The measurement error prediction method of voltage transformer based on TCN-RBF

Accurate prediction of voltage transformer error is an important guarantee for the stable and economic operation of the power grid. This paper presents a measurement error prediction method for voltage transformers based on time convolution and radial basis neural networks (TCN-RBF). Firstly, the ensemble empirical mode decomposition is used to decompose the original error sequence into multiple components. Secondly, a TCN prediction model was established based on each component and historical error sequence to achieve error prediction. To further improve the prediction progress, the RBF model was used to further fit the predicted results with the environmental parameter data and confirm the final predicted value. The operation error of the transformer in a substation is tested. The test results show that compared with LSTM, SVM, BP, and other prediction models, this method has better performance in error prediction.


Introduction
As the hub connecting the primary and secondary systems, the voltage transformer is the key data source of power grid status monitoring, relay protection, and electric energy measurement, and its measurement accuracy directly affects the stable and economic operation of the power grid [1] .
The traditional periodic verification has difficulties in power outages, complicated operations, and failure to detect metering errors during equipment operation, which is easy to cause transaction disputes [2] .With the rapid development of machine learning algorithms, data-driven prediction models play an important role in error prediction.Zhang established the error state prediction model based on Q-ARMA to explore the measurement error state prediction method of voltage transformer [3] .Li et al. selected the main influencing factors of ratio difference and angle difference by transferring entropy and then realized transformer error prediction through a wavelet neural network [4] .Zhou et al. improved the LSTM model by introducing strategies such as bidirectional memory and depth feature extraction to improve the prediction accuracy of voltage transformer measurement errors [5] .The existing forecasting methods do not comprehensively consider the error cycle and environmental factors, resulting in low accuracy and insufficient stability of error prediction.To solve this problem, a measurement error prediction method of voltage transformer based on time convolution and radial basis neural network (TCN-RBF) is proposed in this paper.Error components and historical errors are input into the TCN for error prediction, and then the predicted results and historical environmental parameters are input into the RBF network to further fit and complete the model construction.
Experimental results show that the proposed method has higher prediction accuracy than TCN, LSTM, SVM, and BP models.

TCN
Temporal Convolutional Network (TCN) is an effective model for time series prediction, which mainly includes extended causal convolution and residual connection [6] .We suppose that the given input sequence is  1 ,  2 , . . ., and   , and the expected prediction is  1 ,  2 , . .., and   .The relationship between the predicted input and output sequences is: where   is the predicted value of the input sequence and   is the historical sequence.
(1) Expansion of causal convolution TCN uses Dilated Causal Convolution (DCC) to increase the receptive field without significantly increasing the computational cost.DCC is a convolution operation that performs skip operations on input sequences [7] .The equation is as follows: where () is the convolution result of i elements in the sequence; ℎ() is the filter;  is the amplification factor when =1 (standard causal convolution).
(2) Residual block The residual block activates the output prediction result through linear transformation and function and inputs x into the residual block.The output o of the residual block is:

RBF
A radical basic function (RBF) neural network is a typical structure of a local approximation network consisting of input, implicit, and output layers [8] .It is assumed that the RBF neural network has n input nodes and L hidden nodes, and the Gaussian function is chosen as the basis function of the hidden layer.The activation function of RBF can be expressed as: 2 )  =1 (5)   where  = ( 1 ,  2 , . . .,   )  represents the input vector,   ′ is the output of the m-th output node, m (m = 1, 2, . . ., M) represents the number of output nodes,   is the weight between the -th hidden layer node and the -th output node, L is the number of implicit nodes, and   and   are the center and width of the basis function of the L-th implicit node respectively.

Measurement error prediction method of voltage transformer based on TCN-RBF
The specific framework of the voltage transformer error prediction model based on CN-RBF is shown in Figure 1.

EEMD error decomposition
The EEMD model is used to decompose the historical error, and the decomposed modal component   and residual component  are obtained [9] .
where   () represents the  −TH component of the error sequence; () is the residual component and  is the timing length.

TCN error prediction
The feature sequence X is composed of historical error components and historical error sequences of n lengths, which are taken as input data of the TCN model.The input X can be expressed as:  Based on Input X, the error prediction sequence of the transformer is obtained through the TCN.

RBF error fitting
The prediction results of TCN output are fused with the characteristics of historical temperature, historical humidity, and historical load at the corresponding time to form input vector B, which is input into the RBF model.RBF model consists of 4 input units, 10 hidden neurons, and 1 output unit.
′ = ∑    =1 −  (10)   where   (P=1, 2,...,  and  is the prediction step length) represents the first input vector, y is the transformer error prediction result obtained by RBF network,   is the weight of   , and b is the threshold of the output layer.

Results & discussion
The test data set of this experiment is selected from the error data of monitoring and evaluation of a substation from July 1, 2020 to January 20, 2022, with a total of about 550 sampling points, including the transformer ratio difference and its environmental parameters.In this paper, the error data of the first 520 days are used as training data, and the data of the last 30 days are used as a test set.

Evaluation indicators
To verify the precision of the model proposed in this paper, Mean Absolute Percentage Error (MAPE) and Root Mean Squared Error (RMSE) are adopted as evaluation indexes for error prediction accuracy [10]   .The calculation equations are as follows: (11) where  is the predicted quantity,   ′ is the prediction results of the model, and  ̂ is the true value of the error.

Error decomposition based on EEMD
The results of the original error sequence after EEMD decomposition are shown in Figure 2. To verify the effectiveness of modal decomposition for error prediction, the prediction results of TCN, EMD-TCN, and EEMD-TCN models are compared and analyzed.The results are shown in Table 1.As can be seen from the prediction results in Table 1, the EEMD-TCN model has the highest prediction accuracy and is most suitable for error prediction under the current data.Compared with the TCN model, the EEMD-TCN model has a decrease of 8.82% and 13.4% in the two indexes, respectively, indicating that the prediction effect can be effectively improved through modal decomposition.Compared with the EMD-TCN model, the evaluation and prediction indexes of EEMD-TCN are reduced by 4.47% and 7.78%, which further indicates that EEMD can eliminate noise interference more effectively than EMD.

Comparison of results based on different prediction models
To test the validity and accuracy of the prediction model, the input parameters are constructed based on the error components and historical errors.The input parameters are input into TCN, TCN-RBF, LSTM, SVM, BP, and other models for error prediction, and the error prediction results of different models are compared and analyzed, as shown in Figure 3 and Table 2.
From Figure 3 and Table 2, it can be seen that compared with the TCN model, the prediction accuracy of index   and index   decreased by 17.8% and 3.7% respectively, indicating that the error fluctuation was affected by environmental factors.Considering environmental parameters in the model, the prediction accuracy could be effectively improved.Comparing the prediction results of the TCN-RBF model with those of LSTM, SVM, BP, and other prediction models, the TCN-RBF model shows obvious advantages in terms of indicators, mainly because the TCN model has better feature extraction ability in time series.

Conclusions
In this paper, a measurement error prediction method for a voltage transformer based on TCN-RBF is proposed.Multiple error components are obtained by decomposition of the error sequence, and the error components and historical errors are predicted by the TCN model.Considering the influence of the environment on the error, the RBF model is used to further fit the prediction error and the environmental parameters, and the accurate prediction of the error is obtained.By comparing with TCN, LSTM, SVM, BP, and other models, TCN-RBF has a good effect on an index and realizes the accurate prediction of voltage transformer error.

Figure 2 .
Figure 2. Decomposition results of ensemble empirical mode.

Figure 3 .
Figure 3. Prediction results of different prediction models.

Table 1 .
Comparison of the results of different prediction models.

Table 2 .
Comparison of the results of different prediction models.