Adaptive Inertia Weight Research Articles

Research on hybrid strategy Particle Swarm Optimization algorithm and its applications

The increasing complexity and high-dimensional nature of real-world optimization problems necessitate the development of advanced optimization algorithms. Traditional Particle Swarm Optimization (PSO) often faces challenges such as local optima entrapment and slow convergence, limiting its effectiveness in complex tasks. This paper introduces a novel Hybrid Strategy Particle Swarm Optimization (HSPSO) algorithm, which integrates adaptive weight adjustment, reverse learning, Cauchy mutation, and the Hook-Jeeves strategy to enhance both global and local search capabilities. HSPSO is evaluated using CEC-2005 and CEC-2014 benchmark functions, demonstrating superior performance over standard PSO, Dynamic Adaptive Inertia Weight PSO (DAIW-PSO), Hummingbird Flight patterns PSO (HBF-PSO), Butterfly Optimization Algorithm (BOA), Ant Colony Optimization (ACO), and Firefly Algorithm (FA). Experimental results show that HSPSO achieves optimal results in terms of best fitness, average fitness, and stability. Additionally, HSPSO is applied to feature selection for the UCI Arrhythmia dataset, resulting in a high-accuracy classification model that outperforms traditional methods. These findings establish HSPSO as an effective solution for complex optimization and feature selection tasks.

Improved krill swarm algorithm and its application in structural optimization

This paper improves and perfects the standard Krill herd algorithm (KH), which has the defects of slow convergence speed, insufficient calculation accuracy and easy to fall into local optimal solution for complex problems. An improved Krill herd algorithm (SDEKH) integrating improved differential evolution operator and S-type adaptive inertia weight is proposed. SDEKH, KH and other intelligent algorithms are compared and tested through a variety of standard test functions to verify the excellent performance of SDEKH; SDEKH is used to optimize the design of truss structure. By comparing the optimization results with other methods, it is verified that the optimization efficiency and accuracy of SDEKH are improved, which provides a more efficient and accurate method for the optimization design of engineering structures.

A three-stage prediction model for firm default risk: An integration of text sentiment analysis

Predicting firm default risk is vital for financial institutions to avert significant economic losses, making the enhancement of its prediction precision both imperative and intricate. This research introduces a three-stage prediction model, including association rule algorithm (ARA), support vector machine (SVM) and modified particle swarm optimization algorithm (MPSO). Features selected by ARA are used as inputs for SVM, and penalty parameter and kernel parameter of SVM is optimized by MPSO that uses the adaptive inertia weight. The importance of text sentiment variables are emphasized to predict firm default risk. In the first stage, feature selection seeks to curtail the dimensions of both financial and non-financial variables. The empirical findings validate the efficacy of the ARA, revealing a strong correlation between text sentiment and default risk. The subsequent two stages deploy the SVM, refined by the MPSO, to predict the default risk. Compared with renowned models, the proposed model displays superior prediction precision and a reduced computational overhead. This research furnishes a potent instrument for regulators and firms alike, aiding in mitigating prospective default risks and forestalling broader economic upheavals.

Air supply subsystem efficiency optimization for fuel cell power system with layered control method

Efficient and stable operation is critical for the large-scale commercialization of the proton exchange membrane fuel cell power system. The effective control and optimization of the operating conditions, such as oxygen excess ratio and cathode pressure of the air supply system, is a solution to improve the overall system efficiency. This work proposes a novel layered control method to achieve rapid and stable control of the operating conditions. The control structure in this paper consists of the optimization and control layers. A two-dimensional objective optimization function for the optimization layer is established to characterize the system efficiency based on theoretical analysis and experimental testing on the fuel cell power generation process and air supply system power consumption pattern. Then, a modified salp swarm algorithm with adaptive inertia weight is proposed to quickly and accurately obtain the optimal operating conditions for the maximum efficiency under different load current densities. Meanwhile, the local optimal solutions are avoided by introducing mutation operations. For the control layer, a third-order state space equation is developed to accurately describe the operating characteristics of the air supply system according to its operating principles. A feedback linearization-based sliding mode controller is designed to achieve rapid and stable control of the optimal working conditions outputted from the optimization layer. Finally, the fuel cell system was tested in the lab and verified on the fuel cell city buses. The results show that the system's operating efficiency is improved by 0.6 %–2.6 % at different current densities, and the hydrogen consumption of all three city buses is reduced by more than 5 %. The optimization effect was enhanced significantly. Therefore, the layered control method is effective in solving the optimization and control problems of the fuel cell power system.

A new multi-objective-stochastic framework for reconfiguration and wind energy resource allocation in distribution network incorporating improved dandelion optimizer and uncertainty

Improving the reliability and power quality of unbalanced distribution networks is crucial for ensuring consistent and reliable electricity supply. In this research, multi-objective optimization of unbalanced distribution networks reconfiguration integrated with wind turbine allocation (MORWTA) is implemented considering uncertainties of networks load, and also wind power incorporating a stochastic framework. The multi-objective function is defined by the minimization of power loss, voltage sag (VS), total harmonic distortion (THD), voltage unbalance (VU), energy not-supplied (ENS), system average interruption frequency index (SAIFI), system average interruption duration index (SAIDI), and momentary average interruption frequency (MAIFI). A new improved dandelion optimizer (IDO) with adaptive inertia weight is recommended to counteract premature convergence to identify decision variables, including the optimal network configuration through opened switches and the best location and size of wind turbines in the networks. The stochastic problem is modeled using the 2m + 1 point estimate method (PEM) combined with K-means clustering, taking into account the mentioned uncertainties. The proposed stochastic methodology is implemented on three modified 33-bus, and unbalanced 25-, and 37-bus distribution networks. The results demonstrated that the MORWTA enhanced all study objectives in comparison to the base networks. The results also demonstrated that the IDO had superior capability to solve the deterministic- and stochastic-MORWTA in comparison to the conventional DO, grey wolf optimizer (GWO), particle swarm optimization (PSO), and arithmetic optimization algorithm (AOA) in terms of achieving greater objective value. Moreover, the results demonstrated that when the stochastic-MORWTA model is considered, the power loss, VS, THD, VU, ENS, SAIFI, SAIDI, and MAIFI are increased by 18.35%, 9.07%, 10.43%, 12.46%, 11.90%, 9.28%, 12.16% and 14.36%, respectively for 25-bus network, and also these objectives are increased by 12.21%, 10.64%, 12.37%, 9.82%, 14.30%, 12.65%, 12.63% and 13.89%, respectively for 37-bus network compared to the deterministic-MORWTA model, which is related to the defined uncertainty patterns.

Bushing fault diagnosis based on SVM and the improved sparrow search algorithm

Abstract In order to address the issue of low precision in traditional bushing fault diagnosis, a bushing fault diagnosis method based on an improved sparrow search algorithm (ISSA) and support vector machine (SVM) is proposed in this paper. Firstly, the bushing vibration signals are extracted by wavelet packet, and the feature vectors are used as inputs for the SVM. In view of the impact of support vector machine parameters on the model, a sparrow search algorithm is proposed for intelligent optimization. To prevent reaching a local optimum, adaptive inertia weight is added based on the original approach. The final bushing fault diagnosis model is established by training. Comparison experiments with three fault diagnosis models, SSA-SVM, PSO-SVM, and SVM, found that the proposed method achieves complete diagnosis in a shorter time, and the diagnostic accuracy rate is 96.5%, which verifies the feasibility and effectiveness of the model.

An improved particle swarm optimisation algorithm for optimum placement and sizing of biogas-fuelled distributed generators for seasonal loads in a radial distribution network

Optimal allocation is critical for effective planning and operation of grid connected distributed generator(s) (DGs) subject to efficient optimisation approach(es). In this paper, an improved particle swarm optimisation (IPSO) algorithm based on weighted randomised acceleration coefficients and adaptive inertia weight for dynamic movement of the particles towards an optimal solution is proposed. The technique is expected to determine optimal placement, sizing and power factor of biogas-fuelled distributed generators (B-DGs such as internal combustion engine (ICE) and fuel cells (FC)) in a radial distribution network. The proposed IPSO is applied to simultaneously optimise three technical objectives namely total active power loss (TAPL), voltage deviation (VD) and voltage stability index (VSI) considering load growth and seasonal voltage dependence. The proposed IPSO is validated using an IEEE 33 bus radial distribution network to evaluate its effectiveness through various combinations of B-DGs for different scenarios and cases with a maximum of 3 B-DGs. Some of the key findings indicate that operating 3 ICEs at optimal power factor reduces TAPL by 93.52 %, reduces VD by 99.62 % and improves VSI by 23.32 % compared to 61.80 % TAPL reduction, 94.77 % VD reduction and 17.89 % VSI improvement for 3 FCs operating at unity power factor. The proposed IPSO gives better results than the standard PSO in terms of solution quality, convergence speed and statistical results. It is also perceived to be better compared to most of the recent state-of-the-art optimization algorithms found in literature.

A novel data-driven IBA-ELM model for SOH/SOC estimation of lithium-ion batteries

Lithium-ion batteries (Libs) have become the best power source for electric vehicles due to their high energy density and long cycle life. The state estimation is directly related to the safe and efficient use of Libs. In our previous research, a Bat Algorithm-Extreme Learning Machine (BA-ELM) model was constructed and promising results were achieved for state of health (SOH) estimation. To increase the engineering application value of BA-ELM for SOH/state of charge (SOC) estimation, optimization mechanism of improving the global exploration and local exploitation capabilities of the model was studied. First, three improved methods including a conversion mechanism of Artificial Bee Colony algorithm employing bees and scout bees, a mechanism for adaptive adjustment of the search frequency of bats according to the Euclidean distance difference, and an adaptive inertia weight coefficient method were proposed to construct a novel Improved Bat Algorithm (IBA)-ELM model. Second, a battery life degradation test platform was built to obtain the degradation dataset for Libs. The variation rules of nine battery variables and cycles were revealed. Four high-quality SOH-related features were extracted. Thus, the SOH estimation performance of six models was evaluated. Third, a Libs charge–discharge dataset under driving conditions was collected and processed. Four models were trained based on 72,048 samples from four conditions. Further, SOC estimation tests were conducted on 35,071 samples from completely different conditions. The results show that the root mean square error (RMSE) and mean absolute error (MAE) of SOH estimation by IBA-ELM decreased by 21.93 % and 23.79 %, respectively, compared to those by BA-ELM. The RMSE and MAE of SOC estimation decreased by more than 10 %. Error, radar, and violin plots demonstrate that the IBA-ELM could accurately estimate SOH/SOC in larger scale datasets, simultaneously exhibiting better generalization performance.

Enhancing Air Quality Prediction with an Adaptive PSO-Optimized CNN-Bi-LSTM Model

Effective air quality prediction models are crucial for the timely prevention and control of air pollution. However, previous models often fail to fully consider air quality’s temporal and spatial distribution characteristics. In this study, Xi’an City is used as the study area. Data from 1 January 2019 to 31 October 2020 are used as the training set, while data from 1 November 2020 to 31 December 2020 are used as the test set. This paper proposes a multi-time and multi-site air quality prediction model for Xi’an, leveraging a deep learning network model based on APSO-CNN-Bi-LSTM. The CNN model extracts the spatial features of the input data, the Bi-LSTM model extracts the time series features, and the PSO algorithm with adaptive inertia weight (APSO) optimizes the model’s hyperparameters. The results show that the model achieves the best results in terms of MAE and RMSE. Compared to the PSO-SVR, BPTT, CNN-LSTM, and GA-ACO-BP models, the MAE improved by 9.375%, 6.667%, 2.276%, and 4.975%, while the RMSE improved by 8.371%, 8.217%, 6.327%, and 5.293%. These significant improvements highlight the model’s accuracy and its promising application prospects.

Adaptive chaotic dynamic learning-based gazelle optimization algorithm for feature selection problems

Feature Selection (FS) is considered a crucial procedure for eliminating unnecessary features from datasets. FS is considered a challenging problem that is difficult to solve efficiently due to its combinatorial nature. As the problem size increases, so does the computation time. Recently, researchers have been focusing on metaheuristic FS algorithms. Therefore, this paper introduces an adaptive chaotic dynamic gazelle optimization algorithm (ACD-GOA) with enhanced elite Strategy for FS problems. The ACD-GOA is a novel enhanced version of the recently published GOA, incorporating multiple improved strategies to enhance its search capabilities and convergence speed. The initialization phase and iteration initialization adopt the dynamic opposition learning strategy to avoid premature convergence. Additionally, several enhancement strategies are employed to improve the efficiency of the standard GOA. Adaptive inertia weight and sigmoid function are used to enhance search efficiency. Furthermore, enhanced elite and exchanging information strategies are implemented to maintain population diversity and avoid local solutions, respectively. Experimental evaluations are conducted using various functions, including the twelve CEC2022 standard functions and fourteen FS benchmark datasets. The performance of ACD-GOA is compared with several other metaheuristic algorithms. Statistical tests such as Friedman and Wilcoxon signed-rank tests are used to analyze the experimental data. The experimental results highlight the proposed algorithm’s exceptional capacity to overcome problems associated with local minima and expedite the convergence process. The suggested algorithm has been extensively compared to state-of-the-art algorithms, demonstrating significant breakthroughs. The reported accuracy of the proposed algorithm ranges from 0.78 to 1.00 with K-NN classifier.

Transient frequency response test and measurement error prediction of DCTV based on adaptive inertial weight improved ACO

AbstractA temporary frequency response test and measurement error prediction method of direct current voltage transformer (DCTV) based on artificial intelligence (AI) is proposed. Firstly, the frequency characteristic of direct current (DC) side voltage of DCTV is analyzed. On this basis, a DCTV transient Frequency Response testing method based on transient alternating current (AC) & DC superposition was developed. Then, the method of voltage sudden change and phase correction is used to achieve transient process DCTV response time testing. Finally, the ant colony optimization (ACO) algorithm was improved by combining an adaptive inertia weight improvement strategy, achieving accurate prediction of the Measurement Error of DCTV. The proposed AI based DCTV transient Frequency Response testing and Measurement Error prediction method were compared and analyzed with the other three methods through simulation experiments. Compared to the other three comparison methods, the maximum transformation error in the evaluation indicators of mean squared error (MSE), root mean squared error (RMSE), and mean absolute error (MAE) decreased by 0.006, 0.0119, and 0.0085, respectively, while the maximum phase error decreased by 0.2794, 0.3004, and 0.2823, respectively.

Multi-subswarm cooperative particle swarm optimization algorithm and its application

Particle swarm optimization (PSO) is a classical evolutionary algorithm and has been widely used to solve continuous optimization problems. However, the performance of the PSO algorithm is highly sensitive to its control parameters and learning strategies. To address this problem, a multi-subswarm cooperative particle swarm optimization algorithm (MSC-PSO) is proposed in this paper. In MSC-PSO, firstly, a population muti-subswarm division method is designed to enhance the global exploration capability. Secondly, a level-based particle adaptive inertia weight strategy is introduced to adjust control parameters. Finally, a level-based update learning mechanism is used for particles adaptive selection learning strategy. The performance of the MSC-PSO is evaluated by the CEC2020 test suite, and five heuristic algorithms are compared with the MSC-PSO in the experiments. The experimental results demonstrate that the proposed MSC-PSO algorithm has significant advantages in terms of convergence speed and solving optimal solutions. Furthermore, the proposed MSC-PSO algorithm is used to solve the plant protection unmanned aerial vehicles (UAV) path planning problem. The experimental results proved that the proposed MSC-PSO algorithm effectively solves the real-world UAV path planning problem, and achieves at most a 34 % reduction in round-trip distance and a 24.1 % reduction in total non-spraying time compared to classical PSO.

Exploring microstructure evolution and machine-learning methods based on SCAT-CIWOA-BP-DMM theory during hot deformation of 56Ni–32Ti–12Hf alloy

Hot compression simulation tests were carried out on a 56Ni–32Ti–12Hf alloy at temperatures ranging from 800 to 1000 °C and strain rates ranging from 0.001 to 1 s−1. The study aimed to explore the hot deformation behavior and microstructure evolution mechanism of the alloy. Two flow stress prediction models were developed: a strain-compensated Arrhenius constitutive (SCAT) model based on phenomenology and a chaotic mapping adaptive inertia weight whale optimization algorithm-optimized back propagation neural network (CIWOA-BPNN) model based on machine learning. In comparison to the SCAT model, the CIWOA-BPNN model demonstrates higher prediction accuracy, improved error correlation coefficient, decreased average relative error, and lower root mean square error and mean absolute error. The hot processing map of the 56Ni–32Ti–12Hf alloy was developed using dynamic material modeling (DMM) theory and flow stress data predicted by the CIWOA-BPNN model. The optimal hot processing range was found to be between temperatures of 875–975 °C and strain rates of 0.001–1 s−1. The study found that within the optimal processing interval, the softening mechanisms observed were discontinuous dynamic recrystallization (DDRX) and continuous dynamic recrystallization (CDRX). An increase in power dissipation efficiency (η) led to an increase in the dynamic recrystallization (DRX) fraction from 27.1 % to 41.9 %, while the substructure fraction decreased from 34.2 % to 19.8 %. The decrease in the number of dislocations around the DDRX grains further validated the accuracy of the identified optimum machining interval in this research.

Inversion of 2D Magnetotelluric (MT) Data with Axial Anisotropy using Adaptive Particle Swarm Optimization (PSO)

In the presence of anisotropic resistivity structures, Magnetotelluric (MT) data inversion based on an isotropic resistivity model will be insufficient. Recent work on anisotropic MT inversions is primarily conducted using gradient-based, deterministic methods which may converge towards local minima and in general, suffer from ill-posedness of the inversion problem. Alternatively, meta-heuristical methods, such as Particle Swarm Optimization (PSO), are able to find the global minima of cost function. At the same time, they are easy to implement and do not require access to the dense Jacobian (and Hessian) matrix as needed by deterministic inversion methods. In this paper we propose PSO inversion for the anisotropic MT inversion problem using an adaptive inertia weights control strategy to balance exploration and exploitation during the PSO search. Higher order differential regularization is introduced with a Gaussian filter, where the conventional first order, Occam-like regularization term is replaced by a variance term. We investigate the performance of the proposed PSO inversion for an isotropic as well as an axial anisotropic synthetic case to demonstrate improvements over an Occam-like regularization. For the COPROD2 benchmark data set, the proposed PSO inversion can not only reproduce the extend of the North American Central Plains (NACP) conductive anomaly, but also to predict an anisotropy ratio of two orders of magnitude, which is in accordance with laboratory findings.

Application of Improved Sparrow Search Algorithm to Flexible Job Shop Scheduling Problem

When the reality of energy saving and reduction of enterprises’ emissions are a concern, research on the investigation of establishing mathematical models to optimize the production time has been studied. In this article, an enhanced sparrow optimization method is proposed. First, two-layer coding is employed for workpieces and machines according to the model requirements. Secondly, the three-dimensional chaotic mapping scheme is presented to improve the population heterogeneity of the algorithm, and the adaptive inertia weight balance algorithm is implemented to offset the speed of the convergence and its probability. Finally, the Cauchy mutation scheme is adopted to help the algorithm jump out of the local optimum. Simulated data is run to check the superiority of the proposed method. So, through the simulations and comparisons of 10 kinds of test datasets, the outcomes suggest that the solution quality of the enhanced sparrow optimization method has been effectively advanced, and its good global optimization ability is shown, which can provide scheduling strategies for workshop productions. One of the successes of the ISSA algorithm is its superior search accuracy.

MECSBO: Multi‐strategy enhanced circulatory system based optimisation algorithm for global optimisation and reliability‐based design optimisation problems

AbstractThe Circulatory System Based Optimisation (CSBO) stands as a nascent metaheuristic optimisation algorithm known for its proficiency in tackling global optimisation problems. The authors introduce the Multi‐strategy Enhanced CSBO (MECSBO), an algorithm designed for global optimisation and Reliability‐based Design Optimisation (RBDO). MECSBO integrates adaptive inertia weight, golden sine operator and chaos strategy to augment the convergence capacity and efficiency of the original CSBO. Furthermore, MECSBO‐based RBDO algorithm is presented to address RBDO problem. The comparative analysis utilising standard real‐world benchmark functions has been carried out to validate the effectiveness of the proposed MECSBO. Several RBDO problems, including three typical numerical examples and three engineering cases, are used to show abilities of the proposed MECSBO‐based RBDO algorithm. The results demonstrated that MECSBO is outperformed comparing to the state‐of‐the‐art algorithms in terms of accuracy, efficiency, and robustness in RBDO problems.

Layout of Detection Array Based on Multi-Strategy Fusion Improved Adaptive Mayfly Algorithm in Bearing-Only Sensor Network.

The various applications of bearing-only sensor networks for detection and localization are becoming increasingly widespread and important. The array layout of the bearing-only sensor network seriously impacts the detection performance. This paper proposes a multi-strategy fusion improved adaptive mayfly algorithm (MIAMA) in a bearing-only sensor network to perform layout planning on the geometric configuration of the optimal detection. Firstly, the system model of a bearing-only sensor network was constructed, and the observability of the system was analyzed based on the Cramer-Rao Lower Bound and Fisher Information Matrix. Then, in view of the limitations of the traditional mayfly algorithm, which has a single initial population and no adaptability and poor global search capabilities, multi-strategy fusion improvements were carried out by introducing Tent chaos mapping, the adaptive inertia weight factor, and Random Opposition-based Learning. Finally, three simulation experiments were conducted. Through comparison with the Particle Swarm Optimization (PSO) algorithm, Mayfly Algorithm (MA), and Genetic Algorithm (GA), the effectiveness and superiority of the proposed MIAMA were validated.

An Enhanced IHHO-LSTM Model for Predicting Online Public Opinion Trends in Public Health Emergencies

Social networks accelerate information communication in public health emergencies. Some negative information may cause an outbreak of public opinion crisis. Accurately predicting online public opinion trends can help the relevant departments take timely and effective measures to cope with risks. Therefore, this research proposes a prediction model incorporating the swarm intelligence optimization algorithm and the deep learning method. In this model, we improve the Harris Hawks Optimization (HHO) algorithm by introducing the Cauchy distribution function, the stochastic contraction exponential function, and the adaptive inertia weight. Then we utilize the improved HHO (IHHO) algorithm to optimize the hyperparameters of the deep learning method LSTM, including the learning rate and the number of neurons in the hidden layer. Finally, we construct the IHHO-LSTM model to make predictions in three public health emergencies. The experiments verify that the proposed model outperforms other single and hybrid models. The MAPE values reduce by 78.34%, 54.46%, and 46.42% relative to the average values of the three single models. Compared with the mean values of the two hybrid models, the MAPE values decrease by 47.69%, 18.45%, and 5.78%. The IHHO-LSTM model can be applied to public opinion early warning and reversal identification, providing a reference in public opinion management.

Stability-Considered Lane Keeping Control of Commercial Vehicles Based on Improved APF Algorithm

Regarding the lane keeping system, path tracking accuracy and lateral stability at high speeds need to be taken into account especially for commercial vehicles due to the characteristics of larger mass, longer wheelbase and higher mass center. To improve the performance mentioned above comprehensively, the control strategy based on improved artificial potential field (APF) algorithm is proposed. In the paper, time to lane crossing (TLC) is introduced into the potential field function to enhance the accuracy of path tracking, meanwhile the vehicle dynamics parameters including yaw rate and lateral acceleration are chosen as the repulsive force field source. The lane keeping controller based on improved APF algorithm is designed and the stability of the control system is proved based on Lyapunov theory. In addition, adaptive inertial weight particle swarm optimization algorithm (AIWPSO) is applied to optimize the gain of each potential field function. The co-simulation results indicate that the comprehensive evaluation index respecting lane tracking accuracy and lateral stability is reduced remarkably. Finally, the proposed control strategy is verified by the HiL test. It provides a beneficial reference for dynamics control of commercial vehicles and enriches the theoretical development and practical application of artificial potential field method in the field of intelligent driving.

Time–jerk optimal trajectory planning for industrial robots with coupled interpolation function selection

AbstractIn the contemporary field of optimal trajectory planning for industrial robots, it is customary to construct trajectories through the manual predefinition of interpolation functions. Unfortunately, this method frequently overlooks the influence of the interpolation function itself on the optimization objectives, resulting in suboptimal outcomes. To remedy this limitation, an optimal trajectory planning method with coupled interpolation function selection is proposed, in which the total task time and the integral squared jerk are defined as optimization objectives. This method minimizes the optimization objectives while also factoring in the optimal interpolation function, and avoiding subjective interference. To address the aforementioned biobjective optimization problem better, an Improved MultiObjective Golden Eagle Optimizer is introduced. Population diversity and the ability to escape local optima are enhanced through the incorporation of Chaotic Mapping, Opposition‐Based Learning, Differential Evolution, and adaptive inertia weight strategy into the algorithm. The superiority of the algorithm is validated through a series of simulations on 17 benchmark functions. In the context of the robotic stirring operation within the automated block cast charging process, the proposed method is utilized to derive the time–jerk optimal trajectory. The results demonstrate the effectiveness of the proposed method.