Multi-objective Particle Swarm Optimization Method Research Articles

Multi-objective optimization of stepped spillway and stilling basin dimensions

Abstract In the present paper, an optimization procedure is proposed for stepped spillway design dimensions, which leads to maximum energy dissipation rate and minimum construction cost, considering independently the chute cost and stilling basin cost. Three independent objective functions are thus simultaneously satisfied. The procedure involves four main tools: the multi-objective particle swarm optimization method (MOPSO) to find Pareto solutions in one run, the K-means clustering algorithm to reduce the size of the obtained non-dominated solutions, the pseudo-weight vector approach (PWV) to facilitate the decision making and to select some adequate solutions, and finally, CFD simulations to analyze the retained optimal solutions. The suitability of the proposed procedure is tested through an example of application. As results, a set of 20 solutions with different satisfaction levels is found and compared to existing solutions. A multi-objective optimization problem may have many different solutions; the originality of the present work lies in the proposed procedure, which explores several possible solutions and reduces their number to give help to the decision making. Furthermore, an approximate expression of spillway total cost is also derived as a function of flow energy dissipation rate.

An Online Data-Driven Multi-Objective Optimization of a Permanent Magnet Linear Synchronous Motor

In this article, an online data-driven multi-objective optimization is proposed to suppress the undesired thrust ripple and improve the average thrust of the permanent magnet linear synchronous motor (PMLSM). A two-loop optimization process is adopted, and the first loop is designed to achieve a higher accuracy of the Kriging models by adding points based on expected improvement (EI) criteria. Through the first loop, the Kriging models achieve a significant increase in the prediction accuracy from 76% to 99.7%. Then, the second loop runs to optimize the PMLSM by using the Kriging assisted multi-objective particle swarm optimization (MOPSO) method. The results show that the optimized PMLSM successfully achieve a 64% reduction in the thrust ripple and a 6.6% increase in the average thrust compared with the original PMLSM.

Optimised MOPSO with the grey relationship analysis for the multi-criteria objective energy dispatch of a novel SOFC-solar hybrid CCHP residential system in the UK

In the quest to achieve home comfort with the highest possible efficiency, there is an increasing interest in combined cooling, heating and power (CCHP) systems. These can be fuelled by natural gas and potentially by hydrogen enriched natural gas and ultimately by hydrogen. The optimised designs largely depend on the energy dispatch algorithms that take into account aspects of economic and environmental impacts as well as system efficiency. This paper details a new algorithm to optimise the operation of a novel hybrid solid oxide fuel cell (SOFC)-solar hybrid CCHP residential system. The proposed algorithm is based on the multi-objective particle swarm optimization (MOPSO) and the grey relationship analysis (GRA) (named as MOPSO-GRA) with the capability of resolving the objective energy dispatch issues within the system by effectively avoiding local optimum problem. More specifically, the grey incidences are first integrated into MOPSO to analyse the degree of closeness between non-ideal solutions and the ideal solution. Then, the relationship of degree is introduced to give objective energy dispatch in order to maximise the efficiency of energy utilization while minimising the system capital cost, operating costs, maintenance costs, fuel costs and emissions. Finally, the new algorithm is validated on the proposed new design of CCHP system. Ten cases are studied to evaluate the technical, economic and environmental performance of the MOPSO-GRA algorithm when being applied to an SOFC-based CCHP system under both the grid-connected and the island modes. A comparison is made with the conventional MOPSO method and system structure, and the impact of a plug-in EV is also evaluated. Based on a detailed cost and emissions analysis, the results indicate the environmental and economic advantages, in addition to the higher efficiency of the proposed methodology and system structure. The impact of these results and observations leads to the promotion of intelligent FC-based multi-energy system technologies for residential use.

Parameter study and optimization design of a hat oblique tunnel portal

The present work focuses on the aerodynamic problems resulting from a high-speed train (HST) passing through a tunnel. Numerical simulations were employed to obtain the numerical results, and they were verified by a moving-model test. Two responses, [Formula: see text] (coefficient of the peak-to-peak pressure of a single fluctuation) and[Formula: see text] (pressure value of micro-pressure wave), were studied with regard to the three building parameters of the portal-hat buffer structure of the tunnel entrance and exit. The MOPSO (multi-objective particle swarm optimization) method was employed to solve the optimization problem in order to find the minimum [Formula: see text] and[Formula: see text]. Results showed that the effects of the three design parameters on [Formula: see text] were not monotonous, and the influences of[Formula: see text] (the oblique angle of the portal) and [Formula: see text] (the height of the hat structure) were more significant than that of[Formula: see text] (the angle between the vertical line of the portal and the hat). Monotonically decreasing responses were found in [Formula: see text] for [Formula: see text] and[Formula: see text]. The Pareto front of [Formula: see text] and[Formula: see text]was obtained. The ideal single-objective optimums for each response located at the ends of the Pareto front had values of 1.0560 for [Formula: see text] and 101.8 Pa for[Formula: see text].

Nonlinear normal modes and optimization of a square root nonlinear energy sink

Since the vibration mitigation of nonlinear energy sinks (NESs) needs multi-objective optimization, this paper aims to address this issue with efficient multi-objective particle swarm optimization (MOPSO) methods. A non-polynomial NES model, namely a square root NES (SRNES), is targeted for multi-objective optimization and dynamical analysis on an impulsively excited linear oscillator (LO) and compared with the corresponding polynomial model. Different MOPSO methods are developed to balance the mass and dissipation efficiency of SRNESs by utilizing weighted ranking, external archive, and non-dominated sorting. And nonlinear normal modes (NNMs) are extended for the analysis of SRNESs. The optimization results show the efficient targeted energy transfer (TET) and high robustness of the bistable SRNES in different LO initial and damping conditions. Specifically, more than 95% energy absorption can be easily witnessed, even in a small mass ratio (around 0.003), but the amplitude of the SRNES can be very large. The principal manifestation of the optimal TET is 1:1 resonance capture along with considerable low-frequency components of the SRNES. The optimization results of the SRNESs and the linear vibration absorbers show high similarities, while with a large mass ratio, the SRNESs possess broader range of spring stiffness. In the comparison of the two NES models, the differences are found in potential energy surfaces, NNMs, in-well motion, and vibration attenuation performance. Besides, the SRNESs can be divided into two groups in terms of the conservative dynamics, which can be explained by the polynomial model. The frequency component of in-phase NNMs dominates in the conservative and dissipative dynamics of the SRNESs, which lead to the transition of NNMs in the dissipative dynamics. In summary, the efficient MOPSO methods can be designed for different needs of vibration suppression and structure cost. Furthermore, it is important to directly analyze the non-polynomial nonlinear vibration, because transforming non-polynomial models into polynomial ones may lead to unreasonable or imprecise results. Finally, the NES damping structure and the NES performance in small mass conditions should be enhanced.

Model-based production optimization under geological and economic uncertainties using multi-objective particle swarm method

Optimization of the water-flooding process in the oilfields is inherently subject to several uncertainties arising from the imperfect reservoir subsurface model and inadequate data. On the other hand, the uncertainty of economic conditions due to oil price fluctuations puts the decision-making process at risk. It is essential to handle optimization problems under both geological and economic uncertainties. In this study, a Pareto-based Multi-Objective Particle Swarm Optimization (MOPSO) method has been utilized to maximize the short-term and long-term production goals, robust to uncertainties. Some modifications, including applying a variable in the procedure of leader determination, namely crowding distance, a corrected archive controller, and a changing boundary exploration, are performed on the MOPSO algorithm. These corrections led to a complete Pareto front with enough diversity on the investigated model, covering the entire solution space. Net Present Value (NPV) is considered the first goal that represents the long-term gains, while a highly discounted NPV (with a discount rate of 25%) has been considered short-term gains since economic uncertainty risk grows with time. The proposed optimization method has been used to optimize water flooding on the Egg benchmark model. Geological uncertainty is represented with ensembles, including 100 model realizations. The k-means clustering method is utilized to reduce the realizations to 10 to reduce the computing cost. The Pareto front is obtained from Robust Optimization (RO) by maximizing average NPV over the ensembles, as the conservative production plan. Results show that optimization over the ensemble of a reduced number of realizations by the k-means technique is consistent with all realizations’ ensembles results, comparing their cumulative density functions. Furthermore, 10 oil price functions have been considered to form the economic uncertainty space. When SNPV and LNPV are optimized, considering uncertainty in oil price scenarios, the Pareto front’s production scenarios are robust to oil price fluctuations. Using the robust Pareto front of LNPV versus SNPV in both cases, one can optimize production strategy conservatively and update it according to the current reservoir and economic conditions. This approach can help a decision-maker to handle unexpected situations in reservoir management.

Milling of Graphene Reinforced Ti6Al4V Nanocomposites: An Artificial Intelligence Based Industry 4.0 Approach.

The studies about the effect of the graphene reinforcement ratio and machining parameters to improve the machining performance of Ti6Al4V alloy are still rare and incomplete to meet the Industry 4.0 manufacturing criteria. In this study, a hybrid adaptive neuro-fuzzy inference system (ANFIS) with a multi-objective particle swarm optimization method is developed to obtain the optimal combination of milling parameters and reinforcement ratio that lead to minimize the feed force, depth force, and surface roughness. For achieving this, Ti6Al4V matrix nanocomposites reinforced with 0 wt.%, 0.6 wt.%, and 1.2 wt.% graphene nanoplatelets (GNPs) are produced. Afterward, a full factorial approach was used to design experiments to investigate the effect of cutting speed, feed rate, and graphene nanoplatelets ratio on machining behaviour. After that, artificial intelligence based on ANFIS is used to develop prediction models as the fitness function of the multi-objective particle swarm optimization method. The experimental results showed that the developed models can obtain an accurate estimation of depth force, feed force, and surface roughness with a mean absolute percentage error of 3.87%, 8.56%, and 2.21%, respectively, as compared with experimentally measured outputs. In addition, the developed artificial intelligence models showed 361.24%, 35.05%, and 276.47% less errors for depth force, feed force, and surface roughness, respectively, as compared with the traditional mathematical models. The multi-objective optimization results from the new approach indicated that a cutting speed of 62 m/min, feed rate of 139 mm/min, and GNPs reinforcement ratio of 1.145 wt.% lead to the improved machining characteristics of GNPs reinforced Ti6Al4V matrix nanocomposites. Henceforth, the hybrid method as a novel artificial intelligent method can be used for optimizing the machining processes with complex relationships between the output responses.

A multi-objective optimization-based model for the deployment of reclosers and remote-controlled switches using NSGA2 and entropy weighted TOPSIS method

<p>Since they are fast, remote-controlled, automated and intelligent, reclosers<br />and switches are an inevitable solution for improving the reliability of<br />intelligent electrical distribution networks at optimal cost. However, their<br />location and coordination have great effects on the protection and automation<br />strategies of complex electrical distribution networks against multiple<br />unpredictable faults. Which requires a flexible and multi-criteria optimization<br />method. In this article, we propose a multi-objective method based on an<br />analytical model by considering the fault rate, restoration times, outage cost<br />and coordination between devices. The non-dominated genetic sorting<br />algorithm II was proposed to obtain the optimal Pareto solutions, and a<br />technique of performance control by similarity with the ideal solution was<br />used to classify them. The objective criteria weights are based on the entropy<br />method which allows solutions to be obtained and better classified with the<br />minimum of subjectivity. The IEEE33 and IEEE13 bus test networks were<br />used to verify the method. The results obtained are compared to a binary<br />multi-objective particle swarm optimization method and the results show that<br />the proposed method reduces the overall costs, reduces the undelivered<br />energy of the system and improves the reliability of the service.</p>

Multi-Objective Optimal Feedback Controls for Under-Actuated Dynamical System

This paper presents a study of optimal control design for a single-inverted pendulum (SIP) system with the multi-objective particle swarm optimization (MOPSO) algorithm. The proportional derivative (PD) control algorithm is utilized to control the system. Since the SIP system is nonlinear and the output (the pendulum angle) cannot be directly controlled (it is under-actuated), the PD control gains are not tuned with classical approaches. In this work, the MOPSO method is used to obtain the best PD gains. The use of multi-objective optimization algorithm allows the control design of the system without the need of linearization, which is not provided by using classical methods. The multi-objective optimal control design of the nonlinear system involves four design parameters (PD gains) and six objective functions (time-domain performance indices). The Hausdorff distances of consecutive Pareto sets, obtained in the MOPSO iterations, are computed to check the convergence of the MOPSO algorithm. The MOPSO algorithm finds the Pareto set and the Pareto front efficiently. Numerical simulations and experiments of the rotary inverted pendulum system are done to verify this design technique. Numerical and experimental results show that the multi-objective optimal controls offer a wide range of choices including the ones that have comparable performances to the linear quadratic regulator (LQR) control.

Improving wind turbine blade based on multi-objective particle swarm optimization

This paper studies a new method for optimizing the design of wind turbine blades. Compared with the existing blade design methods, the proposed method not only considers the structural strength and stiffness of the blade but also considers the noise and power generation efficiency of the blade. The method utilizes a multi-objective particle swarm optimization method and the finite volume method in combination to meet the strength and stiffness requirements of the wind turbine blade, improve its aerodynamic performance and reduce its noise. In the study, the geometries of the blade used by a 2 MW wind turbine are taken as the initial parameters of the target blade and MATLAB and ANSYS are employed to perform the optimization and finite element analysis based performance calculations. Then an intelligent optimization algorithm was developed for achieving a quiet and efficient wind turbine blade. In such a multi-objective optimization algorithm, both structural strength, stiffness, noise reduction, and aerodynamic performance of the blade are taken as objective functions. The simulation results have shown that through optimization, the blade noise was reduced by 3.1 dB and the power coefficient was increased by 6.9%. Moreover, it is found that the blade’s structural strength and stiffness are also improved after optimization. This implies that the proposed algorithm is also helpful to further reduce the manufacturing materials and costs of wind turbine blades.

Dynamic modeling and trajectory tracking control method of segmented linkage cable-driven hyper-redundant robot

The dynamics modeling and trajectory optimization of a segmented linkage cable-driven hyper-redundant robot (SL-CDHRR) become more challenging, since there are multiple couplings between the active cables, passive cables, joints and end-effector. To deal with these problems, this paper proposes a dynamic modeling and trajectory tracking control methods for such type of CDHRR, i.e., SL-CDHRR. First, the multi-coupling kinematics equation (i.e., cable-joint-end) of the hyper-redundant robot is derived. Then, according to the transmission characteristics of the hybrid active/passive segmented linkage, the dynamic equation of series–parallel coupling is derived. It consists of parallel-active dynamics and series-passive dynamics. Furthermore, using the tension of active cables and the pose of the end-effector as optimization indicators, a trajectory tracking framework was constructed by the combination of dynamic feedforward control and PD control. The multi-objective particle swarm optimization method is used to achieve the simultaneous optimization of the energy indicator and control accuracy indicator during the trajectory tracking process. Finally, a MATLAB/SimMechanics co-simulation system is built, and the proposed methods are verified by the built co-simulation system.

An Efficient Economic-Statistical Design of Simple Linear Profiles Using a Hybrid Approach of Data Envelopment Analysis, Taguchi Loss Function, and MOPSO

Statistically constrained economic design for profiles usually refers to the selection of some parameters such as the sample size, sampling interval, smoothing constant, and control limit for minimizing the total implementation cost while the designed profiles demonstrate a proper statistical performance. In this paper, the Lorenzen-Vance function is first used to model the implementation costs. Then, this function is extended by the Taguchi loss function to involve intangible costs. Next, a multi-objective particle swarm optimization (MOPSO) method is employed to optimize the extended model. The parameters of the MOPSO are tuned using response surface methodology (RSM). In addition, data envelopment analysis (DEA) is employed to find efficient solutions among all near-optimum solutions found by MOPSO. Finally, a sensitivity analysis based on the principal parameters of the cost function is applied to evaluate the impacts of changes on the main parameters. The results show that the proposed model is robust on some parameters such as the cost of detecting and repairing an assignable cause, variable cost of sampling, and fixed cost of sampling.

Optimal energy management of microgrid using advanced multi-objective particle swarm optimization

Purpose The purpose of this paper is to propose an optimal energy management (OEM) method using intelligent optimization techniques applied to implement an optimally hybrid heat and power isolated microgrid. The microgrid investigated combines renewable and conventional power generation. Design/methodology/approach Five bio-inspired optimization methods include an advanced proposed multi-objective particle swarm optimization (MOPSO) approach which is comparatively applied for OEM of the implemented microgrid with other bio-inspired optimization approaches via their comparative simulation results. Findings Optimal multi-objective solutions through Pareto front demonstrate that the advanced proposed MOPSO method performs quite better in comparison with other meta-heuristic optimization methods. Moreover, the proposed MOPSO is successfully applied to perform 24-h OEM microgrid. The simulation results also display the merits of the real time optimization along with the arbitrary of users’ selection as to satisfy their power requirement. Originality/value This paper focuses on the OEM of a designed microgrid using a newly proposed modified MOPSO algorithm. Optimal multi-objective solutions through Pareto front demonstrate that the advanced proposed MOPSO method performs quite better in comparison with other meta-heuristic optimization approaches.

Robust Optimization of Planar Constrained Mechanical System with Varying Joint Clearance Size Based on Sensitivity Analysis

This paper is devoted to a general methodological study on sensitivity analysis and robust optimization for a planar crank-slider mechanism in presence of joint clearances and random parameters and investigate the effects of parameter uncertainty on optimization results when joint clearance sizes are constantly changing due to wear. The first-order sensitivity analysis based on the response surface proxy model is performed. Then, a multiobjective robust optimization algorithm based on sensitivity analysis is carried out to reduce the undesirable effects of joint clearances and random parameters. In the algorithm, a multiobjective robust optimization model derived from the mean and variance of the objective function is constructed. Here, the objective function is defined based on the consideration of reducing the contact force generated at all clearance joints. Additionally, in order to balance computational accuracy and efficiency in the multiobjective robust optimization process, high-precision Kriging agent models are established. The optimum values of design variables are determined by combining Monte Carlo sampling and multiobjective particle swarm optimization method. By combining the Baumgarte approach with Lankarani–Nikravesh contact force model and Coulomb friction model, the dynamic equations of the planar multibody system with clearance joints are established. The uniform probability distribution is applied for characterizing random parameters. Simulation results show that the influence of design variable variations on the objective function changes in relation to the joint clearance size, but their relative influence degree on the objective function will not vary with the size of joint clearances. Moreover, the optimal solution selected on the Pareto front will affect the average levels and peak fluctuations of the dynamic responses in multibody systems.

Multi-objective robust design of vehicle structure based on multi-objective particle swarm optimization

With the continuous spread of COVID-19 epidemic, the strict control of personnel makes it a problem to optimize the design of vehicle parameters after field measurement. The energy absorption characteristics and deformation mode of the front structure of the vehicle determine the acceleration or force response of the vehicle body during the impact, which plays an important role in occupant protection. The traditional multi-objective optimization method is to transform multi-objective problems into single objective optimization problems through weighted combination, objective planning, efficiency coefficient and other methods. This method requires a strong prior knowledge. The purpose of this paper is to combine the experimental design with the Multi-objective Particle Swarm Optimization (MPSO) method to achieve the optimization of the crash worthiness of automobile structure. This method can effectively overcome the defect of low precision caused by the conventional response surface method in the whole design space. In this paper, the multi-objective particle swarm optimization method is applied to the research of Crash worthiness optimization of automobile structure, which expands the application field of the multi-objective particle swarm optimization method, and also has a very big role in the optimization of other complex systems. It can be seen from the experiment that the speed of multi-objective particle swarm optimization is much faster than that of other methods. Only 100 iterations can get the relative better results. The case study on the front structure of a car shows that the method has a good result. It is of great significance to apply the method to the optimization design of the crash worthiness of the car structure to improve the crash safety of the car under the influence of COVID-19 epidemic.

Optimal Design of Standalone Photovoltaic System Based on Multi-Objective Particle Swarm Optimization: A Case Study of Malaysia

This paper presents a multi-objective particle swarm optimization (MOPSO) method for optimal sizing of the standalone photovoltaic (SAPV) systems. Loss of load probability (LLP) analysis is considered to determine the technical evaluation of the system. Life cycle cost (LCC) and levelized cost of energy (LCE) are treated as the economic criteria. The two variants of the proposed PSO method, referred to as adaptive weights PSO ( A W P S O c f ) and sigmoid function PSO ( S F P S O c f ) , are implemented using MATLAB software to the optimize the number of PV modules in (series and parallel) and number of the storage battery. The case study of the proposed SAPV system is executed using the hourly meteorological data and typical load demand for one year in a rural area in Malaysia. The performance outcomes of the proposed A W / S F P S O c f methods give various configurations at desired levels of LLP values and the corresponding minimum cost. The performance results showed the superiority of S F P S O c f in terms of accuracy is selecting an optimal configuration at fitness function value 0.031268, LLP value 0.002431, LCC 53167 USD, and LCE 1.6413 USD. The accuracy of A W / S F P S O c f methods is verified by using the iterative method.

Improving Prediction Intervals Using Measured Solar Power with a Multi-Objective Approach

Prediction Intervals are pairs of lower and upper bounds on point forecasts and are useful to take into account the uncertainty on predictions. This article studies the influence of using measured solar power, available at prediction time, on the quality of prediction intervals. While previous studies have suggested that using measured variables can improve point forecasts, not much research has been done on the usefulness of that additional information, so that prediction intervals with less uncertainty can be obtained. With this aim, a multi-objective particle swarm optimization method was used to train neural networks whose outputs are the interval bounds. The inputs to the network used measured solar power in addition to hourly meteorological forecasts. This study was carried out on data from three different locations and for five forecast horizons, from 1 to 5 h. The results were compared with two benchmark methods (quantile regression and quantile regression forests). The Wilcoxon test was used to assess statistical significance. The results show that using measured power reduces the uncertainty associated to the prediction intervals, but mainly for the short forecasting horizons.

A Self-Organizing Evolutionary Method to Model and Optimize Correlated Multiresponse Metrics for Additive Manufacturing Processes

Abstract The use of robust multiresponse constrained optimization techniques in which multiple-objective responses are involved is becoming a crucial part in additive manufacturing (AM) processes. Common and popular techniques, in most cases, rely on the assumption of independent responses. In practice, however, many of the desired quality characteristics can be correlated. In this work, we propose a technique based on three ingredients: hybrid self-organizing (HSO) method, desirability function (DF), and evolutionary algorithms to analyze, model, and optimize the multiple correlated responses for the fused deposition modeling (FDM) process, one of the most popular AM technologies. The multiobjective functions are formulated by employing the HSO method and DF, where structural integrity and process efficiency metrics are considered for the data-driven correlated multiresponse models. Subsequently, layer thickness, nozzle temperature, printing speed, and raster angles are taken as process parameters (decision variables). The operational settings and capabilities for the FDM machine are defined as boundary constraints. Different EA algorithms, the nondominated sorting genetic algorithm, and the multiobjective particle swarm optimization method, are then deployed to model the AM criteria accordingly to extract the Pareto-front curve for the correlated multiresponse functions. FDM experimental design and data collection for the proposed method are provided and used to validate our approach. This study sheds light on formulating robust and efficient data-driven modeling and optimizations for AM processes.

Integrated sourcing and inventory decisions considering sources’ disruptions with a hybrid simulation-MOPSO-TOPSIS approach: A pharmaceutical case study

In this research, the integrated sourcing and inventory policy problem in a pharmaceutical distribution company is investigated. In order to select the superior solution, a new tool is introduced. Sourcing is one of the most critical issues in pharmaceutical industry. In addition, drug inventory shortages can cause irreparable humanitarian crises. However, only a limited number of studies has been focused on integrated sourcing and inventory policy of drugs so far. In real-world problems, it is difficult to calculate the exact cost of inventory shortage such as company reputation and humanitarian crises. To overcome this obstacle, in this study, the number of shortage is considered as a separate objective. Likewise, demands of the distributors and breakdowns of suppliers are stochastic, and due to the complicated nature of the problem is difficult to calculate the objective function by using classic methods. So, simulation is used for estimating the objectives of the problem. It’s been proved that the problem of this study is NP-Hard. Therefore, a metaheuristic multi-objective particle swarm optimization (MOPSO) method is used to find the optimal solution. To test the reliability of the model and the proposed algorithm, a real drug distributing problem is used and after estimating a Pareto front, the best answer is chosen by The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method.

Design optimization of electrostatic sensor electrodes via MOPSO

An ideal electrostatic sensor with optimized electrodes is required for solid particles (SPs) velocity determination. The simplicity and reliability of the multi-objective particle swarm optimization (MOPSO) imparted further impetus to the accurate determination of SPs velocity using electrostatic sensors. This paper reports a new design of electrostatic sensors via MOPSO, where the sensors geometry is determined using response surface methodology (RSM). Besides, electrode sensing factors including spatial sensitivity and statistical error of electrostatic sensors are considered for the optimization. Parameters such as electrode dimensions, velocity, and distribution of particles in the pipeline are found to greatly influence the electrodes sensing performance. Electrodes with different physical sizes were applied to find their optimal value to attain minimum statistical error and maximum spatial sensitivity, which were optimization targets. Following RSM and MOPSO methods, an electrode’s optimum length of 0.6 cm and thickness of 0.5 cm has been achieved. It is established that the electrodes spatial sensitivity distribution appears uniform due to their size optimization. Furthermore, very thin electrodes with short length are determined to be prerequisite for sensitivity enhancement. Present study finds a theoretical basis for the electrostatic electrodes optimal design.