# Mathematical Modeling Method of the Improved Genetic Algorithm for Random Power Fluctuation

Authors: Fu Zhu
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

In order to solve the problem that the traditional genetic algorithm has a slow search speed and is easy to fall into the local optimal solution, a mathematical modeling method of an improved genetic algorithm in random power fluctuation is proposed. Drawing on the idea of a genetic algorithm (GA) and using the randomness and stability trend of cloud droplets of the normal cloud model, the author proposes a new genetic algorithm, cloud genetic algorithm (CGA). CGA is implemented by the Y-condition cloud generator of the normal cloud model to realize the cross operation, and the basic cloud generator realizes the mutation operation. Finally, the power function optimization experiment and the IIR digital filter optimization design are carried out, and the standard GA, NQGA, CAGA, and LARES algorithms are carried out compared. The experimental results show that the IIR digital filter designed by CGA has the smallest maximum ripple (Ap) in the passband, which is 0.342, and the largest minimum attenuation (As) in the stopband, which is 34.27. It can be observed that the overall performance is better. The validity of the algorithm is proved, and it has a certain reference and application value.


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## Body

## 1. Introduction

The evolutionary process of a disease is usually the result of crossovers and mutations of chromosomes. Based on the principles of genetics and evolution in nature, many scientists have developed various types of coding to provide solutions and have to create and use different types of genetics to simulate it. Many different genetic operators have been developed to mimic the genetic properties of organisms in different environments. In this way, different encodings and different genetic operators create different genetic algorithms [1].Genetic algorithms are a global and advanced scientific method that follows the theoretical process of mutation in the body. During the research process, you can gain and compile knowledge of the research area and refine and manage the search process to achieve a good solution. The function of the genetic algorithm is based on the principle of optimal survival, and of the various solutions, a set of optimal solutions is developed [2]. At each stage of the genetic algorithm, the value of identity is based on the value of the physical body in the problem, and the process of reconstruction provided by natural genetics then creates a new approximate concept. These processes lead to the evolution of the population, and new individuals will adapt to a new environment rather than the old one, just as changes in nature.The genetic algorithm is a global research heuristic algorithm used to solve the best performance in the field of computer science and is a kind of mutation that can be overcome. The quality of the process is always there which easily falls into the minimum values. Figure1 shows the flow of the genetic algorithm. Genetic algorithms are widely used in bioinformatics, phylogeny, Chinese science, engineering, economics, chemistry, manufacturing, mathematics, physics, pharmacometrics, and other fields [3].Figure 1
Genetic algorithm flow.


## 2. Literature Review

In practical applications, the standard genetic algorithm has many defects in maintaining population diversity, convergence accuracy, and convergence speed, which limit the development and application of genetic algorithms. Therefore, researchers have improved the genetic algorithm from the aspects of parameter improvement and optimization, scheme adjustment, hybrid genetic algorithm, and neighborhood topology improvement. In terms of parameter improvement and optimization and scheme adjustment, Cao, Y. et al. proposed a new improved genetic algorithm, its fitness function can change with the individual state, at the same time, the mutation operation is adjusted in the genetic algorithm, and it is found that the performance of the improved genetic algorithm has been significantly improved [4]. Duan, Li et al. deal with the large dimension of the optimization problem, and an improved genetic algorithm is proposed [5]. Peng, D. et al. optimized and adjusted the operation operator of the genetic algorithm and adaptively processed the crossover and mutation operations in the genetic algorithm, at the same time, the corresponding formula is adjusted reasonably, and the elite retention strategy is optimized to adopt a more reasonable and effective scheme [6]. In the research on combining with other algorithms, Liu, Q. et al. combined the standard genetic algorithm with the particle swarm algorithm, the improved algorithm is specially reserved for the excellent individuals, and the particle swarm algorithm further optimizes the excellent individuals; the experiment found that the combination of the two algorithms can effectively improve the performance of the algorithm [7]. Yan, F. et al. proposed a new hybrid algorithm, the algorithm combines the genetic algorithm with the adaptive particle swarm algorithm and adjusts the selection strategy reasonably, at the same time, the operation operator of the genetic algorithm and the particle update rule of the adaptive particle swarm optimization algorithm are integrated, and the design method of the fitness function is improved; the experimental results show that the improved algorithm greatly improves the solution efficiency of the optimization problem [8].


## 3. Research Methods

### 3.1. Cloud Theory

Cloud models are models of variations of the uncertainty of the positive terms expressed in key words and their data representations, which often involve uncertainties and uncertainty of the context in the target world or human data. And the two are completely integrated together for the qualitative and quantitative combination of information processing provides a powerful means [9].


#### 3.1.1. Basic Concepts

Definition 1.
LetT be the language value on the universe u and map CTx:u⟶0,1,∀x∈u,x⟶CTx, then the distribution of CTx on u is called the membership cloud of T or cloud for short. When CTx obeys a normal distribution, it is called a normal cloud model.
The normal cloud model is a random number set that follows the normal distribution law and has a stable tendency and is characterized by three values: expected valueEx, entropy En, and superentropy He. Expected value Ex: the point in the number domain space that can best represent this qualitative concept, reflecting the position of the cloud’s center of gravity. Entropy En: on the one hand, it reflects the acceptable range of language values in the number domain space; on the other hand, it also reflects the probability that a point in the number domain space can represent the language value and the randomness of cloud droplets representing qualitative concepts. It reveals the correlation between ambiguity and randomness. Hyperentropy He: it is the uncertainty measure of entropy, namely the entropy of entropy, which reflects the cohesion of the uncertainty of all points representing the language value in the number domain space, that is, the cohesion of cloud droplets [10].


#### 3.1.2. Basic Cloud Generator

Algorithm 1: Basic normal cloud generator.
INPUT: {Ex, En, He}, n //Numerical features and cloud droplet countOUTPUT: {(x1, μ1), …, (xn, μn)}// ncloud dropletsFORi = 1 to n//Generate a normal random number with expected valueEn and variance HeEn′ = RANDN En,Hexi = RANDN Ex,En′μi=e−xi−Ex2/2En′2dropxi,μi //Generate the i-th cloud drop


#### 3.1.3.X Condition Cloud Generator

Given the three digital featuresEx,En,He of the cloud and a specific value x0 on the universe of discourse u, a cloud droplet drop x0,μi is generated, this kind of a cloud generator is called the X-condition cloud generator.Algorithm 2: X Condition Cloud Generator.
INPUT: {Ex,En,He}, n, x0OUTPUT: {x0,μ1, …, x0,μn}FORi = 1 to nEn′ = RANDN En,Heμi=e−xi−Ex2/2En′2dropx0,μi


#### 3.1.4. Y-Condition Cloud Generator

Given the three digital featuresEx,En,He of the cloud and a certain degree of certainty μ0, a cloud droplet drop xi,μ0 is generated, and such a cloud generator is called a Y-conditional cloud generator.Algorithm 3: Y Condition Cloud Generator.
INPUT:Ex,En,He,n,μ0OUTPUT: {x1,μ0, …, xn,μ0}FORi = 1 to nEn′ = RANDN En,Hexi=Ex±En′−2lnμ0dropxi,μ0


### 3.2. Cloud Genetic Algorithm

In the continuous variable space, there is a neighborhood around the global optimal solution. In this neighborhood, with the optimal solution as the center, the value of the objective function approaches the value from far to near. When the fitness of the current solution is large, the search should be carried out in a smaller neighborhood; otherwise, it should be searched in a larger neighborhood [11]. In this way, the region where the optimal solution is located can be positioned step by step, and finally, the optimal solution can be approached. Taking the search for the “black box“ of an aircraft as an example, the density of the aircraft wreckage is regarded as fitness. On the contrary, the smaller the scattering density of debris (the smaller the fitness), the smaller the range, and the black box is less likely to exist and should be searched in a larger range [12].Combined with the idea of a genetic algorithm, the cloud genetic algorithm follows the concept of crossover and mutation operation of GA, the crossover operation is realized by the Y-condition cloud generation algorithm of the normal cloud model, and the mutation operation is realized by the basic cloud generation algorithm. Since the normal cloud model has the characteristics of randomness and stability tendency, randomness can maintain the diversity of individuals to avoid the search falling into the local extremum, while the stable tendency can also protect the superior individual and then adaptively locate the global optimal value. The CGA is encoded with real numbers and is individually updated by the cloud model [13].Algorithm 4: Cloud Genetic Algorithm.
(1)
Initialize the population(2)
Calculate fitness(3)
Select operation(4)
Cross(4.1)
Randomly generated or artificially specified degree of certaintyμ(4.2)
Ex=Ff/Ff+Fmxf+Fm/Ff+Fmxm (1)(4.3)
En variable search range/ c1(4.4)
He=En/c2(4.5)
Generate a pair of children by Algorithm3(5)
Variation(5.1)
Ex takes the original individual(5.2)
En= variable search range/ c3(5.3)
He=En/c4 (Note: c1−4 is the control coefficient)(5.4)
Execute Algorithm 1, and generate a random numberTemp, when μ>Temp, update the individual (6) Go to (2) until the stopping condition is metIn formula (1), xf and xm are the parent and parent of the crossover operation, respectively; Ff and Fm correspond to their fitness, respectively. This means that the Ex in the crossover operation is determined by the weighted fitness of both parents and moves closer to the one with the larger fitness [14].


### 3.3. Parameter Influence and Performance Analysis

From the normal cloud generator, we know that the normal cloud is a pan-normal distribution with the characteristics of “more in the middle and less at both ends. “FromEn′～NEn,He2, It is understood that EX=Ex+−2lnμEn, standard deviation D=−2lnμHe. Therefore, the changes of parameters Ex and En affect the horizontal position and steepness of the cloud model, respectively, while He is proportional to the dispersion degree of cloud droplets, and μ is inversely proportional to it, that is, the larger the He is, the greater the degree of dispersion is, and the smaller the μ is (for the position of the cloud, the closer to the foot of the mountain), the more dispersed the cloud droplets are. All cloud droplets fluctuate randomly around the expected curve, and the magnitude of the fluctuation is controlled by [15].


#### 3.3.1. Certainty

The degree of certainty in 4.1 of Algorithm 4 is discussed first. The larger theμ, the closer the cloud droplets are to the top, and the narrower the variable is searched for. In accordance with the finding of the black box phenomenon described in the introduction, we adapt the μ value to the fitness and gradually locate the optimal value. Here, two methods are introduced; one is a deterministic method using a linear function, and the other is a stochastic method using the X-condition cloud generator [16, 17].Algorithm 5.
Deterministic Linear Function Method(1)μ=μmax−Fmax−F′Fmax+Fminμmax−μmin,where Fmax and Fmin represent the global “maximum“ and “minimum“ fitness values of the contemporary population, respectively, F′ is the larger of the fitness of the crossover of two parent individuals, μmax and μmin are the maximum and minimum values of the artificially specified degrees of certainty, such as μmax=0.95 and μmin=0.2.Algorithm 6.
RandomnessX Conditional Cloud Generator Method(2)Ex=Fmax,En=Fmax−Fminc5,He=Enc6,En′=RANDNEn,He,μ=e−F′−Ex2/2En′2.
According to Algorithms5 and 6, individuals with greater fitness have a narrower search range, which is conducive to protecting the pattern of better individuals; at the same time, the individual with the largest fitness has a certainty of 1, and the two subindividuals generated after the crossover operation are both Ex in formula (1), which is not conducive to population diversity. To this end, we add a constraint: when μ≥0.95, μ=0.95. In order to protect the optimal individual, we introduce an optimal retention strategy in the selection operation. According to the “3 σ” rule, c5 takes a value slightly less than 3, and we take c5=2.8. At the same time, it is suggested that 5≤c6≤15, we take c6=10.


#### 3.3.2. En and He

The larger theEn, the larger the horizontal width of cloud coverage so that the individual search range is larger during crossover and mutation operations. According to the “3σ” rule, combined with the speed and accuracy of the evolutionary algorithm, 6≤c13≤3×p (p is the population size) in Algorithm 4 is recommended. As the evolutionary algebra increases, a larger value can be taken [18]. If the He is too large, the “stable tendency” will be lost to a certain extent; if the He is too small, the “randomness” will be lost to a certain extent. It is recommended that c2,4 be within the range of 5 to 15.At the same time, in order to expand the search range in the early stage of evolution and improve the search accuracy in the later stage of evolution, parameters can be adaptively and dynamically adjusted according to algebra and fitness. For example,c1∼4 is given by the sigmoid function or linear function (monotonically increasing) of evolutionary algebra while overcoming the inconvenience of artificial specification. Although En and He are important parameters of the cloud model, in CGA, changes in both can produce the same evolutionary results through changes in Ex and certainty. Therefore, after several generations of evolution, the randomness of Ex and certainty partially masks the difference in evolutionary results caused by their different values [19]. Therefore, different values of c1∼4 within a certain range will not have a significant impact on the final evolutionary performance, but evolutionary differences may always exist, so the selection of parameters requires further research in the future. The author’s experiment takes c1,3=3∗p,c2,4=10.


## 3.1. Cloud Theory

Cloud models are models of variations of the uncertainty of the positive terms expressed in key words and their data representations, which often involve uncertainties and uncertainty of the context in the target world or human data. And the two are completely integrated together for the qualitative and quantitative combination of information processing provides a powerful means [9].


### 3.1.1. Basic Concepts

Definition 1.
LetT be the language value on the universe u and map CTx:u⟶0,1,∀x∈u,x⟶CTx, then the distribution of CTx on u is called the membership cloud of T or cloud for short. When CTx obeys a normal distribution, it is called a normal cloud model.
The normal cloud model is a random number set that follows the normal distribution law and has a stable tendency and is characterized by three values: expected valueEx, entropy En, and superentropy He. Expected value Ex: the point in the number domain space that can best represent this qualitative concept, reflecting the position of the cloud’s center of gravity. Entropy En: on the one hand, it reflects the acceptable range of language values in the number domain space; on the other hand, it also reflects the probability that a point in the number domain space can represent the language value and the randomness of cloud droplets representing qualitative concepts. It reveals the correlation between ambiguity and randomness. Hyperentropy He: it is the uncertainty measure of entropy, namely the entropy of entropy, which reflects the cohesion of the uncertainty of all points representing the language value in the number domain space, that is, the cohesion of cloud droplets [10].


### 3.1.2. Basic Cloud Generator

Algorithm 1: Basic normal cloud generator.
INPUT: {Ex, En, He}, n //Numerical features and cloud droplet countOUTPUT: {(x1, μ1), …, (xn, μn)}// ncloud dropletsFORi = 1 to n//Generate a normal random number with expected valueEn and variance HeEn′ = RANDN En,Hexi = RANDN Ex,En′μi=e−xi−Ex2/2En′2dropxi,μi //Generate the i-th cloud drop


### 3.1.3.X Condition Cloud Generator

Given the three digital featuresEx,En,He of the cloud and a specific value x0 on the universe of discourse u, a cloud droplet drop x0,μi is generated, this kind of a cloud generator is called the X-condition cloud generator.Algorithm 2: X Condition Cloud Generator.
INPUT: {Ex,En,He}, n, x0OUTPUT: {x0,μ1, …, x0,μn}FORi = 1 to nEn′ = RANDN En,Heμi=e−xi−Ex2/2En′2dropx0,μi


### 3.1.4. Y-Condition Cloud Generator

Given the three digital featuresEx,En,He of the cloud and a certain degree of certainty μ0, a cloud droplet drop xi,μ0 is generated, and such a cloud generator is called a Y-conditional cloud generator.Algorithm 3: Y Condition Cloud Generator.
INPUT:Ex,En,He,n,μ0OUTPUT: {x1,μ0, …, xn,μ0}FORi = 1 to nEn′ = RANDN En,Hexi=Ex±En′−2lnμ0dropxi,μ0


## 3.1.1. Basic Concepts

Definition 1.
LetT be the language value on the universe u and map CTx:u⟶0,1,∀x∈u,x⟶CTx, then the distribution of CTx on u is called the membership cloud of T or cloud for short. When CTx obeys a normal distribution, it is called a normal cloud model.
The normal cloud model is a random number set that follows the normal distribution law and has a stable tendency and is characterized by three values: expected valueEx, entropy En, and superentropy He. Expected value Ex: the point in the number domain space that can best represent this qualitative concept, reflecting the position of the cloud’s center of gravity. Entropy En: on the one hand, it reflects the acceptable range of language values in the number domain space; on the other hand, it also reflects the probability that a point in the number domain space can represent the language value and the randomness of cloud droplets representing qualitative concepts. It reveals the correlation between ambiguity and randomness. Hyperentropy He: it is the uncertainty measure of entropy, namely the entropy of entropy, which reflects the cohesion of the uncertainty of all points representing the language value in the number domain space, that is, the cohesion of cloud droplets [10].


## 3.1.2. Basic Cloud Generator

Algorithm 1: Basic normal cloud generator.
INPUT: {Ex, En, He}, n //Numerical features and cloud droplet countOUTPUT: {(x1, μ1), …, (xn, μn)}// ncloud dropletsFORi = 1 to n//Generate a normal random number with expected valueEn and variance HeEn′ = RANDN En,Hexi = RANDN Ex,En′μi=e−xi−Ex2/2En′2dropxi,μi //Generate the i-th cloud drop


## 3.1.3.X Condition Cloud Generator

Given the three digital featuresEx,En,He of the cloud and a specific value x0 on the universe of discourse u, a cloud droplet drop x0,μi is generated, this kind of a cloud generator is called the X-condition cloud generator.Algorithm 2: X Condition Cloud Generator.
INPUT: {Ex,En,He}, n, x0OUTPUT: {x0,μ1, …, x0,μn}FORi = 1 to nEn′ = RANDN En,Heμi=e−xi−Ex2/2En′2dropx0,μi


## 3.1.4. Y-Condition Cloud Generator

Given the three digital featuresEx,En,He of the cloud and a certain degree of certainty μ0, a cloud droplet drop xi,μ0 is generated, and such a cloud generator is called a Y-conditional cloud generator.Algorithm 3: Y Condition Cloud Generator.
INPUT:Ex,En,He,n,μ0OUTPUT: {x1,μ0, …, xn,μ0}FORi = 1 to nEn′ = RANDN En,Hexi=Ex±En′−2lnμ0dropxi,μ0


## 3.2. Cloud Genetic Algorithm

In the continuous variable space, there is a neighborhood around the global optimal solution. In this neighborhood, with the optimal solution as the center, the value of the objective function approaches the value from far to near. When the fitness of the current solution is large, the search should be carried out in a smaller neighborhood; otherwise, it should be searched in a larger neighborhood [11]. In this way, the region where the optimal solution is located can be positioned step by step, and finally, the optimal solution can be approached. Taking the search for the “black box“ of an aircraft as an example, the density of the aircraft wreckage is regarded as fitness. On the contrary, the smaller the scattering density of debris (the smaller the fitness), the smaller the range, and the black box is less likely to exist and should be searched in a larger range [12].Combined with the idea of a genetic algorithm, the cloud genetic algorithm follows the concept of crossover and mutation operation of GA, the crossover operation is realized by the Y-condition cloud generation algorithm of the normal cloud model, and the mutation operation is realized by the basic cloud generation algorithm. Since the normal cloud model has the characteristics of randomness and stability tendency, randomness can maintain the diversity of individuals to avoid the search falling into the local extremum, while the stable tendency can also protect the superior individual and then adaptively locate the global optimal value. The CGA is encoded with real numbers and is individually updated by the cloud model [13].Algorithm 4: Cloud Genetic Algorithm.
(1)
Initialize the population(2)
Calculate fitness(3)
Select operation(4)
Cross(4.1)
Randomly generated or artificially specified degree of certaintyμ(4.2)
Ex=Ff/Ff+Fmxf+Fm/Ff+Fmxm (1)(4.3)
En variable search range/ c1(4.4)
He=En/c2(4.5)
Generate a pair of children by Algorithm3(5)
Variation(5.1)
Ex takes the original individual(5.2)
En= variable search range/ c3(5.3)
He=En/c4 (Note: c1−4 is the control coefficient)(5.4)
Execute Algorithm 1, and generate a random numberTemp, when μ>Temp, update the individual (6) Go to (2) until the stopping condition is metIn formula (1), xf and xm are the parent and parent of the crossover operation, respectively; Ff and Fm correspond to their fitness, respectively. This means that the Ex in the crossover operation is determined by the weighted fitness of both parents and moves closer to the one with the larger fitness [14].


## 3.3. Parameter Influence and Performance Analysis

From the normal cloud generator, we know that the normal cloud is a pan-normal distribution with the characteristics of “more in the middle and less at both ends. “FromEn′～NEn,He2, It is understood that EX=Ex+−2lnμEn, standard deviation D=−2lnμHe. Therefore, the changes of parameters Ex and En affect the horizontal position and steepness of the cloud model, respectively, while He is proportional to the dispersion degree of cloud droplets, and μ is inversely proportional to it, that is, the larger the He is, the greater the degree of dispersion is, and the smaller the μ is (for the position of the cloud, the closer to the foot of the mountain), the more dispersed the cloud droplets are. All cloud droplets fluctuate randomly around the expected curve, and the magnitude of the fluctuation is controlled by [15].


### 3.3.1. Certainty

The degree of certainty in 4.1 of Algorithm 4 is discussed first. The larger theμ, the closer the cloud droplets are to the top, and the narrower the variable is searched for. In accordance with the finding of the black box phenomenon described in the introduction, we adapt the μ value to the fitness and gradually locate the optimal value. Here, two methods are introduced; one is a deterministic method using a linear function, and the other is a stochastic method using the X-condition cloud generator [16, 17].Algorithm 5.
Deterministic Linear Function Method(1)μ=μmax−Fmax−F′Fmax+Fminμmax−μmin,where Fmax and Fmin represent the global “maximum“ and “minimum“ fitness values of the contemporary population, respectively, F′ is the larger of the fitness of the crossover of two parent individuals, μmax and μmin are the maximum and minimum values of the artificially specified degrees of certainty, such as μmax=0.95 and μmin=0.2.Algorithm 6.
RandomnessX Conditional Cloud Generator Method(2)Ex=Fmax,En=Fmax−Fminc5,He=Enc6,En′=RANDNEn,He,μ=e−F′−Ex2/2En′2.
According to Algorithms5 and 6, individuals with greater fitness have a narrower search range, which is conducive to protecting the pattern of better individuals; at the same time, the individual with the largest fitness has a certainty of 1, and the two subindividuals generated after the crossover operation are both Ex in formula (1), which is not conducive to population diversity. To this end, we add a constraint: when μ≥0.95, μ=0.95. In order to protect the optimal individual, we introduce an optimal retention strategy in the selection operation. According to the “3 σ” rule, c5 takes a value slightly less than 3, and we take c5=2.8. At the same time, it is suggested that 5≤c6≤15, we take c6=10.


### 3.3.2. En and He

The larger theEn, the larger the horizontal width of cloud coverage so that the individual search range is larger during crossover and mutation operations. According to the “3σ” rule, combined with the speed and accuracy of the evolutionary algorithm, 6≤c13≤3×p (p is the population size) in Algorithm 4 is recommended. As the evolutionary algebra increases, a larger value can be taken [18]. If the He is too large, the “stable tendency” will be lost to a certain extent; if the He is too small, the “randomness” will be lost to a certain extent. It is recommended that c2,4 be within the range of 5 to 15.At the same time, in order to expand the search range in the early stage of evolution and improve the search accuracy in the later stage of evolution, parameters can be adaptively and dynamically adjusted according to algebra and fitness. For example,c1∼4 is given by the sigmoid function or linear function (monotonically increasing) of evolutionary algebra while overcoming the inconvenience of artificial specification. Although En and He are important parameters of the cloud model, in CGA, changes in both can produce the same evolutionary results through changes in Ex and certainty. Therefore, after several generations of evolution, the randomness of Ex and certainty partially masks the difference in evolutionary results caused by their different values [19]. Therefore, different values of c1∼4 within a certain range will not have a significant impact on the final evolutionary performance, but evolutionary differences may always exist, so the selection of parameters requires further research in the future. The author’s experiment takes c1,3=3∗p,c2,4=10.


## 3.3.1. Certainty

The degree of certainty in 4.1 of Algorithm 4 is discussed first. The larger theμ, the closer the cloud droplets are to the top, and the narrower the variable is searched for. In accordance with the finding of the black box phenomenon described in the introduction, we adapt the μ value to the fitness and gradually locate the optimal value. Here, two methods are introduced; one is a deterministic method using a linear function, and the other is a stochastic method using the X-condition cloud generator [16, 17].Algorithm 5.
Deterministic Linear Function Method(1)μ=μmax−Fmax−F′Fmax+Fminμmax−μmin,where Fmax and Fmin represent the global “maximum“ and “minimum“ fitness values of the contemporary population, respectively, F′ is the larger of the fitness of the crossover of two parent individuals, μmax and μmin are the maximum and minimum values of the artificially specified degrees of certainty, such as μmax=0.95 and μmin=0.2.Algorithm 6.
RandomnessX Conditional Cloud Generator Method(2)Ex=Fmax,En=Fmax−Fminc5,He=Enc6,En′=RANDNEn,He,μ=e−F′−Ex2/2En′2.
According to Algorithms5 and 6, individuals with greater fitness have a narrower search range, which is conducive to protecting the pattern of better individuals; at the same time, the individual with the largest fitness has a certainty of 1, and the two subindividuals generated after the crossover operation are both Ex in formula (1), which is not conducive to population diversity. To this end, we add a constraint: when μ≥0.95, μ=0.95. In order to protect the optimal individual, we introduce an optimal retention strategy in the selection operation. According to the “3 σ” rule, c5 takes a value slightly less than 3, and we take c5=2.8. At the same time, it is suggested that 5≤c6≤15, we take c6=10.


## 3.3.2. En and He

The larger theEn, the larger the horizontal width of cloud coverage so that the individual search range is larger during crossover and mutation operations. According to the “3σ” rule, combined with the speed and accuracy of the evolutionary algorithm, 6≤c13≤3×p (p is the population size) in Algorithm 4 is recommended. As the evolutionary algebra increases, a larger value can be taken [18]. If the He is too large, the “stable tendency” will be lost to a certain extent; if the He is too small, the “randomness” will be lost to a certain extent. It is recommended that c2,4 be within the range of 5 to 15.At the same time, in order to expand the search range in the early stage of evolution and improve the search accuracy in the later stage of evolution, parameters can be adaptively and dynamically adjusted according to algebra and fitness. For example,c1∼4 is given by the sigmoid function or linear function (monotonically increasing) of evolutionary algebra while overcoming the inconvenience of artificial specification. Although En and He are important parameters of the cloud model, in CGA, changes in both can produce the same evolutionary results through changes in Ex and certainty. Therefore, after several generations of evolution, the randomness of Ex and certainty partially masks the difference in evolutionary results caused by their different values [19]. Therefore, different values of c1∼4 within a certain range will not have a significant impact on the final evolutionary performance, but evolutionary differences may always exist, so the selection of parameters requires further research in the future. The author’s experiment takes c1,3=3∗p,c2,4=10.


## 4. Results Analysis

### 4.1. Typical Electric Power Function Optimization

16 typical functions are used for electric power function optimization experiments. F1–F5 are used to verify the new quantum genetic algorithm (NQGA) proposed by the author, and F2, F3, F6–F16 are used to verify the LARES algorithm proposed by the author. At the same time, it is compared with the cloud adaptive genetic algorithm (CAGA) proposed for edge cloud joint resource allocation based on ant colony optimization and the genetic algorithm [20].The comparison between F1–F5 and NQGA is shown in Table1 (100 independent experiments), and the comparison between F2, F3, F7–F14, and the LARES algorithm (30 independent experiments) and the results are shown in Table 2. Table 1 shows the optimization results of the NQGA algorithm, and Table 2 shows the optimization results of the GA and LARES algorithms. The average algebra refers to the difference between the current optimal fitness and the reference optimal fitness, which is less than the average of multiple independent experiments of evolutionary algebra at 10−3 [21].Table 1
Performance comparison of CGA with NQGA and CAGA.
FunctionReferenceBest valueAverage valueMean algebraNQGACAGACGANQGACAGACGANQGACAGACGAF108.1080e − 0054.3368e − 0171.3147e − 0090.0005274.3368e − 0171.4949e − 00717.8026.133.60F200.000035416.3056e − 0129.6253e − 0090.000433703.4925e − 0112.5749e − 00867.89161.7039.97F333.0001213.0000003.0000003.000585513.0000003.000000190.1375.540.43F401.8627e − 0051.1369e − 0122.2204e − 0160.000242151.7896e − 061.3174e − 007329.13535.8080.23F5−1.03161.031581−1.031628−1.031628−1.0313833−1.031628−1.03162879.8283.5457.43Table 2
Performance comparison of GA, LARES, CAGA, and CGA.
FunctionReferenceBest valueAverage valueMean algebraGALARESCAGACGACAGACGACAGACGAF200.001030.000516.3056e − 0129.6253e−0093.49251e − 0112.5749e − 008161.7039.96F333.037133.000003.000003.000003.000003.0000075.540.43F60.9981.248881.131030.9980040.9980040.9980040.998004116.2109.83F709.4239250.000325.9009e − 0048.9394e – 0067.4938e − 0041.4909e − 00541015.6237249.8F877.001447.000647.000007.000007.000007.0000019.1819.37F9−186.73−185.588−186.604−186.7309−186.7309−186.6914−186.62678663.48906.53F1011.005591.007861.0000011.001501.0005847374.92482.37F1100.010580.002472.4399e − 0091.4891e − 0103.6361e − 0075.8789e − 009987.2526.1F1200.000790.000642.3842e − 0063.4605e − 0052.3842e − 0060.00014340.910746.37F1300.001530.001531.1921e − 0054.1921e − 0051.1921e − 0050.00013194.710642.73F140001.7217e − 0191.1809e − 0173.3200e − 0172.5825e − 01723.677.67F1500.192750.089270.153153.5051e − 0080.196050.01170200000170477.1F16014.532984.96543.280325.1994e − 0078.405761.8529e − 00620000015027.3CGA uses a population-based variation of the system and method of homogeneity and randomization of weather patterns [22]. The stability tendency can protect the better individuals and realize the adaptive positioning of the optimal value.The stochastic performance preserves individual diversity, which improves the ability of the algorithm to prevent falling into local extremum, and enables CGA to better maintain the balance between exploration and production. CGA can control “research” and “control”. As can be observed from Tables 1 and 2, CGA significantly outperforms NQGA and LARES algorithms in evolution speed, robustness, and ability to avoid getting stuck in local optima [23]. Since both NQGA and LARES algorithms outperform the traditional GA, improved genetic algorithm (IGA), and optimal preserving genetic algorithm (OMGA), CGA is better than GA, IGA, and OMGA. Compared with CAGA, CGA achieves better accuracy for the rest of the functions except for the average solution accuracy of F1, F2, F9, F12, and F13. Except for F8 and F12–F14, the convergence speed of CGA for other functions is significantly faster than that of CAGA. Therefore, the overall performance of CGA is better than that of CAGA [24].


### 4.2. IIR Digital Filter Optimization Design

The following uses CGA to directly optimize the design of the IIR digital filter in the frequency domain.The IIR digital filter adopts the cascade structure of the second-order section, that is, the following formula:(3)Hz=A∏k=1N1+akz−1+bkz−21+ckz−1+dkz−2.Example 1.
Design a 6th-order band-pass IIR digital filter with the following formula:(4)Hdejω=0,0⩽ω⩽0.28π,0.72π⩽ω⩽π,1,0.32π⩽ω⩽0.68π.
The IIR digital filter designed by CGA is shown in formula (5), and its frequency response is shown in Figures 2 and 3.(5)Hz=0.12116×1+1.4745z−1+z−21−0.94906z−1+0.83404z−2×1−1.4731z−1+z−21−0.00073464z−1+045374z−2.
Table3 compares the performance of CGA with the other four algorithms. As can be observed from Table 3, the IIR digital filter designed by CGA has the smallest passband maximum fluctuation (Ap) of 0.342 and the largest stopband minimum attenuation (As) of 34.27, which shows that the overall performance is better [25].Figure 2
Amplitude-frequency characteristic of the IIR digital filter.Figure 3
Logarithmic amplitude-frequency characteristics of the IIR digital filter.Table 3
Filter performance of each algorithm design.
AlgorithmOMGAIGAIqgaNQGACGAAp/dB0.6470.57640.7320.35460.342As/dB24.0031.2328.39533.5834.27


## 4.1. Typical Electric Power Function Optimization

16 typical functions are used for electric power function optimization experiments. F1–F5 are used to verify the new quantum genetic algorithm (NQGA) proposed by the author, and F2, F3, F6–F16 are used to verify the LARES algorithm proposed by the author. At the same time, it is compared with the cloud adaptive genetic algorithm (CAGA) proposed for edge cloud joint resource allocation based on ant colony optimization and the genetic algorithm [20].The comparison between F1–F5 and NQGA is shown in Table1 (100 independent experiments), and the comparison between F2, F3, F7–F14, and the LARES algorithm (30 independent experiments) and the results are shown in Table 2. Table 1 shows the optimization results of the NQGA algorithm, and Table 2 shows the optimization results of the GA and LARES algorithms. The average algebra refers to the difference between the current optimal fitness and the reference optimal fitness, which is less than the average of multiple independent experiments of evolutionary algebra at 10−3 [21].Table 1
Performance comparison of CGA with NQGA and CAGA.
FunctionReferenceBest valueAverage valueMean algebraNQGACAGACGANQGACAGACGANQGACAGACGAF108.1080e − 0054.3368e − 0171.3147e − 0090.0005274.3368e − 0171.4949e − 00717.8026.133.60F200.000035416.3056e − 0129.6253e − 0090.000433703.4925e − 0112.5749e − 00867.89161.7039.97F333.0001213.0000003.0000003.000585513.0000003.000000190.1375.540.43F401.8627e − 0051.1369e − 0122.2204e − 0160.000242151.7896e − 061.3174e − 007329.13535.8080.23F5−1.03161.031581−1.031628−1.031628−1.0313833−1.031628−1.03162879.8283.5457.43Table 2
Performance comparison of GA, LARES, CAGA, and CGA.
FunctionReferenceBest valueAverage valueMean algebraGALARESCAGACGACAGACGACAGACGAF200.001030.000516.3056e − 0129.6253e−0093.49251e − 0112.5749e − 008161.7039.96F333.037133.000003.000003.000003.000003.0000075.540.43F60.9981.248881.131030.9980040.9980040.9980040.998004116.2109.83F709.4239250.000325.9009e − 0048.9394e – 0067.4938e − 0041.4909e − 00541015.6237249.8F877.001447.000647.000007.000007.000007.0000019.1819.37F9−186.73−185.588−186.604−186.7309−186.7309−186.6914−186.62678663.48906.53F1011.005591.007861.0000011.001501.0005847374.92482.37F1100.010580.002472.4399e − 0091.4891e − 0103.6361e − 0075.8789e − 009987.2526.1F1200.000790.000642.3842e − 0063.4605e − 0052.3842e − 0060.00014340.910746.37F1300.001530.001531.1921e − 0054.1921e − 0051.1921e − 0050.00013194.710642.73F140001.7217e − 0191.1809e − 0173.3200e − 0172.5825e − 01723.677.67F1500.192750.089270.153153.5051e − 0080.196050.01170200000170477.1F16014.532984.96543.280325.1994e − 0078.405761.8529e − 00620000015027.3CGA uses a population-based variation of the system and method of homogeneity and randomization of weather patterns [22]. The stability tendency can protect the better individuals and realize the adaptive positioning of the optimal value.The stochastic performance preserves individual diversity, which improves the ability of the algorithm to prevent falling into local extremum, and enables CGA to better maintain the balance between exploration and production. CGA can control “research” and “control”. As can be observed from Tables 1 and 2, CGA significantly outperforms NQGA and LARES algorithms in evolution speed, robustness, and ability to avoid getting stuck in local optima [23]. Since both NQGA and LARES algorithms outperform the traditional GA, improved genetic algorithm (IGA), and optimal preserving genetic algorithm (OMGA), CGA is better than GA, IGA, and OMGA. Compared with CAGA, CGA achieves better accuracy for the rest of the functions except for the average solution accuracy of F1, F2, F9, F12, and F13. Except for F8 and F12–F14, the convergence speed of CGA for other functions is significantly faster than that of CAGA. Therefore, the overall performance of CGA is better than that of CAGA [24].


## 4.2. IIR Digital Filter Optimization Design

The following uses CGA to directly optimize the design of the IIR digital filter in the frequency domain.The IIR digital filter adopts the cascade structure of the second-order section, that is, the following formula:(3)Hz=A∏k=1N1+akz−1+bkz−21+ckz−1+dkz−2.Example 1.
Design a 6th-order band-pass IIR digital filter with the following formula:(4)Hdejω=0,0⩽ω⩽0.28π,0.72π⩽ω⩽π,1,0.32π⩽ω⩽0.68π.
The IIR digital filter designed by CGA is shown in formula (5), and its frequency response is shown in Figures 2 and 3.(5)Hz=0.12116×1+1.4745z−1+z−21−0.94906z−1+0.83404z−2×1−1.4731z−1+z−21−0.00073464z−1+045374z−2.
Table3 compares the performance of CGA with the other four algorithms. As can be observed from Table 3, the IIR digital filter designed by CGA has the smallest passband maximum fluctuation (Ap) of 0.342 and the largest stopband minimum attenuation (As) of 34.27, which shows that the overall performance is better [25].Figure 2
Amplitude-frequency characteristic of the IIR digital filter.Figure 3
Logarithmic amplitude-frequency characteristics of the IIR digital filter.Table 3
Filter performance of each algorithm design.
AlgorithmOMGAIGAIqgaNQGACGAAp/dB0.6470.57640.7320.35460.342As/dB24.0031.2328.39533.5834.27


## 5. Conclusion

The author uses the randomness and stable tendency of cloud droplets of the normal cloud model, combined with the idea of crossover and mutation of the genetic algorithm, the cross operation is realized by the cloud model’s Y-condition cloud generation algorithm, the basic cloud generator algorithm realizes the mutation operation, and the evolution process is skillfully completed, and a new cloud genetic algorithm is proposed. CGA utilizes the advantages of cloud structure to create security, randomness, and population-based transformations. One way to better sustainability is to protect people, be aware of the changing nature of the best value, and preserve self-diversity, thus improving the process’s ability against local rates. Population-based changes in integrated processes, some of the most common weather patterns, have resulted in strong research in small communities and people with small-scale research in large communities. Therefore, CGA can better maintain the balance between “exploration” and “mining” and has better search performance. Through the electric power function optimization experiment, it can be observed that the CGA algorithm is not only feasible but also outperforms GA, IGA, OMGA, NQGA, LARES, and CAGA. Not only the evolutionary generation is small, the evolution speed is improved, but the ability to obtain the optimal value is also strong, and the average value of multiple experiments is closer to the optimal value, so it has better robustness. The results of the optimal design of the IIR digital filter also prove that the algorithm has good application value. Subsequent work has focused on theoretical evidence of algorithmic integration, parametric analysis, algorithm improvement, and other applications of engineering optimization.



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*Source: 1889952-2022-08-25.xml*