A method for measuring gas volume fraction of gas-liquid two-phase flow

Abstract

A kind of gas-liquid two-phase flow gas volume fraction measurement method, local gas rate signal at each position is collected by array optical fiber probe sensor, corresponding standard gas volume fraction is measured by flowmeter simultaneously, and sample data set is constructed;BP neural network is constructed, the weight and threshold of BP neural network are optimized using particle swarm optimization algorithm, and T weak predictors are obtained;T weak predictors are combined according to the following formula to obtain strong predictor, which is used to obtain the final gas volume fraction prediction value.The optical fiber probe sensor used in the application can accurately detect the gas phase and liquid phase medium in gas-liquid two-phase flow, provide basic data for accurate measurement, the weight and threshold of BP neural network are optimized using particle swarm optimization algorithm, help the network to jump out of local optimum, improve the global search ability of model, thereby improve the measurement accuracy, AdaBoost algorithm combines multiple BP neural network weak predictors optimized by PSO into a strong predictor, greatly improves the prediction accuracy and stability of model.

Description

Technical Field

[0001] The invention belongs to the field of measurement technology, and in particular relates to a method for measuring the gas volume fraction of a gas-liquid two-phase flow. Background Art

[0002] Multiphase flow is frequently encountered in the chemical, nuclear, and petroleum industries. Gas-liquid two-phase flow, as a typical multiphase flow system, exhibits complex fluid dynamics, known as flow patterns, due to complex interfacial effects and the relative motion caused by two dispersed phases with different physical properties. Different flow behaviors can affect sensor performance in flow parameter measurements. A challenging issue in flow parameter measurement is measuring the gas volume fraction, which is associated with the spatiotemporal structure of the flow pattern.

[0003] Fiber-optic probe sensors are a common tool for measuring gas-phase parameters in multiphase flows due to their resistance to electromagnetic interference, ease of array creation, and high sensitivity. However, a single fiber-optic probe can only measure the gas fraction in a local cross-section. Therefore, most studies have employed fiber-optic probe arrays to increase the number of measurement points, and then use the average value of these measurement points to estimate the cross-sectional gas fraction.

[0004] However, the multiphase flow structure has temporal and spatial complexity, and the measurement environment will change dynamically. Especially in annular and stirring flows, there will be a certain degree of error between the measured values ​​at different times and the true values. Simply adding the measured values ​​of each probe and averaging them cannot effectively eliminate this error. At the same time, due to the large gap between the gas-liquid two-phase flow and the difference in inherent properties, slippage will occur between the phases, resulting in a difference between the gas volume fraction (that is, the ratio of gas flow rate to total flow rate) and the cross-sectional gas content (that is, the area ratio of the gas phase to the pipe cross section). Therefore, there is a complex nonlinear relationship between the local gas fraction and the gas volume fraction, and it is difficult to directly establish an accurate model between the two. This shows that there are great challenges in relying on a single sensor to achieve accurate measurement of the gas volume fraction. Summary of the Invention

[0005] The technology to be solved by the present invention is to overcome the shortcomings of the existing technology and provide a method for measuring the gas volume fraction of a gas-liquid two-phase flow with a reasonable design and high measurement accuracy.

[0006] The technical solution adopted to solve the above technical problems is: a method for measuring the gas volume fraction of a gas-liquid two-phase flow, comprising the following steps:

[0007] Step 1. Use an array fiber optic probe sensor to collect local gas void fraction signals at various locations on the cross section. Simultaneously, use a flow meter to measure the corresponding standard gas volume fraction. The local gas void fraction signals at each location are processed to obtain local gas void fraction data. Each acquisition takes s, and the acquisition is repeated N times. The local gas void fraction data and the corresponding gas volume fraction at each location collected each time constitute a sample data set, thus forming a sample data set containing N sample data.

[0008] Step 2. Construct a BP neural network and use the particle swarm optimization algorithm to optimize the weights and thresholds of the BP neural network to obtain a weak predictor;

[0009] Step 2.1. Combine the weights and thresholds of the BP neural network into particle codes;

[0010] Step 2.2. Randomly generate a particle swarm in the solution space as the initial solution, where each particle represents a set of weights and thresholds;

[0011] Step 2.3. Evaluate the performance of each particle using the fitness function. Iteratively update the particle's velocity and position based on the evaluation results. During the update, the particle continuously approaches the optimal solution region. When the preset number of iterations is reached or the convergence condition is met, the optimized weights and thresholds are obtained, thus obtaining a weak predictor.

[0012] Step 3. Use the AdaBoost algorithm to train the weak predictor to obtain a strong predictor;

[0013] Step 3.1. Randomly select m sample data from the sample data set as the training set;

[0014] Step 3.2. Assign the same initial weight to each sample in the training set. These weights are used in the subsequent weighted training process.

[0015] Step 3.3. In each round of training, use the current sample weights to train the weak predictor to generate the weak predictor for this round. Determine the classification error rate of the weak predictor on the training set, and then derive the weight of the prediction sequence of the weak predictor based on the classification error rate.

[0016] Step 3.4. Update the sample weights for the next round of training based on the prediction results of the weak learner in this round and the weights of the prediction sequence. Increase the weights of misclassified samples and decrease the weights of correctly classified samples. Normalize the updated weights to ensure that the sum of all sample weights is 1.

[0017] Step 3.5. After a preset T rounds of training, generate T weak predictors;

[0018] Step 4. Combine T weak predictors according to the following formula to obtain a strong predictor to obtain the final gas volume fraction prediction value,

[0019]

[0020] Where F(x) is the predicted value of the final gas volume fraction obtained for the input local gas content x at each location, and f t (x) is the gas volume fraction prediction value of the local gas content x at each input position by the weak predictor in the tth round, α t is the weight of the prediction sequence of the weak predictor in round t.

[0021] Preferably, the method for processing the local gas content signal at each position in step 1 to obtain the local gas content data is:

[0022] Step 1.1. First, perform photoelectric conversion and analog-to-digital conversion on the reflected light signal of the array fiber probe sensor to obtain a digital signal;

[0023] Step 1.2. Convert the digital signal into a binary rectangular wave signal using a dynamic threshold adjustment method;

[0024] Step 1.2.1. Set two adjustable parameters, namely the maximum value a max and the minimum value a min , and give them two initial values ​​respectively;

[0025] Step 1.2.2. The current sampled signal amplitude a n Compared with the previously sampled signal amplitude a n-1 For comparison, a n >a n-1 , then a max Updated to a n , indicating that the current sampling signal amplitude is greater than the previous sampling signal amplitude, and the signal is close to the range of the gas phase signal, so the maximum value is updated; a n <a n-1 , then a min Updated to a n , indicating that the current sampling signal amplitude is smaller than the previous sampling signal amplitude, and the signal is close to the range of the water phase signal, so the minimum value is updated; a n =a n-1 , then keep a max and a min The original state remains unchanged;

[0026] Step 1.2.3. Determine the true value of the current sampling point according to the following formula (1) (2):

[0027] a n >a min +y (1)

[0028] a n <amax -y (2)

[0029] Where y is a variable that depends on the signal noise and is adjusted according to the noise level of the actual signal to ensure effective signal conversion;

[0030] If equation (1) is true, the output is 1, indicating that the signal at the current sampling point is close to the gas phase signal;

[0031] If equation (2) is true, the output is 0, indicating that the signal at the current sampling point is close to the water phase signal;

[0032] If neither equation (1) nor (2) is satisfied, the output of this sampling point is the same as the previous sampling point, maintaining the continuity of the signal;

[0033] Step 1.2.3. Determine the local gas void fraction based on the binary rectangular wave signal;

[0034] Statistics show that the total amount of data collected at the position j of the optical fiber probe array sensor within the acquisition time s is N tj , where the number of data outputted as 1 is N gj , represents the gas phase signal, and the local gas content x at the position j of the fiber probe array sensor is j for:

[0035]

[0036] Where R is the total number of probes.

[0037] Preferably, the BP neural network includes an input layer, a hidden layer, and an output layer. The neuron set of the input layer is: x = (x1, x2, ..., x j ,…,x R ), R is the number of neurons in the input layer, 1≤j≤R, x j represents the local gas content at position j of the optical fiber probe;

[0038] The number of neurons a in the hidden layer is determined according to the following formula:

[0039]

[0040] Where l is the number of neurons in the output layer, b is an integer between 0 and 10, which is used to adjust the complexity of the model;

[0041] The number of neurons in the output layer is l=1, and the neurons in the output layer represent the gas volume fraction of the gas-liquid two-phase flow.

[0042] Preferably, the method for iteratively updating the particle speed and position according to the evaluation results in step 2.3 is:

[0043]

[0044] Where, is the velocity of the dth dimension of the i-th particle at the k+1th iteration, w is the inertia weight, which is used to control the attenuation of the particle velocity. is the velocity of the dth dimension of the ith particle at the kth iteration, c1 and c2 are learning factors used to control the speed at which the particle moves to the individual optimal position and the global optimal position, and is the random number at the kth iteration, used to increase the randomness of particle movement. is the individual optimal position of the dth dimension of the i-th particle at the k-th iteration, is the global optimal position of the dth dimension of the i-th particle at the k-th iteration, is the position of the d-th dimension of the i-th particle at the k-th iteration, is the position of the d-th dimension of the i-th particle at the k+1-th iteration.

[0045] The beneficial effects of the present invention are as follows:

[0046] The fiber optic probe sensor used in the present invention has the characteristics of high sensitivity and resistance to electromagnetic interference. It can accurately detect the gas and liquid phase media in the gas-liquid two-phase flow, providing basic data for precise measurement. The particle swarm optimization (PSO) algorithm is used to optimize the weights and thresholds of the BP neural network, helping the network to escape the local optimum and enhance the global search capability of the model, thereby improving measurement accuracy. The AdaBoost algorithm combines multiple PSO-optimized BP neural network weak predictors into a strong predictor, further improving the prediction accuracy and stability of the model.

[0047] The present invention uses fiber optic probe sensors to collect data, which can adapt to complex gas-liquid two-phase flow patterns such as bubbly flow and slug flow, ensuring stable measurement under different flow conditions. The number of fiber optic probes can be increased as needed, and the algorithm model can be further optimized to adapt to different measurement requirements and complex environments. It has extremely high application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 This is a principle block diagram of the gas volume fraction measurement method of gas-liquid two-phase flow of the present invention.

[0049] Figure 2 This is the planar position distribution diagram of the optical fiber probes of the array optical fiber probe sensor in the experiment.

[0050] Figure 3 Schematic diagram of the comparison results between the measured and predicted gas volume fractions in the experiment.

[0051] Figure 4 Schematic diagram of the relative error results of the SVM model and the PSO-BP-AdaBoost model of the present invention.

[0052] Figure 5 Schematic diagram of the relative error results of the BP neural network and the PSO-BP-AdaBoost model of the present invention.

[0053] Figure 6 Schematic diagram of the relative error results of the PSO-BP neural network and the PSO-BP-AdaBoost model of the present invention.

[0054] Figure 7 This is a comparison chart of the error indicators of the PSO-BP-AdaBoost model of the present invention and the existing model. DETAILED DESCRIPTION

[0055] The present invention will be further described in detail below with reference to the accompanying drawings and examples, but the present invention is not limited to the following embodiments.

[0056] Example

[0057] exist Figure 1 In this embodiment, a method for measuring the gas volume fraction of a gas-liquid two-phase flow comprises the following steps:

[0058] Step 1. Use an array fiber optic probe sensor to collect local gas void fraction signals at various locations on the cross section, and simultaneously use a flow meter to measure the corresponding standard gas volume fraction. The local gas void fraction signals at each location are processed to obtain local gas void fraction data. Each acquisition time is s, and the acquisition is performed N times. The local gas void fraction data at each location collected each time and the corresponding measured gas volume fraction constitute a sample data set, forming a sample data set containing N sample data.

[0059] The method for processing the local gas content signal at each location to obtain the local gas content data is as follows:

[0060] Step 1.1. First, perform photoelectric conversion and analog-to-digital conversion on the reflected light signal of the array fiber probe sensor to obtain a digital signal;

[0061] Step 1.2. Convert the digital signal into a binary rectangular wave signal using a dynamic threshold adjustment method;

[0062] Step 1.2.1. Set two adjustable parameters, namely the maximum value a max and the minimum value a min , and give them two initial values ​​respectively;

[0063] Step 1.2.2. The current sampled signal amplitude a nThe signal amplitude a n-1 Comparison is made, a n >a n-1 Then a max is updated to a n , indicating that the current sampling signal amplitude is greater than the previous sampling signal amplitude, the signal is close to the range of air signal, so update the maximum value; a n <a n-1 Then a min is updated to a n , indicating that the current sampling signal amplitude is less than the previous sampling signal amplitude, the signal is close to the range of water signal, so update the minimum value; a n =a n-1 Then a max and a min remain unchanged in the original state;

[0064] Step 1.2.3. Determine the true value of the current sampling point according to the following formula (1) (2):

[0065] a n >a min +y (1)

[0066] a n <a max -y (2)

[0067] In the formula, y is a variable, which depends on the signal noise, and is adjusted according to the noise level of the actual signal to ensure the effective conversion of the signal;

[0068] If formula (1) is true, the output is 1, indicating that the signal of the current sampling point is close to the air signal;

[0069] If formula (2) is true, the output is 0, indicating that the signal of the current sampling point is close to the water signal;

[0070] If neither formula (1) nor (2) is true, the output of the sampling point is the same as the previous sampling point, maintaining the continuity of the signal;

[0071] Step 1.2.3. Determine the local gas holdup based on the binary rectangular wave signal;

[0072] The total amount of data collected at the position j of the array optical fiber probe sensor of the array optical fiber probe sensor in the collection time s is N tj , of which the number of data with output 1 is N gj , indicating the air signal, and the local gas holdup x j at the position j of the array optical fiber probe sensor of the array optical fiber probe sensor is:

[0073]

[0074] Where R is the total number of optical fiber probes in the array optical fiber probe sensor.

[0075] Step 2. Construct a BP neural network and use the particle swarm optimization algorithm to optimize the weights and thresholds of the BP neural network to obtain a weak predictor;

[0076] The BP neural network includes an input layer, a hidden layer, and an output layer. The neuron set of the input layer is: x = (x1, x2, ..., x j ,…,x R ), R is the number of neurons in the input layer, i.e. the total number of fiber probes in the array fiber probe sensor, 1≤j≤R, x j represents the local gas content at position j of the optical fiber probe;

[0077] The number of neurons a in the hidden layer is determined by the following formula:

[0078]

[0079] Where l is the number of neurons in the output layer, b is an integer between 0 and 10, which is used to adjust the complexity of the model;

[0080] The number of neurons in the output layer is l=1, and the neurons in the output layer represent the gas volume fraction of the gas-liquid two-phase flow.

[0081] Step 2.1. Combine the weights and thresholds of the BP neural network into particle codes;

[0082] Step 2.2. Randomly generate a particle swarm in the solution space as the initial solution, where each particle represents a set of weights and thresholds;

[0083] Step 2.3. Evaluate the performance of each particle through the fitness function, and iteratively update the particle's speed and position according to the evaluation results.

[0084]

[0085] Where, is the velocity of the dth dimension of the i-th particle at the k+1th iteration, w is the inertia weight, which is used to control the attenuation of the particle velocity. is the velocity of the dth dimension of the ith particle at the kth iteration, c1 and c2 are learning factors used to control the speed at which the particle moves to the individual optimal position and the global optimal position, and is the random number at the kth iteration, used to increase the randomness of particle movement. is the individual optimal position of the dth dimension of the i-th particle at the k-th iteration, is the global optimal position of the dth dimension of the i-th particle at the k-th iteration, is the position of the d-th dimension of the i-th particle at the k-th iteration, is the position of the d-th dimension of the i-th particle at the k+1-th iteration.

[0086] During the update, the particles continuously approach the optimal solution area. When the preset number of iterations is reached or the convergence condition is met, the optimized weights and thresholds are obtained, thereby obtaining a weak predictor.

[0087] Step 3. Use the AdaBoost algorithm to train the weak predictor to obtain a strong predictor;

[0088] Step 3.1. Randomly select m sample data from the sample data set obtained in step 1 as the training set;

[0089] Step 3.2. Assign the same initial weight to each sample data in the training set. The initial weight is These weights are used in the subsequent weighted training process;

[0090] Step 3.3. In each round of training, use the current sample weights to train the weak predictor to generate the weak predictor for this round. Determine the classification error rate of the weak predictor on the training set, and then derive the weight of the prediction sequence of the weak predictor based on the classification error rate.

[0091] Step 3.4. Update the sample weights for the next round of training based on the prediction results of the weak learner in this round and the weights of the prediction sequence. Increase the weights of misclassified samples and decrease the weights of correctly classified samples. Normalize the updated weights to ensure that the sum of all sample weights is 1.

[0092] Step 3.5. After a preset T rounds of training, generate T weak predictors;

[0093] Step 4. Combine T weak predictors according to the following formula to obtain a strong predictor to obtain the final gas volume fraction prediction value,

[0094]

[0095] Where F(x) is the predicted value of the final gas volume fraction obtained for the input local gas content x at each location, and f t (x) is the gas volume fraction prediction value of the local gas content x at each input position by the weak predictor in the tth round, α t is the weight of the prediction sequence of the weak predictor in round t.

[0096] experiment

[0097] In order to verify the beneficial effects of the present invention, the inventors used the technical solution of Example 1 (hereinafter referred to as the PSO-BP-AdaBoost model) to conduct a gas volume fraction measurement experiment for gas-liquid two-phase flow:

[0098] The total number R of optical fiber probes in the array optical fiber probe sensor of this experiment is 7. Figure 2 The voltage responses of 7 fiber optic probes were recorded using a data acquisition card at a sampling frequency of 2kHz, and the corresponding standard gas volume fraction was measured using a flow meter. Each acquisition lasted 5 seconds, and 480 acquisitions were made. After processing, 480 sets of local gas content data at 7 positions were obtained. The local gas content data at 7 positions collected each time and the corresponding measured gas volume fraction constituted a sample data, forming a total of 480 sample data. The sample data were divided into a training set (including 440 sample data) and a test set (including 40 sample data).

[0099] After training the PSO-BP-AdaBoost model using the training set, the 40 samples in the test set are used to verify the effectiveness of the trained model. Figure 3 , comparing the measured gas volume fraction and the predicted gas volume fraction. As can be seen from the figure, the predicted value of the present invention is highly close to the measured value, and the data points overlap well with each other.

[0100] The relative errors of the prediction results of the present invention (hereinafter referred to as PSO-BP-AdaBoost model) are compared with those of the existing models, namely support vector machine (SVM), BP neural network, and PSO-BP neural network. Figure 4 、 Figure 5 、 Figure 6 The relative error of the PSO-BP-AdaBoost model is 0.14%, that of the SVM is 4.37%, that of the BP neural network is 4.14%, and that of the PSO-BP neural network is 1.09%. The relative error of the proposed method is at the lowest level and has good robustness.

[0101] The performance of the present invention is compared with that of the existing model, where the four indicators are: mean absolute error (MAE), root mean square error (RMSE), mean absolute percentage error (MAPE), and maximum error (ME). MAE, RMSE, and MAPE all reflect the deviation between the model prediction and the actual situation. The smaller the value, the closer the model prediction is to the true value, and the better the performance of the model. ME is the maximum absolute value of the difference between the predicted value and the actual value, reflecting the error degree of the model in the worst case. Comparison data such as Figure 7 ,It can be seen that the present invention outperforms the other three models in key evaluation ,indicators such as MAE, RMSE, MAPE and ME, showing significant ,advantages.

Claims

1. A method for measuring the gas volume fraction of a gas-liquid two-phase flow, characterized in that: The following steps are involved: Step 1. Use an array fiber optic probe sensor to collect local gas void fraction signals at various locations on the cross section. Simultaneously, use a flow meter to measure the corresponding standard gas volume fraction. The local gas void fraction signals at each location are processed to obtain local gas void fraction data. Each acquisition takes s, and the acquisition is repeated N times. The local gas void fraction data and the corresponding gas volume fraction at each location collected each time constitute a sample data set, thus forming a sample data set containing N sample data. Step 2. Construct a BP neural network and use the particle swarm optimization algorithm to optimize the weights and thresholds of the BP neural network to obtain a weak predictor; Step 2.

1. Combine the weights and thresholds of the BP neural network into particle codes; Step 2.

2. Randomly generate a particle swarm in the solution space as the initial solution, where each particle represents a set of weights and thresholds; Step 2.

3. Evaluate the performance of each particle using the fitness function. Iteratively update the particle's velocity and position based on the evaluation results. During the update, the particle continuously approaches the optimal solution region. When the preset number of iterations is reached or the convergence condition is met, the optimized weights and thresholds are obtained, thus obtaining a weak predictor. Step 3. Use the AdaBoost algorithm to train the weak predictors to obtain T preset weak predictors; Step 3.

1. Randomly select m sample data from the sample data set as the training set; Step 3.

2. Assign the same initial weight to each sample in the training set. These weights are used in the subsequent weighted training process. Step 3.

3. In each round of training, use the current sample weights to train the weak predictor to generate the weak predictor for this round. Determine the classification error rate of the weak predictor on the training set, and then derive the weight of the prediction sequence of the weak predictor based on the classification error rate. Step 3.

4. Update the sample weights for the next round of training based on the prediction results of the weak learner in this round and the weights of the prediction sequence. Increase the weights of misclassified samples and decrease the weights of correctly classified samples. Normalize the updated weights to ensure that the sum of all sample weights is 1. Step 3.

5. After a preset T rounds of training, generate T weak predictors; Step 4. Combine T weak predictors according to the following formula to obtain a strong predictor to obtain the final gas volume fraction prediction value, Where F(x) is the predicted value of the final gas volume fraction obtained for the input local gas content x at each location, and f t (x) is the gas volume fraction prediction value of the local gas content x at each input position by the weak predictor in the tth round, α t is the weight of the prediction sequence of the weak predictor in round t.

2. The method for measuring the gas volume fraction of gas-liquid two-phase flow according to claim 1, characterized in that: The method for obtaining local gas content data by processing the local gas content signal at each position in step 1 is as follows: Step 1.

1. First, perform photoelectric conversion and analog-to-digital conversion on the reflected light signal of the array fiber probe sensor to obtain a digital signal; Step 1.

2. Convert the digital signal into a binary rectangular wave signal using a dynamic threshold adjustment method; Step 1.2.

1. Set two adjustable parameters, namely the maximum value a max and the minimum value a min , and give them two initial values ​​respectively; Step 1.2.

2. The current sampled signal amplitude a n Compared with the previously sampled signal amplitude a n-1 For comparison, a n >a n-1 , then a max Updated to a n , indicating that the current sampling signal amplitude is greater than the previous sampling signal amplitude, and the signal is close to the range of the gas phase signal, so the maximum value is updated; a n <a n-1 , then a min Updated to a n , indicating that the current sampling signal amplitude is smaller than the previous sampling signal amplitude, and the signal is close to the range of the water phase signal, so the minimum value is updated; a n =a n-1 , then keep a max and a min The original state remains unchanged; Step 1.2.

3. Determine the true value of the current sampling point according to the following formula (1) (2): to n >a min +y (1) to n max -and (2)​ Where y is a variable that depends on the signal noise and is adjusted according to the noise level of the actual signal to ensure effective signal conversion; If equation (1) is true, the output is 1, indicating that the signal at the current sampling point is close to the gas phase signal; If equation (2) is true, the output is 0, indicating that the signal at the current sampling point is close to the water phase signal; If neither equation (1) nor (2) is satisfied, the output of this sampling point is the same as the previous sampling point, maintaining the continuity of the signal; Step 1.2.

3. Determine the local gas void fraction based on the binary rectangular wave signal; Statistics show that the total amount of data collected at the position j of the optical fiber probe array sensor within the acquisition time s is N tj , where the number of data with output 1 is N gj , represents the gas phase signal, and the local gas content x at the position j of the fiber probe array sensor is j for: Where R is the total number of probes.

3. The method for measuring the gas volume fraction of gas-liquid two-phase flow according to claim 1, characterized in that: The BP neural network includes an input layer, a hidden layer, and an output layer. The neuron set of the input layer is: x=(x1, x2, ..., x j ,…,x R ), R is the number of neurons in the input layer, 1≤j≤R, x j represents the local gas content at the position j of the optical fiber probe; The number of neurons a in the hidden layer is determined according to the following formula: Where l is the number of neurons in the output layer, b is an integer between 0 and 10, which is used to adjust the complexity of the model; The number of neurons in the output layer is l=1, and the neurons in the output layer represent the gas volume fraction of the gas-liquid two-phase flow.

4. The method for measuring the gas volume fraction of gas-liquid two-phase flow according to claim 1, characterized in that: The method for iteratively updating the particle speed and position according to the evaluation results in step 2.3 is: Where, is the velocity of the dth dimension of the i-th particle at the k+1th iteration, w is the inertia weight, which is used to control the attenuation of the particle velocity. is the velocity of the dth dimension of the ith particle at the kth iteration, c1 and c2 are learning factors used to control the speed at which the particle moves to the individual optimal position and the global optimal position, and is the random number at the kth iteration, used to increase the randomness of particle movement. is the individual optimal position of the dth dimension of the i-th particle at the k-th iteration, is the global optimal position of the dth dimension of the i-th particle at the k-th iteration, is the position of the d-th dimension of the i-th particle at the k-th iteration, is the position of the d-th dimension of the i-th particle at the k+1-th iteration.

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