Intelligent computing and analysis of breakdown voltage of rod-rod long air gap

The discharge characteristics of air gap is one of the important factors to determine the insulation level which is crucial for electrical engineering construction. In order to calculate the 50% breakdown voltage of rod-rod air gap under different meteorological conditions and different gap distances, we take the positive standard switching impulse test data of rod-rod air gap under different meteorological conditions as samples. Based on training set, a breakdown voltage prediction model of rod-rod air gap under different meteorological conditions was established by cuckoo search optimized weighted support vector regression (CS-ω-SVR). The results show that the average absolute percentage error of the test set is 3.6%, which verifies the validity of this computing method. The method can provide reference for the computation of air gap breakdown voltage under different altitudes and extreme meteorological conditions.


INTRODUCTION
The discharge characteristic of long air gap play a significance role in the external insulation design of high-voltage electrical equipment. And changes of gap structures and meteorological conditions will have a greater impact on it [1]. As one of the typical electrode structures, the effective calculation method for the breakdown voltage of rod-rod air gap is beneficial to research the discharge characteristics of typical air gap under various conditions. Many factors such as gap structures, voltage wave forms, atmospheric conditions and so on can affect air gap breakdown voltage. Due to the intricate non-linear relationship between them, the breakdown voltage usually cannot be accurately calculated by empirical formulas based on traditional statistical methods [2]. Under extreme atmospheric conditions, the accuracy of these traditional calculation methods is further reduced. However, the machine learning algorithms can deal with the complex relationship. Appropriate machine learning algorithms can be used to establish the rod-rod long air gap breakdown voltage prediction model. And this kind of prediction model can solve the shortcomings of traditional calculation methods effectively.
Early researchers mainly used artificial neural network algorithms to construct the air gap breakdown voltage prediction model. And they achieved good prediction results in a small range of test sets [3]. However, the number of parameters of neural network is larger, and the performance on small sample 2 data is often not as good as SVM. The SVM is a training model basis for the principle of structural risk minimization, that can avoid overfitting effectively by introducing regularization item. The air gap discharge data is multi-dimensional, non-linear and small sample data, and SVM has good regression performance for this kind of data, which can establish a prediction model with high accuracy for such data [4]. At present, the intelligent calculation model of air gap is mostly aimed at the short gap, while the long air gap is the usual insulation form in real engineering application. And most of the researches on breakdown voltage prediction of long air gap only focus on the rod-plane gap, and other typical electrode structure gaps are rarely involved.
In order to obtain the required samples for modeling, a series of rod-rod air gap switching impulse test are conducted in UHV DC test base of State Grid Corporation of China. And 70 samples were obtained from spring to winter in Beijing. In this article, four features are selected as the input variables, and the 50% breakdown voltage of rod-rod long air gap is taken as the forecast target. Firstly, the grey relation degree between input variables and the breakdown voltage is calculated. The breakdown voltage prediction model rod-rod long air gap under positive standard switching impulse voltage is established on account of a weighted support vector regression algorithm optimized by cuckoo search. And the reserved test set is used to verify the validity of the model. Last, the trained model is used to research the discharge characteristics under extreme temperature and humidity.

Grey Relation Analysis
The grey relation analysis is an algorithm to determine the relation between factors through the geometry similarity of the change curves of various factors in the system, and the relation between different factors is quantitatively described by grey relation degree [5]. The basic steps are as follows: (1) Normalize the original data (or standardization, etc.). (3) Calculate the relation coefficient ( 0 ( ), ( )) of each corresponding point of reference sequence and comparison sequence: (4) Calculate the grey relation degree ( 0 , ) of reference sequence and comparison sequence:

CS-ω-SVR Model
The SVR is a supervised learning method. Its main idea is to map input variables to a high-dimensional space through the kernel function, and then the simple multiple linear regression is carried out in the highdimensional feature space [6]. Give the training sample ( 1 , 1 ), ( 2 , 2 ),•••, ( , ). We hope to learn a hyperplane to make ( ) and as close as possible: In (3), is the regression coefficient and is the bias term. Give a small positive number . If there is the following formula: Then the SVR problem can be formalized as: Where, is the regularization constant, and is -insensitive loss function: Introduce the non-negative slack variable: , ′ and Lagrange multiplier , ′ , , ′ ≥ 0： Calculate the partial derivatives of , , and ′ , make them equal to 0, and transform the original problem into its dual problem. Then introduce the kernel function to map the original data to a highdimensional plane, and obtain the hyperplane expression after solving: The role of kernel function is to avoid the difficulty in calculating the inner product of vectors in highdimensional space and RBF is selected as the kernel function of SVR in this paper: From the above derivation, it is not difficult to find that the values of two parameters cannot be directly obtained from the sample. And we use Cuckoo search (CS) algorithm to search the optimal and because it has a good global optimization ability [7].
To improve the prediction precision of models, the grey relation degree of each input variable and breakdown voltage analyzed by grey relation analysis is used as the weight of variables in this paper. And it is applied in the calculation of kernel function to construct a weighted SVR model optimized by CS algorithm.

Error Analysis Method
In this paper, we select two error indexes to quantitative the accuracy of predicted results.
(1) Absolute percentage error (APE) (2) Mean absolute percentage error (MAPE) Among the tow formulas, is the test values, is the predicted values, and stands for the number of samples. The smaller the two indexes are, the more accurate the prediction results are.

Intelligent Computation and Analysis of the prediction model
In this paper, 70 groups of long rod-rod air gap test data under different meteorological conditions are taken as the samples. Four features of rod-rod gap distance (d/m), temperature (t/℃), air pressure (p/kPa) and absolute humidity (h/g/m 3 ) are selected as the input variables to construct the prediction model of 50% breakdown voltage of long rod-rod air gap based on CS-ω-SVR. According to the sample distribution of samples, respectively select 2 groups of samples under high temperature, 7 groups of samples under low temperature and 5 groups of samples at about 20℃ as the test set, and others as the training set.
First, all samples after normalization are analyzed based on the grey relation analysis method. And the parameter of ξ=0.5 is set to obtain the grey relation degree between four features and the 50% breakdown voltage of rod-rod gap, as shown in Table Ⅰ. Take the grey relation degree in Table Ⅰ as the weight of each input variable to calculate the RBF kernel function, and take the mean square error of 5-fold cross-validation as the fitness function to search the optimal parameter of SVR by CS. The parameter setting of CS is: iteration = 100, number of nests = 20 and random elimination probability = 0.25. Input the training set to obtain the optimal parameter: = 50.00 and = 0.34.
Under the condition that the optimal model is obtained, the prediction result of the training set is shown in Table Ⅱ. It can be indicated that the maximum of of the test sample is 8.8% and the is 3.6%. From the comparison between the prediction and test results of Figure 1, it can also be directly seen that the prediction results are pretty close to the test values, which verifies the effectiveness of breakdown voltage intelligent calculation method of long rod-rod air gap proposed in this article.

Calculation of Breakdown Voltage of Rod-rod Gap at the Extreme Atmospheric Conditions
The 50% breakdown voltage prediction model of rod-rod long air gap constructed above can be used to calculate the breakdown voltage at various conditions. Subject to laboratory conditions, it is difficult to obtain discharge characteristics of rod-rod gap at some extreme atmospheric conditions by real test. In this section, we calculate the breakdown voltage of rod-rod air gap under three extreme atmospheric conditions based on the intelligent model and draw the corresponding discharge curve.
Taking the test condition of 70 samples into consideration, we select three extreme atmospheric conditions listed in Table Ⅲ according to the climate characteristics of Beijing. In the table, Rh (%) is the relative humidity. By fitting of the predicted voltage computed by trained model, the discharge curve under three conditions is drawn as Figure 2.
Through the analysis of discharge curve of rod-rod gap under 3 extreme atmospheric conditions, the results show that the 50% breakdown voltage under high temperature and high humidity is the minimum. And the difference on 50% breakdown voltage under low-temperature drying and low-temperature wetting is not large, which is consistent with the rule obtained from the test. The larger the gap distance, the less obvious the influence of meteorological conditions on discharge characteristics. In other words, the difference of discharge characteristics under different conditions is gradually reduced as gap distance grows. When the gap distance is 2 m, the maximum difference on 50% breakdown voltage under high temperature and humidity condition and 50% breakdown voltage predicted under the other three meteorological conditions is about 13.1%, and when the gap distance increases to 6 m, the maximum difference is about 1.9%.  Figure 2. Switching impulse discharge curve of rod-rod gap at three extreme atmospheric conditions

CONCLUSION
In this paper, the rod-rod gap distance, temperature, air pressure and absolute humidity are taken as the input variables to construct the prediction model of 50% positive switching impulse breakdown voltage based on CS-ω-SVR algorithm, with the following conclusions obtained: