مدل برنامه ریزی عدد صحیح مختلط میانگین واریانس فضایی برای تجزیه و تحلیل ریسک بازار انرژی

The paper presents a short-term market risk model based on the Markowitz mean-variance method for spatial electricity markets. The spatial nature is captured using the correlation of geographically separated markets and the consideration of wheeling administration. The model also includes transaction costs and other practical constraints, resulting in a mixed integer programming (MIP) model. The incorporation of those practical constraints makes the model more attractive than the traditional Markowitz portfolio model with continuity. A case study is used to illustrate the practical application of the model. The results show that the MIP portfolio efficient frontier is neither smooth nor concave. The paper also considers the possible extension of the model to other energy markets, including natural gas and oil markets.

There have been many studies on risk and portfolio selection based on the Markowitz mean-variance (MV) method (Markowitz, 1952 and Markowitz, 1959). Due to its robustness and ease of implementation (i.e. using Quadratic Programming), the MV method has also found wide applications in the financial industry worldwide. Despite the fact that semivariance (SV) has been used as a risk measure (e.g., Markowitz, 1959 and Porter, 1974), the MV method is still a powerful tool because of its computational advantage. Besides, there is only a minor difference between the efficient frontiers of the MV and SV portfolios (Michaud, 1998). As pointed out by Markowitz, the successful application of the SV method depends on finding the joint probability distribution of a portfolio and this has been a serious computational problem. As a result, only approximate methods are used for estimating the portfolio semivariance (Markowitz et al., 1993 and Nawrocki, 1991). The MV method is not perfect—it cannot take into consideration fixed transaction fees and other fixed costs. As a result, the MV method can only achieve sub-optimal solutions in portfolio selection and risk quantification. Portfolio models based on MV often overstate diversification and the problem is partly induced by ignoring fixed transaction costs. The CAPM (Capital Asset Pricing Model—Sharpe, 1964) and the APT (Arbitrage Pricing Theorem—Ross, 1976) models also have the same problem. Fortunately, there have been attempts to resolve the problem. Proportional transaction costs have been discussed by many authors (e.g. Pogue, 1970 and Chen et al., 1971). For portfolio risk analysis considering option pricing and transaction costs, Leland (1985) introduced the notion of a break-even volatility and the idea was furthered by Whalley and Wilmott (1993). However, all these studies do not incorporate integer formulations. The MIP models are the most appropriate choice for portfolio risk analysis when fixed transaction costs and minimum transaction lots are considered. In fact, there have been a few portfolio studies with MIP formulations. For example, Bertsimas et al. (1999) described an MIP portfolio model based on the maximization of expected returns, Kellerer et al. (2001) discussed an MIP model with fixed transaction costs and the minimum transaction lots. However, no MIP portfolios, with a market risk formulation, have been used for spatial energy markets and this makes our study very meaningful. In the following, we will first focus our portfolio risk study on short-term electricity markets with a special attention paid to a spatial market setting and will discuss the extension of the model to other energy markets. 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 (Billinton and Allan, 1996). The risk in power production costing due to load uncertainty and plant availability has also been addressed extensively (Breipohl et al., 1992 and Douglas et al., 1998). 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 these risks has yet to be researched. Producers need to assess different market risks in the deregulated environment. For example, a large power producer in a region may assume an oligopolistic strategy. The producer would face the risk that other producers may not use the same oligopolistic strategy and may be undercut by its opponents. On the other hand, a small producer or a competitive fringe may seldom adopt a gaming strategy (Hogan, 1997) and it may primarily be concerned with the risk associated with market prices. Note that in economic literature a competitive fringe assumes that its decision would not affect market prices. This paper focuses on modeling the short-term, spatial market risk for fringe producers or bigger producers without adopting oligopolistic strategies. There have been several academic studies on non-spatial risk assessment for deregulated electricity markets. For example, Andrews, 1995, David, 1996 and Siddiqi, 2000 addressed power project valuation risk. Bjorgan et al. (1999) analyzed market risk using Pareto optimality for the trade-off between the expected value and variance. Sheblé (1998) presented a method of risk analysis based on the decision tree method. In practice on the other hand, 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, the MV, the VaR (Value-at-Risk—Jorion, 2000) and other methods have been under investigation and in use. Nonetheless, the spatial nature of the electricity market risks has not explicitly modeled either in academics or in practice. Compared with non-spatial risk models, the spatial model can be more efficient in diversifying market risks. Market risk analysis based on a joint probability distribution of several risk factors can be very demanding. These risk factors may include load forecast errors, generation availability, the behavior of market players (e.g. market gaming), etc. Even with perfect weather forecasts, load can still be uncertain due to randomness of consumption (Yu et al., 1996a and Yu et al., 1996b). This paper proposes an approach that avoids the calculation of the joint distribution of the various market risk factors. Our research shows that market risk assessment for competitive producers in a spatial market setting can be facilitated using the Markowitz MV method. In this paper, the model is formulated to minimize the variance of a short-term electrical energy portfolio, 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. Since the model is for spatial markets, a lower bound on net profit is used for trading off market risk and expected profit among different markets. Fuel prices are first assumed to be given before the short-term energy portfolio selection so that the production cost functions are deterministic and the model will be extended to the case when fuel prices are random. 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), spinning reserve, regulation, etc. Each power pool may have slightly different definitions for the products. However, the definitions are close enough for power pools that are next to each other geographically, such as the New York Power Pool (NYPP) and the Pennsylvania–Jersey–Maryland (PJM) Power Pool. A case study is conducted based on the market data of NYPP and PJM. The results show that the MIP portfolio efficient frontier has flat segments due to the inclusion of the production and transaction minimum lots. In short, the efficient frontier is neither smooth nor concave compared with the Markowitz MV efficient frontier with continuity. The proposed model is a general mathematical formulation that should not be confused with any specific application model with certain parameters estimated from historical data. The use of the model in practice can be tricky because the individual variances, such as the variances of market prices used for building up the portfolio variance, are often not exactly known in advance. A common practice is the estimation of the variances based on recent historical data. A drawback of using historical data for estimating portfolio variance is that the method does not provide a measure of absolute worst loss. Exceptions and accidences can occur. Model evaluation, such as back testing, should be conducted whenever possible. In particular, when there is a major structure change in market conditions, the individual variances estimated from historical data may not be good for building up the portfolio variance. In other words, the model based on historical data may not be suitable for the situations with major market structure changes. The structured estimation used in this paper can partially resolve the problem but with limited usefulness due to the small sample size. More samples will be available as time proceeds. In general, stress testing, coupled with Monte Carlo simulation can be used for model evaluation and stability analysis. For example, different stress scenarios can be created. Monte Carlo simulations can be done for each scenario with variance estimates for future use. However, this paper does not intend to go into details of stress testing because our focus is the general mathematical formulation. There are five major sections of the paper. Section 2 presents the risk model for short-term electricity markets with detailed explanations of the major constraints. Section 3 discusses potential algorithms for solving the problem. Section 4 illustrates the use of the model via a case study. Section 5 discusses the extension of the model to include random fuel prices and the extension of the model to other energy markets. Finally, Section 6 concludes the paper.

This paper presents a short-term, spatial market risk model based on the Markowitz MV method for deregulated electricity markets. The model is for assessing the risk of profit making of competitive power producers in a multi-pool market setting. The spatial nature of the problem is captured through the use of inter-pool correlation and wheeling administration. The model also includes practical constraints such as transaction costs and wheeling contracting, leading to a MIP formulation. The case study shows the successful application of the model. An interesting observation is that the Markowitz MV efficient frontier is neither smooth nor concave mainly due to the addition of those physical constraints of the plants and fixed costs. The model can readily be extended to other energy markets with certain modifications as discussed in the paper.