An optimal solution to unit commitment problem of realistic integrated power system involving wind and electric vehicles using chaotic slime mould optimizer

Dhawale, Dinesh; Kamboj, Vikram Kumar; Anand, Priyanka

doi:10.1186/s43067-023-00069-2

Journal of Electrical Systems and Information Technology

Table 2 Chaotic functions [14]

From: An optimal solution to unit commitment problem of realistic integrated power system involving wind and electric vehicles using chaotic slime mould optimizer

Sr. No	Chaotic name	Mathematical description
1	Chebyshev	$y_{i + 1} \, = \,\cos \left( {\cos^{ - 1} \left( {y_{i} } \right)} \right)$
2	Iterative	$y_{i + 1\,} = \,{\text{Sin}}\left( {{\raise0.7ex\hbox{${a\,\pi }$} \!\mathord{\left/ {\vphantom {{a\,\pi } {y_{i} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${y_{i} }$}}} \right)\,,\,a = 0.7$
3	Sinusoidal	$y_{i + 1} \, = ax_{i} \,{\text{Sin}}\left( {\pi xi} \right)\,;\,a\, = \,2.3$
4	Sine	$y_{i + 1} \, = \,\frac{a}{4}\,\,{\text{Sin}}\left( {\pi yi} \right)\,,a = 4\,$
5	Circle	$y_{i + 1} \, = \,\bmod \,(y_{i} \, + \,b\, - \left( {{\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a {2\pi }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${2\pi }$}}} \right)\,{\text{Sin}}\left( {2\pi y_{i} } \right)\,,\,\left. 1 \right)\,;\,a = 0.5,\,b = 0.2$
6	Piecewise	\(\left\{ {\begin{array}{*{20}c} {{\raise0.7ex\hbox{${y_{i} }$} \!\mathord{\left/ {\vphantom {{y_{i} } p}}\right.\kern-0pt} \!\lower0.7ex\hbox{$p$}}} & {0 \le \,y_{i\,} \langle \,p} & {} \\ {{\raise0.7ex\hbox{${\left( {y_{i} \, - p} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {y_{i} \, - p} \right)} {\left( {0.5 - p} \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {0.5 - p} \right)}$}}} & {p \le \,y_{i\,} \,\langle \,0.5} & {} \\ \begin{gathered} {\raise0.7ex\hbox{${\left( {1 - p - y_{i} } \right)}$} \!\mathord{\left/ {\vphantom {{\left( {1 - p - y_{i} } \right)} {\left( {0.5 - p} \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {0.5 - p} \right)}$}} \hfill \\ {\raise0.7ex\hbox{${\left( {1 - y_{i} } \right)}$} \!\mathord{\left/ {\vphantom {{\left( {1 - y_{i} } \right)} p}}\right.\kern-0pt} \!\lower0.7ex\hbox{$p$}} \hfill \\ \end{gathered} & \begin{gathered} 0.5 \le \,y_{i} \,\langle \,1 - p \hfill \\ 1 - p\, \le \,y_{i\,\,} \langle \,1\, \hfill \\ \end{gathered} & {,p = \,0.4} \\ \end{array} } \right.\)
7	Gauss/mouse	$\left\{ {\begin{array}{*{20}c} {1,} & {y_{i} \, = 0} \\ {\frac{1}{{\bmod \left( {y_{i\,} ,1} \right)}}} & {{\text{otherwise}}} \\ \end{array} } \right.$
8	Singer	$y_{i + 1} \, = \,\mu \,\left( {7.86\,y_{i} - \,23.3\,y_{i}^{2} \, + 28.75\,y_{i}^{3} \, - 13.301875\,y_{i}^{4} } \right)\,,\,\mu \, = 1.07$
9	Logistic	$y_{i + 1\,} \, = \,a\,y_{i} \,\left( {1 - y_{i} } \right)\,,\,a = 4\,$
10	Tent	$y_{i + 1} = \,\left\{ {\begin{array}{*{20}c} {\left( {y_{i} /0.7} \right),} & {y_{i} \, < \,0.7} \\ {\left( {10/3} \right)\left( {1 - y} \right),} & {y_{i} \, \ge \,0.7} \\ \end{array} } \right.$

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Sr. No	Chaotic name	Mathematical description
1	Chebyshev	\(y_{i + 1} \, = \,\cos \left( {\cos^{ - 1} \left( {y_{i} } \right)} \right)\)
2	Iterative	\(y_{i + 1\,} = \,{\text{Sin}}\left( {{\raise0.7ex\hbox{${a\,\pi }$} \!\mathord{\left/ {\vphantom {{a\,\pi } {y_{i} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${y_{i} }$}}} \right)\,,\,a = 0.7\)
3	Sinusoidal	\(y_{i + 1} \, = ax_{i} \,{\text{Sin}}\left( {\pi xi} \right)\,;\,a\, = \,2.3\)
4	Sine	\(y_{i + 1} \, = \,\frac{a}{4}\,\,{\text{Sin}}\left( {\pi yi} \right)\,,a = 4\,\)
5	Circle	\(y_{i + 1} \, = \,\bmod \,(y_{i} \, + \,b\, - \left( {{\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a {2\pi }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${2\pi }$}}} \right)\,{\text{Sin}}\left( {2\pi y_{i} } \right)\,,\,\left. 1 \right)\,;\,a = 0.5,\,b = 0.2\)
6	Piecewise	\(\left\{ {\begin{array}{*{20}c} {{\raise0.7ex\hbox{${y_{i} }$} \!\mathord{\left/ {\vphantom {{y_{i} } p}}\right.\kern-0pt} \!\lower0.7ex\hbox{$p$}}} & {0 \le \,y_{i\,} \langle \,p} & {} \\ {{\raise0.7ex\hbox{${\left( {y_{i} \, - p} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {y_{i} \, - p} \right)} {\left( {0.5 - p} \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {0.5 - p} \right)}$}}} & {p \le \,y_{i\,} \,\langle \,0.5} & {} \\ \begin{gathered} {\raise0.7ex\hbox{${\left( {1 - p - y_{i} } \right)}$} \!\mathord{\left/ {\vphantom {{\left( {1 - p - y_{i} } \right)} {\left( {0.5 - p} \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {0.5 - p} \right)}$}} \hfill \\ {\raise0.7ex\hbox{${\left( {1 - y_{i} } \right)}$} \!\mathord{\left/ {\vphantom {{\left( {1 - y_{i} } \right)} p}}\right.\kern-0pt} \!\lower0.7ex\hbox{$p$}} \hfill \\ \end{gathered} & \begin{gathered} 0.5 \le \,y_{i} \,\langle \,1 - p \hfill \\ 1 - p\, \le \,y_{i\,\,} \langle \,1\, \hfill \\ \end{gathered} & {,p = \,0.4} \\ \end{array} } \right.\)
7	Gauss/mouse	\(\left\{ {\begin{array}{*{20}c} {1,} & {y_{i} \, = 0} \\ {\frac{1}{{\bmod \left( {y_{i\,} ,1} \right)}}} & {{\text{otherwise}}} \\ \end{array} } \right.\)
8	Singer	\(y_{i + 1} \, = \,\mu \,\left( {7.86\,y_{i} - \,23.3\,y_{i}^{2} \, + 28.75\,y_{i}^{3} \, - 13.301875\,y_{i}^{4} } \right)\,,\,\mu \, = 1.07\)
9	Logistic	\(y_{i + 1\,} \, = \,a\,y_{i} \,\left( {1 - y_{i} } \right)\,,\,a = 4\,\)
10	Tent	\(y_{i + 1} = \,\left\{ {\begin{array}{*{20}c} {\left( {y_{i} /0.7} \right),} & {y_{i} \, < \,0.7} \\ {\left( {10/3} \right)\left( {1 - y} \right),} & {y_{i} \, \ge \,0.7} \\ \end{array} } \right.\)