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.

We discuss the almost-sure convergence of a broad class of sampling algorithms for multi-stage 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.

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 di¤erent levels of risk aversion is compared with the risk-neutral policy.

We prove the almost-sure convergence of a class of sampling-based nested decomposition algorithms for multistage stochastic convex programs in which the stage costs are general convex functions of the decisions and uncertainty is modelled by a scenario tree. As special cases, our results imply the almost-sure convergence of stochastic dual dynamic programming, cutting-plane and partial-sampling (CUPPS) algorithm, and dynamic outer-approximation sampling algorithms when applied to problems with general convex cost functions.Keywords: stochastic programming; dynamic programming; stochastic dual dynamic programming algorithm; Monte-Carlo sampling; Benders decomposition MSC2000 subject classification: Primary: 90C14; secondary: 90C39 OR/MS subject classification: Primary: stochastic programming; secondary: dynamic programming Having general convex stage costs does not preclude the use of cutting plane algorithms to attack these problems. Kelley's cutting plane method (Kelley [7]) was originally devised for general convex objective functions, and can be shown to converge to an optimal solution (see, e.g., Ruszczyński [14, Theorem 7.7]), although on some instances this convergence might be very slow (Nesterov [9]). Our goal in this paper is to extend the convergence result of Ruszczyński [14] to the setting of multistage stochastic convex programming.The most well-known application of cutting planes in multistage stochastic programming is the stochastic dual dynamic programming (SDDP) algorithm of Pereira and Pinto [10]. This algorithm constructs feasible dynamic programming (DP) policies using an outer approximation of a (convex) future cost function that is computed using Benders cuts. The policies defined by these cuts can be evaluated using simulation and their performance measured against a lower bound on their expected cost. This provides a convergence criterion that may be applied to terminate the algorithm when the estimated cost of the candidate policy is close enough to its lower bound. The SDDP algorithm has led to a number of related methods (Chen and Powell [1], Donohue [2], Donohue and Birge [3], Hindsberger and Philpott [6], Philpott and Guan [11]) that are based on the same essential idea but that seek to improve the method by exploiting the structure of particular applications. We call these methods DOASA (dynamic outer-approximation sampling algorithms), but they are now generically named SDDP methods.SDDP methods are known to converge almost surely on a finite scenario tree when the stage problems are linear programs. The first formal proof of such a result was published by Chen and Powell [1], who derived this for their cutting-plane and partial-sampling (CUPPS) algorithm. This proof was extended by Linowsky and Philpott [8] to cover other SDDP algorithms. The convergence proofs in Chen and Powell [1] and Linowsky and Philpott [8] make use of an unstated assumption regarding the independence of sampled random variables and convergent subsequences of algorithm iterates. This assumption was identifi...

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