Spatial energy market risk analysis using the semivariance risk measure

Electrical Power and Energy Systems 29 (2007) 600–608 www.elsevier.com/locate/ijepes

Spatial energy market risk analysis using the semivariance risk measure Zuwei Yu

*

Energy Center, Purdue University, Room 334, 500 Central Drive, West Lafayette, IN 47907, USA Received 6 October 2004; accepted 12 December 2006

Abstract The paper concentrates on the analysis of semivariance (SV) as a market risk measure for market risk analysis of mean–semivariance (MSV) portfolios. The advantage of MSV over variance as a risk measure is that MSV provides a more logical measure of risk than the MV method. In addition, the relationship of the SV with the lower partial movements is discussed. A spatial risk model is proposed as a basis of risk assessment for short-term energy markets. Transaction costs and other practical constraints are also included. A case study is provided to show the successful application of the model.  2007 Elsevier Ltd. All rights reserved. Keywords: Electricity markets; Lower partial movements; Portfolio; Risk; MV; MSV

1. Introduction There have been many studies on risk in portfolio selection and practical applications based on the Markowitz mean–variance (MV) method [1,2]. The most important aspect of the MV method is that it introduces an important concept of portfolio efficient frontier on which each point is an efficient portfolio point with the variance minimized at a given level of return expectation. After the introduction of a risk-free asset, Sharpe proposed the capital asset pricing model (CAPM) [3] that has also attracted extensive attention in academics but with limited applications in practice [26]. The logical connection of CAPM and MV is explained in detail in [3]. The problem with the MV method is that variance may not be a proper risk measure because it contains the effect of return deviations above the mean. By common sense, portfolio returns above the mean should be regarded as beneficial not as risk. Hence, using variance as a risk measure does not make a good logical sense, as argued by many. A simple improvement over the MV method for measuring risk is *

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to use deviations below the mean, and lower semivariance (SV) is hence a simple and good choice for the purpose. The SV method, however, has attracted less attention in practice due to more computation requirement. The other problem with the traditional MV method is that it cannot take into consideration fixed transaction cost, which usually over estimates the mean return of a portfolio. Since the fixed transaction cost is a lumpy sum no matter how big the transaction deal is, a binary integer variable is needed for properly modeling it. If a portfolio does not contain asset, the binary variable is zero, the fixed transaction cost is not counted for after optimization. Since the model contains both integer and continuous variables, it is called the mixed integer programming (MIP). The SV is also called the second lower partial movement (LPM) and is one of the several downside risk measures [10]. Other downside risk measures may include the absolute downside deviation (also called semideviation (SDV) or the first LPM), etc. The SDV as a risk measure has a computation advantage but cannot take into consideration correlations of returns on assets. LPM plays an important role in our discussion of downside risk measures. Therefore, it is appropriate that the definition of the k-degree LPM be provided:

Z. Yu / Electrical Power and Energy Systems 29 (2007) 600–608

601

List of primary symbols Indices g h i j k m n Mk N Ngi Sm TH

wcinh generation unit index time in hour or other appropriate unit power pool index (e.g., PJM, NYPOOL) an alias of i degree of freedom in lower partial movement calculation an alias of n energy product index (n = 1 for electricity in MWh, 2 for spinning reserve, 3 for regulation, etc.) total number of pools of interest total number of commodity products in a market number of generation unit in pool i set of commodity products with minimum contracting requirement short-term planning horizon.

Parameters fcin one time fixed charge ($) for transacting product n in pool i (=0 if it is not required) Mmax total budget limit ($ plus other resources, e.g., fuel reserve, etc.) pcinh proportional charge ($ per unit per time) for transacting product n in pool i in time h (can be zero) Pmaxig capacity upper limit of generation unit (i,g) Pminig minimum production level in MW SDVinh semideviation of prices below the expected price linh

LPM ¼

Z

s k

ðs  RÞ dF ðRÞ; 1

where R is a stochastic process such as the return of a portfolio, s is a target value that an investor would use for measuring his/her preference of risk. F(R) is the cumulative probability distribution of R. s is the expected return of a portfolio for both the SDV(k = 1) and the SV(k = 2). LPM is closely related to the value at risk (VaR) measure where s is defined as a percentile on the lower tail, say, 1% of the probability distribution. When k P 1, LPM is used for measuring the risk preference of those risk averters. However, the k-degree LPM must be correctly related to the standard statistical movements of the distribution where investors have a preference for higher values of odd movements (skewness) and a dislike of higher values of even movements (variance, kurtosis and the like). In short, LPM is used for measuring an investor’s risk attitude towards the below-target returns. The motivation for the research is based on the reasoning that the variance of a portfolio return may not be an appropriate measure of risk. It is logical to think that variance is a measure of uncertainty rather than risk and only

contracted sales of product n in h committed before h = 1 whijnh wheeling charge in $ per unit product in time h (can be zero for some product) Wmax a sufficiently large positive real number (e.g., e + 10) ld desired net profit in $. The Markowitz efficient frontier is obtained by solving the model with various values of ld, an adjustable parameter qijnmh correlation coefficient between the prices of market products winh and wjmh in time h stcigh generation unit start-up cost in $.

Variables CT total cost or expenditure in $ CPigh production cost of unit (i,g), excluding start-up or shut-down cost Iijh a binary variable for wheeling negotiation, 1 = wheeling allowed, 0= no wheeling allowed. Iijh = 0 can be used to disallow cross-pool sales of certain energy products (e.g., regulation) Uigh a binary variable for generation unit on/off status w1ingh product n from unit g in pool i and time h w2ijnh product n originated in i, wheeled to j in time h winh product n bid in pool i and in time h Yinh a binary variable for selling product n in pool i in time h linh the expected price ($ per unit product) for product winh.

the part of the variance with returns less than the mean return is relevant to risk. A variance also includes the effect of the returns greater than the mean return, and a rational investor should love but not to avert higher returns. Therefore, the part of the variance reflecting greater returns than the mean should not be regarded as risk. It is generally agreed that this argument offers a logical thinking and it is consistent with the framework of VaR measure being widely used in the financial industry [4]. Note that VaR is also a downside risk measure. The paper is centered on the minimization of SV subject to practical constraints currently overlooked by many portfolio software packages. However, models based on the minimization of VaR and SDV are not excluded as alternatives. In the proposal, we do not assume symmetry in distribution and normality of returns. We also incorporate integers, fixed and proportional transaction costs, and other practical constraints. We do not propose CAPM as a risk management tool for electricity markets because CAPM is for efficient markets where the characteristics do not apply for electricity markets with imperfections such as games.

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Z. Yu / Electrical Power and Energy Systems 29 (2007) 600–608

For symmetrical distributions of asset returns, the MSV and MV only have a minor difference between their efficient frontiers for small values of the variance of portfolio returns [5]. However, for energy portfolios, especially with heavy option derivatives, asset returns are well known to be asymmetrical. The MSV method is hence a more general and better choice than the MV method for energy portfolio optimization. The following example shows how skewed the returns of the PJM electricity market can be for a gas turbine plant. Assume the plant has an average heat rate of 11 MMBTU/MWh, a variable O&M cost of $1/MWh and a capacity of 50 MW. Suppose that the peak plant ran from hour 15 to hour 18 at full capacity. A simple estimation of the discrete distribution of the expected returns for the plant is illustrated in Table 1 given a natural gas price of $2/MWh in August 1998. The market price for the plant to make nonnegative returns is about $23/MWh ignoring the start-up cost. The energy weighted average price is about $55.5/MWh and the root mean square error is about $29.8/MWh. It can be seen from Table 1 that the return distribution is seriously skewed and with a fat tail to the higher price range. Furthermore, the proposed model is extended to include the spatial nature of the portfolio optimization with downside risks. The introduction of integers will make portfolio selection more practical. Portfolio selection models with integers are often NP-complete and very difficult to solve. However, small portfolios can be solved within a short time using the nested LaGrange relaxation (LR) method and even the Brunch & Bound method. We will show that the objective function of the MSV portfolio model is in the form of multiplication of variables and a simple decomposition using the LR may not work well. In case there is a need to solve large energy portfolio problems, algorithms can be developed for computation time reduction. For example, several algorithms can be used to speed up com-

Table 1 Discrete distribution of returns on production of a gas turbine (Estimation is based on the observed electricity prices in PJM, August 1998) Price range in$/ MWh

No. of occurrence

Cumulative occurrence

Relative frequency

[17, 23) [23, 25) [25, 30) [30, 35) [35, 40) [40, 45) [45, 50) [50, 55) [55, 60) [60, 65) [65, 70) [100, 105) [120, 130) [130, 140) [680, 690) [990, 1000)

16 6 8 18 17 16 22 5 4 3 3 2 1 1 1 1

22 30 48 65 81 103 108 112 115 118 120 121 122 123 124

.1290 .0484 .0654 .1452 .1371 .1290 .1774 .0403 .0323 .0242 .0242 .0161 .0081 .0081 .0081 .0081

putation, including the modified Benders decomposition (MBD) method, the genetic algorithms (GAs), etc. The remainder of the paper is arranged as follows. Section 2 introduces the Markowitz MV, discusses the Markowitz MSV method, and presents the MSV model with transaction costs. Section 3 presents the spatial risk model with detailed explanations for the major constraints, and it also discusses potential algorithms for solving the problem. Section 4 illustrates the use of the model via a case study. And Section 5 concludes the paper. 2. A background on downside risk measures We first discuss the MV risk measure before going to the topic of downside risks. 2.1. The Markowitz mean–variance method Risk is related an old concept of diminishing marginal utility that can be traced back to the Bernoulli cousins in early 18th century [6]. The first systematic work on economic decision making with utility maximization under uncertainty is from Von Neumann and Morgenstern [7]. Their theory of games and economic behavior laid a foundation for later research in many areas, including financial engineering. The Markowitz MV method for portfolio selection was seminal because it first provided the concept of portfolio efficiency in a definite form of utility function – the quadratic function as defined below [1,2] U ðRÞ ¼ R  AR2 ;

ð1Þ

where R is the return on N (N P 1) risky assets (or energy commodities) of a portfolio. ‘‘A’’, a positive parameter, reflects the degree of risk aversion in the Markowitz MV method. U(Æ) stands for utility. Taking expectations of (1), we have E½U ðRÞ ¼ EðRÞ  AEðR2 Þ ¼ lR  Al2R  Ar2R

ð2Þ

lR, restricted to be positive, and r2R are the mean and variance of the portfolio return. Eq. (2) is parsimonious and can offer desirable solutions for a set of stochastically non-dominant assets to diversify risks given that transaction fees and other fixed costs are negligible. (For the topic of stochastic dominance, readers are referred to [27–31].) Eq. (2) states that the expected utility is negatively proportional to the variance, indicating risk aversion. It is easy to verify that Eq. (2) satisfies the condition of diminishing marginal utility with respect to the mean return: oE½U ðRÞ=olR ¼ 1  2AlR :

ð3Þ

Different variants of the Markowitz model exist but are not enumerated in the interest of brevity. The well-known CAPM can be regarded as an extension of the Markowitz MV model.

Z. Yu / Electrical Power and Energy Systems 29 (2007) 600–608

2.2. The mean–semivariance method The MV method is not without problems. The first problem is that it cannot take into consideration fixed transaction fees and other fixed costs. As a result, the MV method is for sub-optimal solutions in portfolio selection. Portfolio models based on the Markowitz MV often overstate diversification and the problem is partly induced by ignoring fixed transaction costs. Perhaps the most controversial part of the MV method is why the portfolio variance is treated as a measure of risk. It is illogical to include the upper SV in the portfolio risk measure. As a matter of fact, Markowitz himself discussed the use of SV in 1959 and admitted that the MSV may produce better portfolios than those based on the MV (see p. 194 of [1]). The SV ðr2R Þ belongs to the category of downside risk measures. Downside risk measures have a long history. Fox example, Roy described the downside risk measures as early as in 1952 [8]. Many have researched different versions of the downside risks since then. One possible modification to the MV method is presented below. Define U ðRÞ ¼ R  AR2 with R 2 fRjR < lR g, it follows that E½U ðRÞ ¼ EðRÞ  AEðR2 Þ ¼ lR  Al2R  Ar2R ; ð4Þ R l1 R lN where PlR ¼ w1 1 x1 f ðx1 Þ dx1 þ    þ wN 1 xN f ðxN Þ dxN ¼ Nn¼1 wn ln , with r2R ¼ r2R =2 for a symmetrical distribution and to be determined for an asymmetrical distribution. ln is the (per unit) mean rate of return on asset n in the portfolio and wn the weight selected (number of units/ shares) for asset n. It is important to make sure that lR > 0 to avoid the situation of non-diminishing marginal utility. f(xn) is the probability distribution of the per unit rate of return on asset n. Eq. (4) has about half of the variance (depending on the specific definition). The concern with the negative marginal utility is, though not completely resolved, relieved somewhat because of the fact that lR < lR in the MSV method (we will assume that lR ¼ alR , with 0 < a < 1.) The (sub)optimal portfolio selection model without transaction costs and minimum lots, based on the MSV method, can be defined below. E½U ðRÞ ¼ lR  Al2R  Ar2R

Max

Subject to XN wn C n 6 Mmax; n¼1 lR ¼

N X

wn ln > 0;

ð5Þ

ð5:1Þ ð5:2Þ

n¼1

lR ¼

N X

603

equality as described in the standard literature. The advantage of the inequality formulation is that an investor (or power producer) can choose not to invest (or produce) or to invest a portion of the planned budget if the short-run business is not profitable. In other words, this formulation avoids a forced investment of the total planned budget. In model (5), Mmax is total cash of an investor, say, at banks and in hand. ln can be an estimate based on past data. Cn is the unit price for asset n at the beginning of the planning horizon for the portfolio. Proportional transaction costs of buying asset n can be incorporated in Cn. One alternative for approximating the SV can be found in [9], in which Markowitz and his co-authors used actual historical returns for the model. Their mathematical definition for the SV estimation is #  )2 (" T N X X 1 r2R ffi S E ðW Þ ¼ Rnt wn  E ; ð6Þ T t¼1 n¼1 where E is the desired mean return of a portfolio, W is the vector of assets weights to be determined, and PN  ½ n¼1 Rnt wn  E is SDV, with " # ( ) N N X X SDV ¼ E  Rnt wn \ Rnt wn < E : n¼1

n¼1

Note that t is measured in the past but not in the future. An alternative definition can be found in [10] it is also one of the early studies using the critical line method to compute the MV efficient frontier when the portfolio does not include integers. Define qmn the correlation coefficient between returns of assets m and n, we have r2R ffi S E ðW Þ ¼

N X N X

wm wn SDV m SDV n qmn ;

ð7Þ

m¼1 n¼1

PT where SDV n ¼ T1 t¼1 ðE  Rnt Þ for Rnt < E. Note that time t is measured in history too. The advantages of the MSV model over the MV model lie in concept and model performance but not in computation time. Markowitz indicated the MSV method would require more computation time. This is especially true when the joint distribution of the asset returns is to be found (there are several methods of joint distribution estimation, including the fast Fourier transformation (FFT) technique. However, they all would add considerable computation time to a mixed integer portfolio). Problem (5) can be redefined in (8). Both models are approximately equivalent but not exactly the same. In fact, models (5) and (8) would produce results that would have some difference. However, model (8) is a more frequently used in literature due to its ease of implementation.

wn ln ;

ð5:3Þ

Min

for all n:

ð5:4Þ

Subject to

r2R

ð8Þ

n¼1

wn P 0

It is important to note that the total budget constraint (5.1) is total budget limit that is an inequality instead of an

N X n¼1

wn C n 6 Mmax

ð8:1Þ

604

lR ¼

Z. Yu / Electrical Power and Energy Systems 29 (2007) 600–608 N X

wn ln P ld :

ð8:2Þ

n¼1

Note that the efficient frontier can be computed by varying the value of ld in (8). Typically, too high a value set for ld will cause infeasibility to problem (8). The maximum value of ld cannot be greater than the return of the asset whose return is the greatest of all assets. The higher the value of ld it is desired, the higher value of the risk represented by the SV in the objective function. The methods for computing the Markowitz efficient frontier can be found in [9,10,25]. 2.3. The MSV method with transaction costs Traditional textbook portfolio and risk models, including the Markowitz MV, the CAPM and the APT (arbitrage pricing theorem [11]) models, ignore transactions costs and minimum lots required by many securities and commodities. As a result, these models have been subject to wide criticism. Fortunately, there have been attempts to resolve the problem. Proportional transaction costs have been discussed in [12,13]. Later on, some introduced a model for determining the optimal number of securities in a portfolio taken into consideration fixed transaction costs [14]. However, this work was criticized because some believed that the approach is based on the CAPM that assumes no fixed costs [15]. This criticism is relevant because the CAPM is an equilibrium pricing approach for frictionless financial markets. For portfolio selection considering option pricing and transaction costs, the notion of a break-even volatility was introduced in [16] and the idea was furthered in [17]. There are also studies on portfolio selection with transaction costs in discrete and continuous time settings [18– 20]. There can be a long list of studies on portfolio selection with transaction costs. However, the majority of the studies assume that investors adopt a strategy of buying and holding without integer formulation. Integer programming models are the most appropriate choice for portfolio selection when fixed transaction costs and minimum lots are considered. Typical works based on the Markowitz MV method while incorporating mixed integer programming can be found in [21,22]. There have also been commercial software packages based on the integer quadratic programming technique for portfolio selection with claims of considerable reduction of trading costs (e.g. [23,24]). However, these works are nearly all based on the portfolio variance evaluation approach deduced from the Markowitz MV method. Thus, they all have the same problem: aversion of obtaining higher rate of returns as discussed earlier. To overcome the problem, we propose the following MSV portfolio model with transaction costs and minimum lots. Let pcn and fcn be the proportional and fixed transaction cost respectively, wnmin the minimum lot of trade for asset n in terms of investment dollars and Sm the set of assets with minimum lot requirement such that Sm 6 N. Define Yn (a

binary variable) as a decision variable for contracting (Yn = 1) or rejecting (Yn = 0) commodity or asset n. The optimal portfolio selection model with transaction costs and minimum lots, based on the MSV method, can be defined as follows: Min

r2R

ð9Þ

Subject to N X ðwn C n þ wn pcn þ fcn Y n Þ 6 Mmax;

ð9:1Þ

n¼1 N X ðwn ln  wn pcn  fcn Y n Þ P ld ;

ð9:2Þ

n¼1



wn

P W nmin

8Y n ¼ 1;

¼0

8Y n ¼ 0;

8n 2 Sm:

ð9:3Þ

The last constraint can also be expressed as: Wmax Yn P wn P wnminYn,"n 2 Sm. Wmax is a sufficiently large positive number. A paper based on this model has been published by Yu [43]. This model will provide a basis for a spatial energy market risk assessment described next. 3. The proposed spatial portfolio model with semivariance In regulated power systems, risks are primarily associated with system planning and operation. They include the risk of capacity shortages due to under planning, load uncertainty and system failures [32]. The risk in power production costing due to load uncertainty and plant availability has also been addressed extensively [33,34]. As deregulation unfolds, power producers are facing more risks than before. To name a few, there can be operating risk, credit risk, market risk, legal risk, etc. A grand unification model including all the risks has yet to be researched. This paper is centered on the modeling of market risk. A short-term electrical energy portfolio can be defined as: A power producer product mix that may include fuel purchase or sale, power to purchase or sale, spinning reserve purchase or sale, capacity reserve purchase or sale in competitive markets. Market risks can be related to fuel price volatility, electricity price volatility, etc. The risk related to fuel availability is ignored in the interest of brevity. The product mix is optimized by minimizing the overall volatility of the products subject to a certain level of return (e.g., profit), which will result in a ‘‘portfolio’’. Producers need to assess different market risks in the deregulated environment. For example, a dominant power producer in a region may assume an oligopolistic strategy in power production. The dominant producer would face the risk that other producers may not use the same strategy of non-cooperative gaming and may be undercut by its opponents. On the other hand, a small producer or a competitive fringe may seldom adopt a gaming strategy [35] and it may primarily be concerned with the risk associated with market prices. Note that a competitive fringe may

Z. Yu / Electrical Power and Energy Systems 29 (2007) 600–608

assume that its decision would not affect market prices. This paper focuses on modeling the spatial market risk for the fringe producers or bigger producers without adopting oligopolistic strategies. There have been several academic studies of risk assessment for deregulated electricity markets. For example, Refs. [36–38] address power project valuation risk. Bjorgan et al. [39] discusses market risk using Pareto optimality for the trade-off between the expected value and variance. Sheble [40] presents a method of risk analysis based on the decision tree method. All the studies have made contributions to the area. Nonetheless, the spatial nature of the market risks has not been explicitly modeled. Currently, power producers in North America are intensifying their effort in managing market risks. Since there was not much tradition of market risk management in the traditional regulated environment, they have been turning to financial engineers from financial industry for assistance. As a result, VaR, a risk measure from the financial industry, has been used extensively by power producers and energy traders. VaR is based on the finding of probability distribution of certain valuables (e.g., returns or prices) and on the estimate of percentile of the lower tail [4]. However, finding spatial distributions has been proved very difficulty in practice. Market risk analysis based on an exact distribution analysis can be very demanding. The distribution is related to many risk factors, such as load forecast errors, generation availability, the behavior of market traders (e.g., market gaming), etc. Even with perfect weather forecasts, load can still be uncertain due to randomness of consumption [41,42]. This paper proposes an approach that avoids the estimate of the joint distribution of the various market risk factors. Our research shows that market risk assessment for competitive market players in a spatial market setting can be facilitated using the Markowitz SV method. In this paper, the model is formulated to minimize the SV of short-term electrical energy portfolios, subject to major practical constraints, transaction costs, wheeling administration, etc. Correlations of product prices across power pools are used for capturing the spatial nature of the pools. The wheeling administration is not conducted through a power flow model but through contracting with independent system operators (ISOs) or regional transmission organizations (RTOs) in an iterative manner. The reason is simple: a market player may not have the necessary information for simulating inter-pool power flows. The competitive bidding strategy is modeled as a lower bound on profit. Since the model is for spatial markets, the lower bound can be used for trading off market risk and expected profit among different markets. Fuel prices are assumed to be given before the short-term energy portfolio selection so that production cost functions are deterministic. Forced outages of plants can be simulated using the Monte Carlo method but will not be conducted in this paper. The proposed risk model is for multiple commodity products that include electricity (real power per unit time),

605

spinning reserve, regulation, etc. Each power pool may have slightly different definitions for these products but we will ignore the difference in definitions for simplifying the problem. 3.1. The mathematical model With the symbols in mind which are listed at the beginning of the paper, the mathematical model is then presented below. r2R

Min

ð10Þ

Subject to ð10:1Þ

CT 6 Mmax; TH X Mk X N X winh linh  CT P ld ; h¼1

i¼1

winh ¼

Ngi X

w1ingh þ

Mk X

w2jinh 

( TH X Mk N Mk X X X h¼1

þ

i¼1

Ngi X

n¼1

w2ijnh

8i; n; h;

j¼1

CP igh U igh þ

ð10:3Þ !

w2ijnh whijnh I ijnh þ winh pcinh þ fcin Ysinh :

j¼1

g¼1

Ngi X

) stcigh U igh ð1  U igh Þ ;

ð10:4Þ

g¼1

WmaxY in P win P win min Y n Ngi X

X

j¼1

g¼1

CT ¼

ð10:2Þ

n¼1

8n 2 Sm;

ð10:5Þ

wingh 6 Pmaxig U igh

8i; g; h;

ð10:6Þ

wi1gh 6 Pminig U igh

8i; g; h;

ð10:7Þ

n¼1 2 X n¼1 Ngi Mk X N Mk X X X ðwinh þ wcinh Þ 6 Pmaxig i¼1

n¼1

Ysinh P Y inh  Y inðh1Þ ; winh 6 Y inh Wmax;

i¼1

8h;

ð10:8Þ

g¼1

Ysinh P 0;

Minimum up and down times of units; Initial conditions:

ð10:9Þ ð10:10Þ ð10:11Þ ð10:12Þ

The initial conditions include the initial status of generation units and committed contracts before h = 1. Production costs are quadratic functions of real power production: CP igh þ ¼ c0ig þ c1ig w1i1gh þ c2ig w12i1gh . Constraint (10.7) may not be required for units that serve only as backup units. In (10.3), (w2jinh  w2ijnh) is the net interpool balancing of product n from the same producer. The optimal solution will guarantee that w2jinh · w2ijnh = 0. Constraint (10.10) is designed to ensure that only one fixed cost is charged for contracting a product in the day ahead markets (this formulation is good for a daily demand profile with only one peak. For multiple peaks of a daily demand profile, a more complicated equation is needed). While other constraints are straight forward, constraint (10.2) deserves more explanation. We know that ld is adjustable for the trade-off of risk and profit. There must be a ld that maximizes the profit for the producer, possibly with a higher risk. The maximum value of ld cannot be

606

Z. Yu / Electrical Power and Energy Systems 29 (2007) 600–608

higher than the product of the total capacity of the producer and the greatest return among the commodity products, minus its total cost. It is clear that the model matches the expected prices of different market products with its operating cost for different risk levels, assuming its decision does not affect market prices (i.e., it is a competitive fringe). The objective function is an extension of Eq. (7) and is defined below. r2R ¼

Mk X mk X N X N X TH X i¼1

j¼1 m¼1

n

winh wjmh SDV inh SDV jmh qijnmh :

h¼1

ð11Þ It is also possible to estimate the time series of the SV processes using the generalized autoregressive hetero-skedastic (GARCH) models. Firstly, let us define the SV for the first hour (h = 1) below: r2R ðh ¼ 1Þ ¼

Mk X mk X N X N X i¼1

win1 wjm1 SDV in1 SDV jm1 qijmn1 ;

j¼1 m¼1 n¼1

ð12Þ where c(h,h1) is the auto-correlation coefficient across time. Assume that the energy product price processes are Markov, the conditional SV can then be approximated by r2R ðh þ 1Þ ffi r2R ðhÞ½1  c2ðhþ1;hÞ  (see [33]). This approximation is consistent with the GARCH(1,1) model (see [4, pp. 187–197]). Sum over all these conditional SVs over the TH time periods, we have another estimate of the SV of the portfolio. The coefficients can be estimated using standard statistical procedures. Note that the estimation of SDVinh, SDVjmh and qijnmh must use a structured data set in history: the data for the same hour of the days in history with similar demand levels. For example, for h = 17, we need to use historical T days’ data at hour 17 for the estimate. However, if we use the data at hour 8 to estimate the correlation coefficient of hour 17, we may lead to bad estimate because of the difference in demand patterns. Small MIP portfolio problems can be solved using the Brunch & Bound or dynamic programming (DP) algorithms. Large problems can be solved using the LaGrange relaxation (LR) or heuristic algorithms (e.g., genetic algorithm) [43]. 4. A case study The case study involves two power pools: the New York power pool (NYPP) and the PJM power pool For illustrative purpose. Due to a lack of enough data for parameter estimation, we only consider two products: the day-ahead energy and spinning reserve in the case study. The energy prices in NYPP are zonal prices of New York City while the spinning reserve prices are NYPP pool wide because there are no zonal spinning reserve prices reported. The energy prices in PJM are PJM–NYPP interface prices and the PJM spinning (operating) reserve prices are also

pool wide, daily average data (no hourly data available). The SDVs of energy and spinning reserve prices are estimated using the pool wide data with daily peak demand levels around 40,000 MW (i.e., ±4% of this number) in PJM and 25,000 MW (±4% as well) in NYPP in summer time (we can call these as stratified or conditional estimate). That is, we assume that the demand levels around 40,000 MW in PJM and 25,000 MW in NYPP coincide in the portfolio horizon (i.e., the next day). This is just for illustrative purpose and other assumptions can also be made for different case studies. Looking at history, the probability is zero for a daily peak of 40,000 MW in PJM to coincide with a daily peak of 25,000 MW in NYPP so we have to use a band of (±4%) for obtaining enough data points. The estimate is done using the structured data of summer 2000 and summer 2001 before August 3 for both markets. The assumed proportional transaction costs are listed in Table 2 and the estimated parameters are listed in Table 4. Some data, such as qiinnh = qjjnnh = 1 and qijmnh = qjinmh, are not listed for saving space. Furthermore, the producer is assumed to have two peaking units located in PJM but close to the eastern NYPP, each with a capacity of 100 MW and minimum up/down times of one hour (see Table 3). It is also assumed that wcinh ¼ 0, a $2/MWh and a $1/MWh are charged for inter-pool wheeling and intrapool transmission usage respectively, both for energy but not for spinning reserve. Lastly, it is assumed that interpool wheeling is feasible for the case study. Only peak hours, 13–18, are used for the case study because the two generators are peaking units. However, the model is general for any time horizon. The major results are summarized in Table 5. As it is expected, the SV of the portfolio remains constant for a wide range of the adjustable values of ld, from zero to about $1050. This is caused by the fact that no generation unit can sell any spinning reserve before it produces energy at the minimum production level. It is well known that the Markowitz mean–variance efficient frontier is often smooth for a portfolio with continuity. However, the Markowitz MSV efficient frontier of this case study is neither smooth Table 2 Assumed transaction costs ($) pc11h

pc21h

fc11h

fc21h

Others

1

1.1

50

60

0

Table 3 Generation unit data

c0 ($) c1 ($/MWh) c2 ($/MWh/MW) Minimum up/down time

Unit 1

Unit 2

30 18 .002 1/1

40 20 .002 1/1

Z. Yu / Electrical Power and Energy Systems 29 (2007) 600–608 Table 4 Parameters estimated

607

MSV: Semivariance 154200

Hour

13

14

15

16

17

18

l11h l12h l21h l22h SDV11h SDV12h SDV21h SDV22h q1112h q1211h q1212h q1221h q1222h q2212h

70.4 0.37 110 4.2 30.3 0.14 36.1 1.96 0.1 0.92 0.4 0.2 0.19 0.33

88.5 0.37 135.4 4.5 38.7 0.14 44.34 2.26 0.19 0.99 0.47 0.22 0.22 0.39

63 0.37 102.7 5.46 16.2 0.14 22.6 3.44 0.005 083 0.77 0.15 0.15 0.75

62.3 0.37 98.6 5.43 16.1 0.14 20.2 3.43 0.023 0.844 0.74 0.19 0.185 0.74

56.2 0.37 86.8 5.22 13.6 0.14 15.7 3.34 0.006 0.84 0.68 0.05 0.054 0.69

51.1 0.37 70.4 4.03 13.2 0.14 13.4 0.96 0.05 0.94 0.2 0.01 0.01 0.30

MSV: Semivariance

154150 154100 154050 154000 153950 153900 153850 153800

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Mean points

nor concave due to the effect of various constraints such as the minimum production levels and fixed costs. Noting that only 15 semivariance calculations are listed in Table 5 in the interest of brevity, and a plot of the points is provided in Fig. 1. In the figure, the ‘‘mean point 1’’ represents the actual mean return of 10, and so on. We believe that these 15 data points are enough for illustrating the nature of the model. For these 15 data points, only unit one is committed with a minimum production of 30 MW because its production cost is lower than the other generation unit. The remaining capacity of unit one is offered largely to the NYPP markets as spinning reserve because of the negative correlations between the prices of energy in PJM and the prices of spinning reserve in NYPP. This negativity of correlation between product prices in the markets tends to reduce the SV of the portfolio, as pointed out by Markowitz and many other authors. In order to compare with the MSV efficient frontier, the MV variance calculations from [43] are also listed in Table 5. As can be seen, the average magnitude of the variances is much greater than that of the semivariances, which may

Table 5 Sample efficient frontier results Runs

Mean return ld

MV: variance r2R

MSV: semivariance r2R

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

10 100 500 940 950 1000 1010 1020 1040 1050 1060 1070 1080 1090 1100

736,146 736,146 736,146 736,170 736,322 736,322 736,590 737,042 737,042 739,082 739,082 739,082 739,082 739,082 739,082

153,948 153,948 153,948 153,948 153,948 153,948 153,948 153,948 153,948 153,950 153,963 153,990 154,031 154,087 154,156

Fig. 1. A plot of the sample portfolio semivariances.

indicate that the MV risk measure is overly stated. From Table 5, we can see there are flat segments for both the MV and the MSV frontiers, e.g., the SV remains constant for mean returns from 10 to 1040. As we discussed earlier in the paper, this would reduce the portfolio decision choices because only the right end point of each flat segment can be regarded as ‘‘efficient’’ – it simply gives the largest mean/risk ratio. Compared with the concave and continuous MV (CCMV) efficient frontier, where for the same range of mean returns corresponding to the flat segment (say, 10–1040), the choices for picking up a portfolio are infinite on the CCMV efficient frontier. In this sense, the MIP portfolio can be much more easily applicable and successful in practical applications. 5. Conclusions The paper presents a spatial energy market risk model based on the Markowitz MSV method. The model is for assessing the risk of profit making of power producers in a multi-pool market setting. The model also includes practical constraints such as transaction costs and wheeling contracts, leading to a mixed integer formulation. The case study shows the successful application of the model. An interesting observation is that the Markowitz MSV efficient frontier is neither smooth nor concave mainly due to the addition of those physical constraints of the plants and fixed costs. References [1] Markowitz HM. Portfolio selection, efficient diversification of investment. New York: Wiley; 1959. [2] Markowitz HM. Portfolio selection. J Financ 1952;7:77–91. [3] Sharpe WF. Capital market prices: a theory of market equilibrium under conditions of risk. J Financ 1964;19:425–42. [4] Jorion P. Value at risk. 2nd ed. New York: McGraw-Hill; 2000. p. 167–8.

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