# Improved Whale Optimization Algorithm and Low-Energy Consumption Design of Circuit Breaker

Authors: Guodong Li; Yunhe Tian; Min Xie
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In the transmission and distribution system, the high-energy consumption of the low-voltage circuit breaker easily leads to the decrease of the distribution efficiency of the system. Therefore, on the basis of calculating the internal power consumption of circuit breakers, combining with several factors affecting the energy consumption of circuit breakers, the low-energy consumption mathematical model of circuit breakers is built, and an improved whale optimization algorithm (LA-WOA) is proposed for optimization. The Lévy flight mechanism and the adaptive weight factor are introduced into the traditional whale algorithm to enhance the global and local search ability of the algorithm and also improve the convergence speed and accuracy of the algorithm. The test function simulation experiment proves that the improved algorithm has better performance than other algorithms. Furthermore, the improved LA-WOA algorithm is used to iteratively solve the low-energy consumption mathematical model of the circuit breaker. The experimental results show that the improved algorithm obtains a lower-energy consumption value, which can effectively improve the design efficiency and accuracy of the circuit breaker.


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## Body

## 1. Introduction

With the gradual expansion of the power supply and distribution market and the rapid development of smart grid and new energy, it is necessary to optimize the energy conservation control of low-voltage power distribution system. Smart distribution networks can rationally utilize resources, achieve the balance of energy supply, improve the utilization rate of power resources, and reduce energy loss [1, 2]. At the same time, with an annual increase of users’ electricity consumption, as the basic control component of the distribution system, and the circuit breaker can bring huge economic benefit to the whole power network by reducing its energy loss. So, it is imperative to study the energy consumption of circuit breakers. During the working process of the circuit breaker, there are various losses in the internal structure of the circuit breaker. The current flowing through the conductor will inevitably produce corresponding energy loss, which converts a part of the energy of the power supply into heat, which causes the temperature of internal parts to rise, and then produces a series of negative effects [3, 4]. In the main circuit structure of the circuit breaker, many factors such as the number of contacts, contact resistance, and phase offset angle will affect the energy consumption of the circuit breaker itself [5]. Therefore, in order to reduce the energy loss of the circuit breaker, it is necessary to obtain the optimal energy consumption value and related parameters in combination with a good-performance swarm intelligence optimization algorithm [6].Swarm intelligence optimization algorithm has global optimization performance, which can be used to solve the optimal solution of multidimensional complex problems in engineering practice. It is an important branch of intelligent optimization algorithms and has good optimization performance and ability [7]. There are many common swarm intelligence optimization algorithms, such as Particle Swarm Optimization (PSO) [8], Differential Evolution Algorithm (DE) [9], Grey Wolf Optimizer (GWO) [10], the relatively novel Naked Mole-rat Algorithm (NMR) [11], and so on, and they have been widely used in many fields. Whale Optimization Algorithm (WOA) [12] is a new metaheuristic intelligent optimization algorithm proposed by Mirjalili and Lewis in 2016 according to the unique hunting behavior of humpback whales. WOA has the characteristics of simple parameters, easy adjustment of parameters, and strong search ability [13]. However, when solving complex and high-dimensional problems, the algorithm has some defects, such as slow convergence speed and easiness to fall into local optimum. In WOA, the convergence factor a in the algorithm cannot accurately reflect the complex nonlinear search process of the algorithm, so it lacks the ability to adjust global search and local search. And in the later iteration process, the population distribution will shrink due to the existence of individuals with higher fitness values in the population, which will lead to the stagnation of the algorithm optimization process. At present, many scholars have improved the whale optimization algorithm and applied it to many fields. In order to solve the problems of slow optimization speed and low convergence accuracy in the later stage of the standard WOA. Yan et al. [14] used the Lévy flight mechanism and mutation operator in the algorithm to improve the global search ability of the algorithm. The convergence speed and accuracy of the algorithm have been effectively improved. Jin et al. [15] conducted a comprehensive survey of WOA, which used random Gaussian distribution operation to initialize the population and increase its diversity. At the same time, the weight factor was introduced to improve the stability of the algorithm, and the effectiveness of the algorithm is verified by simulation experiments. Oliva et al. [16] proposed a chaotic whale optimization algorithm (CWOA) for the problem of solar cell parameter estimation, which uses chaotic mapping to adaptively adjust the relevant parameters of the standard whale optimization algorithm. During the entire iteration process, the algorithm’s global search ability is improved. Experiments have proved that the improved algorithm has good performance and higher accuracy in estimating solar cell parameters.Research on energy loss optimization technology in low-voltage circuit breaker. Szulborski et al. [17] artificially protect low-voltage distribution lines from the influence of short circuit and overload, establish a 3D model of the circuit breaker, use ANSYS software for simulation calculation, and conduct transient thermal simulation of the current path of the circuit breaker. Through the experimental analysis of different current values, the current density and ohmic energy consumption value in the current path are obtained. Thus, the propagation mode of heat in the modeling current path of circuit breaker is determined. This model has important guiding significance for thermal loss behavior analysis under current flow condition. Dai et al. [18] used an adaptive particle swarm optimization algorithm to study the energy loss performance of low-voltage circuit breaker. By analyzing the internal circuit structure of the circuit breaker and many factors that affect the energy loss, they built a low-energy mathematical model and used the improved algorithm to solve the model iteratively. It can effectively meet the low-energy consumption design requirements of circuit breakers.To sum up, the convergence factora in the whale optimization algorithm is improved by Lévy flight mechanism in this paper to enhance the conversion ability of the algorithm between global search and local search, and the adaptive weight factor is introduced into the position update formula of the algorithm to improve the convergence speed of the solution in the later stage of the algorithm. A new low-energy consumption mathematical model is constructed based on the factors affecting the energy consumption of circuit breakers, and the improved whale optimization algorithm is used to solve the model iteratively. The minimum energy consumption of circuit breakers is taken as the optimization objective, which can effectively improve the product quality and work efficiency of circuit breakers. Therefore, it is of great theoretical and practical significance to apply the improved whale optimization algorithm to the low-energy optimization design of circuit breakers.


## 2. Energy Consumption Analysis of Circuit Breakers

In the power distribution system, the high-voltage network can transmit electric energy over long distances to the transformer end, through the transformer, bus, main circuit breaker, bus bar, and then to the branch circuit breaker, and finally to the terminal load [19]. During the whole process, all levels of equipment consume about 30% to 40% of the total electric energy. The loss of electric energy is not the useless work done by current, but caused by electrical resistance, skin effect, proximity effect, and the loss of electric energy is also generated [20].According to the standard of GB14048.2-2008, the energy consumption of the circuit breaker is measured under the condition of current rating steady-state temperature, and the measurement circuit is as short as possible. In order to make the measurement accurate, the voltmeter should be connected in parallel at two ports of the circuit breaker’s inlet and outlet current. Figure1 is the schematic diagram of energy consumption measurement in the main circuit of circuit breaker under the condition of three-phase current.Figure 1
Schematic diagram of energy consumption measurement in the main circuit of circuit breaker.According to the international electrical standards, the calculation expression of the internal power consumption of a low-voltage circuit breaker is(1)W=∑k=1k=pΔUIncosφ,(2)ΔU=In×Z,(3)Z=R+X.In the formula,P is the phase series, K is the series, ΔU is the voltage drop (V), I is the rated current of the main circuit of the circuit breaker (A), cosφ is the phase deflection angle of the internal circuit, Z is the impedance, and X is the reactance.The influence of reactance on impedance is negligible when the energy consumption of resistance is analyzed. Therefore, according to equations (1)–(3), the internal power consumption expression can be transformed into(4)W=∑k=1k=pIn2Rcosφ.According to formula (4), to reduce power consumption, it is necessary to reduce the resistance of the contact. Therefore, the main circuit contact adopts a multistage parallel structure, and other factors affecting energy consumption are further analyzed and modeled.The contact resistance of the circuit breaker is mainly the contact resistance between static and static contacts, and its expression is(5)R=ρ×lS.The contact resistance volume is expressed as(6)V=Sl=S×RSρ=k1RS2,where S is the cross-sectional area of the contact resistance, l is the length of the contact resistance, and k1 is the reciprocal of the resistivity.In the internal structure of the circuit breaker, there is resistance loss in the connecting wire, and the heat from the terminal to the connecting wire can also be dissipated through the surface of the connecting wire. The heat expression is(7)Q=Tterminal−Tt.amod.lc.B.Ac.Set the number of contacts ast and t > 1; then, the total impedance can be expressed as(8)1Z=∑i=2t1Ri=∑i=2t1R.Through further analysis of circuit breaker energy consumption, the influence of contact resistance volume, resistance thermal loss, and other factors are taken into consideration, respectively. Then, the mathematical model of low-energy consumption of new circuit breaker can be constructed as follows:(9)f1=λ1∑k=1k=pIn2cosφ1∑i=2i=n1/R+λ2t∑i=2i=nk1RS2+λ3Tterminal−Tt.amod.lc.B.Ac.Among them,n, cosφ, R, and Ac are all variables, and λ1, λ2, and λ3 are control coefficients that affect energy consumption factors.When the value of the evaluation functionf1 is the smallest, the energy consumption design of the circuit breaker is considered to be optimal. Aiming at the above model, an improved whale optimization algorithm is used to solve the value of each parameter in the model and the corresponding power consumption value of the circuit breaker itself.


## 3. Improved Whale Optimization Algorithm

The whale optimization algorithm is a metaheuristic optimization algorithm that simulates the hunting behavior of humpback whales. Humpback whales spit out bubbles in a spiral shape during predation and form a bubble net, as shown in Figure2.Figure 2
Schematic diagram of bubble net predation.In the whale optimization algorithm, each whale acts as the search agent of the population and is regarded as a candidate solution. Among them, the hunting process of humpback whales is divided into three stages: encircling prey, bubble-net attacking, and search for prey [21, 22].In view of the disadvantages of the traditional whale optimization algorithm, such as slow convergence speed, low precision, and easiness to fall into the local optimum during the optimization process, the algorithm is improved in many aspects.


### 3.1. Lévy Flight Mechanism

In the traditional WOA algorithm, the coefficient vectorA is the random value of the interval −a,a, which is used to balance the global and local search ability of the algorithm. The global search is chosen when A≥1 while the local search is chosen when A<1, where a is the convergence factor of the algorithm, which decreases from 2 to 0 in the iteration process. In the iterative process of the standard WOA algorithm, the coefficient vector A is limited, which easily leads to the local optimum of the algorithm, thus affecting the convergence accuracy of the algorithm. Therefore, the Lévy flight mechanism is introduced to improve the ability of the algorithm to jump out of the local optimum.The Lévy flight mechanism is a widely used Gaussian stochastic distribution model, whose step size is random, which is very similar to the trajectory of biological activities in nature [23, 24]. Therefore, Lévy flight is widely used in the measurement and simulation of random and pseudorandom natural phenomena. The expression of coefficient vector A improved by Lévy flight mechanism is as follows:(10)A=2aLevyλ−a.Among them, Lévy (λ) means that it obeys the Lévy distribution with parameter λ; namely,(11)Levy∼u=t−λ.Mantegna’s algorithm is used to simulate the Lévy distribution, and the flight path step size following the Lévy distribution is generated. Its expression is as follows:(12)s=μv1/β,where μ and v follow the normal distribution of parameters σμ and σν; that is,(13)μ∼N0,σμ2,v∼N0,σμ2,σμ=Γ1+βsinπβ/2Γ1+β/2β2β−1/2,σv=1.To reduce the computation of the whole algorithm,β is usually valued at 1.5.


### 3.2. Adaptive Weighting Factor

To solve the problem of slow convergence rate in the later stage of the tradition whale optimization algorithm, the adaptive weight factor is introduced to improve the position update process of the WOA algorithm and speed up the encircling of humpback whales to prey. The expression of the adaptive weight factor is(14)ω=1−et/Tmax−1e−1.Then, the position update formula of the improved whale optimization algorithm can be expressed as follows:(15)Xt+1=ω⋅X∗t−A⋅DA<1,P<0.5,(16)Xt+1=ω⋅Xrandt−A⋅DA≥1,P<0.5,(17)Xt+1=D′⋅ebl⋅cos2πl+1−ωX∗tP≥0.5.By introducing the Lévy flight mechanism and the adaptive weight factor, the convergence speed and search ability of the algorithm can be accelerated, and the algorithm can be effectively prevented from falling into the local optimal, thus improving the overall convergence speed and accuracy of the algorithm.The specific implementation steps of the improved whale optimization algorithm (LA-WOA) are shown in Table1.Table 1
LA-WOA algorithm steps.
LA-WOA algorithm steps(1) Set relevant parameters of the algorithm and initialize the population(2) Calculate the fitness value of each individual and record the current optimal individual and location(3) While (t < maximum number of iterations)fori = 1 to N do(4) Update the value of the convergence factora(5) Update the value of the coefficient vectorA according to the Lévy flight mechanism, and update the values of C and l, and randomly generate P in [0, 1]if (P<0.05) doif (|A|<1)) do(6) Update the current position of the individual whale according to formula (15)else if (|A|≥1) do (2)(7) Randomly search for prey according to formula (16)end ifelse if (P≥0.05) do(8) Spiral update position according to formula (17)end(9) Calculate the fitness value of the population and update the optimal solution individual and positiont=t+1end


### 3.3. Simulation Experiment Analysis

In the simulation experiment, in order to verify the performance of LA-WOA, 18 test functions [25] are selected for the simulation experiment, as shown in Table 2, where F1–F7 are single-peak functions with only one optimal value, which can be used to verify the convergence performance of the algorithm. F8–F13 are fixed-dimensional multipeak functions, F14–F18 are mixed-dimensional multipeak functions, including multiple local optimal solutions. The experimental environment is as follows: MATLAB R2018a, Win10 (64-bit), and the PC processor is Intel® Core™ i7-7500U. We analyze and compare LA-WOA with particle swarm algorithm, grey wolf optimization algorithm, and traditional whale optimization algorithm. The algorithm parameters are set as follows: the population number N = 30, the maximum number of iterations Max_iteration = 1000, and the logarithmic spiral shape constant b = 1 is defined.Table 2
Test functions.
FunctionV no.RangefminF1x=∑i=1nxi230[−100, 100]0F2x=∑i=1nxi+∏i=1nxi30[−10, 10]0F3x=∑i=1n∑j−1ixj230[−100, 100]0F4x=maxixi,1≤i≤n30[−100, 100]0F5x=∑i=1n−1100xi+1−xi22+xi−1230[−30, 30]0F6x=∑i=1nxi+0.5230[−100, 100]0F7x=∑i=1nixi4+random0,130[−1.28, 1.28]0F8x=∑i=1n−xisinxi30[−500, 500]−418 .9829 ×5F9x=∑i=1nxi2−10cos2πxi+1030[−5.12, 5.12]0F10x=−20exp−0.21/n∑i=1nxi2−exp1/n∑i=1ncos2πxi+20+e30[−32, 32]0F11x=1/4000∑i=1nxi2−∏i=1ncosxi/i+130[−600, 600]0F12x=π/n10sinπy1+∑i=1n−1yi−121+10sin2πyi+1+yn−12+∑i=1nuxi,10,100,4yi=1+xi+1/4uxi,a,k,m=kxi−am,xi>a0,−a<xi<ak−xi−am,xi<−a30[−50, 50]0F13x=0.1sin23πx1+∑i=1nxi−121+sin23πxi+1+xn−121+sin22πxn+∑i=1nuxi,5,100,430[−50, 50]0F14x=1500+∑j=1251/j+∑i=12x1−aij6−12[−65, 65]1F15x=∑i=111ai−xibi2+bix2/bi2+bix3+x424[−5, 5]0.0030F16x=−∑i=15X−aiX−aiT+ci−14[0, 10]−10 .1532F17x=−∑i=17X−aiX−aiT+ci−14[0, 10]−10 .4028F18x=−∑i=110X−aiX−aiT+ci−14[0, 10]–10 .5363Experimental simulation is carried out on 18 test functions, respectively. Particle swarm optimization algorithm (PSO), grey wolf optimizer (GWO), traditional whale algorithm (WOA), and improved LA-WOA algorithm are used for the iterative solution. Simulation results of test functions are shown in Figure3.Figure 3
Test function simulation results.As can be seen from Figure3, through the test function experiment and simulation, the improved LA-WOA algorithm has higher optimization speed and precision compared with other algorithms. It also achieves the best performance. Therefore, this paper has a positive impact on the improvement of the traditional whale optimization algorithm.To further verify the effectiveness of the algorithm, the experiment is repeated 30 times for 10 test functions, and the maximum, minimum, average, and standard deviation of each algorithm are obtained. The optimization results obtained for the test function are shown in Table3.Table 3
Test results of different algorithms.
FunctionAlgorithmMaximum valueMinimum valueAverageStandard deviationF1PSO1.00E + 046.44E − 162.00E + 034.22E + 03GWO3.53E − 586.17E − 611.04E − 581.11E − 58WOA1.37E − 1506.29E − 1641.69E − 1514.33E − 151LA-WOA0.000E + 000.000E + 000.000E + 000.000E + 00F2PSO7.01E + 011.00E + 013.81E + 011.93E + 01GWO3.65E − 341.83E − 351.09E − 349.97E − 35WOA8.64E − 1027.14E − 1138.64E − 1032.73E − 102LA-WOA4.97E − 3040.00E + 006.36E − 3050.00E + 00F3PSO5.70E + 042.00E + 043.59E + 041.20E + 04GWO5.82E − 151.07E − 189.71E − 161.92E − 15WOA3.93E + 043.06E + 032.42E + 041.12E + 04LA-WOA0.00E + 000.00E + 000.00E + 000.00E + 00F4PSO4.50E + 012.24E + 013.43E + 017.26E + 00GWO9.31E − 143.34E − 161.93E − 142.70E − 14WOA9.14E + 011.65E − 014.23E + 013.00E + 01LA-WOA0.00E + 000.00E + 000.00E + 000.00E + 00F5PSO3.09E + 037.07E + 014.58E + 029.39E + 02GWO2.72E + 012.51E + 012.63E + 018.43E − 01WOA2.85E + 012.64E + 012.73E + 015.79E − 01LA-WOA1.24E − 017.43E − 051.89E − 023.74E − 02F6PSO2.00E + 041.02E − 124.02E + 037.01E + 03GWO1.00E + 002.12E − 056.77E − 013.56E − 01WOA6.36E − 017.80E − 031.81E − 012.12E − 01LA-WOA2.48E − 031.33E − 094.65E − 048.10E − 04F7PSO1.35E + 017.28E − 023.09E + 005.11E + 00GWO1.52E − 034.31E − 048.59E − 043.39E − 04WOA4.75E − 031.86E − 041.57E − 031.62E − 03LA-WOA7.67E − 052.15E − 053.86E − 052.13E − 05F8PSO−7.32E + 03− 9.37E + 03−8.45E + 037.53E + 02GWO−5.09E + 03− 7.01E + 03−5.95E + 036.05E + 02WOA−8.56E + 03−1.26E + 04−1.09E+041.56E + 03LA-WOA−1.26E + 04−1.26E + 04−1.26E + 043.09E − 02F9PSO1.77E + 029.56E + 011.37E + 022.95E + 01GWO4.53E + 000.00E + 004.53E − 011.43E + 00WOA0.00E + 000.00E + 000.00E + 000.00E + 00LA-WOA0.00E + 000.00E + 000.00E + 000.00E + 00F10PSO2.02E + 011.67E − 051.45E + 018.01E + 00GWO2.93E − 141.15E − 141.69E − 145.09E − 15WOA7.99E − 158.88E − 163.73E − 152.25E − 15LA-WOA8.88E − 168.88E − 168.88E − 160.00E + 00F11PSO7.79E − 021.03E − 131.76E − 022.45E − 02GWO2.86E − 020.00E + 003.97E − 039.33E − 03WOA0.00E + 000.00E + 000.00E + 000.00E + 00LA-WOA0.00E + 000.00E + 000.00E + 000.00E + 00F12PSO1.82E + 021.37E − 071.89E + 015.73E + 01GWO6.06E − 022.54E − 024.27E − 021.27E − 02WOA8.50E − 031.11E − 033.48E − 032.62E − 03LA-WOA2.38E − 047.47E − 083.75E − 057.52E − 05F13PSO7.19E + 005.78E − 022.23E + 002.73E + 00GWO1.10E + 002.02E − 015.04E − 012.34E − 01WOA8.61E − 013.65E − 022.68E − 012.61E − 01LA-WOA6.86E − 048.01E − 091.50E − 042.08E − 04F14PSO9.98E − 019.98E − 019.98E − 010.00E + 00GWO1.08E + 019.98E − 014.32E + 004.51E + 00WOA1.08E + 019.98E − 012.86E + 003.21E + 00LA-WOA9.98E − 019.98E − 019.98E − 012.97E − 09F15PSO2.26E − 025.57E − 045.10E − 038.65E − 03GWO2.04E − 023.07E − 046.61E − 039.50E − 03WOA7.21E − 043.97E − 045.61E − 041.09E − 04LA-WOA4.87E − 043.23E − 043.66E − 046.10E − 05F16PSO−2.68E + 00−1.02E + 01−5.63E + 002.58E + 00GWO−5.10E + 00−1.02E + 01−9.14E + 002.13E + 00WOA−5.06E + 00−1.02E + 01−8.62E + 002.46E + 00LA-WOA−9.70E + 00−1.01E + 01−1.00E + 011.71E − 01F17PSO−2.75E + 00−1.04E + 01−7.58E + 003.71E + 00GWO−1.04E + 01−1.04E + 01−1.04E + 012.14E − 04WOA−5.09E + 00− 1.04E + 01−8.27E + 002.74E + 00LA-WOA−9.93E + 00−1.04E + 01−1.03E + 011.47E − 01F18PSO−2.42E + 00− 1.05E + 01−8.42E + 003.47E + 00GWO−5.13E + 00−1.05E + 01−1.00E + 011.71E + 00WOA−3.84E + 00−1.05E + 01−8.73E + 002.81E + 00LW-WOA−1.04E + 01−1.05E + 01−1.05E + 016.26E − 02It can be seen from Table3 that for the 18 test functions, the optimal value, maximum value, average value, and standard deviation of LA-WOA proposed in this paper are significantly better than those of PSO, GWO, and WOA, and the optimal value of 0 is obtained in several test functions. Therefore, the performance of the improved LA-WOA is stable, and the optimization speed and precision are greatly improved.


## 3.1. Lévy Flight Mechanism

In the traditional WOA algorithm, the coefficient vectorA is the random value of the interval −a,a, which is used to balance the global and local search ability of the algorithm. The global search is chosen when A≥1 while the local search is chosen when A<1, where a is the convergence factor of the algorithm, which decreases from 2 to 0 in the iteration process. In the iterative process of the standard WOA algorithm, the coefficient vector A is limited, which easily leads to the local optimum of the algorithm, thus affecting the convergence accuracy of the algorithm. Therefore, the Lévy flight mechanism is introduced to improve the ability of the algorithm to jump out of the local optimum.The Lévy flight mechanism is a widely used Gaussian stochastic distribution model, whose step size is random, which is very similar to the trajectory of biological activities in nature [23, 24]. Therefore, Lévy flight is widely used in the measurement and simulation of random and pseudorandom natural phenomena. The expression of coefficient vector A improved by Lévy flight mechanism is as follows:(10)A=2aLevyλ−a.Among them, Lévy (λ) means that it obeys the Lévy distribution with parameter λ; namely,(11)Levy∼u=t−λ.Mantegna’s algorithm is used to simulate the Lévy distribution, and the flight path step size following the Lévy distribution is generated. Its expression is as follows:(12)s=μv1/β,where μ and v follow the normal distribution of parameters σμ and σν; that is,(13)μ∼N0,σμ2,v∼N0,σμ2,σμ=Γ1+βsinπβ/2Γ1+β/2β2β−1/2,σv=1.To reduce the computation of the whole algorithm,β is usually valued at 1.5.


## 3.2. Adaptive Weighting Factor

To solve the problem of slow convergence rate in the later stage of the tradition whale optimization algorithm, the adaptive weight factor is introduced to improve the position update process of the WOA algorithm and speed up the encircling of humpback whales to prey. The expression of the adaptive weight factor is(14)ω=1−et/Tmax−1e−1.Then, the position update formula of the improved whale optimization algorithm can be expressed as follows:(15)Xt+1=ω⋅X∗t−A⋅DA<1,P<0.5,(16)Xt+1=ω⋅Xrandt−A⋅DA≥1,P<0.5,(17)Xt+1=D′⋅ebl⋅cos2πl+1−ωX∗tP≥0.5.By introducing the Lévy flight mechanism and the adaptive weight factor, the convergence speed and search ability of the algorithm can be accelerated, and the algorithm can be effectively prevented from falling into the local optimal, thus improving the overall convergence speed and accuracy of the algorithm.The specific implementation steps of the improved whale optimization algorithm (LA-WOA) are shown in Table1.Table 1
LA-WOA algorithm steps.
LA-WOA algorithm steps(1) Set relevant parameters of the algorithm and initialize the population(2) Calculate the fitness value of each individual and record the current optimal individual and location(3) While (t < maximum number of iterations)fori = 1 to N do(4) Update the value of the convergence factora(5) Update the value of the coefficient vectorA according to the Lévy flight mechanism, and update the values of C and l, and randomly generate P in [0, 1]if (P<0.05) doif (|A|<1)) do(6) Update the current position of the individual whale according to formula (15)else if (|A|≥1) do (2)(7) Randomly search for prey according to formula (16)end ifelse if (P≥0.05) do(8) Spiral update position according to formula (17)end(9) Calculate the fitness value of the population and update the optimal solution individual and positiont=t+1end


## 3.3. Simulation Experiment Analysis

In the simulation experiment, in order to verify the performance of LA-WOA, 18 test functions [25] are selected for the simulation experiment, as shown in Table 2, where F1–F7 are single-peak functions with only one optimal value, which can be used to verify the convergence performance of the algorithm. F8–F13 are fixed-dimensional multipeak functions, F14–F18 are mixed-dimensional multipeak functions, including multiple local optimal solutions. The experimental environment is as follows: MATLAB R2018a, Win10 (64-bit), and the PC processor is Intel® Core™ i7-7500U. We analyze and compare LA-WOA with particle swarm algorithm, grey wolf optimization algorithm, and traditional whale optimization algorithm. The algorithm parameters are set as follows: the population number N = 30, the maximum number of iterations Max_iteration = 1000, and the logarithmic spiral shape constant b = 1 is defined.Table 2
Test functions.
FunctionV no.RangefminF1x=∑i=1nxi230[−100, 100]0F2x=∑i=1nxi+∏i=1nxi30[−10, 10]0F3x=∑i=1n∑j−1ixj230[−100, 100]0F4x=maxixi,1≤i≤n30[−100, 100]0F5x=∑i=1n−1100xi+1−xi22+xi−1230[−30, 30]0F6x=∑i=1nxi+0.5230[−100, 100]0F7x=∑i=1nixi4+random0,130[−1.28, 1.28]0F8x=∑i=1n−xisinxi30[−500, 500]−418 .9829 ×5F9x=∑i=1nxi2−10cos2πxi+1030[−5.12, 5.12]0F10x=−20exp−0.21/n∑i=1nxi2−exp1/n∑i=1ncos2πxi+20+e30[−32, 32]0F11x=1/4000∑i=1nxi2−∏i=1ncosxi/i+130[−600, 600]0F12x=π/n10sinπy1+∑i=1n−1yi−121+10sin2πyi+1+yn−12+∑i=1nuxi,10,100,4yi=1+xi+1/4uxi,a,k,m=kxi−am,xi>a0,−a<xi<ak−xi−am,xi<−a30[−50, 50]0F13x=0.1sin23πx1+∑i=1nxi−121+sin23πxi+1+xn−121+sin22πxn+∑i=1nuxi,5,100,430[−50, 50]0F14x=1500+∑j=1251/j+∑i=12x1−aij6−12[−65, 65]1F15x=∑i=111ai−xibi2+bix2/bi2+bix3+x424[−5, 5]0.0030F16x=−∑i=15X−aiX−aiT+ci−14[0, 10]−10 .1532F17x=−∑i=17X−aiX−aiT+ci−14[0, 10]−10 .4028F18x=−∑i=110X−aiX−aiT+ci−14[0, 10]–10 .5363Experimental simulation is carried out on 18 test functions, respectively. Particle swarm optimization algorithm (PSO), grey wolf optimizer (GWO), traditional whale algorithm (WOA), and improved LA-WOA algorithm are used for the iterative solution. Simulation results of test functions are shown in Figure3.Figure 3
Test function simulation results.As can be seen from Figure3, through the test function experiment and simulation, the improved LA-WOA algorithm has higher optimization speed and precision compared with other algorithms. It also achieves the best performance. Therefore, this paper has a positive impact on the improvement of the traditional whale optimization algorithm.To further verify the effectiveness of the algorithm, the experiment is repeated 30 times for 10 test functions, and the maximum, minimum, average, and standard deviation of each algorithm are obtained. The optimization results obtained for the test function are shown in Table3.Table 3
Test results of different algorithms.
FunctionAlgorithmMaximum valueMinimum valueAverageStandard deviationF1PSO1.00E + 046.44E − 162.00E + 034.22E + 03GWO3.53E − 586.17E − 611.04E − 581.11E − 58WOA1.37E − 1506.29E − 1641.69E − 1514.33E − 151LA-WOA0.000E + 000.000E + 000.000E + 000.000E + 00F2PSO7.01E + 011.00E + 013.81E + 011.93E + 01GWO3.65E − 341.83E − 351.09E − 349.97E − 35WOA8.64E − 1027.14E − 1138.64E − 1032.73E − 102LA-WOA4.97E − 3040.00E + 006.36E − 3050.00E + 00F3PSO5.70E + 042.00E + 043.59E + 041.20E + 04GWO5.82E − 151.07E − 189.71E − 161.92E − 15WOA3.93E + 043.06E + 032.42E + 041.12E + 04LA-WOA0.00E + 000.00E + 000.00E + 000.00E + 00F4PSO4.50E + 012.24E + 013.43E + 017.26E + 00GWO9.31E − 143.34E − 161.93E − 142.70E − 14WOA9.14E + 011.65E − 014.23E + 013.00E + 01LA-WOA0.00E + 000.00E + 000.00E + 000.00E + 00F5PSO3.09E + 037.07E + 014.58E + 029.39E + 02GWO2.72E + 012.51E + 012.63E + 018.43E − 01WOA2.85E + 012.64E + 012.73E + 015.79E − 01LA-WOA1.24E − 017.43E − 051.89E − 023.74E − 02F6PSO2.00E + 041.02E − 124.02E + 037.01E + 03GWO1.00E + 002.12E − 056.77E − 013.56E − 01WOA6.36E − 017.80E − 031.81E − 012.12E − 01LA-WOA2.48E − 031.33E − 094.65E − 048.10E − 04F7PSO1.35E + 017.28E − 023.09E + 005.11E + 00GWO1.52E − 034.31E − 048.59E − 043.39E − 04WOA4.75E − 031.86E − 041.57E − 031.62E − 03LA-WOA7.67E − 052.15E − 053.86E − 052.13E − 05F8PSO−7.32E + 03− 9.37E + 03−8.45E + 037.53E + 02GWO−5.09E + 03− 7.01E + 03−5.95E + 036.05E + 02WOA−8.56E + 03−1.26E + 04−1.09E+041.56E + 03LA-WOA−1.26E + 04−1.26E + 04−1.26E + 043.09E − 02F9PSO1.77E + 029.56E + 011.37E + 022.95E + 01GWO4.53E + 000.00E + 004.53E − 011.43E + 00WOA0.00E + 000.00E + 000.00E + 000.00E + 00LA-WOA0.00E + 000.00E + 000.00E + 000.00E + 00F10PSO2.02E + 011.67E − 051.45E + 018.01E + 00GWO2.93E − 141.15E − 141.69E − 145.09E − 15WOA7.99E − 158.88E − 163.73E − 152.25E − 15LA-WOA8.88E − 168.88E − 168.88E − 160.00E + 00F11PSO7.79E − 021.03E − 131.76E − 022.45E − 02GWO2.86E − 020.00E + 003.97E − 039.33E − 03WOA0.00E + 000.00E + 000.00E + 000.00E + 00LA-WOA0.00E + 000.00E + 000.00E + 000.00E + 00F12PSO1.82E + 021.37E − 071.89E + 015.73E + 01GWO6.06E − 022.54E − 024.27E − 021.27E − 02WOA8.50E − 031.11E − 033.48E − 032.62E − 03LA-WOA2.38E − 047.47E − 083.75E − 057.52E − 05F13PSO7.19E + 005.78E − 022.23E + 002.73E + 00GWO1.10E + 002.02E − 015.04E − 012.34E − 01WOA8.61E − 013.65E − 022.68E − 012.61E − 01LA-WOA6.86E − 048.01E − 091.50E − 042.08E − 04F14PSO9.98E − 019.98E − 019.98E − 010.00E + 00GWO1.08E + 019.98E − 014.32E + 004.51E + 00WOA1.08E + 019.98E − 012.86E + 003.21E + 00LA-WOA9.98E − 019.98E − 019.98E − 012.97E − 09F15PSO2.26E − 025.57E − 045.10E − 038.65E − 03GWO2.04E − 023.07E − 046.61E − 039.50E − 03WOA7.21E − 043.97E − 045.61E − 041.09E − 04LA-WOA4.87E − 043.23E − 043.66E − 046.10E − 05F16PSO−2.68E + 00−1.02E + 01−5.63E + 002.58E + 00GWO−5.10E + 00−1.02E + 01−9.14E + 002.13E + 00WOA−5.06E + 00−1.02E + 01−8.62E + 002.46E + 00LA-WOA−9.70E + 00−1.01E + 01−1.00E + 011.71E − 01F17PSO−2.75E + 00−1.04E + 01−7.58E + 003.71E + 00GWO−1.04E + 01−1.04E + 01−1.04E + 012.14E − 04WOA−5.09E + 00− 1.04E + 01−8.27E + 002.74E + 00LA-WOA−9.93E + 00−1.04E + 01−1.03E + 011.47E − 01F18PSO−2.42E + 00− 1.05E + 01−8.42E + 003.47E + 00GWO−5.13E + 00−1.05E + 01−1.00E + 011.71E + 00WOA−3.84E + 00−1.05E + 01−8.73E + 002.81E + 00LW-WOA−1.04E + 01−1.05E + 01−1.05E + 016.26E − 02It can be seen from Table3 that for the 18 test functions, the optimal value, maximum value, average value, and standard deviation of LA-WOA proposed in this paper are significantly better than those of PSO, GWO, and WOA, and the optimal value of 0 is obtained in several test functions. Therefore, the performance of the improved LA-WOA is stable, and the optimization speed and precision are greatly improved.


## 4. Circuit Breaker Optimization Design and Experimental Analysis

### 4.1. Design Scheme

The new mathematical model of circuit breaker energy consumption is a complex multidimensional optimization evaluation function. The improved LA-WOA is used to iteratively solve the low energy consumption model of the circuit breaker. The specific steps are as follows:Step 1.
Initialize 50 groups of parameters related to circuit breaker energy consumption, including contact resistance value, cosine value of phase offset Angle, number of contacts, and conductor cross-sectional area.Step 2.
Establish the corresponding fitness function according to the circuit breaker energy consumption model.Step 3.
Iterative update solution, adaptive adjustment of LA-WOA parameters.Step 4.
Determine whether the maximum number of iterations has been achieved. If not, return to Step 3 and continue the iterative solution. If the maximum number of iterations has been achieved, the optimal energy consumption value will be output.


### 4.2. Experimental and Comparative Analysis

The following uses adaptive particle swarm algorithm (APSO) [26], adaptive genetic algorithm (AGA) [27], heterogeneous differential evolution algorithm (HDE) [28], GWO algorithm, WOA algorithm, and LA-WOA algorithm to iteratively solve the low-energy consumption optimization model. The influencing parameters such as contact resistance value R, contact number t, phase deflection angle cosine value cosφ, and conductor cross-sectional area Ac are set as independent variables, and two types of circuit breakers are selected for simulation experiment analysis.Set the parameter value range of circuit breaker A: single contact resistance value 8 × 10−5Ω ≤ R ≤ 10−4Ω; phase offset angle cosine value 0.3 ≤ cosφ ≤ 0.32; number of contacts 6 ≤ t ≤ 13; and the conductor cut area is 4 × 10−6 m2 ≤ R ≤ 8 × 10−6 m2.Five algorithms are used to simulate and solve the energy consumption model of circuit breaker A with a rated current of 4000A. The simulation result is shown in Figure4.Figure 4
Circuit breaker A energy consumption iteration curve.The data in parentheses in the figure, respectively, indicate the number of iterations, the optimal fitness value of the function, and the number of iterations when the optimal value is reached. It can be seen from Figure4 that when the number of iterations of the improved LA-WOA is 18, the lowest energy consumption value of 235.5W can be obtained, which has the highest convergence accuracy compared to other algorithms.Set the parameter value range of circuit breaker B: single contact resistance value 10−4Ω ≤ R ≤ 1.8 × 10−4Ω, phase offset angle cosine value 0.3 ≤ cosφ ≤ 0.32, number of contacts 9 ≤ t ≤ 25, and the conductor cut area is 6 × 10−6 m2 ≤ R ≤ 10 × 10−6 m2.Five algorithms are used to simulate and solve the energy consumption model of circuit breaker B with a rated current of 4000A. The simulation results are shown in Figure5.Figure 5
Circuit breaker B energy consumption iteration curve.It can be seen from Figure5 that both GWO algorithm and WOA algorithm reach the optimal value when the number of iterations is 2. When the number of iterations is 31, the improved LA-WOA can obtain the minimum energy consumption value of 245.5W, which has the highest convergence accuracy compared with the other algorithms.As shown in Table4, using LA-WOA, in the circuit breaker low-energy optimization process, it overcomes the shortcomings of the traditional algorithm that the optimization purpose is not strong, and the lowest energy consumption value can be obtained in the optimization process. Compared with this series of HSW6-type circuit breaker (energy consumption is 790W), the energy consumption loss of the circuit breaker is significantly reduced, and by optimizing the contact resistance value, the HSW6-type circuit breaker is based on the new multicircuit, the main circuit is always. The resistance is further reduced, thereby reducing the temperature rise of the main circuit. Therefore, the power consumption of the main circuit inside the circuit breaker is effectively reduced.Table 4
Simulation design parameters of a three-phase circuit breaker with a rated current of 4000 A.
ModelCircuit breaker ACircuit breaker BSingle contact resistance value (Ω)8 × 10−51.5 × 10−4Phase offset cosine value0.3100.308Number of contacts1120Conductor cross-sectional area (m2)6 × 10−69 × 10−6Circuit breaker power consumption (W)235.5245.5In summary, the improved LA-WOA has the convergence speed and accuracy in the process of iteratively solving the low-energy consumption mathematical model of the circuit breaker, and the global optimal solution can be obtained. It effectively reduces the amount of consumables required to design the circuit breaker and reduces the loss of internal energy consumption of the circuit breaker. Compared with other optimization algorithms, its optimization effect is significant. By calculating the relevant parameter values of the new circuit breaker optimization evaluation model, it is applied to the production of the HSW6 model circuit breaker prototype.


## 4.1. Design Scheme

The new mathematical model of circuit breaker energy consumption is a complex multidimensional optimization evaluation function. The improved LA-WOA is used to iteratively solve the low energy consumption model of the circuit breaker. The specific steps are as follows:Step 1.
Initialize 50 groups of parameters related to circuit breaker energy consumption, including contact resistance value, cosine value of phase offset Angle, number of contacts, and conductor cross-sectional area.Step 2.
Establish the corresponding fitness function according to the circuit breaker energy consumption model.Step 3.
Iterative update solution, adaptive adjustment of LA-WOA parameters.Step 4.
Determine whether the maximum number of iterations has been achieved. If not, return to Step 3 and continue the iterative solution. If the maximum number of iterations has been achieved, the optimal energy consumption value will be output.


## 4.2. Experimental and Comparative Analysis

The following uses adaptive particle swarm algorithm (APSO) [26], adaptive genetic algorithm (AGA) [27], heterogeneous differential evolution algorithm (HDE) [28], GWO algorithm, WOA algorithm, and LA-WOA algorithm to iteratively solve the low-energy consumption optimization model. The influencing parameters such as contact resistance value R, contact number t, phase deflection angle cosine value cosφ, and conductor cross-sectional area Ac are set as independent variables, and two types of circuit breakers are selected for simulation experiment analysis.Set the parameter value range of circuit breaker A: single contact resistance value 8 × 10−5Ω ≤ R ≤ 10−4Ω; phase offset angle cosine value 0.3 ≤ cosφ ≤ 0.32; number of contacts 6 ≤ t ≤ 13; and the conductor cut area is 4 × 10−6 m2 ≤ R ≤ 8 × 10−6 m2.Five algorithms are used to simulate and solve the energy consumption model of circuit breaker A with a rated current of 4000A. The simulation result is shown in Figure4.Figure 4
Circuit breaker A energy consumption iteration curve.The data in parentheses in the figure, respectively, indicate the number of iterations, the optimal fitness value of the function, and the number of iterations when the optimal value is reached. It can be seen from Figure4 that when the number of iterations of the improved LA-WOA is 18, the lowest energy consumption value of 235.5W can be obtained, which has the highest convergence accuracy compared to other algorithms.Set the parameter value range of circuit breaker B: single contact resistance value 10−4Ω ≤ R ≤ 1.8 × 10−4Ω, phase offset angle cosine value 0.3 ≤ cosφ ≤ 0.32, number of contacts 9 ≤ t ≤ 25, and the conductor cut area is 6 × 10−6 m2 ≤ R ≤ 10 × 10−6 m2.Five algorithms are used to simulate and solve the energy consumption model of circuit breaker B with a rated current of 4000A. The simulation results are shown in Figure5.Figure 5
Circuit breaker B energy consumption iteration curve.It can be seen from Figure5 that both GWO algorithm and WOA algorithm reach the optimal value when the number of iterations is 2. When the number of iterations is 31, the improved LA-WOA can obtain the minimum energy consumption value of 245.5W, which has the highest convergence accuracy compared with the other algorithms.As shown in Table4, using LA-WOA, in the circuit breaker low-energy optimization process, it overcomes the shortcomings of the traditional algorithm that the optimization purpose is not strong, and the lowest energy consumption value can be obtained in the optimization process. Compared with this series of HSW6-type circuit breaker (energy consumption is 790W), the energy consumption loss of the circuit breaker is significantly reduced, and by optimizing the contact resistance value, the HSW6-type circuit breaker is based on the new multicircuit, the main circuit is always. The resistance is further reduced, thereby reducing the temperature rise of the main circuit. Therefore, the power consumption of the main circuit inside the circuit breaker is effectively reduced.Table 4
Simulation design parameters of a three-phase circuit breaker with a rated current of 4000 A.
ModelCircuit breaker ACircuit breaker BSingle contact resistance value (Ω)8 × 10−51.5 × 10−4Phase offset cosine value0.3100.308Number of contacts1120Conductor cross-sectional area (m2)6 × 10−69 × 10−6Circuit breaker power consumption (W)235.5245.5In summary, the improved LA-WOA has the convergence speed and accuracy in the process of iteratively solving the low-energy consumption mathematical model of the circuit breaker, and the global optimal solution can be obtained. It effectively reduces the amount of consumables required to design the circuit breaker and reduces the loss of internal energy consumption of the circuit breaker. Compared with other optimization algorithms, its optimization effect is significant. By calculating the relevant parameter values of the new circuit breaker optimization evaluation model, it is applied to the production of the HSW6 model circuit breaker prototype.


## 5. Conclusion

In this paper, an improved whale optimization algorithm is proposed and applied to the low-energy consumption optimization design of low-voltage circuit breaker.(1)
The Lévy flight mechanism and weight factor are introduced into the traditional whale optimization algorithm to effectively improve the convergence speed and accuracy of the algorithm. The experimental simulation is conducted by using the test function. The results show that the improved LA-WOA has better performance and higher stability compared with other algorithms.(2)
In order to solve the problems of high-energy consumption and serious energy loss of low-voltage circuit breakers, a mathematical model of low-energy consumption was built according to the circuit breaker energy consumption formula and combined with several factors affecting energy consumption. The improved LA-WOA is used for iterative optimization of the model, and the results show that the improved LA-WOA can get a lower-energy consumption value.The improved LA-WOA can not only be used in the low energy consumption optimization design of circuit breakers, but also can extend the design methods and ideas to engineering simulation optimization calculations in other fields. Thus, it can better replace the traditional experience estimation and a large number of prototype manufacturing methods, which is of great significance to improving the product quality and work efficiency of low-voltage circuit breaker.



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*Source: 8349258-2022-05-27.xml*