where Rmax is a positive scalar. The previous problem join with the marketing process constraints is a linear programming problem. 1.3.4. EXPECTED INCOME AND CVaR INCOME CONTRAINTS The introduction of a risk constraint gives rationality to the expected value utility function. The risk measure more known is the Value at Risk -VaR- that corresponds to the superior limit of an interval for the losses associated with a portfolio at a given probability level. The introduction of VaR constraints has been studied widely (Anderson and Ursayev 1999, Uryasev 2000). It has been demonstrated that the appropriate form to consider VaR constraints is using the Conditional Valueat-Risk (CVaR) risk measure. CVaR is the expected loss exceeding Value-at-Risk and it is also known as Mean Excess, Mean Shortfall, or Tail VaR. The basic problem can be formulated as { Maximize d(X) + h r(X|Yh) /NE | (X) } (23) where (X) represents for expected revenue since it is less than (X). (X) represents the income level that it can be exceeded with a probability . is the lower bound for (X), that can be written as (X) = (X) - (1-)-1 h=1,NE Maximum[0, (X)-f(X|Yh)]/NE (X) can be expressed by a set of linear inequations (X) = (X) - (1-)-1NEh=1,NE h h(X) - f(X|Yh) h h0 h (25a) (25b) (25c) (24) where h represents the income deficit with respect to (X) if is taken the decision X and occurs the random condition Yh. The model can be written as { Maximize d(X) + h r(X|Yh) /NE | (X) - (1-)-1NEh=1,NE h h(X) - d(X) - r(X|Yh) h h0 h } The previous problem join with the marketing process constraints is a linear programming problem. 1.4. EQUILIBRIUM PRICES (26a) (26b) (26c) (26d) The model generates information about the opportunity cost/benefit of the different negotiation modalities; then it can be used to determine the equilibrium price of purchase/sale electricity for each modality. This price corresponds to an indifferent price that maintains the utility level of the agent, and serves as reference to establishes the convenience or not of a negotiation. The prices, economic variables, are obtained from the dual variables of the model. It should be consider that these prices are function of the quantity and that if we wished to have a demand curve it should be done a sensibility analysis. 2. EXPERIMENTAL RESULTS 6 Below it is presented a hypothetical case of a buyer agent that evaluates an electricity purchase competitive process in which participate multiple seller agents. The planning horizon is one year and the numbers are related with a "realistic" case in the Colombian Electricity Market. As additional condition, it is assumed that the agent limits the total of annual purchases to his annual demand (which is assumed deterministic), that implies tdb CLPt,d,b tdb DEMt,d,b 2.1. 2.1.1. PARAMETERS ELECTRICITY DEMAND (27) In this case the demand is considered as a deterministic parameter. The buyer agent attends a demand that has only one day type with the a typical load curve . Additionally it is known the aggregate monthly demand for the planning horizon. The hourly demand is calculated combining the aggregate monthly demand with the typical load curve. Alternatively, the hourly demand can be calculated taking into account the contracts that attends the agent. 1 MW 270 260 0.5 250 0 1 7 13 19 Hour 240 01/01/00 05/01/00 09/01/00 Month Figure 1. Normalized Load Curve Figure 2. Aggregate Demand 2.1.2. OFFERS The buyer agent has received quotes of eleven sellers in five modalities. The numerical details of the quotes are omitted, but they are available in Velasquez (2001). TABLE 1. OFFERS MODALITIES SELLER CHBG CHVG CRLG CTFG EBSG EEBG EPMG EPSG GCLG ISGG TRMG Annual Blocks Monthly Blocks Annual Blocks Modulated Monthly Blocks Modulated Free Blocks X X X X X X X X X X X X X X X X X X X X X X X X X 2.1.3. SPOT PRICE The information about the future variability of the spot price is summarized in ten synthetic series. As reference, the price average in the spot market is 53345 $/MWh (Colombian pesos by megawatt-hour), superior to the price average in the long term offers, near to 40000 $/MWh. 7 $/MHh 175000 150000 125000 100000 75000 50000 25000 0 Figure 3. Spot Price Synthetic Series 2.2. RESULTS OPTMER is used to determine the optimum policy of agent purchases. The following utility functions were analyzed: maximum cost, expected cost, maximum regret and expected cost with CVaR cost constraint. The results are presented in two stages: initially some detailed results of two traditional utility functions, the expected cost and the maximum cost, are compared; thereinafter, global results are presented for the others utility functions. 2.2.1. MAXIMUM COST VERSUS EXPECTED COST 2.2.1.1. TOTAL COSTS Below it is presented the comparative analysis of the results obtained for these cases. The monetary unit is thousands of Colombian pesos ($**3) TABLE 2. TOTAL COSTS ($**3) LONG TERM SPOT M ARKET STATISTIC TOTAL PURCHASES NET SALES UTILITY FUNCTION: MAXIMUM COST MEAN 77159.4 43632.3 33527.2 DEVIATION 0 5615.8 5615.8 MAXIMUM 77159.4 56835.2 37588.1 MINIMUM 77159.4 39571.3 20324.2 UTILITY FUNCTION: EXPECTED COST MEAN 834960 1076067 -241107 DEVIATION 0 462877.7 462877.7 MAXIMUM 834960 1707938 455757 MINIMUM 834960 379203 -872978 Below the distributions of the costs are presented % 100 MAXIMUM COST 75 50 25 EXPECTED COST 0 -806541 -540794 -275047 -9300 256447 $**3 Figure 4. Costs Distribution. Maximum Cost versus Expected Cost 8 An analysis of the previous graph concludes that the expected cost policy implies risk prone positions. In this case, it can be verified that the results are equivalent to solve a deterministic optimization problem using the expected spot price. Alternatively, minimizing the maximum cost the agent rationalizes his purchases in the long term limiting them to the minimal possible risk, but it is a policy totally risk averse. 2.2.1.2. EQUILIBRIUM PRICES The equilibrium prices, based on the dual variables, indicate balance prices, in the sense that produce an utility equal to zero if negotiations are accomplished at that price in the indicated modality. The equilibrium prices based on the expected cost permit to assume much more risk and therefore they are but large. 80000 $/MWh 42000 MAXIMUM COST EXPECTED COST $/MWh MAXIMUM COST EXPECTED COST 50000 38000 20000 Ene-99 Mar-99 May-99 Jul-99 Sep-99 Nov-99 34000 Ene-99 Mar-99 May-99 Jul-99 Sep-99 Nov-99 Figure 5. Equilibrium Prices - Monthly Blocks Figure 6. Equilibrium Prices - Modulate Monthly Blocks 2.2.1.3. PURCHASES Below the results related to the quantities to negotiate in some modalities are presented. The optimum purchases for annual blocks are in Table 3. TABLE 3. OPTIMUM PURCHASES ANNUAL BLOCKS OFFER POWER PURCHASES SHADOW PRICE QUANTITY-PRICE (MW) ($/MWh) POWER PRICE MINI EXPECTED MINI EXPECTED (MW) $/MWh MAX VALUE MAX VALUE 540 40960 0 540 950 41600 0 950 35211 55854 70 37800 0 70 100 39700 0 100 AGENT CHBG EPMG EPSG ISGG The equilibrium prices are established using the reduced costs, "simplex multipliers", associated with the upper bound of the purchase variables. It is evident the greater arrangement to pay of the expected cost policy. Below the quantities to negotiate for the modality annual modulate blocks are presented. TABLE 4. OPTIMUM PURCHASES MONTHLY BLOCKS OFFER PURCHASES SHADOW PRICE QUANTITY-PRICE DEMAND PERCENTAGE (%) ($/MWh) PERCENTAGE PRICE MINI EXPECTED MINI EXPECTED (%) U$/MWh MAX VALUE MAX VALUE 50 44733 0 0 100 37154 0 0 60 41560 0 0 1324 1355 60 40332 0 0 100 37632 0 0 50 38612 0 0 AGENT CHBG EBSG EEBG EPMG GCLG ISGG 9 The reason of the equilibrium prices so low is that this modality commits all annual demand of the buyer agent, subtracting to him flexibility to capture earnings in other negotiation modalities in the long term market and/or in the spot market. This is coherent with the principle that the portfolio optimality is the diversification and not the concentration, unless the prices offered are totally outside of market. 2.2.2. OTHER UTILITY FUNCTIONS Below the comparative results for all the studied utility functions are presented. The analysis is concentrated in the comparison of the distribution of the total costs. Detailed results are available in Velasquez 2001. 2.2.2.1. MAXIMUM REGRET Below it is presented the cost distribution when is used as decision criterion the maximum regret % 30 20 10 0 -611235 -243451 124333 $**3 Figure 7. Cost Distribution. Utility Function: Maximum Regret Table 5 contains the comparative analysis of the results obtained in the three cases. The monetary unit is millions of Colombian pesos ($**6) RANDOM SCENARIO 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 OPTIMUM COST -885 -21 -16 31 16 -200 -18 -894 -445 -745 -318 390 31 -894 926 TABLE 5. COMPARATIVE ANALYSIS OF UTILITY FUNCTIONS ($**6) REGRET TOTAL COST MINI MAX MAX REGRET EXPECTED VALUE MINI MAX MAX REGRET EXPECTED VALUE MEAN DEVIATION MAXIMUM MINIMUM RANGE 921 41 54 6 20 236 49 929 483 772 351 390 929 6 923 274 43 75 277 40 58 24 277 137 210 142 108 277 24 253 22 68 100 424 66 11 41 21 10 2 77 126 424 2 422 36 20 38 38 36 36 31 35 38 28 34 6 38 20 17 -611 22 59 308 56 -141 7 -618 -308 -534 -176 324 308 -618 926 -864 47 84 456 82 -189 24 -873 -435 -743 -241 463 456 -873 1329 10 The results indicate that the policy of the maximum regret is an intermediate point between the expected cost and the maximum cost. However, in this case, the maximum regret presents a high propensity to the risk. The equilibrium prices are coherent with the previous affirmation. The next graph compares the equilibrium prices for monthly blocks. 80000 $/MWh MAXIMUM COST MAXIMUM REGRET EXPECTED COST 50000 20000 Ene-99 Mar-99 May-99 Jul-99 Sep-99 Nov-99 Figure 8. Equilibrium Prices - Monthly Blocks 2.2.2.2. EXPECTED COST INCLUDING CVAR CONSTRAINT The decision making process based in to minimize the expected cost including CVaR constraint to a given probability level was studied. Experiments with several probability levels and several limits for the CVaR are presented. The results obtained in four cases in those which CVaR limit is fixed in 50 millions of Colombian pesos and the probability level of not to exceed this limit is varied are presented in the next table. RANDOM SCENARIO 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 MEAN DEVIATION MAXIMUM MINIMUM RANGE TABLE 6. COMPARATIVE ANALYSIS WITH FIXED CVAR LIMIT TOTAL COST ($**6) CVAR LIMIT = 50 ($**6) OPTIMUM PROBABILITY OF NOT TO EXCEED LIMIT - - COST 0.0 0.60 0.75 0.95 -885 -864 -295 -130 -13 -21 47 -12 -6 8 -16 84 -2 -1 10 31 456 186 102 50 16 82 28 23 29 -200 -189 -74 -38 8 -18 24 -13 -8 6 -894 -873 -298 -132 -14 -445 -435 -130 -45 13 -745 -743 -257 -123 -21 -318 -241 -87 -36 8 390 463 158 75 21 31 456 186 102 50 -894 -873 -298 -132 -21 926 1329 484 234 71 The results are coherent with the theory: the increase in the probability level reduces the cost volatility. The solution for =0.0 corresponds to the case of minimizing the unrestricted expected cost. Below the comparative analysis of six cases in those which is fixed the probability level and the CVaR limit is varied are presented 11 TABLE 7. COMPARATIVE ANALYSIS WITH PROBABILITY LEVEL FIXED TOTAL COST ($**6) = 0.95 RANDOM MINI EXPECTED CVAR LIMIT ($**6) SCENARIO MAX VALUE 38 40 50 75 100 200 1988 36 34 22 -13 -85 -148 -372 -864 1989 20 14 14 8 5 8 7 47 1990 38 25 22 10 10 20 30 84 1991 38 38 40 50 75 100 200 456 1992 36 32 32 29 27 30 38 82 1993 36 30 26 8 -20 -39 -93 -189 1994 31 22 19 6 1 1 -3 24 1995 35 33 22 -14 -86 -149 -376 -873 1996 38 38 31 13 -24 -59 -179 -435 1997 28 26 15 -21 -87 -143 -332 -743 MEAN 34 29 24 8 -18 -38 -108 -241 DEVIATION 6 8 8 21 54 86 199 463 MAXIMUM 38 38 40 50 75 100 200 456 MINIMUM 20 14 14 -21 -87 -149 -376 -873 RANGE 17 24 26 71 162 249 576 1329 Again, the results are coherent with the theory, when the CVaR limit is reduced the solution tends to the maximum cost solution, when the limit increases the solution tends to the expected cost solution. 3. CONCLUSIONS It can be concluded that: The decisions based on the simple analysis of the expected cost put on serious danger the financial stability of the agents. This fact includes the use of stochastic optimization models in the decision making process. The policy of minimizing the maximum cost seems to extremely risk averse. The weakness of the policy of the maximum regret is that does not control the risk level, which it is a result of the optimization process The incorporation of CVaR constraints in the policy of optimizing the expected costs is an interesting alternative, since it permits to integrate the optimization of the expected revenue of the decisions with the risk level that the decision maker wishes and/or can to assume. It can be asserted that assigning the appropriate parameters this alternative represents anyone of the others utility functions, and therefore covers all the cases, since it handles explicitly the risk levels and the expected utility that imply the decisions. Finally, the results prove that the existing theory for the strategic managing of financial risks is valid and coherent mathematically, and it can be extended to the electricity markets composed by a spot market and a long term market. REFERENCES Andersson, F. and Uryasev, S. "Credit Risk Optimization with Conditional Value-at-Risk Criterion". Research Report #99-9, Center for Applied Optimization, Dept. of Industrial and Systems Engineering, University of Florida, 1999. Cariño D. R. and Ziemba W. T. “Formulation of the Rusell-Yasuda Kasai Financial Planing Model”. Operations Research Vol No. 4 (1998). 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