Differential Evolution Variants Research Articles

State of Charge Estimation for Lithium Battery in Shipboard DC Power Grid Based on Differential Evolutionary Algorithm

More and more attention has been paid to ships with a DC power grid. State-of-charge (SOC) estimation is a pivotal and challenging assignment for lithium-ion batteries in such ships. However, the precision of SOC estimation is strongly connected with the system parameters. To better identify these parameters in lithium-ion batteries, a differential evolution (DE) algorithm was introduced into this paper as the optimizer. Initially, a first-order RC equivalent circuit model (ECM) was created to characterize the battery’s dynamic behavior. Following this, to estimate open-circuit voltage (OCV) throughout the entire dynamic process, a math model of optimization was established to minimize inaccuracies between the real and estimated terminal voltages. Moreover, estimated SOC values were obtained through OCV-SOC mappings and were contrasted against the true SOC values. The findings manifested the efficacy of the presented structure and technique in comparison with various frequently-cited DE variants.

A novel economic dispatch of energy-renewable multi-area power systems with group-based differential evolution

Spinning reserve capacity can handle the effects of load and renewable energy output forecasting errors. To ensure the safe and stable operation of the power systems and enhance economic benefits, it is meaningful to develop reasonable reserve capacity model for multi-area power systems. In this paper, a calculation method for reserve capacity, based on multi-area joint reserve capacity, is proposed to handle the uncertainty of renewable energy output. Multi-area interconnected economic dispatch with renewable energy, based on multi-area joint reserve capacity, can comprehensively consider the characteristics of power generation resources in each areas, and then reasonably allocate reserve capacity in each areas. Furthermore, to solve the multi-area interconnected economic dispatch with renewable energy, a sorted grouping-based adaptive multi-strategy differential evolution algorithm is designed, which divides the population into four groups according to the current convergence and balance, then adopts several new mutation strategies, and uses adaptive schemes for each group of populations separately. Notably, the sorted grouping-based adaptive multi-strategy differential evolution algorithm has advantages over several differential evolution variants and several well-known algorithms. Finally, the proposed method is compared with individual reserve capacity, which proves that the superiority of the proposed method in reserve capacity programming.

Competitive Elimination Improved Differential Evolution for Wind Farm Layout Optimization Problems

The wind farm layout optimization problem (WFLOP) aims to maximize wind energy utilization efficiency under different wind conditions by optimizing the spatial layout of wind turbines to fully mitigate energy losses caused by wake effects. Some high-performance continuous optimization methods, such as differential evolution (DE) variants, exhibit limited performance when directly applied due to WFLOP’s discrete nature. Therefore, metaheuristic algorithms with inherent discrete characteristics like genetic algorithms (GAs) and particle swarm optimization (PSO) have been extensively developed into current state-of-the-art WFLOP optimizers. In this paper, we propose a novel DE optimizer based on a genetic learning-guided competitive elimination mechanism called CEDE. By designing specialized genetic learning and competitive elimination mechanisms, we effectively address the issue of DE variants failing in the WFLOP due to a lack of discrete optimization characteristics. This method retains the adaptive parameter adjustment capability of advanced DE variants and actively enhances population diversity during convergence through the proposed mechanism, preventing premature convergence caused by non-adaptiveness. Experimental results show that under 10 complex wind field conditions, CEDE significantly outperforms six state-of-the-art WFLOP optimizers, improving the upper limit of power generation efficiency while demonstrating robustness and effectiveness. Additionally, our experiments introduce more realistic wind condition data to enhance WFLOP modeling.

External archive guided radial-grid multi objective differential evolution

Differential evolution (DE) is a robust evolutionary algorithm for solving single-objective and multi-objective optimization problems (MOPs). While numerous multi-objective DE (MODE) variants exist, prior research has primarily focused on parameter control and mutation operators, often neglecting the issue of inadequate population distribution across the objective space. This paper proposes an external archive-guided radial-grid-driven differential evolution for multi-objective optimization (Ar-RGDEMO) to address these challenges. The proposed Ar-RGDEMO incorporates three key components: a novel mutation operator that integrates a radial-grid-driven strategy with a performance metric derived from Pareto front estimation, a truncation procedure that employs Pareto dominance in conjunction with a ranking strategy based on shifted similarity distances between candidate solutions, and an external archive that preserves elite individuals using a clustering approach. Experimental results on four sets of benchmark problems demonstrate that the proposed Ar-RGDEMO exhibits competitive or superior performance compared to seven state-of-the-art algorithms in the literature.

An Experimental Comparison of Self-Adaptive Differential Evolution Algorithms to Induce Oblique Decision Trees

This study addresses the challenge of generating accurate and compact oblique decision trees using self-adaptive differential evolution algorithms. Although traditional decision tree induction methods create explainable models, they often fail to achieve optimal classification accuracy. To overcome these limitations, other strategies, such as those based on evolutionary computation, have been proposed in the literature. In particular, we evaluate the use of self-adaptive differential evolution variants to evolve a population of oblique decision trees encoded as real-valued vectors. Our proposal includes (1) an alternative initialization strategy that reduces redundant nodes and (2) a fitness function that penalizes excessive leaf nodes, promoting smaller and more accurate decision trees. We perform a comparative performance analysis of these differential evolution variants, showing that while they exhibit similar statistical behavior, the Single-Objective real-parameter optimization (jSO) method produces the most accurate oblique decision trees and is second best in compactness. The findings highlight the potential of self-adaptive differential evolution algorithms to improve the effectiveness of oblique decision trees in machine learning applications.

A framework of integrated differential evolution variants based on adaptive relay mode for global optimization

Differential evolution (DE) is highly competitive in single-objective real parameter optimization. However, there are still some problems with differential evolutionary variants in optimizing complex multimode functions, such as poor solution accuracy and slow convergence. To address these problems, an optimization framework of integrated DE variants based on adaptive relay mode (fDE-ARM) is proposed. In this framework, different DE variants are integrated through two adaptive relay mechanisms to give full play to the advantages of the algorithms, thereby improving the performance of the whole algorithm. For the first adaptive relay mode, when the currently executed algorithm is judged to have converged, the population is updated by Gaussian random walk, and then the optimization is continued through the relay algorithm. For the second adaptive relay mode, the relay condition is determined by the average fitness improvement rate. If the condition is met, the relay algorithm takes over the optimization. At the same time, if the diversity is lower than the threshold, to improve the exploration, the roulette wheel method is used to select some individuals to perform the Gaussian random walk. In addition, a feedback mechanism is introduced in the relay process to avoid false switching. To verify the performance of the proposed algorithm, extensive simulations are performed on CEC2005, CEC2014, CEC2017, and CEC2021 benchmark functions. In addition, three engineering problems are used to test the performance. Compared with some newly proposed optimization algorithms, fDE-ARM is statistically superior to the comparison algorithms in terms of solution accuracy, convergence speed, and stability.

Towards efficient identification of fractional-order systems

This study focuses on the efficient identification of fractional-order chaotic systems, particularly under challenging conditions where the time step, iteration time, orders, parameters, initial values, and system structure are all unknown. Unlike previous work, our approach emphasizes the complexities introduced by the unknown time step and iteration time. We introduce a novel differential evolution (DE) variant, termed global and local search using success-history based parameter adaptation for differential evolution (GL-SHADE), to tackle this problem in both noiseless and noisy environments. Comprehensive simulations show that GL-SHADE outperforms other comparative algorithms by delivering both higher precision and faster convergence under these uncertain conditions, highlighting the potential of advanced DE techniques in complex system identification.

Elite-guided Resampling and Multi-mutation based Differential Evolution with exponential crossover for numerical optimization

In the existing Differential Evolution (DE) research, there are two main crossover schemes: exponential crossover and binomial crossover. Most researchers believe that binomial crossover is skilled in handling numerical optimization problems while exponential crossover has the advantage in handling optimization problems with certain correlation between adjacent variables. However, we discover that DE variants with exponential crossover can achieve performance comparable to advanced ones utilizing binomial crossover in optimization applications, provided that an appropriate crossover rate CR and associated parameter control are established. In this paper, an Elite-guided Resampling and Multi-mutation based DE with exponential crossover (ERM-DE) is raised to fill the blank in this field, and its main highlights are as follows: First, a novel structure with two stages is raised in our ERM-DE algorithm. The first stage is elite-guided global opposition learning based resampling, and the other is novel exponential crossover DE evolution stage. Second, a Novel Parameter Control (NPC) technique involving scale factor F and crossover rate CR is raised. Different from applying the fitness-based parameter control, the adaptation schemes of the parameters F and CR in our ERM-DE algorithm are fitness-independent. In addition, this NPC technique will also be selectively applied to the trial vector generation strategy DE/target-to-pbest/1/exp with or without optional archive. Third, a novel population Diversity Enhancement Mechanism (DEM) is proposed, which can enhance population diversity by recalculating individuals in a stagnant state. Finally, a time stamp mechanism is posed to deal with outdated inferior and historical solutions in the external archive. A Large test suite with 88 benchmark functions from CEC2013, CEC2014, and CEC2017 test suites is used for algorithm validation and the experimental results indicate that our ERM-DE demonstrates competitiveness with several state-of-the-art DE variants.

CIR-DE: A chaotic individual regeneration mechanism for solving the stagnation problem in differential evolution

Stagnant evolution is a problem frequently encountered by the population in differential evolution (DE). Aiming at the stagnation phenomenon, a comprehensive interpretation is provided in this paper. Our experiment confirms that the individuals that continuously stop evolving can be classified into two categories: global and local stagnant individuals, whose causes and exhibited characteristics are associated with the search behavior of the population. Based on the above findings, we propose a chaotic individual regeneration framework (CIR) for DEs. In the CIR-DE, a monitor is designed to recognize different types of stagnant individuals by evaluating the whole population’s convergence speed and specific individual’s location. Besides, two chaotic regeneration techniques are proposed to guide the above two types of individuals away from stagnation using the knowledge from solution and objective spaces. The CIR framework is implemented in nine representative DEs and tested in the CEC 2014, CEC 2017, CEC 2022 theoretical benchmarks and five real-world problems. The results reveal that our framework can significantly improve original DEs’ performance and alleviate stagnation in both theoretical and practical scenarios. The CIR framework also shows strong competitiveness compared to the other stagnation-related frameworks and the state-of-the-art DE variants.

A triple population adaptive differential evolution

The Differential Evolution (DE) algorithm is one of the most efficient algorithms for complex numerical optimization. However, the nature of differential mutation and crossover hinders the individuals from a major change and always guides them toward their superior neighbors. There's a lack of useful directional information to help the population escape from early convergence. To solve the above problem, this paper proposes a novel Triple-population-based Adaptive Differential Evolution (TPADE) to enhance the evolutionary efficiency in solving various complex numerical optimization problems. First, a population division method with symmetrical linear reduction is designed to divide the parent population of each iteration into three sub-populations of different sizes, i.e., superior sub-population, medium sub-population, and inferior sub-population. Each sub-population adopts distinct differential mutation and crossover operators to maintain balanced search directions. Second, a superior-trial-preserved selection mechanism is proposed to screen useful directional information to guide the next iteration of evolution. Third, an effective parameter adaptation strategy is designed with the linear population size reduction strategy to avoid redundant search. Experiments are then conducted to show that the TPADE exhibits well performance compared with eleven state-of-the-art DE variants, CEC winners, and their variants on the CEC'2014, CEC'2017, and CEC'2022 benchmark suites. The C++ source code of TPADE can be downloaded from https://github.com/DoubleGong/TPADE.

A novel hybrid adaptive differential evolution for global optimization

Differential Evolution (DE) stands as a potent global optimization algorithm, renowned for its application in addressing a myriad of practical engineering issues. The efficacy of DE is profoundly influenced by its control parameters and mutation strategies. In light of this, we introduce a refined DE algorithm characterized by adaptive parameters and dual mutation strategies (APDSDE). APDSDE inaugurates an adaptive switching mechanism that alternates between two innovative mutation strategies: DE/current-to-pBest-w/1 and DE/current-to-Amean-w/1. Furthermore, a novel parameter adaptation technique rooted in cosine similarity is established, with the derivation of explicit calculation formulas for both the scaling factor weight and crossover rate weight. In pursuit of optimizing convergence speed whilst preserving population diversity, a sophisticated nonlinear population size reduction method is proposed. The robustness of each algorithm is rigorously evaluated against the CEC2017 benchmark functions, with empirical evidence underscoring the superior performance of APDSDE in comparison to a host of advanced DE variants.

An improved differential evolution with adaptive population allocation and mutation selection

In order to improve the performance of differential evolution (DE) for different optimization problems, an improved DE with adaptive population allocation and mutation selection (iDE-APAMS) is proposed. First, different mutation strategies are selected to constitute the exploration strategy pool and the exploitation strategy pool. The exploration strategy pool is mainly used for global search to increase the population diversity; the exploitation strategy pool is primarily used for local search to accelerate the convergence and improve the solution accuracy. Second, different mutation strategies obtain population resources dynamically through cooperation and competition. Mutation strategies in the same strategy pool first compete for population resources with mutation strategies in the other strategy pool through cooperation, and in this way the population is allocated between the two strategy pools. Then within each strategy pool, different mutation strategies compete with each other to obtain corresponding population resources. Third, mutation scale factor and crossover rate are adaptively adjusted based on population diversity and fitness improvement. Finally, the Levy random walk is applied to individuals with better fitness at the later stage of the iteration to avoid falling into local optima and further improve the accuracy of the solution. The iDE-APAMS is compared with 4 classical DE variants and 11 state-of-the-art algorithms on the CEC2013, CEC2014, and CEC2017 benchmark test suites respectively, and it is also tested on 22 real-world optimization problems in CEC2011. The results show that iDE-APAMS can improve the convergence of the algorithm and the stability of the solution, and it is statistically significantly better than the comparison algorithms.

HAPI-DE: Differential evolution with hierarchical archive based mutation strategy and promising information

Differential Evolution (DE), as a population-based meta-heuristic global optimization technique, has shown excellent performance in handling optimization problems in continuous spaces. Despite its effectiveness, the DE algorithm suffers from shortcomings such as complexity of parameter selection and limitations of the mutation strategy. Therefore, this paper presents a new strategy for generating trial vectors based on a hierarchical archive, which integrates promising information during evolution with current populations to obtain a good perception of the objective landscape. Moreover, to mitigate mis-scaling by scale factor, an adaptive parameter generation mechanism with hierarchical selection (APSH) is proposed. Furthermore, a novel population diversity metric technique and a restart mechanism based on wavelet functions is introduced in this paper. Comparative experiments were conducted to evaluate the performance of the proposed algorithm using 100 benchmark functions from the CEC2013, CEC2014, CEC2017, and CEC2022 test suites. The results demonstrate that the HAPI-DE algorithm outperforms or is on par with 6 recent powerful DE variants. Additionally, HAPI-DE was utilized in parameter extraction for the photovoltaic model STP6-120/36. The findings suggest that our algorithm, HAPI-DE, demonstrates competitiveness when compared to the 6 other DE variants.

Dynamic Mutation Strategy Selection in Differential Evolution Using Perturbed Adaptive Pursuit

Diverse mutant vectors play a significant role in the performance of the Differential Evolution (DE). A mutant vector is generated using a stochastic mathematical equation, known as mutation strategy. Many mutation strategies have been proposed in the literature. Utilizing multiple mutation strategies with the help of an adaptive operator selection (AOS) technique can improve the quality of the mutant vector. In this research, one popular AOS technique known as perturbation adaptive pursuit (PAP) is integrated with the DE algorithm for managing a pool of mutation strategies. A community-based reward criterion is proposed that rewards the cumulative performance of the whole population. The proposed approach is called ‘Dynamic Mutation Strategy Selection in Differential Evolution using Perturbed Adaptive Pursuit (dmss-DE-pap)’. The performance of dmss-DE-pap is evaluated over the 30D and 50D optimization problems of the CEC 2014 benchmark test suite. Results are competitive when compared with other state-of-the-art evolutionary algorithms and some recent DE variants.

Photovoltaic model parameters identification using diversity improvement-oriented differential evolution

Fast and accurate parameter identification of the photovoltaic (PV) model is crucial for calculating, controlling, and managing PV generation systems. Numerous meta-heuristic algorithms have been applied to identify unknown parameters due to the multimodal and nonlinear characteristics of the parameter identification problems. Although many of them can obtain satisfactory results, problems such as premature convergence and population stagnation still exist, influencing the optimization performance. A novel variant of Differential Evolution, namely, Diversity Improvement-Oriented Differential Evolution (DIODE), is proposed to mitigate these deficiencies and obtain reliable parameters for PV models. In DIODE, an adaptive perturbation strategy is employed to perturb current individuals to mitigate premature convergence by enhancing population diversity. Secondly, a diversity improvement mechanism is proposed, where information on the covariance matrix and fitness improvement of individuals is used as a diversity indicator to detect stagnant individuals, which are then updated by the intervention strategy. Lastly, a novel parameter adaptation strategy is employed to maintain a sound balance between exploration and exploitation. The proposed DIODE algorithm is applied to parameter identification problems of six PV models, including single, double, and triple diode and three PV module models. In addition, a large test bed containing 72 benchmark functions from CEC2014, CEC2017, and CEC2022 test suites is employed to verify DIODE’s overall performance in terms of optimization accuracy. Experiment results demonstrate that DIODE can secure accurate parameters of PV models and achieve highly competitive performance on benchmark functions.

E-procurement optimization in supply chain: A dynamic approach using evolutionary algorithms

The increasing dynamism of global markets, coupled with the occurrence of unpredictable events, has introduced substantial challenges in formulating efficient supply chain strategies. The inherent dynamic nature of logistic networks necessitates a departure from traditional supply chain methodologies. This study proposes an advanced solution for dynamic e-procurement utilizing evolutionary algorithms (EAs). In conventional supply chains involving buyers and suppliers, a critical challenge is identifying cost-efficient suppliers capable of fulfilling consumer demands amidst fluctuating prices and quantities. Traditional optimization techniques often fail to perform effectively under these dynamic conditions. Moreover, detecting changes during the optimization process is an additional hurdle in dynamic optimization problems. Recent advancements have demonstrated the efficacy of EAs in solving a variety of real-world dynamic optimization issues.This research introduces a novel evolutionary algorithmic framework that integrates the Hybrid Multipopulational Reinitialization Strategy (HMRS), with a proposed hybrid change detection mechanism (named Smirnov-based Multi-sensor Detection Mechanism (SMDM)) to address the dynamic e-procurement problems. The proposed framework enhances the algorithm’s adaptability and responsiveness to real-time changes within the e-procurement environment. By effectively detecting and responding to these variations, the framework aims to optimize procurement processes, ensuring efficiency and robustness in managing fluctuating requirements and conditions inherent to dynamic e-procurement scenarios. The empirical analysis presented underscores the superiority of Differential Evolution (DE) variants over Genetic Algorithm (GA) variants within the procurement context. The detailed empirical study validates the effectiveness of the proposed dynamic approach in addressing the challenges associated with dynamic e-procurement. Considering real-world parameter fluctuations, the proposed approach demonstrates significant resilience, positioning it as a robust and efficient solution for optimizing the e-procurement process and adeptly managing the complexities of the supply chain environment.

Differential Evolution Algorithm with Three Mutation Operators for Global Optimization

Differential evolution algorithm is a very powerful and recently proposed evolutionary algorithm. Generally, only a mutation operator and predefined parameter values of differential evolution algorithm are utilized to solve various optimization problems, which limits the performance of the algorithm. In this paper, six commonly used mutation operators are divided into three categories according to their own features. A mutation pool is established based on the three categories. A parameter pool with three predefined values is designed. During evolution, three mutation operators are randomly chosen from the three categories, and three parameter values are also randomly selected from the parameter pool. The three groups of mutation operators and parameter values are employed to produce trial vectors. The proposed algorithm makes good use of different mutation operators. Three recently proposed differential evolution variants and three non-differential evolution algorithms are used to make comparisons on the 29 testing functions from CEC. The experimental results have demonstrated that the proposed algorithm is very competitive. The proposed algorithm is utilized to solve three real applications, and the results are superior.

Large-scale power system multi-area economic dispatch considering valve point effects with comprehensive learning differential evolution

The role of multi-area economic dispatch (MAED) in power system operation is increasingly significant. It is a non-linear and multi-constraint problem with many local extremes when considering the valve point effects, posing challenges in obtaining a globally optimal solution, especially for large-scale systems. In this study, an improved variant of differential evolution (DE) called CLDE based on comprehensive learning strategy (CLS) is proposed to solve this problem. Three improved strategies are employed to enhance the performance of CLDE. (1) A CLS-based guided mutation strategy is proposed, in which learning exemplars constructed by competent individuals are used to generate mutant vectors to prevent the searching away from global optimum and speed up convergence. (2) A time-varying increasing crossover rate is devised. It can endow CLDE with a larger probability at the later stage to help individuals escape from local extremes. (3) A CLS-based crossover strategy is presented. Trial vectors directly utilize the information from learning exemplars for evolving, which can ensure the search efficiency and population diversity. CLDE is applied to six MAED cases. Compared with DE, it approximately consumes 32 %, 35 %, 10 %, 22 %, 62 %, and 20 % of evaluations to attain comparable results, saves 126.2544$/h, 81.8173$/h, 152.0660$/h, 360.7907$/h, 65.5757$/h, and 1732.8544$/h in fuel costs on average, and exhibits improvements of 34.77 %, 1.80 %, 0.00 %, 76.09 %, 95.15 %, and 16.76 % in robustness, respectively. Moreover, it also outperforms other state-of-the-art algorithms significantly in statistical analysis. Furthermore, the effects of improved strategies on CLDE are thoroughly investigated.

Generation of black-box adversarial attacks using many independent objective-based algorithm for testing the robustness of deep neural networks

Deep neural networks (DNNs) have become increasingly ubiquitous in our daily lives, finding applications in areas such as image recognition, voice recognition, and natural language processing. However, a growing concern revolves around the vulnerability of DNNs to adversarial examples—malicious inputs that can compromise the safety and accuracy of their outcomes. Existing studies predominantly fall into two categories: white-box and black-box techniques. White-box techniques require detailed internal information about the model, often focusing on gradient-based methods. In contrast, black-box techniques, which emulate real-world scenarios more closely, only rely on input–output knowledge. This study focuses specifically on black-box strategies, with key contributions from Single-Objective Variant of Differential Evolution (Pixel-SOO) and Multi-Objective Variant of Differential Evolution (Pixel-MOO) algorithms. While these approaches show promise, they suffer from drawbacks like long execution times and being unable to generate instances for adversarial purposes. To address these challenges, we introduce a novel archive-based Many Independent Objective (MIO) algorithm for using the first time in this context. Our proposed algorithm identifies the most vulnerable image pixels through the MIO algorithm, enabling efficient label flip attacks with minimized attempts. Furthermore, we balance exploration and exploitation by incorporating an adaptive parameter mechanism. The effectiveness of the proposed algorithm is assessed by employing VGG (VGG16 and VGG19) and ResNet (ResNet50, ResNet101, and ResNet152) architectures, both of which are convolutional neural network (CNN) models. The success criterion for the algorithms is to minimize the number of pixel changes while achieving high flip rates with a minimal number of requests to the network. A comprehensive analysis of the experimental results reveals that our algorithm consistently outperforms Pixel-SOO and Pixel-MOO, exhibiting an average speedup of three times compared to Pixel-SOO and seven times compared to Pixel-MOO. In most runs, adversarial attacks are generated with fewer pixel changes than Pixel-MOO and Pixel-SOO. In addition, our findings are openly accessible on GitHub to ensure transparency and reproducibility and encourage future research efforts.

Serial multilevel-learned differential evolution with adaptive guidance of exploration and exploitation

Recent years have witnessed a surge in the development of multilevel variants of differential evolution (DE), significantly enhancing the performance of DE algorithms. However, systematically guiding algorithms to strike a balance between exploration and exploitation within a parallel multilevel structure remains a challenge. In response to this challenge, we propose serial multilevel-learned differential evolution (SMLDE) with adaptive guidance for exploration and exploitation. This algorithm establishes a tightly connected multilevel-learned structure and an adaptive current best level. It also incorporates a combination of strategies including single iterative adaption, Cauchy perturbation, and iterative constraint strategy into each of the adapted levels, thus enhancing inter-component connections and dynamically balancing exploration and exploitation. To validate its effectiveness, we conduct ablation experiments and visualized analyses of exploration and exploitation to demonstrate the reliable strength of the multilevel-learned structure. The experimental results comparing SMLDE with 15 state-of-the-art algorithms using the IEEE Conference on Evolutionary Computation (CEC) 2017 benchmark test sets across various dimensions showcase its superior performance. Additionally, its remarkable results on the CEC2011 benchmark test and two real-world engineering optimization problems underscore the robustness and effectiveness of SMLDE.