Andrew B. Philpott

Optimal Offer Construction in Electricity Markets

In this paper we study strategies for generators making offers into electricity markets in circumstances where both the demand for electricity and the behaviour of competing generators is unknown but can be represented by a probability distribution. Given this probability distribution, we derive necessary optimality conditions for a broad class of supply offer curves. We show how these conditions can be used to construct an optimal solution for a simple example. We also consider the case in which a generator is restricted in the number of prices at which power can be offered.

Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion

We consider the incorporation of a time-consistent coherent risk measure into a multi-stage stochastic programming model, so that the model can be solved using a SDDP-type algorithm. We describe the implementation of this algorithm, and study the solutions it gives for an application of hydro-thermal scheduling in the New Zealand electricity system. The performance of policies using this risk measure at different levels of risk aversion is compared with the risk-neutral policy.

On the convergence of stochastic dual dynamic programming and related methods

We discuss the almost-sure convergence of a broad class of sampling algorithms for multistage stochastic linear programs. We provide a convergence proof based on the finiteness of the set of distinct cut coefficients. This differs from existing published proofs in that it does not require a restrictive assumption.

Using Supply Functions for Offering Generation into an Electricity Market

In this paper, we study strategies for generators making offers into electricity markets in circumstances where demand is unknown in advance. We concentrate on a model with smooth supply functions and derive conditions under which a single supply function can represent an optimal response to the offers of the other market participants over a range of demands. In order to apply this approach in practice, it may be necessary to approximate the supply functions of other players. We derive bounds on the loss in revenue that occurs in comparison with the exact supply function response, when a generator uses an approximation both for its own supply function and for the supply functions of other players. We also demonstrate the existence of symmetric supply-function equilibria.

Optimizing demand-side bids in day-ahead electricity markets

We present a model of a purchaser of electricity in Norway, bidding into a wholesale electricity pool market that operates a day ahead of dispatch. The purchaser must arrange purchase for an uncertain demand that occurs the following day. Deviations from the day-ahead purchase are bought in a secondary market at a price that differs from the day-ahead price by virtue of regulating offers submitted by generators. Under an assumption that arbitrageurs are absent in these markets, we study conditions under which the purchaser should bid their expected demand and examine the two-period game played between a single generator and purchaser in the presence of a competitive fringe. In all our models, it is found that purchasers have an incentive to underbid their expected demand, and so the day-ahead prices will be below expected real-time prices. We also derive conditions on the optimal demand curve that purchasers should bid if the behavior of the other participants is unknown but can be modeled by a market distribution function.

A Single-Settlement, Energy-Only Electric Power Market for Unpredictable and Intermittent Participants

We discuss a stochastic-programming-based method for scheduling electric power generation subject to uncertainty. Such uncertainty may arise from either imperfect forecasting or moment-to-moment fluctuations, and on either the supply or the demand side. The method gives a system of locational marginal prices that reflect the uncertainty, and these may be used in a market settlement scheme in which payment is for energy only. We show that this scheme is revenue adequate in expectation.

Hydroelectric reservoir optimization in a pool market

Abstract.For a price-taking generator operating a hydro-electric reservoir in a pool electricity market, the optimal stack to offer in each trading period over a planning horizon can be computed using dynamic programming. However, the market trading period (usually 1 hour or less) may be much shorter than the inherent time scale of the reservoir (often many months). We devise a dynamic programming model for such situations in which each stage represents many trading periods. In this model, the decision made at the beginning of each stage consists of a target mean and variance of the water release in the coming stage. This decomposes the problem into inter-stage and intra-stage subproblems.

Hydro-electric unit commitment subject to uncertain demand

Abstract We consider the problem of scheduling daily hydro-electricity generation in a river valley. Each generating station in this river valley has a number of turbines which incur fixed charges on startup and have a generation efficiency which varies nonlinearly with flow. With appropriate approximations the problem of determining what turbine units to commit in each half hour of the day can be formulated as a large mixed-integer linear programming problem. In practice the generation required from this group of stations in each half hour is often different from that forecast. We investigate the impact of this uncertainty on the unit commitment by using an optimization-based heuristic to give an approximate solution to the stochastic problem.

Inexact Cuts in Benders Decomposition

Benders decomposition is a well-known technique for solving large linear programs with a special structure. In particular, it is a popular technique for solving multistage stochastic linear programming problems. Early termination in the subproblems generated during Benders decomposition (assuming dual feasibility) produces valid cuts that are inexact in the sense that they are not as constraining as cuts derived from an exact solution. We describe an inexact cut algorithm, prove its convergence under easily verifiable assumptions, and discuss a corresponding Dantzig--Wolfe decomposition algorithm. The paper is concluded with some computational results from applying the algorithm to a class of stochastic programming problems that arise in hydroelectric scheduling.

Continuous-time flows in networks

We consider a class of maximum flow problems formulated in a directed network where the arc flows vary as Lebesgue-measurable functions of time, and storage is allowed at the nodes of the network. A max flow-min cut result of Anderson, Nash and Philpott [1] is extended to cover the case where each arc has a traversal time. The ideas of the paper are then applied to derive some simple results on emptying networks at least cost.