We present JuDGE.jl, an open-source Julia package for solving multistage stochastic capacity expansion problems using Dantzig-Wolfe decomposition. Models for JuDGE.jl are built using JuMP, the algebraic modelling language in Julia, and solved by repeatedly applying mixed-integer programming. We illustrate JuDGE.jl by formulating and solving a toy knapsack problem, and demonstrate the performance of JuDGE.jl on problems of increasing size in comparison with a deterministic equivalent model.

This paper studies risk in the context of a stochastic auction designed to facilitate the integration of renewable generation in electricity markets. We model market participants who are risk averse, when their risk aversion is reflected through coherent risk measures. We uncover a closed form characterization of the optimal pre-commitment behaviour for a given real-time policy, with arbi- trary risk aversion: when participants cannot trade risk, generators provide less pre-commitment than when participants are risk-neutral, alternatively, when participants trade a rich set of financial instruments, generators provide more pre-commitment than when they are risk-neutral.

In this paper we take a heuristic dynamic programming approach to solve a multistage stochastic optimization of energy consumption for a large manufacturer who is a price-making major consumer of electricity. We introduce a mixed-integer program that co-optimizes consumption bids and interruptible load reserve (ILR) offers for such a major consumer over a finite time horizon. By utilizing Lagrangian methods, we decompose our model by approximately pricing the constraints that link the stages together. We construct look-up tables in the form of consumption-utility curves, which our model uses to determine optimal consumption levels. We also present heuristics, in order to tackle the non-convexities within our model, and improve the accuracy of our policies. In the second part of the paper, we present stochastic solution methods for our model, with both stage-wise dependent and independent uncertainty. In addition, we reduce the size of our model's scenario tree by utilizing a tailor-made scenario clustering method. Furthermore, we conduct an experiment for a major consumer in the New Zealand Electricity Market and present numerical results.

We present a new algorithm for solving linear multistage stochastic programming problems with objective function coefficients modeled as a stochastic process. This algorithm overcomes the difficulties of existing methods which require discretization. Using an argument based on the finiteness of the set of possible cuts, we prove that the algorithm converges almost surely. Finally, we demonstrate the practical application of the algorithm on a hydro-bidding example with the spot-price modeled as an auto-regressive process.

In this paper, we present an algorithm for solving multistage stochastic minimax dynamic programmes. We begin by presenting linear-programming upper and lower bound representations of saddle functions -- extending outer representations from traditional convex cutting plane algorithms. Our algorithm is similar to many stochastic dual dynamic programming (SDDP) type algorithms. Bounding functions are iteratively updated in order to compute cost-to-go functions; however special consideration must be taken in ensuring the validity of the cost-to-go bounding functions (which are now saddle functions) over the domain. We apply the theory developed in this paper to multistage risk-averse optimisation. Through the dual representation of coherent risk measures, we represent risk-averse stochastic optimisation problems as minimax stochastic dynamic programmes, and provide two formulations for these problems: the nested risk formulation, and the end-of-horizon formulation. Finally, we demonstrate our algorithm on a risk-averse portfolio optimisation problem; providing numerical results and a discussion.

Multistage stochastic programming problems are an important class of optimisation problems, especially in energy planning and scheduling. These problems and their solution methods have been of particular interest to researchers in stochastic programming recently. Because of the large scenario trees that these problems induce, current solution methods require random sampling of the tree in order to build a candidate policy. Candidate policies are then evaluated using Monte Carlo simulation. Under certain sampling assumptions, theoretical convergence is obtained almost surely. In practice, convergence of a given policy requires a statistical test and is only guaranteed at a given level of confidence. In this paper, we present a deterministic algorithm to solve these problems. The main feature of this algorithm is a deterministic path sampling scheme during the forward pass phase of the algorithm which is guaranteed to reduce the bound gap at all the nodes visited. Because policy simulation is no longer required, there is an improvement in performance over traditional methods for problems in which a high level of confidence is sought.

Transport constraints limit competition and arbitrageurs' possibilities of exploiting price differences between commodities in neighbouring markets. We analyze a transport-constrained network with local demand shocks, where spatially distributed oligopoly producers compete with supply functions, as in wholesale electricity markets. Uniqueness and existence results are proven, and we are able to explicitly solve for symmetric supply-function equilibria in some special cases.

Supply function equilibrium models are used to study electricity market auctions with uncertain demand. We study the effects on supply function equilibrium of a system tax on the observed benefits of suppliers. Such a tax provides an incentive for agents to alter their offers to avoid the tax. We show how this surprisingly can lead to lower prices in equilibrium. The model is extended to a setting in which the agents are taxed on the benefits accruing to them from a transmission line expansion (in order to help fund the line). In these circumstances we study how incentives for agents to alter their bids varies with the relative size of the capacity expansion.

In 2013 Transpower NZ commissioned a new HVDC link to transfer electrical power between the North and South Islands of New Zealand. This was a substantial and prolonged undertaking, requiring ca. 400 in-situ capability tests. Transpower elected to perform these tests ``live", without suspending the normal operation of the wholesale electricity market. Instead, Transpower's trading team attempted to create suitable flow conditions for each test by entering into innovative financial derivative contracts with power generation firms. We created a stochastic dynamic programming model to handle the contingent scheduling of the tests; its most important random variable was the state of water storage available to hydropower plants.

The high penetration of renewables envisaged for future power systems will significantly increase the need for flexible operational measures and generation technologies, whose associated investment decisions must be properly planned in the long term. To achieve this, expansion models will need to incorporate unit commitment constraints, which can result in large scale MILP problems that require significant computational resources to be solved. In this context, this paper proposes a novel Dantzig–Wolfe decomposition and a column generation approach to reduce the computational burden and overcome intractability. We demonstrate through multiple case studies that the proposed approach outperforms direct application of commercial solvers, significantly reducing both computational times and memory usage. Using the Chilean power system as a reference case, we also confirm and highlight the importance of considering unit commitment constraints in generation expansion models.

Risk has always been a dominant part of financial decision making in any industry. Recently models, tools and computational techniques have been developed so that we can effectively incorporate risk in optimal decision policies. The focus of this paper is on electricity markets, where much of the inherent risk falls on the retail sector. We introduce a three-stage model of an electricity market where firms can choose to enter the retail market, then enter into retail contracts, and finally purchase electricity in a wholesale market to satisfy their contracts. We explicitly assume that firms are risk-averse in this model. We demonstrate how the behaviour of firms change with risk-aversion, and use the example of an asset-swap policy over a transmission network to demonstrate the importance of modeling risk-aversion in determining policy outcomes.

Supply function equilibrium models are used to study electricity market auctions with uncertain demand. We study the effects on supply function equilibrium of a system tax on the observed benefits of suppliers. Such a tax provides an incentive for agents to alter their offers to avoid the tax. We show how this surprisingly can lead to lower prices in equilibrium. The model is extended to a setting in which the agents are taxed on the benefits accruing to them from a transmission line expansion (in order to help fund the line). In these circumstances we study how incentives for agents to alter their bids varies with the relative size of the capacity expansion.

A model to determine the optimal consumption level and associated reserve offer for a large consumer in a co-optimised electricity market is presented. The method uses numerical simulation along with a full representation of the New Zealand electricity market dispatch model. Uncertainty is introduced through the use of stochastic demand sampling. We approach this process in three phases: phase one contains simulations to determine potential energy and reserve prices under uncertainty. Phase two uses a dynamic programming method, adapted from a generator model, to determine the optimal reserve offer. Phase three is the repetition of phase one with the optimal reserve offer intact. The model has been applied to a user in New Zealand and initial results have been presented. The model approached a theoretical maximum profitability when used as an input to a site curtailment response strategy.

The correspondence of competitive partial equilibrium with a social optimum is well documented in the welfare theorems of economics. These theorems can be applied to single-period electricity pool auctions in which price-taking agents maximize profits at competitive prices, and extend naturally to standard models with locational marginal prices. In hydro-thermal markets where the auctions are repeated over many periods, agents seek to optimize their current and future profit accounting for future prices that depend on uncertain inflows. This makes the agent problems multistage stochastic optimization models, but perfectly competitive partial equilibrium still corresponds to a social optimum when all agents are risk neutral and share common knowledge of the probability distribution governing future inflows. The situation is complicated when agents are risk averse. In this setting we show under mild conditions that a social optimum corresponds to a competitive market equilibrium if agents have time-consistent dynamic coherent risk measures and there are enough traded market instruments to hedge inflow uncertainty. We illustrate some of the consequences of risk aversion on market outcomes using a simple two-stage competitive equilibrium model with three agents.

Transport constraints limit competition and arbitrageurs' possibilities of exploiting price differences between commodities in neighbouring markets. We analyze a transport-constrained network with local demand shocks, where spatially distributed oligopoly producers compete with supply functions, as in wholesale electricity markets. Uniqueness and existence results are proven, and we are able to explicitly solve for symmetric supply-function equilibria in some special cases.

Supply function equilibrium models are used to study electricity market auctions with uncertain demand. We study the effects on supply function equilibrium of a system tax on the observed benefits of suppliers. Such a tax provides an incentive for agents to alter their offers to avoid the tax. We show how this surprisingly can lead to lower prices in equilibrium. The model is extended to a setting in which the agents are taxed on the benefits accruing to them from a transmission line expansion (in order to help fund the line). In these circumstances we study how incentives for agents to alter their bids varies with the relative size of the capacity expansion.

The correspondence of competitive partial equilibrium with a social optimum is well documented in the welfare theorems of economics. These theorems can be applied to single-period electricity pool auctions in which price-taking agents maximize profits at competitive prices, and extend naturally to standard models with locational marginal prices. In hydro-thermal markets where the auctions are repeated over many periods, agents seek to optimize their current and future profit accounting for future prices that depend on uncertain inflows. This makes the agent problems multistage stochastic optimization models, but perfectly competitive partial equilibrium still corresponds to a social optimum when all agents are risk neutral and share common knowledge of the probability distribution governing future inflows. The situation is complicated when agents are risk averse. In this setting we show under mild conditions that a social optimum corresponds to a competitive market equilibrium if agents have time-consistent dynamic coherent risk measures and there are enough traded market instruments to hedge inflow uncertainty. We illustrate some of the consequences of risk aversion on market outcomes using a simple two-stage competitive equilibrium model with three agents.

Supply function equilibrium models are used to study electricity market auctions with uncertain demand. We study the effects on supply function equilibrium of a system tax on the observed benefits of suppliers. Such a tax provides an incentive for agents to alter their offers to avoid the tax. We show how this surprisingly can lead to lower prices in equilibrium. The model is extended to a setting in which the agents are taxed on the benefits accruing to them from a transmission line expansion (in order to help fund the line). In these circumstances we study how incentives for agents to alter their bids varies with the relative size of the capacity expansion.

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 SDDP, CUPPS and DOASA when applied to problems with general convex cost functions.

In this paper we present a simulation tool for large consumers to optimise their consumption and reserve offers in a security constrained electricity market. We utilise a replica of the NZEM market clearing software, vSPD which has security constrained generation and transmission and accurately recreates final prices. We use a series of small optimal power flow to illustrate how security constraints can influence prices. We study a five year time horizon to investigate the occurrence of security constrained pricing behaviour in the NZEM final prices. A large energy consumer may utilize our tool to assess their impact upon the prices, taking into account the instances of security constraints. We expect this approach to be extensi- ble to other markets although we note that information surrounding the underlying market structure will heavily influence the viability

We develop a Supply Function Equilibria model to assess the impact of uncompetitive Contingency Reserve markets on participant behaviour in a market where Transmission is a risky asset. Companies may submit energy and reserve offers with the requirement for reserve specified by the co-optimised dispatch of energy assets subject to the reserve offers. The results of this model suggest that withholding reserve may be an optimal strategy. A generator at the sending end of a reserve constrained transmission line should self withhold in order to minimise price differences between the nodes. A preliminary empirical assessment suggests that one provider systematically offered less combined energy and reserve capacity to the market during periods of reserve constraints.

The correspondence of competitive partial equilibrium with a social optimum is well documented in the welfare theorems of economics. These theorems can be applied to single-period electricity pool auctions in which price-taking agents maximize profits at competitive prices, and extend naturally to standard models with locational marginal prices. In hydro-thermal markets where the auctions are repeated over many periods, agents seek to optimize their current and future profit accounting for future prices that depend on uncertain inflows. In this setting perfectly competitive partial equilibrium corresponds to a social optimum when all agents share common knowledge of the probability distribution governing future inflows. The situation is complicated when agents are risk averse. We illustrate some of the consequences of risk aversion on market outcomes using simple two-stage competitive equilibrium models in which agents are endowed with coherent risk measures. In this setting we show that welfare is optimized in a competitive market if there are enough traded market instruments to hedge inflow uncertainty but might not be if these are missing.

Electricity markets face a substantial amount of uncertainty. Traditionally this un- certainty has been due to varying demand. With the integration of larger proportions of volatile renewable energy, this added uncertainty from generation must also be faced. Conventional electricity market designs cope with uncertainty by running two markets: a day ahead or pre-dispatch market that is cleared ahead of time, followed by a real-time balancing market to reconcile actual realizations of demand and available generation. In such markets, the day ahead market clearing process does not take into account the distri- bution of outcomes in the balancing market. Recently an alternative so-called stochastic settlement market has been proposed (see e.g. Pritchard et al. [5] and Bouffard et al. [2]). In such a market, the ISO co-optimizes pre-dispatch and spot in one single settlement market. In this paper we consider simplified models for three types of market clearing mecha- nisms. We demonstrate that under the assumption of symmetry, our simplified stochastic programming market clearing is equivalent to a two period single settlement (TS) system that takes count of deviation penalties in the second stage. These however differ from a TS model that dispenses with deviation penalties and has been (and continues to be) in use in New Zealand (NZTS). Our models are targeted towards analyzing imperfectly compet- itive markets. We will construct Nash equilibria of the resulting games for the introduced market clearing mechanisms and compare them under the assumptions of symmetry and in an asymmetric example.

We consider a consumer of a resource, such as electricity, who must pay a per unit charge to procure the resource, as well as a peak demand charge. We will present an efficient linear programming formulation for the demand response of such a consumer who could be a price taker, industrial or commercial user of electricity that has some ability to self-generate. We will establish a monotonicity result that indicates fuel supply of S, utilized for self generation, may be spent in successive steps adding to S in total.

Introducing wind generation into an electricity market can incur an extra cost resulting from the volatile nature of wind. To reduce this cost, an alternative stochastic market clearing mechanism is proposed in the literature [1, 2, 3]. However, implementing a stochastic market clearing can also impose some extra cost on the market. Therefore, it is essential to estimate the efficiency gain resulting from implementing a stochastic market clearing mechanism. We describe the result of an empirical study to quantify value of a stochastic clearing mechanism for the New Zealand electricity market. We extend our analysis for possible larger wind integration in the future.

Transport constraints limit competition and arbitrageurs' possibilities of exploiting price differences between commodities in neighbouring markets. We analyze a transportation network where oligopoly producers compete with supply functions under uncertain demand, as in wholesale electricity markets. For networks with a radial structure, we show that symmetric supply function equilibria (SFE) can be determined from an exogenous market integration function. Existence of such equilibria (SFE) is ensured if demand shocks are sufficiently evenly distributed. The market integration function simplifies to a constant for uniform multi-dimensional nodal demand shocks, and then we can explicitly solve for SFE.

Electricity markets face a substantial amount of uncertainty. Traditionally this uncertainty has been due to varying demand. With the integration of larger proportions of volatile renewable energy, this added uncertainty from generation must also be faced. Conventional electricity market designs cope with uncertainty by running two markets: a day ahead or pre-dispatch market that is cleared ahead of time, followed by a real-time balancing market to reconcile actual realizations of demand and available generation. In such markets, the day ahead market clearing process does not take into account the distribution of outcomes in the balancing market. Recently an alternative so-called stochastic settlement market has been proposed. In such a market, the ISO co-optimizes pre-dispatch and spot in one single settlement market. In this paper we consider simplified models for three types of market clearing mechanisms. We demonstrate that under the assumption of symmetry, our simplified stochastic programming market clearing is equivalent to a two period single settlement (TS) system that takes count of deviation penalties in the second stage. These however differ from a TS model that dispenses with deviation penalties and has been (and continues to be) in use in New Zealand (NZTS). Our models are targeted towards analyzing imperfectly competitive markets. We will construct Nash equilibria of the resulting games for the introduced market clearing mechanisms and compare them under the assumptions of symmetry and in an asymmetric example.

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 SDDP, CUPPS and DOASA when applied to problems with general convex cost functions.

In this paper, we first present a market environment with a conventional two settlement mechanism. We show that when we add some wind generation to the system, the steady-state market conditions yield lower social and consumer welfare and higher use of fossil fuels. We also present results of a counterfactual stochastic settlement market which improves social and consumer welfare after the introduction of new intermittent generation. Thus, we conclude that the choice of market mechanism is a critical factor for capturing the benefits of large-scale wind integration. We also introduce a method to compute analytical equilibria of games in which the payoff functions of players depend on the optimal solution to an optimization problem with inequality constraints.

In this paper we will discuss the implications of reserve constraints acting upon energy prices in a co-optimised electricity market. We will identify five situations and illustrate them using simplified linear programs based upon the Scheduling, Pricing and Dispatch model used in New Zealand. We will then use the insights gleaned from our analysis to screen the empirical data from the 2008-2010 calendar years. We will identify over 9000 constrained periods in total and theoretically demonstrate, in a simplified setting, how a generator may utilise these reserve constraints to profit. This demonstration will illustrate the primary cash flows between participants and identify the possibility for a integrated participant to exert market power to extract rentals.

We consider the effect that a tax on observed profits has on supplier strategies in supply-function equilibrium. In some circumstances such a tax can make supply offers more competitive, decrease prices, and give greater efficiency.

The behaviour of DC Load-flow formulations when they are used in economic dispatch and nodal pricing models is discussed. It is demonstrated that non-negative prices in these models are sufficient to guarantee global optimality of any local optimum, even if the feasible region is not convex, and so a negative nodal price is an indicator of a possible loss in optimality. It is also discuss the possible effect that negative prices might have on algorithms that assume this convexity.

In 2025 electricity production from wind power should constitute nearly 50 % of electricity demand in Denmark. In this paper we look at optimal expansion of the transmission network in order to integrate 50 % wind power in the system, while minimising total fixed investment cost and expected cost of power generation. We allow for active switching of transmission elements to eliminate negative effects of Kirchhoffs voltage law. Results show that actively switching transmission lines may yield a better utilisation of transmission networks with large- scale wind power and increased wind power penetration. Furthermore, transmission switching is likely to affect the optimal line capacity expansion plan.

We consider the application of Dantzig-Wolfe decomposition to stochastic integer programming problems arising in the capacity planning of electricity transmission networks that have some switchable transmission elements. The decomposition enables a column-generation algorithm to be applied, which allows the solution of large problem instances. The methodology is illustrated by its application to a problem of determining the optimal investment in switching equipment and transmission capacity for an existing network. Computational tests on IEEE test networks with 73 nodes and 118 nodes confirm the efficiency of the approach.

We introduce the concept of an offer distribution function to analyze randomized offer curves in multi-unit procurement auctions. We characterize mixed-strategy Nash equilibria for pay-as-bid auctions where demand is uncertain and costs are common knowledge; a setting for which pure-strategy supply function equilibria typically do not exist. We generalize previous results on mixtures over horizontal offers as in Bertrand-Edgeworth games, and we also characterize novel mixtures over partly increasing supply functions. We show that the randomization can cause considerable production inefficiencies.

We consider dynamic programming (DP) approximations to hydro-electric reservoir scheduling problems. The first class of approximate DP methods uses decomposition and multimodeling heuristics to produce policies that can be expressed as the sum of one-dimensional Bellman functions. This heuristic allows us to take into account non-convexities (appearing in models with head-effect) by solving a MIP at each time stage. The second class of methods uses cutting planes and sampling. It is able to provide multi-dimensional policies. We show that the cutting plane methods will produce better policies than the first DP approximation on two convex problem formulations of different types. Modifying the cutting plane method to approximate the effect of reservoir head level on generation also yields better results on problems including these effects. The results are illustrated using tests on two river valley systems.

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.

We discuss some ways of formulating quantile-type models for forecasting variations in wind power in the short term (within a few hours). Such models predict quantiles of the conditional distribution of the wind power available at some future time using information presently available. A natural reference for models of this kind is a "probabilistic persistence" quantile forecast whose only input is the present wind power. Using data from some New Zealand wind farms, we find that more complex quantile models can readily improve on probabilistic persistence in resolution but not in sharpness. The most valuable model inputs, apart from the present power, are found to be real-time air pressure measurements and a power total-variation indicator.

In recent years there has been a move in the majority of industrialized countries to invest in renewable resources for the production of energy. This move has come about as people worldwide are more aware of negative effects of fossil fuel sources of energy on the environment including the release of greenhouse gases such as CO2. Utilization of renewable sources of energy, for instance harnessing wind power in electricity production, is deemed to be reducing the use of fossil fuels and hence results in the reduction of CO2. Mechanisms that promote and facilitate utilization of renewable sources of energy are being developed. In particular, recently stochastic programming market clearing mechanisms have been suggested that would seemingly allow for a more efficient use of wind energy hence reduction of fossil fuel use, that ultimately would result in a reduction of CO2. In this paper we will examine the steady state behaviour of participants in an electricity market to fully analyze the hypothesis that the stochastic programming market clearing mechanism is less fossil fuel (and hence CO2) intensive than a conventional two settlement market through some simple examples.

The behaviour of DC Load-flow formulations when they are used in economic dispatch and nodal pricing models is discussed. It is demonstrated that non-negative prices in these models are sufficient to guarantee global optimality of any local optimum, even if the feasible region is not convex, and so a negative nodal price is an indicator of a possible loss in optimality. It is also discuss the possible effect that negative prices might have on algorithms that assume this convexity.

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.

Electricity market designs that decentralize decision making for participants can lead to inefficiencies in the presence of nonconvexity or missing markets. This has been shown in the case of unit-commitment problems that can make a decentralized market equilibrium less efficient than a centrally-planned solution. Less attention has been focused on systems with large amounts of hydro-electric generation. We describe the results of an empirical study of the New Zealand wholesale electricity market that attempts to quantify production efficiency losses by comparing market outcomes with a counterfactual central plan. (Updated October 11, 2010).

The inter-island HVDC line is a major transmission line in New Zealand, as it is the only link between its two islands. It enables the transfer of electricity between the South Island and the North Island. Because these transfers are generally beneficial to both the generators of the South Island and the consumers of the North Island, South Island generators are currently charged for the cost of the HVDC line based on a peak charge. We investigate an alternative scheme based on auctioning physical flow rights. Using a simplified supply-function equilibrium model we show that this is welfare optimizing in a perfectly competitive setting, but can result in inefficient dispatch and loss of rights revenue if generators bid for capacity strategically.

This paper (to appear in the Wiley Encyclopedia on OR/MS) gives an elementary account of techniques for modeling uncertainty in optimization problems.

Using the concept of market-distribution functions, we derive general optimality conditions for discriminatory divisible-good auctions, which are also applicable to Bertrand games and non-linear pricing. We introduce the concept of offer distribution function to analyze randomized offer curves, and characterize mixed-strategy Nash equilibria for pay-as-bid auctions where demand is uncertain and costs are common knowledge; a setting for which pure-strategy supply function equilibria typically do not exist. We generalize previous results on mixtures over horizontal offers as in Bertrand-Edgeworth games, but more importantly we characterize novel mixtures over partly increasing supply functions.

While operations research is utilized across all sectors of wholesale electricity markets, it is most widely and intensely used in the generation sector. We review the operations of a wholesale electricity market and provide a detailed treatment of optimization in the generation sector.

The idea of rearranging generation assets amongst firms to improve competition has once again surfaced in a recent report on improvements to the New Zealand Electricity Market. We show with examples that rearranging assets, either with asset divestiture to a new firm, or asset swaps between existing firms, may actually reduce competition in electricity markets. Our examples emphasize features that are particular to electricity, such as seasonality and transmission constraints. These results warn that applying economic rules of thumb to electricity markets may lead to erroneous conclusions.

Using the concept of market-distribution functions, we derive general optimality conditions for discriminatory divisible-good auctions, which are also applicable to Bertrand games and non-linear pricing. We introduce the concept of offer distribution function to analyze randomized offer curves, and characterize mixed-strategy Nash equilibria for pay-as-bid auctions where demand is uncertain and costs are common knowledge; a setting for which pure-strategy supply function equilibria typically do not exist. We generalize previous results on mixtures over horizontal offers as in Bertrand-Edgeworth games, but more importantly we characterize novel mixtures over partly increasing supply functions.

The behaviour of DC Load-flow formulations when they are used in economic dispatch and nodal pricing models is discussed. It is demonstrated that non-negative prices in these models are sufficient to guarantee global optimality of any local optimum, even if the feasible region is not convex, and so a negative nodal price is an indicator of a possible loss in optimality. It is also discuss the possible effect that negative prices might have on algorithms that assume this convexity.

The welfare of electricity producers and consumers depends on congestion in the transmission grid, generation costs that consist mainly of fuel costs, and strategic behavior. We formulate a game theoretic model of an oligopolistic electricity market where generation costs are derived from a fuel supply network. The game consists of a fuel dispatcher that transports fuels at minimum cost to meet generator demands, generators that maximize profit in Cournot competition, and an independent system operator (ISO) that sets nodal prices to balance electricity supply with linear demand functions. We prove the existence of an equilibrium. If fuel supplies are unlimited, the same equilibria hold in a simplified version of the game in which each generator optimizes its fuel acquisition from the network. In some very simple examples under different assumptions about the rationality of generators with respect to ISO decisions, paradoxical effects on total welfare can occur from expanding either electricity transmission capacity or the transportation capacity of low-cost fuel. We find some instances in which the paradox occurs only under bounded rationality of the generators, others where it occurs only if the generators are fully rational, and still others where it occurs to different degrees under the two rationality assumptions.

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 which 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 examine the effect of introducing a carbon tax on electricity generation. We model this by way of a two generator Cournot game over a two node electricity network. We find that within the electricity system, emissions of carbon dioxide can increase after a carbon tax is introduced.

We discuss the almost-sure convergence of a broad class of sampling algorithms for multi-stage stochastic linear programs. Although the convergence of methods of this type is part of the stochastic programming folklore, we provide an explicit 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.

This tutorial is a PDF version of the Introduction to Stochastic Programming tutorial that is provided on the COSP site http:stoprog.org

We develop a generalized Nash equilibrium model with two players to compare the effects of using coincident-peak transmission charges with anytime-peak transmission charges. Players are assumed to be able to shift load between periods with a cost that grows quadratically with the amount shifted. When the shifting costs are large compared with peak charges, the model has a unique equilibrium. Coincident-peak charging and anytime-peak charging give different outcomes when the peak load for one purchaser does not coincide with the coincident peak. Coincident-peak charges favour purchasers whose peaks do not coincide with the system peak. They are more effective than anytime-peak charges at decreasing peak loads and therefore lowering peak charges.

We consider electricity pool markets in radial electricity transmission networks in which the lines have no transmission losses, but have transmission capacities. At each node there is a strategic generator submitting generation quantities to the pool. Prices are determined by a linear competitive fringe at each node. We derive necessary and sufficient conditions on the line capacities that ensure that the unconstrained one-shot Cournot equilibrium remains an equilibrium in the constrained network. These conditions are characterized by a convex polyhedral set.

This paper provides a technique based on stochastic programming to optimally solve the electricity procurement problem faced by a large consumer. Supply sources include bilateral contracts, a limited amount of self-production and the pool. Risk aversion is explicitly modeled using the CVaR methodology. Results from a realistic case study are provided and analyzed.

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.

We present a model for the design of a minimum-cost, survivable electricity distribution network, which generalizes to telecommunications, logistics and other network types. We formulate this problem as a two-stage stochastic mixed-integer program in which first-stage decisions expand capacity, and recourse deicisions configure and operate the network so as to be feasible under various scenarios corresponding to individual link failures.

We discuss the effects of setting penalty costs on artificial variables in electricity dispatch software. It is shown under a feasibility assumption that a choice of these can be made to give no shortfalls in grid security and energy, but a possible shortfall in spinning reserve.

We consider an electricity generator making offers of energy into an electricity pool market over a horizon of several trading periods (typically a single trading day). The generator runs a set of generating units with given start-up costs, shut-down costs and operating ranges. At the start of each trading period the generator must submit to the pool system operator a new supply curve defining quantities of offered energy and the prices at which it wants these dispatched. The amount of dispatch depends on the supply curve offered along with the offers of the other generators and market demand, both of which are random, but do not change in response to the actions of the generator we consider. After dispatch the generator determines which units to run in the current trading period to meet the dispatch. The generator seeks a supply function that maximizes its expected profit. We describe an optimization procedure based on dynamic programming that can be used to construct optimal offers in successive time periods over a fixed planning horizon.

Interconnecting distinct electricity markets by adding a new transmission line affects the outcomes in these markets. We examine the effects of interconnection using market distribution functions. We give analytical formulae for computing market outcomes when the uncertain events in the markets being connected are statistically independent, and show by example how to compute these outcomes when these events are correlated.

We describe a general multi-stage stochastic integer-programming model for planning discrete capacity expansion of production facilities. A scenario tree represents uncertainty in the model. Variable splitting leads to two forms of this model: the first allows multiple expansions of each facility over the planning horizon while the second allows at most one. Dantzig-Wolfe decomposition of either split-variable model results in a binary master problem that solves easily, as its linear-programming relaxation tends to yield integer solutions. For each scenario-tree node, the decomposition defines a subproblem that may be viewed as a single-period, deterministic capacity-expansion problem. An effective solution procedure results as long as the subproblems solve efficiently, and the procedure incorporates a good ﾂ“duals stabilization schemeﾂ”. We present computational results for a model to plan the capacity expansion of a real-world electricity distribution network given uncertain future demand. The largest problem we solve to optimality has 6 stages and 243 scenarios corresponding to a deterministic equivalent with a quarter of a million binary variables.

We consider a purchaser of electricity, 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. We study conditions under which the retailer should bid their expected demand, and derive conditions on the optimal demand curve that they should bid if the behaviour of the other participants is unknown, but can be modelled by a market distribution function.

In an electricity pool market, each generator is required to submit a supply function (offer stack), indicating how much power it will generate as a function of the price. For a generator operating a hydro-electric reservoir, 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. The application of the model to a real generation system is described.

The paper presents a convergence proof for a broad class of sampling algorithms for multi-stage stochastic linear programs in which the uncertain parameters occur only in the constraint right-hand sides. This class includes SDDP, AND, ReSa, and CUPPS. We show under an independence assumption on the sampling procedure that the algorithms converge with probability 1.

We describe a mixed integer programming model for scheduling mechanical pulp production with uncertain electricity prices.

The New Zealand transmission grid operator Transpower uses special constraints in the SPD dispatch software to ensure that in the event of a single transmission line failure, remaining circuits are not over-loaded. We discuss a mechanism by which these constraints might be able to be relaxed by making use of interruptible load.

We consider a simple game-theoretical model in which an electricity retailer and a network owner offer incentives to consumers to shift load from a peak period to an off-peak period. Using a simple example we compare the market outcomes from collusion with those from the equilibrium of a non-cooperative game, and examine the behaviour in this game when it is repeated in a situation in which agents have imperfect information.

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 in the context of optimization.

This paper studies financial transmission rights in electricity pool markets with nodal pricing, when these rights are to be allocated by an auction mechanism. We prove that simultaneous feasibility entails revenue adequacy in a general framework of convex optimization, and show by counterexample how this result might fail in the absence of convexity. A market distribution function approach is used to investigate the effects on electricity offering behaviour when participants hold financial transmission rights, and the implications of this for the auction design are discussed. The paper also discusses the incentives provided by financial transmission rights for encouraging investment in network transmission capacity.

In an electricity pool market the market distribution function gives the probability that a generator offering a certain quantity of power at a certain price will not be dispatched all of this quantity by the pool. It represents the uncertainty in a pool market associated with the offers of the other agents as well as demand. We present a general Bayesian update scheme for market distribution functions. To illustrate the approach a particular form of this procedure is applied to real data obtained from a New Zealand electricity generator.

We consider a generator making offers of energy into an electricity pool market. For a given time period, it must submit an offer stack, consisting of a fixed number of quantities of energy and prices at which it wants these quantities dispatched. We assume that the generator cannot offer enough power to substantially affect the market price, so the optimal response would be to offer energy at marginal cost. However, the market rules do not permit an arbitrary function, so the problem is to find an offer stack approximating marginal cost in a way that maximizes its profit. We give optimality conditions for this problem and derive an optimization procedure based on dynamic programming. This procedure is illustrated by applying it to several examples with different costs of production.

In a nodal spot market for electricity, there may be circumstances in which generators may wish to offer energy at negative prices, e.g. to avoid being shut down for a short period. Such behaviour can create some severe computational difficulties for the system dispatcher. The ``must-run dispatch auction" is a system used to handle such cases in New Zealand; we construct an equilibrium model for generators' behaviour in the auction under stochastic demands. An interesting feature of the auction is that its outcome may be economically sub-optimal even under very idealized assumptions of perfect competition.

We consider wholesale electricity market pools in which generators must offer supply functions that are centrally dispatched. Each generator seeks a supply function to offer to the spot market, so as to maximise expected return. We give conditions under which a supply function exists that optimises return for every demand realisation. We also analyse the case in which the behaviour of the competition can be modelled by an appropriate probability distribution, and derive optimality conditions for the optimal supply-function offer in this case. The paper concludes with some remarks on applying the theory to the case where each generator must offer a limited number of prices in their stack.

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 can be used to construct an optimal solution for a simple example. We also consider the case where a generator is restricted in the number of prices at which power can be offered.

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.

We consider the problem of offering electricity produced by a series of hydroelectric reservoirs to a pool-type central market. The market model is a simplified version of the New Zealand wholesale electricity market, with prices modelled by a stochastic process. The demand for electricity is not explicitly modelled. The hydroelectric generator is assumed to be unable to influence market prices (i.e. to be a price-taker). We discuss the resulting stochastic dynamic program, methods for its solution, and the explicit optimal offer curves that it produces.

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.

Spot prices of electricity are determined in New Zealand (and a number of other electricity markets in the world) using a linear programming model to construct a dispatch schedule to meet metered loads at the nodes of the transmission network. The linear program seeks to minimise in each half hour the delivered cost of the energy, as represented by the prices that generators offer their power to the market, while accounting for transmission losses, network constraints and spinning reserve constraints. The ex-post electricity price at any given node of the transmission system is given by the shadow price of the energy balance constraint at optimality. We discuss the results of some experiments carried out with a small electricity pricing model developed in the Department of Engineering Science. The prices obtained from these linear programming models have some interesting properties. Some of these properties, although counterintuitive, have rational explanations. Other properties are less benign, and arise in circumstances when a linear programming model is an inappropriate approximation of the true load flow problem. We shall explain these effects, and discuss some possible remedies.