CN106446467A - Optimal configuration method of fault current limiter based on adaptive particle swarm algorithm - Google Patents

Abstract

The invention discloses an optimal configuration method of a fault current limiter based on an adaptive particle swarm algorithm. The method includes the following steps: 1, calculating short circuit current of each branch circuit, and selecting a standard-exceeding node; 2, building a mathematical model of an FCL optimization configuration problem, 3, conducting optimization configuration on the FCL by the adaptive particle swarm algorithm. According to the optimal configuration method, a more accurate optimal computation can be conducted on installation impedance of the FCL, the installation impedance value is optimized on the premise that fault current is limited, so that a reference is provided for the actual installation of the FCL in an electric power system.

Description

Optimal Configuration Method based on the fault current limiter of APSO algorithm

Technical field

The present invention relates to Power System Analysis technical field, more specifically a kind of optimization of fault current limiter are joined The method that puts.

Background technology

The developing rapidly of modern power systems, to increase unit capacity, improve electric pressure, expand electrical network scale and big Power System Interconnection is main characteristics, and accordingly, system short-circuit levels of current is also constantly increasing.Fault current limiter is used as FACTS A member of family, is widely used in recent years in terms of fault current limitation.Due to typically the more commonly used series resonance-type fault Known to the structure people of current limiter are, but during practical application, when the current-limiting impedance difference of fault current limiter is installed, to circuit Or even the short circuit current restriction effect of whole electrical network all has a significant impact.Therefore optimum installation is found using particle swarm optimization algorithm Impedance is the effective way for solving the problem.

The impedance selection mode of fault current limiter can be divided into two classes at present：The first is distributing rationally based on Sensitivity Method Method, mainly by drawing sensitivity of the exceeded self-impedance of short circuit current to these branch impedance parameters, according to sensitivity Size, therefrom choose a plurality of candidate's branch road for installing FCL, then calculating be optimized to candidate's branch road, realize the installation of SFCL Position, the distributing rationally of quantity and resistance value；It is for second distributing rationally based on PSO algorithm, mainly by basic PSO algorithm is being optimized configuration to the installation impedance of fault current limiter；It is adapted to choose optimum based on the method for sensitivity Installation site, the algorithm types of introducing are many, more complicated；Optimal Configuration Method based on basic PSO algorithm is fairly simple, but In particle optimization process, particle is during optimal particle is followed, and as particle becomes closer to optimal particle, speed is increasingly Slowly, strong homoplasy is shown, particle may be absorbed in local optimum quickly, cause Premature Convergence, the optimum impedance for obtaining Sequence may not be optimal solution.

Content of the invention

The present invention is for avoiding the weak point existing for above-mentioned prior art, provides a kind of based on the calculation of self adaptation population The Optimal Configuration Method of the fault current limiter of method, counts to can be compared accurate optimization to the installation impedance of FCL Calculate, optimize on the premise of fault current limiting and resistance value is installed, so as to offer reference during power system actual installation FCL.

In order to achieve the above object, the technical solution adopted in the present invention is：

A kind of Optimal Configuration Method of the fault current limiter based on APSO algorithm of the present invention, the fault Demand limiter is series resonant-type fault current limiter, and produces current-limiting impedance value X when fault is short-circuited, described short Road fault type is three-phase symmetrical short trouble；It is characterized in, the Optimal Configuration Method is carried out as follows：

Short circuit current on step 1, the circuit for calculating the fault current limiter to be installed and its each branch road, and choose The exceeded node of short circuit current：

Step 1.1, the circuit in normal state fault current limiter to be installed being located carry out Load flow calculation, Obtain each node voltage and each branch current；

Step 1.2, the bus admittance matrix of fault current limiter place circuit to be installed is obtained, and calculate and wait to pacify Fill the fault current limiter place circuit and its perimeter circuit on all nodes three short circuit current, so as to find out n The exceeded node of individual short circuit current；

Step 1.3, obtain when being short-circuited fault the short circuit current of the branch road connected by the exceeded node of j-th short circuit current I_j, and by j-th short circuit current I_jAction message electric current I with the fault current limiter_brMake comparisons, choose and meet I_j≥λI_br The c bar branch road connected by the exceeded node of all short circuit currents of condition；Choose from the c bar branch road for meeting condition to meet to install and want The N bar branch road that asks the branch road as the fault current limiter to be installed；λ represents coefficient of reliability；1≤j≤n；

Step 2 rationally, founding mathematical models are distributed to the fault current limiter：

Step 2.1, object function f (x) that the fault current limiter distributes rationally is set it is：

In formula (1), Z_FCLI () represents the resistance value of i-th fault current limiter, N represents the installation fault current limit The number of device processed；

Arranging short circuit current constraints is：

I_j≤I_{j max}(2)

In formula (2), I_{j max}Represent current limliting desired value；

Arranging node voltage constraints is：

V_{r min}≤V_r≤V_{r max}(3)

In formula (3), V_rRepresent the node voltage of r-th node；V_{r min}And V_{r max}Respectively r-th node voltage V_rUpper Lower limit；n_bAll node numbers on the circuit being located for the fault current limiter to be installed and its perimeter circuit；1≤r≤ n_b；

Step 2.2, each constraints is processed：

Voltage out-of-limit value V of all nodes is set_limFor：

Arrange all nodes short circuit current get over limit value be:

The population trip of the optimal configuration algorithm of the fault current limiter is obtained using the penalty shown in formula (6) Fitness function is：

Min F (x)=min f (x)+K (V_lim+I_lim) (6)

In formula (6), K is penalty factor；

Step 3, configuration is optimized to the fault current limiter by APSO algorithm：

Step 3.1, current-limiting impedance value X is regarded as one group of discrete variable, and b position is carried out to one group of discrete variable Binary coding；So that each branch road to be installed all has 2^b- a kind of impedance is selected, so as to define the dimension D for installing impedance population =N (b+1)；

Step 3.2, random initializtion install impedance population, and randomly generate N number of impedance particle of installing for { X₁,X₂,…, X_t,…,X_N}；X_tRepresent t-th installation impedance particle；1≤t≤N；

Studying factors are set as c₁And c₂, minimum crossover probability be_min, population scale be that M, Particles Moving velocity interval are [V_min,V_max]=[- 1,1]；Particle position scope is [X_min,X_max]=[0,5], maximum iteration time is G_max, k represents current Iterationses；

Step 3.3, fitness evaluation：

Firstization iterationses k=1, with t-th installation impedance particle X of kth generation_tItself is as kth t-th individuality of generation most Excellent positionThe initial adaptive value of each installation impedance particle in colony is calculated, and obtains global optimum of the kth for population Position gbest_k；

Step 3.4, by formula (7) update kth generation t-th installation impedance particle X_tSpeedObtain t-th of+1 generation of kth Impedance particle X is installed_tSpeed

In formula (7), w represents that speed updates coefficient；For t-th installation impedance particle X of kth generation_tPosition, r₁And r₂For [0,1] random number in interval；

T-th installation impedance particle X of kth generation is updated using formula (8)_tPositionObtain the t-th installation resistance of+1 generation of kth Anti- particle X_tPosition

In formula (8), rand () is for being evenly distributed on the random function in [0,1] interval；For position judgment letter Number, and have：

Step 3.5, the probability P according to formula (10) calculating cross and variation operation, if meet P ＞ R_min, then 3.6 are gone to step, 3.9 are otherwise gone to step；

P=μ+Re σ (10)

In formula (10), μ and σ is the regulation parameter of aberration rate, and Re is global optimum in G_maxIn secondary iterative process continuously not Update or update unconspicuous algebraically；

Step 3.6, in whole population each install impedance particle cross and variation operation is carried out according to condition；

Step 3.6.1, using formula (11) obtain kth generation t-th installation impedance particle X_tPositionWith the kth generation overall situation most Excellent position gbest_kBetween Euclidean distance

Distance threshold Δ φ is obtained using formula (12)：

Δ φ=(1-k/G_max)^m×(X_max-X_min) (12)

In formula (12), m is regulation parameter；

Step 3.6.2, judgementWhether set up, if so, then using formula (13) to kth t-th impedance particle of generation X_tPositionCrossover operation is carried out, the kth t-th impedance grain X of generation after being intersected_tPosition

In formula (13),Represent the position of kth a-th impedance particle of generation；Represent a-th resistance of kth generation after intersecting The position of anti-particle；E is the random number in (0,1) interval；1≤a≤N；And a ≠ t；

Calculate kth t-th impedance particle X of generation_tAdaptive value and intersect after kth generation t-th impedance particle X_tAdaptation Value, and the position of impedance particle corresponding to adaptive optimal control value is chosen as the position of kth t-th impedance particle of generation, it is designated as

Step 3.6.3, judgementWhether setting up, such as sets up, then mutation operation is carried out using formula (14), become Kth t-th impedance particle X of generation after different_tPosition

In formula (14),Represent the kth a-th impedance particle X of generation after variation_aPosition；α is the weights of variation；

Calculate kth t-th impedance particle X of generation_tAdaptive value and variation after kth generation t-th impedance particle X_tAdaptive value, And the position of impedance particle corresponding to adaptive optimal control value is chosen as kth t-th impedance particle X of generation_tPosition, be designated as

Step 3.7, by kth for global optimum position gbest_kRespectively to the direction movement of the bound of particle position scope One small step delta, mobile number of times is the selection in new optimal particle group q, so as to obtain new optimal particle group of the scale for C Adaptive value highest is installed the position of impedance particle and replaces global optimum position gbest_k, so as to obtain new global optimum's particle position Put gbest_k′；

Step 3.8, calculating complete all fitness for installing impedance particles in intersection and the population after mutation operation, are used in combination The global optimum position gbest_kTo replace the position of the worst installation impedance particle of adaptive value；So as to complete kth for population Renewal；

Step 3.9, k+1 is assigned to k, and 3.4 order of return to step is executed, until k=G_maxTill；So as to obtain most Excellent installation impedance particleWith the optimum resistance value that installs corresponding to impedance particle Installation impedance as the fault current limiter so that install the three-phase on the c bar branch road after N number of fault current limiter Short circuit current can meet short circuit current constraints, all node voltages can meet node voltage on the perimeter circuit of c bar branch road Constraints, and object function f (x) minimum.

Compared with the prior art, beneficial effects of the present invention are embodied in：

1st, the present invention is optimized using improved APSO algorithm, with avoiding Premature Convergence, improve grain Son optimizes the advantage of quality, installs the optimal solution of impedance sequence so as to rapidly and accurately obtain fault current limiter, and fault is limited Stream device (FCL) has great importance in the restriction of bulk power grid short circuit current.

2nd, the present invention need to only refresh admittance square using the method for bus admittance matrix calculating short circuit current in fault Battle array, it is to avoid the complicated electric power networks of analysis.

3rd, the present invention is by using the addition of penalty so that the voltage of each node and branch current are unlikely to exceed Normal range and affect power supply quality.

4th, the present invention is optimized just for the impedance sequence of N platform fault current limiter, it is to avoid while optimizing multiple targets The low shortcoming of the time length that causes, precision.

5th, the present invention is not continuously updated or updates unconspicuous algebraically, adaptively according to convergence of algorithm situation Determine global adaptation probability, the differential evolution particle cluster algorithm after improvement can have stronger global optimization ability in the early stage, Optimal speed is ensure that, in the later stage, be there is stronger local optimum ability, the optimization quality of algorithm is improve, is enhanced algorithm and exist Global optimization ability and avoid balance between local optimum.

Description of the drawings

Fig. 1 is the structure chart of series resonance-type fault current limiter involved in the present invention；

Fig. 2 a is the equivalent circuit diagram that fault current limiter involved in the present invention is normally run in circuit；

Fig. 2 b is the equivalent circuit diagram that fault current limiter involved in the present invention puts into limited current state in short circuit；

Fig. 3 is that FCL involved in the present invention installs impedance optimized algorithm flow chart.

Specific embodiment

In the present embodiment, fault current limiter structure is as shown in figure 1, put into current limliting operation after normal operating condition and action As shown in Figure 2 a and 2 b, before input, electric capacity and inductance form series resonance to the equivalent circuit diagram of state, and equiva lent impedance is several Being zero, after tapping into circuit, only having current-limiting inductance to connect, note induction reactance value is that X, major parameter is the current limliting that system is linked into during fault Impedance X.

As shown in figure 3, a kind of Optimal Configuration Method of the fault current limiter based on APSO algorithm, can The optimal solution for installing impedance sequence rapidly and accurately being obtained, current limliting precision is improved so as to reach, saves the purpose of current limliting cost.Tool Saying for body is to carry out as follows：

Short circuit current on step 1, the circuit for calculating the fault current limiter to be installed and its each branch road, and choose The exceeded node of short circuit current：

Step 1.1, the circuit in normal state fault current limiter to be installed being located carry out Load flow calculation, Obtain each node voltage and each branch current；

Step 1.2, the bus admittance matrix of fault current limiter place circuit to be installed is obtained, and calculate and wait to pacify Fill the fault current limiter place circuit and its perimeter circuit on all nodes three short circuit current, so as to find out n The exceeded node of individual short circuit current；

Complicated Three-phase Power Systems short circuit current is generally calculated using computer, using utilizing node in the present invention The method of admittance matrix, is initially formed bus admittance matrix, according to definition obtain in network the self-impedance of each node and node it Between mutual impedance, and then the starting time transient current of node and the branch current of each branch road when obtaining short circuit.For putting in FCL Calculation of short-circuit current afterwards equally adopts the method, it is assumed that mutual impedance itself is Z_ijI, j node between put into current-limiting impedance be Z_fFCL, the method that now only need to utilize equivalent circuit, be equivalent between i, j node an impedance in parallel be_fBranch road, Oneself, the transadmittance of concept transfer i, j are only needed, it is to avoid repeat to set up admittance matrix.

Step 1.3, obtain when being short-circuited fault the short circuit current of the branch road connected by the exceeded node of j-th short circuit current I_j, and by j-th short circuit current I_jAction message electric current I with the fault current limiter_brMake comparisons, choose and meet I_j≥λI_br The c bar branch road connected by the exceeded node of all short circuit currents of condition；Choose from the c bar branch road for meeting condition to meet to install and want The N bar branch road that asks the branch road as the fault current limiter to be installed；λ represents coefficient of reliability；λ ∈ [1.1～ 1.3]；1≤j≤n；

Here it is to formulate the quantity and position for installing current limiter according to the actual load situation of grid nodes.

Step 2 rationally, founding mathematical models are distributed to the fault current limiter：

Step 2.1, object function f (x) that the fault current limiter distributes rationally is set it is：

In formula (1), Z_FCLI () represents the resistance value of i-th fault current limiter, N represents the installation fault current limit The number of device processed；

Arranging short circuit current constraints is：

I_j≤I_{j max}(2)

In formula (2), I_{j max}Represent current limliting desired value；

Arranging node voltage constraints is：

V_{r min}≤V_r≤V_{r max}(3)

In formula (3), V_rRepresent the node voltage of r-th node；V_{r min}And V_{r max}Respectively r-th node voltage V_rUpper Lower limit；n_bAll node numbers on the circuit being located for the fault current limiter to be installed and its perimeter circuit；1≤r≤ n_b；

In actual motion, as the optimization to installing resistance value is only from from the point of view of limiting short-circuit current, can The node voltage in electrical network or line current level can be made to be in an abnormal state of affairs.If not adding certain constraint Condition, may be negatively affected to line powering.

Step 2.2, each constraints is processed：

Voltage out-of-limit value V of all nodes is set_limFor：

Arrange all nodes short circuit current get over limit value be:

The population trip of the optimal configuration algorithm of the fault current limiter is obtained using the penalty shown in formula (6) Fitness function is：

Min F (x)=min f (x)+K (V_lim+I_lim) (6)

In formula (6), K is penalty factor；

But if directly to node voltage V_rWith line current I_jConstrain plus numerical value, not only increase Load flow calculation Amount of calculation, and easily make result of calculation unreasonable, so using the concept of the penalty function inside algorithm, by each constraints Value processed so as to become functional form, build a specific population trip fitness function with restrictive condition.

Step 3, configuration is optimized to the fault current limiter by APSO algorithm：

Step 3.1, current-limiting impedance value X is regarded as one group of discrete variable, and b position is carried out to one group of discrete variable Binary coding；So that each branch road to be installed all has 2^b- a kind of impedance is selected, so as to define the dimension D for installing impedance population =N (b+1)；

Impedance particle is installed binary coding is adopted, number of units is such as installed and is 2, First installation impedance-encoded 1,1, 0,1,0 }, second is that { 0,0,0,1,0 }, then representing has 2 per platform FCL⁵- 1 grade of impedance, and this two install resistance values be respectively 26 Ω and 2 Ω.

Step 3.3, fitness evaluation：

Firstization iterationses k=1, with t-th installation impedance particle X of kth generation_tItself is as kth t-th individuality of generation most Excellent positionThe initial adaptive value of each installation impedance particle in colony is calculated, and obtains global optimum of the kth for population Position gbest_k；

Step 3.4, by formula (7) update kth generation t-th installation impedance particle X_tSpeedObtain t-th of+1 generation of kth Impedance particle X is installed_tSpeed

In formula (7), w represents that speed updates coefficient；For t-th installation impedance particle X of kth generation_tPosition, r₁And r₂For [0,1] random number in interval；

For t-th installation impedance particle X of kth generation_tPosition, r₁And r₂For the random number in [0,1] interval；

T-th installation impedance particle X of kth generation is updated using formula (8)_tPositionObtain the t-th installation resistance of+1 generation of kth Anti- particle X_tPosition

In formula (8), rand () is for being evenly distributed on the random function in [0,1] interval；For position judgment letter Number, and have：

Similar with fundamental particle colony optimization algorithm, the initial value for installing impedance particle is fixed tentatively as individual optimum, by adapting to Spend appraisement system iteration is updated, reach optimization purpose, accelerated factor c₁Optimal particle is flown to for adjusting particle itself Step-length, accelerated factor c₂Flight step-length of the particle to global optimum position can be adjusted, more big then step-length is bigger, takes c₁=c₂, Span is between 0～4；

Step 3.5, the probability P according to formula (10) calculating cross and variation operation, if meet P ＞ R_min, then 3.6 are gone to step, 3.9 are otherwise gone to step；

P=μ+Re σ (10)

In formula (10), μ and σ is the regulation parameter of aberration rate, and Re is global optimum in G_maxIn secondary iterative process continuously not Update or update unconspicuous algebraically；

If convergence in population speed is stagnated, or continuously some generations do not update, Re value is by accumulative increase, then the tune to population Section probability is then increased.

Step 3.6, in whole population each install impedance particle cross and variation operation is carried out according to condition；

Step 3.6.1, using formula (11) obtain kth generation t-th installation impedance particle X_tPositionWith the kth generation overall situation most Excellent position gbest_kBetween Euclidean distance

Distance threshold Δ φ is obtained using formula (12)：

Δ φ=(1-k/G_max)^m×(X_max-X_min) (12)

In formula (12), m is regulation parameter；

Step 3.6.2, judgementWhether set up, if so, then using formula (13) to kth t-th impedance particle of generation X_tPositionCrossover operation is carried out, the kth t-th impedance grain X of generation after being intersected_tPosition

In formula (13),Represent the position of kth a-th impedance particle of generation；Represent a-th resistance of kth generation after intersecting The position of anti-particle；E is the random number in (0,1) interval；1≤a≤N；And a ≠ t；

Calculate kth t-th impedance particle X of generation_tAdaptive value and intersect after kth generation t-th impedance particle X_tAdaptation Value, and the position of impedance particle corresponding to adaptive optimal control value is chosen as the position of kth t-th impedance particle of generation, it is designated as

Step 3.6.3, judgementWhether setting up, such as sets up, then mutation operation is carried out using formula (14), become Kth t-th impedance particle X of generation after different_tPosition

In formula (14),Represent the kth a-th impedance particle X of generation after variation_aPosition；α is the weights of variation；

Calculate kth t-th impedance particle X of generation_tAdaptive value and variation after kth generation t-th impedance particle X_tAdaptive value, And the position of impedance particle corresponding to adaptive optimal control value is chosen as kth t-th impedance particle X of generation_tPosition, be designated as

This example according to the distance of each particle and global optimum's particle, to assemble in population serious particle introduce intersect and Mutation operator, strengthens the mobility of particle, it is to avoid Premature Convergence or be absorbed in local optimum in undesirable region.

Step 3.7, by kth for global optimum position gbest_kRespectively to the direction movement of the bound of particle position scope One small step delta, mobile number of times is the selection in new optimal particle group q, so as to obtain new optimal particle group of the scale for C Adaptive value highest is installed the position of impedance particle and replaces global optimum position gbest_k, so as to obtain new global optimum's particle position Put gbest_k′；

Step 3.8, calculating complete all fitness for installing impedance particles in intersection and the population after mutation operation, are used in combination The global optimum position gbest_kTo replace the position of the worst installation impedance particle of adaptive value；So as to complete kth for population Renewal；

Step 3.9, k+1 is assigned to k, and 3.4 order of return to step is executed, until k=G_maxTill；So as to obtain most Excellent installation impedance particleWith the optimum resistance value that installs corresponding to impedance particle Installation impedance as the fault current limiter so that install the three-phase on the c bar branch road after N number of fault current limiter Short circuit current can meet short circuit current constraints, all node voltages can meet node voltage on the perimeter circuit of c bar branch road Constraints, and object function f (x) minimum.

Step 4, change optimized algorithm relevant parameter carry out multiple optimizing,

Step 4.1 changes accelerated factor c₁, c₂, the value of adaptive probability parameter μ and σ, meter is optimized again according to step 3 Calculate, new one group resistance value obtained,

To in test keep μ and σ value will in an order of magnitude,

Step 4.2, the new one group of particle to obtaining carry out fitness evaluation, the particle fitness ratio for producing with upper step Relatively, between choose optimal solution.

Claims (1)

1. a kind of Optimal Configuration Method of the fault current limiter based on APSO algorithm, the fault current limitation Device is series resonant-type fault current limiter, and produces current-limiting impedance value X, the short trouble class when fault is short-circuited Type is three-phase symmetrical short trouble；It is characterized in that, the Optimal Configuration Method is carried out as follows：

Short circuit current on step 1, the circuit for calculating the fault current limiter to be installed and its each branch road, and choose short circuit The exceeded node of electric current：

Step 1.1, the circuit in normal state fault current limiter to be installed being located carry out Load flow calculation, obtain Each node voltage and each branch current；

Step 1.2, the bus admittance matrix of acquisition fault current limiter place circuit to be installed, and calculate institute to be installed The circuit at fault current limiter place and its three short circuit current of all nodes on perimeter circuit is stated, short so as to find out n The exceeded node of road electric current；

Step 1.3, obtain when being short-circuited fault the short circuit current I of the branch road connected by the exceeded node of j-th short circuit current_j, and By j-th short circuit current I_jAction message electric current I with the fault current limiter_brMake comparisons, choose and meet I_j≥λI_brCondition The c bar branch road connected by the exceeded node of all short circuit currents；The N for meeting installation requirement is chosen from the c bar branch road for meeting condition Bar branch road the branch road as the fault current limiter to be installed；λ represents coefficient of reliability；1≤j≤n；

Step 2 rationally, founding mathematical models are distributed to the fault current limiter：

Step 2.1, object function f (x) that the fault current limiter distributes rationally is set it is：

$f\left(x\right)=min\underset{i=1}{\overset{N}{\Σ}}{Z}_{FCL}\left(i\right)---\left(1\right)$

In formula (1), Z_FCLI () represents the resistance value of i-th fault current limiter, N represents the installation fault current limiter Number；

Arranging short circuit current constraints is：

I_j≤I_jmax(2)

In formula (2), I_jmaxRepresent current limliting desired value；

Arranging node voltage constraints is：

V_rmin≤V_r≤V_rmax(3)

In formula (3), V_rRepresent the node voltage of r-th node；V_rminAnd V_rmaxRespectively r-th node voltage V_rBound； n_bAll node numbers on the circuit being located for the fault current limiter to be installed and its perimeter circuit；1≤r≤n_b；

Step 2.2, each constraints is processed：

Voltage out-of-limit value V of all nodes is set_limFor：

$V_{\lim }=\underset{r=1}{\overset{n_b}{\mathrm{\Σ}}}max\{V_{rmax}-V_r,V_r-V_{rmin},0\}---\left(4\right)$

Arrange all nodes short circuit current get over limit value be:

$I_{\lim }=\underset{j=1}{\overset{n}{\Σ}}max\{I_{jmax}-I_j,0\}---\left(5\right)$

The population trip for being obtained the optimal configuration algorithm of the fault current limiter using the penalty shown in formula (6) is adapted to Function is：

MinF (x)=minf (x)+K (V_lim+I_lim) (6)

In formula (6), K is penalty factor；

Step 3, configuration is optimized to the fault current limiter by APSO algorithm：

Step 3.1, current-limiting impedance value X is regarded as one group of discrete variable, and carries out b position two to one group of discrete variable entering System coding；So that each branch road to be installed all has 2^b- a kind of impedance is selected, so as to define the dimension D=N for installing impedance population (b+1)；

Step 3.2, random initializtion install impedance population, and randomly generate N number of impedance particle of installing for { X₁,X₂,…, X_t,…,X_N}；X_tRepresent t-th installation impedance particle；1≤t≤N；

Studying factors are set as c₁And c₂, minimum crossover probability be_min, population scale be M, Particles Moving velocity interval be [V_min, V_max]=[- 1,1]；Particle position scope is [X_min,X_max]=[0,5], maximum iteration time is G_max, k represents current iteration time Number；

Step 3.3, fitness evaluation：

Firstization iterationses k=1, with t-th installation impedance particle X of kth generation_tItself is used as t-th individual optimum position of kth generation PutThe initial adaptive value of each installation impedance particle in colony is calculated, and obtains global optimum position of the kth for population gbest_k；

Step 3.4, by formula (7) update kth generation t-th installation impedance particle X_tSpeedObtaining+1 generation of kth installs for t-th Impedance particle X_tSpeed

$v_t^{(k+1)}={\mathrm{wv}}_t^{\left(k\right)}+c_1{r}_1({\mathrm{pbest}}_t^{\left(k\right)}-x_t^{\left(k\right)})+c_2{r}_2({\mathrm{gbest}}_k-x_t^{\left(k\right)})---\left(7\right)$

In formula (7), w represents that speed updates coefficient；For t-th installation impedance particle X of kth generation_tPosition, r₁And r₂For [0,1] Random number in interval；

T-th installation impedance particle X of kth generation is updated using formula (8)_tPositionObtain t-th installation impedance grain of+1 generation of kth Sub- X_tPosition

$x_t^{(k+1)}=\left\{\begin{array}{c}0,rand\left(\right)\&GreaterEqual;Sig\left(v_t^{(k+1)}\right)\\ 1,rand\left(\right)<Sig\left(v_t^{(k+1)}\right)\end{array}\right.---\left(8\right)$

In formula (8), rand () is for being evenly distributed on the random function in [0,1] interval；For position judgment function, and Have：

$Sig\left(v_t^{(k+1)}\right)=\frac{1}{1+\exp (-v_t^{(k+1)})}---\left(9\right)$

Step 3.5, the probability P according to formula (10) calculating cross and variation operation, if meet P ＞ R_min, then 3.6 are gone to step, otherwise Go to step 3.9；

P=μ+Re σ (10)

In formula (10), μ and σ is the regulation parameter of aberration rate, and Re is global optimum in G_maxContinuously do not update in secondary iterative process Or update unconspicuous algebraically；

Step 3.6, in whole population each install impedance particle cross and variation operation is carried out according to condition；

Step 3.6.1, using formula (11) obtain kth generation t-th installation impedance particle X_tPositionWith kth for global optimum position Put gbest_kBetween Euclidean distance

$l_t^{\left(k\right)}=\sqrt{\underset{d=1}{\overset{D}{\&Sigma;}}{\&lsqb;x_t^{\left(k\right)}-{\mathrm{gbest}}_k\&rsqb;}^2}---\left(11\right)$

Distance threshold Δ φ is obtained using formula (12)：

Δ φ=(1-k/G_max)^m×(X_max-X_min) (12)

In formula (12), m is regulation parameter；

Step 3.6.2, judgementWhether set up, if so, then using formula (13) to kth t-th impedance particle X of generation_t's PositionCrossover operation is carried out, the kth t-th impedance grain X of generation after being intersected_tPosition

$\left\{\begin{array}{c}{x}_t^{\&prime;\left(k\right)}=x_t^{\left(k\right)}e+x_a^{\left(k\right)}(1-e)\\ x_a^{\&prime;\left(k\right)}=x_t^{\left(k\right)}(1-e)+x_a^{\left(k\right)}e\end{array}\right.---\left(13\right)$

In formula (13),Represent the position of kth a-th impedance particle of generation；Represent the kth a-th impedance grain of generation after intersecting The position of son；E is the random number in (0,1) interval；1≤a≤N；And a ≠ t；

Calculate kth t-th impedance particle X of generation_tAdaptive value and intersect after kth generation t-th impedance particle X_tAdaptive value, and The position of the impedance particle corresponding to selection adaptive optimal control value is designated as the position of kth t-th impedance particle of generation

Step 3.6.3, judgementWhether setting up, such as sets up, then mutation operation is carried out using formula (14), after being made a variation Kth generation t-th impedance particle X_tPosition

$\left\{\begin{array}{c}{x}_t^{\&prime;\&prime;\left(k\right)}=x_t^{\left(k\right)}+{(1-\frac{k}{G_{max}})}^{\mathrm{\&alpha;}}(X_{max}-x_a^{\left(k\right)})\\ x_a^{\&prime;\&prime;\left(k\right)}=x_t^{\left(k\right)}-{(1-\frac{k}{G_{max}})}^{\mathrm{\&alpha;}}(x_a^{\left(k\right)}-X_{\min })\end{array}\right.---\left(14\right)$

In formula (14),Represent the kth a-th impedance particle X of generation after variation_aPosition；α is the weights of variation；

Calculate kth t-th impedance particle X of generation_tAdaptive value and variation after kth generation t-th impedance particle X_tAdaptive value, and select The position of impedance particle corresponding to adaptive optimal control value is taken as kth t-th impedance particle X of generation_tPosition, be designated as

Step 3.7, by kth for global optimum position gbest_kDirection to the bound of particle position scope moves one respectively Small step delta, mobile number of times is that selection in new optimal particle group adapts to q, so as to obtain new optimal particle group of the scale for C Value highest is installed the position of impedance particle and replaces global optimum position gbest_k, so as to obtain new global optimum's particle position gbest′_k；

Step 3.8, calculate complete to intersect and mutation operation after population in all fitness for installing impedance particles, and with described Global optimum position gbest_kTo replace the position of the worst installation impedance particle of adaptive value；So as to complete kth for population more Newly；

Step 3.9, k+1 is assigned to k, and 3.4 order of return to step is executed, until k=G_maxTill；So as to obtain optimum peace Dress impedance particleUsing the optimum resistance value that installs corresponding to impedance particle as The installation impedance of the fault current limiter so that install the three-phase shortcircuit on the c bar branch road after N number of fault current limiter Electric current can meet short circuit current constraints, all node voltages can meet node voltage constraint on the perimeter circuit of c bar branch road Condition, and object function f (x) minimum.