M. M. Ruiz

The Exact Solution of the Environmental/Economic Dispatch Problem

In this paper, the exact analytical solution for the environmental/economic dispatch (EED) optimization problem is presented for the first time. The EED, which simultaneously satisfies multiple contradictory criteria, is stated as a multiobjective optimization problem (MOP). Our paper has improved several aspects of a previous analytical approach. First, we take into account the unit capacity constraints in the exact formulae. Second, we obtain the set of compromise solutions known as Pareto optimal solutions and, third, our treatment of transmission losses satisfies the power balance constraint. In contrast with the known heuristic methods used in the literature, which only provide a reasonable solution (suboptimal, nearly global optimal), our method provides the global solution. Moreover, our method can obtain the Pareto optimal set under different loading conditions. The performance of the proposed technique is validated using a standard test system.

Mathematical modelling of the combined optimization of a pumped-storage hydro-plant and a wind park

Abstract The new Spanish regulations allow wind farms to go to the market to sell the energy generated by their facilities. In the case of over- or under-supply, other producers must reduce or increase their production to resolve the so-called deviation, thereby incurring financial losses. Faced with this situation, wind farms have several options. In this paper we consider one promising method: the combined optimization of a pumped-storage hydro-plant and a wind park. First, we are going to present the mathematical modelling of the resulting problem. Second, we shall solve the optimization problem using Optimal Control techniques and finally we shall present several examples of the combined optimization and analyse which strategy is the best one possible.

An optimization problem in deregulated electricity markets solved with the nonsmooth maximum principle

In this paper, the new short-term problems that are faced by a generation company in a deregulated electricity market are addressed and an optimization algorithm is proposed. Our model of the spot market explicitly represents the price of electricity as an uncertain exogenous variable. We consider a very complex problem of hydrothermal optimization with pumped-storage plants, so the problem deals with non-regular Lagrangian and non-holonomic inequality constraints. To obtain a necessary minimum condition, the problem was formulated within the framework of nonsmooth analysis using the generalized (or Clarkes) gradient and the Nonsmooth maximum principle. The optimal control problem is solved by means of an algorithm implemented in the commercial software package Mathematica. Results of the application of the method to a numerical example are presented.

New developments in the application of Pontryagin's Principle for the hydrothermal optimization

In this paper we have developed a much simpler theory than previous ones that resolves the problem of the optimization of hydrothermal systems. The problem involves non-holonomic inequality constraints. In particular, we have established a necessary condition for the stationary functions of the functional. We shall use Pontryagin’s Minimum Principle as the basis for proving this theorem, setting out our problem in terms of optimal control in continuous time, with the Lagrange-type functional. This theorem allows us to elaborate the optimization algorithm that leads to the determination of the optimal solution of the hydrothermal system. We generalize the problem, taking into account a cost associated with the water, to then set out and solve the corresponding Bolza’s problem. Finally, we present an example employing the algorithm developed for this purpose with the ‘Mathematica’ package.

A quasi-linear algorithm for calculating the infimal convolution of convex quadratic functions

In this paper we present an algorithm of quasi-linear complexity to exactly calculate the infimal convolution of convex quadratic functions. The algorithm exactly and simultaneously solves a separable uniparametric family of quadratic programming problems resulting from varying the equality constraint.

An algorithm for bang-bang control of fixed-head hydroplants

This paper deals with the optimal control (OC) problem that arise when a hydraulic system with fixed-head hydroplants is considered. In the frame of a deregulated electricity market, the resulting Hamiltonian for such OC problems is linear in the control variable and results in an optimal singular/bang–bang control policy. To avoid difficulties associated with the computation of optimal singular/bang–bang controls, an efficient and simple optimization algorithm is proposed. The computational technique is illustrated on one example.

Real-time optimization of wind farms and fixed-head pumped-storage hydro-plants

Renewable energies and, in particular, wind power have come to the forefront in the electricity market in recent years. The main drawback of wind power generation, however, is the major difficulty in forecasting its production. For this reason, when wind farms go to the market, they are very often made to pay penalties for the deviations between forecasting and actual production. In this paper we shall analyse whether real-time compensation of wind power plant deviation penalties is profitable by means of the coordinated optimization of the wind power plant with a large pumped-storage hydro-plant. We shall model the pumped storage plant in great detail and make use of optimal control techniques to carry out the optimization. We shall analyse the most relevant recent papers on the subject and compare them with our technique. We shall also analyse another possible solution based on compensation carried out a posteriori, instead of in real time.

Algorithm for calculating the analytic solution for economic dispatch with multiple fuel units

The problem of economic dispatch with multiple fuel units has been widely addressed via different techniques using approximate methods due to the exponential complexity of full enumeration in the underlying combinatory problem. A method has recently been outlined by Min et al. (2008)[12], that allows the problem to be solved in an exact way in polynomial time. In this paper, we present an alternative technique and take this idea further, studying and comparing two algorithms of polynomial complexity: basic recurrence and divide-and-conquer. Moreover, we provide the exact solution to the problem by Lin and Viviani (1984)[1], that constitutes the traditional test for all approximate methods and present a comprehensive survey of several heuristic approaches.

An environmentally constrained economic dispatch: CFBC boilers in the day-ahead market

Abstract This paper presents an environmentally constrained economic dispatch algorithm for a hydrothermal system within the framework of a competitive and deregulated electricity market. The optimization problem of a firm in a competitive market is described, the objective function of which can be defined as profit maximization, and we consider that thermal plants are constrained to technical and environmental restrictions. An optimal control technique is applied and Pontryagins theorem is employed. The proposed algorithm is implemented using the Mathematica© package and is applied to a sample system.

The operation of infimal/supremal convolution in mathematical economics

The infimal convolution operation arises in mathematical economics in the analysis of several problems. In this paper we first present a survey and summarize two previous papers by the authors on the classic firms cost-minimization problem. Moreover, we present a new application: the analytical solution of the utility maximization problem which we shall obtain applying the supremal convolution operation.