Narcís Nabona

An Oligopoly Model for Medium-Term Power Planning in a Liberalized Electricity Market

We address the problem of finding optimal medium- term generation policies for a specific generation company by modeling the supply side of a liberalized electricity market. The model assumes a noncooperative oligopoly and determines the joint optimal generation policies of all market generators, taking into account hydro, market, and system uncertainties. We propose an endogenous function of market price with respect to load duration where the choice of fuel and technology influences both the average and range of variation of the medium-term market price. We assume an inelastic demand represented by the load-duration curve, which is matched using the Bloom and Gallant formulation. This accounts for unit outages without using scenarios, which are reserved for modeling other uncertainties such as a latent price variable and the hydro inflows. The equilibrium is solved using the Nikaido-Isoda relaxation algorithm, which enables a series of multistage cubic stochastic programming models to be solved. In order to deal with the large number of load matching constraints, we use a heuristic which allows us to generate only those constraints that will presumably be active at the optimal solution. The model is calibrated to the Spanish electricity market using historical price and generation data.

Long-Term Thermal Power Planning at VEW Energie Using a Multi-Interval Bloom and Gallant Method

Bloom and Gallant have proposed an elegant model for finding the optimal thermal schedule subject to matching the load duration curve and general linear constraints. Their method is based on a linear program with some linear equality constraints and many linear inequality constraints. There are well-documented specialized linear programming algorithms that are able to solve this problem. This paper proposes the extension of the Bloom and Gallant model to several successive intervals with constraints on generation encompassing different intervals. A procedure for finding an initial feasible point which is an essential part of the solution procedure is described here, together with details of the modeling of several operational constraints and computational results with real test cases.

On the first-order estimation of multipliers from Kuhn-Tucker systems

Abstract The minimization of a nonlinear function with linear and nonlinear constraints and simple bounds can be performed by minimizing an augmented Lagrangian function that includes only the nonlinear constraints subject to the linear constraints and simple bounds. It is then necessary to estimate the multipliers of the nonlinear constraints and variable reduction techniques can be used to carry out the successive minimizations. The viability of estimating the multipliers of the nonlinear constraints from the Kuhn–Tucker system is analyzed and an acceptability test on the residual of the estimation is put forward. The computational performance of the procedure is compared with that of the inexpensive Hestenes–Powell multiplier update. Scope and purpose It is possible to minimize a nonlinear function with linear and nonlinear constraints and simple bounds through the successive minimization of an augmented Lagrangian function including only the nonlinear constraints subject to the linear constraints and simple bounds. This method is particularly interesting when the linear constraints are flow conservation equations, as there are efficient techniques for solving nonlinear network problems. Regarding the successive estimation of the multipliers of the nonlinear constraints there is some doubt as to whether using the Kuhn–Tucker system could improve upon the inexpensive Hestenes–Powell update, especially considering that the Kuhn–Tucker system with partial augmented Lagrangians may not always lead to an acceptable multiplier estimation. Clarifying the computational efficiency of the multiplier update when there are linear or nonlinear side constraints is also a necessary previous step regarding the comparison between partial augmented Lagrangian techniques and either primal partitioning techniques for linear side constraints or projected Lagrangian methods in the case of nonlinear side constraints.

Optimum long-term hydrothermal coordination with fuel limits

Optimizing the thermal production of electricity in the long term, once the maintenance schedules have been decided, means optimizing both the fuel procurement policies and the use of fuels for generation in each thermal unit throughout the time period under study. A fundamental constraint to be satisfied at each interval into which the long time period is subdivided is the covering of its load duration curve with thermal and the stochastic hydrogeneration. A new procedure to optimize this problem is proposed. It is based on the use of a power-energy function for each interval, which changes with the deterministic and stochastic hydrogeneration and with thermal generation. Through this function the generation duration curves that come from the load duration curves of all intervals, are matched. >

A heuristic for the long-term electricity generation planning problem using the Bloom and Gallant formulation

Abstract Long-term power planning is a stochastic problem often confronted by electrical utilities in liberalized markets. One can model it for profit maximization—using market-price estimation functions for each interval—by posing it as a quadratic programming problem with some linear equalities and an exponential number of load-matching linear inequality constraints. In order to avoid handling all the inequalities when one is attempting to solve the problem, column generation methods have been employed herein. In this paper, we describe the foundations and implementation of a heuristic that tries to iteratively guess the active set of constraints at the optimizer, alongside a normal quadratic programming solution used at each iteration. The two methods are compared and the heuristic procedure is shown to be more efficient.

Renewable Energies in Medium-Term Power Planning

The medium-term generation planning over a yearly horizon for generation portfolios including hydro generation and non-dispatchable renewables such as wind power and solar photovoltaic generation reveals how the penetration of these technologies reduces the share of other technologies and how it changes the profits and the profit spread in liberalized electricity markets. The matching of the load duration curves in different periods and the forced outages of generation units are here addressed through probabilistic methods and a heuristic is employed to guess which of the load-matching constraints may be active at the solution. There are well-established models for hydro generation in medium-term planning, but specific models are necessary to account for wind power and photovoltaic generation. A proposal for these is made in this work, which is suitable for use in the load duration curve matching through probabilistic methods. Moreover the stochasticity of the renewable sources requires the use of stochastic programming employing scenario trees, here developed using quasi-Monte Carlo techniques. Medium-term pumping schemes are also considered. Several realistic cases will be solved for two behavioral principles of a pure pool market: endogenous cartel and equilibrium.

Medium-term load matching in power planning

The medium-term planning for a company participating in a pure pool market can be considered under different behavioral principles: cartel, endogenous cartel with respect to hydro generation, and equilibrium. One of the important aspects of the model is the medium-term load matching, where load is represented by the load duration curves of each period into which the medium-term horizon is subdivided. The probabilistically exact Bloom and Gallant formulation of load matching could be employed, but its exponential number of load-matching inequality constraints limits its use to small cases. A heuristic has been proposed where only a small subset of the load-matching constraints, hopefully containing the active ones at the optimizer, is employed. This heuristic does not perform totally well with the more complex models, as those of endogenous cartel and equilibrium behavior, and a more comprehensive heuristic procedure is presented here. A well-known alternative method of medium-term load matching is the multi-block approximation to the load duration curve. The three load-matching techniques mentioned above are here described and are compared, and computational results for realistic systems are analyzed for the different behavioral models, showing the advantages of each procedure.

Warmstarting for interior point methods applied to the long-term power planning problem

The long-term planning of electricity generation in a liberalised market using the Bloom and Gallant model can be posed as a quadratic programming (QP) problem with an exponential number of linear inequality constraints called load-matching constraints (LMCs) and several other linear non-LMCs. Direct solution methods are inefficient at handling such problems and a heuristic procedure has been devised to generate only those LMCs that are likely to be active at the optimiser. The problem is then solved as a finite succession of QP problems with an increasing, though still limited, number of LMCs, which can be solved efficiently using a direct method, as would be the case with a QP interior-point algorithm. Warm starting between successive QP solutions helps then in reducing the number of iterations necessary to reach the optimiser. The warm start technique employed herein is an extension of Gondzio and Grotheys approach to quadratic programming problems. We also propose how to initialise new variables in the problem to which a warm start technique is applied. This study shows that warm starting requires on average 50% fewer iterations than a cold start in the test cases solved. The reduction in computation time is smaller, however.

Primal-dual interior point method for multicommodity network flows with side constraints and comparison with alternative methods

This document presents a primal-dual interior point algorithm for the solution of large multicommodity network flow problems with or without side constraints. The method exploits the structure of the problem and uses a preconditioned conjugate gradient solver. The algorithm has been implemented for the case of pure multicommodity problems (without side constraints), and some computational results are presented, comparing the performance of the code developed with alternative ones.

Long-Term Hydrothermal Coordination with Natural Inflows Modelled through a PDF and Simulation Results

SummaryThe long-term hydrothermal coordination of electricity generation has to optimize many variables tied to stochastic parameters. Multicommodity network flows in a replicated reservoir hydronetwork with multicommodity water inflows is one of the possible ways to model and optimize the long-term coordination. In the existing literature on this methodology there are several hypotheses and simplifications that need validation and analysis, and this is the prime purpose of this work. Simulation tests show that the method is sound, and indicate ways to improve the algorithm. The advantages and limitations of taking more or less water commodities are also analyzed.