CN110633494A - Multi-objective optimization design method of Swiss rectifier based on NSGA-II algorithm - Google Patents

Multi-objective optimization design method of Swiss rectifier based on NSGA-II algorithm Download PDF

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CN110633494A
CN110633494A CN201910729666.3A CN201910729666A CN110633494A CN 110633494 A CN110633494 A CN 110633494A CN 201910729666 A CN201910729666 A CN 201910729666A CN 110633494 A CN110633494 A CN 110633494A
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颜景斌
沈云森
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Harbin University of Science and Technology
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Abstract

The invention provides a Swiss rectifier multi-objective optimization design method based on NSGA-II algorithm, belonging to the technical field of rectifier parameter optimization design. Step 1, determining optimized components and modeling power and volume; step 2, defining a target function; step 3, initializing a population; step 4, selecting; step 5, selecting a binary tournament; step 6, crossing; step 7, mutation; step 8, merging the parent population and the child population to obtain a first generation population; step 9, forming a population by utilizing the parent population and the child population; step 10, reaching a maximum evolution algebra gen; and 11, outputting the Pareto optimal solution front edge and the parameter matrix, obtaining corresponding decision variable values, namely solutions of optimization problems, according to the selected target performance indexes, and selecting and designing components according to the parameters.

Description

Multi-objective optimization design method of Swiss rectifier based on NSGA-II algorithm
Technical Field
The invention relates to a Swiss rectifier multi-objective optimization design method based on an NSGA-II algorithm, and belongs to the technical field of rectifier parameter optimization design.
Background
In recent years, with the wide application of power electronic devices, more and more researches and optimization technologies about converters are paid high attention by researchers at home and abroad, and various high-efficiency and novel rectifier topological structures and digital control are generated at the same time, so that electric vehicles and new energy technologies are rapidly developed; the rectifier system has the obvious performance characteristics of high frequency, high efficiency, high power density, high power factor, high reliability and the like, and along with the appearance and application of various novel electromagnetic materials, electronic components and transformation technologies, the rectifier system develops towards the direction of small volume, high efficiency and low cost.
However, since the Swiss rectifier has nonlinear elements such as inductors, the rectifier system is a strong nonlinear system, and the parameters and performances of components to be optimized are more, and the performances conflict with each other and are restricted with each other, and the improvement of one performance is usually at the cost of the reduction of the other performance; therefore, the Swiss rectifier which is a complex optimization problem with multiple objectives, uncertainty, nonlinearity and multiple parameters is difficult to realize in the traditional optimization method, and the accuracy of each parameter can not be ensured by judging each parameter by experience; how to select a proper device by a rigorous mathematical method and process the multi-objective optimization problem with discrete variables, such as a rectifier, so that the comprehensive performance of the system is optimal, and the method has important significance on the reliable operation of the system and the sustainable development of energy; therefore, a multi-objective optimization method for the Swiss rectifier is imperative.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a Swiss rectifier multi-objective optimization design method based on an NSGA-II algorithm, which is beneficial to the optimization design of Swiss rectifier parameters, so that the efficiency and the power density of the rectifier are improved. The technical scheme is as follows:
a Swiss rectifier multi-objective optimization design method based on NSGA-II algorithm comprises the following steps:
step 1, taking a direct current inductor L, an output capacitor C, a switching tube IGBT and a Diode in a basic topology circuit of a Swiss rectifier as optimization variables influencing performance indexes of the rectifier, and carrying out power and volume modeling on the direct current inductor L, the output capacitor C, the switching tube IGBT and the Diode;
step 2, defining an objective function: measuring the performance index of the Swiss rectifier by taking the efficiency and the power density of the Swiss rectifier as objective functions;
step 3, initializing a population: setting the population size as N, the maximum evolution algebra as gen, and the number of decision variables as V; then, real number coding is carried out on the decision variables, and the upper limit and the lower limit of each parameter variable are input by taking the established component database as a constraint condition; randomly generating an N-dimensional decision matrix, taking a V +1 th column and a V +2 th column of the decision matrix as objective function values, taking the last two columns as non-dominant layer numbers and crowding degree distances, and enabling one parameter vector to correspond to one individual or be called a chromosome;
step 4, selection: setting two parameters niAnd Si，niGoverning the number of individuals i, S, for all individuals in a populationiFor an individual set dominated by the individual i, judging the quality of a solution of the matrix and sequencing operation by adopting rapid non-dominated sequencing;
step 5, selecting the binary championship: setting the championship size to be 2 and the matching pool size to be N/2; randomly selecting two individuals in an initial population, comparing the non-dominant grades of the two individuals, and putting the low-grade individual into a matching pool; if the grades are the same, comparing the crowdedness distance, and keeping the distance larger; if the non-domination grade and the crowding degree distance are the same, randomly selecting one individual to be reserved, and repeating the step until the number of the individuals in the matching pool is N/2;
step 6, crossing: firstly, randomly selecting two individuals in a current generation population as parent individuals of cross operation, carrying out cross operation on genes on matched chromosomes to generate a pair of new chromosomes, and repeating the cross operation to form a new generation population; wherein the current generation population is a parent generation population; the specific operation process is as follows: setting the gene length of chromosome as L, randomly selecting an integer value K as a cross position within the range of [0, L ], and mutually exchanging genes of [ K, L ] parts at the cross position by matching chromosomes, thereby forming a new pair of chromosomes, namely individuals;
and 7, mutation: defining a mutation operator, carrying out small-probability replacement on the gene of an individual, and acting the mutation operator on a population to change the gene of a part of individuals in the population to generate a new allele, and marking the population at the moment as an offspring population;
step 8, merging the parent population obtained in the step 6 and the child population obtained in the step 7, and marking the merged population as a first generation population (gen is 1) with the size of the population being N;
step 9, taking the first generation population obtained in the step 8 as a parent population, then carrying out cross variation on the parent population to obtain a child population, and then combining the parent population and the child population in the step to form a population, wherein the size of the population is 2N;
step 10, processing the population formed in the step 9 by using the operation process of the step 4, selecting N individuals as a new parent population, and generating a new child population P through the steps 6 and 7t+1(ii) a Repeating the step 9 until the maximum evolution algebra gen is reached;
and 11, outputting the Pareto optimal solution front edge and the parameter matrix, obtaining corresponding decision variable values, namely solutions of optimization problems, according to the selected target performance indexes, and selecting and designing components according to the parameters.
Further, the power and volume modeling process in step 1 includes:
calculating the power of an inductor, the power P of the inductorLComprises the following steps:
Figure BDA0002160101270000021
wherein u isLIs the input voltage of an inductor, IrmsEffective value of current flowing through the inductor, fswTo the switching frequency, VLIs the volume of the inductor;
calculating the coil volume of the inductor; the coil volume VclComprises the following steps:
wherein, ω iswIs the winding width, dwIs the winding depth;
calculating the volume V of the conductorcdComprises the following steps: vcd＝kpfVclWherein k ispfIs the fill factor of the coil and,
Figure BDA0002160101270000031
n is the number of turns of the coil,
Figure BDA0002160101270000032
is the cross-sectional area of the conductor,
calculating core volume
Figure BDA0002160101270000033
Comprises the following steps:
Figure BDA0002160101270000034
then the volume of the inductor VLComprises the following steps:
Figure BDA0002160101270000035
wherein l represents the core length;
and calculating the on-state loss of the IGBT of the switching tube as follows:
Figure BDA0002160101270000036
wherein, VGEIs the threshold voltage of IGBT, IIGBTIs the magnitude of the current flowing through the IGBT, rCEIs an on-state equivalent resistance, M is a modulation ratio,
Figure BDA0002160101270000037
expressed as a voltage current phase angle;
the switching losses of the IGBT are:
Figure BDA0002160101270000038
wherein E issw(on)And Esw(off)Energy lost for switching on and off the IGBT once, respectively, INRated operating current for the rectifier, uCENAt a rated operating voltage of upnOutputting a voltage for the rectifier;
and (3) calculating the total loss of the IGBT as follows: pIGBT＝Pcond,IGBT+Psw,IGBT；
The conduction loss of the diode is:
Pon,Diode＝VFIFD
wherein, VFIs the forward conduction voltage drop of the diode, IFThe current is positive on-state current, D is duty ratio, and the duty ratios of different diodes are obtained according to 4 conducting circuit states;
the off-state loss of the diode is:
Poff,Diode＝VRIR(1-D)
wherein, VRFor its reverse pressure drop, IRIs a diode reverse leakage current;
the switching losses of the diodes are:
Figure BDA0002160101270000039
wherein, VfpAnd VrpRespectively forward and reverse peak voltage of the diode, IfpAnd IrpRespectively the forward and reverse peak currents through the diode, tfpFor its forward recovery time, tbIs its reverse current fall time;
the total diode loss is then: pDiode＝Pon,Diode+Poff,Diode+Psw,Diode。
Further, the specific process of defining the objective function in step 2 includes:
defining an objective function, taking the efficiency and the power density of the Swiss rectifier as the objective function, wherein the efficiency eta is as follows:
Figure BDA0002160101270000041
wherein, P0For output power, the power density ρ is:
Figure BDA0002160101270000042
wherein, P0To output power, PlossFor total loss, PlFor total loss of inductance, PIGBTFor total IGBT losses, PDiodeIs the total Diode loss; vGeneral assemblyIs the total volume of the component, VLIs the volume of the inductor, VIGBTIs the IGBT volume, VDiodeIs the diode volume, VCIs the output capacitance volume, C is the inductance; the power density rho is an objective function
Further, the specific process of judging the quality and the ranking of the solution in the fourth step includes:
step 1, selecting n in the populationiIndividuals of 0, i.e. non-dominant optimal individuals, and put into the set F1Performing the following steps;
step 2, pair set F1Respectively finding out a dominant individual set S corresponding to iiTo SiOf the population, the number n of dominant individuals j dominating all individuals in the populationjPerforming a subtraction operation, i.e. nj＝nj-1, excluding the influence of the individual i dominating j if n j0, then put the individual j into the set F2Performing the following steps;
then set F1For the first layer non-dominant set, for F1The number of non-dominant layers of all individuals in (A) is recorded as Rank1, F2The Chinese is marked as Rank 2;
step 3, repeating the operations of the step 1 and the step 2, and performing non-dominated sorting on all individuals in the population, wherein the number of non-dominated layers is from low to high, and the smaller the Rank value is, the better the individuals are;
in the iterative process of the NSGA-II algorithm, the population size is always N, so that in the selection operation, individuals in the same Rank layer need to be selected or rejected, and a crowdedness distance C is defined, namely the sum of the distance differences of two adjacent individuals on each sub-target function, and C isk＝(fk+1,1-fk-1,1)+(fk-1,2-fk+1,2) Wherein f isk+1,1、fk-1,1、fk-1,2And fk+1,2Respectively expressed as target functions corresponding to (k +1,1), (k-1,2) and (k +1, 2);
the individuals with large crowding degree distance are better than the individuals with small crowding degree distance, the non-dominated layer number and the crowding degree distance are used as the last two elements of the decision vector, and the quality of the solution and the sorting operation are judged according to the non-dominated layer number and the crowding degree distance.
The invention has the beneficial effects that:
the multi-objective optimization design method of the Swiss rectifier based on the NSGA-II algorithm, provided by the invention, is used for carrying out multi-objective optimization on the complex problems of the Swiss rectifier, such as multi-objective, uncertainty, nonlinearity and multi-parameter, solves the problem of device selection of each element selected by experience, can select a performance index through a Pareto frontier obtained by the optimization algorithm, improves the efficiency to 95.28-96.86%, and has the power density of 4.48-6.05 KW/dm3The overall performance of the rectifier is optimized.
Drawings
FIG. 1 is a circuit topology of a Swiss rectifier;
FIG. 2 is a diagram of four turn-on circuits of a Swiss rectifier;
FIG. 3 is a flow chart of the NSGA-II algorithm;
FIG. 4 is a schematic diagram of an elite selection strategy;
FIG. 5 is a diagram of a DC inductor UI core structure;
FIG. 6 is a diagram of an output capacitor structure;
fig. 7 is a Pareto optimal solution front.
Detailed Description
The present invention will be further described with reference to the following specific examples, but the present invention is not limited to these examples.
Example 1:
step 1, performing power and volume modeling on the optimized component; FIG. 5 is a diagram of the UI core structure of a DC inductor L, the power P of the inductorLComprises the following steps:
Figure BDA0002160101270000051
wherein u isLIs the input voltage of an inductor, IrmsEffective value of current flowing through the inductor, fswTo the switching frequency, VLIs an inductive volume.
Coil volume V of inductorclComprises the following steps:
Figure BDA0002160101270000052
volume of conductor VcdComprises the following steps: vcd＝kpfVclWherein k ispfIs the fill factor of the coil and,
Figure BDA0002160101270000053
n is the number of turns of the coil,is conductor cross section area, magnetic core volumeComprises the following steps:
then VLComprises the following steps:
the on-state loss of the switching tube IGBT is as follows:
wherein, VGEIs the threshold voltage of IGBT, IIGBTIs the magnitude of the current flowing through the IGBT, rCEM is the modulation ratio for the on-state equivalent resistance. The switching losses of the IGBT are:
Figure BDA0002160101270000062
Esw(on)and Esw(off)The energy lost for switching on and off the IGBT once, respectively.
Then the total IGBT losses are: pIGBT＝Pcond,IGBT+Psw,IGBT。
The conduction loss of the diode is:
Pon,Diode＝VFIFD
VFis the forward conduction voltage drop of the diode, IFFor forward on-state current, D is the duty cycle, and the duty cycles of the different diodes can be obtained from the 4 conducting circuit states shown in fig. 2.
The off-state loss of the diode is:
Poff,Diode＝VRIR(1-D)
wherein, VRFor its reverse pressure drop, IRIs a diode reverse leakage current.
The switching losses of the diodes are:
Figure BDA0002160101270000063
Vfpand VrpRespectively forward and reverse peak voltage of the diode, IfpAnd IrpRespectively the forward and reverse peak currents through the diode, tfpFor its forward recovery time, tbIts reverse current fall time.
The total diode loss is then: pDiode＝Pon,Diode+Poff,Diode+Psw,Diode。
Step 2, defining an objective function, taking the efficiency and the power density of the Swiss rectifier as the objective function, wherein the efficiency eta is as follows:
Figure BDA0002160101270000064
wherein, P0For output power, set to 5KW, the power density ρ is:
Figure BDA0002160101270000065
step 3, initializing a population, setting the population size to be 200, setting the maximum evolution algebra to be 200, setting the number of decision variables to be 39, carrying out real number coding on the decision variables, taking the established component database as a constraint condition, inputting the upper limit and the lower limit of each parameter variable, randomly generating a 200 x 43 decision matrix, setting the 40 th column and the 41 th column of the matrix to be target function values, setting the last two columns to be non-dominated layer numbers and crowding degree distances, and setting one parameter vector to correspond to one individual or be called a chromosome.
Step 4, selecting, adopting rapid non-dominated sorting, and setting two parameters niAnd Si，niGoverning the number of individuals i, S, for all individuals in a populationiA set of individuals governed by an individual i;
firstly, n in the population is selectediIndividuals of 0, i.e. non-dominant optimal individuals, and put into the set F1Performing the following steps;
for F1Respectively finding out a dominant individual set S corresponding to iiTo SiIndividual j, pair n ofjPerforming a subtraction operation, i.e. nj＝nj-1, excluding the influence of the individual i dominating j if n j0, then put the individual j into the set F2Performing the following steps;
then set F1For the first layer non-dominant set, for F1The number of non-dominant layers of all individuals in (A) is recorded as Rank1, F2The Chinese is marked as Rank 2;
repeating the above operations, and performing non-dominated sorting on all individuals in the population, wherein the number of non-dominated layers is from low to high, and the smaller the Rank value is, the better the individuals are;
in the iterative process of the NSGA-II algorithm, the population size is always N, so that individuals in the same Rank layer need to be selected and rejected in the selection operation, and a crowdedness distance C is defined, namely the sum of the distance differences of two adjacent individuals on each sub-target function, namely: ck＝(fk+1,1-fk-1,1)+(fk-1,2-fk+1,2)；
The individuals with large crowding degree distance are better than the individuals with small crowding degree distance, and the non-dominated layer number and the crowding degree distance are used as the last two elements of a decision vector and are used for judging the quality of a solution and carrying out sorting operation.
Step 5, selecting a binary tournament, setting the tournament size to be 2 and the matching pool size to be 100; randomly selecting two individuals in an initial population, comparing the non-dominant grades, and putting the low-grade individuals into a matching pool; if the grades are the same, comparing the crowdedness distance, and keeping the distance larger; if the non-dominant grade and the crowding degree distance are the same, one individual is randomly selected for reservation, and the operations are repeated until 100 individuals in the matching pool are obtained.
Step 6, crossing, namely randomly selecting two individuals from the current generation population as parent individuals for crossing operation, and performing crossing operation on genes on matched chromosomes to generate a pair of new chromosomes; the specific operation process is as follows: assuming that the gene length of the chromosome is L, in the range of [0, L ], an integer value K is randomly selected as a crossing position, and the matching chromosomes mutually exchange the genes of the [ K, L ] part at the crossing position, thereby forming a new pair of chromosomes, namely individuals.
And 7, mutation, namely defining a mutation operator, carrying out small-probability replacement on the genes of the individuals, and acting the mutation operator on the population to change the genes of some individuals in the population to generate new alleles.
And 8, combining the parent-child population subjected to the operations in the steps 5 to 7, and marking as a first generation population (gen is 1).
Step 9, fig. 4 is a schematic diagram of elite selection strategy, the offspring population and the parent population are combined through step 6 and step 7, the population size is 400, 200 individuals are selected as a new parent population through step 4, and a new offspring population P is generated through step 6 and step 7t+1(ii) a And repeating the step 9 until the maximum evolution algebra is reached to 200.
Step 10, as shown in FIG. 7, the Pareto optimal solution front edge is output after 200 generations, the front edge and the parameter matrix are output, the efficiency eta is 95.28-96.86%, and the power density rho is 4.48-6.05 KW/dm3And selecting the performance index in the range to obtain corresponding decision variable values, namely, the solution of the optimization problem, and selecting and designing components according to various parameters.
Although the present invention has been described with reference to the preferred embodiments, it should be understood that various changes and modifications can be made therein by those skilled in the art without departing from the spirit and scope of the invention as defined in the appended claims.

Claims (4)

1. A Swiss rectifier multi-objective optimization design method based on NSGA-II algorithm is characterized by comprising the following steps:
step 1, taking a direct current inductor L, an output capacitor C, a switching tube IGBT and a Diode in a basic topology circuit of a Swiss rectifier as optimization variables influencing performance indexes of the rectifier, and carrying out power and volume modeling on the direct current inductor L, the output capacitor C, the switching tube IGBT and the Diode;
step 2, defining an objective function: measuring the performance index of the Swiss rectifier by taking the efficiency and the power density of the Swiss rectifier as objective functions;
step 3, initializing a population: setting the population size as N, the maximum evolution algebra as gen, and the number of decision variables as V; then, real number coding is carried out on the decision variables, and the upper limit and the lower limit of each parameter variable are input by taking the established component database as a constraint condition; randomly generating an N-dimensional decision matrix, taking a V +1 th column and a V +2 th column of the decision matrix as objective function values, taking the last two columns as non-dominant layer numbers and crowding degree distances, and enabling one parameter vector to correspond to one individual or be called a chromosome;
step 4, selection: setting two parameters niAnd Si，niGoverning the number of individuals i, S, for all individuals in a populationiJudging the quality of the solution of the matrix and the sorting operation for the individual set dominated by the individual i by adopting rapid non-dominated sorting;
step 5, selecting the binary championship: setting the championship size to be 2 and the matching pool size to be N/2; randomly selecting two individuals in an initial population, comparing the non-dominant grades of the two individuals, and putting the low-grade individual into a matching pool; if the grades are the same, comparing the crowdedness distance, and keeping the distance larger; if the non-domination grade and the crowding degree distance are the same, randomly selecting one individual to be reserved, and repeating the step until the number of the individuals in the matching pool is N/2;
step 6, crossing: firstly, randomly selecting two individuals in a current generation population as parent individuals of cross operation, carrying out cross operation on genes on matched chromosomes to generate a pair of new chromosomes, and repeating the cross operation to form a new generation population; wherein the current generation population is a parent generation population;
and 7, mutation: defining a mutation operator, carrying out small-probability replacement on the gene of an individual, and acting the mutation operator on a population to change the gene of a part of individuals in the population to generate a new allele, and marking the population at the moment as an offspring population;
step 8, merging the parent population obtained in the step 6 and the child population obtained in the step 7, and marking the merged population as a first generation population (gen is 1) with the size of the population being N;
step 9, taking the first generation population obtained in the step 8 as a parent population, then carrying out cross variation on the parent population to obtain a child population, and then combining the parent population and the child population in the step to form a population, wherein the size of the population is 2N;
step 10, processing the population formed in the step 9 by using the operation process of the step 4, selecting N individuals as a new parent population, and generating a new child population P through the steps 6 and 7t+1(ii) a Repeating the step 9 until the maximum evolution algebra gen is reached;
and 11, outputting the Pareto optimal solution front edge and the parameter matrix, obtaining corresponding decision variable values, namely solutions of optimization problems, according to the selected target performance indexes, and selecting and setting components according to the parameters.
2. The optimal design method according to claim 1, wherein the power and volume modeling process of step 1 comprises: calculating the power of an inductor, the power P of the inductorLComprises the following steps:
Figure FDA0002160101260000021
wherein u isLIs the input voltage of an inductor, IrmsEffective value of current flowing through the inductor, fswTo the switching frequency, VLIs the volume of the inductor;
calculating the coil volume of the inductor; the coil volume VclComprises the following steps:
Figure FDA0002160101260000022
wherein, ω iswIs the winding width, dwIs the winding depth;
calculating the volume V of the conductorcdComprises the following steps: vcd＝kpfVclWherein k ispfIs the fill factor of the coil and,
Figure FDA0002160101260000023
n is the number of turns of the coil,
Figure FDA0002160101260000024
is the cross-sectional area of the conductor,
calculating core volume
Figure FDA0002160101260000025
Comprises the following steps:then the volume of the inductor VLComprises the following steps:
Figure FDA0002160101260000027
wherein l represents the core length;
and calculating the on-state loss of the IGBT of the switching tube as follows:
Figure FDA0002160101260000028
wherein, VGEIs threshold voltage of IGBT，IIGBTIs the magnitude of the current flowing through the IGBT, rCEIs an on-state equivalent resistance, M is a modulation ratio,
Figure FDA0002160101260000029
expressed as a voltage current phase angle;
the switching losses of the IGBT are:
Figure FDA00021601012600000210
wherein E issw(on)And Esw(off)Energy lost for switching on and off the IGBT once, respectively, INRated operating current for the rectifier, uCENAt a rated operating voltage of upnOutputting a voltage for the rectifier;
and (3) calculating the total loss of the IGBT as follows: pIGBT＝Pcond,IGBT+Psw,IGBT；
The conduction loss of the diode is:
Pon,Diode＝VFIFD
wherein, VFIs the forward conduction voltage drop of the diode, IFThe current is positive on-state current, D is duty ratio, and the duty ratios of different diodes are obtained according to 4 conducting circuit states;
the off-state loss of the diode is:
Poff,Diode＝VRIR(1-D)
wherein, VRFor its reverse pressure drop, IRIs a diode reverse leakage current;
the switching losses of the diodes are:
Figure FDA0002160101260000031
wherein, VfpAnd VrpRespectively forward and reverse peak voltage of the diode, IfpAnd IrpRespectively the forward and reverse peak currents through the diode, tfpFor its forward recovery time, tbIs its reverse current fall time;
the total diode loss is then: pDiode＝Pon,Diode+Poff,Diode+Psw,Diode。
3. The optimal design method according to claim 1, wherein the specific process of defining the objective function in step 2 comprises: defining an objective function, taking the efficiency and the power density of the Swiss rectifier as the objective function, wherein the efficiency eta is as follows:
wherein, P0For output power, the power density ρ is:
wherein, P0To output power, PlossFor total loss, PlFor total loss of inductance, PIGBTFor total IGBT losses, PDiodeIs the total Diode loss; vGeneral assemblyIs the total volume of the component, VLIs the volume of the inductor, VIGBTIs the IGBT volume, VDiodeIs the diode volume, VCThe volume of the output capacitor, the total volume and the volume of each component are shown, and C is an inductor; the power density ρ is an objective function.
4. The optimal design method according to claim 1, wherein the specific process of judging the quality and the ranking of the solutions in the fourth step comprises:
step 1, selecting n in the populationiIndividuals of 0, i.e. non-dominant optimal individuals, and put into the set F1Performing the following steps;
step 2, pair set F1Respectively finding out a dominant individual set S corresponding to iiTo SiOf the population, the number n of dominant individuals j dominating all individuals in the populationjPerforming a subtraction operation, i.e. nj＝nj-1, excluding the influence of the individual i dominating j if nj0, then put the individual j into the set F2Performing the following steps;
then set F1For the first layer non-dominant set, for F1The number of non-dominant layers of all individuals in (A) is recorded as Rank1, F2The Chinese is marked as Rank 2;
step 3, repeating the operations of the step 1 and the step 2, and performing non-dominated sorting on all individuals in the population, wherein the number of non-dominated layers is from low to high, and the smaller the Rank value is, the better the individuals are;
in the iterative process of the NSGA-II algorithm, the population size is always N, so that in the selection operation, individuals in the same Rank layer need to be selected or rejected, and a crowdedness distance C is defined, namely the sum of the distance differences of two adjacent individuals on each sub-target function, and C isk＝(fk+1,1-fk-1,1)+(fk-1,2-fk+1,2) Where k is any point in the solution front edge, fk+1,1、fk-1,1、fk-1,2And fk+1,2Respectively expressed as target functions corresponding to (k +1,1), (k-1,2) and (k +1, 2); the individuals with large crowding degree distance are better than the individuals with small crowding degree distance, the non-dominated layer number and the crowding degree distance are used as the last two elements of the decision vector, and the quality of the solution and the sorting operation are judged according to the non-dominated layer number and the crowding degree distance.
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