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March  2021, 14(3): 1161-1180. doi: 10.3934/dcdss.2020227

## Comparison of modern heuristics on solving the phase stability testing problem

 Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Trojanova 13,120 00 Prague 2, Czech Republic

* Corresponding author: Tomáš Smejkal

Received  January 2019 Revised  November 2019 Published  March 2021 Early access  December 2019

In this paper, we are concerned with the phase stability testing at constant volume, temperature, and moles ($VTN$-specification) of a multicomponent mixture, which is an unconstrained minimization problem. We present and compare the performance of five chosen optimization algorithms: Differential Evolution, Cuckoo Search, Harmony Search, CMA-ES, and Elephant Herding Optimization. For the comparison of the evolution strategies, we use the Wilcoxon signed-rank test. In addition, we compare the evolution strategies with the classical Newton-Raphson method based on the computation times. Moreover, we present the expanded mirroring technique, which mirrors the computed solution into a given simplex.

Citation: Tomáš Smejkal, Jiří Mikyška, Jaromír Kukal. Comparison of modern heuristics on solving the phase stability testing problem. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1161-1180. doi: 10.3934/dcdss.2020227
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##### References:
Geometric interpretation of mirroring into the feasible simplex
Global minimum of the $\mathrm{TPD}$ function in the $cT$-space and the number of successful runs of each evolution heuristic. The red line represents the phase boundary (above the line the global minimum is zero). Example 1: mixture C$_1$-C$_3$
Global minimum of the $\mathrm{TPD}$ function in the $cT$-space and the number of successful runs of each evolution heuristic. The red line represents the phase boundary (above the line the global minimum is zero). Example 2: mixture N$_2$-CO$_2$-C$_1$-PC$_i$
The advantages and disadvantages of the chosen evolution algorithms
 advantages disadvantages DE ● good convergence properties ● parameter tuning is necessary ● strong theoretical analysis ● easy to stuck in a local minimum CS ● supports local and global search ● small precision ● easy to hybridize ● no theoretical analysis HS ● simple implementation ● slow convergence ● small population ● small precision CMA-ES ● no curse of dimensionality [2] ● harder implementation ● in-variance properties ● a lot of parameters EHO ● hard to stuck in a local minimum ● slow convergence ● fewer parameters ● fixed parameters
 advantages disadvantages DE ● good convergence properties ● parameter tuning is necessary ● strong theoretical analysis ● easy to stuck in a local minimum CS ● supports local and global search ● small precision ● easy to hybridize ● no theoretical analysis HS ● simple implementation ● slow convergence ● small population ● small precision CMA-ES ● no curse of dimensionality [2] ● harder implementation ● in-variance properties ● a lot of parameters EHO ● hard to stuck in a local minimum ● slow convergence ● fewer parameters ● fixed parameters
Parameters of the Peng-Robinson equation of state used in Examples 1–2
 Component $T_{\mathrm{crit}}$ [K] $P_{\mathrm{crit}}$ [MPa] $\omega$ [-] C$_1$ 190.40 4.60 0.0110 C$_3$ 369.80 4.25 0.1530 CO$_2$ 304.14 7.375 0.2390 N$_2$ 126.21 3.390 0.0390 PC$_1$ 333.91 5.329 0.1113 PC$_2$ 456.25 3.445 0.2344 PC$_3$ 590.76 2.376 0.4470 C$_{12+}$ 742.58 1.341 0.9125
 Component $T_{\mathrm{crit}}$ [K] $P_{\mathrm{crit}}$ [MPa] $\omega$ [-] C$_1$ 190.40 4.60 0.0110 C$_3$ 369.80 4.25 0.1530 CO$_2$ 304.14 7.375 0.2390 N$_2$ 126.21 3.390 0.0390 PC$_1$ 333.91 5.329 0.1113 PC$_2$ 456.25 3.445 0.2344 PC$_3$ 590.76 2.376 0.4470 C$_{12+}$ 742.58 1.341 0.9125
The binary interaction coefficients between all components in Example 2
 Component N$_2$ CO$_2$ C$_1$ PC$_1$ PC$_2$ PC$_3$ C$_{12+}$ N$_2$ 0.000 0.000 0.100 0.100 0.100 0.100 0.100 CO$_2$ 0.000 0.000 0.150 0.150 0.150 0.150 0.150 C$_1$ 0.100 0.150 0.000 0.035 0.040 0.049 0.069 PC$_1$ 0.100 0.150 0.035 0.000 0.000 0.000 0.000 PC$_2$ 0.100 0.150 0.040 0.000 0.000 0.000 0.000 PC$_3$ 0.100 0.150 0.049 0.000 0.000 0.000 0.000 C$_{12+}$ 0.100 0.150 0.069 0.000 0.000 0.000 0.000
 Component N$_2$ CO$_2$ C$_1$ PC$_1$ PC$_2$ PC$_3$ C$_{12+}$ N$_2$ 0.000 0.000 0.100 0.100 0.100 0.100 0.100 CO$_2$ 0.000 0.000 0.150 0.150 0.150 0.150 0.150 C$_1$ 0.100 0.150 0.000 0.035 0.040 0.049 0.069 PC$_1$ 0.100 0.150 0.035 0.000 0.000 0.000 0.000 PC$_2$ 0.100 0.150 0.040 0.000 0.000 0.000 0.000 PC$_3$ 0.100 0.150 0.049 0.000 0.000 0.000 0.000 C$_{12+}$ 0.100 0.150 0.069 0.000 0.000 0.000 0.000
Computation times in seconds for Examples 1–2
 Example 1 Example 2 Newton-Raphson 0.99 11.55 Differential Evolution 35.87 995.03 Cuckoo Search 78.55 394.63 Harmony Search 210.72 862.48 CMA-ES 26.48 408.91 Elephant Herding Optimization 500.30 1777.72
 Example 1 Example 2 Newton-Raphson 0.99 11.55 Differential Evolution 35.87 995.03 Cuckoo Search 78.55 394.63 Harmony Search 210.72 862.48 CMA-ES 26.48 408.91 Elephant Herding Optimization 500.30 1777.72
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