Evolutionary algorithms for multi-objective energetic and economic optimization in thermal system design
Introduction
The optimization of energy system design consists of modifying the system structure and component design parameters according to one or more specified design objectives [1]. In general, multiple objectives are involved in the design process [2]: thermodynamic (e.g., maximum efficiency, minimum fuel consumption), economic (e.g., minimum cost per unit of time, maximum profit per unit of production) and environmental (e.g., limited emissions, minimum environmental impact). However, most of the analyses performed in the past consider either only the thermodynamic objective or only the economic one.
In the field of thermoeconomics [3], [4], [5], [6], [7], [8], [9], [10], design optimization aims at minimizing the total levelized cost of the system products, which implicitly includes thermodynamic information in the fuel cost rate through the fuel exergy flow rate. Various methodologies have been suggested in the literature as ways to pursue this objective, based on different approaches [3], [6], [7]. An example comparison of four different thermoeconomic methodologies to the design optimization of a cogeneration plant has been presented as a test case, known as the CGAM problem [11], [12], [13], [14], [15]: this represents a paradigmatic application of a single objective optimization.
The aim of the present work is to enlarge the perspective of traditional thermoeconomic analysis, by applying a multi-objective approach in order to determine the complete spectrum of solutions that satisfies the economic objective as well as the energetic one. These objectives usually compete with each other, so that it is impossible to find a solution that simultaneously satisfies both of them. Therefore, it is necessary to find the set of optimal solutions that lead to the highest values of the efficiency at fixed costs, or to the lowest costs of production at fixed efficiencies. Note that the same spectrum of optimal solutions could be obtained by a single-objective approach weighting explicitly or implicitly the two objectives into an overall single-objective function and varying the weight coefficients. This procedure, however, would require a much higher computational effort since an optimization run is needed per each combination of the weight coefficients, whereas a multi-objective approach gets the final solution in a single run of the optimization algorithm. Moreover, a “pure” single-objective approach that considers only the economic objective or only the thermodynamic one would search just for one of the two extreme points on the spectrum of optimal solutions: one corresponds to the minimum of the cost objective function, the other to the maximum of the efficiency objective function. In particular, only the former solution is considered in the CGAM problem, but this choice of design variables excludes other sets of optimal design variables corresponding to plants having higher efficiencies at the expense of higher total costs. Using a multi–objective approach, the decision maker is able to choose from the entire set of solutions having the lowest costs at higher efficiencies. This also means taking into account higher technology plants that save more fuel than those with minimum costs or, from the other viewpoint, plants that involve minimum expenditures at a given efficiency.
The Pareto approach to multi-objective optimization is used here to find the optimal set of design variables, since the concepts of Pareto dominance and optimality are straightforward tools for determining the best trade-off solutions among conflicting objectives. An evolutionary algorithm is then chosen to carry out the search for the Pareto optimal solution, because evolutionary optimization techniques already deal with a set of solutions (a “population”) to pursue their task, so a multi-objective Pareto-based evolutionary algorithm is able to make the population converge to the entire set of optimal solutions in a single run. The algorithm proposed in this paper features an innovative diversity preserving mechanism [16] that helps solutions spread over the Pareto optimal set. This mechanism introduces an additional evaluation step to improve the exploration of the search space, assigning a better rank to high performance solutions and genetically diverse ones as well. To perform the optimization of the CGAM test case, the algorithm’s routines are interfaced with a MATLAB-Simulink1 model in which the thermodynamic and economic equations of the CGAM problem are implemented and solved, returning the values of the objectives for a given set of decision variables.
Section snippets
Pareto approach to multi-objective optimization
A multi-objective optimization problem requires the simultaneous satisfaction of a number of different and often conflicting objectives. These objectives are characterized by distinct measures of performance that may be (in)dependent and/or incommensurable. In thermal system design, for example, the energetic efficiency and the money spent per unit of time may have little dependence on each other (the cost for purchase of equipment and maintenance does not depend on energetic efficiency
The CGAM problem
The CGAM problem [11] deals with the search for the optimal design parameters of a cogeneration plant producing electrical energy and steam for a set of fixed demands. The two objectives are the maximization of exergetic efficiency and the minimization of the total cost rate. The installation is made up of a basic gas turbine cycle with regeneration and of a heat recovery steam generator for saturated steam production. The plant produces an electrical power of 30 MW and 14 kg/s of saturated
Results
The first two runs of the optimization algorithm were performed to investigate the effect of the constraint on the temperature (>105°C) of the exhaust gases. The results of these two cases are shown in Fig. 7, Fig. 8, Fig. 9, Fig. 10, where dots and triangles represent the optimal solutions with and without this constraint, respectively.
Conclusions
An evolutionary-based procedure was presented for the exergetic and economic design optimization of thermal systems.
The proposed multi-objective evolutionary algorithm was shown to be a powerful and effective tool in finding the set of the best trade-off solutions between the two competing objectives for the choice of design parameters in the CGAM cogeneration plant. The mathematical complexity of the optimization problem (five decision variables, two objective functions and several constraints
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