Multi-objective modeling, uncertainty analysis, and optimization of reversible solid oxide cells

International Journal of Energy and Environmental Engineering https://doi.org/10.1007/s40095-018-0269-5 ORIGINAL RESEARCH Multi‑objective modeling, uncertainty analysis, and optimization of reversible solid oxide cells Zahra Salehi1 · Iman Gholaminezhad2 Received: 31 October 2017 / Accepted: 7 March 2018 © The Author(s) 2018 Abstract Reversible solid oxide cells can provide efficient and cost-effective scheme for electrical-energy storage applications. However, this technology faces many challenges from material development to system-level operational parameters , which should be tackle for practical purposes. Accordingly, this study focuses on developing novel robust artificial intelligence-based blackbox models to optimize operational variables of the system. A genetic-programming algorithm is used for Pareto modeling of reversible solid oxide cells in a multi-objective fashion based on experimental input–output data. The robustness of the obtained optimal model evaluated using Monte Carlo simulations technique. An optimization study adopted to optimize the operating parameters, such as temperature and fuel composition using a differential evolution algorithm. The objective functions that have been considered for Pareto multi-objective modeling process are training error and model complexity. In addition, the discrepancy between maximum and minimum output voltage in the whole operation of the system is chosen as the optimization process objective function. The robustness of the optimal trade-off model is shown in terms of statistical indices for varied uncertainty levels from 1 to 10%. The optimized operational condition based on the suggested model reveals optimal intermediate temperature of 762 °C and fuel mixture of about 29% H2, 25% H2O, and 14% CO. Keywords Reversible solid oxide cell · Multi-objective · Genetic programming · Pareto · Monte Carlo simulations Abbreviations X Random variable fX (x) Probability density function FX (x) Cumulative distribution function μ (X) Mean σ2 (X) Variance N Number of samples f(x) Probability distribution Xi  ith design variable xi Mole fraction of species I Current density (A cm−2) T Temperature (°C) Vout Output voltage (V) R2 Correlation coefficient OCV Open circuit voltage (V) * Iman Gholaminezhad 1 Department of Materials Science and Engineering, School of Engineering, Shiraz University, Shiraz, Iran 2 Department of Materials and Life Chemistry, Kanagawa University, 3‑27‑1, Rokkakubashi, Kanagawa‑ku, Yokohama, Kanagawa 221‑8686, Japan P Power density (W cm−2) ηtot,i Total overpotential in i mode of operation Objfun Objective function Introduction As part of the efforts to develop new energy conversion systems, there is great interest of using standalone or hybrid renewable energy systems that can help meeting the future demand [1]. In this regard, reversible solid oxide cells receiving increasing scientific and industrial interest. Reversible solid oxide cells (RSOCs) are single-unit, all solid-state, electrochemical devices that can operate in both the fuel cell (SOFC) and electrolysis (SOEC) mode, thus acting as flexible energy conversion and storage systems, particularly to store intermittent renewable energy, such as wind or solar [2–4]. A reversible fuel cell can take advantage of excess electrical grid capacity during off-peak hours to produce hydrogen fuel, to be utilized later during periods of high electrical demand [5, 6]. Artificial intelligence (AI) techniques, such as artificial neural networks (ANNs), support vector machines (SVM), 13 Vol.:(0123456789) International Journal of Energy and Environmental Engineering and genetic programming (GP), are useful methods for black-box modeling of electrochemical systems [7–11]. GP uses the concept of evolutionary computing based on Darwinian theory and natural selection to search over complex space of models to find the global optimum one [12]. There are just few studies on fuel cell system modeling using GP, while there is not any paper on RSOCs using AI techniques. Chakraborty [13] used GP for static and dynamic modeling of solid oxide fuel cells. He well showed the superiority of GP compared to radial basis function neural networks in various modeling approaches. Chakraborty [14] also applied GP for modeling and simulation of SOFC output voltage versus fuel utilization behavior. Nazari [15] utilized GP for output voltage prediction of PEM fuel cells. In his study, variety of input parameters, such as current density, fuel cell temperature, anode and cathode humidification temperature, operating pressure, fuel cell type, and oxidant flow rate, are considered. There are also various optimization studies on SOFCs from various operational and microstructural aspects based on mathematical and artificial intelligence models. Bozorgmehri and Hamedi [16] proposed a neural network model of anode-supported SOFC. They used a genetic algorithm to optimize the neural network model to improve the performance of SOFC. Behzadi and Roshandel [17] implemented multi-objective optimization of SOFC stack by considering the effects of fuel utilization and hydrogen cost. Quddus et al. [18] implemented multi-objective optimization for oxidative coupling of methane using genetic algorithms by considering maximization of power and C2 selectivity and also minimization of the production of undesired side products (COx ). Borji et al. [19] optimized the performance of an anode-supported methane fed SOFC by obtaining a tradeoff between system efficiency and output power considering pre-reforming rate, fuel utilization, air ratio, average current density, and steam-to-carbon ratio as design variables. More recently, Gholaminezhad et al. [20] applied a multi-objective optimization and uncertainty analysis of methane fed SOFCs for maximum power density and efficiency performance achievement. This is while, there is not any work regarding optimization of reversible solid oxide cells in the literature. In this work, a differential evolution algorithm is used to optimize a developed genetic-programming-based RSOC model for maximum performance. Uncertainty analysis of the obtained optimum design also is important for practical purposes due to various sources of uncertainties in real operation of the system. Such uncertainty analysis can be accomplished by sensitivity analysis tools and sampling methods such as Monte Carlo simulations (MCSs) [21, 22] and Latin hypercube sampling (LHS) [23]. MCS is a direct and simple numerical method for uncertainty quantification and is used in this research for 13 stochastic analysis of obtained optimum design solutions [21]. It generates random samples considering pre-defined probabilistic distributions for uncertain parameters. In this study, a multi-objective genetic-programming algorithm is deployed for modeling reversible solid oxide cells considering various operational parameters. The objective functions that have been considere (...truncated)