A Real-Coding Population-Based Incremental Learning Evolutionary Algorithm for Multi-Satellite Scheduling

: With the increasing demand for earth observation in various ﬁelds, remote satellites play an important role in ground information assurance. Apparently, the effective scheduling and utilization of multi-satellite resources determine the quality and efﬁciency of information acquisition. In this paper, focusing on the problem of centralized multi-satellite scheduling, we establish a mathematical model of satellite scheduling with complex constraints of load and platform operation. We also propose a real-coding Population Incremental Based Learning (PBIL) algorithm to solve the multi-satellite scheduling problem. The real-coding format can greatly shorten the coding length compared to the traditional PBIL algorithm with binary coding so that the computational efﬁciency is improved. Additionally, we design a value probability matrix, correction coefﬁcient and mutation operator to guide better evolution and avoid early convergence. Finally, we take some numerical examples to verify the real-coding PBIL algorithm for multi-satellite scheduling. The performance of the algorithm is analyzed by comparing it with binary-coding PBIL and the Genetic Algorithm (GA). Additionally, the inﬂuence of key parameters on algorithm performance, such as probability correction coefﬁcient, is also analyzed.


Introduction
With the development of space technology and the popularity of satellite applications, the number and types of Earth observation satellites are increasing, playing quite an important role in the fields of economy, military and people's livelihood [1]. Facing the growing demand for large-scale and diversified tasks of users from all walks of life, multisatellite scheduling for constellations has become a hot research issue at home and abroad.
The Multi-Satellite Scheduling Problem (MSSP) refers to the process by which the satellite control department allocates resources, including satellites, payloads and ground stations, etc., to meet the routine needs or urgent requests of earth observation. The process must satisfy various constraints, maximizing the benefits of satellite missions during the satellite in-orbit operation [2,3].
Obviously, multi-satellite scheduling involves not only satellite payload resources but also ground management resources to ensure their normal operation. It can be seen that there are many constraints in multi-satellite scheduling. Therefore, reasonable models and efficient scheduling methods are of great significance for describing and solving problems such as the MSSP.
As for the multi-satellite scheduling methods, due to the Non-deterministic Polynomial (NP) characteristics of scheduling problems, most of the current research uses heuristic methods to solve the problem [4], mainly including Tabu Search (TS) [5], Ant

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Considering the load and platform operation of satellites in reality, we establish a mathematical model of satellite scheduling with complex constraints, and the related mathematical statement is given.

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We propose the real-coding Population-Based Incremental Learning (PBIL) algorithm to solve the MSSP. Compared to the traditional PBIL algorithm with binary coding, the real-coding format can greatly shorten the coding length so that the computational efficiency is improved. Additionally, we design the value probability matrix, correction coefficient and mutation operator to guide better evolution and avoid early convergence.

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We conduct some numerical simulations and analyses to verify the real-coding PBIL algorithm for the MSSP. The performance of the algorithm is analyzed by comparing it with binary-coding PBIL and Genetic Algorithm (GA). Additionally, the influence of key parameters on algorithm performance, such as probability correction coefficient, is analyzed.
We note that a shorter conference version of this paper appeared in Wireless and Satellite Systems-11th EAI International Conference (WiSATS 2020) [14]. Our initial conference paper neglected to provide the mathematical expressions of constraints in the multi-satellite scheduling model and did not address the influence of key parameters or coding formats on the performance of the PBIL algorithm. This manuscript addresses these issues and provides additional comparisons and analyses to verify the efficiency and quality of algorithms for multi-satellite scheduling in our study.
The organization of this paper is as follows: Section 2 provides a mathematical model of MSSP and gives the related mathematical statement of constraints; Section 3 proposes the real-coding PBIL algorithm for satellite schedules and explains a detailed framework of our method; the simulation details and results are elaborated on and analyzed in Section 4; lastly, the conclusion is presented in Section 5.

Symbol Definitions
For the convenience of description, the relevant symbol definitions are given, as shown in Table 1.

D rm
The duration of the satellite S m observing the target t r .

V rm
The benefit of the target t r observed by the satellite S m .

M ij
Time required for attitude maneuvering, i.e., satellite attitude adjustment time between missions.
DS m f ree Satellite S m currently available storage capacity.

DS rm
The satellite S m storage space capacity that will be occupied by the observation task for the target t r .
T gk , T dk The start time of the illumination period of the kth circle and the start time of the shadow period of the kth circle.
P 0 Ground shadow period constant load power. P C Charging power.
Battery parameters. Where E B is the battery capacity, η D is the discharge controller or discharge regulator efficiency, η C is the charge controller or charge regulator efficiency, η Wh is the watt-hour efficiency of the battery pack.

Problem Description and Basic Assumptions
The multi-satellite scheduling problem can be described as that M satellites cooperatively observing R targets in a scheduling cycle so that the objective function is optimal. The final output of the scheduling plan is mainly the allocation scheme of the observation mission. For a satellite, the distribution result can be expressed as a six-element array as follows: [m, r, ST rm , ET rm , D rm , V rm ] Considering the actual satellite system, some reasonable simplifications and basic assumptions for multi-satellite scheduling problems are as follows: (1) The observation of each target by the satellite requires a certain image scanning time; that is, the observation activity has a certain duration; (2) The satellite resources involved in the scheduling process are the satellites with sideswing capability carrying only one space-borne remote sensor; (3) The satellite needs to maintain a stable attitude during the execution of the observation mission. After completing an observation task, it needs to adjust the posture so that the observation task for the next target can be performed smoothly. From the start of the attitude adjustment to the stable attitude of the satellite, the time taken for this process is the satellite attitude adjustment time (also called the attitude maneuver stabilization time).

Task Time Constraints
A satellite can only observe one target at a time; that is, each satellite-borne remote sensor can only perform one observation task at any time. The mathematical expression is shown in Equation (1).
In Equation (1), t 1 , t 2 ∈ T, if the targets t 1 , t 2 occupy the satellite S m and the observation task for the target t 1 is followed by the task for the target t 2 , then f t 1 t 2 = 1; otherwise, f t 1 t 2 = 0. For the two adjacent observations of t 1 , t 2 , the next observation task can be performed only after the current task t 1 ends and the necessary attitude maneuver is completed; the attitude maneuver angle must not exceed 25 • .
The task start time and end time for each target t r shall be within its corresponding visible window, as shown in Equation (2).

Data Storage Constraints
Due to the limited storage space on the satellite, the data size between the two missions of the satellite cannot exceed the capacity of the storage device. The mathematical expression is shown in Equation (3).
In Equation (3), x rm = 1 indicates that the satellite S m observes the target t r , and x rm = 0 indicates that the satellite S m does not observe the target t r .

Energy Constraints
Energy constraints are manifested in the following two aspects: (1) the discharge depth of the battery for each discharge activity cannot exceed 20%; (2) the satellite must achieve the energy balance of the circle during each illumination ground period. That is, the discharge energy of the battery pack during the grounding can be fully replenished during the subsequent illumination period.
The mathematical expression is shown in Equation (4), and the meaning of the symbols is explained in Table 1.

Objective Function
In this paper, the optimization goal is to maximize the benefits of observation, the objective function Q 1 is as shown in Equation (5).

Real-Coding PBIL Algorithm for Multi-Satellite Scheduling
The real-coding PBIL algorithm for multi-satellite scheduling is mainly based on the evolutionary framework, including the key part of encoding, population initialization, fitness function, value probability matrix, population update, etc.

Encoding
When using the traditional PBIL algorithm for multi-satellite scheduling, the binarycoding format is often used, as shown in Figure 1.
Each bit of the chromosome represents an observation time window corresponding to a target, and its value is 0 or 1, which indicates whether the time window is selected to arrange the observation task. The length of the chromosome is the number of visible time windows for all satellites toward all targets. However, in the actual multi-satellite scheduling process, the number of satellites in a constellation and observation targets will be large, the planning interval will be long and the number of visible time windows will be correspondingly larger. If the binary-coding format is used to deal with the large-scale satellite scheduling problem, the chromosome will be so long that it will take a lot of time for each bit of the chromosome to do the constraints check, which will result in very low algorithm efficiency.
To enhance the computational efficiency of the algorithm, a real-coding format is elaborated in this paper, as shown in Figure 2.  In the real-coding format, each bit of chromosome represents a target. For one target in the overall target set , it corresponds to the i-th bit of the chromosome.

Range of Values
Assuming that the total number of visible time windows for the target is , each time window is numbered, corresponding from 1 to . Each bit of the chromosome represents an observation time window corresponding to a target, and its value is 0 or 1, which indicates whether the time window is selected to arrange the observation task. The length of the chromosome is the number of visible time windows for all satellites toward all targets.
However, in the actual multi-satellite scheduling process, the number of satellites in a constellation and observation targets will be large, the planning interval will be long and the number of visible time windows will be correspondingly larger. If the binary-coding format is used to deal with the large-scale satellite scheduling problem, the chromosome will be so long that it will take a lot of time for each bit of the chromosome to do the constraints check, which will result in very low algorithm efficiency.
To enhance the computational efficiency of the algorithm, a real-coding format is elaborated in this paper, as shown in Figure 2.

Real-Coding PBIL Algorithm for Multi-Satellite Scheduling
The real-coding PBIL algorithm for multi-satellite scheduling is mainly based on the evolutionary framework, including the key part of encoding, population initialization, fitness function, value probability matrix, population update, etc.

Encoding
When using the traditional PBIL algorithm for multi-satellite scheduling, the binarycoding format is often used, as shown in Figure 1.
Each bit of the chromosome represents an observation time window corresponding to a target, and its value is 0 or 1, which indicates whether the time window is selected to arrange the observation task. The length of the chromosome is the number of visible time windows for all satellites toward all targets. However, in the actual multi-satellite scheduling process, the number of satellites in a constellation and observation targets will be large, the planning interval will be long and the number of visible time windows will be correspondingly larger. If the binary-coding format is used to deal with the large-scale satellite scheduling problem, the chromosome will be so long that it will take a lot of time for each bit of the chromosome to do the constraints check, which will result in very low algorithm efficiency.
To enhance the computational efficiency of the algorithm, a real-coding format is elaborated in this paper, as shown in Figure 2. In the real-coding format, each bit of chromosome represents a target. For one target in the overall target set , it corresponds to the i-th bit of the chromosome.
Assuming that the total number of visible time windows for the target is , each time window is numbered, corresponding from 1 to . In the real-coding format, each bit of chromosome represents a target. For one target i in the overall target set I, it corresponds to the i-th bit of the chromosome.
Assuming that the total number of visible time windows for the target i is n i , each time window is numbered, corresponding from 1 to n i . If the value of the i-th gene position of the chromosome is w i , one of the natural numbers from 1 to n i , it means that the time window w i of target i is selected to complete the observation task of this target. Thus the mapping relationship between chromosomes and problem search space points is established.

Fitness Function
In the PBIL algorithm, the fitness function represents the direction of evolution. It determines which individual will be chosen to learn and generate the probability of value to guide the generation of offspring. Generally, different fitness functions can be established according to different optimization goals.
In this paper, the objective function in the mathematical model is directly taken as the fitness function.

Value Probability Matrix
In the real-coding PBIL algorithm, the probability is in the form of a matrix. The initial probability matrix is shown in Figure 3. If the value of the i-th gene position of the chromosome is , one of the natural numbers from 1 to , it means that the time window of target is selected to complete the observation task of this target. Thus the mapping relationship between chromosomes and problem search space points is established.

Fitness Function
In the PBIL algorithm, the fitness function represents the direction of evolution. It determines which individual will be chosen to learn and generate the probability of value to guide the generation of offspring. Generally, different fitness functions can be established according to different optimization goals.
In this paper, the objective function in the mathematical model is directly taken as the fitness function.

Value Probability Matrix
In the real-coding PBIL algorithm, the probability is in the form of a matrix. The initial probability matrix is shown in Figure 3.  A column of the probability matrix corresponds to a gene position of a chromosome; in other words, it corresponds to an observation target. For the target in the overall target set , it has observable time windows in total. This means that the value of the i-th gene position of the chromosome has selections. The matrix is the probability of value selections in the algorithm. The j-th row of the i-th column of the probability matrix represents the probability of selecting the j-th value of the i-th gene position. The probability matrix is initialized as shown in Equation (6), which ensures that each value of each gene position has the same probability at the beginning of evolution. The matrix is visually represented in Figure 3.

Range of Values
Among the new populations of each generation, the individual with the highest fitness value is selected for learning, and the probability matrix is updated, which will guide the optimization direction for the population updating. A column of the probability matrix corresponds to a gene position of a chromosome; in other words, it corresponds to an observation target. For the target i in the overall target set I, it has n i observable time windows in total. This means that the value of the i-th gene position of the chromosome has n i selections.
The matrix P is the probability of value selections in the algorithm. The j-th row of the i-th column of the probability matrix P represents the probability of selecting the j-th value of the i-th gene position. The probability matrix P is initialized as shown in Equation (6), which ensures that each value of each gene position has the same probability at the beginning of evolution. The matrix P is visually represented in Figure 3.
Among the new populations of each generation, the individual A with the highest fitness value is selected for learning, and the probability matrix P is updated, which will guide the optimization direction for the population updating. After the optimal individual A is generated, the update formula of the probability P is as shown in Equation (7).
Obviously, the probability of the j-th value of the i-th gene position is increased. In order to keep the sum of the probabilities of all the values of the i-th gene position as 1, it is necessary to normalize the probability matrix. The normalization formula is as shown in Equation (8).
Take the probability matrix update of the second generation as an example. The update of the probability matrix is shown in Figure 4. After the optimal individual is generated, the update formula of the probability is as shown in Equation (7).
Obviously, the probability of the j-th value of the i-th gene position is increased. In order to keep the sum of the probabilities of all the values of the i-th gene position as 1, it is necessary to normalize the probability matrix. The normalization formula is as shown in Equation (8).
Take the probability matrix update of the second generation as an example. The update of the probability matrix is shown in Figure 4.  As the evolution process progresses, the probability of each value will deviate differently from the initial probability. Additionally, the offspring generated according to the probability update will be more likely to be highly adaptable.

Mutation Operator
When the fitness value between individuals in the population is not much different, it means that the diversity of the population genes is low, and there may be a situation of evolutionary stagnation, resulting in insufficient global search and falling into a local optimum.
Therefore, in this paper, for each individual generated from the probability matrix, it decides whether to perform a mutation operation according to a small mutation probability. When it is selected for mutation, a certain bit in the chromosome is randomly selected, and the locus is randomly reassigned. The mutation operation is completed, as shown in Figure 5. As the evolution process progresses, the probability of each value will deviate differently from the initial probability. Additionally, the offspring generated according to the probability update will be more likely to be highly adaptable.

Mutation Operator
When the fitness value between individuals in the population is not much different, it means that the diversity of the population genes is low, and there may be a situation of evolutionary stagnation, resulting in insufficient global search and falling into a local optimum. Therefore, in this paper, for each individual generated from the probability matrix, it decides whether to perform a mutation operation according to a small mutation probability. When it is selected for mutation, a certain bit in the chromosome is randomly selected, and the locus is randomly reassigned. The mutation operation is completed, as shown in Figure 5.

Population Update
The update of the population is under the guidance of the probability matrix. With the real-coding format, the way of population updates is slightly different from that with binary coding.
Similar to the roulette selection strategy, the probability of different values of the same gene position is sequentially accumulated to obtain cumulative probabilities. Then, a random number between 0 and 1 is generated.
The value corresponding to the smallest cumulative probability greater than r is the value of the genetic position. In this way, new individuals generation and population updates can be completed.

End Condition
In this paper, the end condition is the number of evolutions determined by some numerical experiments in which the population fitness value is not significantly improved in the late stage of evolution, and the algorithm stops when the population completes these iterations. Usually, the number of iterations will be related to the size of the population. The larger the population is, the larger number of iterations is, and vice versa.

Algorithm Steps
The algorithm flow chart is shown in Figure 6.

Population Update
The update of the population is under the guidance of the probability matrix. With the real-coding format, the way of population updates is slightly different from that with binary coding.
Similar to the roulette selection strategy, the probability of n different values of the same gene position is sequentially accumulated to obtain n cumulative probabilities. Then, a random number r between 0 and 1 is generated.
The value corresponding to the smallest cumulative probability greater than r is the value of the genetic position. In this way, new individuals generation and population updates can be completed.

End Condition
In this paper, the end condition is the number of evolutions determined by some numerical experiments in which the population fitness value is not significantly improved in the late stage of evolution, and the algorithm stops when the population completes these iterations. Usually, the number of iterations will be related to the size of the population. The larger the population is, the larger number of iterations is, and vice versa.

Algorithm Steps
The algorithm flow chart is shown in Figure 6.

Population Update
The update of the population is under the guidance of the probability matrix. With the real-coding format, the way of population updates is slightly different from that with binary coding.
Similar to the roulette selection strategy, the probability of different values of the same gene position is sequentially accumulated to obtain cumulative probabilities. Then, a random number between 0 and 1 is generated.
The value corresponding to the smallest cumulative probability greater than r is the value of the genetic position. In this way, new individuals generation and population updates can be completed.

End Condition
In this paper, the end condition is the number of evolutions determined by some numerical experiments in which the population fitness value is not significantly improved in the late stage of evolution, and the algorithm stops when the population completes these iterations. Usually, the number of iterations will be related to the size of the population. The larger the population is, the larger number of iterations is, and vice versa.

Algorithm Steps
The algorithm flow chart is shown in Figure 6.  Step 1. Encode each task according to the real-coding format, and initialize the probability matrix.
Step 2. According to the above update method, use the probability matrix to guide the generation of the new population.
Step 3. According to the constraints in the mathematical model, each of the genetic positions of each chromosome in the population, that is, each task is checked for conflict. A task that does not pass the conflict check will be abandoned; that is, the value of this gene position of the chromosome will be set as zero.
Step 4. The fitness value of each individual is calculated to obtain the best individual with the highest fitness value.
Step 5. If the end condition is satisfied, shift to Step 6. Otherwise, according to the best individual generated in the previous step, update and normalize the probability matrix and shift to Step 2.
Step 6. The algorithm ends. Obtain the best individual and output the corresponding mission planning scheme.

Experimental Results and Analysis
In this paper, the model and algorithm proposed for multi-satellite scheduling are verified by numerical examples as follows. By designing simulation scenarios and typical examples, the performance of the real-coding PBIL algorithm is analyzed by comparing it with binary-coding PBIL and the Genetic Algorithm (GA). Additionally, the influence of key parameters on algorithm performance, such as the probability correction coefficient, is analyzed.

Simulation Scenario
First of all, we build the simulation scenario, including satellites, targets, ground stations, etc.

The Satellites
In this paper, the number of satellites is set to 10, remote satellites S1 to S10 are established in STK, and the distribution of the satellites in STK is shown in Figure 7.
Encode each task according to the real-coding format, and initialize the probability matrix.
According to the above update method, use the probability matrix to guide the generation of the new population.
According to the constraints in the mathematical model, each of the genetic positions of each chromosome in the population, that is, each task is checked for conflict. A task that does not pass the conflict check will be abandoned; that is, the value of this gene position of the chromosome will be set as zero.
The fitness value of each individual is calculated to obtain the best individual with the highest fitness value.
If the end condition is satisfied, shift to Step 6. Otherwise, according to the best individual generated in the previous step, update and normalize the probability matrix and shift to Step 2.
The algorithm ends. Obtain the best individual and output the corresponding mission planning scheme.

Experimental Results and Analysis
In this paper, the model and algorithm proposed for multi-satellite scheduling are verified by numerical examples as follows. By designing simulation scenarios and typical examples, the performance of the real-coding PBIL algorithm is analyzed by comparing it with binary-coding PBIL and the Genetic Algorithm (GA). Additionally, the influence of key parameters on algorithm performance, such as the probability correction coefficient, is analyzed.

Simulation Scenario
First of all, we build the simulation scenario, including satellites, targets, ground stations, etc.

The Satellites
In this paper, the number of satellites is set to 10, remote satellites S1 to S10 are established in STK, and the distribution of the satellites in STK is shown in Figure 7.

The Targets
A number of observation target points are randomly established globally and assigned to each target a benefit value. The distribution of the targets in STK is shown in Figure 8.

The Targets
A number of observation target points are randomly established globally and assigned to each target a benefit value. The distribution of the targets in STK is shown in Figure 8.

The Ground Stations
After the targets are observed by the satellites and the information is obtained, the data will be temporarily stored in the onboard device. When communication conditions permit, the data will be passed back to the ground station. In this paper, 12 ground stations are set up to simulate the completion of digital missions. The distribution of the ground stations in STK is shown in Figure 9.

The Ground Stations
After the targets are observed by the satellites and the information is obtained, the data will be temporarily stored in the onboard device. When communication conditions permit, the data will be passed back to the ground station. In this paper, 12 ground stations are set up to simulate the completion of digital missions. The distribution of the ground stations in STK is shown in Figure 9.

Typical Results of Multi-Satellite Scheduling Based on Real-Coding PBIL Algorithm
The average results of the 10 experiments are shown in Table 2, and one of the typical results is shown in Figure 10.

Objective function Number of targets
Average optimal fitness value Average running time (s)

The Ground Stations
After the targets are observed by the satellites and the information is obtained, the data will be temporarily stored in the onboard device. When communication conditions permit, the data will be passed back to the ground station. In this paper, 12 ground stations are set up to simulate the completion of digital missions. The distribution of the ground stations in STK is shown in Figure 9.

Typical Results of Multi-Satellite Scheduling Based on Real-Coding PBIL Algorithm
The average results of the 10 experiments are shown in Table 2, and one of the typical results is shown in Figure 10. Table 2. Results of multi-satellite scheduling.

Objective function Number of targets
Average optimal fitness value Average running time (s)

Typical Results of Multi-Satellite Scheduling Based on Real-Coding PBIL Algorithm
The average results of the 10 experiments are shown in Table 2, and one of the typical results is shown in Figure 10. Table 2. Results of multi-satellite scheduling.

Simulation Scene Settings Average Optimal Fitness Value
Objective function Number of targets Average optimal fitness value Average running time (s) Electronics 2022, 11, x FOR PEER REVIEW 11 of 16 (a) (b) Figure 10. This is the typical simulation results of multi-satellite scheduling based on a real-coding PBIL algorithm: (a) Evolution curve of the algorithm, a description of the optimization process, in which x-coordinate is iterations, and y-coordinate is fitness (corresponding to the scheduling results); (b) Gantt chart of multi-satellite scheduling results, in which x-coordinate is time, and y-coordinate is satellite ID, each colored rectangle represents a satellite mission, with a number next to it is the ground targets ID observed by that mission.
As shown in the table and figure above, it can be seen that the real-coding PBIL algorithm can effectively solve multi-satellite scheduling problems. As shown in the evolution curve in Figure 9a, it converges around generation 80 and has high computational efficiency. Figure 10. This is the typical simulation results of multi-satellite scheduling based on a real-coding PBIL algorithm: (a) Evolution curve of the algorithm, a description of the optimization process, in which x-coordinate is iterations, and y-coordinate is fitness (corresponding to the scheduling results); (b) Gantt chart of multi-satellite scheduling results, in which x-coordinate is time, and y-coordinate is satellite ID, each colored rectangle represents a satellite mission, with a number next to it is the ground targets ID observed by that mission.
As shown in the table and figure above, it can be seen that the real-coding PBIL algorithm can effectively solve multi-satellite scheduling problems. As shown in the evolution curve in Figure 9a, it converges around generation 80 and has high computational efficiency.

Influence of Probability Correction Coefficient on PBIL
In the real-coding PBIL algorithm for multi-satellite scheduling, the value probability matrix affects the direction of evolution, and the probability correction coefficient will affect the change of the value probability matrix so as to have a certain impact on the solution accuracy and calculation efficiency.
In this section, we analyze the influence of different probability correction coefficients on the performance of the real-coding PBIL algorithm. For 50 targets with different probability correction coefficients (X ∈ [0, 1]), 10 simulation experiments are carried out. The statistical results of the real-coding PBIL algorithm are shown in Table 3 and Figure 11. The following data are the average results of 10 simulation experiments.  It can be seen from the above table and figure that when the probability correction coefficient X is 0, the value probability matrix has no role in guiding the evolution direction but a random optimization process, so the average running time is long, and the quality of the solution is low. After adding the probability correction coefficient for guidance, the optimization ability of the algorithm is significantly enhanced, and the average running time is greatly reduced.
With the increase of coefficient X, the average running time is basically stable, but the average optimal fitness value generally shows a downward trend, especially between 0.1 and 0.5. The possible reason is that the increase of the correction probability coefficient leads to a rapid evolution process falling into a local optimum, and at the same time, a It can be seen from the above table and figure that when the probability correction coefficient X is 0, the value probability matrix has no role in guiding the evolution direction but a random optimization process, so the average running time is long, and the quality of the solution is low. After adding the probability correction coefficient for guidance, the optimization ability of the algorithm is significantly enhanced, and the average running time is greatly reduced.
With the increase of coefficient X, the average running time is basically stable, but the average optimal fitness value generally shows a downward trend, especially between 0.1 and 0.5. The possible reason is that the increase of the correction probability coefficient leads to a rapid evolution process falling into a local optimum, and at the same time, a large probability correction coefficient also makes the algorithm's optimization and exploration ability low, thus affecting the quality of solutions.
Hence, we can know that the real-coding PBIL algorithm can achieve optimal performance when X = 0.1.

Influence of Coding Format on PBIL
The encoding format is the main difference between the real-coding PBIL algorithm and the traditional PBIL with binary coding. Different encoding formats usually determine the length of chromosomes and the size of the probability matrix, thus having a significant effect on the efficiency of the evolution process.
In this paper, we implement two encoding formats of binary coding and real coding for the PBIL algorithm and solve the multi-satellite scheduling problem under the same conditions, respectively. Ten simulation experiments are carried out, and the statistical results are shown in Table 4.
As can be seen from the table above, the chromosome length of real coding is 50, which is equal to the quantities of targets. In contrast, the length of binary coding is 729, about 14 times that of real coding. Obviously, the chromosome length has a significant effect on the efficiency of constraint checking and the evolution process; therefore, shorter chromosome lengths can improve search speed. This is consistent with the results in the table that the running time of real-coding PBIL is significantly shorter than that of binary-coding PBIL. The simulation results show that the running time of binary-coding PBIL is about 1.5 times that of real-coding PBIL, which indicates that the encoding format plays an important role in the performance of the PBIL algorithm. More notably, real-coding PBIL can obtain higher optimal fitness values; that is, it can obtain higher quality solutions than binary-coding PBIL for multi-satellite scheduling.

Comparison of PBIL and GA
According to the optimization goal in Section 2, we carry out the comparison experiments of the real-coding PBIL algorithm and real-coding Genetic Algorithm for solving multi-satellite scheduling under the same conditions. The Evolutionary curves of 10 simulation experiments for 50 targets are shown in Figure 12, and the statistical results of different target quantities are shown in Table 5 and Figure 13. All the data as follows are the average of the results of 10 simulation experiments. than binary-coding PBIL for multi-satellite scheduling.

Comparison of PBIL and GA
According to the optimization goal in Section 2, we carry out the comparison experiments of the real-coding PBIL algorithm and real-coding Genetic Algorithm for solving multi-satellite scheduling under the same conditions. The Evolutionary curves of 10 simulation experiments for 50 targets are shown in Figure 12, and the statistical results of different target quantities are shown in Table 5 and Figure 13. All the data as follows are the average of the results of 10 simulation experiments.   As can be seen from the figure above, to solve the multi-satellite scheduling problem under the same conditions, the PBIL algorithm converges rapidly around iteration 80, while the GA algorithm converges after iteration 100, indicating that the PBIL algorithm has faster computational performance than the GA algorithm.
In addition, the evolution curve of 10 simulation experiments shows that the evolution curves obtained by the PBIL algorithm have small fluctuation, while that obtained by the GA algorithm have significant fluctuation and difference, indicating that the PBIL algorithm is also better than the GA algorithm in terms of computational stability.
For different target quantity scales, the ability of PBIL and GA for solving multi-satellite scheduling is further compared and analyzed.  As shown in the table and the figure above, the running time of the PBIL algorithm is obviously much shorter than that of the genetic algorithm, where the computational efficiency is almost doubled, and most of the optimal fitness values are slightly higher than that of GA.
Therefore, considering algorithm stability, computational efficiency and optimization quality, etc., the PBIL algorithm is superior to the genetic algorithm in solving satellite mission planning problems to some extent.

Conclusions
In this paper, a number of theoretical analyses and numerical experiments were conducted to find a better method to solve centralized multi-satellite scheduling problems.
Firstly, considering the load and platform operation of satellites in reality, we established a mathematical model of satellite scheduling with complex constraints, and the related mathematical statement was given.
Secondly, the real-coding Population-Based Incremental Learning (PBIL) algorithm As can be seen from the figure above, to solve the multi-satellite scheduling problem under the same conditions, the PBIL algorithm converges rapidly around iteration 80, while the GA algorithm converges after iteration 100, indicating that the PBIL algorithm has faster computational performance than the GA algorithm.
In addition, the evolution curve of 10 simulation experiments shows that the evolution curves obtained by the PBIL algorithm have small fluctuation, while that obtained by the GA algorithm have significant fluctuation and difference, indicating that the PBIL algorithm is also better than the GA algorithm in terms of computational stability.
For different target quantity scales, the ability of PBIL and GA for solving multisatellite scheduling is further compared and analyzed.
As shown in the table and the figure above, the running time of the PBIL algorithm is obviously much shorter than that of the genetic algorithm, where the computational efficiency is almost doubled, and most of the optimal fitness values are slightly higher than that of GA.
Therefore, considering algorithm stability, computational efficiency and optimization quality, etc., the PBIL algorithm is superior to the genetic algorithm in solving satellite mission planning problems to some extent.

Conclusions
In this paper, a number of theoretical analyses and numerical experiments were conducted to find a better method to solve centralized multi-satellite scheduling problems.
Firstly, considering the load and platform operation of satellites in reality, we established a mathematical model of satellite scheduling with complex constraints, and the related mathematical statement was given.
Secondly, the real-coding Population-Based Incremental Learning (PBIL) algorithm was proposed to solve the MSSP. Compared to the traditional PBIL algorithm with binary coding, the real-coding format can greatly shorten the coding length so that the computational efficiency is improved. Additionally, we designed the value probability matrix, correction coefficient and mutation operator to guide better evolution and avoid early convergence.
Furthermore, some numerical simulations and analyses were conducted to verify the real-coding PBIL algorithm for multi-satellite scheduling. Additionally, we analyzed the influence of key parameters such as the probability correction coefficient on algorithm performance. The performance of the algorithm was analyzed by comparing it with binarycoding PBIL and Genetic Algorithm (GA). According to the numerical results, it can be seen that when solving multi-satellite scheduling problems, in terms of the algorithm efficiency, solution quality and task completion rate, the real-coding PBIL algorithm is superior to other algorithms in our study.

Applications and Limitations
Concerning the actual satellite scheduling process, the real-coding PBIL algorithm might play a crucial part to help the Satellite Control Center make decisions precisely and quickly. Generally, users call for demands regarding earth observation and send these requirements to the Satellite Control Center. Then, the Center will aggregate all the demands to obtain the targets to be observed and make a reasonable scheduling plan within a specific period for satellites to meet the observation needs. The planning process is difficult to rely on manual completion, while the algorithm we proposed will be helpful. According to the information on targets, satellites and ground stations, it is easy to obtain the visible time window set of satellite to target. Just input the window set, and the algorithm will output the satellite scheduling result, that is, which target is to be observed by which satellite at which time.
In this paper, we carry out simulations to verify the algorithm dealing with 10 satellites and 50-100 targets. The results show that the algorithm displays good performance in both optimal speed and quality. However, the scenarios in our paper are far from the large-scale scenario with vast quantities of satellites and targets as the reality. Whether the algorithm can deal with large-scale satellite scheduling problems well still needs further discussions and verifications. Another limitation of our algorithm is that it might not be suitable for multi-satellite scheduling problems requiring the revisiting observation of targets. Due to the real-coding format, each target can be observed, at most, once in a scheduling period. Our approach might be applicable to situations with a large number of observation demands while the satellite resources are limited, rather than the scenarios with repeated observation requirements for a specific area.

Future Works
Concerning future work, two main paths could be taken. At first, the expansion of the satellite scheduling model can be more applicable to various situations in reality, such as multi-satellite scheduling with different payload types, large-scale constellations scheduling and multi-satellite scheduling for moving target search and tracking, etc.
In addition, we will continue to study algorithms for multi-satellite scheduling problems. The internal mechanism of the search algorithm will be further explored, and a more effective method for multi-satellite scheduling will be put forward, even considering the integration of artificial intelligence theory to achieve an end-to-end intelligent satellite scheduling process.