Risk analysis of bidding strategies in an electricity pay as bid auction: A new theorem

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Abstract

Considering the uncertainties in the power market, the bidding problem has an important role for the power supplier to reach his goals, and using risk management methods to protect against the market risk is unavoidable. Thus, in this paper, the bidding decision making problem is formulated from a supplier’s viewpoint in a spot market. The spot market works based on a pay as bid auction or a discriminatory price auction. The market clearing price (MCP) is uncertain, and we consider a probabilistic model for it. Regarding the literature of the bidding problem and forecasting methods of MCP, a normal probability density function, pdf (N(μm, σm)), is a proper distribution for the MCP. The statistical parameters of MCP vary on different times (peak and off peak), and considering the concept of supplier risk, their effects on the supplier expected benefit and expected sell from selling energy will be discussed analytically. An important section of this research work concerns the optimal bidding strategy when μm and σm vary in different conditions of the power market. Thus, the coefficient of variation index (CV), as a proper measure, mathematically defined as σm divided by μm, is introduced to measure the market risk index. In this paper, the CV index is used to analyze and manage the supplier risk and introduce the optimal strategy. Then, for a constant amount of the CV as a theorem, it is proved that: (1) the maximum of the expected sell occurs at a constant level of the supplier risk and (2) the optimal bid price linearly depends on the standard deviation of the MCP. This theorem is generalized for the case that the expected value of the supplier benefit is considered as an objective function in the bidding process. Some numerical examples are presented, and application of the proposed theorem is discussed.

Introduction

Electricity utility industries of many countries are undergoing a fundamental transformation from regulated and monopolistic to a deregulated and competitive industry. The economic restructuring of the power industry has been bringing new problems to the operation and planning of power systems. In this environment, electricity as a commodity can be sold or purchased through the spot market or bilateral transactions. The core of the deregulation is introducing competition among power suppliers. In the new regime, in contrast to the regulated environment, individual suppliers and transmission companies have to make decisions independently and with incomplete knowledge about their competitors. From a supplier’s viewpoint, pricing electricity is a major issue, and he is able to increase his profit by choosing a suitable strategy in the bidding process.

Because of various uncertainties in the power market such as load variation and rival behavior, bidding strategies should be designed and analyzed. Based on these strategies, the supplier could make optimal decisions and control the risk in the uncertain and variable environment of the market.

In recent years, some researches have been done on bidding strategies for competitive suppliers. In Ref. [1], a dynamic model for strategic bidding is proposed. In this dynamic model, three generators using current available information about the MCP exercise their market power to maximize profit in the repetition game. The optimal bidding strategy based on a linear supply/demand function model is utilized to maximize social welfare [2]. In Refs. [3], [4], [5], the bidding strategy is formulated as a two level optimization problem where, at the first level, a market participant tries to maximize his profit, and at the second level, the independent system operator, ISO, solves the optimal power flow problem to minimize the total system cost. In Ref. [6], generation scheduling and suppliers’ bidding are formulated and solved using the Lagrangian relaxation method. Monte Carlo simulation for calculating the expected benefit and optimization formulation to maximize it are proposed for solving the bidding problem in Ref. [7]. In Ref. [8], the bidding decision making problem and the short term generation scheduling are combined and solved in the electricity market where the objective function is defined as the multiplication of the bid price acceptance probability and positive profit probability. The MCP is modeled as normal probability distribution function (pdf).

The concept of risk and decision making in an uncertain environment is not new to the power system. However, new challenges arise in the power market operation. In this uncertain environment, to solve the bidding decision making problem, it is necessary to assess risk from the supplier’s viewpoint. In Ref. [9], by obtaining a probability distribution function for the MCP, the market risk is analyzed. A state of the art summary of risk assessment in the energy market is reported in Ref. [10], and applications of Value-at-Risk, conditional Value-at-Risk and hedging technique using market scenarios are illustrated. In Ref. [11], to measure and manage market risk, a method based on stochastic optimization is developed. A risk model in Ref. [12] is proposed to assess risk considering transaction costs and other practical constraints where the semi-variance (SV) as a market risk index is introduced. A self scheduling model that considers simultaneous profit and risk is proposed in Ref. [13] and is formulated as a mixed integer programming problem. In Ref. [14] considering the pdf of MCP, the bidding problem is solved analytically based on two different supplier risk criteria i.e. variance of the supplier’s benefit and acceptance probability of the bid price. The bidding problem is generalized to consider a multi-step supplier bid function. The optimal bidding strategies corresponding to both risk indexes are compared numerically.

The power market in Iran was established in October 2003. The pricing rule in this power market is based on the pay as bid (PAB) auction or discriminatory auction. In this auction, in sharp contrast with the uniform auction, generators do not receive the same price. In other words, the owner of each accepted bid receives as his price the amount of his offer. The ith bid is accepted if its corresponding price (ρi) is smaller than the MCP. Because of uncertainties in the power market such as load fluctuations and competitors’ behavior, the MCP is considered to be a random variable [9]. Thus, the bidding problem is affected by the MCP risk, and there is some extent of risk for the supplier to accept his bid.

In this paper, the bidding decision making problem from the supplier’s viewpoint and risk analysis are studied and formulated in a PAB auction. In Section 2, the probabilistic nature of the MCP and the effects of different market parameters on it are discussed. Furthermore, as a proper model, a normal distribution function for the MCP is proposed. In Section 3, a mathematical method is developed to determine the expected benefit and the expected sell for supplier i versus his bid price and corresponding risk level. To analyze the supplier’s risk and compare various strategies for different time intervals, a new approach is developed. In this approach, different market uncertainties are aggregated in the pdf parameters of the MCP. Therefore, μm and σm can vary in different time intervals. Thus, in Section 4, the effects of these parameters on the expected sell of the ith bidder using the coefficient of variation index, called CV index, as a market risk measure, are studied. Also, in this section, two applicable results in supplier risk analysis are obtained. As a theorem, it is proved for a constant amount of CV index: (1) the maximum of the expected sell occurs at a constant level of supplier risk and (2) the optimal bid price linearly depends on the standard deviation of the MCP. This theorem is generalized for the case that the expected benefit is considered as a supplier’s objective function. In Section 5, some applicable results of the proposed theorem to find the optimal risk strategy for the supplier in different conditions of the power market and to analyze the supplier risk for multi-step bidding are illustrated. The relevant conclusions are presented in Section 6.

Section snippets

MCP uncertainty and modeling

The MCP is an important factor in the power market. It depends not only upon the system supply and demand but also upon the participants’ behavior. Thus, due to inherent uncertainties in the power market, such as the demand for electricity, contingencies in the power system and competitors’ behavior, the MCP will fluctuate frequently and intensively. The uncertain fluctuation of MCP exposes every participant to some extent of risk, which should not be ignored because it directly affects the

Bidding strategy

In this section, the bidding strategy is formulated for a power supplier in the electricity market. The supplier submits his bid with the price (ρ) and quantity (G) at which he is going to sell electricity for each hour of the next day. The bids are received by the system operator, whose duty is to dispatch the generators from the lowest offers, considering the technical constraints, until the market demand is supplied. The price of the last winning unit is defined as the market clearing price

Power supplier’s risk analysis

Because of the market uncertainties, each market participant is exposed to market risk. One of the important parts of the market risk is price risk, or precisely, MCP risk. Based on the discussion in Section 2, the MCP is considered as a normally random variable with two statistical parameters, i.e. μm and σm. Regarding load variation and change of participants’ behavior in different periods, the mean value of MCP changes according to the supply and demand balance. In addition, the standard

Results

The suppliers have different abilities to accept risk. They are often categorized as risk indifferent, risk averse or risk taker. According to these three categories, just as an example the risk axis is divided equally into three intervals, as Fig. 4 represents. The risk averse supplier likes to bid low price and takes a few risk levels in interval [0, 0.33]. The risk indifferent supplier bids prices higher than the risk averse supplier, and as a result, his risk level increases in interval

Conclusion

In this paper, the decision making problem is studied from the supplier’s viewpoint using risk analysis in an electricity PAB auction. By modeling the MCP as a normally random variable and introducing the CV index, the power supplier can assess and measure the MCP risk. Regarding load variation and change of participants’ behavior in different periods, the mean value of the MCP changes according to the supply and demand balance. In addition, the standard deviation of the MCP for different

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