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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  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$_i$
 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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