Repulsive firefly algorithm-based optimal switching device placement in power distribution systems

1 Introduction

The switching devices in power distribution systems are usually installed using a feeder automation mechanism to isolate fault sections.This method is quite helpful for quickly recovering non-fault sections in the power supply,effectively improving power supply reliability,and controlling user power shortage cost [1-3].With an increase in the number of switching device installations,the costs associated with equipment investment,maintenance,and operation have substantially increased [4-6].Power distribution systems in China are characterized by numerous loads,branches,and segments,making it impossible to realize global setting of the switching devices [7-9].In this context,research on the application of artificial intelligencebased optimization methods to determine the optimal location of switching devices is ongoing [10-14].

Abdelaziz et al.used a Tabu list with a variable size according to the system size and proposed a distribution system reconfiguration method based on a modified Tabu search (TS) algorithm [15].Golestani et al.proposed an optimal switching placement method using a linear fragmented particle swarm optimization (PSO) algorithm and established a novel linear model for probability calculation of switch type offer for each candidate location [16].Ranjan et al.proposed an optimal switching placement method based on a genetic algorithm (GA) and a PSO algorithm and obtained the solution to a given objective function to enhance the reliability of a distribution system [17].

In this paper,we analyze the technical characteristics of switching device placement in the planning of power distribution systems and take the comprehensive reliability costs of the switching devices as the maximum objective function.For an effective and efficient decision-making with regard to switching device placement,a repulsive firefly algorithm (RFA)-based optimal switching device placement method is proposed.In this method,the influence of territorial repulsion during firefly courtship is considered,thus realizing the interactions of learning and competition among fireflies in the newly proposed RFA.We employ the sigmoid function to perform normalization tasks and represent the probability that switching devices are in a closed state.Finally,a binary optimal solution to the 0-1 planning configuration optimization problem for switching device placement is obtained using the RFA.The results of a simulation conducted on IEEE RBTS-BUS6 and actual feeder lines confirm the feasibility and efficiency of the proposed method.

2 Mathematical modeling for switching device placement

In a power distribution system,the greater the number of switching devices,the higher the power supply reliability [17,18].However,the costs associated with equipment investment,operation,and maintenance will significantly increase with an increase in the number of switching devices,as shown in Fig.1.

Fig.1 Reliability-cost curves of the switching devices in a power distribution system

As shown in Fig.1,curve (a)～(c) represent the relationships between power supply reliability and equipment investment cost of switching devices under different switching device placement strategies.In these cases,although the power supply reliability is improved,the equipment investment cost (associated with the switching devices) increases.Curve (d)～(f) represent the relationship between power supply reliability and user power shortage cost under different settings.Similar to the previous cases,although the user power shortage cost is reduced,the equipment investment cost increases.

Notably,even if the number of installed switching devices is fixed,the installation position significantly affects the power supply reliability and user power shortage cost.Therefore,considering the power supply reliability,equipment investment cost,and user power shortage cost,the optimal switching device placement problem in power distribution systems becomes a trade-off optimization problem.To solve this problem,we take the comprehensive reliability costs of the switching devices as the maximum objective function and solve the optimization problem as follows:

Here,P is the difference in the economic benefit between distribution networks without switching devices and those with switching devices,L0 is the user power shortage cost without switching devices,IC is the equipment investment cost,MC is the operation and maintenance cost,LC is the user power shortage cost,DC is the equipment scrap cost,R is the average service availability index,R0 is the lowest reliability constraint of the power distribution system,N is the switching device number,Ni is the user number at load point i,and Ui is the average outage time at load point i.The reliability constraints help introduce the average service availability index (ASAI) to verify whether the electricity supply and demand meet the lowest reliability request of the power distribution system.

The equipment investment cost,operation and maintenance cost,user power shortage cost,and equipment scrap cost are generally calculated as follows [1,8].

(1) Equipment investment cost:

Here,CS is the present value of the switching device investment cost,i is the discount rate,and p represents the economic life of the switching devices.

(2) Operation and maintenance cost:

Here,C0 is the proportion of operation and maintenance cost in the equipment investment cost.

(3) User power shortage cost:

Here,WENS is the system power shortage,and K is the electricity production ratio.

(4) Equipment scrap cost:

Here,D0 is the proportion of equipment scrap cost in the equipment investment cost.

3 Improved firefly algorithm considering repulsion forces

The firefly algorithm,proposed by Yang et al.in 2008,has been a widely-used artificial bionic optimization search algorithm [19,21]; it is based on how fireflies attract surrounding ones with their own light in the courtship period.In the optimization process of the algorithm,fireflies are randomly initialized in the solution space of the target maximum objective function.The more superior the target firefly behaves,the more attractive its luminance to the others.The influences of distance and communication media are considered to obtain the final solution for the target optimization function.The attractiveness between fireflies is inversely proportional to the distance and light absorption ability of the communication media.Therefore,the attractiveness between fireflies Xm and Xn can be defined as:

Here,β0 is the maximal attractiveness between fireflies,which is only related to the objective function values of the fireflies Xm and Xn,γ is the light absorption parameter,which is normally set to γ=1 as a constant,and rmn is the distance between the fireflies Xm and Xn.

In the t-th iteration,if firefly Xm(t) is the most attractive one for the target firefly Xn(t),the subsequent position of firefly Xn(t) after the movement can be expressed as:

Here,the parameter α～N(0,1) is a random step factor satisfying a normal distribution,and K represents the disturbance level.

To further improve the global optimization search ability of the conventional firefly algorithm,the RFA was proposed.The proposed algorithm considers the influence of territorial repulsion [22] during firefly courtship,thus realizing the interactions of learning and competition among the fireflies.The set of male fireflies was assumed as and the female set was assumed as the repulsive neighborhood of the former set can be described as If two male fireflies enter the same repulsive neighborhood during the optimization process,the disadvantageous one (i.e.,the one with a lower objective function value) would escape along the diameter direction,as shown in Fig.2.To perform a quick search of the entire solution space,we introduce a gender-exchange mechanism to ensure that the optimal firefly in the repulsive neighborhood is male and avoid early slipping into the local optimal solution.

Fig.2 Diagram of repulsive infiuence between firefiies in RFA

Based on the discussions above,the steps involved in the proposed RFA are as follows:

Table1 Steps involved in the repulsive firefly algorithm

Inputs Set the maximal attractiveness β0,repulsive neighborhood radius rrep,disturbance level K,step factor α,convergence threshold ε,and iteration threshold T.Randomly select the firefly set 1 W { }M N j j W +== ,which Step 1X={ }M i i contains male firefly set 1 Y { }N j j = .Y =X = and female firefly set 1 For female firefly Yk(t),search the most attractive firefly 1 1 N M Step 2 and calculate the position of female firefly Yk(t+1) after the iteration:( 1) ( ) ( ( ) ( ))k k k W t X t Y t Y t=( ) { ( )} { ( )} / ( )∈∪=i i j j k Y t Y t W t Y t K + = + - +β α (8)For male firefly Xk(t),search the most attractive firefly 1 1 M N W t X t Y t X t=( ) { ( )} { ( )} / ( )∈∪ and calculate the position of male firefly Xk(t+1) after the iteration:( 1) ( ) ( ( ) ( ))k k k i i j j k =X t X t W t X t K + = + - +β α (9)Determine whether the intersection ( ( 1),) X k rep δ X t r+ ∩ Step 3between repulsive neighborhood of male firefly Xk(t+1) and female firefly set is empty or not.If there exists ( ) ( ( 1),) X l k rep X t X t r∈+∩δ and the target function value F(Xl(t)) of Xl(t) is more optimal,the position of male firefly Xk(t+1) is updated as follows:( 1) ( 1) ( ( 1) ( ))k ′ + ← + + + -γ X t X t X t X t (10)and turn to Step 2; otherwise,directly turn to Step 4.k k l

If the following convergence conditions are satisfied:|max(F( ( 1)) max(F( ( )) | ＆＆arg max(F( ( 1)) arg max(F( ( ))X t X t X t X t k + - ＜+ ==k ε k k k k Step 4 (11)or the condition t≥T is satisfied,output the most optimal firefly; otherwise,determine whether the repulsive neighborhood of male firefly 1{ ( ,)}M i rep i δ X r = has more optimal target value than that of female firefly and then exchange the gender with the most optimal female firefly and return to Step 2.Output The most optimal firefly.

4 Realization of optimal switching device placement based on RFA

The optimal switching device placement task can be treated as a 0-1 planning configuration optimization problem [23].Inspired by Kennedy et al.[24-26],we obtain the binary optimal solution for the proposed RFA as follows.

First,we assume firefly movement vectors,such as β( Z -Yk ),β( Z -Xk ),and γ( X k(t + 1) - X l(t )),shown in (8)-(10),as V = [ v1 ,...,vL].We then employ the Sigmoid function to perform normalization tasks to represent the probability that the switching devices are in a closed state [25,26]:

Based on the comparison of the closure probability P(vi) and random variables,we can realize the binarization of the firefly movement vectors and obtain the firefly movement vector D = [ d1 ,...,d L] of the binary RFA,with the following condition:

Here, θ～U(0,1) denotes the random variable under the uniform distribution U(0,1).Fig.3 shows the implementation of the RFA-based switching device placement optimization for power distribution systems.

5 Experimental simulation and result analysis

To verify the performance of the newly proposed RFA for the optimal switching device placement in a power distribution system,the TS algorithm [15,27],GA [16,28],and PSO algorithm [17,29] were introduced as the control group.

Fig.3 Flowchart of RFA-based switching device placement optimization for power distribution systems

We take the equipment cost of each switching device as 25,000 yuan.We assume the economic life of a switching device as 20 years,the discount rate as 10%,the proportion of operation and maintenance cost in the equipment investment cost as 3%,and the proportion of equipment scrap cost in the equipment investment cost as 5%.The electricity production ratio is set as 9.59 yuan/kWh,the average failure rates of the main feeder line and branch line are 0.1 and 0.01 per kilometer per year,respectively,the average repair times of the main feeder line and branch line are 4 and 10 h,the operation time of a switching device is 20 min,and the minimum power supply reliability rate is 99.98%.

5.1 Case A:Optimal switching device placement in the case of IEEE RBTS-BUS6 dataset

The aforementioned algorithms were employed to optimize the placement of the switching devices in a 10 kV part of the IEEE RBTS-BUS6 system [5].Similar placement optimization results were obtained for this small-scale system,while the average time consumed by the RFA was less than those by the TS algorithm,GA,and PSO algorithm.Table 2 lists the placement optimization results of the switching devices.

Table2 Optimal switching device placement in a 10 kV part of the IEEE RBTS-BUS6 system

Switching device position Benefit/yuan per year 7 0.0097 0.354*104 1.1113*104 1.4653*104 0.4835*104 21 0.0115 0.354*104 1.3577*104 1.7117*104 0.6656*104 29,31 0.0033 0.708*104 1.0100*104 2.5618*104 1.5518*104 ASAI increment/%Switching cost/yuan per year User power shortage cost/yuan per year Total cost/yuan per year

As shown in the optimization results,four switching devices were installed in sections 7,21,29,and 31 in the optimized strategy,and the equipment investment cost increased by approximately 14,000 yuan.With the optimized switching device placement,the ASAI increased by 2%,and the power supply reliability was enhanced with certain economic benefits.

5.2 Case B:Optimal switching device placement for actual feeder lines in a certain area

In this case,we used the four algorithms for the placement optimization of the switching devices,as shown in Fig.4.Table 3 and 4 list the length and load data of the feeder lines,respectively.

Fig.4 Schematic of actual feeder lines

Since the placement optimization results of the switching devices are largely the same,we first compared the power distribution systems before and after the optimization,as listed in Table 5.Table 6 gives the optimal placement comparison of the switching devices in the power distribution system.

Table3 Actual length data of feeder lines

Beginning End Length/km Beginning End Length/km 1 2 0.403 10 0.583 2 3 0.120 14 16 0.400 2 4 0.524 14 17 0.195 4 5 0.480 17 19 0.200 4 6 0.610 19 20 0.200 6 8 0.220 19 21 0.204 6 10 0.263 21 22 0.140

Table4 Actual load data of feeder lines

Beginning End Load/MW Beginning End Load /MW 2 3 192 23 25 243 4 5 201 14 15 658 6 7 134 14 16 201 6 8 201 17 18 383 6 9 128 19 20 639 10 11 201 21 22 255 10 12 639 23 24 255 10 13 361 / / /

Table5 Comparison of power distribution system before and after optimization

Benefit/yuan per year Original / 99.9833 0 65,869 65,869 /Optimized 6-10,10-14,17-19 99.9952 1.0619 22,012 32,631 33.238 State Switching device No.ASAI/% Switching device cost Power shortage cost/yuan per year Total cost/yuan per year

Table6 Comparison of optimal switching device placement in a power distribution system

Algorithm Switching device ASAI/% Total cost/yuan per year Not convergent ratio in 100 trials/(%)TS [6] 6-10,10-14,17-19 99.9952 32,631 15,241(4) 3.61691 1 GA [7] 6-10,10-14,17-19 99.9952 32,631 20,194(5) 2.33782 6 PSO [8] 6-10,10-14,17-19 99.9952 32,631 23,115(5) 5.93548 0 RFA 6-10,10-14,17-19 99.9952 32,631 3,414 0.36446 0 Iteration(generation)Running time/(s)

According to the details in Table 2 and RFA-based optimization results listed in Table 6,if we install switching devices between 6-10,10-14,and 17-19,the equipment investment cost would increase by 10,600 yuan per year,while the power supply reliability would increase by 0.0119%,the user power shortage cost would reduce by 43.8 k,and the total cost would drop by 33.2 k.

The four algorithms could obtain optimal solutions,and the generations of these algorithms,in which the optimal solution was obtained,were basically the same.However,the optimal solution probability of the GA is slightly lower,indicating a higher probability for GA to obtain a local optimal solution.The RFA gave the highest probability and was least time-consuming.

6 Conclusion

To improve the power supply reliability of distribution networks and reduce the power consumption loss of customers,a so-called repulsive firefly algorithm (RFA)-based optimal switching device placement method is proposed.In this method,the influence of territorial repulsion during firefly courtship is considered,thus realizing the interactions of learning and competition among fireflies in the newly proposed RFA.We employed the Sigmoid function to perform normalization tasks to represent the probability that switching devices are in a closed state.Finally,a binary optimal solution to the 0-1 planning configuration optimization problem for switching device placement is obtained using the RFA.

The proposed algorithm was applied to IEEE RBTSBUS6 and an actual feeder line.With optimization,the reliability index of the power distribution system could be improved,and some economic benefits were obtained.The simulation results confirmed the feasibility and efficiency of the proposed method.In future research,the team will explore the application of RFA algorithm in setting value calculation of relay protection and distribution network automation control.

Acknowledgements

This work was supported by the State Grid Science and Technology Project “Research on Technology System and Applications Scenarios of Artificial Intelligence in Power System” (No.SGZJ0000KXJS1800435); Key Technology Project of State Grid Shanghai Municipal Electric Power Company “Research and demonstration of Shanghai power grid reliability analysis platform”; Key Technology Project of China Electric Power Research Institute “Research on setting calculation technology of power grid phase protection based on Artificial Intelligence” (JB83-19-007).