Conditional Value-at-Risk Based Mid-Term Generation Operation Planning in Electricity Market Environment Gang Lu, Fushuan Wen, C.Y. Chung, Member, IEEE, and K.P. Wong, Fellow, IEEE
Abstract—In the electricity market environment, it is very important for generation companies (GENCOs) to make the optimal mid-term generation operation planning (MTGOP) which includes the trading strategies in the spot market and the contract market as well as the suitable unit maintenance scheduling (UMS). In making the decision of MTGOP, GENCOs are subject to risk due to uncertain factors, and hence should manage the inevitable risk rationally. Given this background, a new MTGOP model is first developed for a GENCO as a price taker so as to maximize its profit and minimize its risk measured by the Conditional Value-at-Risk (CVaR). In this model, the bilateral physical contracts are taken into consideration, together with the transmission congestion and the operation constraints of generating units. Then, a solving method is given by integrating the Genetic Algorithm and the Monte Carlo method. Finally, a numerical example is used to show the features of the proposed method.
Index Terms—Conditional Value-at-Risk; contract market; mid-term generation operation planning; spot market; unit maintenance scheduling I. INTRODUCTION
T
HE worldwide power industry restructuring makes generation companies (GENCOs) the market participants, and basically changes the nature of dispatching and operation planning for generating units. In the new market environment, GENCOs play a more active role in dispatching and operation planning. GENCOs may trade energy in the spot market and the contract market. In addition, the unit maintenance scheduling (UMS) problem also owns new features and faces new challenges. Determining the UMS is now a strategic issue of a GENCO with profit maximization as the objective. The ISO then checks the submitted UMSs by GENCOs and coordinates some of them, fairly, for maintaining the power Manuscript received on March 31, 2007. This work is jointly supported by The Hong Kong Polytechnic University under the joint-supervision scheme of PhD students with Zhejiang University (Project number: G-U096), by National Natural Science Foundation of China (No. 70673023) and by Special Fund of the National Basic Research Program of China (No. 2004CB217905). Gang Lu is both with College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China and with Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong, China (email:
[email protected]). Fushuan Wen is both with Department of Electrical Engineering, South China University of Technology, Guangzhou 510640, China, and College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China (phone and fax: 86-20-87114258; email:
[email protected]). C.Y. Chung and K.P. Wong are with Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong, China (email:
[email protected];
[email protected]).
system reliability, as discussed in [1]. An optimal UMS for a GENCO should be the one with minimum outage loss. Obviously, to minimize the outage loss associated with the UMS, continuous maintenance should be carried out. Hence, the problem of developing the optimal strategy of UMS is actually to determine the optimal starting time of the maintenance work. In the new market environment, the traditional mid-term generation operation planning (MTGOP) has been transformed from the centralized determination aiming at minimizing the total cost of energy supply into the decision-making of each individual GENCO for maximizing its own profit. For a GENCO, its MTGOP strategy includes how to sell energy in the spot market, how to sign contracts in the contract market and how to arrange the unit maintenance in the future specified period. These three sub-strategies are closely related, and may have significant impacts on the GENCO’s future profit. Up to now, research work on MTGOP is mainly for traditional power systems [2-4], and only limited research work has been done for the new market environment [5-9]. In [5-7], the well-known game theory is employed. Specifically, a gaming model is developed and the equilibrium state found in [5]. A more general game equilibrium model between the Cournot model and the perfect competition model is built in [6] with bilateral contracts taken into account. In [7], a stochastic gaming equilibrium model is developed with the scenario tree employed for dealing with random factors. However, the methods developed in [5-7] are all based on such an assumption that the production costs of GENCOs are public information, and surely this is not true in the market environment. A stochastic optimization method is developed for GENCOs as price takers in [8] for solving the MTGOP with risk management by optimizing contracting quantities. In [9], a stochastic optimization model of MTGOP is developed for Nordic hydro plants modeled as price takers, and the variance employed to measure the risk concerned. In the methods developed in [5-9], UMS was overlooked and the impact of the transmission congestion on the contract transaction was not considered as well. In the electricity market environment, there are many uncertain factors associated with the MTGOP problem such as the fluctuating spot prices in the future, and as the result, risk management must be considered in the MTGOP decision-making for a rational GENCO. Given this background, the Conditional Value-at-Risk (CVaR) technique [10] was employed to measure the risk associated.
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CVaR was recently applied to solve some problems in electricity markets [11, 12], and good outcomes achieved. In this work, a new mid-term generation operation planning model for GENCOs modeled as price takers is developed. The objective is to maximize the expected profit of selling energy and to minimize the risk associated. The major features of the developed MTGOP method here have the following points: 1) UMS is considered; 2) strategies in both the spot and contract markets are included; 3) transmission capacity constraints and the operating constraints of generating units are considered; 4) the CVaR is utilized for measuring the risk associated. II. MATHEMATICAL MODEL A. Basic Assumptions In this study, it is assumed that the time span of the studied MTGOP is one year. For the convenience of presentation, we suppose that a GENCO owns only one unit which must be arranged for maintenance within the next one year. It should be stressed that the developed framework here could be extended to general cases. In the market environment, the future spot price and contract price could be modeled as random variables and their probability distributions can be estimated based on historical electricity price data and other factors such as the future load forecasting, as stated in [13, 14]. It is assumed that the contract price is monthly based and the spot market is organized hourly. As stated before, both the hourly spot price p and the monthly contract price cl could be described by some probability density functions, such as the normal distribution function as follows: t T ° pt ,i ~ N ( Pt ,i , Vt ,i ) (1) ® t [l1 , l2 ] °¯ct cl ~ N ( Pl , Vl ) where i denotes nodes; t is the trading interval (assumed as one hour here); T is the total trading intervals in the specified one year; l is the month, and l1 and l2 are the sequence position of the starting and ending hours of the month l in T respectively; N is the normal distribution function; ȝ and V are expectation value and variance, respectively. The generation company is assumed to have a quadratic production cost curve in each trading interval: C (qt ) b2 b1qt b0 qt 2 (2) where b0, b1 and b2 are the coefficients of production cost function ($/MW2h, $/MWh, $/h); qt is the output power of the unit in t. In the following, only the situation that the GENCO is the price taker is considered.
B. Profit of the GENCO 1) Income from Contract Transaction: Contracts may be of various kinds and could be flexible, e.g. contracts for difference, unilateral contracts, and bilateral contracts. Only bilateral physical contracts are investigated in this paper.
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Physical contracts may lead to transmission congestion, and then the corresponding congestion charge and congestion surplus need to be considered. For example, the PJM (Pennsylvania-New Jersey-Maryland) ISO requires the participants to declare whether the submitted contracts may be regulated or not, and the California ISO requires the participants to consider the possible transmission congestion in contract transaction with the congestion charge involved. Since different forms of bilateral contracts exist, it is not easy to formulate them by a uniform model. Hence, we have to model the specific forms of contracts studied in the MTGOP formulation. For example, in the PJM market, there are three forms of contracts[15]: the first kind is the fixed one which are implemented no matter how high the congestion charge will be in the future; the second is called the schedulable ones and could be regulated; the third kind is the so-called “transaction up to congestion”, i.e., a congestion-controlled price no more than a specified value (currently 25$/MWh) could be executed, and the contract will be cancelled if congestion charge is higher than this price. In the following, the third kind is taken as the example. Accordingly, the income from contracts transaction for the GENCO could be formulated: T
B1
¦ (c Q t
c ,t
f t ) max{0, F gt }/( F gt ) (3)
t 1
where ct is the contract price; Qc,t is the contracting quantities in t; ft is the part of congestion charge borne by the GENCO; F ($/MWh) is the submitted congestion-controlled price; gt ($/MWh) is the congestion charge per MWh. In different electricity markets, the congestion charge may be computed by different ways. In a LMP (Locational Marginal Pricing) based electricity market, LMP difference represents the congestion charge. How to deal with the congestion charge needs to be specified in advance, and the cost could be borne by the GENCOs [16], customers or both. Here, it is assumed that the GENCO is responsible for a share of the congestion charge according to the following formula: ft
O Qc ,t gt
O Qc ,t max{ pt ,2 pt ,1 , 0}
(4)
where pt,1 and pt,2 are the respective spot prices at the generation side and the customer side corresponding to the contract transaction; Ȝ is a congestion coefficient in [0, 1] which denotes the share of the congestion charge for the GENCO. 2) Income from Selling Energy in the Spot Market: The GENCO’s income from the spot market is: T
B2
¦ (q
t
Qc ,t ) pt ,1
(5)
t 1
3) Unit Maintenance Scheduling Model: During the maintenance interval Tƍ, the unit is out of operation and the corresponding income is 0: °qt 0 t T c [24 (a 1) 1, 24 (a M 1)] (6) ® °¯Qc ,t 0 where a is the starting date of the maintenance represented as the sequence number from Jan. 1 to Dec. 31; M is the maintenance duration (days).
2007 IEEE Congress on Evolutionary Computation (CEC 2007)
4) Profit of Selling Energy: From (2)-(6), the GENCO’s profit of selling energy could then be formulated as (7): T
B1 B2 ¦ C (qt )
S
(t T ' )
(7)
t 1
C. Daily Generation Operation Scheduling Model The profit ʌ is transformed according to (3) and (5) as follows: T
¦
S
[ q t p t ,1 C ( q t )]
t 1,tT ' T
¦
[ Q c , t p t ,1 ( c t Q c , t f t )
t 1, t T '
m a x {0 , F g t } ] (F gt )
The impact of UMS on the GENCO’s profit includes two aspects: one is the unit outage during the maintenance duration, and in this period the generator concerned could not generate to make profit; and the other is that the UMS will affect the operation state of the unit in some relevant hours before and after the maintenance work due to the minimal on/off state constraints. Because the time span of the MTGOP is quite long, for decreasing the computation burden and being consistent with the day-ahead dispatching widely employed in the worldwide electricity markets, the annually generation operation scheduling is carried out on a daily basis. The daily generation operation scheduling (DGOS) could be formulated as follows: max
S0
24
¦ [q p
t ,1
t
t 1
24
C (qt )] ¦ xt (1 xt 1 )Cs
(8)
t 1
s.t. qxt d qt d qxt
(9)
qt 1 'q d qt d qt 1 'q (10) minimal on-line time: Ton (11) (12) minimal off-line time: Toff where ʌ0 represents the profit function of the DGOS; x is the sate variable of unit and x=1 denotes the unit is online and x=0 offline; Cs is the start-up cost; q and q are the lower and
conditions within a specified period. In other words, the cases under abnormal conditions are not taken into account. This means that we only know the probability of the event that the loss will exceed this VaR but do not know how much the loss will be when the event happens. Although VaR is a very popular method for measuring risk, it is not a coherent risk measurement and has undesirable mathematical characteristics such as lack of subadditivity and convexity[10]. Moreover, different methods for calculating VaR may lead to quite different results. Hence CVaR is proposed for describing the conditional expectation of the losses above VaR. This method could measure the potential risk better on abnormal conditions. Although derived from VaR, CVaR owns better properties such as coherency and convexity [10]. Although mainly applied to finance domain, CVaR is also feasible in others. Here, we apply the basic principle of CVaR in [10] to measure the decision-making risk of the MTGOP. Here, the risk is defined as the difference of the actual profit less than the expected profit. With E(ʌ) representing the expected profit, the risk r(ʌ) is E(ʌ)-ʌ when E(ʌ)>ʌ. According to the definition, the VaR associated with MTGOP under the confidence level ȕ on the normal market condition is as follows: (13) VaRE (y ) min{E (S ) S c M (y, S c) t E }
M (y , S c)
³
f (S )d S
(14)
S tS c
where y is a decision vector that includes the starting date of the unit maintenance and the monthly contracting quantities for the MTGOP, and profit ʌ is correlated with y and random variables such as the spot price; ij(y, ʌƍ) is the accumulation distribution function of the loss associated with y; f(ʌ) is the probability density function of ʌ as shown in Fig.1.
-
upper limits of the generation output, respectively; ¸q and + ¸q are the maximal down/up ramp rate constraints in each trading interval. The optimal DGOS depends on the spot price, the production cost and the operation constraints of the units. Actually, when the spot price is low, the output power may decrease, even down to zero, and the contracted power could be fulfilled through purchasing energy from the spot market. D. Risk Analysis Based on CVaR Many risk measurement methods are available, such as variance and Value-at-Risk (VaR)[17]. However, these methods have some disadvantages. The widely employed expectation-variance model in electricity market analysis utilizes variance of profit to measure risk, and treats winning and loss equally. Surely, this is not reasonable. VaR represents that a risk portfolio has the maximum possible loss with respect to a certain confidence level on normal market
Fig.1 Probability density function of profit ʌ
Under assumption S Ec
max{S c M (y , S c) t E } , CVaR
that loss exceeds VaR is as follows: CVaR(y ) E[S E (S ) S t E (S ) S Ec ]
(15)
By introducing the following function in [10], we could simplify the calculation of CVaR: FE (y , S c) E (S ) S c (1 E ) 1 ³ >S c S @ f (S )d S (16)
where [ʌƍ-ʌ]+ means max(0, ʌƍ-ʌ). According to Theorem 1 and Theorem 2 in [10], the following properties are satisfied for (16): FE (y, S c) °CVaR E (y ) min S c (17) ® CVaR E (y ) min FE (y , S c) °¯ min yY ( y ,S c )Y u where Y is the feasible set of y.
2007 IEEE Congress on Evolutionary Computation (CEC 2007)
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From (17), it is known that minimizing CVaR is equivalent to minimizing (16), and hence optimizing CVaR and calculating VaR could be done at the same time, i.e. S c(y * ) S Ec (y * ) , and y* is the optimal decision vector that minimizes CVaR under the confidence level ȕ. According to (13) and the formula S Ec max{S c M (y , S c) t E } , it is known that VaR is E(ʌ)-ʌƍȕ. Furthermore, the integral in (16) actually represents the expectation value operation, and hence FE (y , S c) could be approximated by simulation as follows: FE (y , S c)
Z
E (S ) S c Z 1 (1 E ) 1 ¦ [S c S ]
(18)
z 1
where the total number of sample z amounts to Z. E. Mid-term Generation Operation Planning Model Hence, the MTGOP for the GENCO is formulated as: max E (S ) ° Z (19) ® 1 1 ° min E (S ) S c Z (1 E ) ¦ [S c S ] ¯ z 1 Generally speaking, maximizing the expectation value of the profit and minimizing the risk are two contradictory objectives. By introducing a risk-avoiding coefficient J (0 d J d 1), which indicates the degree of risk aversion for the GENCO, we could transform (19) into (20): Z
max
E (S ) J {E (S ) S c Z 1 (1 E ) 1 ¦ [S c S ] } (20) z 1
This is a mixed integer stochastic optimization problem with three kinds of variables: a) the decision variables, i.e., both the discrete variable (the starting date of maintenance) and continuous ones (the contracting quantities); b) qt; c) the known stochastic variables, i.e. the spot prices and the contract prices. Recognizing the complexity of the problem, the Genetic Algorithm (GA) and the Monte-Carlo simulation method are employed. III. SOLUTION PROCEDURE The MTGOP can be optimized by the GA based on the Monte Carlo samples of the stochastic variables. In this simulation process, ʌ and ʌƍ are closely related to DGOS that could also be solved by the GA. Accordingly, the solution framework includes two-level GA based on the Monte Carlo simulation, i.e. the upper level GA for solving the MTGOP and the lower level GA for solving the DGOS. The solving procedure mainly includes the following three steps: Step 1): Sample the spot price and the contract price according to their distribution functions for Z times respectively. Step 2): Get the Z annual generation operation scheduling scenarios by optimizing the DGOS repeatedly according to the Z samples of the spot price. The DGOS model represents a mixed integer programming problem. The GA method for
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optimizing unit commitment in [18] is applied here to solve DGOS, just with constraint (9) added. Step 3): The stochastic optimization problem of (20) is solved by the well-developed GA as detailed below: a) Coding. The employed coding structure could be expressed as [a " Qc ,l "] , with both the integer variable a and the real variables Qc,l included. Here, l represents the sequence of months. b) Fitness calculation. Rank the ʌj’s obtained from the Z simulation results for each individual in a descending order and designate the ʌj at j= «ª Z E »º place as the ʌƍ according to (17), where symbol «ª»º denotes the upward rounding operator. Then, calculate the fitness function using formula (20) for every individual. c) Elite strategy. Retain some best solutions in each generation and directly pass them to the next generation for reproduction. d) Crossover operation. Two father individuals x1 and x2, are randomly selected such that their fitness values satisfy 0
