# New Hybrid Algorithms for Prediction of Daily Load of Power Network
## 4 Experiments and Results  
In the section, we use 29 benchmark functions to test. These classical functions, listed in Tables 1-4, are used by many researchers [56; 57]. _Space_ is the boundary of its search range, \(D_{im}\) represents the dimension of the function, and \(f_{min}\) indicates the optimum of it.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Space** & \(\mathbf{D}_{im}\) & \(\mathbf{f}_{\mathbf{min}}\) \\ \hline \(f_{1}(x)=\sum_{i=1}^{n}x_{i}^{2}\) & \([-100,100]\) & 30 & 0 \\ \(f_{2}(x)=\sum_{i=1}^{n}|x_{i}|+\prod_{i=1}^{n}|x_{i}|\) & \([-10,10]\) & 30 & 0 \\ \(f_{3}(x)=\sum_{i=1}^{n}(\sum_{i=1}^{n}x_{i}^{2})\) & \([-100,100]\) & 30 & 0 \\ \(f_{4}(x)=max_{i}(\sum_{i=1}^{n}x_{i}^{2})\) & \([-100,100]\) & 30 & 0 \\ \(f_{5}(x)=\sum_{i=1}^{n-1}[100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2}]\) & \([-30,30]\) & 30 & 0 \\ \(f_{6}(x)=\sum_{i=1}^{n}[(x_{i}+0.5)]^{2}\) & \([-100,100]\) & 30 & 0 \\ \(f_{7}(x)=\sum_{i=1}^{n}x_{i}^{4}+random[0,1)\) & \([-1.28,1.28]\) & 30 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Unimodal benchmark functions.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Space** & \(\mathbf{D}_{im}\) & \(\mathbf{f}_{\mathbf{min}}\) \\ \hline \(f_{8}(x)=\sum_{i=1}^{n}-x_{i}sin(\sqrt{|x_{i}|})\) & \([-500,500]\) & 30 & \(-12\),569 \\ \(f_{9}(x)=\sum_{i=1}^{n}[x_{i}^{2}-10 cos(2\pi x_{i})+10]\) & \([-5.12,5.12]\) & 30 & 0 \\ \(f_{10}(x)=-20 exp(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}})-exp(\frac{1} {n}\sum_{i=1}^{n}\) & \([-32,32]\) & 30 & 0 \\ \(\cos(2\pi x_{i}))+20+e\) & & & \\ \(f_{11}(x)=\frac{1}{4000}\sum_{i=1}^{n}x_{i}^{2}-\prod_{i=1}^{n}cos(\frac{x_{i} }{\sqrt{i}})+1\) & \([-600,600]\) & 30 & 0 \\ \(f_{12}(x)=\frac{\pi}{n}\{10 sin(\pi y_{1})+\sum_{i=1}^{n-1}(y_{i}-1)^{2}[1+10 sin^{2}(\pi}\) & & \\ \(y_{i+1})]+(y_{n}-1)^{2}\}+\sum_{i=1}^{n}u(x_{i},10,100,4)\) & & \\ \(y_{i}=1+\frac{x_{i}+1}{4}u(x_{i},a,k,m)=\begin{cases}k(x_{i}-a)^{m}\quad x_{i}>a \\ 0\quad\quad\quad\quad\quad-a<x_{i}<a\\ k(-x_{i}-a)^{m}\quad x_{i}<-a\end{cases}\) & & \\ \(f_{13}(x)=0.1\{sin^{2}(3\pi x_{1})+\sum_{i=1}^{n}(x_{i}-1)^{2}[1+sin^{2}(3\pi x _{i}\\ +1)]+(x_{n}-1)^{2}[1+sin^{2}(2\pi x_{n})]\}+\sum_{i=1}^{n}u(x_{i},10,100,4)\) & \([-50,50]\) & 30 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Multimodal benchmark functions.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Space** & \(\mathbf{D}_{im}\) & \(\mathbf{f}_{\mathbf{min}}\) \\ \hline \(f_{14}(x)=(\frac{1}{500}\sum_{j=1}^{25}\frac{1}{j+\sum_{i=1}^{2}(x_{i}-a_{ij}) ^{6}})^{-1}\) & \([-65,65]\) & 2 & 1 \\ \(f_{15}(x)=\sum_{i=1}^{11}[a_{i}-\frac{x_{1}(b_{i}^{2}+b_{i}x_{2})}{b_{i}^{2}+b_ {i}x_{3}+x_{4}}]^{2}\) & \([-5,5]\) & 4 & 0.00030 \\ \(f_{16}(x)=4x_{1}^{2}-2.1x_{1}^{4}+\frac{1}{3}x_{1}^{6}+x_{1}x_{2}-4x_{2}^{2}+4x_ {2}^{4}\) & \([-5,5]\) & 2 & \(-1.0316\) \\ \(f_{17}(x)=(x_{2}-\frac{5}{4\pi^{2}}x_{1}^{2}+\frac{5}{\pi}x_{1}-6)^{2}+10(1- \frac{1}{8\pi})cosx_{1}+10\) & \([-5,5]\) & 2 & 0.398 \\ \(f_{18}(x)=[1+(x_{1}+x_{2}+1)^{2}\) & & \\ \((19-14x_{1}+3x_{2}^{2}-14x_{2}+6x_{1}x_{2}+3x_{2}^{2})]\) & & \\ \(\times[30+(2x_{1}-3x_{2})^{2}\times(18-32x_{1}+12x_{1}^{2}+48x_{2}\) & \([-2,2]\) & 2 & 3 \\ \(-36x_{1}x_{2}+27x_{2}^{2})]\) & & \\ \(f_{19}(x)=-\sum_{i=1}^{4}c_{i}exp(-\sum_{j=1}^{6}a_{ij}(x_{j}-p_{ij})^{2})\) & \([1,3]\) & 3 & \(-3.86\) \\ \(f_{20}(x)=-\sum_{i=1}^{4}c_{i}exp(-\sum_{j=1}^{6}a_{ij}(x_{j}-p_{ij})^{2})\) & \([0,1]\) & 6 & \(-3.32\) \\ \(f_{21}(x)=-\sum_{i=1}^{5}[(X-a_{i})(X-a_{i})^{T}+c_{i}]^{-1}\) & \([0,10]\) & 4 & \(-10.1532\) \\ \(f_{22}(x)=-\sum_{i=1}^{4}[(X-a_{i})(X-a_{i})^{T}+c_{i}]^{-1}\) & \([0,10]\) & 4 & \(-10.4028\) \\ \(f_{23}(x)=-\sum_{i=1}^{4}[(X-a_{i})(X-a_{i})^{T}+c_{i}]^{-1}\) & \([0,10]\) & 4 & \(-10.5363\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Fixed-dimension multimodal benchmark functions.