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基于透镜成像学习策略的灰狼优化算法

龙文,  伍铁斌,  唐明珠,  徐明,  蔡绍洪

文章导航 > 自动化学报  > 2020 >  46(10): 2148-2164
龙文, 伍铁斌, 唐明珠, 徐明, 蔡绍洪.基于透镜成像学习策略的灰狼优化算法.自动化学报, 2020, 46(10): 2148-2164 doi: 10.16383/j.aas.c180695
引用本文: 龙文, 伍铁斌, 唐明珠, 徐明, 蔡绍洪.基于透镜成像学习策略的灰狼优化算法.自动化学报, 2020, 46(10): 2148-2164 doi: 10.16383/j.aas.c180695
shu
Long Wen, Wu Tie-Bin, Tang Ming-Zhu, Xu Ming, Cai Shao-Hong. Grey wolf optimizer algorithm based on lens imaging learning strategy. Acta Automatica Sinica, 2020, 46(10): 2148-2164 doi: 10.16383/j.aas.c180695
Citation: Long Wen, Wu Tie-Bin, Tang Ming-Zhu, Xu Ming, Cai Shao-Hong. Grey wolf optimizer algorithm based on lens imaging learning strategy. Acta Automatica Sinica, 2020, 46(10): 2148-2164 doi: 10.16383/j.aas.c180695
shu

基于透镜成像学习策略的灰狼优化算法

doi: 10.16383/j.aas.c180695
  • 1.

    贵州财经大学贵州省经济系统仿真重点实验室 贵阳 550025

  • 2.

    湖南人文科技学院能源与机电工程学院 娄底 417000

  • 3.

    长沙 理工大学能源与动力工程学院 长沙 410114

  • 4.

    贵州财经大学数学与 统计学院 贵阳 550025

基金项目: 

国家自然科学基金 61463009

贵州省科学技术基金 [2020]1Y012

湖南省教育厅重点项目 19A254

湖南省自然科学基金 2020JJ4382

详细信息
    作者简介:

    龙文  贵州财经大学教授. 2011年获得中南大学信息科学与工程学院控制科学与工程专业博士学位.主要研究方向为智能优化算法, 数据挖掘及应用. E-mail: longwen227@mail.gufe.edu.cn

    伍铁斌  湖南人文科技学院教授. 2014年获得中南大学信息科学与工程学院控制科学与工程专业博士学位.主要研究方向为智能优化算法, 复杂生产过程建模与控制. E-mail: wutiebin81@csu.edu.cn

    唐明珠  长沙理工大学副教授. 2011年获得中南大学信息科学与工程学院控制科学与工程专业博士学位.主要研究方向为故障诊断, 机器学习. E-mail: tmz@csust.edu.cn

    徐明  贵州财经大学教授. 2016年获得云南大学物理科学与技术学院系统分析与集成专业博士学位.主要研究方向为智能优化算法, 数据挖掘及应用. E-mail: xuming@mail.gufe.aedu.cn

    通讯作者:

    蔡绍洪  贵州财经大学教授.主要研究方向为大数据挖掘, 智能优化算法, 复杂系统分析及应用.本文通信作者. E-mail: caish@mail.gufe.edu.cn

Grey Wolf Optimizer Algorithm Based on Lens Imaging Learning Strategy

  • 1.

    Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 550025

  • 2.

    Department of Energy and Electrical Engineering, Hunan University of Humanities, Science and Technology, Loudi 417000

  • 3.

    School of Energy Power and Engineering, Changsha University of Science and Technology, Changsha 410114

  • 4.

    School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025

Funds: 

National Natural Science Foundation of China 61463009

Science and Technology Foundation of Guizhou Province [2020]1Y012

Key Projects of Education Department of Hunan Province 19A254

Natural Science Foundation of Hunan Province 2020JJ4382

More Information
    Author Bio:

    LONG Wen  Professor at Guizhou University of Finance and Economics. He received his Ph. D. degree in control science and engineering from the School of Information Science and Engineering, Central South University in 2011. His research interest covers intelligence optimization algorithm, data mining and their applications

    WU Tie-Bin  Professor at Hunan University of Humanities, Science and Technology. He received his Ph. D. degree in control science and engineering from the School of Information Science and Engineering, Central South University in 2014. His research interest covers intelligence optimization algorithm, complex production process modeling and control

    TANG Ming-Zhu  Associate professor at Changsha University of Science and Technology. He received his Ph. D. degree in control science and engineering from the School of Information Science and Engineering, Central South University in 2011. His research interest covers fault diagnosis and machine learning

    XU Ming  Professor at Guizhou University of Finance and Economics. He received his Ph. D. degree in system analysis and integration from the School of Physical Science and Technology, Yunnan University in 2016. His research interest covers intelligence optimization algorithm, data mining and their applications

    Corresponding author: CAI Shao-Hong  Professor at Guizhou University of Finance and Economics. His research interest covers big data mining, intelligence optimization algorithm, complex system analysis and their applications. Corresponding author of this paper

    摘要
  • 摘要: 在灰狼优化算法中, $ {{\pmb C}} $是一个重要的参数, 其功能是负责算法的勘探能力.目前, 针对参数$ {{\pmb C}} $的研究工作相对较少.另外, 在算法进化过程中, 群体中其他个体均向$\alpha$、$\beta$和$\delta$所在区域靠近以加快收敛速度.然而, 算法易陷入局部最优.为解决以上问题, 本文提出一种改进的灰狼优化算法(Lens imaging learning grey wolf optimizer algorithm, LIL-GWO).该算法首先分析了参数$ {{\pmb C}} $的作用, 提出一种新的参数$\pmb C$策略以平衡算法的勘探和开采能力; 同时, 分析了灰狼优化算法后期个体均向决策层区域聚集, 从而导致群体多样性较差, 提出一种基于光学透镜成像原理的反向学习策略以避免算法陷入局部最优.对LIL-GWO算法的收敛性进行了证明.选取12个通用的标准测试函数和30个CEC 2014测试函数进行实验, 在相同的适应度函数评价次数条件下, LIL-GWO算法在总体性能上优于基本GWO算法、改进GWO算法和其他比较算法.最后, 将LIL-GWO算法应用于辨识光伏模型的参数, 获得了满意的结果.
    Abstract: In the grey wolf optimizer algorithm, $\pmb C$ is an important parameter. The exploration capability of GWO mainly depends on the parameter $\pmb C$. At present, few of researchers are aiming at parameter $\pmb C$ in GWO algorithm. Many issues need to be settled. In addition, during the evolution process, the other individuals in the population move towards to the $\alpha$, $\beta$ and $\delta$ which are to accelerate convergence. However, the algorithm is easy to trap in the local optima. In this paper, an improved GWO algorithm called LIL-GWO is proposed to solve these problems. The proposed algorithm firstly analyzes the role of parameter $\pmb C$ and presents a new parameter $\pmb C$ strategy to balance between exploration and exploitation of GWO. Secondly, at the end of the GWO algorithm, all individuals assemble into the decision-making region which is resulted in poor population multiplicity. A new opposition-learning strategy based on optical lens imaging principle is proposed to help the population jump out of a local optimum. A theoretical proof of convergence for LIL-GWO algorithm is provided. Simulation experiments were conducted on the 12 widely used benchmark test functions and 30 benchmark test functions from the CEC 2014. The overall performance of LIL-GWO algorithm is much better than the basic GWO algorithm, several improved GWO algorithms, and other compared algorithms with the same number of fitness evaluations (FEs). Finally, LIL-GWO is applied to identify the parameters of PV model and the satisfied results are obtained.
    Recommended by Associate Editor WANG Ding
    1)  本文责任编委  王鼎
    • HTML全文

  • 图  1  光的凸透镜成像原理图

    Fig.  1  The convex lens image of light

    下载: 全尺寸图片 幻灯片

    图  2  GWO算法对两个测试函数的收敛曲线

    Fig.  2  Convergence curves of GWO for two test functions

    下载: 全尺寸图片 幻灯片

    图  3  式(4)和式(11)计算得到的${{\pmb C}}$值

    Fig.  3  ${{\pmb C}}$ values calculated by Eqs. (4) and (11)

    下载: 全尺寸图片 幻灯片

    图  4  GWO算法求解Sphere函数$(D = 2)$时30个种群个体分布情况

    Fig.  4  Population distribution observed at various stages in GWO for solving Sphere function $(D =2)$

    下载: 全尺寸图片 幻灯片

    图  5  基于透镜成像的反向学习策略示意图

    Fig.  5  Opposition learning strategy based on lens image

    下载: 全尺寸图片 幻灯片

    图  6  6种算法对6个代表性测试函数的收敛曲线

    Fig.  6  Convergence curves of six algorithms for six representative test functions

    下载: 全尺寸图片 幻灯片

    图  7  单二极管模型的结构

    Fig.  7  Structure of single diode model

    下载: 全尺寸图片 幻灯片

    图  8  单二极管模型的测试数据和LIL-GWO计算数据的比较

    Fig.  8  Comparison of the measured data and calculated data obtained by LIL-GWO for the single diode model

    下载: 全尺寸图片 幻灯片

    表  1  12个标准测试函数

    Table  1  Twelve benchmark test functions

    函数名函数表达式搜索空间
    Sphere$ f_1 ({{\pmb X}}) =\sum\nolimits_{i = 1}^D {x_i^2 }$$[-100, 100]$
    Schwefel's 2.22$ f_2 ({{\pmb X}}) =\sum\nolimits_{i = 1}^D {{\rm{|}}x_i {\rm{|}}}+\prod\nolimits_{i = 1}^D {|x_i |}$$[-10, 10]$
    Schwefel's 2.21$ f_3 ({{\pmb X}}) = \max _i \{|x_i |, 1\le x_i\le D\}$$[-100, 100]$
    Rosenbrock$ f_4 ({{\pmb X}}) = \sum\nolimits_{i = 1}^D {[100{\kern 1pt} {\kern 1pt} (x_{i + 1} - x_i^2 )^2 + (x_i - 1)^2 ]}$[-30, 30]
    Sum-Power$f_5 ({{\pmb X}}) = \sum\nolimits_{i=1}^D {|x_i |} ^{(i + 1)}$[-1, 1]
    Elliptic $f_6 ({{\pmb X}}) =\sum\nolimits_{i =1}^D {(10^6)^{(i - 1)/(n - 1)} x_i^2 }$[-100, 100]
    Rastrigin$f_7 ({{\pmb X}}) =\sum\nolimits_{i =1}^D {[x_i^2- 10\cos (2\pi x_i) + 10]}$$[-5.12, 5.12]$
    Ackley$ f_8 ({{\pmb X}}) =- 20\exp\left({ - 0.2\sqrt {{\textstyle{1 \over D}}\sum\nolimits_{i =1}^D {x_i^2 } } } \right) - \exp \left({{\frac{1}{D}}\sum\nolimits_{i =1}^D {\cos (2\pi x_i)} } \right) + 20 + e $$[-32, 32]$
    Griewank$ f_9 ({{\pmb X}}) = {\textstyle{1\over {4\, 000}}}\sum\nolimits_{i = 1}^D {x_i^2-\prod\nolimits_{i = 1}^D {\cos \left({{\textstyle{{x_i } \over{\sqrt i }}}} \right)} }+ 1$$[-600, 600]$
    Alpine$ f_{10} ({{\pmb X}}) =\sum\nolimits_{i= 1}^D {|x_i \sin (x_i) + 0.1x_i |} $$[-10, 10]$
    Levy$ f_{11} ({{\pmb X}}) =\sum\nolimits_{i = 1}^D {(x_i- 1)^2 [1 + \sin ^2 (3\pi x_{i + 1})] + \sin ^2 (3 \pi x_1) + |x_D- 1|} [1 + \sin ^2 (3\pi x_D)] $$[-10, 10]$
    Stretched V-sine$ f_{12} ({{\pmb X}}) =\sum\nolimits_{i = 1}^{D - 1} {(x_i^2+ 2x_{i + 1}^2)^{0.25}\cdot ((\sin 50(x_i^2+ x_{i + 1}^2)^{0.1})^2+ 1)} $$[-10, 10]$
    下载: 导出CSV

    表  2  LIL-GWO与其他5种算法对12个测试函数的结果比较

    Table  2  Comparisons of LIL-GWO and other five algorithms for 12 test functions

    函数统计结果GWOmGWOWAGWOAIGWOEEGWOLIL-GWO
    平均值$ 1.36 \times 10^{ - 29} $$ 7.89 \times 10^{ - 44} $$ 7.66 \times 10^{ - 35} $$ 3.62 \times 10^{ - 42} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_1$标准差$ 1.62 \times 10^{ - 29} $$ 8.68 \times 10^{ - 44} $$ 1.01 \times 10^{ - 34} $$ 3.89 \times 10^{ - 42} $00
    排名635411
    平均值$ 4.87 \times 10^{ - 18} $$ 1.16\times 10^{ - 26} $$ 6.83 \times 10^{ - 21} $$ 1.38 \times 10^{ - 25} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_2$标准差$ 2.58 \times 10^{ - 18} $$ 6.14\times 10^{ - 27} $$ 5.18 \times 10^{ - 21} $$ 1.64 \times 10^{ - 25} $00
    排名635411
    平均值$ 1.89 \times 10^{ - 7} $$ 4.48\times 10^{ - 12} $$ 6.97 \times 10^{ - 9} $$ 3.17 \times 10^{ - 12} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_3$标准差$ 8.81 \times 10^{ - 8} $$ 6.38\times 10^{ - 12} $$ 5.29 \times 10^{ - 9} $$ 6.34 \times 10^{ - 12} $00
    排名645311
    平均值$ 2.72 \times 10^{ 1} $$ {\bf{2.71 \times 10}}^{\pmb{1}} $$ 2.75 \times 10^{ 1} $$ 2.78 \times 10^{ 1} $$ 2.90 \times 10^{ 1} $$ 2.89 \times 10^{ 1} $
    $f_4$标准差$ 9.99 \times 10^{ - 1} $$ 6.53\times 10^{ - 1} $$ 9.75 \times 10^{ - 1} $$ 1.13 \times 10^{ 0} $$ 5.58 \times 10^{ - 3} $$ 7.43 \times 10^{ - 2} $
    排名213465
    平均值$ 1.44 \times 10^{ - 101} $$ 4.69\times 10^{ - 152} $$ 4.08 \times 10^{ - 121} $$ 4.89 \times 10^{ - 154} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_5$标准差$ 1.49 \times 10^{ - 101} $$ 5.30\times 10^{ - 152} $$ 5.77 \times 10^{ - 121} $$ 2.80 \times 10^{ - 154} $00
    排名645311
    平均值$ 9.01 \times 10^{ - 26} $$ 3.12\times 10^{ - 40} $$ 1.14 \times 10^{ - 31} $$ 6.74 \times 10^{ - 40} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_6$标准差$ 2.81 \times 10^{ - 25} $$ 3.34\times 10^{ - 40} $$ 9.06 \times 10^{ - 32} $$ 1.24 \times 10^{ - 39} $00
    排名635411
    平均值$ 2.08 \times 10^{ 0} $$ {\pmb{0}} $$ 4.54 \times 10^{ - 14} $$ {\pmb{0}} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_7$标准差$ 4.64 \times 10^{ 0} $0$ 2.54 \times 10^{ - 14} $000
    排名615111
    平均值$ 6.84 \times 10^{ - 14} $$ 1.23\times 10^{ - 14} $$ 3.29 \times 10^{ - 14} $$ 1.37 \times 10^{ - 14} $$ {\pmb{8.88 \times 10}}^{ - {\pmb{16}}} $$ {\pmb{8.88 \times 10}}^{ - {\pmb{16}}} $
    $f_8$标准差$ 1.08 \times 10^{ - 14} $$ 3.89\times 10^{ - 15} $$ 1.97 \times 10^{ - 15} $$ 3.18 \times 10^{ - 15} $00
    排名635411
    平均值$ 8.52 \times 10^{ - 13} $$ 4.44\times 10^{ - 16} $$ 4.17 \times 10^{ - 15} $$ {\pmb{0}} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_9$标准差$ 3.03 \times 10^{ - 13} $0$ 1.69 \times 10^{ - 15} $000
    排名645111
    平均值$ 5.25 \times 10^{ - 4} $$ 1.09\times 10^{ - 24} $$ 1.05 \times 10^{ - 4} $$ 9.70 \times 10^{ - 22} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_{10}$标准差$ 5.34 \times 10^{ - 4} $$ 2.18\times 10^{ - 24} $$ 2.35 \times 10^{ - 4} $$ 2.13 \times 10^{ - 21} $00
    排名635411
    平均值$ 2.59 \times 10^{ - 31} $$ 4.64\times 10^{ - 45} $$ 6.40 \times 10^{ - 36} $$ 2.32 \times 10^{ - 44} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_{11}$标准差$ 1.34 \times 10^{ - 31} $$ 7.73\times 10^{ - 45} $$ 8.76 \times 10^{ - 36} $$ 5.97 \times 10^{ - 44} $00
    排名635411
    平均值$ 6.12 \times 10^{ - 8} $$ 3.21\times 10^{ - 12} $$ 3.82 \times 10^{ - 9} $$ 1.75 \times 10^{ - 11} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_{12}$标准差$ 1.74 \times 10^{ - 8} $$ 2.24\times 10^{ - 12} $$ 1.37 \times 10^{ - 9} $$ 1.33 \times 10^{ - 11} $00
    排名635411
    平均排名5.66672.91674.83333.33331.41671.3333
    最终排名635421
    下载: 导出CSV

    表  3  LIL-GWOLIL-GWO与其他7种算法对12个函数的结果比较

    Table  3  Comparisons of LIL-GWOLIL-GWO and other seven algorithms for 12 test functions

    函数统计结果CMA-ESIPSOODEGABCETLBOIWOAISCALIL-GWO
    平均值$ 8.64 \times 10^{ - 11} $$ 2.82\times 10^{ - 16} $$ 2.68 \times 10^{ - 49} $$ 4.00\times 10^{ - 16} $$ 2.70 \times 10^{ - 119} $$ {\pmb{0}} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_1$标准差$ 3.83 \times 10^{ - 11} $$5.07 \times 10^{ - 16} $$ 2.50 \times 10^{ - 49} $$ 3.76\times 10^{ - 16} $$ 4.29 \times 10^{ - 119} $000
    排名86574111
    平均值$ 2.03 \times 10^{ - 5} $$4.03\times 10^{ - 3} $$ 3.86 \times 10^{ - 31} $$ 2.59\times 10^{ -7} $$ 1.19 \times 10^{ - 60} $$ 2.77 \times 10^{ - 267} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_2$标准差$ 1.15 \times 10^{ - 5} $$ 8.05\times10^{ - 3} $$ 4.00 \times 10^{ - 31} $$ 1.98 \times 10^{-8} $$ 5.87 \times 10^{ - 61} $000
    排名78564311
    平均值$ 1.38 \times 10^{ - 4} $$2.13\times 10^{ 0} $$ 1.47 \times 10^{ - 2} $$ 1.16\times 10^{ -1} $$ 2.35 \times 10^{ - 36} $$ 6.35 \times 10^{ - 95} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_3$标准差$ 2.59 \times 10^{ - 5} $$ 7.32\times10^{ - 1} $$ 2.66 \times 10^{ - 3} $$ 2.27 \times 10^{ -2} $$ 1.82 \times 10^{ - 36} $$ 1.26 \times 10^{ - 94} $00
    排名58674311
    平均值$ {\pmb{1.83 \times 10}}^{\pmb{1}}$$ 7.98 \times 10^{ 1}$$ 2.81 \times 10^{ 1} $$ 2.86 \times 10^{ 1} $$ 2.50 \times 10^{ 1} $$ 2.87 \times 10^{ 1} $$ 2.90 \times 10^{ 1} $$ 2.89 \times 10^{ 1} $
    $f_4$标准差$ 3.56 \times 10^{ - 1} $$ 5.57\times10^{ 1} $$ 3.45 \times 10^{ - 1} $$ 1.66\times 10^{ - 1} $$ 2.65 \times 10^{ - 1} $$ 5.87 \times 10^{ - 2} $$ 4.55 \times 10^{ - 1} $$ 7.43 \times 10^{ - 2} $
    排名18342576
    平均值$ 3.91 \times 10^{ - 10} $$2.17\times 10^{ - 34} $$ 8.51 \times 10^{ - 149} $$ 4.02\times 10^{ - 42} $$ 3.26 \times 10^{ - 278} $$ {\pmb{0}} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_5$标准差$ 4.00 \times 10^{ - 10} $$ 4.21\times10^{ - 34} $$ 8.47 \times 10^{ - 149} $$ 6.81 \times10^{- 42} $0000
    排名87564111
    平均值$ 2.89 \times 10^{ - 3} $$5.28\times 10^{ - 8} $$ 1.91 \times 10^{ - 44} $$ 1.86\times 10^{ -12} $$ 1.18 \times 10^{ - 115} $$ {\pmb{0}} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_6$标准差$ 2.58 \times 10^{ - 3} $$6.99\times 10^{ - 8} $$ 2.23 \times 10^{ - 44} $$ 1.02\times 10^{ -12} $$ 1.30 \times 10^{ - 115} $000
    排名87564111
    平均值$ 1.26 \times 10^{ 2} $$ 2.57 \times10^{ 1} $$ 1.14 \times 10^{ - 14} $$ 9.62 \times 10^{ -15} $$ 8.96 \times 10^{ 0} $$ {\pmb{0}} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_7$标准差$ 6.85 \times 10^{ 1} $$ 1.54\times 10^{ 0} $$ 2.54 \times 10^{ - 14} $$1.88 \times 10^{ - 14} $$ 6.17 \times 10^{ 0} $000
    排名87546111
    平均值$ 2.41 \times 10^{ - 6} $$7.10\times 10^{ - 7} $$ 1.07 \times 10^{ - 14} $$ 3.81\times 10^{ - 14} $$ 2.66 \times 10^{ - 15} $$ {\pmb{8.88\times 10}}^{ - {\pmb{16}}} $$ {\pmb{8.88 \times 10}}^{ -{\pmb{16}}} $$ {\pmb{8.88 \times 10}}^{ - {\pmb{16}}}$
    $f_8$标准差$ 6.98 \times 10^{ - 7} $$ 1.54\times10^{ - 6} $$ 1.85 \times 10^{ - 15} $$ 2.16 \times 10^{-15} $$ 9.93 \times 10^{ - 16} $000
    排名87564111
    平均值$ 6.60 \times 10^{ - 11} $$2.33\times10^{ - 1} $$ 2.44 \times 10^{ - 16} $$ 6.21 \times 10^{ - 16} $$ {\pmb{0}} $$ {\pmb{0}} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_9$标准差$ 1.32 \times 10^{ - 11} $$ 1.30\times10^{ - 1} $$ 1.45 \times 10^{ - 16} $$ 9.63\times 10^{ - 16} $0000
    排名78561111
    平均值$ 9.71 \times 10^{ - 6} $$6.32\times 10^{ - 5} $$ 9.53 \times 10^{ 0} $$ 1.35\times 10^{ - 6} $$ 2.21 \times 10^{ - 61} $$ 7.28 \times 10^{ - 262} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_{10}$标准差$ 2.94 \times 10^{ - 6} $$ 7.25\times10^{ - 5} $$ 7.79 \times 10^{ 0} $$ 2.42 \times 10^{ -6} $$ 2.97 \times 10^{ - 61} $000
    排名67854311
    平均值$ 4.59 \times 10^{ - 5} $$1.19\times 10^{ 1} $$ 2.38 \times 10^{ - 49} $$ 4.58\times 10^{ - 15} $$ 2.62 \times 10^{ - 120} $$ {\pmb{0}} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_{11}$标准差$ 4.27 \times 10^{ - 5} $$ 8.43\times10^{ 0} $$ 3.10 \times 10^{ - 49} $$ 6.63 \times 10^{ -15} $$ 3.04\times 10^{ - 120} $000
    排名78564111
    平均值$ 1.49 \times 10^{ - 1} $$1.53\times 10^{ 1} $$ 3.51 \times 10^{ - 15} $$ 8.06\times 10^{ - 2} $$ 2.08 \times 10^{ - 28} $$ {\pmb{0}} $$ {\pmb{0}} $$ {\pmb{0}} $
    $f_{12}$标准差$ 5.87 \times 10^{ - 3} $$ 6.60\times10^{ 0} $$ 2.45 \times 10^{ - 15} $$ 1.85 \times 10^{ -2} $$ 1.35 \times 10^{ - 28} $000
    排名78564111
    平均排名6.66677.41675.16675.75003.75001.91671.50001.4167
    最终排名78564321
    下载: 导出CSV

    表  4  LIL-GWO与其他7种算法的统计检验结果比较

    Table  4  Statistical test results of LIL-GWO and other seven algorithms

    算法$ {\rm{R}}^ +$$ {\rm{R}}^ -$$p$-value$\alpha = 0.05$$\alpha = 0.1$
    LIL-GWO versus CMA-ES67.011.0$ 2.8848 \times 10^{ - 4} $YesYes
    LIL-GWO versus IPSO78.00.0$ 4.5561 \times 10^{ - 4} $YesYes
    LIL-GWO versus ODE67.011.0$ 1.3461 \times 10^{ - 3} $YesYes
    LIL-GWO versus GABC66.012.0$ 7.0988 \times 10^{ - 4} $YesYes
    LIL-GWO versus ETLBO62.016.0$ 5.7382 \times 10^{ - 3} $YesYes
    LIL-GWO versus IWOA40.038.0$ 3.1412 \times 10^{ - 1} $NoNo
    LIL-GWO versus ISCA39.538.5$ 9.8936 \times 10^{ - 1} $NoNo
    下载: 导出CSV

    表  5  两种算法对12个函数的实验结果比较

    Table  5  Experimental results of two algorithms for functions

    函数OBL-GWOLIBL-GWO
    平均值标准差平均值标准差
    $f_{1}$$ 2.37 \times 10^{ - 35} $$ 3.66 \times 10^{ - 35} $ 00
    $f_{2}$$ 1.27 \times 10^{ - 18} $$ 7.90 \times 10^{ - 19} $ 00
    $f_{3}$$ 2.50 \times 10^{ - 33} $$ 4.36 \times 10^{ - 33} $ 00
    $f_{4}$$ {\bf{2.87 \times 10}}^{\pmb{1}}$$ 1.55 \times 10^{ - 1} $$ 2.89 \times 10^{ 1} $$ 5.83 \times 10^{ - 2} $
    $f_{5}$$ 1.16 \times 10^{ - 145} $$ 1.60 \times 10^{ - 145} $ 00
    $f_{6}$$ 1.46 \times 10^{ - 28} $$ 2.05 \times 10^{ - 28} $ 00
    $f_{7}$ 00 00
    $f_{8}$$ 4.44 \times 10^{ - 15} $0$ {\pmb{8.88 \times10}}^{\bf{-16}} $0
    $f_{9}$ 00 00
    $f_{10}$$ 5.91 \times 10^{ - 19} $$ 8.32 \times 10^{ - 19} $ 00
    $f_{11}$$ 2.90 \times 10^{ - 34} $$ 5.73 \times 10^{ - 34} $ 00
    $f_{12}$$ 1.68 \times 10^{ - 8} $$ 1.46 \times 10^{ - 8} $ 00
    下载: 导出CSV

    表  6  8种算法对CEC2014测试集30个函数的实验结果比较

    Table  6  Comparisons of eight algorithms for 30 test functions from CEC 2014

    函数统计结果HS-SAPBACoDEMoABCDGS-TLBOHSCALM-BBOLIL-GWO
    平均误差$ 1.16 \times 10^{ 7} $$3.50\times 10^{ 7} $$ 1.21 \times 10^{ 7} $$ 2.81 \times 10^{7} $$ 1.04 \times 10^{ 7} $$ 3.50 \times 10^{ 7} $${\bf{1.01 \times 10}}^{\bf{7}} $$ 2.59 \times 10^{ 7} $
    F1标准差$ 7.89 \times 10^{ 4} $$2.16 \times 10^{ 7} $$ 4.48 \times 10^{ 6} $$ 1.01\times 10^{ 7} $$ 8.61 \times 10^{ 6} $$ 2.49 \times 10^{ 7} $$ 3.81 \times 10^{ 6} $$ 4.30 \times 10^{ 6} $
    排名37462713
    平均误差$ {\bf{1.38 \times 10}}^{\bf{4}} $$ 3.05\times 10^{ 8} $$ 1.89 \times 10^{ 7} $$ 2.88\times10^{ 4} $$ 4.59 \times 10^{ 6} $$ 1.95 \times 10^{ 7} $$ 5.34 \times 10^{ 4} $$ 1.02 \times 10^{ 7} $
    F2标准差$ 1.34 \times 10^{ 4} $$ 1.89\times10^{ 8} $$ 9.45 \times 10^{ 6} $$ 4.11 \times 10^{ 4} $$ 1.11 \times 10^{ 7} $$ 5.49 \times 10^{ 7} $$ 2.14 \times 10^{4} $$ 2.71 \times 10^{ 6} $
    排名18624735
    平均误差$ 6.31 \times 10^{ 3} $$ 6.97\times10^{ 3} $$ 4.16 \times 10^{ 3} $$ 1.06 \times 10^{ 4} $$ {\bf{1.44 \times 10}}^{\bf{1}} $$ 3.10 \times 10^{ 4} $$ 1.64 \times 10^{ 4} $$ 2.24 \times 10^{ 4} $
    F3标准差$ 6.06 \times 10^{ 3} $$ 3.96\times10^{3} $$ 1.89 \times 10^{ 3} $$ 3.66 \times 10^{ 3} $$ 1.68 \times 10^{ 1} $$ 1.36 \times 10^{ 4} $$ 1.71 \times 10^{ 4} $$ 2.72 \times 10^{ 4} $
    排名34251867
    平均误差$ 1.11 \times 10^{ 2}$$ 5.78 \times10^{ 2} $$ 1.44 \times 10^{ 2} $$ 1.59 \times 10^{ 2} $$ 1.46 \times 10^{ 2} $$ 2.03 \times 10^{ 2} $$ {\bf{9.99 \times 10}}^{\bf{1}}$$ 2.38 \times 10^{ 2} $
    F4标准差$ 4.01 \times 10^{ 1} $$ 3.62\times10^{ 1} $$ 1.55 \times 10^{ 1} $$ 2.76\times 10^{ 1} $$ 3.78 \times 10^{ 1} $$ 6.69 \times 10^{ 1} $$ 2.85 \times 10^{ 1} $$ 4.57 \times 10^{ 1} $
    排名28354617
    平均误差$ 2.00 \times 10^{ 1} $$ 5.21\times10^{ 2} $$ 2.10 \times 10^{ 1} $$ 2.04\times 10^{ 1} $$ 2.10 \times 10^{ 1} $$ 2.00 \times 10^{ 1} $$ {\bf{3.06 \times 10}}^{\bf{0}}$$ 2.04 \times 10^{ 1} $
    F5标准差$ 3.01 \times 10^{ - 4} $$ 5.26\times10^{ - 2} $$ 6.56 \times 10^{ - 2} $$ 3.53 \times 10^{- 2} $$ 4.34 \times 10^{ - 2} $$ 2.28 \times 10^{ - 3} $$ 7.87 \times 10^{ - 1} $$ 6.42 \times 10^{ - 2} $
    排名28646214
    平均误差$ 1.31 \times 10^{ 1} $$ 6.16\times10^{ 2} $$ 5.57 \times 10^{ 1} $$ 3.78 \times 10^{1} $$ 1.67 \times 10^{ 1} $$ 3.23 \times 10^{ 1} $$ 1.69 \times 10^{ 1} $$ {\bf{1.25 \times 10}}^{\bf{1}} $
    F6标准差$ 2.12 \times 10^{ 0} $$ 2.40\times10^{ 0} $$ 2.67 \times 10^{ 0} $$ 2.65 \times 10^{ 0} $$ 3.45 \times 10^{0} $$ 3.27 \times 10^{0} $$ 3.12 \times 10^{0} $$ 2.10 \times 10^{0} $
    排名28763541
    平均误差$ {\bf{1.52 \times 10}}^{\bf{-2}} $$ 7.04 \times10^{ 2} $$ 1.20 \times 10^{ 0} $$ 5.72 \times 10^{ -1} $$ 1.01 \times 10^{ 0} $$ 1.79 \times 10^{ 0} $$ 1.76 \times 10^{ -1} $$ 7.66 \times 10^{ 0} $
    F7标准差$ 1.63 \times 10^{ -2} $$ 1.72 \times 10^{ 0} $$ 7.20 \times 10^{ - 2} $$1.36 \times 10^{ - 1} $$ 1.50 \times 10^{ 0} $$ 2.19 \times 10^{ 0} $$ 8.56 \times 10^{ -2} $$ 4.50 \times 10^{ 0} $
    排名18534627
    平均误差$ {\bf{4.10 \times 10}}^{\bf{-5}} $$ 8.56\times 10^{ 2} $$ 2.30 \times 10^{ 2} $$ 1.26\times 10^{ 1} $$ 7.67 \times 10^{ 1} $$ 1.71 \times10^{2} $$ 5.53 \times 10^{ 1} $$ 3.49 \times 10^{ 1} $
    F8标准差$ 8.13 \times 10^{ - 6} $$ 1.48\times10^{ 1} $$ 1.45 \times 10^{ 1} $$ 1.74 \times 10^{ 0} $$ 2.45 \times 10^{ 1} $$ 3.46 \times 10^{ 1} $$ 3.78 \times 10^{ 2} $$ 1.20 \times 10^{ 1} $
    排名18725643
    平均误差$ 6.71 \times 10^{ 1} $$ 1.01\times10^{ 3} $$ 3.80 \times 10^{ 2} $$ 2.58 \times 10^{ 2} $$ 9.84 \times 10^{ 1} $$ 2.80 \times 10^{ 2} $$ 7.66 \times 10^{ 1} $$ {\bf{6.02 \times 10}}^{\bf{1}} $
    F9标准差$ 1.52 \times 10^{ 1} $$ 1.33\times10^{1} $$ 1.89\times 10^{ 1} $$ 2.83\times 10^{ 1} $$ 3.08\times 10^{ 1} $$ 5.16\times 10^{ 1} $$ 1.61\times 10^{ 1} $$ 8.59\times 10^{ 0} $
    排名28754631
    平均误差$ {\bf{2.01 \times 10}}^{\bf{-1}} $$ 1.89\times 10^{ 3} $$ 7.26 \times 10^{ 3} $$ 2.29\times 10^{ 2} $$ 2.39 \times 10^{ 3} $$ 2.66 \times 10^{ 3} $$ 1.26 \times 10^{ 4} $$ 1.72 \times 10^{ 3} $
    F10标准差$ 4.66 \times 10^{ - 2} $$ 3.68\times10^{ 2} $$ 3.84 \times 10^{ 2} $$ 1.07 \times 10^{ 2} $$ 4.71 \times 10^{2} $$ 5.34 \times 10^{2} $$ 1.16 \times 10^{4} $$ 1.81 \times 10^{2} $
    排名14725683
    平均误差$ {\bf{1.99 \times 10}}^{\bf{3}} $$ 4.49\times 10^{ 3} $$ 1.21 \times 10^{ 4} $$ 5.74\times 10^{ 3} $$ 3.93 \times 10^{ 3} $$ 4.13 \times 10^{ 3} $$ 1.23 \times 10^{4} $$ 2.64 \times 10^{ 3} $
    F11标准差$ 4.34 \times 10^{ 2} $$ 5.35\times10^{ 2} $$ 4.27 \times 10^{ 2} $$ 3.27 \times 10^{ 2} $$ 5.45\times 10^{ 2} $$ 5.35\times 10^{ 2} $$ 3.42\times 10^{ 2} $$ 3.12\times 10^{ 2} $
    排名15763482
    平均误差$ 2.46 \times 10^{ - 2} $$1.20\times 10^{ 3} $$ 2.47 \times 10^{ 0} $$ 4.71\times 10^{ - 1} $$ 2.75 \times 10^{ 0} $$ 5.11 \times 10^{ - 1} $$ {\bf{1.11 \times 10}}^{\bf{-2}} $$ 3.20 \times 10^{ - 1} $
    F12标准差$ 1.26 \times 10^{ - 2} $$ 1.43\times10^{ -1} $$ 2.74 \times 10^{ - 1} $$ 5.73 \times 10^{ -2} $$ 2.62 \times 10^{ - 1} $$ 2.56 \times 10^{ - 1} $$ 1.75 \times 10^{ - 18} $$ 3.19 \times 10^{ - 1} $
    排名28657413
    平均误差$ 5.24 \times 10^{ - 1} $$1.30\times 10^{ 3} $$ 6.53 \times 10^{ -1} $$ 4.51\times 10^{ - 1} $$ 4.71 \times 10^{ -1} $$ 4.81 \times 10^{ - 1} $$ 6.55 \times 10^{ - 1} $$ {\bf{3.40 \times 10}}^{\bf{-1}} $
    F13标准差$ 1.04 \times 10^{ - 1} $$ 9.49\times10^{ -2} $$ 6.56 \times 10^{ - 2} $$ 4.11 \times 10^{ -2} $$ 1.31 \times 10^{ - 1} $$ 1.17 \times 10^{ - 1} $$ 1.56 \times 10^{ - 1} $$ 5.48 \times 10^{ - 2} $
    排名58623471
    平均误差$ 4.15 \times 10^{ - 1} $$1.40\times 10^{ 3} $$ 4.31 \times 10^{ -1} $$ 2.98\times 10^{ - 1} $$ {\bf{2.88 \times 10}}^{\bf{-1}} $$ 3.08 \times 10^{ - 1} $$ 6.20 \times 10^{ - 1} $$ 4.10 \times 10^{ - 1} $
    F14标准差$ 2.29 \times 10^{ - 1} $$ 4.57\times10^{ -2} $$ 8.50 \times 10^{ - 2} $$ 2.50 \times 10^{ -2} $$ 4.92 \times 10^{ - 2} $$ 5.64 \times 10^{ - 2} $$ 2.96 \times 10^{ - 1} $$ 2.68 \times 10^{ - 2} $
    排名58621374
    平均误差$ 1.64 \times 10^{ 1} $$ 1.52\times10^{ 3} $$ 3.78 \times 10^{ 1} $$ 3.14\times 10^{ 1} $$ 3.75 \times 10^{ 1} $$ 9.80 \times 10^{ 1} $$ 1.55 \times 10^{ 1} $$ {\bf{1.68 \times 10}}^{\bf{0}} $
    F15标准差$ 1.17 \times 10^{ 1} $$ 3.36\times10^{ 0} $$ 2.26 \times 10^{ 0} $$ 6.02 \times 10^{ 0} $$ 2.19 \times 10^{ 1} $$ 3.02 \times 10^{ 1} $$ 5.50 \times 10^{ 0} $$ 4.92 \times 10^{ - 1} $
    排名38645721
    平均误差$ 1.42 \times 10^{ 1} $$ 1.63\times10^{ 3} $$ 2.28 \times 10^{ 1} $$ 1.97\times 10^{ 1} $$ 1.11 \times 10^{ 1} $$ 1.27 \times 10^{ 1} $$ 1.08 \times 10^{ 1} $$ {\bf{1.03 \times 10}}^{\bf{1}} $
    F16标准差$ 7.83 \times 10^{ - 1} $$ 3.78\times10^{ - 1} $$ 3.26 \times 10^{ - 1} $$ 4.02 \times 10^{ - 1} $$ 6.62 \times 10^{ - 1} $$ 5.01 \times 10^{ - 1} $$ 5.84 \times 10^{ -1 } $$ 9.04 \times 10^{ - 1} $
    排名58763421
    平均误差$ 2.09 \times 10^{ 6} $$ 3.40\times10^{ 6} $$ 1.81 \times 10^{ 5} $$ 1.01\times 10^{ 7} $$ {\bf{1.67 \times 10}}^{\bf{5}} $$ 1.48 \times 10^{ 6} $$ 1.46 \times 10^{ 6} $$ 1.29 \times 10^{ 6} $
    F17标准差$ 1.31 \times 10^{ 6} $$ 2.12\times10^{ 6} $$ 1.24 \times 10^{ 5} $$ 4.96 \times 10^{ 6} $$ 2.13 \times 10^{ 5} $$ 1.21 \times 10^{ 6} $$ 9.34 \times 10^{ 5} $$ 1.23 \times 10^{ 6} $
    排名67281543
    平均误差$ 6.16 \times 10^{ 3} $$ 1.70\times10^{ 6} $$ 3.62 \times 10^{ 3} $$ 9.92\times 10^{ 3} $$ {\bf{8.71 \times 10}}^{\bf{2}} $$ 7.67 \times 10^{ 3} $$ 2.90 \times 10^{ 3} $$ 4.07 \times 10^{ 5} $
    F18标准差$ 6.22 \times 10^{ 3} $$ 1.06\times10^{ 6} $$ 2.31 \times 10^{ 3} $$ 9.94 \times 10^{ 3} $$ 1.02 \times 10^{ 3} $$ 6.70 \times 10^{ 3} $$ 4.27 \times 10^{ 3} $$ 7.27 \times 10^{ 5} $
    排名48361527
    平均误差$ 1.89 \times 10^{ 1} $$ 1.91\times10^{ 3} $$ 3.62 \times 10^{ 1} $$ 3.33\times 10^{ 1} $$ 2.71 \times 10^{ 1} $$ 5.33 \times 10^{ 1} $$ 5.19 \times 10^{ 3} $$ {\bf{1.84 \times 10}}^{\bf{1}} $
    F19标准差$ 2.46 \times 10^{ 1} $$ 5.23\times10^{ 0} $$ 1.08 \times 10^{ 1} $$ 1.06 \times 10^{ 1} $$ 2.86 \times 10^{ 1} $$ 3.63 \times 10^{ 1} $$ 5.67 \times 10^{ 3} $$ 3.78 \times 10^{ 0} $
    排名27543681
    平均误差$ 6.77 \times 10^{ 3} $$8.77\times 10^{ 3} $$ 5.04 \times 10^{ 2} $$ 3.96\times 10^{ 4} $$ {\bf{4.28 \times 10}}^{\bf{2}} $$ 3.93 \times 10^{ 4} $$ 2.61 \times 10^{ 4} $$ 1.76 \times 10^{ 4} $
    F20标准差$ 5.09 \times 10^{ 3} $$ 4.31\times10^{ 3} $$ 3.17 \times 10^{ 2} $$ 1.29 \times 10^{ 4} $$ 1.77 \times 10^{2} $$ 2.20 \times 10^{4} $$ 1.56 \times 10^{4} $$ 9.12 \times 10^{3} $
    排名34281765
    平均误差$ 5.27 \times 10^{ 5} $$ 4.39\times10^{ 5} $$ {\bf{2.12 \times 10}}^{\bf{4}} $$ 7.30\times 10^{ 6} $$ 2.20 \times 10^{ 4} $$ 3.54 \times 10^{ 5} $$ 1.11 \times 10^{6} $$ 3.27 \times 10^{ 5} $
    F21标准差$ 3.59 \times 10^{ 5} $$ 3.40\times10^{ 5} $$ 1.61 \times 10^{ 4} $$ 4.36 \times 10^{ 6} $$ 2.22\times 10^{ 4} $$ 3.48\times 10^{ 5} $$ 7.95\times 10^{ 5} $$ 2.86\times 10^{ 5} $
    排名65182473
    平均误差$ 4.88 \times 10^{2} $$ 2.53\times10^{ 3} $$ 1.44 \times 10^{ 3} $$ 1.14\times 10^{ 3} $$ {\bf{3.14 \times 10}}^{\bf{2}} $$ 9.47 \times 10^{ 2} $$ 1.88 \times 10^{ 3} $$ 1.69 \times 10^{ 3} $
    F22标准差$ 9.15 \times 10^{ 1} $$ 1.08\times10^{ 2} $$ 1.59 \times 10^{ 2} $$ 1.89 \times 10^{2} $$ 1.41 \times 10^{ 2} $$ 3.31 \times 10^{ 2} $$ 2.04 \times 10^{ 2} $$ 2.13 \times 10^{ 2} $
    排名28541376
    平均误差$ 3.16 \times 10^{ 2} $$2.60\times 10^{ 3} $$ 3.55 \times 10^{ 2} $$ 3.57\times 10^{ 2} $$ 3.15 \times 10^{ 2} $$ 3.29 \times 10^{ 2} $$ 4.11 \times 10^{ 2} $$ {\bf{2.00 \times 10}}^{\bf{2}} $
    F23标准差$ 5.74 \times 10^{ - 1} $$ 3.88\times10^{ 1} $$ 1.77 \times 10^{ - 1} $$ 7.30 \times 10^{ 0} $$ 4.43 \times 10^{ - 1} $$ 7.51 \times 10^{ 0} $$ 6.43 \times 10^{ 1} $0
    排名38562471
    平均误差$ 2.31 \times 10^{ 2} $$2.61\times 10^{ 3} $$ 2.83 \times 10^{ 2} $$ 2.71\times 10^{ 2} $$ {\bf{2.00 \times 10}}^{\bf{2}} $$ 2.78 \times 10^{ 2} $$ 1.48 \times 10^{ 4} $$ {\bf{2.00 \times 10}}^{\bf{2}} $
    F24标准差$ 5.17 \times 10^{ 0} $$ 3.98\times10^{ 0} $$ 1.80 \times 10^{ 0} $$ 1.78 \times 10^{ 0} $$ 9.68 \times 10^{ - 4} $$ 3.11 \times 10^{ 1} $$ 8.37 \times 10^{ 3} $0
    排名37641581
    平均误差$ 2.05 \times 10^{ 2} $$ 2.70\times10^{ 3} $$ 2.18 \times 10^{ 2} $$ 2.22\times 10^{ 2} $$ 2.02 \times 10^{ 2} $$ 2.23 \times 10^{ 2} $$ 5.29 \times 10^{ 2} $$ {\bf{2.00 \times 10}}^{\bf{2}} $
    F25标准差$ 1.51 \times 10^{ 0} $$ 1.41\times10^{ 0} $$ 1.94 \times 10^{ 0} $$ 2.80 \times 10^{ 0} $$ 3.62 \times 10^{ 2} $$ 9.39 \times 10^{ 0} $$ 4.37 \times 10^{ 1} $0
    排名38645721
    平均误差$ 1.38 \times 10^{2} $$ 2.70\times10^{ 3} $$ 1.04 \times 10^{ 2} $$ 1.01\times 10^{ 2} $$ 1.10 \times 10^{ 2} $$ 1.00 \times 10^{ 2} $$ {\bf{2.13 \times 10}}^{\bf{0}} $$ 1.00 \times 10^{ 2} $
    F26标准差$ 4.84 \times 10^{ 1} $$ 1.93\times10^{ 1} $$ 1.82 \times 10^{ 1} $$ 6.75 \times 10^{ - 2} $$ 3.15 \times 10^{ 1} $$ 1.63 \times 10^{ - 1} $$ 3.46 \times 10^{ 0 } $0
    排名58763421
    平均误差$ 6.69 \times 10^{ 2} $$ 3.14\times10^{ 3} $$ 1.28 \times 10^{ 3} $$ 1.08\times 10^{ 3} $$ 7.94 \times 10^{ 2} $$ 4.27 \times 10^{ 2} $$ {\bf{1.96 \times 10}}^{\bf{2}} $$ 2.00 \times 10^{ 2} $
    F27标准差$ 1.63 \times 10^{ 2} $$ 6.53\times10^{ 1} $$ 1.47 \times 10^{ 2} $$ 3.78 \times 10^{ 2} $$ 2.15 \times 10^{ 2} $$ 1.96 \times 10^{ 1} $$ 1.04 \times 10^{ 2} $0
    排名48765312
    平均误差$ 1.03 \times 10^{ 3} $$ 3.90\times10^{ 3} $$ 1.92 \times 10^{ 3} $$ 2.15 \times 10^{ 3} $$ 1.43 \times 10^{ 3} $$ 3.49 \times 10^{ 3} $$ 1.94 \times 10^{ 3} $$ {\bf{2.00 \times 10}}^{\bf{2}} $
    F28标准差$ 1.21 \times 10^{ 2} $$ 2.16\times10^{ 2} $$ 1.26 \times 10^{ 2} $$ 3.42 \times 10^{ 2} $$ 4.37 \times 10^{ 2} $$ 5.48 \times 10^{ 2} $$ 5.49 \times 10^{ 2} $0
    排名28463751
    平均误差$ 1.40 \times 10^{ 3} $$ 3.59\times10^{ 4} $$ 2.00 \times 10^{ 4} $$ 3.32\times 10^{ 3} $$ 3.08 \times 10^{ 6} $$ 5.44 \times 10^{ 5} $$ 1.98 \times 10^{ 7} $$ {\bf{2.00 \times 10}}^{\bf{2}} $
    F29标准差$ 4.27 \times 10^{ 2} $$ 1.64\times10^{ 5} $$ 7.15 \times 10^{ 3} $$ 1.46 \times 10^{ 3} $$ 4.99 \times 10^{ 6} $$ 2.61 \times 10^{ 6} $$ 3.96 \times 10^{ 6} $0
    排名25437681
    平均误差$ 4.63 \times 10^{ 3} $$ 1.55\times10^{ 4} $$ 1.97 \times 10^{ 4} $$ 1.61\times 10^{ 4} $$ 6.47 \times 10^{ 3} $$ 2.49 \times 10^{ 4} $$ 6.96 \times 10^{ 6} $$ {\bf{2.00 \times 10}}^{\bf{2}} $
    F30标准差$ 2.32 \times 10^{3} $$ 4.24\times10^{ 3} $$ 2.00 \times 10^{ 3} $$ 4.10 \times 10^{ 3} $$ 3.43 \times 10^{3} $$ 2.26 \times 10^{4} $$ 1.03 \times 10^{7} $0
    排名24653781
    平均排名2.93337.03335.03334.73333.26675.16674.63333.0000
    最终排名18653742
    下载: 导出CSV

    表  7  不同算法对单二极管模型的最优辨识参数比较

    Table  7  Comparison among different algorithms on single diode model

    算法$ I_{ph} $ (A)$ I_{sd}\, (\mu$A)$R_S\, (\Omega)$$R_{sh}\, (\Omega)$$n$RMSE
    BSA0.76090.377490.035856.52661.4970$ 1.0398 \times 10^{ - 3}$
    IBSA0.76070.355020.036158.20121.4907$ 1.0092 \times 10^{ - 3}$
    LBSA0.76060.346180.036259.09781.4881$ 1.0143 \times 10^{ - 3}$
    CLPSO0.76080.343020.036154.19651.4873$ 9.9633 \times 10^{ - 3}$
    BLPSO0.76070.366200.035960.28451.4939$ 1.0272 \times 10^{ - 3}$
    DE-BBO0.76050.324770.036455.26271.4817$ 9.9922 \times 10^{ - 4}$
    GOTLBO0.76080.329700.036353.36641.4833$ 9.8856 \times 10^{ - 3}$
    LIL-GWO0.76080.323630.036453.79671.4814$ 9.8604 \times 10^{ - 4}$
    下载: 导出CSV
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