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Article Excerpt
The market distribution function is a probabilistic device that can be used to model the randomness in dispatch and clearing price that generators in electricity-pool markets must take account of when submitting offers. We discuss techniques for estimating the market distribution function, and ways of measuring the quality of these estimators, using both classical statistical approaches and an expected-foregone-revenue approach.
Key words: market distribution function; electricity market; estimation
MSC2000 subject classification: Primary: 62G05, 91B26; secondary: 90B50, 91B70
OR/MS subject classification: Primary: economics--econometrics, industries- electric; secondary: probability-stochastic model applications, statistics-nonparametric
History: Received May 6, 2005; revised December 12, 2005.
1. Introduction. A common form of electricity market structure that has emerged in recent years is the electricity-pool market. Examples are the Pennsylvania-Jersey-Maryland (PJM) market, the Nordpool, and the electricity markets in New Zealand and Australia. In electricity-pool markets, generators submit offers of generation to a central system operator, who dispatches some of the offered generation in such a way as to minimize the total cost of the power generated. Each generation offer takes the form of a supply function or offer stack, which specifies the price at which a generator is prepared to offer a certain amount of power to the market. Although the exact form of the offer stack varies with the market design, typically these can be represented by nondecreasing piecewise-constant functions, consisting of a finite number of tranches of power.
The prices at which the generated power is offered are at the discretion of the generator. If the price asked is too high, then they are unlikely to be dispatched and so will earn no revenue. On the other hand, the price should not under normal circumstances be chosen to be less than the marginal cost of generation. In a perfectly competitive environment, in which the choice of offer stack from an individual generator has no effect on the price, a stack that offers power at its marginal cost maximizes the profit of the generator. However, most electricity markets have a small number of large generating companies whose offers have an effect on the clearing price and quantity that they are dispatched. In this setting, generators are faced with the question of what offer stack to submit in order to maximize their profit.
With perfect information on the demand for electricity, and the offers of other generators, it is possible for a generator to compute a stack that maximizes its profit. For some simple cases it is possible to compute Nash equilibria for the one-shot game where generators each choose an optimal supply function to submit, assuming that the others do not change their (optimal) offers. Although of interest in a theoretical sense, supply-function equilibria are extremely difficult to compute for all but the simplest models, and are of limited use in practical hour-by-hour trading operations.
The optimal stacks to submit to the market on an hour-by-hour basis will be computed with little information on the offers of the other generators, and with imperfect forecasts of demand. This makes the profit R associated with an offer stack a function of the random quantity Q that the generator is dispatched, as well as the (random) clearing price P. In this circumstance the generator might seek an offer stack to maximize the expectation of R with respect to some probability distribution.
An approach to this problem is described in Anderson and Philpott [1], where a market distribution function [PSI](q, p) is defined as the probability of a generator not being fully dispatched if it were to offer a single quantity q at price p. In that paper, it is shown that the expected profit from offering a stack defined by a curve s is
E[R] = [[integral].sub.s] R(q, p)d[PSI](q, p).
As an aside, we note that the market distribution function is not, as one might expect from the name, solely a property of the market. Rather, it describes the market situation faced by a particular generator. Thus, there is a different market distribution function for each generator.
The market distribution function in Anderson and Philpott [1] is assumed to be continuous. Under further smoothness assumptions on [PSI] it is possible to derive optimality criteria for s. In Anderson and Xu [3], the theory is extended to the case of discontinuous [PSI]. For this case, the existence of an optimal offer s is not guaranteed, and [member of]-optimal solutions are sought instead. The present paper mostly considers this more general case, in which market distribution functions will not be assumed to have any continuity or smoothness properties except where this is explicitly stated.
In this paper we develop a methodology based on maximum-likelihood estimation for constructing estimates of [PSI](q, p) from historical dispatch data. The layout of the paper is as follows. In the next section, we provide an intuitive derivation of the market distribution function, using an approach that makes the estimation procedure straightforward. In [section]3 we describe the estimation technique, and in [section]4 we study the classical statistical properties of this estimator. In particular, we show that our estimator is consistent, asymptotically unbiased, and obeys a central limit theorem. Some numerical investigation of the performance of the estimator is conducted in [section]5.
2. Market distribution functions. Market distribution functions have been formally defined in Anderson and Philpott [1]. In this paper, we adopt a slightly different formulation that will provide the structure to make a statistical analysis more straightforward.
In Anderson and Philpott [1], an offer stack is modelled as a parameterized curve. We will regard an offer stack s as being defined by a continuous planar offer curve: a connected, totally ordered (with respect to the order ([q.sub.1], [p.sub.1]) [greater than or equal to] ([q.sub.2], [p.sub.2]) [??] [q.sub.1] [greater than or equal to] [q.sub.2] and [p.sub.1] [greater than or equal to] [p.sub.2]), subset of the plane. That is, an increasing curve. It will often be convenient, if x and y are points on the offer curve s, to write x [less than or equal to] y if they are so ordered. An offer stack is an offer curve consisting of a union of horizontal and vertical lines.
For a given offer curve s, the distribution of the point of dispatch (Q, P) on s can be described by its cumulative distribution function [[PSI].sub.s]. That is, [[PSI].sub.s](q, P) = P((Q, P) [less than or equal to] (q, p)) for all (q, p) on s. The problem of optimizing expected generator returns then becomes:
(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].
Fortunately, in many circumstances, including the setting of (a single trading period in) an electricity market, it is possible to simplify the description of the dispatch distribution by specifying a single function [PSI], defined on the whole (q, p) plane, such that [[PSI].sub.s](q, P) = [PSI](q, P) for every s and (q, p) on s. This is clearly desirable, as it would simplify the optimization problem (1) to
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].
The function [PSI] is easily recognized to be the market distribution function of Anderson and Philpott [1].
The extension of [[PSI].sub.s] to the whole (q, p) plane is made possible by the following argument. Suppose the market chooses points of dispatch in the following way. First, a random nonincreasing right-continuous function V(q) is generated. The point of dispatch (Q, P) chosen for any offer curve s will then be the point where s intersects the graph of p = V(q). (The total order on s allows us to write this more precisely as
(Q, P)= arg min{(q, p) on s |V(q) [less than or equal to] p}.
The minimum is achieved because {(q, p) | V(q) [less than or equal to] p} is a closed set. This...
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