# 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

- Wang, S.; Zhang, N.; Wu, L.; Wang, Y. Wind speed forecasting based on the hybrid ensemble empirical mode decomposition and GA-BP neural network method. Renew. Energy
**2016**, 94, 629–636. [Google Scholar] [CrossRef] - Zhou, W.; Lou, C.; Li, Z.; Lu, L.; Yang, H. Current status of research on optimum sizing of stand-alone hybrid solar–wind power generation systems. Appl. Energy
**2010**, 87, 380–389. [Google Scholar] [CrossRef] - Bhaskar, K.; Singh, S. AWNN-assisted wind power forecasting using feed-forward neural network. IEEE Trans. Sustain. Energy
**2012**, 3, 306–315. [Google Scholar] [CrossRef] - Liu, H.; Tian, H.Q.; Chen, C.; Li, Y.f. A hybrid statistical method to predict wind speed and wind power. Renew. Energy
**2010**, 35, 1857–1861. [Google Scholar] [CrossRef] - Wang, J.; Zhou, Y. Multi-objective dynamic unit commitment optimization for energy-saving and emission reduction with wind power. In Proceedings of the 2015 5th International Conference on Electric Utility Deregulation and Restructuring and Power Technologies (DRPT), Changsha, China, 26–29 November 2015; pp. 2074–2078. [Google Scholar]
- Wang, C.N.; Le, T.M.; Nguyen, H.K.; Ngoc-Nguyen, H. Using the Optimization Algorithm to Evaluate the Energetic Industry: A Case Study in Thailand. Processes
**2019**, 7, 87. [Google Scholar] [CrossRef] - Hu, P.; Pan, J.S.; Chu, S.C.; Chai, Q.W.; Liu, T.; Li, Z.C. New Hybrid Algorithms for Prediction of Daily Load of Power Network. Appl. Sci.
**2019**, 9, 4514. [Google Scholar] [CrossRef] - Morris, G.M.; Goodsell, D.S.; Halliday, R.S.; Huey, R.; Hart, W.E.; Belew, R.K.; Olson, A.J. Automated docking using a Lamarckian genetic algorithm and an empirical binding free energy function. J. Comput. Chem.
**1998**, 19, 1639–1662. [Google Scholar] [CrossRef] - Pan, J.; McInnes, F.; Jack, M. Application of parallel genetic algorithm and property of multiple global optima to VQ codevector index assignment for noisy channels. Electron. Lett.
**1996**, 32, 296–297. [Google Scholar] [CrossRef] - Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput.
**2002**, 6, 182–197. [Google Scholar] [CrossRef] - Shieh, C.S.; Huang, H.C.; Wang, F.H.; Pan, J.S. Genetic watermarking based on transform-domain techniques. Pattern Recognit.
**2004**, 37, 555–565. [Google Scholar] [CrossRef] - Huang, H.C.; Pan, J.S.; Lu, Z.M.; Sun, S.H.; Hang, H.M. Vector quantization based on genetic simulated annealing. Signal Process.
**2001**, 81, 1513–1523. [Google Scholar] [CrossRef] - Eberhart, R.; Kennedy, J. A new optimizer using particle swarm theory. In Proceedings of the Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan, 4–6 October 1995; pp. 39–43. [Google Scholar]
- Wang, H.; Sun, H.; Li, C.; Rahnamayan, S.; Pan, J.S. Diversity enhanced particle swarm optimization with neighborhood search. Inf. Sci.
**2013**, 223, 119–135. [Google Scholar] [CrossRef] - Shi, Y.; Eberhart. Particle swarm optimization: Developments, applications and resources. In Proceedings of the 2001 Congress on Evolutionary Computation, Seoul, Korea, 27–30 May 2001; Volume 1, pp. 81–86. [Google Scholar]
- Chang, J.F.; Roddick, J.F.; Pan, J.S.; Chu, S. A parallel particle swarm optimization algorithm with communication strategies. J. Inf. Sci. Eng.
**2005**, 21, 809–818. [Google Scholar] - Sun, C.L.; Zeng, J.C.; Pan, J.S. An improved vector particle swarm optimization for constrained optimization problems. Inf. Sci.
**2011**, 181, 1153–1163. [Google Scholar] [CrossRef] - Wang, J.; Ju, C.; Ji, H.; Youn, G.; Kim, J.U. A Particle Swarm Optimization and Mutation Operator Based Node Deployment Strategy for WSNs; International Conference on Cloud Computing and Security; Springer: Boston, MA, USA, 2017; pp. 430–437. [Google Scholar]
- Liu, W.; Wang, J.; Chen, L.; Chen, B. Prediction of protein essentiality by the improved particle swarm optimization. Soft Comput.
**2018**, 22, 6657–6669. [Google Scholar] [CrossRef] - Wang, J.; Ju, C.; Kim, H.j.; Sherratt, R.S.; Lee, S. A mobile assisted coverage hole patching scheme based on particle swarm optimization for WSNs. Clust. Comput.
**2019**, 22, 1787–1795. [Google Scholar] [CrossRef] - Storn, R.; Price, K. Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim.
**1997**, 11, 341–359. [Google Scholar] [CrossRef] - Qin, A.K.; Huang, V.L.; Suganthan, P.N. Differential evolution algorithm with strategy adaptation for global numerical optimization. IEEE Trans. Evol. Comput.
**2008**, 13, 398–417. [Google Scholar] [CrossRef] - Meng, Z.; Pan, J.S.; Kong, L. Parameters with adaptive learning mechanism (PALM) for the enhancement of differential evolution. Knowl.-Based Syst.
**2018**, 141, 92–112. [Google Scholar] [CrossRef] - Meng, Z.; Pan, J.S.; Tseng, K.K. PaDE: An enhanced Differential Evolution algorithm with novel control parameter adaptation schemes for numerical optimization. Knowl.-Based Syst.
**2019**, 168, 80–99. [Google Scholar] [CrossRef] - Das, S.; Suganthan, P.N. Differential evolution: A survey of the state-of-the-art. IEEE Trans. Evol. Comput.
**2010**, 15, 4–31. [Google Scholar] [CrossRef] - Penas, D.R.; Banga, J.R.; González, P.; Doallo, R. Enhanced parallel differential evolution algorithm for problems in computational systems biology. Appl. Soft Comput.
**2015**, 33, 86–99. [Google Scholar] [CrossRef] - Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey wolf optimizer. Adv. Eng. Softw.
**2014**, 69, 46–61. [Google Scholar] [CrossRef] - Mirjalili, S. How effective is the Grey Wolf optimizer in training multi-layer perceptrons. Appl. Intell.
**2015**, 43, 150–161. [Google Scholar] [CrossRef] - Pan, J.S.; Dao, T.K.; Chu, S.C.; Nguyen, T.-T. A novel hybrid GWO-FPA algorithm for optimization applications. In Proceedings of the International Conference on Smart Vehicular Technology, Transportation, Communication and Applications, Kaohsiung, Taiwan, 6–8 November 2017; pp. 274–281. [Google Scholar]
- Song, X.; Tang, L.; Zhao, S.; Zhang, X.; Li, L.; Huang, J.; Cai, W. Grey Wolf Optimizer for parameter estimation in surface waves. Soil Dyn. Earthq. Eng.
**2015**, 75, 147–157. [Google Scholar] [CrossRef] - El-Fergany, A.A.; Hasanien, H.M. Single and multi-objective optimal power flow using grey wolf optimizer and differential evolution algorithms. Electr. Power Components Syst.
**2015**, 43, 1548–1559. [Google Scholar] [CrossRef] - Meng, Z.; Pan, J.S.; Xu, H. QUasi-Affine TRansformation Evolutionary (QUATRE) algorithm: A cooperative swarm based algorithm for global optimization. Knowl.-Based Syst.
**2016**, 109, 104–121. [Google Scholar] [CrossRef] - Liu, N.; Pan, J.S.; Xue, J.Y. An Orthogonal QUasi-Affine TRansformation Evolution (O-QUATRE). In Proceedings of the 15th International Conference on IIH-MSP in Conjunction with the 12th International Conference on FITAT, Jilin, China, 18–20 July 2019; Volume 2, pp. 57–66. [Google Scholar]
- Pan, J.S.; Meng, Z.; Xu, H.; Li, X. QUasi-Affine TRansformation Evolution (QUATRE) algorithm: A new simple and accurate structure for global optimization. In Proceedings of the International Conference on Industrial, Engineering and Other Applications of Applied Intelligent Systems, Morioka, Japan, 2–4 August 2016; pp. 657–667. [Google Scholar]
- Meng, Z.; Pan, J.S. QUasi-Affine TRansformation Evolution with External ARchive (QUATRE-EAR): An enhanced structure for differential evolution. Knowl.-Based Syst.
**2018**, 155, 35–53. [Google Scholar] [CrossRef] - Weber, M.; Neri, F.; Tirronen, V. Shuffle or update parallel differential evolution for large-scale optimization. Soft Comput.
**2011**, 15, 2089–2107. [Google Scholar] [CrossRef] - Pooranian, Z.; Shojafar, M.; Abawajy, J.H.; Abraham, A. An efficient meta-heuristic algorithm for grid computing. J. Comb. Optim.
**2015**, 30, 413–434. [Google Scholar] [CrossRef] - Yang, Z.; Li, K.; Niu, Q.; Xue, Y. A novel parallel-series hybrid meta-heuristic method for solving a hybrid unit commitment problem. Knowl.-Based Syst.
**2017**, 134, 13–30. [Google Scholar] [CrossRef] - Mussi, L.; Daolio, F.; Cagnoni, S. Evaluation of parallel particle swarm optimization algorithms within the CUDA™ architecture. Inf. Sci.
**2012**, 181, 4642–4657. [Google Scholar] [CrossRef] - Schutte, J.F.; Reinbolt, J.A.; Fregly, B.J.; Haftka, R.T.; George, A.D. Parallel global optimization with the particle swarm algorithm. Int. J. Numer. Methods Eng.
**2004**, 61, 2296–2315. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pan, J.S.; Dao, T.K.; Nguyen, T.-T. A Novel Improved Bat Algorithm Based on Hybrid Parallel and Compact for Balancing an Energy Consumption Problem. Information
**2019**, 10, 194–215. [Google Scholar] - Alba, E.; Luque, G.; Nesmachnow, S. Parallel metaheuristics: Recent advances and new trends. Int. Trans. Oper. Res.
**2013**, 20, 1–48. [Google Scholar] [CrossRef] - Xue, X.; Chen, J. Optimizing ontology alignment through hybrid population-based incremental learning algorithm. Memetic Comput.
**2019**, 11, 209–217. [Google Scholar] [CrossRef] - Lalwani, S.; Sharma, H.; Satapathy, S.C.; Deep, K.; Bansal, J.C. A Survey on Parallel Particle Swarm Optimization Algorithms. Arab. J. Sci. Eng.
**2019**, 44, 2899–2923. [Google Scholar] [CrossRef] - Madhuri, D.K.; Deep, K. A state-of-the-art review of population-based parallel meta-heuristics. In Proceedings of the Nature & Biologically Inspired Computing, Coimbatore, India, 9–11 December 2009; pp. 9–11. [Google Scholar]
- Liao, D.Y.; Wang, C.N. Neural-network-based delivery time estimates for prioritized 300-mm automatic material handling operations. IEEE Trans. Semicond. Manuf.
**2004**, 17, 324–332. [Google Scholar] [CrossRef] - Jeng-Shyang, P.; Lingping, K.; Tien-Wen, S.; Pei-Wei, T.; Waclav, S. α-fraction first strategy for hierarchical wireless sensor networks. J. Internet Technol.
**2018**, 19, 1717–1726. [Google Scholar] - Nguyen, T.T.; Pan, J.S.; Dao, T.K. An Improved Flower Pollination Algorithm for Optimizing Layouts of Nodes in Wireless Sensor Network. IEEE Access
**2019**, 7, 75985–75998. [Google Scholar] [CrossRef] - Pan, J.S.; Lee, C.Y.; Sghaier, A.; Zeghid, M.; Xie, J. Novel Systolization of Subquadratic Space Complexity Multipliers Based on Toeplitz Matrix-Vector Product Approach. IEEE Trans. Very Large Scale Integr. (VLSI) Syst.
**2019**. [Google Scholar] [CrossRef] - Xue, X.; Chen, J.; Yao, X. Efficient User Involvement in Semiautomatic Ontology Matching. IEEE Trans. Emerg. Top. Comput. Intell.
**2018**. [Google Scholar] [CrossRef] - Pan, J.S.; Kong, L.; Sung, T.W.; Tsai, P.W.; Snášel, V. A clustering scheme for wireless sensor networks based on genetic algorithm and dominating set. J. Internet Technol.
**2018**, 19, 1111–1118. [Google Scholar]

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 |

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