Wind power forecasting method based on improved sparrow algorithm and support vector machine optimization

By improving the Sparrow algorithm to optimize the support vector machine, the C-SSA-SVM ultra-short-term wind power power prediction model was constructed, which solved the problem of low prediction accuracy of the stroke wind power in the existing technology, and achieved higher prediction accuracy and effect.

CN119765299APending Publication Date: 2025-04-04YUNNAN ELECTRIC POWER TESTING & RES INST (GRP) CO LTD
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Patent Information

Application Number
CN202411830907.0
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2024-12-12
Publication Date
2025-04-04

AI Technical Summary

Technical Problem

In the prior art, the prediction accuracy of ultra-short-term wind power power through the Sparrow algorithm is not high enough, and it is difficult to effectively solve the volatility and randomness of wind power power.

Method used

The improved Sparrow algorithm is used to optimize the support vector machine, and the C-SSA-SVM ultra-short-term wind power power prediction model is constructed, and the accuracy of the prediction model is improved by iteratively finding the optimal kernel parameter g and the penalty factor c.

Benefits of technology

By improving the Sparrow algorithm to optimize the support vector machine, the ultra-short-term prediction accuracy of wind power is significantly improved, and the wind power can be predicted more accurately, improving the prediction effect.

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Abstract

The invention discloses a wind power prediction method based on an improved sparrow algorithm to optimize a support vector machine, comprising the following steps: collecting meteorological data and output power data of a wind farm, processing the meteorological data and the output power data of the wind farm to obtain a data set; using the sparrow algorithm to iteratively optimize a kernel parameter g and a penalty factor c of a support vector machine, and constructing an ultra-short-term wind power prediction model based on C-SSA-SVM; inputting the data set into the ultra-short-term wind power prediction model to predict wind power; optimizing model parameters by combining the improved sparrow algorithm with a support vector machine, and then predicting wind power by using the short-term wind power prediction model, with high prediction accuracy.
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Description

Technical Field

[0001] The invention relates to the technical field of wind power prediction, and in particular to a wind power prediction method based on an improved sparrow algorithm to optimize a support vector machine. Background Art

[0003] In the field of new energy, the output power of solar and wind power generation is greatly affected by weather and environment, and has obvious volatility and randomness. Because changes in wind speed and sunlight intensity directly affect the power generation power, this uncertainty brings challenges to the security of the power grid and the absorption of new energy. Power forecasting can improve the stability of the power grid, optimize power system scheduling and energy storage management, reduce wind and solar power abandonment, support electricity market transactions, improve operation and maintenance efficiency, extend equipment life, promote technological innovation and data application, etc. Therefore, in the field of new energy power generation, power forecasting is crucial.

[0004] However, in the prior art, the prediction accuracy of ultra-short-term wind power by the sparrow algorithm is not high enough. Therefore, it is necessary to propose a wind power prediction method based on an improved sparrow algorithm to optimize a support vector machine. Summary of the invention

[0005] In view of the above-mentioned prior art, the present invention provides a wind power prediction method based on an improved sparrow algorithm to optimize a support vector machine, mainly to solve the technical problems existing in the above-mentioned background technology.

[0006] To achieve the above object, the technical solution of the embodiment of the present invention is implemented as follows:

[0007] The wind power prediction method based on the improved sparrow algorithm and support vector machine optimization includes the following steps:

[0008] Collecting meteorological data and output power data of the wind farm, and processing the meteorological data and the output power data of the wind farm to obtain a data set;

[0009] The sparrow algorithm is used to iteratively optimize the kernel parameter g and penalty factor c of the support vector machine, and an ultra-short-term wind power prediction model based on C-SSA-SVM is constructed.

[0010] The data set is input into the ultra-short-term wind power prediction model to predict the wind power.

[0011] Optionally, collecting meteorological data and output power data of the wind farm, and processing the meteorological data and the output power data of the wind farm to obtain a data set includes:

[0012] Meteorological data are collected to construct a meteorological feature vector, output power data of the wind farm are collected to construct a power feature vector, the Pearson correlation coefficient between the meteorological feature vector and the power feature vector is calculated, and the meteorological feature vector and the power feature vector with the highest Pearson correlation coefficient are used to construct a cross feature sequence to obtain a data set including the meteorological feature vector, the power feature vector and the cross feature sequence.

[0013] Optionally, the Pearson correlation coefficient between the meteorological characteristic vector and the power characteristic vector is calculated, and the expression is:

[0014] New Feature=V×H

[0015] Among them, V is wind speed and H is humidity.

[0016] Optionally, the process further includes normalizing the data set, wherein the formula for the normalization process is:

[0017]

[0018] Among them, x max is the maximum value in the data set, x min is the minimum value in the data set, and x is the current value.

[0019] Optionally, the sparrow algorithm is used to iteratively optimize the kernel parameter g and the penalty factor c of the support vector machine, including:

[0020] S1: Define the initial population sparrow position x, the dimension dim of the sparrow population, the maximum number of iterations maxeval, the population size n, and the number of discoverers, guards, and joiners in the population, and use Logistic mapping to initialize the initial position of each individual. The formula is:

[0021] x n+1 = r·x n ·(1-x n )

[0022] Among them, r is the control parameter, x n is the current value, x n+1 is the next value;

[0023] S2: Define the maximum and minimum values ​​of the kernel parameter g and penalty factor c of the support vector machine, and use the kernel parameter and penalty factor to represent the upper and lower limits of the sparrow search. The upper limit is ub, and the lower limit is lb. The search range A = ub-lb. At the same time, the kernel parameter g and penalty factor c are used to represent the initial fireworks position x = rand*(ub-lb)+lb;

[0024] S3: Introduce the fitness function. The fitness function formula is:

[0025]

[0026] Among them, R rmse is the root mean square error of the support vector machines g and c.

[0027] S4: Calculate the fitness of each sparrow and find the optimal sparrow fitness;

[0028] S5: Determine the termination condition and whether the maximum number of iterations reached in the termination condition is met. If it is met, output the minimum fitness value min fitness and the current optimal sparrow individual Xbest, and output the optimal bestc and bestg at the same time, otherwise continue to execute S1-S4.

[0029] Optionally, the update formula of the discoverer position in the population is as follows:

[0030]

[0031] in, is the value of the jth dimension of the i-th sparrow during iteration, R2 is the alarm value, ST is the safety threshold, Q is a random number that obeys the normal distribution, and L is a 1×m-dimensional matrix in which each element is 1.

[0032] Optionally, the position update formula of the sentinels in the population is as follows:

[0033]

[0034] Among them, x p is the optimal position occupied by the discoverer; x worst is the current worst position; A is a 1×m matrix.

[0035] Optionally, the position update formula of the joiner in the population is as follows:

[0036]

[0037] Among them, X best is the global optimal position, β is a step size control parameter, K is a random number indicating the direction of the sparrow's movement, f is the fitness value, and f i is the fitness value of the current position, f w is the global worst fitness value, and ε is a minimum constant.

[0038] The beneficial effects of the present invention are as follows: the present invention provides a wind power prediction method based on an improved sparrow algorithm to optimize a support vector machine, obtains a data set including a meteorological feature vector, a power feature vector and a cross feature sequence by processing meteorological data and power data of a wind farm, optimizes the parameters of an ultra-short-term wind power prediction model by using the sparrow algorithm and a support vector machine, and trains the data set by using an ultra-short-term wind power prediction model of C-SSA-SVM to obtain a trained model, thereby being able to predict wind power; the prediction accuracy is high and the effect is good. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 A schematic flow chart of a wind power prediction method based on an improved sparrow algorithm to optimize a support vector machine provided in an embodiment of the present invention;

[0040] Figure 2 A process diagram of a wind power prediction method based on an improved sparrow algorithm to optimize a support vector machine provided in an embodiment of the present invention;

[0041] Figure 3 This is a simulation test result diagram of an ultra-short-term wind power prediction model based on an improved sparrow algorithm to optimize a support vector machine provided in an embodiment of the present invention;

[0042] Figure 4 is another simulation test result diagram of the ultra-short-term wind power prediction model based on the improved sparrow algorithm to optimize the support vector machine provided in an embodiment of the present invention;

[0043] Figure 5 It is a Pearson correlation analysis diagram of a group of meteorological data and power data of a wind power prediction method based on an improved sparrow algorithm to optimize a support vector machine provided in an embodiment of the present invention;

[0044] Figure 6 It is another set of Pearson correlation analysis diagrams of meteorological data and power data in the wind power prediction method based on the improved sparrow algorithm to optimize the support vector machine provided in the embodiment of the present invention. DETAILED DESCRIPTION

[0045] The technical solution of the present invention is further elaborated in detail below in conjunction with the drawings and specific embodiments of the specification. Unless otherwise defined, all technical and scientific terms used herein have the same meaning as those commonly understood by technicians in the technical field of the present invention. The terms used in the specification of the present invention herein are only for the purpose of describing specific embodiments and are not intended to limit the present invention. In the following description, the expression "some embodiments" is related to a subset of all possible embodiments, but it should be understood that "some embodiments" can be the same subset or different subsets of all possible embodiments, and can be combined with each other without conflict.

[0046] In the following description, a large number of specific details are provided to provide a more thorough understanding of the present invention. However, it is apparent to those skilled in the art that the present invention can be implemented without one or more of these details. In other examples, in order to avoid confusion with the present invention, some technical features well known in the art are not described.

[0047] It should be understood that the present invention can be implemented in different forms and should not be interpreted as being limited to the embodiments proposed herein. On the contrary, providing these embodiments will make the disclosure thorough and complete, and the scope of the present invention will be fully conveyed to those skilled in the art. And the purpose of the terms used herein is only to describe specific embodiments and is not intended to be a limitation of the present invention. When used herein, the singular forms of "one", "one" and "said / the" are also intended to include plural forms, unless the context clearly indicates another way. It should also be understood that the terms "compose" and / or "include" when used in this specification determine the presence of the features, integers, steps, operations, elements and / or parts, but do not exclude the presence or addition of one or more other features, integers, steps, operations, elements, parts and / or groups. When used herein, the term "and / or" includes any and all combinations of the relevant listed items.

[0048] It should also be noted that when an element is referred to as being "fixed to" another element, it may be directly on the other element or there may be a central element. When an element is considered to be "connected to" another element, it may be directly connected to the other element or there may be a central element at the same time. The terms "vertical", "horizontal", "inside", "outside", "left", "right" and similar expressions used herein are for illustrative purposes only and are not intended to be the only implementation method.

[0049] In order to fully understand the present invention, a detailed structure will be proposed in the following description to illustrate the technical solution proposed by the present invention. The optional embodiments of the present invention are described in detail as follows, but in addition to these detailed descriptions, the present invention may also have other implementations.

[0050] Example

[0051] Please refer to the attached Figure 1-2 The present application provides a wind power prediction method based on an improved sparrow algorithm to optimize a support vector machine, comprising the following steps:

[0052] Collecting meteorological data and output power data of the wind farm, and processing the meteorological data and the output power data of the wind farm to obtain a data set;

[0053] The sparrow algorithm is used to iteratively optimize the kernel parameter g and penalty factor c of the support vector machine, and an ultra-short-term wind power prediction model based on C-SSA-SVM is constructed.

[0054] The data set is input into the ultra-short-term wind power prediction model to predict the wind power.

[0055] As an optional implementation manner, meteorological data and output power data of a wind farm are collected, and the meteorological data and the output power data of the wind farm are processed to obtain a data set, including:

[0056] Collect meteorological data to construct a meteorological feature vector, collect output power data of the wind farm to construct a power feature vector, calculate the Pearson correlation coefficient between the meteorological feature vector and the power feature vector, construct a cross feature sequence from the meteorological feature vector and the power feature vector with the highest Pearson correlation coefficient, and obtain a data set including the meteorological feature vector, the power feature vector and the cross feature sequence;

[0057] Specifically, meteorological data is collected, and the data collection density is one set every 10 minutes. Each data set is meteorological data for one week, that is, 1008 sets of meteorological data; the meteorological data includes wind speed data, wind direction data, temperature data, and humidity data, and the wind speed characteristic vector, wind direction characteristic vector, temperature characteristic vector, and humidity characteristic vector are constructed. The output power data of the wind farm in the same time period as the meteorological data is collected to construct a power characteristic vector; and a 6-element window function is used to process the wind speed characteristic vector, wind direction characteristic vector, temperature characteristic vector, humidity characteristic vector, and power characteristic vector, and the data points that differ from the mean by more than 2 times are defined as outliers and deleted to improve data quality and reduce the impact of outliers on the ultra-short-term wind power prediction model;

[0058] Then, the correlation analysis between meteorological data and power characteristics is used to help identify which meteorological factors have a significant impact on power output. Specifically, the Pearson correlation coefficient is calculated for the power feature vector with the wind speed feature vector, wind direction feature vector, temperature feature vector and humidity feature vector to measure the degree of linear correlation between the two variables. For feature vectors with high Pearson correlation coefficients, it means that these meteorological features have a strong linear relationship with wind power. The two groups of feature vectors with the highest Pearson correlation coefficients are used to construct a cross feature sequence, that is, the wind speed feature vector and the humidity feature vector are selected, and the cross feature sequence is constructed through the wind speed feature vector and the humidity feature vector. The formula is:

[0059] New Feature=V×H

[0060] Among them, V is wind speed and H is humidity;

[0061] This cross - feature sequence is a newly generated feature that can capture more complex relationships between meteorological data and power data, thereby improving the performance of the prediction model. The final dataset includes the original meteorological feature vectors, power feature vectors, and the newly generated cross - feature sequence.

[0062] As an optional implementation, it also includes normalizing the dataset. The formula for the normalization process is:

[0063]

[0064] where x max is the maximum value in the dataset, x min is the minimum value in the dataset, and x is the current value.

[0065] As an optional implementation, the sparrow algorithm is used to iteratively optimize the kernel parameter g and penalty factor c of the support vector machine, including:

[0066] S1: Define the initial population sparrow position x, the dimension dim of the sparrow population, the maximum number of iterations maxeval, the population size n, and the number of discoverers, guardians, and joiners in the population. The dimension dim of the sparrow population is 2, the maximum number of iterations maxeval is 2000, the population size n is 300, the warning value R2 is 0.4, and the initial position of each individual is initialized using the Logistic map. Through multiple iterations, a chaotic sequence within the range [0, 1] can be obtained. The formula is:

[0067] x n+1 = r · x n · (1 - x n )

[0068] where r is the control parameter (usually taking values between 3.57 < r < 4), x n is the current value, and x n+1 is the next value;

[0069] S2: Define the maximum and minimum values of the kernel parameter g and penalty factor c of the support vector machine. Use the kernel parameter and penalty factor to represent the upper and lower bounds of the sparrow optimization. The upper bound is denoted as ub, the lower bound is denoted as lb, the optimization range A = ub - lb, and at the same time, use the kernel parameter g and penalty factor c to represent the initial firework position x = rand * (ub - lb)+lb;

[0070] S3: Introduce the fitness function. The formula for the fitness function is:

[0071]

[0072] where R rmseis the root mean square error of the support vector machines g and c.

[0073] S4: Calculate the fitness of each sparrow and find the optimal sparrow fitness;

[0074] Specifically, the fitness function is used to update the position of individuals. During the iteration process, the position of individuals will be adjusted accordingly according to the value of the fitness function. Individuals with higher fitness values ​​will be retained by the algorithm and used in subsequent iterations, while individuals with lower fitness values ​​may be eliminated or updated in the hope of finding a better solution.

[0075] S5: Determine the termination condition and whether the maximum number of iterations reached in the termination condition is met. If it is met, output the minimum fitness value min fitness and the current optimal sparrow individual Xbest, and output the optimal bestc and bestg at the same time, otherwise continue to execute S1-S4.

[0076] As an optional implementation, the formula for updating the position of the discoverer in the population is as follows:

[0077]

[0078] in, is the value of the jth dimension of the i-th sparrow during iteration, R2 is the alarm value, R2∈[0,1], ST is the safety threshold, ST∈[0.5,1.0], Q is a random number that obeys the normal distribution, and L is a 1×m-dimensional matrix in which each element is 1.

[0079] As an optional implementation, the position update formula of the sentinels in the population is as follows:

[0080]

[0081] Among them, x p is the optimal position occupied by the discoverer; x worst is the current worst position; A is a 1×m matrix, in which each element is randomly assigned to 1 or -1, A + =A T (AA T ) -1 .

[0082] As an optional implementation, the position update formula of the joiner in the population is as follows:

[0083]

[0084] Among them, X bestis the global optimal position, β is a step size control parameter, a random number that obeys Gaussian distribution, K is a random number that represents the direction of the sparrow's movement, K∈[-1,1], f is the fitness value, f i is the fitness value of the current position, f w is the global worst fitness value, ε is a minimum constant, and ε is 0.01.

[0085] The meteorological characteristic vector and the power characteristic vector are obtained by processing the collected meteorological data and the output power data of the wind farm, and the cross-characteristic sequence is obtained by calculating the Pearson correlation coefficient between each meteorological characteristic vector and the power characteristic vector, and finally a data set including the meteorological characteristic vector, the power characteristic vector and the cross-characteristic sequence is obtained, and the data set is input into the ultra-short-term wind power prediction model constructed based on the improved sparrow algorithm to optimize the support vector machine, and the data set is trained to obtain a trained ultra-short-term wind power prediction model, so that the wind power in the next 7 hours can be predicted; the wind power can be accurately predicted by this method with high prediction accuracy; in order to verify the prediction accuracy of the method of the present invention, the original data (50 groups of meteorological data and power data) are used to carry out wind power prediction and simulation tests on the trained ultra-short-term wind power prediction model;

[0086] Table 1 50 sets of meteorological data and power data

[0087]

[0088]

[0089] Table 2 Prediction results of ultra-short-term wind power prediction model based on C-SSA-SVM 1

[0090] SVM SSA-SVM C-SSA-SVM RMSE / MW 3.1626 2.5307 2.3554 MAPE / % 13.6519% 6.2279% 3.2456% <![CDATA[R 2 ]]> 0.9894 0.9805 0.9986

[0091] Table 3 Prediction results of ultra-short-term wind power prediction model based on C-SSA-SVM 2

[0092] SVM SSA-SVM C-SSA-SVM RMSE / MW 4.2511 3.6419 2.4804 MAPE / % 7.9248% 5.4820% 3.5766% <![CDATA[R 2 ]]> 0.9857 0.9880 0.9940

[0093] Please refer to the attached Figure 3-6 The prediction results in the above table are as follows Figure 3-4 As shown, the Pearson correlation coefficient of the data is Figure 5-6 The original data were input into SVM (support vector machine), SSA-SVM (support vector machine optimized by sparrow algorithm) and C-SSA-SVM (support vector machine optimized by improved sparrow algorithm) for simulation test. The simulation results of wind power prediction value and actual value are shown in the figure below. Figure 3 , Figure 4As shown, it can be seen from the figure that the ultra-short-term wind power prediction model of C-SSA-SVM provided in the embodiment of the present invention can better realize the ultra-short-term prediction of wind power; the evaluation indicators in Table 2 and Table 3 are RMSE (mean square error), MAPE (mean absolute percentage error), and R2 (coefficient of determination); RMSE is an indicator to measure the difference between the predicted value and the actual value of the model, which can be used to measure the prediction accuracy of continuous variables. The smaller the RMSE, the higher the prediction accuracy of the model; MAPE is another indicator to measure the difference between the predicted value and the actual value, which can be used to measure the relative error of the predicted value. The smaller the MAPE, the higher the prediction accuracy of the model; R2 is used to measure the degree of fit of the model to the data, and its value is between 0 and 1. The closer the value is to 1, the better the prediction effect of the model; according to the prediction results of Table 2 and Table 3, compared with the other two models, the ultra-short-term wind power prediction model based on C-SSA-SVM provided by the present invention has a smaller RMSE, a smaller MAPE, and a larger one, indicating that the ultra-short-term wind power prediction model based on C-SSA-SVM provided by the present invention has a higher prediction accuracy and a better prediction effect of wind power.

[0094] The present invention integrates the sparrow algorithm with the support vector machine to optimize parameters, and proposes a prediction model based on C-SSA-SVM. Experimental and simulation results show that the method proposed in the present invention has a high accuracy rate for the training of small sample data sets and the prediction of single data, and can effectively solve the problems of slow and poor prediction of the prediction model of the support vector machine optimized by the traditional optimization algorithm, and provides a new idea for some extreme cases where prediction results need to be obtained quickly.

[0095] The above are only specific embodiments of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art can easily think of changes or substitutions within the technical scope disclosed by the present invention, which should be included in the protection scope of the present invention. The protection scope of the present invention should be based on the protection scope of the claims.

Claims

1. A wind power prediction method based on an improved sparrow algorithm to optimize support vector machine, characterized in that: The following steps are involved: Collecting meteorological data and output power data of the wind farm, and processing the meteorological data and the output power data of the wind farm to obtain a data set; The sparrow algorithm is used to iteratively optimize the kernel parameter g and penalty factor c of the support vector machine, and an ultra-short-term wind power prediction model based on C-SSA-SVM is constructed. The data set is input into the ultra-short-term wind power prediction model to predict the wind power.

2. The wind power prediction method based on the improved sparrow algorithm and support vector machine optimization according to claim 1 is characterized in that: Collecting meteorological data and output power data of the wind farm, and processing the meteorological data and the output power data of the wind farm to obtain a data set, including: Meteorological data are collected to construct a meteorological feature vector, output power data of the wind farm are collected to construct a power feature vector, the Pearson correlation coefficient between the meteorological feature vector and the power feature vector is calculated, and the meteorological feature vector and the power feature vector with the highest Pearson correlation coefficient are used to construct a cross feature sequence to obtain a data set including the meteorological feature vector, the power feature vector and the cross feature sequence.

3. The wind power prediction method based on the improved sparrow algorithm and support vector machine optimization according to claim 2 is characterized in that: The Pearson correlation coefficient between the meteorological characteristic vector and the power characteristic vector is calculated as follows: New Feature=V×H Among them, V is wind speed and H is humidity.

4. The wind power prediction method based on the improved sparrow algorithm and support vector machine optimization according to claim 2 is characterized in that: It also includes normalizing the data set, and the formula for the normalization is: Among them, x max is the maximum value in the data set, x min is the minimum value in the data set, and x is the current value.

5. The wind power prediction method based on the improved sparrow algorithm and support vector machine optimization according to claim 1 is characterized in that: The sparrow algorithm is used to iteratively optimize the kernel parameter g and penalty factor c of the support vector machine, including: S1: Define the initial population sparrow position x, the dimension dim of the sparrow population, the maximum number of iterations maxeval, the population size n, and the number of discoverers, guards, and joiners in the population, and use Logistic mapping to initialize the initial position of each individual. The formula is: x n+1 =r·x n ·(1-x n ) Among them, r is the control parameter, x n is the current value, x n+1 is the next value; S2: Define the maximum and minimum values ​​of the kernel parameter g and penalty factor c of the support vector machine, and use the kernel parameter and penalty factor to represent the upper and lower limits of the sparrow search. The upper limit is ub, and the lower limit is lb. The search range A = ub-lb. At the same time, the kernel parameter g and penalty factor c are used to represent the initial fireworks position x = rand*(ub-lb)+lb; S3: Introduce the fitness function. The fitness function formula is: Among them, R rmse is the root mean square error of the support vector machines g and c. S4: Calculate the fitness of each sparrow and find the optimal sparrow fitness; S5: Determine the termination condition and whether the maximum number of iterations reached in the termination condition is met. If it is met, output the minimum fitness value min fitness and the current optimal sparrow individual Xbest, and output the optimal bestc and bestg at the same time, otherwise continue to execute S1-S4.

6. The wind power prediction method based on the improved sparrow algorithm and support vector machine optimization according to claim 5 is characterized in that: The formula for updating the position of the discoverer in the population is as follows: in, is the value of the jth dimension of the i-th sparrow during iteration, R2 is the alarm value, ST is the safety threshold, Q is a random number that obeys the normal distribution, and L is a 1×m-dimensional matrix in which each element is 1.

7. The wind power prediction method based on improved sparrow algorithm optimization support vector machine according to claim 5 is characterized in that: The formula for updating the position of the sentinels in the population is as follows: Among them, x p is the optimal position occupied by the discoverer; x worst is the current worst position; A is a 1×m matrix.

8. The method for wind power prediction based on improved sparrow algorithm and support vector machine optimization according to claim 5 is characterized in that: The formula for updating the position of the joiner in the population is as follows: Among them, X best is the global optimal position, β is a step size control parameter, K is a random number indicating the direction of the sparrow's movement, f is the fitness value, and f i is the fitness value of the current position, f w is the global worst fitness value, and ε is a minimum constant.