Detection of partial discharge patterns in hybrid high voltage power transmission lines based on parallel recognition method

Feature analysis of PD signal

Considering the special attributions of the PD signal, the article proposes the utilization of wavelet packet decomposition (WPD) to perform time-frequency analysis on the PD signal of HHVPTLs and then analyze the features of the PD signal. When WPD is applied to analyze PD signals’ multi-resolution, the Hilbert space $L^2\left(\mathrm{R}\right)=\underset{j\in z}{\oplus }{W}_j$ is employed to define. Then, L²(R) decomposition is performed based on distinct pairs of scale factors j to obtain the orthogonal sum of wavelet subspace W_j(j ∈ z) of wavelet function Ψ(t). The frequency resolution can be improved by binary frequency division of the wavelet subspace (Huang et al., 2023; Han et al., 2022). The new subspace ${\Omega }_j^n$ is employed to characterize the scale subspace Γ_j and the wavelet subspace Λ_j in a uniform way. Equation (1) presents this setting. (1)${\displaystyle \left\{\begin{array}{c}{\Omega }_j^0={\Gamma }_j\hfill \\ {\Omega }_j^1={\Lambda }_j.\hfill \end{array}\right.}$

Then, the decomposition of the new subspace ${\Omega }_j^n$ can be employed to unify the Γ_j+1 = Γ_j⊕Λ_j obtained by the orthogonal decomposition of L²(R)defined by (2)${\displaystyle {\Omega }_{{}_{j+1}}^0={\Omega }_{{}_j}^0\oplus {\Omega }_{{}_j}^1.}$

The subspace $U_j^n$ is set to be the wavelet subspace of the function u_n(t) and the subspace ${\Omega }_j^{2n}$ is set to be the wavelet subspace of the function u_2n(t). Then, the dual scale equation is constructed and defined (Zhang et al., 2019) by (3)${\displaystyle \left\{\begin{array}{c}{u}_{2n}\left(t\right)=\sqrt{2}\sum _{k\in z}A\left(k\right)\cdot u_n\left(2t-k\right)\hfill \\ u_{2n+1}\left(t\right)=\sqrt{2}\sum _{k\in z}B\left(k\right)\cdot u_n\left(2t-k\right)\hfill \end{array}\right..}$ where A(k) and B(k) represents the low-pass filter and high-pass filter coefficients, respectively, and are orthogonal to each other where Eq. (3) is transformed and rewritten by (4)${\displaystyle \left\{\begin{array}{c}{u}_0\left(t\right)=\sqrt{2}\sum _{k\in z}A\left(k\right)\cdot u_0\left(2t-k\right)\hfill \\ u_1\left(t\right)=\sqrt{2}\sum _{k\in z}B\left(k\right)\cdot u_0\left(2t-k\right)\hfill \end{array}\right..}$

Equation (4) presents functions that are reduced to obtain the corresponding scale function φ(t) and the corresponding wavelet function Ψ(t), respectively in the multi-resolution analysis. The dual scale equation is constructed by (5)${\displaystyle \left\{\begin{array}{c}\Psi \left(t\right)=\sqrt{2}\sum _{k\in z}A\left(k\right)\cdot \Psi \left(2t-k\right)\hfill \\ \phi \left(t\right)=\sqrt{2}\sum _{k\in z}B\left(k\right)\cdot \Psi \left(2t-k\right)\hfill \end{array}\right..}$

By the equivalence of Eqs. (3) and (4), the condition of n ∈ Z is deduced. Then, Eq. (2) can be rewritten by (6)${\displaystyle {\Omega }_{{}_{j+1}}^n={\Omega }_{{}_j}^{2n}\oplus {\Omega }_{{}_j}^{2n+1}.}$

The essence of the WPD is to further decompose the signal into two parts, namely, low frequency and high frequency, respectively. Then, $s_j^n\left(t\right)\in U_j^n$ isset as follows: (7)${\displaystyle s_j^n\left(t\right)=\sum \underset{l}{\overset{j,n}{r}}{u}_n\left(2^{j}t-l\right).}$

By Eq. (7), $r_l^{j+1,n}$ is resolved to obtain $r_l^{j,2n}$ and $r_l^{j,2n+1}$defined by (8)${\displaystyle \left\{\begin{array}{c}{r}_l^{j,2n}=\sum _k{\alpha }_{k-2l}\cdot r_k^{j+1,n}\hfill \\ r_l^{j,2n+1}=\sum _k{\beta }_{k-2l}\cdot r_k^{j+1,n}\hfill \end{array}\right..}$

Equation (8) is the WPD algorithm, which makes it possible to divide the PD signal into different WDs, which have the same bandwidth and are connected back and forth. The number of signal points contained is folded into half of the previous layer.

The PD data of HVPTLs from the Kaggle featured prediction competition is utilized. The dataset is normalized and categorized into two cases, namely, normal and faulty. The original signals of the two states and the results of the decomposition of the original signals implementing the WPD are presented in Fig. 1.

Figure 1 depicts that the PD signal composition of HHVPTLs is extremely complex, and it is difficult to guarantee higher accuracy by simply running the diagnosis manually. So, the extraction of the features of the PD signal is required to run a further diagnosis.

Feature extraction of PD signal

Figure 1 depicts that the PD signal energy of HHVTPLs is mainly concentrated in the fundamental frequency ranging from 0∼10,000 Hz. To swiftly grab the PD signal features of HHVPTLs, the article adopts the main frequency energy share of the PD signal obtained by the WPD as the core feature index. It adopts the assessment index of the conventional mechanical signal in Table 1 as the main attribute index of the PD signal to jointly establish the feature vector of the PD signal of HHVPTLs.

Since the frequency of the PD signal is primarily concentrated in the first decomposition node, the expression for the energy share of the main frequency V ₁ is defined by (9)${\displaystyle {\mathrm{V}}_1=\frac{\left|x_{1,n}^2\right|}{\sum _{i=1}^7\sum _n^N\left|x_{i,n}^2\right|}}$ where E₀, N and x_i,n denote the fundamental frequency energy, the length of the signal data sample, and the amplitude of each discrete point after the signal was reconstructed.

The final attribute vector of the PD signal of HHVPTLs that can be obtained consists of 12 total features denoted by V ₁ through V ₁₂.

BPNN

The essence of BPNN processes error terms between the actual output value and the predicted value by implementing the gradient descent algorithm. Namely, the derivative of the errors is resolved to tune the network weights and the threshold and the correction along the negative gradient of the derivative is continuously updated until the output error meets the preset requirements. When compared with the conventional algorithms, the advantage of the BPNN is that a deterministic mathematical model is no longer needed. The training stage is embedded in the model, and the corrected network can be obtained by tuning the parameters repeatedly (Zhan et al., 2022; Ou et al., 2022). The structure of BPNN is shown in Fig. 2.

The input layer in Fig. 2 contains n neurons that define the input vector, the hidden layer contains l neurons, the hidden layer contains m neurons, and the output vector is defined as the last layer of the network. The link weights between the input layer and the hidden layer are defined as w_ij, the link weights between the hidden layer and the output layer are defined as w_jk, the threshold value of the neurons contained in the hidden layer is set to α_j(j = 1, 2, ⋅⋅⋅, l), and the threshold value of the neurons contained in the output layer is set to β_k(j = 1, 2, ⋅⋅⋅, l).

In the implementation of the BPNN, the training stage first determines network weights, and thresholds are continuously corrected during the training stage to transform the network into a prediction function. The steps are presented below.

Initialize the model parameters. The number of neurons in each layer is set according to the input and output vectors of the model, the link weights, the threshold of neurons in the hidden layer, and the threshold of neurons in the output layer are initialized, and the learning rate and the corresponding excitation function of the model are set.

Calculation of hidden layer output. The output vector H of the hidden layer is calculated based on the input vector of the model, the link weights between the input layer and the hidden layer, and the threshold of the hidden layer. (10)${\displaystyle H_{\mathrm{j}}=Function\left(\sum _{i=1}^n{w}_{ij}{x}_i-{\alpha }_j\right),j=1,2\cdot \cdot \cdot ,l.}$

Equation (10) represents the excitation function of the hidden layer.

Calculation of output layer output. Similar to , the network output vector O is calculated by employing the output of the hidden layer, the link weights of both the hidden layer and the output layer between each other, and the threshold value of the output layer. Equation (11) presents the output layer. (11)${\displaystyle O_k=\sum _{j=1}^l{H}_j{w}_{jk}-{\beta }_k,k=1,2\cdot \cdot \cdot ,m.}$

Calculation of network error. The expected error E_k is obtained by calculating the difference between the expected output O_k of the model and the desired output Y_k. Equation (12) presents the error term. (12)${\displaystyle E_k={\mathrm{Y}}_k-O_k,k=1,2,\cdot \cdot \cdot ,m.}$

Correction of network weights. The link weights of the network are corrected according to the network error. (13)${\displaystyle w_{ij}^{\prime }=w_{ij}+\xi H_j\left(1-H_j\right)x_i\sum _{k=1}^m{w}_{jk}{E}_k}$ (14)${\displaystyle w_{jk}^{\prime }=w_{jk}+\xi H_j{E}_k}$ where ξ denotes the learning rate, $w_{ij}^{\prime }$ and w_ij denote the updated weight at the next iteration, $w_{jk}^{\prime }$ and w_jk represent the updated weight at the next iteration.

Correction of network threshold. The network threshold is corrected according to the network error, where ${\alpha }_i^{\prime }$ and ${\beta }_k^{\prime }$ denote the threshold parameters to be updated at the next iteration. (15)${\displaystyle {\alpha }_i^{\prime }={\alpha }_i+\xi H_j\left(1-H_j\right)x_i\sum _{k=1}^m{w}_{jk}{E}_k}$ (16)${\displaystyle {\beta }_k^{\prime }={\beta }_k+E_k.}$

Determine whether the output meets the termination condition and terminate the iteration if the preset score is reached, otherwise, return to .

Improvement of the BPNN based on orthogonal dyadic learning dung beetle optimization

Since the actual operation of the BPNN is mainly found by the network parameters, such as the number of nodes and connection weights, the final scores of the network parameters will have an impact on the BPNN’s results. To ensure that the BPNN has a good search performance and avoids falling into local optimums, an optimization algorithm needs to be introduced to optimize network parameters.

The dung beetle optimization (DBO) is the recently developed algorithm working on population intelligence optimization proposed in 2022. It is mainly derived from the habits of dung beetle life and has a robust search capability and high convergence efficiency. Thus, the DBO algorithm is employed to optimize BPNN in the manuscript.

DBO employs tentacle navigation when rolling to ensure that the dung ball keeps a straight line forward during the rolling process, and this behavior is needed in the simulation to permit the dung beetle to advance in the search space in the set direction and assume that light intensity affects the selection of dung beetle’s forward path. Then, the position of the dung beetle during the forward process can be defined by (17)${\displaystyle x_i\left(t+1\right)=x_i\left(t\right)+\lambda k{x}_i\left(t-1\right)+\mu \Delta x}$ (18)${\displaystyle \Delta x=\left|x_i\left(t-1\right)-x_{worst}\right|}$

where t characterizes the number of iterations so far, x_i(t) characterizes the position of the i-th dung beetle at the t-th iteration, and k ∈ (0, 0.2) characterizes the deflection factor, which is usually set to a constant value. λ taking either −1 or 1, and µdenotes a constant value in (0-1), x_worst characterizes the local worst position, Δx representing mainly used regulation for light intensity.

When a dung beetle encounters an obstacle and cannot move forward, it needs to re-roll to reorient itself to develop a new route. To model the rolling orientation behavior, the new direction is resolved by utilizing the tangent function defined by. (19)${\displaystyle x_i\left(t+1\right)=x_i\left(t\right)+tan\theta \left|x_i\left(t\right)-x_i\left(t-1\right)\right|.}$

In Eq. (19), θ ∈ [0, π] characterizes the angle of deflection and x_i(t) − x_i(t + 1) represents the difference between the anterior and posterior positions of the i-th dung beetle at distinct iteration cycles.

In addition, female, juvenile, and thieving dung beetles are defined in the population with position expressions as follows: (20)${\displaystyle x_{fdb\text{\_}i}\left(t+1\right)=x_{best}+{\varpi }_1\left(x_{fdb\text{\_}i}\left(t\right)-l{b}^{\mathit{\ast }}\right)+{\varpi }_2\left(x_{fdb\text{\_}i}\left(t\right)-u{b}^{\mathit{\ast }}\right)}$ (21)${\displaystyle x_{jdb\text{\_}i}\left(t+1\right)=x_{best}+{\sigma }_1\left(x_{jdb\text{\_}i}\left(t\right)-l{b}^c\right)+{\sigma }_2\left(x_{jdb\text{\_}i}\left(t\right)-l{b}^c\right)}$ (22)${\displaystyle x_{tdb\text{\_}i}\left(t+1\right)=x_{bf}+\eta \cdot \xi \cdot \left(\begin{array}{c}\left|x_{tdb\text{\_}i}\left(t\right)-x_{best}\right|\hfill \\ +\left|x_{tdb\text{\_}i}\left(t\right)-x_{bf}\right|\hfill \end{array}\right).}$

To enhance the plurality of the dung beetle population, orthogonal contrastive learning is implemented to enhance it. Firstly, the orthogonal variation operator is introduced to perturb the optimal position in the population so that the joiner of the stealer masters from the variation solution of the optimal position of the finder, which can increase the population diversity with the ability to jump out of the local optimum, then Eq. (22) can be rewritten as (23)${\displaystyle x_{tdb\text{\_}i}\left(t+1\right)=x_{bf}+\hslash \cdot \eta \cdot \xi \cdot \left(\begin{array}{c}\left|x_{tdb\text{\_}i}\left(t\right)-x_{best}\right|\hfill \\ +\left|x_{tdb\text{\_}i}\left(t\right)-x_{bf}\right|\hfill \end{array}\right).}$

In Eq. (17), $\hslash =\left\{\begin{array}{cc}\hfill -1,\hfill & \hfill r<0.5\hfill \\ \hfill 1,\hfill & \hfill else\hfill \end{array}\right.$r¡ 0 is defined.

Contrastive learning is a commonly implemented strategy to jump out of the position found as the local optimal solution. In the detection of the DBO, the dung beetles in the optimal position move toward the worst solution, and the dung beetles in other positions move toward the optimal score. Although searching for the worst solution position can avoid falling into the local optimum to some extent, this is not conducive to the convergence of the population. In contrast, contrastive learning not only helps individuals to escape from the current position quickly, but also the contrastive position is more likely to have a better fitness score than the current worst position, so the contrastive learning strategy with these beneficial properties is introduced into the position update formula for rolling directional advance, and Eq. (19) is improved as follows: (24)${\displaystyle x_i\left(t+1\right)=\left\{\begin{array}{c}lb+ub-\delta x_i\left(t\right),f\left[x_i\left(t\right)\right]\le f\left[x_i\left(t-1\right)\right]\hfill \\ x_i\left(t\right)+\delta \cdot tan\theta \left|x_i\left(t\right)-x_i\left(t-1\right)\right|,else\hfill \end{array}\right..}$

In Eq. (24): δ and β represent random numbers following N(0,1), lb and ub denote the lower and upper limits of the merit search, respectively.

Then, the overall optimization process of the IDBO-BPNN is presented as follows:

Step 1: Collect the local discharge data of HVPTLs, divide the training set and test set according to phase and state, and input the training set into the model for learning.

Step 2: Initialization of the BPNN network parameters, DBO population, and algorithm parameters.

Step 3: Solve all individual fitness scores based on DBO according to the objective function.

Step 4: Update the dung beetle positions and determine if they are out of bounds by utilizing the orthogonal opposition learning algorithm.

Step 5: Update the optimal position of the dung beetle and the fitness score.

Step 6: Repeat the above steps until the iteration limit is reached; if not, then skip to step 2; otherwise, output the global optimal solution and fitness score to the BPNN.

Step 7: Output the local release pattern recognition result of HHVPTLs.

Data source and experimental environment

The data sources are extracted from the Kaggle featured prediction competition. The data are initialized to obtain 8,188 sets of normal PD signals and 8,188 sets of faulty PD signals, each with a measurement time of 20ms, containing 800,000 points, and a line operating frequency of 50 Hz. Each group of signals in the data set will experience a complete three-phase AC power supply operating cycle, and all three phases are measured simultaneously.

A method to recognize the local discharge pattern of the HVVTPL is proposed in the article. Tests were conducted on a computer whose experimental environment is shown in Table 2.

Experimental results

To test the accuracy and efficiency of the proposed IDBO-BPNN based on the data set composed of 12,600 subsets of data in the form of the PD signal of HVPTLs, normal and faulty data constitute half of the total number of the dataset. The training and test sets are divided according to the ratio of 7:2. The dataset is utilized as the input to the IDBO-BPNN to train the network. Then, the test set is employed to test the trained IDBO-BPNN network to check whether the constructed network recognizes the fault state of PD signals and calculates the accuracy. Then, the outcomes are compared with the computations of the backpropagation neural network (BPNN), support vector machine (SVM), Elman neural network ELM, deep belief network (DBN), particle swarm optimization-backpropagation neural network (PSO-BPNN) gray wolf optimization- backpropogation neural network (GWO-BPNN), genetic algorithm- backpropogation neural network (GA-BPNN), and dung beetle optimization-backpropagation neural network (DBO-BPNN) (Wu et al., 2021; Jiang et al., 2021; Dan et al., 2021). The obtained outcomes are shown in Fig. 3. While 0 indicates a normal PD signal, 1 designates a faulty PD signal.

Figure 3 depicts that the unoptimized BPNN has a relatively low accuracy of 71.8333% in the recognition of PD signals due to the limitations of its structure and the incapability of the iterative algorithm. In contrast, the recognition accuracy of machine learning algorithms such as SVM and ELM, the accuracy reaches only 76.5% and 74.8333%, respectively. In addition, the recognition accuracy of the DBN reaches 82.1667%. Due to their limitations, the above methods are not efficient in extracting features of PD signals, which leads to unsatisfactory recognition accuracy as well. After implementing the standard optimization algorithms such as particle swarm optimization (PSO), gray wolf optimization (GWO), and genetic algorithm (GA) to optimize the BPNN, the recognition accuracy of the obtained PD signal reaches 84.3333%, 86.5%, and 89.6667%, respectively, which is an important enhancement when compared with the unoptimized BPNN. The improvement of the BPNN based on the optimization algorithms is a better approach to attain higher accuracy.

After implementing the DBO to optimize the BPNN, the recognition accuracy of the DBO-BPNN achieves 90.1667%, which is higher when compared with all implemented algorithms, which indicates that the DBO algorithm optimizes the BPNN better when compared with other optimization methods. However, an accuracy error close to 10% is hardly acceptable in engineering implementations. Therefore, the recognition accuracy of the IDBO-BPNN obtained by implementing the DBO is improved with the orthogonal opposition learning algorithm to deal with this issue. Thus, the IDO-BPNN can achieve 95.5%, which is 5.3333% higher than the recognition accuracy of the DBO-BPNN and far more than other recognition algorithms. Therefore, PD signal patterns can be recognized more accurately and present practical performance.