Cuckoo Search: A new nature-inspired optimization method for phase equilibrium calculations

Abstract

In this study, Cuckoo Search is introduced for performing phase equilibrium and stability calculations for the first time. Cuckoo Search is a population-based method that mimics the reproduction strategy of cuckoos. This meta-heuristics have been successfully used for solving some engineering design and optimization problems with promising results. However, this emerging optimization method has not been applied in chemical engineering problems including thermodynamic calculations. This study reports the application of Cuckoo Search and its modified version for phase equilibrium and stability calculations in both reactive and non-reactive systems. Performance of this nature-inspired optimization method has been analyzed using several phase stability, phase equilibrium and reactive phase equilibrium problems. Results show that Cuckoo Search offers a reliable performance for solving these thermodynamic calculations and is better than other meta-heuristics previously applied in phase equilibrium modeling.

Introduction

Phase equilibrium calculations are a relevant step in any simulator that aims to successfully design, develop and analyze an industrial process [1], [2]. In particular, novel chemical industrial processes usually involve intensified operations like reactive distillation and supercritical fluid extraction, which may show a complex phase behavior at different operating conditions. Under these scenarios, accurate and reliable computations of phase property for any system are crucial due to their direct impact on process performance, costs and energy estimations.

Overall, the prediction of phase behavior in a mixture involves the solution of two relevant thermodynamic problems: phase stability (PS) and phase equilibrium (PE) calculations [2]. PS problems involve the determination of whether a thermodynamic system will remain at one phase at a given operating conditions (i.e., temperature, pressure and composition) or split into two or more phases [3]. This type of problems usually precedes the PE problems, which involve the identification and determination of the quantity and composition of the phases at equilibrium at specified conditions [2], [4]. If a chemical reaction is possible between the components of the mixture under analysis, then reactive phase equilibrium calculations (i.e., the simultaneous physical and chemical equilibrium) (rPE) are performed [5]. Basically, PS problems require the global optimization of tangent plane distance function (TPDF) [2], [3], whereas PE and rPE calculations require the corresponding minimization of the Gibbs free energy function [2], [4], [6]. Available literature [1], [2] has highlighted the numerical challenges for performing the global minimization of Gibbs free energy and TPDF. In particular, the characteristics (i.e., number and type) of phases at equilibrium are unknown a priori and the non-convexity of thermodynamic functions, caused by high non-linearity of thermodynamic models, implies the presence of both trivial and local optimal solutions for PE, rPE and PS problems. Objective functions used for phase equilibrium calculations may have comparable local optimal values with the global optimum (e.g., when the mixture operating conditions are near to the phase boundaries or critical conditions), which increases the global optimization complexity [7].

To date, a number of deterministic and stochastic global optimization methods [2], [4], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38] have been proposed and applied to solve PS, PE and rPE problems but they all have their own capabilities and demerits. In particular, most deterministic methods may require large computational time (e.g., branch and bound methods and homotopy continuation approaches) [8], [9], [10], [11], complex calculations (e.g., Jacobian matrix calculations in Newton Interval method) [12], [13], [14], [15], mathematical reformulation and certain assumptions about the nature of the optimization problem in order to obtain a reasonable chance of success. On the other hand, stochastic optimization methods require no prior assumptions or transformation of the objective function and can find the global minimum with high probability in less computational time [2]. Different meta-heuristics have been applied to perform PE, rPE and PS calculations in multicomponent and multireactive systems [4], [7], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38]. They include a variety of point-to-point and population-based meta-heuristics, e.g., Simulated Annealing [4], [26], [28], [32], Genetic Algorithms [4], [7], Tabu Search [24], [35], Random Tunneling [27], Differential Evolution [7], [30], [31], [35], Particle Swarm Optimization [33], [34], [36], Evolutionary Algorithms [29], [37], Harmony Search [38], Firefly Algorithm [37] and other hybrid methods [37]. These stochastic algorithms have been found more promising than the deterministic global optimization methods but they too have their own limitations for finding the global optimum in very challenging problems. In addition, they often enforce a trade-off between speed and reliability in both phase stability and equilibrium calculations. To date, there is no single stochastic method capable of solving all optimization problems of different types and structures [39]. This situation implies that the requirement to develop new meta-heuristics is still a hot research topic in current literature of phase equilibrium modeling.

Nature-inspired meta-heuristic methods are considered as an emerging computing paradigm, which are becoming increasingly popular and promising to solve challenging global optimization problems involved in several real-world applications [40]. These methods are defined as those algorithms derived by mimicking natural phenomena and biological models; they are capable of outperforming other optimization techniques by providing a better numerical performance. In particular, Yang and Deb [41], [42] introduced a new nature-inspired stochastic method called Cuckoo Search (CS). It is a population-based method that mimics the reproduction strategy of cuckoos. This novel meta-heuristics have been used for solving some engineering design and optimization problems with promising results [43], [44]. Recently, a modification of this method called Modified Cuckoo Search (MCS) was introduced by Walton et al. [45] to enable faster convergence without losing the attractive features of traditional CS method. Both strategies have shown improved convergence characteristics with respect to the results obtained for other traditional stochastic optimization methods. Despite these findings, it appears that there are no studies on the solution of optimization problems from chemical engineering, including thermodynamic applications, using this meta-heuristic. In this article, Cuckoo Search is introduced as a new stochastic optimization method for phase equilibrium modeling. The performance of this novel stochastic method has been studied in solving PS, PE and rPE optimization problems. To the best of our knowledge, the application of CS to these thermodynamic calculations has not been reported and for the first time this topic is investigated in the literature. The aim of our study is to identify the relative strengths of CS for phase equilibrium calculations with and without chemical equilibrium. This study shows that CS offers a reliable performance and is better than other metaheuristics for solving these thermodynamic calculations.