Predictive Analysis and Correction Control of CCT for a Power System Based on a Broad Learning System

Abstract

Transient stability is an important factor for the stability of a power system. With improvements in voltage levels, and the expansion of power network scales, the problem of transient stability is particularly prominent. When a power system circuit fails, if the operation time of the relay protection device is higher than the critical clearing time (CCT), the relay protection device cannot cut the fault line in a timely manner. It is essential to forecast and adjust the CCT to improve the stability of the system; therefore, a method is proposed in this paper to predict and evaluate the critical clearing time using the broad learning system (BLS). The sensitivity of the critical clearing time can be easily calculated based on the prediction results of the critical clearing time using BLS. Moreover, the critical clearing time can be modified using the BLS correction control model. The proposed method was verified using a 4-machine 11-node system and a 10-machine 39-node system. According to the experimental results, the proposed model can predict, evaluate, and correct the CCT very well.

1. Introduction

Currently, power grids are becoming increasingly interconnected and the operating environment is becoming more complex [1]. With the development of interconnected systems, the use of new controls to improve normal and emergency operations has brought sustainability issues to the forefront to a greater extent than previous years [2]. The addition of new energy sources to the grid, such as photovoltaics and wind power, makes it more vulnerable [3,4]. When the power grid has serious faults, or when it suffers from large disturbances, the stability of the power grid is negatively impacted, and it can even lead to system collapse. Serious economic losses and safety-related accidents can be avoided if effective safety measures are taken in time [5,6]. A rapid and accurate evaluation of the transient stability, and an adjustment after transient instability, is an important area of research, and it is conducive to the sustainable development of power systems.

Time-domain simulations and the direct method are two traditional methods for transient stability analysis [7]. In essence, the time-domain simulation method obtains the time solution by calculating differential equations of perturbed motion. Then, the stability of the system is judged using the relative angle of each generator rotor [8]. The direct method is also called the Transient Energy Function. It judges the system stability in accordance with the energy function [7,8]. The Lyapunov stability theory is an important theory for the direct method, and it was applied to the study of power systems’ transient stability by Magnusson in 1947. During the development of the direct method, the following factions were formed: Controlling Unstable Equilibrium Point (CUEP) [9], Potential Energy Boundary Surface (PEBS) [10], Boundary of Stability Region Based Controlling Equilibrium Point (BCU) [11], and the Extended Equal Area Criterion (EEAC) [12,13]. At present, the time-domain simulation method is the most mature method for transient stability analysis, and it has been widely used in various fields [14,15,16,17].

Critical clearing time (CCT) is one of the reference parameters for transient stability; it has been researched and analyzed using the direct method or time-domain simulation method. Naoto, Ardyono, and Hironori proposed a method to directly obtain the CCT based on boundary value problems; this is an accurate method, and it can detect the CCT of various unstable modes [18]. Thanh, Surour, Mohamed, and Konstantin described techniques for screening transient stability emergencies without relying on any time-domain simulations. They obtained the algebraic expression of the lower boundary of the critical clearance time. In addition, various methods have been proposed to extend the CCT to enhance the transient stability of power systems [19,20].

With the development of artificial intelligence, it has been applied to the evaluation of power systems’ transient stability. Researchers have developed artificial intelligence methods based on direct methods and time-domain simulations [21,22,23]. In 1989, Sobajic and Pao applied artificial neural networks (ANN) to the evaluation of the CCT of power systems. The results showed that the error between the estimated CCT and the actual CCT was very small [24]. This is the earliest combination of ANN and CCT. In 1997, Hobson and Allen pointed out the limitations of ANN applied to CCT prediction methods [25]. Amjady and Majedi used a new hybrid intelligent system to predict transient stability [26]. Pawlak and Annakkage utilized the least absolute selection and shrinkage operator (LASSO) to predict CCT [27]. Lv, Pawlak, and Annakkage proposed an additive regression model to forecast CCT, which shows that this method has a higher prediction accuracy than LASSO [28]. The abovementioned methods all improve the speed of CCT analysis in the power system; however, there are some shortcomings, such as inadequate fitting, falling into local optimization, and the inability to parallelize and retrain when the system mode changes.

This paper uses the broad learning system (BLS) for the prediction and adjustment of CCT in power systems. BLS is an effective linear regression method. It is a plane network model which aims to solve the shortcomings of large training errors in single-layer feedforward neural networks and in the complex structures of deep neural networks. It also has the potential to engage in incremental learning. As a result of this capability, BLS does not need to completely retrain the network once the network structure is extended.

This paper proposes a prediction and correction model for the CCT using BLS. The prediction model is used to predict the CCT of different fault lines. When the CCT is lower than the operation time of the relay protection device, the correction model is used to adjust the CCT.

The rest of this article is as follows: Section 2 briefly reviews the traditional calculation method for the CCT and introduces the BLS prediction model. Section 3 introduces the optimization model for the CCT. In Section 4, the proposed method is verified using a 4-machine 11-node system and a 10-machine 39-node system. Finally, the conclusion is given in Section 5.

2. Basic Models

2.1. Traditional Calculation Methods of CCT

Critical clearing time is an accepted transient stability measurement. There are extensive studies concerning exact computations using nonlinear analysis, numerical integration, and approximate computation using energy methods [29]. Regarding traditional transient stability calculations, both the direct method and time-domain simulation need to construct a classical mathematical model. The classic model and its parameters are shown below:

$$\left.\begin{array}{c}\frac{d{\delta }_i}{dt}={\omega }_i\\ \frac{d{\omega }_i}{dt}=\frac{1}{M_i}(P_{mi}-P_{ei})\end{array}\right\}$$

(1)

Among them,

$$\begin{gathered} P_{ei}=Re\left({\dot{E}}_i{\dot{I}}_i\right)=Re({\dot{E}}_i\sum _{j=1}^n{Y}_{ij}{\dot{E}}_j) \\ =Re[\sum _{j=1}^n{E}_i{E}_j\mathrm{\angle }{\delta }_{ij}\left(G_{ij}-j{B}_{ij}\right)] \\ =E_i^2{G}_{ii}+\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^n(E_i{E}_j{B}_{ij}\sin {\delta }_{ij}+E_i{E}_j{G}_{ij}\cos {\delta }_{ij}) \end{gathered}$$

(2)

where ${\omega }_i$ is the deviation between the generator speed and synchronous speed, ${\delta }_i$ is the rotor angle of the generator, $M_i$ is the inertia time constant of the generator rotor, $P_{mi}$ is the mechanical power of the generator, $P_{ei}$ is the electromagnetic power of the generator, ${\dot{E}}_i=E_i\mathrm{\angle }{\delta }_i$ is the internal potential of the generator, $Y_{ij}$ is the element in row i and column j of the node admittance matrix, $G_{ij}$ is the element in row i and column j of the conductance matrix, and $B_{ij}$ is the element in row i and column j of the susceptance matrix.

In accordance with the above, the traditional transient stability analysis method needs to build a complex mathematical model. The traditional method for calculating CCT is very time-consuming; it is more suitable for an offline design or planning work than online operations.

2.2. Prediction Model of CCT

This section focuses on introducing the BLS algorithm and CCT prediction model.

2.2.1. Broad Learning System

At present, deep neural networks have been well developed and they are applied in many fields. Common neural networks include the Deep Boltzmann Machine (DBM), Deep Belief Network (DBN), and Convolutional Neural Network (CNN) [30,31,32]. Although the deep neural networks have excellent feature extraction abilities and nonlinear approximation abilities, there are still many problems that occur during application. For example, there are too many hyperparameters, they are time-consuming, and they easily fall into local optimization. In addition, it is time consuming to retrain the deep neural network when the system has a new operation mode. For instances when the deep neural network is encountered above the bottleneck, Chen and Liu proposed the Broad Learning System (BLS) [33,34] which is based on the random vector function-link neural network (RVFLNN) [35].

The differences between the deep learning mode and broad learning system are as follows:

1.

The BLS has fewer layers and a simpler model structure than the deep learning model. Therefore, BLS has the same accuracy as the deep learning method, but it is faster to train. Compared with the popular deep method, RBM, BLS has the potential to be thousands of times faster [33].

2.

With regard to the deep learning mode, it is usually used to increase the number of windows or layers in order to solve problems that cannot be learned well. This method needs to set up the new structure parameters and train from the beginning. BLS avoids retraining the entire network using incremental learning [34].

3.

Deep learning updates weights and adjusts hyperparameters based on back propagation. BLS uses pseudo-inverse methods to calculate weights; this avoids problems such as local optimization and gradient explosion, which are issues faced by the deep learning methods [34,36].

BLS is different from traditional RVFLNN in that it does not directly take initial data for use as an input layer. Compared with RVFLNN, BLS first maps the inputs to build a set of mapped features. These features are used as the input layer of the original RVFLNN [36].

The process of the BLS training model is as follows:

1.

The inputted original data are mapped in order so that they can become the feature nodes of the network;

2.

The weight of the feature is optimized via the autoencoder;

3.

Feature nodes are changed into enhanced nodes via nonlinear functions;

4.

All the feature nodes and enhancement nodes are directly connected to the output;

5.

The weight of the whole network can be calculated using the pseudo-inverse calculation.

BLS only needs to rely on the weight of the pseudo-inverse network; it has the advantages of a fast calculation speed, no gradients disappear, and explosions do not occur.

Assuming that the input data set is $X=\left\{x_i\right|x_i\in R^M,i=1,\dots ,N\}$, the output data set is $Y=\left\{y_i\right|y_i\in R^C,i=1,\dots ,N\}$, where N is the total number of samples in the data set, M is the input dimensions, and C is the output dimensions. We used the following formula:

$$Z_i={\varphi }_i\left(X{W}_{ei}+{\beta }_{ei}\right),i=1,\dots ,n$$

(3)

where ${\varphi }_i$ is the activation function, and $W_{ei}$ and ${\beta }_{ei}$ are the random weights and bias, respectively. Denoting all the feature nodes as $Z^n=[Z_1,Z_2,\dots ,Z_n]$, the enhanced nodes of group m-th can thus be represented using the following formula:

$$Y=[Z_1,\dots ,Z_n|\xi \left(Z^n{W}_{h1}+{\beta }_{h1}\right),\dots ,\xi \left(Z^n{W}_{hm}+{\beta }_{hm}\right)]W^m=[Z^n\left|H^m\right]W^m$$

(4)

where $W^m$ represents the connection weight of the output matrix. Assuming that $A=\left[Z^n\right|H^m]$, in accordance with the pseudo-inverse calculation, $W^m$ can be expressed as:

$$W^m={\left[Z^n\right|H^m]}^{+}Y=A^{+}Y$$

(5)

It also can be expressed as:

$$W^m={(A^{T}A+\lambda I)}^{-}{A}^{T}Y$$

(6)

where $\lambda$ is the regularization coefficient, and I is the unit matrix.

The framework of a typical BLS is shown in Figure 1.

2.2.2. Prediction Model of CCT

In this paper, first, a BLS model suitable for CCT was constructed. When using a BLS model to predict the CCT, a BLS model should be trained first. Then, the trained BLS model should be tested to obtain a CCT prediction model. The process of the BLS prediction model is shown in Figure 2.

The raw data inputted into the BLS model is as follows:

$$X_q={[X_1,X_2,\dots ,X_q]}^T$$

(7)

$$Y_q={[{CCT}_1,\dots {CCT}_i\dots ,{CCT}_q]}^T$$

(8)

where, $X_i$ corresponds with the input parameter of the $i$-th sample, ${CCT}_i$ corresponds with the CCT of the $i$-th sample, and q is the number of samples.

$$X=[{PG}^k,{QG}^K,{PL}^n,{QL}^n,V^m,{\theta }^m]$$

(9)

where, ${PG}^k=\left[{PG}_1,\dots ,{PG}_k\right]$, ${QG}^k=\left[{QG}_1,\dots ,{QG}_k\right]$, ${PL}^n=\left[{PL}_1,\dots ,{PL}_n\right]$, ${QL}^n=\left[{QL}_1,\dots ,{QL}_n\right]$, $V^m=[V_1\dots ,V_m]$, and ${\theta }^m=\left[{\theta }_1,\dots ,{\theta }_m\right]$. Moreover, $PG,QG,PL,QL,V,\theta$ are the generator’s active power that was injected using the node, the generator’s reactive power, load active power, load reactive power, node injection voltage amplitude, and voltage phase angle, respectively. $k,n,m$ are the number of generators, loads, and nodes, respectively.

The traditional methods recalculate the energy function, or time-domain simulation, each time the CCT is calculated. Different from traditional methods, after training, the CCT prediction model, based on BLS, can calculate the corresponding CCT using only the power and voltage data of each node in the system operation. The proposed method does not need to know the parameters of each component, and it is unnecessary to calculate the energy function or time-domain simulation again. It is faster and more suitable for the real-time change power system.

3. Correction Control Model of CCT Base on BLS

In this section, the optimal adjustment model of the CCT is proposed. When the CCT correction control is carried out, the relay protection action time is set as threshold value. In the simulation experiments, which are detailed in this paper, we set the relay protection action time to 0.2 s. It is necessary to correct the control and improve the value of the CCT when the CCT value is lower than the threshold value.

3.1. The Sensitivity of a Single Variable of CCT in Each Generator

When calculating single-variable sensitivity, the sensitivity of the single-variable of CCT can be calculated using the ratio of the small change in CCT to the small change in the variable. Moreover, in terms of practical applications, the simplest and most common method involves changing the system’s running state by changing the active power output of the generator. Therefore, the CCT is increased by changing the active power of the generator when making optimal adjustments. The formulas are as follows:

$$C_i=\frac{∆CCT}{∆{PG}_i}$$

(10)

$$∆CCT=CCT\left({PG}_{i_0}+∆{PG}_i\right)-CCT\left({PG}_{i_0}\right)$$

(11)

where, ${PG}_{i_0}$ represents the initial value of the i-th parameter in the current running state, and $∆{PG}_i$ is the adjustment amount of the i-th parameter; here, the amount was adjusted to 0.1% of the initial value. $CCT\left({PG}_{i_0}\right)$ represents the initial value predicted by BLS, and $CCT\left({PG}_{i_0}+∆{PG}_i\right)$ represents the predicted value of BLS after the parameter changed.

3.2. Optimal Adjustment Model

The CCT sensitivity coefficient equation, which corresponds with each generator, can be expressed as:

$${∆CCT}^k={C_1}^k{∆{PG}_1}^k+{C_2}^k{∆{PG}_2}^k+\dots +{C_n}^k{∆{PG}_n}^k$$

(12)

where n is the total number of system generators. When the CCT of multiple lines needs to be adjusted, k is used to represent the different lines.

To minimize the active power variation of each generator, and to keep the CCT above the threshold, this paper uses the optimization model to correct the CCT. The formula is as follows:

$$min\sum _{i=1}^n{∆{PG}_i}^2$$

(13)

Constraints:

$$\left.\begin{array}{c}{{(PG}_i^k)}^{min}\le {{PG}_{i_0}}^k+{∆{PG}_i}^k\le {{(PG}_i^k)}^{max}\\ {{CCT}_0}^k+{∆CCT}^k\ge {CCT}_{min}\\ {\displaystyle \sum _{i=1}^n}{C_i}^k{∆{PG}_i}^k={∆CCT}^k\end{array}\right\}$$

(14)

where, ${{(PG}_i^k)}^{min}$ is the lower limit of the active power of generator i, ${{(PG}_i^k)}^{max}$ is the upper limit of the active power of generator i, and ${{CCT}_0}^k$ and ${CCT}_{min}$ are the initial CCT and the set CCT threshold, respectively. The adjustment process of the optimal adjustment model is shown in Figure 3.

4. Case Study

In this section, the formation of the data set and the two evaluation indicators of the prediction model will be introduced. Then, the tests that were performed on the BLS prediction model and correction control model will be discussed. This section will discuss the 4-machine 11-node system and 10-machine 39-node system that were used for the experiments. Figure 4 shows diagrams of the two systems, respectively.

4.1. Data Set

To simulate the real-time changing operation mode of the power system, the reference value is multiplied by a random coefficient within a certain range [37,38]. The load power of each node (${PL}^n$ and ${QL}^n$), and the power of each generator node (${PG}^k$ and ${QG}^K$), varies randomly within a certain range, as shown in

Table 1. The variation range of each node parameter.

Data Type	Range of Random Variation
The active power of each generator node	Reference value × (0.8–1.2)
The reactive power of each generator node	Reference value × (0.8–1.2)
The active power of each load node	Reference value × (0.8–1.2)
The reactive power of each load node	Reference value × (0.8–1.2)

. Within the data set, the input state variables include the voltage amplitude and the phase angle. Therefore, the system data set with 4-machines 11-nodes has 36-dimensional input variables and 1-dimensional output variables, and the system dataset with 10-machine and 39-nodes has 122-dimensional input variables and 1-dimensional output variables. The sample data set consists of the training set used to build the BLS model and the test set used to validate the model. The ratio of the training set to the test set is 9:1.

4.2. Evaluation Index

In this paper, the Root Mean Square Error (RMSE) and Symmetric Mean Absolute Percentage Error (SMAPE) were used to evaluate the effectiveness of the model [37]. SMAPE avoids the disadvantage of MAPE, which is that it is not being computable when the output is 0. RMSE was used to evaluate the deviation degree and error distribution of the model, whereas SMAPE was used to evaluate the mean value of the deviation degree error of the model. The calculation formula is expressed as:

$$SMAPE=\frac{100\%}{n}\sum _{k=1}^n\frac{|{\widehat{y}}_k-y_k|}{(\left|{\widehat{y}}_k\right|+\left|y_k\right|)/2}$$

(15)

$$RMSE=\sqrt{\frac{1}{n}\sum _{k=1}^n{({\widehat{y}}_k-y_k)}^2}$$

(16)

where n represents the number of test samples, $y_k$ represents the true value of the k-th sample, and ${\widehat{y}}_k$ represents the predicted value of the k-th sample.

4.3. Prediction and Analysis of a Single Faulty Line

In the power system, when under the same running state, the three-phase short circuit fault of different lines produces different CCTs. Therefore, the prediction model was used to predict the different faulty lines of the 4-machine 11-node system and to verify the performance of the prediction model. Each line had 4050 rows of training set data and 450 rows of test set data. Each row had 36-dimensional input variables and 1-dimensional output variables. The CCT prediction results of different fault lines are shown in

Table 2. Prediction results of the different faulty lines of the 4-machine 11-node power systems.

Line Number	Faulty Line	RMSE	SMAPE	Training Time (s)
1	Lines 4–5	0.0867	6.98	0.0086
2	Lines 4–6	0.0407	4.25	0.0124
3	Lines 5–7	0.0064	2.86	0.0135
4	Lines 6–9	0.0077	4.51	0.0193
5	Lines 7–8	0.0181	4.65	0.0094
6	Lines 8–9	0.0053	0.99	0.0179
7	Lines 8–10*	0.0021	0.85	0.0183
8	Lines 8*–10	0.0170	4.25	0.0166

* indicates the location where the fault occurs.

. As there were two parallel branches of lines 8–10, the two branches of lines 8–10 were tested separately. Lines 8*–10 indicated that the fault occurred on node 8, and lines 8–10* indicated that the fault occurred on node 10. It is assumed that the fault occurred at the end of the line unless specifically stated.

To observe the predictive performance of the BLS model, Figure 5 shows the comparison between the actual CCT and the predicted CCT of each faulty line, where the horizontal axis is the actual value and the vertical axis is the predicted value. The closer the point in the figure is to the blue diagonal line, the smaller the error between the predicted and measured values.

In Figure 5, Figure 5a–h are the BLS model prediction performance diagrams of line 1 to line 8, respectively. As is evident from Table 2, the maximum RMSE value for different faulty lines is line 1, but the maximum value is only 0.0867. This indicates that the deviation degree and error distribution of the proposed BLS model are very small. There is only a small error between the predicted value and the true value. The maximum value of SMAPE is 6.98%, with regard to line 1, which is less than 10%. This indicates that the mean value of the deviation degree error of the model is small. The training time of all lines is less than 0.02 s, and the calculation speed is fast. As shown in Figure 5, the points of all faulty lines are near the diagonal line, with only a few slightly deviating from the blue diagonal line. This means that the error between the predicted value and the real value is small, and the model exhibits a good degree of accuracy, which is consistent with the data in Table 2.

In this paper, a 10-machine 39-node system was used to verify the prediction model again. Each line had 900 rows of training set data and 100 rows of test set data. Each row had 122-dimensional input variables and 1-dimensional output variables. The results are shown in

Table 3. Prediction results of the different faulty lines of the 10-machine 39-node power systems.

Line Number	Faulty Line	RMSE	SMAPE	Training Time (s)
4	Lines 2–25	0.0025	4.31	0.0055
27	Lines 21–22	0.0119	2.62	0.0175
32	Lines 26–28	0.0048	2.21	0.0051
34	Lines 28–29	0.0043	2.55	0.0055

and Figure 6.

In Figure 6, Figure 6a–d are the BLS model prediction performance diagrams of line 4, line 27, line 32, and line 34, respectively. As is evident from Table 3, for different fault lines, the maximum RMSE value is the RMSE value of line 27, but the maximum value is only 0.0119. The maximum value of SMAPE is the RMSE of line 4, with a value of 4.31%, which is less than 10%. The training time of all the lines is less than 0.02 s, and the calculation speed is fast. As is evident from Figure 5, the points of all the faulty lines are near the diagonal line, with only a few points deviating slightly from the blue diagonal line. As shown in Table 3 and Figure 6, regarding the 10-machine 39-node system, the BLS model still has the advantages of a fast training speed and a high degree of precision.

According to the experimental results above, the BLS model has the ability to predict the CCT of a single faulty line well. The error between the predicted value and the real value is small, which shows that the model has a high level of accuracy, and that it can be applied to predict the CCT of a single faulty line.

4.4. Hybrid Faulty Line Prediction and Analysis

The above prediction model is based on the premise that the system first determines the fault line, and then it imports the data set into the BLS model for training and prediction. Under the same operating conditions, imposed by the system, different faulty lines will generate different CCT. To verify that the BLS model can accurately judge the CCT of different lines and make accurate predictions, the above simulation experiment was changed.

The adjustment method is as follows: add one-dimensional data to the same system’s running data to distinguish between different faulty lines.

$$X=[{N,PG}^k,{QG}^K,{PL}^n,{QL}^n,V^m,{\theta }^m]$$

(17)

where N is the line number. Different faulty lines have different N values to distinguish between different fault lines.

The input for the different faulty lines was the same, except for the N values. Then, the input of all fault lines and corresponding CCT were formed into composed o data sets and imported into the BLS model for training and testing to verify the identification ability and prediction accuracy. The simulation results are shown in

Table 4. Prediction results of the two systems.

System Type	RMSE	SMAPE	Training Time (s)
4-machine 11-node power system	0.1099	6.26	0.725
10-machine 39-node power system	0.0349	8.36	0.943

and Figure 7.

In the mixed fault experiment of the two systems, the RMSE values are 0.1099 and 0.0349, respectively. The values of the SMAPE were 6.26% and 8.36%, respectively, and the SMAPE values were all less than 10%. This indicates that the deviation degree and error distribution of the CCT prediction model, based on BLS, were very small. Moreover, there is only a minimal error between the predicted value and the real value. It also shows that the deviation degree of the model was small, which is acceptable. Figure 7 also shows that the CCT prediction model, based on BLS, produces small errors and a high degree of precision. The BLS prediction model can quickly distinguish between different faulty lines and make accurate judgments when different lines fail during the same operation.

In addition, RMSE is used to evaluate the model in Ref. [28]. In Ref. [28], the minimum RMSE is 0.663, thus indicating that the prediction performance of the additive regression model is accurate, and that it has a higher prediction accuracy than Lasso. However, the maximum RMSE value in this paper is only 0.1099, which is smaller than 0.663. This shows that the BLS model is accurate for the prediction of CCT.

4.5. Optimal Adjustment Model Experiment of a Single Faulty Line

The relay protection device cuts the faulty line out when the line fails. The relay protection device needs a reaction time during this process. When the CCT is less than the reaction time of the relay protection device, it cannot remove the faulty line in time. This will affect the stable operation of the power system. Therefore, we propose an optimal adjustment model, based on BLS, which adjusts the CCT to be higher than, or equal to, the operation time of the relay protection device, while ensuring that the generators undergo minimal changes. This paper uses a 4-machine 11-node system to verify the optimal adjustment model of BLS. The adjustment process is shown in Figure 7, which uses faulty lines 6–9 as an example. The initial predicted value of the CCT of faulty lines 6–9 is 0.163255 s, and the single-variable sensitivity of each parameter is shown in

Table 5. Sensitivity data of the 4-machine 11-node power system for faulty lines 6–9.

PG	${\mathit{P}\mathit{G}}_{{\mathit{i}}_0}$	$∆{\mathit{P}\mathit{G}}_{\mathit{i}}$	$\mathit{C}\mathit{C}\mathit{T}\left({\mathit{P}\mathit{G}}_{{\mathit{i}}_0}+∆{\mathit{P}\mathit{G}}_{\mathit{i}}\right)$	$∆\mathit{C}\mathit{C}\mathit{T}$	C
${PG}_1$	2.34	0.00234	0.163192	−0.000063	−0.027
${PG}_2$	1.60	0.00160	0.162825	−0.000430	−0.253
${PG}_3$	0.91	0.00091	0.162906	−0.000349	−0.382
${PG}_{11}$	1.91	0.00191	0.162686	−0.000569	−0.297

.

To visually observe the sensitivity comparison between each parameter, the sensitivity of each parameter is shown in Figure 8. Red represents negative sensitivity, that is, a negative correlation and blue represents positive sensitivity, which is a positive correlation. When the faulty line was cut at 0.2 s, it produced a swing curve caused by the generator rotor, as shown in Figure 9.

Figure 9a shows that the swing curve diverges before adjustment. This indicates that the system was unstable at this time, with a CCT less than 0.2 s. Figure 9b shows that the swing curve fluctuated within a certain range after adjustment. This indicates that the system was stable and the CCT was adjusted to 0.2 s or higher. As is evident from Table 5 and Figure 8, ${PG}_1$ had the lowest degree of sensitivity, whereas ${PG}_3$ had the highest degree of sensitivity. In addition, it is evident from

Table 6. Active power changes after the optimal adjustment of the 4-machine 11-node system for faulty lines 6–9.

PG Type	${\mathit{P}\mathit{G}}_{{\mathit{i}}_0}$	${\mathit{P}\mathit{G}}_{{\mathit{i}}_0}+∆\mathit{P}\mathit{G}$	$∆\mathit{P}\mathit{G}$
${PG}_1$	2.3444	2.3411	−0.0032
${PG}_2$	1.6984	1.6673	−0.0302
${PG}_3$	0.9152	0.8682	−0.0456
${PG}_{11}$	1.9148	1.8783	−0.0335

that ${PG}_3$ exhibited the greatest power change, whereas ${PG}_1$ exhibited the smallest power change. This shows that the BLS optimal adjustment model is effective.

Once again, to prove the universal validity of the BLS optimal adjustment model, fault lines 26–29 in the 10-machine 39-node system were used to verify the model. The sensitivity parameters of the active power of each generator in the 10-machine 39-node system are shown in

Table 7. Sensitivity data of the 10-machine 39-node power system for faulty lines 28–29.

PG	${\mathit{P}\mathit{G}}_{{\mathit{i}}_0}$	$∆{\mathit{P}\mathit{G}}_{\mathit{i}}$	$\mathit{C}\mathit{C}\mathit{T}\left({\mathit{P}\mathit{G}}_{{\mathit{i}}_0}+∆{\mathit{P}\mathit{G}}_{\mathit{i}}\right)$	$∆\mathit{C}\mathit{C}\mathit{T}$	C
${PG}_{30}$	146.3322	0.1463	0.184759	0.000011	0.000076
${PG}_{31}$	444.1773	0.4442	0.184705	−0.000043	−0.000097
${PG}_{32}$	612.6680	0.6127	0.184717	−0.000031	−0.000049
${PG}_{33}$	599.4829	0.5995	0.184423	−0.000325	−0.000540
${PG}_{34}$	420.2291	0.4202	0.184775	0.000027	0.000064
${PG}_{35}$	613.6363	0.6136	0.184512	−0.000236	−0.000390
${PG}_{36}$	532.3037	0.5323	0.184710	−0.000038	−0.000072
${PG}_{37}$	427.4407	0.4274	0.184758	0.000010	0.000024
${PG}_{38}$	687.0777	0.6871	0.183915	−0.000833	−0.001210
${PG}_{39}$	1046.338	1.0463	0.184534	−0.000214	−0.000200

and Figure 10. The predicted value of the initial CCT of faulty lines 28–29 is 0.184748 s. When the CCT was adjusted to 0.2 s, changes in the active power of each generator occurred and are shown in

Table 8. Active power changes after the optimal adjustment of the 10-machine 39-node system for faulty lines 28–29.

PG Type	${\mathit{P}\mathit{G}}_{{\mathit{i}}_0}$	${\mathit{P}\mathit{G}}_{{\mathit{i}}_0}+∆\mathit{P}\mathit{G}$	$∆\mathit{P}\mathit{G}$
${PG}_{30}$	146.3322	148.0647	1.7325
${PG}_{31}$	444.1773	441.9520	−2.2252
${PG}_{32}$	612.6680	611.5317	−1.1362
${PG}_{33}$	599.4829	587.0986	−12.3841
${PG}_{34}$	420.2291	421.6905	1.4615
${PG}_{35}$	613.6363	604.8490	−8.7872
${PG}_{36}$	532.3037	530.6738	−1.6298
${PG}_{37}$	427.4407	427.9833	0.5426
${PG}_{38}$	687.0777	659.4274	−27.6502
${PG}_{39}$	1046.338	1042.7150	−4.6701

. The swing curve of the generator rotor, after the fault line was cut at 0.2 s, is shown in Figure 11.

As is evident from Figure 11, after adjustment, the system ran stably when the faulty line was removed at 0.2 s. With similar results to the 4-machine 11-node system, Table 7 and Table 8 show the validity of the model.

In accordance with the experimental results of the 10-machine 39-node system, the BLS optimal adjustment model can adjust the CCT to a threshold that is higher than, or equal to, 0.2 s for different systems and different fault lines. Therefore, the optimal BLS adjustment model is suitable for the prediction analysis and adjustment of the CCT.

4.6. Optimal Adjustment Model Experiment of Multiple Faulty Lines

Regarding the abovementioned adjustment, the CCT of only one fault line was considered to be lower than the action time of the relay protection device. During the actual operation of the power system, the CCT of many lines is likely to be lower than the operation time of the relay protection device.

Experiments were conducted to verify whether the optimal adjustment model can adjust the CCT of multiple lines to be higher than, or equal to, the operation time of the relay protection device. The following experiments use the 4-machine 11-node system and 10-machine 39-node system.

In the 4-machine 11-node system, the lines with a CCT lower than the threshold were lines 5–7, lines 6–9, and lines 8–10. The changes in the active power parameters of the generator are shown in

Table 9. The changes in the generator’s active power parameters of the 4-machine 11-node system.

PG Type	${\mathit{P}\mathit{G}}_{{\mathit{i}}_0}$	${\mathit{P}\mathit{G}}_{{\mathit{i}}_0}+∆\mathit{P}\mathit{G}$	$∆\mathit{P}\mathit{G}$
${PG}_1$	1.7829	1.7833	−0.0004
${PG}_2$	1.6496	1.6425	−0.0071
${PG}_3$	1.0948	1.0816	−0.0132
${PG}_{11}$	1.6471	1.5091	−0.1380

, and the changes in the CCT of different fault lines are shown in

Table 10. The changes in the CCT of the different fault lines of the 4-machine 11-node system.

Faulty Lines	Actual CCT before Adjustment	Forecast CCT before Adjustment	Actual CCT after Adjustment	Forecast CCT after Adjustment
5–7	0.161	0.158	0.201	0.228
6–9	0.199	0.197	0.226	0.232
8–10	0.164	0.161	0.201	0.205

. The swing curve of the generator rotor, after the fault line was cut at 0.2 s, is shown in Figure 12.

In a 10-machine 39-node system, lines with a CCT lower than the threshold were lines 26–27 and lines 26–28. The changes in the active power parameters of the generator are shown in

Table 11. Active power changes after the optimal adjustment to the 10-machine 39-node system.

PG Type	${\mathit{P}\mathit{G}}_{{\mathit{i}}_0}$	${\mathit{P}\mathit{G}}_{{\mathit{i}}_0}+∆\mathit{P}\mathit{G}$	$∆\mathit{P}\mathit{G}$
${PG}_{30}$	178.2355	185.9705	−7.7350
${PG}_{31}$	515.6403	530.6462	−15.0059
${PG}_{32}$	624.4492	607.3772	17.0720
${PG}_{33}$	531.3189	514.5355	16.7833
${PG}_{34}$	451.6292	461.1221	−9.4929
${PG}_{35}$	456.9221	456.5326	0.3895
${PG}_{36}$	482.4532	476.2677	6.1855
${PG}_{37}$	439.3755	452.4164	−13.0409
${PG}_{38}$	785.4484	694.6076	90.8408
${PG}_{39}$	722.1255	726.9441	−4.8186

, and the changes in the CCT of different fault lines are shown in

Table 12. The changes in CCT of the different fault lines of the 10-machine 39-node system.

Faulty Lines	Actual CCT before Adjustment	Forecast CCT before Adjustment	Actual CCT after Adjustment	Forecast CCT after Adjustment
26–27	0.199	0.184	0.254	0.236
26–28	0.142	0.151	0.203	0.222

. The swing curve, after the fault line was cut at 0.2 s, is shown in Figure 13.

As shown in Table 10 and Table 12, the CCT of the different lines all increased from less than 0.2 s to more than 0.2 s, and the error between the predicted value and the real value was about 10%. In Figure 12, Figure 12a,c,e are the swing curves of lines 5–7, 6–9, and 8–10, which were removed at 0.2 s before adjustment, respectively. Figure 12b,d,f are the swing curves of lines 5–7, 6–9, and 8–10, which were removed at 0.2 s after adjustment, respectively. In Figure 13, Figure 13a,c are the swing curves of lines 26–27 and 26–28, which were removed at 0.2 s before adjustment, respectively. Figure 13b,d are the swing curves of lines 26–27 and 26–28, removed at 0.2 s after adjustment, respectively.

As is evident from Table 9 and Table 11, the power of all the generators changed during the adjustment process. It is evident from Table 10 and Table 12 that the CCT of all lines increased from less than the threshold, which was 0.2, to more than 0.2, no matter the predicted value or the actual value. In addition, the error between the actual value and the predicted value was within 10%.

As shown in Figure 12 and Figure 13, all of the line swing curves diverge before adjustment, thus indicating that the system was in an unstable state. After adjustment, the swing curve fluctuated within a certain range, thus indicating that the system was stable, and that the CCT had been adjusted to 0.2 s or higher.

As per the results of the abovementioned experiments, the model can adjust the CCT of multiple lines to be higher than, or equal to, the operation time of the relay protection device in order to satisfy power system stability.

5. Conclusions

Regarding the transient stability analysis of power systems, CCT has always been an important research object. This paper proposes a prediction and correction method for CCT, based on the BLS. The BLS model is used to predict CCT. The adjustment of the time variable, CCT, is achieved by changing the input variable. This method can improve the speed of CCT calculation and system adjustment compared with traditional methods. The BLS model was tested using a 4-machine 11-node system and a 10-machine 39-node system. The experimental results show that the BLS model can predict the CCT speed quickly, and the training speed can fall below 0.2 s in a single faulty line. It can distinguish between different faulty lines in a hybrid faulty line experiment, and it can produce accurate results, which is effective for different systems.

During the optimal adjustment model experiment, when the CCT is less than the threshold, the BLS optimal adjustment model can calculate the optimum adjustment scheme. Moreover, it can adjust the CCT to a value that is higher than, or equal to, the threshold, thus indicating that the optimal BLS model can be used to adjust the power system and improve the stable operation of the power system. Therefore, the BLS model can satisfy the prediction analysis and adjustment of the CCT, and it confirms that it is a practical analysis method.

The BLS model also has an incremental learning function. When new data is added to the model, the weight of the system can be updated quickly without needing to completely retrain the model; this is suitable for today’s rapidly changing power system network. Therefore, the incremental learning model of BLS should be studied further in the future, so that the BLS model can be better applied to the transient stability analysis of power systems.