Parameter optimization of SVM algorithm for predicting physical parameters of SF6–Cu mixture plasma under local thermodynamic equilibrium

In flexible DC networks, hybrid DC circuit breakers, as one of their important devices, have a wide range of applications.¹ The sulfur hexafluoride circuit breaker is an important part of the 500 kV hybrid DC circuit breaker. In order to better investigate the opening characteristics of the sulfur hexafluoride circuit breaker, the plasma generated by the arc discharge during opening needs to be calculated, and the simulation of the arc plasma requires the thermodynamic properties and transport properties of various component particles as input. Calculation of the thermodynamic properties of plasma (density, enthalpy, specific entropy, specific heat, etc.) is based on the theory of ideal gas dynamics.² Transport coefficients (electrical conductivity, thermal conductivity, viscosity coefficient, etc.) are an important basis for the study of the macroscopic properties of the arc, and the transport coefficients were first calculated by solving the Boltzmann equation using the Chapman–Enskog calculation method.^3–6 That is to say, in order to make the calculation simple, it is usually assumed that the velocity distribution function of the particle is in the first-order Maxwell perturbative approximation and that the velocity distribution function is decomposed into a series of Sonine polynomial forms so that the Boltzmann equations are treated linearly and the computational difficulty is reduced. This method is widely used when in local thermodynamic equilibrium.^7–10 The same Chapman–Enskog method is used for the calculation of thermodynamic and transport properties of non-local thermodynamic equilibrium (NLTE) plasmas,¹¹ which will not be described in detail here. The derivation of physical parameters of arc plasmas is often complex and often requires solving dozens of differential equations, which is not only time consuming but also prone to errors in the calculation process, affecting the final data results. In addition, with the development of intelligent technology, machine learning has been applied in a large number of fields.¹² With the existing database, machine learning can train a large amount of data in the database to finally give the desired prediction. During the process of data training, machine learning can learn and speculate autonomously based on a series of data and predict the unknown data with a certain relationship to the known dataset. According to the existing literature,^13–15 the data predicted using machine learning are very similar to the real results and can completely replace the actual data without the need for precise data, thus allowing complex human calculations and errors to be avoided and saving a lot of time.

To avoid obtaining data by calculating a large number of complex equations, machine learning methods are usually used to predict the existing data, and there are many such studies in China and abroad. The theoretical determination of the electron-impact ionization cross section (Q_ion) for a molecule takes a lot of time. Zhong¹⁶ used the data calculated by support vector machines to predict the more consistent values of the electron-impact ionization cross section well, and the prediction results showed that the machine learning model shows good generalization performance. Yan¹⁷ and other scholars used the small habitat genetic algorithm to solve the problem of difficult parameter selection in the SVM algorithm and used this parameter to predict the output value of wind power well. Álvarez¹⁸ constructed an SVM model to predict solar radiation to solve the problem of energy generation and transmission in some areas without weather stations. Ding¹⁹ used five prediction algorithms in machine learning based on the existing database to predict and calculate the thermodynamic parameters and transport characteristics of sulfur hexafluoride at different pressures and temperatures, along with the prediction analysis of the sulfur hexafluoride–copper gas mixture at different concentrations at the same pressure. According to the prediction results, the SVM prediction algorithm has a strong generalization ability for each parameter. However, in the authors’ prediction result plots, the results between the predicted electrical conductivity, thermal conductivity, and constant pressure specific heat of the mixture for the SVM and the real data differ significantly, which are not as good as those of algorithms such as KNN and GBR. The authors also show that due to the high requirements of the SVM algorithm for hyperparameters, not all algorithms are suitable for data prediction.

Because some data do not meet the prediction accuracy due to the difficulty in selecting the hyperparameters in the SVR algorithm, this paper first uses the particle swarm algorithm and gray wolf algorithm to optimize the hyperparameters of SVR, then uses the optimized hyperparameters to predict the data, and finally compares and analyzes the predicted data before and after the optimization.

In the SVM algorithm, there are three most important hyperparameters, which are kernel, gamma, and C. Among them, kernel is the type of kernel function used in the algorithm, and the default is “RBF;” gamma means the inverse of the dimensionality of the data, so the value of gamma must be greater than 0. The classification effect for the test set becomes worse as the gamma increases, and the classification effect is good for the training set, as well as makes the complexity of the model increase with poorer generalization ability (prediction ability of the unknowns), which results in overfitting. C is the penalty factor of the error term; it characterizes how much importance is attached to the outliers. The larger the C value, the more importance is attached to the outliers, i.e., the more we do not want to lose them. The penalty for error classification increases when C is large and decreases when C is small. As C converges to infinity, it means that the classification error is not allowed to exist; when C converges to 0, it means that it is no longer concerned with whether the classification is correct or not.

The regression module, which is SVR, in the SVM algorithm is robust to outliers and can be easily updated for decision models. Meanwhile, it is not limited by dimensionality and has strong generalization ability and high prediction accuracy, especially when the Gaussian radial basis function (RBF) is chosen as the kernel function. The prediction accuracy of most categories of data is above 85% even without parameter tuning, and SVR is thus selected as the best model among several prediction models. To achieve higher prediction accuracy, two important hyperparameters of SVR need to be optimized, and this step is also the focus and difficulty in the SVR algorithm. Usually, many scholars are used to search for the optimal parameters by grid search and random search, but because there are various forms of setting the parameters, sometimes the optimal parameters are missed or fall into the local optimum, which adversely affects the prediction results.

Reference 19 mentioned that the method used to select the hyperparameters of the SVM algorithm is grid search and that the predicted results using the final hyperparameters do not fit well with the real data. On the one hand, it may be that the SVM algorithm is very sensitive to the hyperparameters and does not select suitable hyperparameters, leading to a large difference in the data. On the other hand, it may be that only KNN and GBR algorithms show optimal prediction performance, as in the case of prediction results for viscosity. However, according to the results, the SVM algorithm can initially show the overall data situation in the prediction of electrical conductivity, thermal conductivity, and constant pressure specific heat for the sulfur hexafluoride–copper gas mixture although the predictions are poorly fitted, thus indicating that appropriate data can also be fitted better using the SVM algorithm. The key lies in finding the optimal hyperparameters, which is usually the focus and difficulty in the SVM prediction algorithm.¹⁷ To deal with this problem, it is particularly important to find out how the values of the RBF kernel parameters gamma and penalty coefficient can be determined quickly and precisely²⁰ and how the feasibility of the optimization algorithm can be analyzed by using the optimization algorithm to first optimize the hyperparameters. By applying the optimized parameters to the SVM algorithm to predict the physical parameters of the mixture, the results of those before the optimization are compared.

With the continuous development of computer technology, the algorithm technology is also constantly updated, and there are already specialized optimization-seeking algorithms to solve such problems. In general, the existing optimization algorithms are basically based on the evolution of Swarm Intelligent (SI) algorithms. Swarm intelligent algorithms originated in the early 1990s when many scholars were inspired by the phenomenon of groups of different species in nature and proposed optimization algorithms using simulations of the behavior of herd animals. Swarm intelligence algorithms can provide the basis for solutions to complex problems without prior experience and in the absence of global information and models.²¹ As an important disciplinary branch of computational intelligence, this algorithm is a bionic class of stochastic search algorithms, which is proposed through the mechanism of genetic evolution and group cooperation behavior in biology. The algorithm has attracted widespread attention from scholars because of its fast search rate as well as the too little initial information to be considered.

Among the swarm intelligence optimization algorithms, each algorithm is basically similar in structure, steps, and principles, and each algorithm differs mainly in the update rules of the algorithm, ranging from those based on simulating the movement step of swarming organisms (e.g., PSO). In addition, there are update rules set according to some algorithmic mechanism (e.g., ACO). Others are, for example, ant colony algorithms, genetic algorithms, and fish swarm algorithms.²² In this paper, the PSO algorithm and the gray wolf algorithm are used for hyperparameter search,^23–28 and they are not described in detail.

In order to validate the merit-seeking effect of the two optimization algorithms, first, the two optimization algorithms are programmed to process the algorithm in which the support vector machine regression module can be inserted. In addition, in the PSO algorithm, its own coefficient of determination (R²) can be used as the fitness value to judge, and the closer the score is to 1, the higher the accuracy is. Since the particle swarm generates new position vectors and velocity vectors for each particle in the population at each update iteration and in support vector machine regression, the two important parameters are gamma and penalty coefficient C. Therefore, gamma and C can be replaced by the position vector of each particle in the population when writing the particle swarm algorithm. In the GWO algorithm, as many practical problems require a minimum solution, the mean square error (MSE) can be used as the fitness value; the smaller the MSE, the greater the prediction accuracy.

To obtain a specific fitness value, actual data are required, and to compare with the results of the paper, the data from the Gas Discharge Plasma Basic Database (GPLAS) provided by the School of Electrical Engineering of Xi’an Jiaotong University are also taken here to verify the effectiveness of each algorithm.

By using the hyperparameter values optimized by PSO and GWO, the optimal parameter values were applied to the regression model of the SVM. The population size of the particle swarm algorithm was 30, the number of iterations was 100, and other specific parameters are set as shown in Table I. Similarly, the values set for the specific structural parameters of the gray wolf algorithm are shown in Table II. The optimal training curves are shown in Figs. 1–6 after the particle swarm algorithm and the gray wolf algorithm were used to find the optimal values of electrical conductivity, thermal conductivity, and Cp parameters of SF₆–Cu for several times.

Correspondingly, the fitted curves of the conductivity parameters of SF₆–Cu without the search and after the search by the two algorithms are shown in Figs. 7–9.

For the thermal conductivity of SF₆–Cu, the fitted curves before and after optimization by the two algorithms are shown in Figs. 10–12.

In addition, the fitted curve of the constant pressure specific heat parameter of SF₆–Cu obtained by applying the optimal result parameters to the SVM regression and the fitted curve with unoptimized hyperparameters are shown in Figs. 13–15.

In order to analyze the statistics and compare the effect of the two optimization algorithms, the data of SVR prediction effect evaluation indices optimized by the two algorithms are produced as shown in Tables III and IV.

The prediction of the viscosity parameter of SF₆–Cu is the same as that described in the literature,¹⁹ where KNN and GBR have the best prediction ability and the other machine learning algorithms have poor prediction ability. It should be noted that the optimal hyperparameters obtained from each run are different since both PSO and GWO algorithms contain random numbers in their update formulas.

The comparison of the fitted curves of the electrical conductivity of SF₆–Cu show that the unoptimized fitted curve have some errors with the actual values in a wide range of temperatures while the prediction of the curves optimized by two optimization algorithms have improved significantly overall. In addition, the effect of GWO algorithm optimization is more obvious. For the fitted curves of the thermal conductivity, although the data before optimization is overall very close to the real value, the prediction of the initial value is not satisfactory, and the data fluctuates a lot; thus, it will have some influence on the actual demand. The fitted curve optimized by two algorithms has a significant improvement in the prediction effect of the initial value part. Except for the fluctuation in a specific small range of temperature intervals, it can overlap well with the actual value, and the optimization effect of PSO and GWO algorithms is similar at this time. For the prediction of constant pressure specific heat of SF₆–Cu, the fitted curve without optimization by the two algorithms can clearly show that the prediction effect of the SVR algorithm is different from the actual value on the whole while the results optimized by the particle swarm and gray wolf algorithms have improved significantly overall. Although the prediction performance is not as good as that of other algorithms, it is close to the prediction limit of SVR for predicting this parameter, and most of the data overlap with the other algorithms proves the feasibility of the optimization for hyperparameters by optimization algorithms.

The running process and results of both particle swarm and gray wolf algorithms show that the gray wolf algorithm has a faster search speed and it has a stronger global search capability than the particle swarm algorithm. In addition, it is able to find the global optimal solution in a larger search space. However, the gray wolf algorithm is more sensitive to the selection of the initial population, and different initial populations often lead to widely different results, which will inevitably fall into the local optimum. The particle swarm algorithm has a faster convergence speed and can find a better solution in a shorter time. As the number of populations and iterations increases, both algorithms have longer computation time but show better performance, so it is necessary to choose the relevant parameters reasonably according to the actual data situation. The final results show that both algorithms can achieve the expected results for the SVR hyperparameters and can fit the actual data better.

In this paper, a method of SVM prediction based on the optimization algorithm is proposed to overcome the difficulty of taking the two important hyperparameters gamma and penalty coefficient C in the SVM algorithm. The parameter values with better fitting results are identified in a large parameter range by using two optimization algorithms, namely, PSO and GWO. The results show that both particle swarm and gray wolf algorithms can eventually find out the hyperparameter values that are suitable for predicting each physical parameter of the SF₆–Cu mixture and can accurately locate the approximate interval range of the hyperparameters for further obtaining the hyperparameters that satisfy the SVR prediction accuracy. In the prediction and optimization of SF₆–Cu physical parameters, the performances of PSO and GWO algorithms are roughly similar, and both have their advantages and disadvantages; Overall, the GWO algorithm is superior. Finally, the GWO algorithm has the advantages of strong global search ability and precision target search, and it can be given priority to the GWO algorithm in future predictions while supplemented by other algorithms to get the best prediction results.

The authors have no conflicts to disclose.

Can Ding: Methodology (equal); Software (equal); Validation (equal). Donghai Yu: Methodology (equal); Software (equal); Validation (equal). Qing Guo: Methodology (equal). Sheng Zhang: Methodology (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.