The problem of distributing gas through a network of pipelines is formulated as a cost minimization subject to nonlinear flow-pressure relations, material balances, and pressure bounds. The solution method is based on piecewise linear approximations of the nonlinear flow-pressure relations. The approximated problem is solved by an extension of the Simplex method. The solution method is tested on real-world data and compared with alternative solution methods.mathematics: piecewise linear programming: natural resources: energy transport, networks: applications to gas transmission network

We consider three models of investments in generation capacity in restructured electricity systems that differ with respect to their underlying economic assumptions. The first model assumes a perfect, competitive equilibrium. It is very similar to the traditional capacity expansion models even if its economic interpretation is different. The second model (open-loop Cournot game) extends the Cournot model to include investments in new generation capacities. This model can be interpreted as describing investments in an oligopolistic market where capacity is simultaneously built and sold in long-term contracts when there is no spot market. The third model (closed-loop Cournot game) separates the investment and sales decision with investment in the first stage and sales in the second stage-that is, a spot market. This two-stage game corresponds to investments in merchant plants where the first-stage equilibrium problem is solved subject to equilibrium constraints. We show that despite some important differences, the open-and closed-loop games share many properties. One of the important results is that the prices and quantities produced in the closed-loop game, when the solution exists, fall between the prices and quantities in the open-loop game and the competitive equilibrium.

We study the equilibria reached by strategic producers in a pool-based network-constrained electricity market. The behavior of each producer is modeled by a mathematical program with equilibrium constraints (MPEC) whose objective is maximizing profit and whose complementarity constraints describe market clearing. The joint solution of all these MPECs constitutes an equilibrium problem with equilibrium constraints (EPEC). The equilibria associated with the EPEC are analyzed by solving the strong stationarity conditions of all MPECs, which can be linearized without approximation by mixed-integer linear programming (MILP) techniques. The resulting mixed-integer linear conditions can be reformulated as an optimization problem that allows establishing diverse objectives to differentiate among alternative equilibria.Index Terms-Electricity pool, equilibria, equilibrium problem with equilibrium constraints (EPEC), locational marginal price (LMP), mixed-integer linear programming (MILP), power producer, offering strategy.

We cast models of the generation capacity expansion type formally developed for the monopoly regime into equilibrium models better adapted for a competitive environment. We focus on some of the risks faced today by investors in generation capacity and thus pose the problem as a stochastic equilibrium model. We illustrate the approach on the problem of the incentive to invest. Agents can be risk neutral or risk averse. We model risk aversion through the CVaR of plants' profit. The CVaR induces risk-adjusted probabilities according to which investors value their plants. The model is formulated as a complementarity problem (including the CVaR valuation of investments). An illustration is provided on a small problem that captures several features of today's electricity world: a choice often restricted to coal and gas units, a peaky load curve because of wind penetration, uncertain fuel prices, and an evolving carbon market. We assess the potential of the approach by comparing energy-only and capacity market organizations in this risky environment. Our results can be summarized as follows: a deterministic analysis overlooks some changes of capacity structure induced by risk, whether in the capacity market or energy-only organizations. The risk-neutral analysis also misses a shift towards less capital-intensive technologies that may result from risk aversion. Last, risk aversion also increases the shortage of capacity compared to the risk-neutral view in the energy-only market when the price cap is low. This may have a dramatic impact on the bill to the final consumer. The approach relies on mathematical programming techniques and can be extended to full-size problems. The results are illustrative and may deserve more investigation.

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