Framework of model selection criteria approximated genetic programming for optimization function for renewable energy systems

Abstract

For the realization of complex renewable energy systems (such as nano-fluids based direct absorption solar collector), an evolutionary system identification method such as genetic programming (GP) can be applied to develop mathematical models/functional relationships between the process parameters. The system complexity is attributed to interaction among the design variables influencing the outputs. There are also uncertainties in the system due to random and unknown variations in the design and response variables. GP suffers from the higher complexity structure of its solutions and non-optimal convergence, which leads to poor fitness values. Therefore, to address these uncertainties and problems, the framework based on the model selection criteria approximated genetic programming (MSC-GP) is proposed for the formulation of geometry design based thermal efficiency and entropy generation optimization function for direct absorption solar collector (DASC) system. In this proposed method, the four mathematical model selection criteria are used as an approximation for objective functions in GP framework for the evaluation of fitting degree and structure of the model. The results based on statistical measures (best fitness, mean fitness, standard deviation of fitness, number of nodes) show that models obtained from the mathematical selection criteria, Predicted Residual error sum of squares (PRESS), have performed the best. Based on Pareto front analysis of PRESS function, it is found that the best objective values and the number of nodes of models (complexity) follows more or less gradually slow increasing trend which is a good symbolic desirable sign of minimal increase of complexity of model with a decrease in objective values as the values of generation increases. The results of the sensitivity analysis show that the main factor affecting the efficiency of DASC is its geometry of the structure. 3-D interaction analysis shows that increasing the thickness, length and reducing the width of the collector can make the system maintain its higher thermal efficiency and a smaller entropy generation, which is useful for the optimized operation of DASC. Non-dominated sorting genetic algorithm-II (NSGA-II) is applied in the acquisition of the optimal geometric settings of DASC system based on the selected models. The optimal settings achieved is 5 cm in length, 5 cm in width, and 2 cm in thickness. Systems when operated using these settings results in a satisfactory performance with 77.8117% in thermal efficiency and 6.0004E+3 in entropy generation).

Introduction

Physical understanding, prediction and optimization of complex renewable energy systems is a difficult problem. To address this problem, an evolutionary system identification method, such as genetic programming (GP), can be applied to develop mathematical models/functional relationships between the process parameters. The obtained process parameters can be further optimized to obtain the approximation state of the system.

Among the complex renewable energy systems, one such system includes solar collectors. One of the most common collectors is the traditional flat surface collector, which absorbs solar energy through black or spectral selective solid surfaces. The absorber transfers heat to the collector through conduction and then transfers energy to the circulating fluid through convection. However, the thermal performance of these types of solar collectors is limited by the absorption capacity of the black surface and how heat is effectively transferred to the workflow. In order to overcome the shortcomings of traditional solid surface collectors, several methods have been proposed. One proposed method is a collector in which solar radiation is absorbed directly by the working fluid instead of a thin layer on the surface. Therefore, in direct absorption solar collector (DASC), the thermal resistance of converting solar energy into thermal energy is reduced. The concept of direct absorption originated in the 1980s to simplify the design of surface absorbers for fluid plates and potentially improve thermal performance through direct radiation absorption [1]. Typical heat carrier fluids used in solar collectors (i.e., water, ethylene glycol, propylene glycol and heat-conducting oil) have been proven to have poor thermophysical properties and deficient absorption properties in solar spectra. Among the four abovementioned liquids, water is considered as the best absorbent of solar energy. However, it only absorbs about 13% of the energy in previous tests, which implies it is still a weak absorbent [2]. Therefore, the improvement of the thermal performance of the DASC system by improving the ability of the liquid to absorb solar energy has become popular. The thermal performance of DASC systems can be improved by implanting small particles in the matrix which absorbs solar energy. Some researchers introduced the use of liquids with black particles (dispersing Indian ink into a mixture of ethylene glycol and water), to absorb the solar radiation [1].

In recent years, several types of nanoparticles have been synthesized and dispersed in the typical matrix. The SU-Pension of nanoparticles is more stable than that of micron or larger particles. Besides, for suspensions containing nanoparticles, blockage and fouling are found to have little effect. Many studies have shown that the radiation properties of alkaline liquids can be improved by adding an appropriate number of nanoparticles (volume less than 1%). Spectral transmittance and absorptivity of various nanofluids are reported [3], [4], [5], [6], [7]. Researchers use mathematical theories, such as the Rayleigh scattering method [7,8], Maxwell Garnett effective medium theory [7], and Mie scattering theory [9], [10], [11].

The experimental and numerical studies on the direct absorption collector show that the collection efficiency can be improved by using Nano-fluids as the fluid medium. In prior works, the performance of the non-centralized direct absorption solar collector is theoretically studied and compared with that of a typical flat plate solar collector [1]. The mixtures of water and aluminum nanoparticles are 10% more efficient than flat collectors. The experimental results of solar collectors based on carbon nanotubes, graphite and silver nanofluids show that the efficiency can be increased by 5% [12,13]. The feasibility of using graphite nanoparticles dispersed in thermosensitive alcohol VP-1 to concentrate DASC in the laboratory is studied. When the volume fraction of graphite is 0.125%, the efficiency of the base solution is increased by about 11%. Lenert and Wang used Graphite-Coated cobalt to enrich the solar absorption column in thermosensitive alcohol VP-1 nanofluids. The experimental results show that under the optimum conditions, the system efficiency exceeds 35% [14]. Lee et al. employed numerical simulation in the prediction of DASC efficiency. Their results indicated that at 0.05% particle volume concentration, the application of gold-plated silica nanoparticles is able to increase the efficiency of DASC by 70%. Bandarra Filho et al. studied the photothermal conversion ability of water-based silver nanofluids under practical conditions. His-studies concluded that when the concentration of nanoparticles is 6.5 ppm, the stored heat energy can be increased by 144% [15]. Luo et al. performed experiments on a tex-atherm oil-enriched direct absorption column. His-studies show that the photo-thermal efficiency of 0.01 Vol. graphite and 0.5 Vol. Al2O3 nanofluids are 22.7% and 17.5% higher than that of conventional coated collectors [16]. The theoretical study of copper-water nanofluids direct absorption solar collector from Pavin et al. [17] shows that the heat transfer performance and entropy production of the collector is improved when the volume fraction is less than 3% [17].

It can be seen that most of the previous studies emphasized the photothermal conversion efficiency of nanofluids in pre-designed DASC without the incorporation of the geometry of the collector. The system complexity is attributed to interaction among the design variables influencing the outputs. There are also uncertainties in the system due to random and unknown variations in the design and response variables. In a study conducted by Gorji and Ranjbar [1], the effect of different geometric shapes on the overall thermal performance of DASC based on nanofluids is evaluated. The fluent simulation in ANSYS and response surface methodology (RSM) is used. The principle functionality of RSM is based on the statistical assumptions and fits well for the data given in the design domain range. In addition, the models formulated from these methods may not work well for the extrapolation/interpolation cases. Artificial neural network is thought to be a powerful method for numerical model analysis. However, the relatively fixed model structure and form may affect the flexibility of model fitting. Meanwhile, over-fitting may occur in the model generation, which negatively affects the calculation and prediction performance of ANN models [18]. In this situation, an evolutionary system identification method, such as GP, can be applied to develop mathematical models/functional relationships between the process parameters.

The evolutionary search in GP is guided by objective functions used in its framework. Based on preliminary studies, it is found that the GP suffers from the high complexity structure of its solutions and non-optimal convergence, which would lead to poor fitness values. The reasonable selection of mathematical criteria can effectively guide the evolutionary search towards global fitness and penalize the highly complex models and ensures satisfactory accuracy simultaneously. To address this uncertainty/problem, the framework based on model selection criteria approximated genetic programming (MSC-GP) is proposed for the formulation of geometry optimization function for the DASC system. In the proposed method of MSC-GP, the four mathematical model selection criteria are used as an approximation for objective functions in GP framework for evaluation of the fitting accuracy and structure of the model. The proposed and formulated model is used for studying the factors affecting the overall heat dissipation performance of the DASC system.

In general, the geometric design shall affect the thermal behavior and performance of the DASC system. Furthermore, the selection of the objective function is the vital factor for the improvement of performance of the generated model. Objective function comprises both empirical error term and complexity term, which has a greater influence on the performance of models. Therefore, this paper proposes a method combined simulation experiment and artificial intelligence (MSC-GP) to study the effect of geometric design on the thermal behavior of DASC systems, which provides a novel strategy for investigating the performance of DASC system. Findings from parametric interaction analysis and optimization in this research are useful in the optimization of functioning of DASC system. Thus, more studies can be done in future to evolve new measures of defining its complexity to improve the MSC-GP method to improve the efficiency of researches on DASC systems.

The rest of the paper is arranged as follows. Section 2 provides a general view of the research problem. Section 3 describes methodology proposed in formulation of model development. Section 4 discusses the results: the performance of GP models by applying different objective functions, fitting performance comparison of models generated by GP, ANN and RSM, evaluation and selection of generated models, sensitivity analysis and parametric analysis, and the optimization settings for the DASC system. Section 5 concludes the paper and recommendations for future work.