多目标动态差分进化算法及其应用研究
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摘要
多目标优化问题广泛存在于实际工程应用和科学研究中,是一类非常重要而又难以解决的复杂问题。作为随机启发式搜索算法,进化计算通过代表整个解集的种群进化,以内在并行的方式搜索,一次优化运行就能够获得多个非劣解,已被成功应用于多目标优化领域,并发展成为一个相对较热的进化多目标优化研究方向。差分进化算法作为当今最有效的随机优化算法之一,是解决多目标优化问题的一种有效工具。本论文主要研究了基于动态更新种群方式的差分进化算法,重点将其拓展到多目标优化领域,并应用于混合动力汽车多目标优化设计和电力系统环境经济负荷多目标优化分配。全文主要工作包括如下几个方面。
论文首先介绍了多目标差分进化算法的研究背景及意义,给出了多目标优化的相关定义,然后回顾和总结了进化多目标优化的发展历程以及多目标差分进化算法的国内外研究现状,并提出了进化多目标优化领域的热点和难点问题。
针对原创DE算法静态更新种群结构不利于提高算法收敛性的不足,引出了一种采用动态更新种群策略的动态差分进化算法(DDE),并利用随机压缩映射原理对其收敛性进行了分析。21个经典Bechmark函数测试实验结果表明,动态更新种群策略大大提高了DE算法的收敛速率,但也一定程度上增加了"best"变异方式的DE算法陷入局部最优的风险。为解决算法全局探索与局部开发之间的平衡,结合不同进化模式DDE的优点,提出了一种基于自适应变异算子的改进DDE算法(SAMDDE)。大量经典Bechmark函数测试实验以及2D IIR滤波器设计实例均证实了改进算法的有效性。
基于DDE/rand/1/bin变异策略无须选择当前种群中的最优个体,能有效保持种群的多样性从而避免早熟收敛的特点,提出了一种新颖的求解多目标优化问题的参数自适应动态差分进化算法。参数自适应策略有效提高了算法的鲁棒性。针对Deb的拥挤距离估计方法很多时候不能准确地测量非占解之间的拥挤程度的不足,结合解在目标空间中的分布熵和Deb拥挤距离,提出了一种称为拥挤熵的拥挤度测量方法。基于拥挤熵的测量方法能更准确地估计非占优解之间的拥挤程度,从而能更有效保持非占优解集的多样性。选用18个进化多目标优化领域的标准测试问题对提出的算法进行了测试。实验结果表明,与NSGA-Ⅱ、SPEA2和MOPSO三种代表性MOEAs相比,MOSADDE(?)够更好地收敛到问题的Pareto最优前沿,且所得非占优解集具有更好的分布性。
为了进一步提高MOSADDE算法的收敛性能和鲁棒性,针对参数重新随机初始化自适应策略的不足,提出了一种具有自学习能力的参数自适应策略。同时,为了使所求得的非占优解在Pareto最优前沿,尤其在高维目标问题的Pareto最优前沿具有更好的散布性,提出了归一化最近邻域距离的拥挤度测量方法。由此,通过引入新的参数自适应策略和拥挤度估计方法,并基于具有快速收敛特性的DDE/best/2/bin变异方式,提出了一种用于求解MOPs的改进参数自适应动态差分进化算法MOSADDE-Ⅱ。选用27个进化多目标优化领域的标准测试问题对提出的算法进行了测试。实验结果证实了以上方法的有效性。
在前面静态多目标差分进化算法基础上,通过引入环境检测算子和新环境下初始种群多样性保持策略,提出了一种求解动态多目标优化问题的动态多目标差分进化算法(dMODDE)。用七个测试问题进行了仿真研究并与其它动态MOEAs进行了比较。实验结果表明,当问题的Pareto最优解和(或)前沿随时间发生变化时,dMODDE能够跟踪到动态变化的Pareto最优前沿,且所获得的Pareto最优解具有良好的多样性和散布性。
基于非占优排序策略和动态差分进化算法,提出了一种求解双层多目标优化问题(BLMOP)的动态差分进化算法。针对BLMOP的特点,设计了一种特殊的进化种群结构。并对最新文献给出的7个测试问题进行了理论分析并用来测试算法的有效性。实验结果表明,所提出的算法能很好地收敛到每个测试问题的Pareto最优前沿,并能保持良好的多样性和宽广性,是一种求解BLMOP的有效方法。
基于MOSADDE-Ⅱ算法,提出了一种同时优化混合动力汽车动力总成部件参数和控制策略参数的方法。基于电动辅助控制策略,将HEV优化设计问题归结为一个非线性约束多目标优化问题,其中优化目标包括油耗FC、HC排放、CO排放和NOx排放等四个指标,约束条件包括PNGV性能标准和电池SOC荷电状态维持等要求。同时,利用模糊集理论,从所获得的Pareto最优解集中提取出最优折衷解。以典型行驶循环工况FTP、ECE-EUDC和UDDS为试验工况,针对一种并联式混合动力轿车进行了离线优化仿真研究。实验结果表明,与基于GA的加权系数法和NSGA-Ⅱ相比较,MOSADDE-Ⅱ具有明显的优越性。
为了进一步检验MOSADDE-Ⅱ在工程优化设计中的有效性,同时还将其应用于电力系统环境经济负荷分配(EED)多目标优化。针对传统EED只考虑发电成本和污染控制成本两个优化目标的不足,通过引入系统损耗,建立了一个包含经济、环境和线损三个目标的EED多目标优化模型。应用MOSADDE-Ⅱ对IEEE30-和118-bus典型测试系统进行了仿真研究,实验结果证实了所提方法的有效性。
论文最后总结了全文的主要工作和创新性的研究成果,并对下一步研究工作进行了展望。
Multi-objective optimization problems are often encountered in many practice engineering applications and science researches, and it is one of class of very important but difficult complex problem to be solved. As stochastic optimization metaheuristics, evolutionary algorithms deal simultaneously with a set of possible solutions (the so-called population) which allows us to find several members of the Pareto optimal set in a single run of the algorithm. Consequently, Evolutionary multi-objective optimization (EMO) has now become a popular and useful field of research and application. Differential evolution (DE), as one of the most powerful stochastic real-parameter optimization algorithms in current use, is a natural candidate to be extended for multi-objective optimization. This dissertation mainly focuses on the researches of differential evolution with dynamic population updating stratey, and its extention for multi-objective optimization. Main results and contribu-tions of this dissertation are as follows:
The research background of multi-objective differential evolution (MODE) is introduced firstly. Then, the relative definitions of multi-objective optimization (MO) are presented, and a breaf histrory of EMO and main multi-objective evolutionary algorithms (MOEAs) are introduced. Furthermore, a review of the basic concepts of DE and a survey of its application to multiobjective are also presented. Thereafhter, the hot and difficult problems in EMO are discussed and the research significance of this dissertation is also provided.
In the view of slower convergence of original DE with static population updating structure, a dynamic population updating stratey is introduced and a novel dynamic differential evolution (DDE) is then proposed. Twenty-one Benchmark test functions are used to evaluate the effectiveness of the proposed DDE. Experiment results show that the dynamic population updating stratey greatly speeds up the DE's convergence speed, but also increases the risk to premature convergence in some extent for the DEs with 'best' mutation strategy. Therefore, to keep the balance between the global explora-tion and local exploitation, an adaptive mutation operator combined with the advantages of strategies of DDE/rand/1/bin and DDE/best/2/bin is proposed. The effectiveness of the modified version is validated by using twenty-one classical Benchmark functions and a specific2D IIR filter design problem.
An approach called MOSADDE to extend the stragety of DDE/rand/1/bin to solve multi-objective optimization problems with an external elitist archive is presented. To preserve the diversity of the Pareto optimality, a more accurate crowding measure method namely crowding entropy is proposed. Moreover, to improve the robustness of DDE, a control parameter self-adaptive strategy is introduced. Therefore, the user does not need to guess the good values for F and CR, which are problem dependent. The proposed approach was validated using eighteen standard test problems currently adopted in the evolutionary multi-objective optimization community. The experiment results and no-parametric statistical results associated indicate that MOSADDE is able to maintain a better spread of solutions and converge in the obtained Pareto-optimal front compared to three representative multi-objective evolutionary algorithms NSGA-Ⅱ, SPEA2and MOPSO.
To further improve the convergence properties of MOSADDE, based on the stratey of DDE/best/2/bin, an improved version MOSADDE-Ⅱ is proposed. The random initialization based parameter self-adaptive strategy in MOSADDE is replaced by a new parameter self-adaptive strategy with self-learning ability. Moreover, in order to preserve a set of nondominated solutions widely distributed along the Pareto front, especially along the Pareto front of high-dimension problem, a progressive comparison truncate operator based on normalized nearest neighbor distance is proposed in the MOSADDE-Ⅱ. This density estimation method is able to accurately reflect the crowding degree for problems with objective functions range between values of different orders of magnitude. The proposed approach was validated using twenty-seven standard test problems currently adopted in the evolutionary multi-objective optimization community.
Based on the previous static MODDE, an approach for the dynamic multi-objective optimization is presented by introducing environment detecting operator and diversity maintaining strategy for inintial population in new enviroment. The effect-iveness of the dynamic multi-objective dynamic differential evolution (dMODDE) is validated against various dynamic MOEAs upon seven Bechmark problems with different characteristics in Pareto optimal front. Experiment results show that the proposed dMODDE is able to well track the Pareto front as it changes with time in dynamic environments.
Based on the nondominate sorting strategy, a multi-objective dynamic differential evolution algorithm for the bi-level multi-objective optimization problems (BLMOP) is proposed. For the characteristics of BLMOPs, a special evolutionary population structure is designed for the bi-level multi-objective dynamic differential evolution (BLMODDE) algorithm. The effectiveness of the BLMODDE is examined on several updated test problems. Experiment results demonstrate that the proposed BLMODDE is able to maintain a good spread of solutions and converge to the Pareto-optimal front of problems. It is suggested that the proposed approach is promising for dealing with BLMOPs.
The application of the MOSADDE-II for the simultaneous optimization of component sizing and control strategy in parallel hybrid electric vehicles (HEVs) is described. Based on an electric assist control strategy, the HEV optimal design problem is formulated as a nonlinear constrained multiobjective problem with competing and noncommensurable objectives of fuel consumption (FC), CO emission, HC emission, and NOx emission. The driving performance requirements are considered constraints. Moreover, fuzzy set theory is employed to extract the best compromise solution. The optimization process is performed over three typical driving cycles including FTP, ECE+EUDC and UDDS that currently used in United States and European community. The results demonstrate the capability of the proposed approach to generate well-distributed Pareto optimal solutions of the HEV multi-objective optimization design problem. The comparison with reported results of GA based weighting sum approaches and NSGA-Ⅱ reveals the superiority of the proposed approach and confirms its potential for optimal HEV design.
To further demonstrate effectiveness of the MOSADDE-Ⅱ for engineering optimization design, it is used for the Environmental/Economic power Dispatch (EED) problem. The EED problem is formulated as a nonlinear constrained multi-objective problem with competing and non-commensurable fuel cost, emission and system loss objectives. Several optimization runs of the proposed approach have been carried out on the IEEE30-and118-bus test system. The comparison with reported results of other MOEAs reveals the superiority of the proposed approach and confirms its potential for solving other power systems multi-objective optimization problems.
Finally, the main innovations of the dissertation are summarized, and then the fields for further research are prospected.
论文首先介绍了多目标差分进化算法的研究背景及意义,给出了多目标优化的相关定义,然后回顾和总结了进化多目标优化的发展历程以及多目标差分进化算法的国内外研究现状,并提出了进化多目标优化领域的热点和难点问题。
针对原创DE算法静态更新种群结构不利于提高算法收敛性的不足,引出了一种采用动态更新种群策略的动态差分进化算法(DDE),并利用随机压缩映射原理对其收敛性进行了分析。21个经典Bechmark函数测试实验结果表明,动态更新种群策略大大提高了DE算法的收敛速率,但也一定程度上增加了"best"变异方式的DE算法陷入局部最优的风险。为解决算法全局探索与局部开发之间的平衡,结合不同进化模式DDE的优点,提出了一种基于自适应变异算子的改进DDE算法(SAMDDE)。大量经典Bechmark函数测试实验以及2D IIR滤波器设计实例均证实了改进算法的有效性。
基于DDE/rand/1/bin变异策略无须选择当前种群中的最优个体,能有效保持种群的多样性从而避免早熟收敛的特点,提出了一种新颖的求解多目标优化问题的参数自适应动态差分进化算法。参数自适应策略有效提高了算法的鲁棒性。针对Deb的拥挤距离估计方法很多时候不能准确地测量非占解之间的拥挤程度的不足,结合解在目标空间中的分布熵和Deb拥挤距离,提出了一种称为拥挤熵的拥挤度测量方法。基于拥挤熵的测量方法能更准确地估计非占优解之间的拥挤程度,从而能更有效保持非占优解集的多样性。选用18个进化多目标优化领域的标准测试问题对提出的算法进行了测试。实验结果表明,与NSGA-Ⅱ、SPEA2和MOPSO三种代表性MOEAs相比,MOSADDE(?)够更好地收敛到问题的Pareto最优前沿,且所得非占优解集具有更好的分布性。
为了进一步提高MOSADDE算法的收敛性能和鲁棒性,针对参数重新随机初始化自适应策略的不足,提出了一种具有自学习能力的参数自适应策略。同时,为了使所求得的非占优解在Pareto最优前沿,尤其在高维目标问题的Pareto最优前沿具有更好的散布性,提出了归一化最近邻域距离的拥挤度测量方法。由此,通过引入新的参数自适应策略和拥挤度估计方法,并基于具有快速收敛特性的DDE/best/2/bin变异方式,提出了一种用于求解MOPs的改进参数自适应动态差分进化算法MOSADDE-Ⅱ。选用27个进化多目标优化领域的标准测试问题对提出的算法进行了测试。实验结果证实了以上方法的有效性。
在前面静态多目标差分进化算法基础上,通过引入环境检测算子和新环境下初始种群多样性保持策略,提出了一种求解动态多目标优化问题的动态多目标差分进化算法(dMODDE)。用七个测试问题进行了仿真研究并与其它动态MOEAs进行了比较。实验结果表明,当问题的Pareto最优解和(或)前沿随时间发生变化时,dMODDE能够跟踪到动态变化的Pareto最优前沿,且所获得的Pareto最优解具有良好的多样性和散布性。
基于非占优排序策略和动态差分进化算法,提出了一种求解双层多目标优化问题(BLMOP)的动态差分进化算法。针对BLMOP的特点,设计了一种特殊的进化种群结构。并对最新文献给出的7个测试问题进行了理论分析并用来测试算法的有效性。实验结果表明,所提出的算法能很好地收敛到每个测试问题的Pareto最优前沿,并能保持良好的多样性和宽广性,是一种求解BLMOP的有效方法。
基于MOSADDE-Ⅱ算法,提出了一种同时优化混合动力汽车动力总成部件参数和控制策略参数的方法。基于电动辅助控制策略,将HEV优化设计问题归结为一个非线性约束多目标优化问题,其中优化目标包括油耗FC、HC排放、CO排放和NOx排放等四个指标,约束条件包括PNGV性能标准和电池SOC荷电状态维持等要求。同时,利用模糊集理论,从所获得的Pareto最优解集中提取出最优折衷解。以典型行驶循环工况FTP、ECE-EUDC和UDDS为试验工况,针对一种并联式混合动力轿车进行了离线优化仿真研究。实验结果表明,与基于GA的加权系数法和NSGA-Ⅱ相比较,MOSADDE-Ⅱ具有明显的优越性。
为了进一步检验MOSADDE-Ⅱ在工程优化设计中的有效性,同时还将其应用于电力系统环境经济负荷分配(EED)多目标优化。针对传统EED只考虑发电成本和污染控制成本两个优化目标的不足,通过引入系统损耗,建立了一个包含经济、环境和线损三个目标的EED多目标优化模型。应用MOSADDE-Ⅱ对IEEE30-和118-bus典型测试系统进行了仿真研究,实验结果证实了所提方法的有效性。
论文最后总结了全文的主要工作和创新性的研究成果,并对下一步研究工作进行了展望。
Multi-objective optimization problems are often encountered in many practice engineering applications and science researches, and it is one of class of very important but difficult complex problem to be solved. As stochastic optimization metaheuristics, evolutionary algorithms deal simultaneously with a set of possible solutions (the so-called population) which allows us to find several members of the Pareto optimal set in a single run of the algorithm. Consequently, Evolutionary multi-objective optimization (EMO) has now become a popular and useful field of research and application. Differential evolution (DE), as one of the most powerful stochastic real-parameter optimization algorithms in current use, is a natural candidate to be extended for multi-objective optimization. This dissertation mainly focuses on the researches of differential evolution with dynamic population updating stratey, and its extention for multi-objective optimization. Main results and contribu-tions of this dissertation are as follows:
The research background of multi-objective differential evolution (MODE) is introduced firstly. Then, the relative definitions of multi-objective optimization (MO) are presented, and a breaf histrory of EMO and main multi-objective evolutionary algorithms (MOEAs) are introduced. Furthermore, a review of the basic concepts of DE and a survey of its application to multiobjective are also presented. Thereafhter, the hot and difficult problems in EMO are discussed and the research significance of this dissertation is also provided.
In the view of slower convergence of original DE with static population updating structure, a dynamic population updating stratey is introduced and a novel dynamic differential evolution (DDE) is then proposed. Twenty-one Benchmark test functions are used to evaluate the effectiveness of the proposed DDE. Experiment results show that the dynamic population updating stratey greatly speeds up the DE's convergence speed, but also increases the risk to premature convergence in some extent for the DEs with 'best' mutation strategy. Therefore, to keep the balance between the global explora-tion and local exploitation, an adaptive mutation operator combined with the advantages of strategies of DDE/rand/1/bin and DDE/best/2/bin is proposed. The effectiveness of the modified version is validated by using twenty-one classical Benchmark functions and a specific2D IIR filter design problem.
An approach called MOSADDE to extend the stragety of DDE/rand/1/bin to solve multi-objective optimization problems with an external elitist archive is presented. To preserve the diversity of the Pareto optimality, a more accurate crowding measure method namely crowding entropy is proposed. Moreover, to improve the robustness of DDE, a control parameter self-adaptive strategy is introduced. Therefore, the user does not need to guess the good values for F and CR, which are problem dependent. The proposed approach was validated using eighteen standard test problems currently adopted in the evolutionary multi-objective optimization community. The experiment results and no-parametric statistical results associated indicate that MOSADDE is able to maintain a better spread of solutions and converge in the obtained Pareto-optimal front compared to three representative multi-objective evolutionary algorithms NSGA-Ⅱ, SPEA2and MOPSO.
To further improve the convergence properties of MOSADDE, based on the stratey of DDE/best/2/bin, an improved version MOSADDE-Ⅱ is proposed. The random initialization based parameter self-adaptive strategy in MOSADDE is replaced by a new parameter self-adaptive strategy with self-learning ability. Moreover, in order to preserve a set of nondominated solutions widely distributed along the Pareto front, especially along the Pareto front of high-dimension problem, a progressive comparison truncate operator based on normalized nearest neighbor distance is proposed in the MOSADDE-Ⅱ. This density estimation method is able to accurately reflect the crowding degree for problems with objective functions range between values of different orders of magnitude. The proposed approach was validated using twenty-seven standard test problems currently adopted in the evolutionary multi-objective optimization community.
Based on the previous static MODDE, an approach for the dynamic multi-objective optimization is presented by introducing environment detecting operator and diversity maintaining strategy for inintial population in new enviroment. The effect-iveness of the dynamic multi-objective dynamic differential evolution (dMODDE) is validated against various dynamic MOEAs upon seven Bechmark problems with different characteristics in Pareto optimal front. Experiment results show that the proposed dMODDE is able to well track the Pareto front as it changes with time in dynamic environments.
Based on the nondominate sorting strategy, a multi-objective dynamic differential evolution algorithm for the bi-level multi-objective optimization problems (BLMOP) is proposed. For the characteristics of BLMOPs, a special evolutionary population structure is designed for the bi-level multi-objective dynamic differential evolution (BLMODDE) algorithm. The effectiveness of the BLMODDE is examined on several updated test problems. Experiment results demonstrate that the proposed BLMODDE is able to maintain a good spread of solutions and converge to the Pareto-optimal front of problems. It is suggested that the proposed approach is promising for dealing with BLMOPs.
The application of the MOSADDE-II for the simultaneous optimization of component sizing and control strategy in parallel hybrid electric vehicles (HEVs) is described. Based on an electric assist control strategy, the HEV optimal design problem is formulated as a nonlinear constrained multiobjective problem with competing and noncommensurable objectives of fuel consumption (FC), CO emission, HC emission, and NOx emission. The driving performance requirements are considered constraints. Moreover, fuzzy set theory is employed to extract the best compromise solution. The optimization process is performed over three typical driving cycles including FTP, ECE+EUDC and UDDS that currently used in United States and European community. The results demonstrate the capability of the proposed approach to generate well-distributed Pareto optimal solutions of the HEV multi-objective optimization design problem. The comparison with reported results of GA based weighting sum approaches and NSGA-Ⅱ reveals the superiority of the proposed approach and confirms its potential for optimal HEV design.
To further demonstrate effectiveness of the MOSADDE-Ⅱ for engineering optimization design, it is used for the Environmental/Economic power Dispatch (EED) problem. The EED problem is formulated as a nonlinear constrained multi-objective problem with competing and non-commensurable fuel cost, emission and system loss objectives. Several optimization runs of the proposed approach have been carried out on the IEEE30-and118-bus test system. The comparison with reported results of other MOEAs reveals the superiority of the proposed approach and confirms its potential for solving other power systems multi-objective optimization problems.
Finally, the main innovations of the dissertation are summarized, and then the fields for further research are prospected.
引文
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[3]Fonseca C M, Flemin, P J. Genetic algorithms for multiobjective optimization: Formulation, discussion, and generalization. In:Proceedings of the Fifth Inter-national Conference on Genetic Algorithms,1993:416-423.
[4]Srinivas N, Deb K. Multi-objective function optimization using non-dominated sorting genetic algorithms. Evolutionary Computation,1994,2(3):221-248.
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