An efficient orthogonal opposition-based learning slime mould algorithm for maximum power point tracking

Abstract

The slime mould algorithm (SMA) is a recent physics-based optimization approach. The main inspiration of the SMA is motivated by the natural oscillating state of the slime mould organisms. In order to boost the performance, several problems must be resolved properly on the original SMA itself. One of these problems is the dilemma of the improper balancing between the exploration and exploitation phases which might deviate the algorithm to be trapped in the local optima. This work introduces a new version of the SMA called mSMA-based on the hybridization of the original SMA with a modified version of the opposition-based learning (mOBL) and the Orthogonal learning (OL) strategies. To assess the performance of the proposed mSMA, it has been evaluated over ten CEC’2020 test suites and three engineering design problems. As the output performance of the thermoelectric generator (TEG) is mainly based on the applied temperatures on the hot and cold sides of the TEG together with the load value. Consequently, in case of either varying the applied temperature or the load, to force the TEG to operate as close as possible to the maximum power point (MPP), a robust maximum power point tracking (MPPT) strategy is highly required. Therefore, an optimized fractional-order (FO) MPPTS is proposed to increase the delivered energy from the TEG. The suggested strategy is based on the FO control approach. The optimal parameters of the optimized fractional MPPTS were identified by the new mSMA. To demonstrate the superiority of mSMA, the results are compared to other well-known algorithms such as the ABC, GSA, PSO, HHO, TSA, GBO, HBO, and the original SMA. The main purpose of the proposed optimal fractional MPPTS is to increase the dynamic response and to remove the oscillations that occurred at the steady-state response. Therefore, the performance of the proposed strategy is compared to two common methods; the incremental resistance and the perturb & observe. The obtained results proved the superiority of the optimized fractional MPPTS in comparison to the other traditional MPPT methods in both the dynamic and steady-state responses.

Appendices

Appendix A: tension/compression spring design problem

The mathematical model of the Tension/compression spring design problem is as follows:

$$\begin{aligned}&\mathbf {x}=\left[ {{x}_{1}}\,{{x}_{2}}\,{{x}_{3}} \right] =\left[ d\,D\,N \right] \\&\text {Minimize } f\left( {\mathbf {x}} \right) =\left( {{x}_{3}}+2 \right) {{x}_{2}}x_{1}^{2} \\&\text {subject to: } \\&{{g}_{1}}\left( {\mathbf {x}} \right) =1-\frac{x_{2}^{3}{{x}_{3}}}{71785x_{1}^{4}}\le 0 \\&{{g}_{2}}\left( {\mathbf {x}} \right) =\frac{4x_{2}^{2}-{{x}_{1}}{{x}_{2}}}{12566\left( {{x}_{2}}x_{1}^{3}-x_{1}^{4} \right) }+\frac{1}{5108x_{1}^{2}}-1\le 0 \\&{{g}_{3}}\left( {\mathbf {x}} \right) =1-\frac{140.45{{x}_{1}}}{x_{2}^{2}{{x}_{3}}}\le 0 \\&{{g}_{4}}\left( {\mathbf {x}} \right) =\frac{{{x}_{1}}+{{x}_{2}}}{1.5}-1\le 0 \\&\text {with } 0.05\le {{x}_{1}}\le 2.0,0.25\le {{x}_{2}}\le 1.3\,\,\, \text {and}, \,2.0\le {{x}_{3}}\le 15.0 \\ \end{aligned}$$

Appendix B: pressure vessel design problem

The mathematical model of the pressure vessel design problem is as follows:

$$\begin{aligned}&\text {Minimize }f(x)=0.6224{{x}_{1}}{{x}_{3}}{{x}_{4}}+1.7781{{x}_{2}} x_{3}^{2}+3.1661x_{1}^{2}{{x}_{4}}+19.84x_{1}^{2}{{x}_{3}} \\&\text {Subject to: }\\&{{g}_{1}}(x)=-{{x}_{1}}+0.0193x \\&{{g}_{2}}(x)=-{{x}_{2}}+0.00954{{x}_{3}}\,\le 0 \\&{{g}_{3}}(x)=-\pi x_{3}^{2}{{x}_{4}}-(4/3)\pi x_{3}^{3}+1,296,000\le 0 \\&{{g}_{4}}(x)={{x}_{4}}-240\le 0 \\&0\le {{x}_{i}}\le 100,\,\,i=1,2 \\&10\le {{x}_{i}}\le 200,\,\,i=3,4 \\ \end{aligned}$$

Appendix C: rolling element bearing design problem

The mathematical model of the rolling element bearing design problem is as follows:

$$\begin{aligned}&\text {Maximum}[C_d(X)]={\left\{ \begin{array}{ll}max(-f_cz^{2/3}D_b^{1.8}) &{} if D_b =\le 25.5\,\text{mm} \\ max(-3.647f_cz^{2/3}D_b^{1.4}) &{} if D_b =\le 25.5\,\text{mm} \end{array}\right. } \\&\text {Subject to: }\\&g_1(x)=\frac{\phi _0}{2\sin ^{-1}(D_b/D_m)}-Z+1\ge 0, \\&g_2(x)=2D_b-K_{D_{min}}(D-d)\ge 0,\\&g_3(x)=K_{D_{max}}(D-d)-2D_b\ge 0,\\&g_4(x)=\zeta B_w-D_b\ge 0,\\&g_5(x)=D_m-0.5(D+d)\ge 0,\\&g_6(x)=(0.5+e)(D+d)-D_m\ge 0,\\&g_7(x)=0.5(D-D_m-D_b)-\zeta D_b\ge 0,\\&g_8(x)=f_1\ge 0.515,\\&g_9(x)=f_0\ge 0.515,\\&where \\&f_c=37.91\left[ 1+\left[ 1.04\left( \frac{1-\gamma }{1+\gamma }\right) ^{1.72}\right] \left( \frac{f_i(2f_0-1)}{f_i-1}\right) ^{0.41}]^{10/3}\right] ^{-0.3},\\&\gamma = \frac{D_b\cos \alpha }{D_m},\\&f_1=\frac{r_1}{D_b}\\&\phi _0=2\pi -20\cos ^{-1}\left[ \frac{(D-d)/2-3(T/4)^2+[D/2-(T/4-D_b)^2] -[d/2+(T/4)]^2]}{[2(D-d/2-3(T/4)][(T/4)-D_b]}\right] \\&T=D-d-2D_b,\\&D=160, d=90, B_w=30,\\&0.5(D+d)\le D_m\le 0.6(D+d),\\&0.15(D-d)\le D_b\le 0.45(D-d),\\&4\le Z\le 50, 0.515 \le f_1\le 0.6, 0.515\le f_0 \le 0.6,\\&0.4 \le K_{D_{min}}\le 0.5, 0.6 \le K_{D_{max}}\le 0.7,\\&0.3\le \epsilon \le 0.4, 0.02\le e \le 0.1, 0.6\le \xi \le 0.85. \end{aligned}$$