Placement optimization of Multi-Type fault current limiters based on genetic algorithm

0 Introduction

As a large number of distributed generators(DGs)from new energy sources are connected to the power grid in a centralized manner,the short-circuit current level soars rapidly,and the breaking margin of the circuit breakers is also substantially reduced.This causes the short-circuit current to exceed the rated breaking current of the circuit breakers in certain locations,posing a threat to the safe and stable operation of the system.Moreover,a high level of short-circuit current significantly reduces the service life of the circuit breakers.In engineering practice,the installation of a fault current limiter(FCL)on a transmission branch with a large short-circuit current is an effective measure to restrain the short-circuit current.The impedance of the FCL is zero when the system operates normally.When a fault occurs,the impedance of the FCL is imposed on the grid and acts to restrain the short-circuit current level so as to relieve the breaking pressure on the circuit breaker.Various types of FCLs are commercially available,such as the resistive-type,inductive-type,superconducting,fluxlock-type,DC reactor-type,and resonance-type FCLs[1-5].Multiple types of FCLs with different suppressing capabilities and economic costs can be found in a single power system.However,the optimum placement of FCLs is challenging,since the economic benefits after current limitation of FCLs is difficult to quantitatively describe,especially in large grid.

Several studies have investigated the location and capacity allocation of FCLs.The main criterion for the placement of FCLs is that the breaking margins of the various buses should be increased to safe levels.To achieve this objective,various optimization schemes have been proposed[6-31].Most studies have determined the best locations for the FCLs based on the minimization of the number and impedance of the relay protection devices according to the maximum short-circuit level of these devices and the coordination between the devices[6-24].Some studies have also considered the reliability[9,11,12,17,23,25-27],FCL power loss[11,17,18,24,25,28,29],and uncertainty of new energy sources[21,30,31].The impact of superconducting FCL(SFCL)placement on DG expansion considering the relay coordination in power systems was analyzed in reference[22].An SFCL placement method for power system protection based on the minimax regret criterion that considers the uncertainty of the DGs was proposed in reference[21].To quickly obtain the global optimal solution to the placement optimization problem,numerous improved algorithms have been proposed.Enumeration methods have been used to analyze the performance of FCLs in small microgrids[32],while in large-scale grids,various sensitivity index methods[29,33-39]have been used to reduce the search space for optimization.A variation sensitivity factor for the branch parameter was proposed in reference[36],in which the preferred candidates for active FCL installations are chosen based on the reduction in the bus fault currents.

Numerous algorithms have also been developed to increase the convergence speed and solution accuracy.The particle swarm optimization(PSO)algorithm[12,23,25,27,34,40-43]along with the genetic algorithm(GA)[13,33,35,36,38,39]are two of the most commonly used methods in the literature.Other algorithms,such as the elitist gravitational search algorithm(EGSA)[44],gravitational search algorithm(GSA)[31],and hierarchical fuzzy logic decision(HFLD)[42],have also been used to improve the performance.The number,location,and impedance of FCLs in a network were determined in reference[25]considering the reliability,power loss,and economical FCL use as the objective functions through PSO.A multi-objective firework algorithm was implemented in reference[26]to obtain the optimum number,impedance,and locations of FCLs considering the bus fault current difference and reliability.The search space was modified using the effectiveness value to sort out the less effective candidate locations to obtain the solution through GA in reference[38].

Placement optimization is a multi-objective problem.Many studies in the literature have solved this type of problem by converting it to a single-objective problem through the definition of weight coefficients,which are empirical values that significantly affect the optimization results,or by using the Pareto method[27,34,37],which makes it difficult to obtain the global optimal solution.In addition,the circuit breaker loss due to FCL placement has seldom been considered in the literature.Therefore,the placement optimization of multi-type FCLs with multiple objectives considering the circuit breaker loss needs to be studied further.

In contrast to earlier studies,in this study,two objectives with the same dimensional measurement unit,namely,the total capital investment cost of the FCLs and the circuit breaker loss related to short-circuit currents,are considered.The circuit breaker loss model accounts for the service life attenuation of the circuit breakers.The main feature of the proposed method is the use of the circuit breaker loss to quantify the current-limiting effect such that the problem of weight selection in a multi-objective problem is avoided.The GA is improved to determine the optimal type,impedance size,and location of multiple FCLs in largescale grids.

The remainder of this paper is organized as follows.The problem and proposed method are formulated in Section 1.A case study on the IEEE 10-generator 39-bus system in New England is reported in Section 2.Finally,the conclusions are presented in Section 3.

1 Methodology

The optimal planning problem for FCLs and the construction of a mathematical model to optimize the placement of the FCLs on the grid and their types of impedance are described in this section.By selecting the optimal grid position and impedance type of the FCLs,the short-circuit current level of the power system is minimized at the lowest cost while relieving the breaking pressure on the circuit breakers and extending the service life of the switches.

1.1 Problem definition

As described above,the aim of this study is to obtain the optimal location and impedance of the FCLs to achieve the best current-limiting performance at the lowest investment.The optimal solution cannot be obtained easily because it is difficult to define the current-limiting effect,and the simultaneous consideration of the investment cost and current-limiting effect makes the problem a multiobjective one.To overcome these difficulties,the use of the circuit breaker opening loss cost to evaluate the currentlimiting effect is proposed in this study.This enables the conversion of a multi-objective problem into a singletarget optimization problem.Therefore,the objective of the problem contains two components,namely the investment cost and circuit breaker loss,which can be formulated mathematically as

where fis the objective function,ffcl is the total capital investment cost of the FCLs,and fbrk is the circuit breaker loss related to short-circuit currents.

The investment cost of the various types of FCLs is,in general,composed of two parts comprising the costs related to the installed reactances and the costs for installing the FCLs,which can be represented as

where Ntrsm is the number of transmission branches,αis the cost-impedance coefficient representing the cost per-unit impedance for each FCL,βis the installation cost for the FCL,and Niand Zi,which constitute the decision variables,represent the number and impedance of the FCLs installed on the ith branch,respectively.

When a short-circuit failure occurs,the short-circuit current triggers relay protection,and the corresponding circuit breaker acts to break the fault current.This significantly shortens the service life of the breaker.Because the service life of a circuit breaker has an exponential dependence on the breaking current,the circuit breaker loss can be represented as

where Nbrk is the number of buses on which the circuit breaker is installed,γis the normal loss cost of the circuit breaker when it is working below its rated breaking current,δis the current sensitivity coefficient representing the influence of the breaking current on the service life of the FCL,and Ijand Ibrk.jrepresent the short-circuit current and rated breaking current of the jth bus,respectively.Equation(3)implies that fbrk.j,the circuit breaker loss of the jth bus,is γwhen Ij= Ibrk.j,which implies that the loss increases when the short-circuit current exceeds the rated breaking current.When Ij= 0,the loss fbrk.jis γ/eδ,which is always a small number and represents the natural attenuation loss of the FCL.

Because only one FCL is allowed for each branch of the transmission lines in engineering practice and there are only a limited number of FCL impedance types,the decision variables of the model are not continuous.Therefore,Niis defined as an integer variable that takes the values of 0 and 1,where 0 indicates that no FCL is installed,and 1 indicates that one FCL is installed,as shown in(4).Ziis defined as an integer variable with finite discrete values as shown in(5),where tis the number of impedance types of the FCLs.The short-circuit current Ijon each circuit breaker bus should be smaller than the rated breaking current to ensure secure operation of the system,as shown in(6).The corresponding symbols are listed in Table 1.

Table 1 Symbols and their explanations in the objective function

SymbolExplanation αCost-impedance coefficient βInstallation cost of each FCL γNormal loss cost of circuit breaker δCurrent sensitivity coefficient εSafety breaking margin of circuit breaker fObjective function ffclInvestment cost of FCLs fbrkLoss of circuit breaker iIndex of transmission branch jIndex of circuit breaker bus bNumber of branches nNumber of nodes Ybb×Tridiagonal matrix of branch admittance ANode-branch correlation matrix IjShort-circuit current of the jth bus Ibrk.jRated breaking current of the jth bus NtrsmNumber of transmission branches

continue

SymbolExplanation NbrkNumber of circuit breaker buses NiNumber of FCLs on the ith branch ZiImpedance of FCL on the ith branch Znn×Nodal impedance matrix

1.2 Short-circuit fault

As shown in equations(3)and(6),the solution of the short-circuit fault current,which is critical to the objective function,requires the circuit breaker losses to be calculated.The short-circuit fault calculation is performed based on a three-phase ground fault,which is the worst case scenario during operation.Because this fault type is 3-phase balanced,the fault condition of the grid can be separated into a combination of a fault grid and a normal grid.By using the nodal impedance matrix and nodal voltage,the short-circuit fault current can be calculated as

where Ijis the short-circuit current of the jth bus and Ejis the voltage before the fault occurs at the jth bus,which can be set to 1.0 p.u.Zjjis the impedance of the Thevenin’s circuit at the jth bus,which can be obtained from the diagonal entries of the nodal impedance matrix constructed by inverting the nodal admittance matrix:

Here,Zn×nis the n×nimpedance matrix,where nis the number of nodes,Ais the n×bnode-branch correlation matrix where bis the number of branches,and Yb×bis the b×btridiagonal matrix of the admittance of the branches.

If the FCLs are activated after the faults,the impedance matrix changes.The total effect of inserting the FCLs into the system can be considered as the addition of a new branch with admittance ΔYiito the system:

Here,Yiiis the ith tridiagonal element in the tridiagonal matrix Y,and Ziis the impedance of the inserted FCL on the corresponding branch with the same index as the decision variables in Equation(2).Therefore,the short-circuit fault current can be expressed in the vector form

where Iis the vector of the nodal short-circuit fault currents,and diag(·)returns a column vector of the diagonal elements of(·).

1.3 Genetic Algorithm Optimization

The calculation of the nodal short-circuit fault current in Equation(10)is a nonlinear and non-differential operator for the vector of branch admittances,which cannot be easily solved using general deterministic or approximation algorithms.Considering the discrete nature of the decision variables in engineering practice,a GA is used in this study to solve the optimization problem.The steps in the GA are shown in Fig.1.

Fig.1 Optimization flowchart

In the coding step,the decision variables are converted into chromosomes in the form of 0-1 bits.Because the decision variables in group Zicorrespond to Niin the ith transmission branch in Equation(2)and the value of Ziis not used when Ni= 0,it is natural to merge the two decision variables into the same bit area.For example,if there are seven available types of FCLs that can be installed on eight transmission branches,three bits are required to code for each transmission branch,where “000” represents the Ni= 0 configuration in which no FCL is installed,and the other encodings represent the Ni= 1 configuration in which only one type of FCL is installed.Therefore,24 bits are required to code the entire chromosome,as presented in Table 2.

Table 2 Coding examples

BranchNumberTypeCodingDecoding 1 0 Z(*)0000 2 1 Z(1)0010.1 3 1 Z(2)0100.2 4 1 Z(3)0110.3 5 1 Z(4)1000.4 6 1 Z(5)1010.5 7 1 Z(6)1100.6 8 1 Z(7)1110.7 Chromosome000001010011100101110111︷■■■■■■■■■■■■■3bit 24bit Decodings(0,0.1,0.2,0.3,0.4,0.5,0.6,0.7)

In the initialization step,the initial chromosome population is generated.To improve the global convergence performance,the size of the population in the GA should generally be proportional to the number of decision variables,and the population should be as diversified as possible.In this study,the initial population of 10Ntrsm chromosomes was generated randomly.

In the evaluation step,the fitness of each chromosome is determined.The fitness evaluation is closely related to the objectives and optimization restrictions,which are the only information used to decide which chromosomes can mate to generate the next generation of the population.Because the performance of the algorithm is highly sensitive to the fitness evaluation,the objective with the constraints shown in Equation(1)can be solved as a series of unconstrained minimization problems for the chromosomes,which can be represented as

where xrepresents the decoding of the chromosome,Fk(·)is the fitness function in the kth iteration,σkis the penalty coefficient,which increases in each iteration(e.g.,by a factor of 10),H(·)is the Heaviside step function,which has a value of zero for negative arguments and one for positive arguments,and Ij(x)is the short-circuit fault current at the jth bus calculated by(10)after the FCLs determined by xare activated.

In the crossover and mutation step,offspring are generated by exchanging information between a number of mates with high fitness values to improve the convergence speed,and the bits are inverted randomly with a small probability to avoid falling into local minima.

In the convergence step,the fitness of the generations is evaluated and the satisfaction of the convergence conditions is checked.The iterations are continued until one of the two convergence conditions set in this study is satisfied.The first condition that is checked is whether the number of iterations has reached the maximum allowed number.If it has,the algorithm is terminated;otherwise,the number of generations in which the best fitness value remains unchanged(changes below a certain threshold value are considered as unchanged)is accumulated.If the second criterion of whether the number of generations in which the best fitness remains unchanged has reached the maximum allowed value is satisfied,the algorithm is terminated.

2 Results and discussion

The IEEE 10-generator 39-bus system in New England is used as an example in this study to evaluate the performance of the proposed method.

2.1 Network Diagram and Device Information

As shown in the network topology in Fig.2,there are 39 buses and 46 branches in the New England system,among which the 34 branches from branches 1 to 34 are transmission lines with the parameters shown in Table A1,the 12 branches from branches 35 to 46 are equivalent circuits to transformers with the parameters shown in Table A2,and the 10 buses from buses 30 to 39 are connected to generators with the parameters shown in Table A3.

Fig.2 IEEE 10-generator 39-bus system

In this case study,the installation of 10 types of FCLs on the 34 branches from branches 1 to 34 was planned.The FCLs types share the same economic parameters but have different technical parameters,as shown in Table 3.To validate the effectiveness of the proposed algorithm,all the nodes in the case system were set to the unified voltage level of 1.0 p.u.,per-unit values were adopted for all electrical parameters,and the three-phase transmission lines were equivalent to single-circuit transmission lines.The short-circuit currents of buses 30 to 39,which are connected to generators,were not considered because the generators are generally connected to isolated-phase enclosed buses on which circuit breakers cannot be installed.The uniform value of 14 p.u.was adopted for the rated breaking current Ibrk.jin all the buses with a safety breaking margin εof no less than 10%.The other main parameters in the case study are shown in Table 4,where the maximum number of stall generations represents the maximum allowed number of generations in which the best fitness remains unchanged,and the function tolerance represents the value below which the fitness is considered to be unchanged.

Table 3 FCLs in this optimization

Number 1TypeReactance/p.u.ZLB-TY-10-10.1 2 ZLB-TY-20-10.2 3 ZLB-TY-30-10.3 4 ZLB-TY-40-10.4 5 ZLB-TY-50-10.5 6 ZLB-TY-60-10.6 7 ZLB-TY-70-10.7 8 ZLB-TY-80-10.8 9 ZLB-TY-90-10.9 10ZLB-TY-100-11.0

Table 4 Main parameters for the case study

ParameterValueUnit α1 CNY/p.u.β1 CNY γ1 CNY δ1—ε10%Ibrk.j14p.u.Ntrsm34—Nbrk29—Population size340—

continue

ParameterValueUnit Crossover rate0.7—Mutation rate0.05—Max generations200—Max stall generations100—Function tolerance10-6—

2.2 Method Comparison

For comparison,the maximum short-circuit fault currents of each bus and their current margins were calculated for the blank group,in which no FCLs were installed,as well as the general multi-objective and single-objective optimization methods which are used as the standard group.As for the general single-objective optimization method,only the investment for the FCLs was considered without accounting for the circuit breaker loss;as for the multi-objective method,the investment for the FCLs and their current limiting effect were combined using empirical weights.The objectives of the two general methods are as follows

where fsingle and fmulti are the objective of the single-and multiobjective optimization method respectively,Ioriginal.jis the original short-circuit current with no FCLs installed,λis the empirical weight of multi-objective which is 1.0 in this example.

Table 5 FCLs in this optimization

Bus Fault current Ij/p.u.and breaking margin εj/%No FCLs Proposed Method Single--objective Multi--objective 115.36 -9.71 5.70 59.31 10.07 28.09 6.72 51.98 214.93 -6.67 7.20 48.54 11.70 16.41 8.31 40.64 314.44 -3.15 7.16 48.86 11.51 17.80 8.22 41.32 414.17 -1.24 6.96 50.29 11.23 19.76 7.96 43.15 514.20 -1.45 6.79 51.53 11.08 20.89 7.76 44.59 614.18 -1.26 6.79 51.48 11.07 20.95 7.76 44.56 713.77 1.636.55 53.25 10.68 23.71 7.47 46.64 814.05 -0.34 6.53 53.35 10.77 23.05 7.47 46.66 915.08 -7.72 5.38 61.60 9.68 30.86 6.17 55.91 1013.59 2.956.83 51.23 10.84 22.54 7.77 44.53

continue

Bus Fault current Ij/p.u.and breaking margin εj/%No FCLs Proposed Method Single--objective Multi--objective 1113.66 2.426.79 51.52 10.85 22.53 7.73 44.80 1210.55 24.66 5.92 57.74 8.79 37.21 6.62 52.69 1313.52 3.416.83 51.25 10.82 22.71 7.76 44.54 1413.85 1.076.96 50.32 11.08 20.89 7.93 43.37 1513.40 4.267.10 49.28 10.97 21.67 8.05 42.50 1613.96 0.317.38 47.26 11.42 18.46 8.38 40.14 1713.78 1.577.18 48.74 11.20 19.99 8.17 41.64 1813.65 2.517.07 49.53 11.07 20.95 8.05 42.48 1911.91 14.91 7.04 49.72 10.15 27.47 7.85 43.93 2011.51 17.80 7.27 48.09 10.04 28.26 8.01 42.76 2112.57 10.24 7.11 49.21 10.54 24.69 7.99 42.95 2212.20 12.89 7.15 48.92 10.37 25.95 7.99 42.96 2312.03 14.09 7.11 49.20 10.26 26.75 7.93 43.35 2413.12 6.277.18 48.69 10.87 22.32 8.11 42.06 2513.76 1.727.01 49.96 11.04 21.14 8.02 42.73 2611.73 16.23 6.42 54.11 9.71 30.65 7.27 48.10 2712.16 13.15 6.61 52.77 10.05 28.25 7.48 46.54 288.82 36.98 5.48 60.88 7.65 45.38 6.07 56.67 298.86 36.73 5.53 60.47 7.69 45.04 6.12 56.27 min8.82 -9.71 5.38 47.26 7.65 16.41 6.07 40.14 max 15.36 36.98 7.38 61.60 11.70 45.38 8.38 56.67 avg 13.06 6.706.72 51.97 10.40 25.32 7.63 45.53 std1.63 11.62 0.58 4.12 0.99 7.11 0.664.70

The current-limiting effects of the FCLs in these groups are compared with those optimized by the proposed method,in which both the circuit breaker loss and FCL investments are considered,in Table 5.A comparison graph is also shown in Fig.3.The detailed installation plans are listed in Table 6.

As shown in Table 5,when no FCLs were installed,the average value and standard deviation of the breaking margins were less than 10% and 11.62,respectively.This implies that the short-circuit currents significantly exceeded the interrupting capabilities of the breakers in some of the buses,and to a dangerous extent in a number of these buses.The breaking margins of eight buses(bus numbers 1-6,8,and 9)were negative,which implies that the shortcircuit currents exceeded the breaking currents of the circuit breakers on these buses(triangle markers above the dashed line in Fig.3).In addition,there were 11 buses(bus numbers 7,10,11,13-18,24,and 25)in which the breaking margins were less than 10%,which is the safety breaking margin ε(triangle markers above the dash-dotted line in Fig.3).Buses 1,9,and 2 were the three most dangerous nodes in which the breaking margins were less than -5%.All three buses are near the empirically installed FCLs.

Fig.3 Comparison of current-limiting effects for different methods

Comparing the current-limiting effects in the four curves shown in Fig.3,the curves for the proposed and general methods were all below the 10% breaking limit line,which implies that the fault current was effectively limited in all methods.The peak at the minimum margin of 16.41% for the general single-objective method was substantially close to the 10% breaking margin line.Thus,the constraints were met with a fitness value of 2.20 at nearly the lowest investment cost of FCLs when the circuit breaker loss of 27.29 CNY was neglected,as shown in Table 6.When the investment for the FCLs and their current limiting effect were combined using empirical weights,the fault currents were curbed significantly in the solution from the general multi-objective method.The average breaking margin of 45.53% was significantly higher than the breaking margin of 10%,and the dispersion of the fault currents was significantly low because of the smaller standard deviation of 4.70 for the breaking margins compared to that when no FCLs were installed.When the circuit breaker loss was considered,the fault currents were very similar to the general multi-objective method.Its average breaking margin of 51.97% was slightly higher,and the dispersion of the fault currents was slightly lower with a smaller standard deviation of 4.12 for the breaking margins.Thus,the probability of breaker damage was significantly reduced and the empirical weight assignment of the general multiobjective method was overcome.

The placements and impedance of the FCLs between the two groups are compared in Table 6.The FCLs were installed on branches 2 and 15 in both solutions.The only difference is that the impedance of 0.8 p.u.in the proposed method was reduced to 0.1 p.u.in the single-objective method which is the smallest available FCL reactance and 0.4 p.u.in the multi-objective method shown in Table 3.All the branches were linked to bus 39,which was connected to a generator with a large capacitance and small transient reactance,as shown in Table 5,and all the branches were also linked to the two most dangerous nodes(buses 1 and 9)mentioned above,which is consistent with the empirical findings.It is obvious that the circuit breaker loss fbrk in the proposed method was reduced by approximately 5.31 CNY and 1.15 CNY compared to that of the single- and multiobjective method with a small increase of approximately 1.4 CNY and 0.7 CNY in the FCL cost ffcl.This implies that the proposed method provides a better solution than that of the general method when the circuit breaker loss is considered.

Table 6 Detailed optimized installation plans

Method SolutionObjective/CNY Branch BusZiFffclfbrk Propsed 21-390.8 20.863.6017.26 159-390.8 Single-objective 21-390.1 24.772.2022.57 159-390.1 Multi-objective 21-390.4 21.312.9018.41 159-390.5 No FCLs——27.290.0027.29

2.3 Sensitivity Analysis

To evaluate the performance of the proposed algorithm,the fitness values over successive iterations of the GA were recorded,and the sensitivities of the four main parameters of the objective were studied.The variation in the best and mean fitness values with the GA iterations is shown in Fig.4.The sensitivity analysis of the optimization is presented in Table 7.

As depicted in Fig.4,approximately 171 generations of a population containing 340 individuals,each with 34 variables,were generated during the execution of the algorithm.The algorithm was terminated under the second convergence criterion when the number of iterations in which the change in the best fitness was below the tolerance reached the maximum allowed value.This implies that the best individual was generated at the 71st generation.The curve in Fig.4 demonstrates the excellent convergence of the algorithm.

Fig.4 Best and mean fitness values for each iteration

Table 7 presents the solutions and objectives of the optimization when the four main parameters were each increased by 1.2 times.The increase in α,which affects the per-unit impedance cost of the FCLs,caused the impedance of the FCL installed on branch 2 to be reduced by 0.1 p.u.to reduce the impedance cost.In this case,the investment cost of the FCLs was increased by 0.2 CNY,circuit breaker loss by 0.11 CNY,and objective by 0.31 CNY.Increasing β,which is the installation cost for each FCL,did not change the solution,which implies that this parameter has a small impact on the solutions.In this case,the investment cost of the FCLs increased by 0.4 CNY,circuit breaker loss remained unchanged,and objective was increased by 0.4 CNY.Increasing γ,which is the normal loss cost of the circuit breakers,caused an additional FCL with the impedance of 0.6 p.u.to be installed on branch 22 and the impedance of the FCL installed on branch 15 to be increased by 0.1 p.u.This is because the increased γsignificantly increased the normal loss of the circuit breakers,which resulted in more FCLs being required to curb the fault currents.In this case,the investment cost of the FCLs increased by 1.7 CNY,circuit breaker loss by 1.63 CNY,and objective by 3.33 CNY.Increasing δ,which is the loss attenuation coefficient of the circuit breakers for the breaking current,caused the impedance of the FCL installed on branch 15 to be increased by 0.1 p.u.Note that the circuit breaker loss was decreased by 1.81 CNY with a small increase of 0.1 CNY in the FCL investment,because the increased δdecreased the breaker loss under a low breaking current,which enhanced the affordability of the FCLs.

Table A1 The transmission data

Branch BusParameters/p.u.FromTorxb1120.00350.04110.6987 21390.00100.02500.7500 3230.00130.01510.2572

continue

Branch BusParameters/p.u.FromTorxb42250.00700.00860.1460 5340.00130.02130.2214 63180.00110.01330.2138 7450.00080.01280.1342 84140.00080.01290.1382 9560.00020.00260.0434 10580.00080.01120.1476 11670.00060.00920.1130 126110.00070.00820.1389 13780.00040.00460.0780 14890.00230.03630.3804 159390.00100.02501.2000 1610110.00040.00430.0729 1710130.00040.00430.0729 1813140.00090.01010.1723 1914150.00180.02170.3660 2015160.00090.00940.1710 2116170.00070.00890.1342 2216190.00160.01950.3040 2316210.00080.01350.2548 2416240.00030.00590.0680 2517180.00070.00820.1319 2617270.00130.01730.3216 2721220.00080.01400.2565 2822230.00060.00960.1846 2923240.00220.03500.3610 3025260.00320.03230.5130 3126270.00140.01470.2396 3226280.00430.04740.7802 3326290.00570.06251.0290 3428290.00140.01510.2490

Table A2 The transformer data

Branch BusParameters/p.u.FromTorxb3512110.00160.04351.006 3612130.00160.04351.006 376310.00000.02501.070 3810320.00000.02001.070 3919330.00070.01421.070 4020340.00090.01801.009 4122350.00000.01431.025 4223360.00050.02721.000 4325370.00060.02321.025 442300.00000.01811.025 4529380.00080.01561.025 4619200.00070.01381.060

Table A3 The generator data

GeneratorBus Parameters/p.u.xdx'dxqx'q 1 300.2000.0600.2000.060 2 312.2660.6972.2660.697 3 322.4950.5312.3700.531 4 332.6200.4362.5800.436 5 346.7001.3206.2001.320 6 352.5400.5002.4100.500 7 362.9500.4902.9200.490 8 372.9000.5702.8000.570 9 382.1060.5702.0500.570 10391.0000.3100.6900.310

Table 7 Results when each parameter is increased by 1.2 times

Parameter SolutionObjective/CNY BranchBusZiFffclfbrk α21-390.7 21.173.8017.37 159-390.8 β21-390.8 21.264.0017.26 159-390.8 γ21-390.8 24.195.3018.89 159-390.9 2216-190.6 δ21-390.8 19.153.7015.45 159-390.9

This simulation considered a typical IEEE system,which is not very common in industrial construction.Only a new construction was considered.However,network expansion is common in industrial applications.Therefore,the inclusion of some unchangeable changes in the chromosome coding should also be considered.This will be investigated in future studies.

3 Conclusion

In this study,the FCL location optimization problem was studied using a GA with limited type choices.A new objective function that considers the circuit breaker loss for the optimization was proposed along with a new coding method based on the type of FCLs rather than the impedance value.This led to a reduction in the length of the chromosome and the rapid convergence of the optimization.The case study of an IEEE 10-generator 39-bus system was simulated.Optimization results were quickly achieved in the simulation.Thus,this method is proven to be effective for the optimum placement problem and has been submitted for a patent in China.

Appendix A

Acknowledgements

This work was supported by State Grid Science and Technology Projects(SGTYHT/17-JS-199)and National Natural Science Foundation of China(51577163).

Declaration of Competing Interest

We declare that we have no conflict of interest.