# Novel Parallel Heterogeneous Meta-Heuristic and Its Communication Strategies for the Prediction of Wind Power

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## Abstract

**:**

## 1. Introduction

- It first proposes parallel heterogeneous model based on PSO and GWO.
- It introduces four new communication strategies to improve the abilities of exploration and exploitation.
- It dynamically changes the members of subgroup from the diversity of the population.

## 2. Preliminaries

#### 2.1. Particle Swarm Optimization

#### 2.2. Grey Wolf Optimizer

#### 2.3. Population-Based Parallelization

#### 2.3.1. Communication Models

- Star model

- Migration model

- Diffusion model

- Hybrid model

#### 2.3.2. Communication Strategies

- Parameters with loosely correlated

- Parameters with strongly correlated

- Parameters with unknown correlation (Hybrid)

## 3. Novel Parallel Heterogeneous Algorithm

#### 3.1. The Model of Parallel Heterogeneous Algorithm

#### 3.2. New Communication Strategies

#### 3.2.1. Communication Strategy with Ranking

#### 3.2.2. Communication Strategy with Combination

Algorithm 1 Combination |

//ngroups.number is the number of subgroups |

for g = 1 : ngroups.number do |

//ngroups.algorithms is the number of meta-heuristics |

for j = 1 : ngroups.algorithms do |

if j == groups(g).algorithm then |

//ngroups.size is the number of subgroup |

for l = 1 : ngroups.size do |

temp(j,t(j)) = groups(g).pop(l); |

t(j) = t(j) + 1; |

end for |

end if |

end for |

end for |

for j = 1 : ngroups.algorithms do |

i = 1; |

temp(j) = SortPopulationByFitness(temp(j)); |

for g = 1 : ngroups.number do |

if j == groups(g).algorithm && i ≤ t(j) then |

groups(g).pop(p(g)) = temp(j,i); |

i = i + 1; |

p(g) = p(g) + 1; |

end if |

end for |

end for |

#### 3.2.3. Communication Strategy with Dynamic Change

Algorithm 2 Dynamic Change |

Sort A //A is the best solutions of the subgroups |

Sort B // B is the virtual group |

for i = 1 : length(A), j = 1 : length(B) do |

if f(A(i)) < B(j) then |

B(j) = A(i); |

++i; |

else |

++j; |

end if |

end for |

#### 3.2.4. Hybrid Communication Strategy

## 4. Experimental Results and Analysis

#### 4.1. Parameters Configuration

#### 4.2. Unimodal Functions

#### 4.3. Multimodal Functions

#### 4.4. Fixed-Dimension Multimodal Functions

#### 4.5. Composite Multimodal Functions

## 5. Application for Wind Power Forecasting

#### 5.1. The Model of Wind Power Forecasting Based on Hybrid Neural Network

#### 5.2. Simulation Results

#### 5.2.1. Data Preprocessing

#### 5.2.2. The Evaluation Performance of Hybrid Model

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Function | Space | D_{im} | f_{min} |
---|---|---|---|

$\left(Sphere\right){f}_{1}\left(x\right)=\phantom{\rule{0ex}{0ex}}{\sum}_{i=1}^{n}{x}_{i}^{2}$ | [−100, 100] | 30 | 0 |

$\left(Schwefe{l}^{\prime}s\phantom{\rule{1.em}{0ex}}function\phantom{\rule{1.em}{0ex}}2.21\right){f}_{2}\left(x\right)=\phantom{\rule{0ex}{0ex}}{\sum}_{i=1}^{n}|{x}_{i}|+{\prod}_{i=1}^{n}\left|{x}_{i}\right|$ | [−10, 10] | 30 | 0 |

$\left(Schwefe{l}^{\prime}s\phantom{\rule{1.em}{0ex}}problem\phantom{\rule{1.em}{0ex}}1.2\right){f}_{3}\left(x\right)=\phantom{\rule{0ex}{0ex}}{\sum}_{i=1}^{n}{\left({\sum}_{j-1}^{i}{x}_{j}\right)}^{2}$ | [−100, 100] | 30 | 0 |

$\left(Schwefe{l}^{\prime}s\phantom{\rule{1.em}{0ex}}function\phantom{\rule{1.em}{0ex}}2.22\right){f}_{4}\left(x\right)=\phantom{\rule{0ex}{0ex}}{max}_{i}\left\{\right|{x}_{i}|,1\le i\le n\}$ | [−100, 100] | 30 | 0 |

$\left(Step\right){f}_{5}\left(x\right)=\phantom{\rule{0ex}{0ex}}{\sum}_{i=1}^{n}{\left([{x}_{i}+0.5]\right)}^{2}$ | [−100, 100] | 30 | 0 |

$\left(Dejon{g}^{\prime}s\phantom{\rule{1.em}{0ex}}noisy\right){f}_{6}\left(x\right)=\phantom{\rule{0ex}{0ex}}{\sum}_{i=1}^{n}i{x}_{i}^{4}+random[0,1)$ | [−1.28, 1.28] | 30 | 0 |

Function | Space | D_{im} | f_{min} |
---|---|---|---|

$\left(Schwefel\right){f}_{7}\left(x\right)=\phantom{\rule{0ex}{0ex}}{\sum}_{i=1}^{n}-{x}_{i}sin\left(\sqrt{|{x}_{i}|}\right)$ | [−500, 500] | 30 | −12,569 |

$\left(Rastringin\right){f}_{8}\left(x\right)=\phantom{\rule{0ex}{0ex}}{\sum}_{i=1}^{n}[{x}_{i}^{2}-10cos\left(2\pi {x}_{i}\right)+10]$ | [−5.12, 5.12] | 30 | 0 |

$\left(Ackley\right){f}_{9}\left(x\right)=\phantom{\rule{0ex}{0ex}}-20exp(-0.2\sqrt{{\displaystyle \frac{1}{n}}{\sum}_{i=1}^{n}{x}_{i}^{2}})\phantom{\rule{0ex}{0ex}}-exp({\displaystyle \frac{1}{n}}{\sum}_{i=1}^{n}cos\left(2\pi {x}_{i}\right))+20+e$ | [−32, 32] | 30 | 0 |

$\left(Griewank\right){f}_{10}\left(x\right)=\phantom{\rule{0ex}{0ex}}{\displaystyle \frac{1}{4000}}{\sum}_{i=1}^{n}{x}_{i}^{2}-{\prod}_{i=1}^{n}cos({\displaystyle \frac{{x}_{i}}{\sqrt{i}}})+1$ | [−600, 600] | 30 | 0 |

$\left(Generalized\phantom{\rule{1.em}{0ex}}penalized\phantom{\rule{1.em}{0ex}}1\right){f}_{11}\left(x\right)=\phantom{\rule{0ex}{0ex}}{\displaystyle \frac{\pi}{n}}\{10sin\left(\pi {y}_{1}\right)+{\sum}_{i=1}^{n-1}{({y}_{i}-1)}^{2}\phantom{\rule{0ex}{0ex}}[1+10{sin}^{2}\left(\pi {y}_{i+1}\right)]+{({y}_{n}-1)}^{2}\}+{\sum}_{i=1}^{n}u({x}_{i},10,100,4)\phantom{\rule{0ex}{0ex}}{y}_{i}=1+{\displaystyle \frac{{x}_{i}+1}{4}}u({x}_{i},a,k,m)=\left(\right)open="\{"\; close>\begin{array}{ll}k{({x}_{i}-a)}^{m}& {x}_{i}a\\ 0& -a{x}_{i}a\\ k{(-{x}_{i}-a)}^{m}& {x}_{i}-a\end{array}$ | [−50, 50] | 30 | 0 |

$\left(Generalized\phantom{\rule{1.em}{0ex}}penalized\phantom{\rule{1.em}{0ex}}2\right){f}_{12}\left(x\right)=\phantom{\rule{0ex}{0ex}}0.1\{{sin}^{2}\left(3\pi {x}_{1}\right)+{\sum}_{i=1}^{n}{({x}_{i}-1)}^{2}\phantom{\rule{0ex}{0ex}}[1+{sin}^{2}(3\pi {x}_{i}+1)]+{({x}_{n}-1)}^{2}[1+\phantom{\rule{0ex}{0ex}}{sin}^{2}\left(2\pi {x}_{n}\right)]\}+{\sum}_{i=1}^{n}u({x}_{i},10,100,4)$ | [−50, 50] | 30 | 0 |

Function | Space | D_{im} | f_{min} |
---|---|---|---|

$\left(Fifth\phantom{\rule{1.em}{0ex}}of\phantom{\rule{1.em}{0ex}}Dejong\right){f}_{13}\left(x\right)=\phantom{\rule{0ex}{0ex}}{(\frac{1}{500}{\sum}_{j=1}^{25}{\displaystyle \frac{1}{j+{\sum}_{i=1}^{2}{({x}_{i}-{a}_{ij})}^{6}}})}^{-1}$ | [−65, 65] | 2 | 1 |

$\left(Kowalik\right){f}_{14}\left(x\right)=\phantom{\rule{0ex}{0ex}}{\sum}_{i=1}^{11}{[{a}_{i}-{\displaystyle \frac{{x}_{1}({b}_{i}^{2}+{b}_{i}{x}_{2})}{{b}_{i}^{2}+{b}_{i}{x}_{3}+{x}_{4}}}]}^{2}$ | [−5, 5] | 4 | 0.00030 |

$(Six-hump\phantom{\rule{1.em}{0ex}}camel\phantom{\rule{1.em}{0ex}}back){f}_{15}\left(x\right)=\phantom{\rule{0ex}{0ex}}4{x}_{1}^{2}-2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4}$ | [−5, 5] | 2 | −1.0316 |

$\left(Branins\right){f}_{16}\left(x\right)=\phantom{\rule{0ex}{0ex}}{({x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}{x}_{1}-6)}^{2}+10(1-\frac{1}{8\pi})cos{x}_{1}+10$ | [−5, 5] | 2 | 0.398 |

$(Goldstein-Price){f}_{17}\left(x\right)=\phantom{\rule{0ex}{0ex}}[1+{({x}_{1}+{x}_{2}+1)}^{2}(19-14{x}_{1}+3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}\phantom{\rule{0ex}{0ex}}+3{x}_{2}^{2}\left)\right]\times [30+{(2{x}_{1}-3{x}_{2})}^{2}\times (18-32{x}_{1}+12{x}_{1}^{2}\phantom{\rule{0ex}{0ex}}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2}\left)\right]$ | [−2, 2] | 2 | 3 |

$\left(Hartman\phantom{\rule{1.em}{0ex}}1\right){f}_{18}\left(x\right)=\phantom{\rule{0ex}{0ex}}-{\sum}_{i=1}^{4}{c}_{i}exp(-{\sum}_{j=1}^{3}{a}_{ij}{({x}_{j}-{p}_{ij})}^{2})$ | [1, 3] | 3 | −3.86 |

$\left(Hartman\phantom{\rule{1.em}{0ex}}2\right){f}_{19}\left(x\right)=\phantom{\rule{0ex}{0ex}}-{\sum}_{i=1}^{4}{c}_{i}exp(-{\sum}_{j=1}^{6}{a}_{ij}{({x}_{j}-{p}_{ij})}^{2})$ | [0, 1] | 6 | −3.32 |

$\left(Shekel\phantom{\rule{1.em}{0ex}}1\right){f}_{20}\left(x\right)=\phantom{\rule{0ex}{0ex}}-{\sum}_{i=1}^{5}{[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}]}^{-1}$ | [0, 10] | 4 | −10.1532 |

$\left(Shekel\phantom{\rule{1.em}{0ex}}2\right){f}_{21}\left(x\right)=\phantom{\rule{0ex}{0ex}}-{\sum}_{i=1}^{7}{[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}]}^{-1}$ | [0, 10] | 4 | −10.4028 |

$\left(Shekel\phantom{\rule{1.em}{0ex}}3\right){f}_{22}\left(x\right)=\phantom{\rule{0ex}{0ex}}-{\sum}_{i=1}^{10}{[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}]}^{-1}$ | [0, 10] | 4 | −10.5363 |

Function | Space | D_{im} | f_{min} |
---|---|---|---|

$\left(CF1\right){f}_{23}\phantom{\rule{0ex}{0ex}}{f}_{1},{f}_{2},{f}_{3},\dots ,{f}_{10}=Sphere\phantom{\rule{1.em}{0ex}}Function\phantom{\rule{0ex}{0ex}}[{\sigma}_{1},{\sigma}_{2},{\sigma}_{3},\dots ,{\sigma}_{10}]=[1,1,1,\dots ,1]\phantom{\rule{0ex}{0ex}}[{\lambda}_{1},{\lambda}_{2},{\lambda}_{3},\dots ,{\lambda}_{10}]=[5/100,5/100,5/100,\dots ,5/100]$ | [−5, 5] | 30 | 0 |

$\left(CF2\right){f}_{24}\phantom{\rule{0ex}{0ex}}{f}_{1},{f}_{2},{f}_{3},\dots ,{f}_{10}=Griewan{k}^{\u2019}s\phantom{\rule{1.em}{0ex}}Function\phantom{\rule{0ex}{0ex}}[{\sigma}_{1},{\sigma}_{2},{\sigma}_{3},\dots ,{\sigma}_{10}]=[1,1,1,\dots ,1]\phantom{\rule{0ex}{0ex}}[{\lambda}_{1},{\lambda}_{2},{\lambda}_{3},\dots ,{\lambda}_{10}]=[5/100,5/100,5/100,\dots ,5/100]$ | [−5, 5] | 30 | 0 |

$\left(CF3\right){f}_{25}\phantom{\rule{0ex}{0ex}}{f}_{1},{f}_{2},{f}_{3},\dots ,{f}_{10}=Griewan{k}^{\u2019}s\phantom{\rule{1.em}{0ex}}Function\phantom{\rule{0ex}{0ex}}[{\sigma}_{1},{\sigma}_{2},{\sigma}_{3},\dots ,{\sigma}_{10}]=[1,1,1,\dots ,1]\phantom{\rule{0ex}{0ex}}[{\lambda}_{1},{\lambda}_{2},{\lambda}_{3},\dots ,{\lambda}_{10}]=[1,1,1,\dots ,1]$ | [−5, 5] | 30 | 0 |

$\left(CF4\right){f}_{26}\phantom{\rule{0ex}{0ex}}{f}_{1},{f}_{2}=Ackle{y}^{\u2019}sFunction,{f}_{3},{f}_{4}=Rastrigi{n}^{\u2019}s\phantom{\rule{1.em}{0ex}}Function,\phantom{\rule{0ex}{0ex}}{f}_{5},{f}_{6}=WeierstrassFunction,{f}_{7},{f}_{8}=Griewan{k}^{\u2019}s\phantom{\rule{1.em}{0ex}}Function\phantom{\rule{0ex}{0ex}}{f}_{9},{f}_{10}=Sphere\phantom{\rule{1.em}{0ex}}Function\phantom{\rule{0ex}{0ex}}[{\sigma}_{1},{\sigma}_{2},{\sigma}_{3},\dots ,{\sigma}_{10}]=[1,1,1,\dots ,1]\phantom{\rule{0ex}{0ex}}[{\lambda}_{1},{\lambda}_{2},{\lambda}_{3},\dots ,{\lambda}_{10}]=\phantom{\rule{0ex}{0ex}}[5/32,5/32,1,1,5/0.5,5/0.5,5/100,5/100,5/100,5/100]$ | [−5, 5] | 30 | 0 |

$\left(CF5\right){f}_{27}\phantom{\rule{0ex}{0ex}}{f}_{1},{f}_{2}=Rastrigi{n}^{\u2019}s\phantom{\rule{1.em}{0ex}}Function,{f}_{3},{f}_{4}=Weierstrass\phantom{\rule{1.em}{0ex}}Function,\phantom{\rule{0ex}{0ex}}{f}_{5},{f}_{6}=Griewan{k}^{\u2019}sFunction,{f}_{7},{f}_{8}=Ackle{y}^{\u2019}s\phantom{\rule{1.em}{0ex}}Function\phantom{\rule{0ex}{0ex}}{f}_{9},{f}_{10}=Sphere\phantom{\rule{1.em}{0ex}}Function\phantom{\rule{0ex}{0ex}}[{\sigma}_{1},{\sigma}_{2},{\sigma}_{3},\dots ,{\sigma}_{10}]=[1,1,1,\dots ,1]\phantom{\rule{0ex}{0ex}}[{\lambda}_{1},{\lambda}_{2},{\lambda}_{3},\dots ,{\lambda}_{10}]=\phantom{\rule{0ex}{0ex}}[1/5,1/5,5/0.5,5/0.5,5/100,5/100,5/32,5/32,5/100,5/100]$ | [−5, 5] | 30 | 0 |

$\left(CF6\right){f}_{28}\phantom{\rule{0ex}{0ex}}{f}_{1},{f}_{2}=Rastrigi{n}^{\u2019}s\phantom{\rule{1.em}{0ex}}Function,{f}_{3},{f}_{4}=Weierstrass\phantom{\rule{1.em}{0ex}}Function,\phantom{\rule{0ex}{0ex}}{f}_{5},{f}_{6}=Griewan{k}^{\u2019}s\phantom{\rule{1.em}{0ex}}Function,{f}_{7},{f}_{8}=Ackle{y}^{\u2019}s\phantom{\rule{1.em}{0ex}}Function\phantom{\rule{0ex}{0ex}}{f}_{9},{f}_{10}=Sphere\phantom{\rule{1.em}{0ex}}Function\phantom{\rule{0ex}{0ex}}[{\sigma}_{1},{\sigma}_{2},{\sigma}_{3},\dots ,{\sigma}_{10}]=[0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1]\phantom{\rule{0ex}{0ex}}[{\lambda}_{1},{\lambda}_{2},{\lambda}_{3},\dots ,{\lambda}_{10}]=[0.1\ast 1/5,0.2\ast 1/5,0.3\ast 5/0.5,0.4\ast 5/0.5,\phantom{\rule{0ex}{0ex}}0.5\ast 5/100,0.6\ast 5/100,0.7\ast 5/32,0.8\ast 5/32,0.9\ast 5/100,1\ast 5/100]$ | [−5, 5] | 30 | 0 |

Algorithm | Communication Strategy | Main Parameters Setting |
---|---|---|

PH-R | Ranking | ${V}_{max}=6;{w}_{max}=0.9;{w}_{min}=0.2;c1=2;c2=2;\phantom{\rule{0ex}{0ex}}beta\in [0.02,0.08];pcR=0.01$ |

PH-C | Combination | ${V}_{max}=6;{w}_{max}=0.9;c1=2;c2=2.$ |

PH-D | Dynamic Change | ${V}_{max}=6;{w}_{max}=0.9;c1=2;c2=2;\phantom{\rule{0ex}{0ex}}group\_size\in [15,60].$ |

PH-H | Hybrid of Ranking and Combination | ${V}_{max}=6;{w}_{max}=0.9;c1=2;c2=2;\phantom{\rule{0ex}{0ex}}beta\in [0.02,0.08];pcR=0.01$ |

Function | PGWO | PH-R | PH-C | PH-D | PH-H | |||||
---|---|---|---|---|---|---|---|---|---|---|

AVG | STSD | AVG | STSD | AVG | STSD | AVG | STSD | AVG | STSD | |

${f}_{1}$ | $1.40\times {10}^{-78}$ | $3.43\times {10}^{-78}$ | $5.29\times {10}^{-113}$ | $2.26\times {10}^{-112}$ | $9.69\times {10}^{-115}$ | $4.46\times {10}^{-114}$ | $1.93\times {10}^{-118}$ | $6.02\times {10}^{-118}$ | $4.34\times {10}^{-131}$ | $1.84\times {10}^{-130}$ |

${f}_{2}$ | $9.44\times {10}^{-47}$ | $1.52\times {10}^{-46}$ | $9.99\times {10}^{-67}$ | $4.79\times {10}^{-66}$ | $3.02\times {10}^{-67}$ | $1.39\times {10}^{-66}$ | $2.78\times {10}^{-70}$ | $9.13\times {10}^{-70}$ | $4.21\times {10}^{-79}$ | $1.49\times {10}^{-78}$ |

${f}_{3}$ | $3.71\times {10}^{-10}$ | $1.92\times {10}^{-9}$ | $8.98\times {10}^{-18}$ | $4.92\times {10}^{-17}$ | $5.02\times {10}^{-15}$ | $2.00\times {10}^{-14}$ | $8.68\times {10}^{-21}$ | $4.30\times {10}^{-20}$ | $3.27\times {10}^{-17}$ | $1.78\times {10}^{-16}$ |

${f}_{4}$ | $1.19\times {10}^{-25}$ | $4.84\times {10}^{-25}$ | $3.74\times {10}^{-42}$ | $1.10\times {10}^{-41}$ | $6.63\times {10}^{-40}$ | $2.12\times {10}^{-39}$ | $3.43\times {10}^{-42}$ | $1.84\times {10}^{-41}$ | $2.28\times {10}^{-47}$ | $8.93\times {10}^{-47}$ |

${f}_{5}$ | $7.80\times {10}^{-1}$ | $1.82\times {10}^{-1}$ | $1.16\times {10}^{0}$ | $3.95\times {10}^{-1}$ | $8.01\times {10}^{-6}$ | $5.50\times {10}^{-6}$ | $2.62\times {10}^{-6}$ | $3.31\times {10}^{-06}$ | $7.54\times {10}^{-2}$ | $3.15\times {10}^{-2}$ |

${f}_{6}$ | $1.24\times {10}^{-4}$ | $8.72\times {10}^{-5}$ | $2.09\times {10}^{-}2$ | $2.14\times {10}^{-2}$ | $1.77\times {10}^{-2}$ | $1.89\times {10}^{-2}$ | $1.85\times {10}^{-}2$ | $1.91\times {10}^{-2}$ | $5.72\times {10}^{-3}$ | $4.74\times {10}^{-3}$ |

${f}_{7}$ | $-1.01\times {10}^{4}$ | $1.66\times {10}^{3}$ | $-8.96\times {10}^{3}$ | $1.43\times {10}^{3}$ | $-8.88\times {10}^{3}$ | $1.03\times {10}^{3}$ | $-9.15\times {10}^{3}$ | $1.48\times {10}^{3}$ | $-9.10\times {10}^{3}$ | $1.18\times {10}^{3}$ |

${f}_{8}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${f}_{9}$ | $4.20\times {10}^{-15}$ | $9.01\times {10}^{-16}$ | $1.72\times {10}^{-15}$ | $1.53\times {10}^{-15}$ | $1.95\times {10}^{-15}$ | $1.66\times {10}^{-15}$ | $2.66\times {10}^{-15}$ | $1.81\times {10}^{-15}$ | $1.72\times {10}^{-15}$ | $1.53\times {10}^{-15}$ |

${f}_{10}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${f}_{11}$ | $2.94\times {10}^{-2}$ | $1.48\times {10}^{-2}$ | $7.39\times {10}^{-2}$ | $4.61\times {10}^{-2}$ | $9.15\times {10}^{-7}$ | $8.13\times {10}^{-7}$ | $1.73\times {10}^{-7}$ | $2.46\times {10}^{-7}$ | $4.68\times {10}^{-3}$ | $1.18\times {10}^{-3}$ |

${f}_{12}$ | $3.49\times {10}^{-1}$ | $1.17\times {10}^{-1}$ | $6.30\times {10}^{-1}$ | $1.86\times {10}^{-1}$ | $5.03\times {10}^{-3}$ | $5.97\times {10}^{-3}$ | $7.00\times {10}^{-3}$ | $8.87\times {10}^{-3}$ | $1.10\times {10}^{-1}$ | $4.31\times {10}^{-2}$ |

${f}_{13}$ | $1.40\times {10}^{0}$ | $1.03\times {10}^{0}$ | $2.45\times {10}^{0}$ | $1.59\times {10}^{0}$ | $1.82\times {10}^{0}$ | $1.19\times {10}^{0}$ | $1.79\times {10}^{0}$ | $8.79\times {10}^{-1}$ | $1.33\times {10}^{0}$ | $6.02\times {10}^{-1}$ |

${f}_{14}$ | $6.05\times {10}^{-4}$ | $3.24\times {10}^{-4}$ | $6.56\times {10}^{-4}$ | $1.41\times {10}^{-4}$ | $3.74\times {10}^{-4}$ | $1.74\times {10}^{-4}$ | $3.23\times {10}^{-4}$ | $4.70\times {10}^{-5}$ | $4.63\times {10}^{-4}$ | $2.09\times {10}^{-4}$ |

${f}_{15}$ | $-1.03\times {10}^{0}$ | $9.79\times {10}^{-10}$ | $-1.03\times {10}^{0}$ | $6.78\times {10}^{-16}$ | $-1.03\times {10}^{0}$ | $6.65\times {10}^{-16}$ | $-1.03\times {10}^{0}$ | $6.78\times {10}^{-16}$ | $-1.03\times {10}^{0}$ | $1.34\times {10}^{-9}$ |

${f}_{16}$ | $3.98\times {10}^{-1}$ | $1.25\times {10}^{-5}$ | $3.98\times {10}^{-1}$ | 0 | $3.98\times {10}^{-1}$ | 0 | $3.98\times {10}^{-1}$ | 0 | $3.98\times {10}^{-1}$ | $9.77\times {10}^{-7}$ |

${f}_{17}$ | $3.00\times {10}^{0}$ | $2.03\times {10}^{-3}$ | $3.00\times {10}^{0}$ | $1.70\times {10}^{-15}$ | $3.00\times {10}^{0}$ | $9.33\times {10}^{-16}$ | $3.00\times {10}^{0}$ | $1.93\times {10}^{-15}$ | $3.00\times {10}^{0}$ | $4.28\times {10}^{-6}$ |

${f}_{18}$ | $-3.85\times {10}^{0}$ | $1.42\times {10}^{-2}$ | $-3.86\times {10}^{0}$ | $2.54\times {10}^{-15}$ | $-3.86\times {10}^{0}$ | $2.67\times {10}^{-15}$ | $-3.86\times {10}^{0}$ | $2.68\times {10}^{-15}$ | $-3.86\times {10}^{0}$ | $6.84\times {10}^{-5}$ |

${f}_{19}$ | $-3.22\times {10}^{0}$ | $7.54\times {10}^{-2}$ | $-3.28\times {10}^{0}$ | $5.46\times {10}^{-2}$ | $-3.31\times {10}^{0}$ | $4.11\times {10}^{-2}$ | $-3.27\times {10}^{0}$ | $6.02\times {10}^{-2}$ | $-3.27\times {10}^{0}$ | $6.02\times {10}^{-2}$ |

${f}_{20}$ | $-9.51\times {10}^{0}$ | $1.01\times {10}^{0}$ | $-9.60\times {10}^{0}$ | $1.55\times {10}^{0}$ | $-9.98\times {10}^{0}$ | $9.31\times {10}^{-1}$ | $-9.30\times {10}^{0}$ | $1.93\times {10}^{0}$ | $-7.24\times {10}^{0}$ | $2.55\times {10}^{0}$ |

${f}_{21}$ | $-9.42\times {10}^{0}$ | $1.56\times {10}^{0}$ | $-1.04\times {10}^{1}$ | $7.44\times {10}^{-9}$ | $-9.44\times {10}^{0}$ | $2.22\times {10}^{0}$ | $-8.76\times {10}^{0}$ | $2.56\times {10}^{0}$ | $-8.33\times {10}^{0}$ | $2.57\times {10}^{0}$ |

${f}_{22}$ | $-9.24\times {10}^{0}$ | $1.39\times {10}^{0}$ | $-1.02\times {10}^{1}$ | $1.24\times {10}^{0}$ | $-8.69\times {10}^{0}$ | $2.66\times {10}^{0}$ | $-8.37\times {10}^{0}$ | $2.69\times {10}^{0}$ | $-8.71\times {10}^{0}$ | $2.58\times {10}^{0}$ |

${f}_{23}$ | $3.49\times {10}^{2}$ | $1.06\times {10}^{2}$ | $1.49\times {10}^{2}$ | $6.93\times {10}^{1}$ | $4.34\times {10}^{1}$ | $8.58\times {10}^{1}$ | $1.33\times {10}^{1}$ | $3.46\times {10}^{1}$ | $1.12\times {10}^{2}$ | $7.17\times {10}^{1}$ |

${f}_{24}$ | $5.64\times {10}^{2}$ | $1.11\times {10}^{2}$ | $2.53\times {10}^{2}$ | $1.62\times {10}^{2}$ | $1.41\times {10}^{2}$ | $1.35\times {10}^{2}$ | $1.75\times {10}^{2}$ | $1.41\times {10}^{2}$ | $2.43\times {10}^{2}$ | $1.43\times {10}^{2}$ |

${f}_{25}$ | $7.27\times {10}^{2}$ | $1.22\times {10}^{2}$ | $5.99\times {10}^{2}$ | $7.54\times {10}^{1}$ | $3.58\times {10}^{2}$ | $7.83\times {10}^{1}$ | $3.79\times {10}^{2}$ | $1.00\times {10}^{2}$ | $5.45\times {10}^{2}$ | $8.87\times {10}^{1}$ |

${f}_{26}$ | $8.97\times {10}^{2}$ | $1.36\times {10}^{1}$ | $6.95\times {10}^{2}$ | $5.30\times {10}^{1}$ | $6.04\times {10}^{2}$ | $1.70\times {10}^{2}$ | $6.22\times {10}^{2}$ | $1.76\times {10}^{2}$ | $7.82\times {10}^{2}$ | $1.08\times {10}^{2}$ |

${f}_{27}$ | $4.61\times {10}^{2}$ | $2.01\times {10}^{2}$ | $1.59\times {10}^{2}$ | $1.41\times {10}^{2}$ | $3.71\times {10}^{1}$ | $3.13\times {10}^{1}$ | $5.32\times {10}^{1}$ | $3.57\times {10}^{1}$ | $1.23\times {10}^{2}$ | $6.95\times {10}^{1}$ |

${f}_{28}$ | $9.01\times {10}^{2}$ | $2.31\times {10}^{0}$ | $9.03\times {10}^{2}$ | $5.63\times {10}^{0}$ | $9.02\times {10}^{2}$ | $2.37\times {10}^{0}$ | $9.02\times {10}^{2}$ | $2.26\times {10}^{0}$ | $9.02\times {10}^{2}$ | $4.90\times {10}^{0}$ |

Algorithm | Accuracy (%) |
---|---|

PH-R | 84.97 |

PH-C | 83.89 |

PH-D | 84.49 |

PH-H | 83.67 |

NN | 73.30 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pan, J.-S.; Hu, P.; Chu, S.-C.
Novel Parallel Heterogeneous Meta-Heuristic and Its Communication Strategies for the Prediction of Wind Power. *Processes* **2019**, *7*, 845.
https://doi.org/10.3390/pr7110845

**AMA Style**

Pan J-S, Hu P, Chu S-C.
Novel Parallel Heterogeneous Meta-Heuristic and Its Communication Strategies for the Prediction of Wind Power. *Processes*. 2019; 7(11):845.
https://doi.org/10.3390/pr7110845

**Chicago/Turabian Style**

Pan, Jeng-Shyang, Pei Hu, and Shu-Chuan Chu.
2019. "Novel Parallel Heterogeneous Meta-Heuristic and Its Communication Strategies for the Prediction of Wind Power" *Processes* 7, no. 11: 845.
https://doi.org/10.3390/pr7110845