A method for optimizing configuration of a flexible direct current current limiter

Abstract

The present application relates to a kind of flexible dc current limiter optimization configuration method, comprising: equivalent flexible current limiter is controllable voltage source, constructs the fault current model of the dc system containing flexible current limiter;By the fault current model, the fault current value of each branch in dc system is calculated;Select several evaluation indexes;According to the several evaluation indexes, the importance evaluation function is constructed by entropy method, analytic hierarchy process;The importance evaluation value of each branch in dc system is calculated, and the n branch with maximum importance evaluation value is selected as candidate branch;With flexible current limiter cost, current limiting effect as objective function, with flexible current limiter number, circuit breaker maximum breaking current as constraint condition, construct multi-objective optimization configuration model;According to the fault current value of each branch, by multi-objective optimization configuration model and NSGA-Ⅱ algorithm multi-objective optimization calculation, obtain Pareto solution;According to the Pareto solution, the installation position and installation capacity of flexible current limiter are determined.

Description

Technical Field

[0001] This invention relates to a method for optimizing the configuration of a flexible DC current limiter, belonging to the field of flexible current limiter optimization configuration. Background Technology

[0002] When a fault occurs in a flexible DC distribution network, the fault current cannot automatically cross zero. Furthermore, due to the low damping of the flexible DC distribution network system, the fault current rises rapidly to its peak value in the initial stage of a fault, making it difficult for traditional mechanical DC circuit breakers to interrupt the fault. A current limiter is a DC system protection device used in conjunction with a circuit breaker to ensure that the circuit breaker reliably interrupts the fault current. Current limiters mainly include inductive current limiters, superconducting resistor current limiters, solid-state current limiters, and flexible current limiters. Among them, a flexible current limiter is connected in series with the DC line during a fault and provides a variable clamping voltage to the fault circuit, thereby flexibly controlling the voltage drop of the converter station's output capacitor and suppressing the fault current. With a sufficiently large capacity, the flexible current limiter can clamp the voltage drop of the converter station's output capacitor to near its rated value, significantly shortening the voltage dynamic recovery time after a fault.

[0003] Flexible current limiters are expensive, and how to use the fewest number of flexible current limiters to ensure that the short-circuit current margin of all nodes in a flexible DC distribution network meets the given requirements is an urgent problem to be solved. Summary of the Invention

[0004] To overcome the problems existing in the prior art, this invention designs an optimized configuration method for flexible DC current limiters. Considering multiple factors such as the number of flexible current limiters installed, installation location, economy, and current limiting effect, the resulting configuration scheme for flexible current limiters can meet the short-circuit current margin requirements in flexible DC distribution networks. This configuration method is highly flexible and has a wide range of applications.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A method for optimizing the configuration of a flexible DC current limiter includes the following steps:

[0007] The flexible current limiter is equivalent to a controllable voltage source, and a fault current model of the DC system containing the flexible current limiter is constructed; the fault current value of each branch in the DC system is calculated using the fault current model.

[0008] Several evaluation indicators are selected; based on the evaluation indicators, an importance evaluation function is constructed using the entropy method and the analytic hierarchy process; the importance evaluation value of each branch in the DC system is calculated, and the n branches with the largest importance evaluation values ​​are selected as candidate branches;

[0009] Using the cost and current limiting effect of flexible current limiters as objective functions, and the number of flexible current limiters and the maximum breaking current of circuit breakers as constraints, a multi-objective optimization configuration model is constructed. Based on the fault current value of each branch, multi-objective optimization calculations are performed using the multi-objective optimization configuration model and the NSGA-II algorithm to obtain the Pareto solution. The installation location and installation capacity of the flexible current limiters are determined based on the Pareto solution.

[0010] Furthermore, the fault current model is expressed by the following formula:

[0011]

[0012] In the formula, In order to be in t The median node voltage at time +1; In order to be in t The node voltage value at time; In order to be in t The intermediate value of the branch current at time +1; Let be the branch current value at time t; and The equations for node voltage variation and branch current variation are provided. h This is the iteration step size.

[0013] Furthermore, the evaluation metric includes the dielectric constant of the transmission line reactance.

[0014] Furthermore, the importance evaluation function is expressed by the formula:

[0015] ;

[0016] In the formula, L k For the first k The importance assessment value of each route; For the first k The first of the lines j Individual indicator values.

[0017] Furthermore, the multi-objective optimization configuration model is expressed by the following formula:

[0018]

[0019]

[0020]

[0021]

[0022]

[0023] In the formula,x For the solution space; U FCL The equivalent voltage of the flexible current limiter; , The lower and upper limits of the equivalent voltage of the flexible current limiter; , The lower and upper limits for the number of flexible current limiters to be installed; F 1. F 2 represents the objective function, where represents the cost and current limiting effect of the flexible current limiter, respectively; N FCL This represents the total number of flexible current limiters installed. U FCL (i) is the first i The equivalent voltage value of a flexible current limiter; a The cost factor is the equivalent voltage. b This is the installation cost coefficient; For the first i When the first end of a branch road fails t c The fault current at any given moment; For the first i When the end of a branch line fails t c The fault current at any given moment; t c The moment when the DC circuit breaker operates; branch road i The rated current; f p This is the penalty function.

[0024] Furthermore, multi-objective optimization calculations are performed using a multi-objective optimization configuration model and the NSGA-II algorithm, specifically as follows:

[0025] Step 1: Select n candidate branches;

[0026] Step 2: Set the equivalent voltage value of the flexible current limiter at the beginning and end of each candidate branch;

[0027] Step 3: Randomized experiments generate initial parent populations. Each population's chromosome is randomly assigned an equivalent voltage value set by a candidate branch. Verify whether each population satisfies the constraints. Calculate the objective function value using a multi-objective optimization configuration model. Based on the objective function value, perform non-dominated sorting and crowding calculation on each population to obtain the Pareto front solution, which is then stored in an external archive.

[0028] Step 4: After selection, crossover, and mutation evolution, a progeny population is generated, and random individuals are generated to replace individuals in the progeny population that do not meet the constraints, thus maintaining the population size.

[0029] Step 5: Merge the parent and child populations, calculate the objective function values ​​of the parent and child populations, perform non-dominated sorting and crowding calculation, solve the Pareto front solution, and update the external archives;

[0030] Step 6: Determine if the maximum number of iterations has been reached. If it has, output the Pareto front solution as the optimal solution; otherwise, return to step 4.

[0031] Compared with the prior art, the present invention has the following features and beneficial effects:

[0032] 1. This invention considers multiple factors such as the number of flexible current limiters, installation location, economy, and current limiting effect. The resulting configuration scheme (installation location and installation capacity) of the flexible current limiter can meet the short-circuit current margin requirements in flexible DC distribution networks. This configuration method is highly flexible and has a wide range of applications.

[0033] 2. Existing technologies only provide fault current calculation methods for resistive and inductive current limiters. This invention equates the flexible current limiter to a controllable voltage source and constructs a fault current model based on this, providing a method for calculating fault current in DC systems containing flexible current limiters.

[0034] 3. This invention constructs and utilizes an importance evaluation function to analyze the importance of each line, and selects candidate branches for installing flexible current limiters based on the line importance, thereby reducing the number of analysis objects in subsequent multi-objective optimization and improving the computational efficiency of multi-objective optimization. Attached Figure Description

[0035] Figure 1 The diagram shows the structural structure and control block diagram of the flexible DC current limiter model according to an embodiment of the present invention.

[0036] Figure 2 This is a simplified model diagram of the converter station according to an embodiment of the present invention.

[0037] Figure 3 This is an 11-node DC system model according to an embodiment of the present invention.

[0038] Figure 4 This provides an optimized Pareto front solution for a flexible current limiter based on the NSGA-II algorithm in this embodiment of the invention.

[0039] Figure 5 This represents the maximum system current (tc=6ms) during inter-electrode short-circuit faults at different locations in this embodiment of the invention.

[0040] Table 1 shows the node parameters of the 11-node model converter station in this embodiment of the invention.

[0041] Table 2 shows the system parameters of the 11-node model in this embodiment of the invention.

[0042] Table 3 shows the 11-node line parameters of an embodiment of the present invention.

[0043] Table 4 shows the line importance index values ​​in the embodiments of the present invention.

[0044] Table 5 shows the optimized solutions for the nine configuration schemes selected in the embodiments of the present invention. Detailed Implementation

[0045] The present invention will now be described in more detail with reference to the embodiments.

[0046] Example 1

[0047] like Figure 1 As shown, a method for optimizing the configuration of a flexible DC current limiter includes the following steps:

[0048] Based on the simplified model of the converter station, and considering the output characteristics of the flexible current limiter, it is equivalent to a controllable voltage source, and a fault current model of the DC system containing the flexible current limiter is constructed. Through the fault current model, the starting fault current and ending fault current of each branch in the DC system are iteratively calculated.

[0049] Using transmission line length, service life, transmission capacity, and reactance dielectric constant as evaluation indicators, an importance evaluation function is constructed using the entropy method and the analytic hierarchy process. The importance evaluation value of each branch in the DC system is calculated, and the branches are sorted from largest to smallest according to their importance evaluation value. The top n branches are selected as candidate branches for installing flexible current limiters.

[0050] A multi-objective optimization configuration model is constructed with the cost and current limiting effect of flexible current limiters as objective functions, and the number of flexible current limiters and the maximum breaking current of circuit breakers as constraints.

[0051] The Pareto solution is obtained by using the multi-objective optimization configuration model and the NSGA-II algorithm to perform multi-objective optimization calculations; the installation location and installation capacity of the flexible current limiter are determined based on the Pareto solution.

[0052] Example 2

[0053] Constructing a fault current model for a DC system with a flexible current limiter includes the following steps:

[0054] After the fault, the differential equation expression of the DC system is:

[0055] ;

[0056] In the formula, U This is a matrix representing the voltage values ​​of each node, with the first n rows of the matrix showing the voltage values ​​of the converter station nodes. I bThe current values ​​of each branch; I0 is the equivalent current source current of the converter station in the DC system; R, L, and C are the branch resistance matrix, branch inductance matrix, and capacitance matrix, respectively, and their expressions are:

[0057] ;

[0058] ;

[0059] ;

[0060] In the formula, R ij , L ij The first two lines on the diagonal of matrices R and L are respectively the first two lines on the diagonal. k The nth element represents the nth element. k a side road ij The resistor and inductor on it; C i For the first i The equivalent capacitance of each converter station.

[0061] The expression for this differential equation can be expanded as follows:

[0062]

[0063]

[0064]

[0065]

[0066]

[0067] In the formula, the matrix subscript r(b+1) represents the element in the (b+1)th row of the matrix; A is ( b +1)×( n The network affinity matrix (+1) represents the relationship between branches and nodes, and its writing rules are as follows:

[0068] ;

[0069] The correlation matrix A is decomposed into A1 and A2 as follows:

[0070] ;

[0071] After the flexible current limiter is activated, the voltage that was originally distributed across the line impedance becomes distributed across both the line impedance and the flexible current limiter. The differential equation expression for the DC system then becomes:

[0072] ;

[0073] This system of equations contains ( N+b+ 1) An equation, with unknowns to be solved. I b and U The total number is ( n+b+ 1), therefore, the following formula is constructed:

[0074] ;

[0075] In the formula, R1 is the resistance path matrix; L1 is the inductance path matrix; R2 and L2 are constructed intermediate matrices; assuming that the transition resistance is 0 when a bipolar short-circuit fault occurs, the voltage at the short-circuit point is 0, therefore, in the left-hand formula... u N+1 Nodes can be represented ( N The voltage drop from +1) to the short circuit point can be obtained using a depth-first search algorithm to find the shortest path from each non-converter station node to the short circuit point. This path is determined by the system's ( b The system consists of 1+1 branches. By filling in the resistance and inductance of the branches contained in the shortest path into the corresponding positions, R1 and L1 can be obtained.

[0076] Combining the above equations, the differential equation expression for a DC system containing a flexible current limiter is as follows:

[0077] ;

[0078] From the above equation, the equations for node voltage variation and branch current variation are:

[0079]

[0080] The iterative expression for the fault current model is:

[0081]

[0082] In the formula, In order to be in t The median node voltage at time +1; In order to be in t The node voltage value at time; In order to be in t The intermediate value of the branch current at time +1; Let be the branch current value at time t; and The equations for node voltage variation and branch current variation are provided. h This is the iteration step size.

[0083] Assuming the fault location is at the beginning of the branch, the fault current at the beginning of each branch in the DC system is calculated iteratively using this iterative expression. Similarly, assuming the fault location is at the end of the branch, the terminal fault current of each branch in the DC system can be iteratively calculated using this iterative expression. .

[0084] Example 3

[0085] In this embodiment, four line indicators are selected as the basis for evaluating the importance of the line: transmission line length, service life, transmission capacity, and reactance betweenness index. The importance assessment value is determined by combining the entropy method and the analytic hierarchy process (AHP) through a complementary approach of subjective and objective methods.

[0086] The expression for the line reactance index is as follows:

[0087] ;

[0088] In the formula, S and L are the sets of power supply nodes and load nodes, respectively; , The first i The rated power of the first power node, the first j Rated power of each load node; For power nodes i to load node j All paths between them can be obtained using a depth-first search algorithm; For power nodes i to load node j The included lines l The number of paths; For the line l The reactance; x d for Middle path d The reactance.

[0089] The comprehensive weighting is composed of the entropy method and the analytic hierarchy process, and its expression is as follows:

[0090] ;

[0091] In the formula, ω AHP For the weights of the analytic hierarchy process, ω EWM For entropy-based weights, ω j As the corresponding indicator weights, N For indicator dimensions.

[0092] The importance evaluation function can be defined as:

[0093] ;

[0094] In the formula,L k For the first k The importance assessment value of each route; For the first k The first of the lines j Individual indicator values.

[0095] Example 4

[0096] In this embodiment, a multi-objective optimization configuration model is constructed with the goals of cost and current limiting effect of flexible current limiters, and the constraints of the number of flexible current limiters and the maximum breaking current of the circuit breaker. The model includes the following steps:

[0097] The multi-objective optimization configuration model can be expressed by the following formula:

[0098] ;

[0099] In the formula, x For the solution space; F ( x Let be the space of the objective function; U FCL The equivalent voltage of the flexible current limiter; and These are the lower and upper limits of the equivalent voltage of the flexible current limiter. Number of flexible current limiters to be installed; and The lower and upper limits for the number of flexible current limiters to be installed; F 1. F 2 are two objective functions.

[0100] The objective function expression is as follows:

[0101] ;

[0102] ;

[0103] ;

[0104] ;

[0105] In the formula, F 1 represents the equivalent cost of the flexible current limiter; F 2 represents the current limiting effect of the flexible current limiter; N FCL This represents the total number of flexible current limiters installed. U FCL For the first i The equivalent voltage value of a flexible current limiter; a The cost factor is the equivalent voltage. b This is the installation cost coefficient; For the first i When the first end of a branch road fails t c The fault current at any given moment; For the first i When the end of a branch line fails t c The fault current at any given moment; t c The moment when the DC circuit breaker operates; branch road i The rated current; f p This is the penalty function.

[0106] Example 5

[0107] In this embodiment, the multi-objective optimization configuration model and the NSGA-II algorithm are used to perform multi-objective optimization calculations, specifically as follows:

[0108] Step 1: Based on the DC system line impedance, power node and load node power parameters, determine the weights of the four evaluation indicators for each line: transmission line length, service life, transmission capacity and reactance dielectric constant. Then, determine the importance evaluation value of each line. Based on the importance evaluation value, select n candidate branches for installing flexible current limiters.

[0109] Step 2: Perform chromosome coding; assign the equivalent voltage value of the flexible current limiter at the beginning and end of each candidate branch to a discrete variable of 0 to 7, which respectively represent the capacity of the configured flexible current limiter. When the discrete variable value is 0, it means that the branch is not equipped with a flexible current limiter.

[0110] Step 3: Randomized experiments generate 500 initial parent populations. The chromosomes of each initial population are randomly assigned the equivalent voltage of the flexible current limiter configured for each branch, covering various installation locations and capacities. Verify whether the populations meet the constraints and calculate the objective function value using the fault current calculation method in Example 2. Based on the objective function value, perform non-dominated sorting and congestion calculation for each population. The Pareto front solution (a series of non-dominated solutions) is obtained from the calculation results and stored in an external archive.

[0111] Step 4: After selection, crossover, and mutation evolution, a progeny population is generated and random individuals are generated to replace individuals in the progeny population that do not meet the constraints (i.e., the number of flexible current limiters and the maximum breaking current of the circuit breaker exceed the range set in Example 4), thus maintaining the population size.

[0112] Step 5: Merge the parent and child populations, calculate the objective function values ​​of the parent and child populations, perform non-dominated sorting and crowding calculation, solve the Pareto front solution, and update the external archives;

[0113] Step 6: Determine if the maximum number of iterations (300) has been reached. If it has, output the Pareto front solution as the optimal solution; otherwise, return to Step 4.

[0114] Example 6

[0115] To verify the effectiveness of the embodiments of the present invention, a specific example will be used below.

[0116] Step 1: Optimize the configuration of the flexible current limiter using an 11-node DC system model derived from the IEEE-14 node distribution network system upgrade. The structural diagram is shown below. Figure 3 As shown in the figure. Node 5 in this system is a balancing node, used to maintain a constant system voltage. The other 10 nodes are power nodes, and the entire system has 3 power supply nodes and 8 load nodes. The specific system parameters are shown in Tables 1 and 2.

[0117] Table 1

[0118]

[0119] Table 2

[0120]

[0121] Step 2: The importance index of the line is analyzed using four parameters: transmission line length, service life of the transmission line, transmission capacity of the transmission line, and dielectric constant of the transmission line. The specific parameter values ​​are shown in Table 3.

[0122] Table 3

[0123]

[0124] Based on historical experience, a judgment matrix is ​​constructed: D=[1, 1 / 2, 1 / 4, 1 / 8; 2, 1, 1 / 2, 1 / 4; 4,2, 1, 1 / 2; 8, 4, 2, 1]. The subjective weights corresponding to the four parameters can be obtained as follows: =[0.0667, 0.1333, 0.2667, 0.5333]; Based on X1~X4, their corresponding objective weights can be calculated as follows: =[0.3007, 0.2379, 0.2927, 0.1687]. The importance assessment values ​​of the route are obtained by combining subjective and objective weights, as shown in Table 4.

[0125] Table 4

[0126]

[0127] Step 3: Set the NSGA-II algorithm parameters as follows: population size and total number of iterations are 100 and 150 respectively; the maximum and minimum equivalent voltages of the flexible current limiter are 7kV and 0kV respectively; and the installation cost coefficient is... b With capacity cost coefficient a The ratio is 5:1; the set breaking time of the installed DC circuit breaker is 6ms, and the maximum breaking current is 7kA; the crossover probability and mutation probability distribution between populations are 0.9 and 0.1, respectively; since the short circuit current is larger the closer the short circuit point is to the converter station, the location of the inter-pole short circuit fault point is taken as the beginning and end of each line.

[0128] Step Four: Figure 4 The figure shows the Pareto front solution for the optimal configuration of a flexible current limiter based on the NSGA-II algorithm. The horizontal and vertical axes in the figure represent the objective function, respectively. F 2 and F The value of 1, with different colors and symbols representing the leading edge solution with varying numbers of flexible current limiters installed, is used. The Pareto leading edge curve shows that the objective function... F 2 and F The objective function is inversely proportional to 1; the more flexible current limiters installed, the higher the objective function becomes. F The smaller the value of 2, the better the rate limiting effect, but its objective function... F The larger the value of 1, the higher the corresponding cost.

[0129] Step 5: Select one solution from each of the Pareto front solutions with different numbers of flexible current limiters installed, representing 9 different optimized solutions (in practice, one optimized solution is selected as the final solution according to actual needs). The specific configuration scheme is shown in Table 5. A value of 0 in the table indicates that no flexible current limiter is configured at that location. Configure flexible current limiters for the 11-node ring network system according to the configuration scheme in Table 5. The inter-pole short-circuit fault locations are set at the beginning and end of each branch. After the fault occurs... t c The maximum short-circuit current for inter-electrode short-circuit faults on different lines at 6ms is as follows: Figure 5 As shown, it can be seen that compared to the fault current without the flexible current limiter, the fault current is greatly reduced after configuring the flexible current limiter. Under the selected configuration schemes, the system fault current can be reduced to a minimum when the circuit breaker is tripped (…). t c =6ms) is less than its maximum breaking current (7kA), and the circuit breaker can quickly disconnect the faulty line, proving its rationality. At the same time, Figure 5 In the configuration scheme, the overall current limiting effect can be determined by the area enclosed by the polygon formed by the maximum current when different lines fail. As the number of flexible current limiters installed increases, the area enclosed by the polygon becomes smaller, which means that the overall current limiting capability is better, and the investment cost also increases accordingly.

[0130] Table 5

[0131]

[0132] Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

Claims

1. A method for optimizing the configuration of a flexible DC current limiter, characterized in that, Includes the following steps: The flexible current limiter is equivalent to a controllable voltage source, and a fault current model of the DC system containing the flexible current limiter is constructed; the fault current value of each branch in the DC system is calculated using the fault current model. Several evaluation indicators are selected; based on these indicators, an importance evaluation function is constructed using the entropy method and the analytic hierarchy process (AHP). These evaluation indicators include transmission line length, service life, transmission capacity, and line reactance. The expression for the line reactance index is: ； In the formula, , These are the sets of power supply nodes and the sets of load nodes, respectively. , The first The rated power of the first power node, the first Rated power of each load node; For power nodes to load node All paths between them are obtained using a depth-first search algorithm; For power nodes to load node The included lines The number of paths; For the line The reactance; for Middle path The reactance; For the line The line reactance index; Calculate the importance assessment value of each branch in the DC system, and select the n branches with the largest importance assessment values ​​as candidate branches; Using the cost and current-limiting effect of flexible current limiters as objective functions, and the number of flexible current limiters and the maximum breaking current of circuit breakers as constraints, a multi-objective optimization configuration model is constructed. Based on the fault current values ​​of each branch, multi-objective optimization calculations are performed using the multi-objective optimization configuration model and the NSGA-II algorithm to obtain the Pareto solution. The installation location and capacity of the flexible current limiters are determined based on the Pareto solution, wherein: The fault current model is expressed by the following formula: In the formula, In order to be in t The median node voltage at time +1; In order to be in t The node voltage value at time; In order to be in t The intermediate value of the branch current at time +1; Let be the branch current value at time t; and The equations for node voltage variation and branch current variation are provided. h This is the iteration step size.

2. The method for optimizing the configuration of a flexible DC current limiter according to claim 1, characterized in that, The importance evaluation function is expressed by the following formula: ； In the formula, L k For the first k The importance assessment value of each route; For the first k The first of the lines j Individual indicator values.

3. The method for optimizing the configuration of a flexible DC current limiter according to claim 1, characterized in that, The multi-objective optimization configuration model is expressed by the following formula: In the formula, x For the solution space; U FCL The equivalent voltage of the flexible current limiter; , The lower and upper limits of the equivalent voltage of the flexible current limiter; , The lower and upper limits for the number of flexible current limiters to be installed; F 1. F 2 represents the objective function, where represents the cost and current limiting effect of the flexible current limiter, respectively; N FCL This represents the total number of flexible current limiters installed. For the first i The equivalent voltage value of a flexible current limiter; a The cost factor is the equivalent voltage. b This is the installation cost coefficient; For the first i When the first end of a branch road fails t c The fault current at any given moment; For the first i When the end of a branch line fails t c The fault current at any given moment; t c The moment when the DC circuit breaker operates; branch road i The rated current; f p This is the penalty function.

4. The method for optimizing the configuration of a flexible DC current limiter according to claim 1, characterized in that, Multi-objective optimization calculations are performed using a multi-objective optimization configuration model and the NSGA-II algorithm, specifically as follows: Step 1: Select n candidate branches; Step 2: Set the equivalent voltage value of the flexible current limiter at the beginning and end of each candidate branch; Step 3: Randomized experiments generate initial parent populations. Each population's chromosome is randomly assigned an equivalent voltage value set by a candidate branch. Verify whether each population satisfies the constraints. Calculate the objective function value using a multi-objective optimization configuration model. Based on the objective function value, perform non-dominated sorting and crowding calculation on each population to obtain the Pareto front solution, which is then stored in an external archive. Step 4: After selection, crossover, and mutation evolution, a progeny population is generated, and random individuals are generated to replace individuals in the progeny population that do not meet the constraints, thus maintaining the population size. Step 5: Merge the parent and child populations, calculate the objective function values ​​of the parent and child populations, perform non-dominated sorting and crowding calculation, solve the Pareto front solution, and update the external archives; Step 6: Determine if the maximum number of iterations has been reached. If it has, output the Pareto front solution as the optimal solution; otherwise, return to step 4.

Images