Abstract
The steady-state security region (SSR) offers robust support for the security assessment and control of new power systems with high uncertainty and fluctuation. However, accurately solving the steady-state security region boundary (SSRB), which is high-dimensional, non-convex, and non-linear, presents a significant challenge. To address this problem, this paper proposes a method for approximating the SSRB in power systems using the feature non-linear converter and improved oblique decision tree. First, to better characterize the SSRB, boundary samples are generated using the proposed sampling method. These samples are distributed within a limited distance near the SSRB. Then, to handle the high-dimensionality, non-convexity and non-linearity of the SSRB, boundary samples are converted from the original power injection space to a new feature space using the designed feature non-linear converter. Consequently, in this feature space, boundary samples are linearly separated using the proposed information gain rate based weighted oblique decision tree. Finally, the effectiveness and generality of the proposed sampling method are verified on the WECC 3-machine 9-bus system and IEEE 118-bus system.
WITH the significant increase in the proportion of renewable energy and power electronic equipment integrated into the power system, the uncertainty and fluctuation of the system have risen dramatically, and the operating mode has become more variable [
The steady-state security region (SSR), as the set of all operating points on the power injection space that satisfy the power flow equation and system security constraints, can provide powerful support for the security assessment and control of power systems with high uncertainty and fluctuation [
The theoretical derivation of the SSR is achieved by combining the decoupled power flow equation with the Leray-Schauder fixed point [
The methods mentioned above theoretically derive the SSR from the power flow equation and its security constraints. However, these methods are often conservative and may fail to include some secure operating points that are of interest to operators. Therefore, there is a need to develop SSR solutions with higher accuracy. In [
Based on the above conclusions, some research works attempt to address the SSRB from a data-driven perspective. In [
Nevertheless, accurately solving the SSRB remains challenging. This is due to the non-linearity of the power flow equation in AC systems and the numerous complex security constraints, making SSRB non-convex and non-linear. With the increasing size of power systems and the growing number of power electronic devices connected to the grid, these problems are further aggravated. As a result, the aforementioned methods often suffer from significant errors. Meanwhile, the methods proposed above are challenging to be applied to solve high-dimensional SSRB.
In recent years, deep neural networks (DNNs) [
Based on the analysis presented above, combined with the SSR property proven by [
1) For the high-dimensional, non-convex, and non-linear SSRB, a novel SSRB approximation framework via the feature non-linear converter and information gain rate based weighted oblique DT (IGR-WODT) is proposed.
2) An improved sampling method is proposed to search for boundary sample pairs which are distributed near the SSRB in order to facilitate the subsequent model to better learn the characteristics of SSRB.
3) The DNN-based model is designed to address the non-linearly separable issue within the dataset, and IGR-WODT is employed to approximate SSRB.
4) The proposed sampling method successfully approximates the high-dimensional SSRB and reduces the error between the approximated boundary and the actual one, which are verified on the WECC 3-machine 9-bus system and IEEE 118-bus system.
The remainder of this paper is organized as follows. Section II first introduces the SSR model of power systems. Section III details high-dimensional SSRB approximation. Section IV presents the case study. Finally, Section V summarizes the paper.
The SSR model of power systems is defined as the set of points satisfying the power flow equation and operating security constraints [
(1) |
where is the static voltage security region satisfying the voltage constraints; is the static generator active output security region satisfying the generator active output constraints; is the static generator reactive output security region satisfying the generator reactive output constraints; is the branch thermal stability security region satisfying the system branch transmission power constraints; is the whole SSR, which shows the intersection of , , , and ; is the state variable vector; is the power injection vector; is the AC power flow equation in power systems; is the set of system buses; is the set of system generators; and are the lower and upper voltage limits of the bus, respectively; and are the lower and upper limits of active output of the generator, respectively; and are the lower and upper limits of reactive output of the generator, respectively; and and are the forward and reverse transmission power limits of the branch connected buses and , respectively.
In the case of approximating local balance of reactive power, only the SSR under active power injection needs to be considered.
(2) |
where ; is the active power injection at each bus; is the reactive power injection at each bus and denotes a constant vector; is the active power output of the generator; and is the load value of the bus.
The SSR in the active power injection space is studied as the set of operating points at which various security constraints are satisfied by the system unit combinations and load demands under a specific reactive power configuration.
As proven in [
In large-scale power systems, SSRB is high-dimensional, non-convex, and non-linear,
As illustrated in

Fig. 1 Framework of high-dimensional SSRB approximation.
The whole work consists of two parts. ① Boundary sample set generation: the aim of this part is to search for a large number of boundary sample sets that are distributed near the SSRB, thereby providing a suitable dataset for the training and testing of the subsequent classification model. To address the issue of unbalanced samples and uncontrolled distances of samples from the boundary in traditional sampling methods, firstly, the generator output and load values are sampled using the improved Latin hypercube sampling (LHS) algorithm to form the initial sample set. Then, the boundary sample search is performed, and a resampling mechanism that considers sampling gaps is proposed to ultimately form the boundary sample set. ② High-dimensional SSRB approximation: this part is based on the generated boundary sample set, training a classification model, and then converting the model into the form of boundary hyperplanes. Given the non-linear separability of the boundary sample set in high-dimensional SSR, firstly, the original power injection space is converted into the feature space by the feature non-linear conversion using DNNs. The samples in the feature space are approximately linearly separable. Secondly, in the feature space, the security region boundary is linearly fitted based on the proposed IGR-WODT. Then, the SSRB is obtained by the trained IGR-WODT and conservatively translated to ensure the security of the boundary. Finally, the fitted SSRB is evaluated and verified on a large number of randomly generated operating points.
In the initial sample set generation process, the generator output is first sampled based on the LHS algorithm. Specifically, we set an nodal power system containing load nodes and generators. The lower and upper limits of the generator output are shown in (3).
(3) |
We first define the sampling number of initial sample set as , and then, we sample the generator output data based on the LHS, as denoted in (4).
(4) |
where is the generator output in the sample.
(5) |
In this paper, samples located within the security region are defined as secure samples and labeled as 0. Conversely, samples outside the security region are defined as insecure samples and labeled as 1. As the proposed sampling method in this paper requires searching for boundary samples based on a significant number of secure samples, it is essential to increase the proportion of secure samples among samples. To this end, this part proposes a method for sampling load values that considers the constraints of the generator output. Specifically, for the sample, the total generator output is , and the upper and lower limits of the generator output at the slacking bus are set as and , respectively. The total sampling value of load nodes is set to be , where the is the load value of load bus in the sample. To meet the output constraints of the generator at the slacking bus without considering the network loss, it is necessary for to satisfy (6).
(6) |
The constraints in (6) enclose a convex hyper-polyhedron in the -dimensional space. To achieve uniform sampling of load values inside the hyper-polyhedron as above, the problem is modeled as sampling the -dimensional vectors in the hyper-polyhedron, as shown in (7).
(7) |
where is the hyperplane normal vector; represent the variables to be solved; and is a constant vector.
Step 1: construct the linear objective function as:
(8) |
The optimal solution , which is the initial value of , is obtained using the linear programming method. is a vertex of the hyper-polyhedron represented by (7), as shown in
Step 2: solve the center of the hyper-polyhedron with . For this purpose, we construct a non-linear optimization problem.
(9) |
where is the row vector of matrix A; and is the feature of b.
Step 3: as shown in
(10) |
where represents the intersection point of the line and the hyperplanes represented by . In fact, among those intersection points, only two are on the convex polytope, denoted as and .
(11) |
where represents the projection of the line direction vector onto ; and and represent that the values of correspond to the indices in where the features are less than and greater than 0, respectively. The positive or negative value of indicates whether the intersection points are located in the direction of or with respect to , respectively, as shown in
(12) |
Step 4: make uniform sampling among the points and , as shown in
(13) |

Fig. 2 Sampling process of initial samples.
Following Steps 1-4, the load values in samples are obtained. The secure samples within samples are selected and formed into the initial sample set, , where , and is the generator output vector of the sample, and is the load value vector of the sample.
As shown in
(14) |

Fig. 3 Boundary sample search and resampling mechanism.
where denotes the two-norm of vectors; and is the boundary distance threshold, which represents the maximum Euclidean distance of boundary sample pairs. This threshold can be set according to engineering requirements, as shown in
Algorithm 1 : boundary sample search |
---|
Input: , ξ, and initial step size 1: Sample power growth direction vectors based on LHS. Each direction vector is normalized to form the direction vector matrix 2: for in do 3: 4: while in the security region do 5: 6: end while 7: , 8: while do 9: 10: 11: if is not in the security region then 12: , 13: else 14: , 15: end if 16: end while 17: Store (, 0), (, 1) in B 18: end for Output: B |
As mentioned in Section II, SSR is uniquely determined, connected, and internally void-free under a given network topology and system component parameters. Therefore, when starting from a secure sample and slowly increasing the injected power in a quasi-steady state, the SSRB must be encountered. Thus, the convergence of the
After the boundary sample set is formed, certain security region boundaries may still have large sampling gaps, which leads to the situation that subsequent models inaccurately estimate the boundaries of these regions [
As shown in
Due to the high-dimensionality, non-convexity, and non-linearity of the SSRB, the boundary dataset generated above is non-linearly separable. This poses a challenge to accurately approximate the boundary using linear classification methods such as DTs. Therefore, this paper proposes a method to convert the boundary samples from the original active power injection space to a 3-dimensional feature space. This is achieved by designing a non-linear feature converter, which renders the sample set linearly separable and simultaneously facilitates visualization. Ultimately, the security region boundary is linearly fitted in the feature space, and the boundary conservative translation is performed to enhance its security. This translation is crucial in achieving a more accurate and reliable SSRB.
DNNs possess an exceptional ability for non-linear approximation. Through multi-layer non-linear transformations, DNNs progressively model the original data and map it onto a space that is nearly linearly separable. In this subsection, we convert the original power injection space into a 3-dimensional feature space using different types of DNNs. Depending on the dataset, diverse DNNs can be chosen to achieve optimal results.
In this subsection, we introduce two types of feature non-linear converters. ① Type 1: a converter based on fully connected neural networks (FCNs). FCN is a simple type of neural network that can serve as a feature non-linear converter when dealing with low-dimensional features and a small number of samples. ② Type 2: a converter based on 1-dimensional CNN (1D-CNN). CNN incorporates convolutional processing of features into DNNs and is more suitable for processing higher dimensional data. For the datasets with higher feature dimensions and larger volumes of data, more complex neural networks such as transformers [
As an illustration, we will use the type 2 converter to demonstrate the conversion of the power injection space into the feature space . The feature non-linear converter based on 1D-CNN is depicted in

Fig. 4 Feature non-linear converter based on 1D-CNN.
As mentioned above, DNNs are used as a feature non-linear converter to enhance the linear separability of the dataset. In this part, the boundary linear approximation using IGR-WODT in the feature space is performed.
IGR-WODT is improved by WODT [
(15) |
where is the sample of ; is the sigmoid function; is the vector of model parameters (including a bias), which needs to be continuously updated during the model training process. When , the sample is classified as belonging to the left subset; when , the sample is classified as belonging to the right subset.
The weights of each sample set are defined as:
(16) |
Therefore, IGR-WODT defines 2 datasets associated with sample weights as:
(17) |
IGR-WODT calculates the empirical entropy of the dataset , and the conditional empirical entropy of the dataset under the division of .
(18) |
where is the number of samples in the dataset , which is equal to ; and is the number of samples in the dataset with the category :
(19) |
(20) |
where is the indicator function; represents the sum of corresponding to the samples of category k in set ; and represents the sum of corresponding to the samples of category k in set .
As shown in (18) and (19), , , , and are all calculated based on the probabilities of the sample belonging to the left or right child node.
IGR-WODT defines the information gain rate as:
(21) |
To avoid overfitting of the model under each node, the regularization term is introduced and the final model objective function expressed as:
(22) |
where is the coefficient of the regularization term .
To optimize the objective function , IGR-WODT employs the improved Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [
Then, we extract the decision parameters, i.e., , from IGR-WODT to construct the SSRB hyperplane in the feature space. Algorithm SA2 in Supplementary Material A provides a detailed depiction. The essence of the Algorithm SA2 lies in identifying all decision paths within IGR-WODT that lead from the root node to the leaf node labeled as 0. These paths are then transformed into a combination of hyperplanes by converting of all the nodes along these paths.
This section validates the effectiveness and generality of the proposed high-dimensional SSRB approximation model on the WECC 3-machine 9-bus system and IEEE 118-bus system using PYPOWER [
As shown in

Fig. 5 WECC 3-machine 9-bus system.
To demonstrate the non-convexity and non-linearity of SSRB in power systems intuitively, we first study the SSRB in the power injection space , i.e., we set as constant values.
The boundary samples are searched according to the method proposed in Section III and shown in

Fig. 6 2-dimensional SSRB samples on WECC 3-machine 9-bus system. (a) Upper boundary samples of node voltages. (b) Lower boundary samples of node voltages. (c) Upper boundary samples of generator outputs. (d) Lower boundary samples of generator outputs. (e) Boundary samples of branch thermal security. (f) Whole SSR.
We further analyze the impact of different setting values of upper and lower voltage limits on the characteristics of the SSRB. When the voltage range is set between 0.98 p.u. and 1.02 p.u., the 2-dimensional SSRB samples is presented, as shown in

Fig. 7 2-dimensional SSRB samples when voltage range is set between 0.98 p.u. and 1.02 p.u..
Compared with
Therefore, even if the boundary is approximated by segmented linear approximation, the error can still be significant, especially in high-dimensional spaces. To address this issue, the original power injection space needs to be converted into a new feature space using the feature non-linear converter, and then the linear approximation of the boundary can be performed.
For the WECC 3-machine 9-bus system, SSRB under active power injection is 5-dimensional. The power injection space is .
We first generate the initial samples using the proposed sampling method. To validate the efficiency of the proposed sampling method, we compare it with the LHS, MCS, importance sampling (IS), and Markov chain Monte Carlo sampling (MCMCS). The proportion of secure samples in the initially generated samples is used as the benchmark for the evaluation. The experiments are performed with five different random seeds, and the results are presented in
Method | Number of secure samples | Proportion of secure samples (%) |
---|---|---|
Proposed | ||
LHS | ||
MCS | ||
IS | ||
MCMCS |
Since the dimension of the SSRB under active power injection space for the WECC 3-machine 9-bus system is 5, an FCN with three hidden layers (64, 16, 3 neurons) and one output layer is chosen as the feature non-linear converter. The Adam optimization algorithm is selected for training. The boundary sample pairs are split into training and validation sets according to 8:2.
After 1000 rounds of training, the optimal model is selected, and feature conversion is performed.

Fig. 8 SSRB and sample distribution in 3-dimensional feature space.
Therefore, the training and validation sets are converted into the feature space by the trained FCN model respectively, and the boundaries are fitted using IGR-WODT. The maximum depth of IGR-WODT is set between 1 and 4, and the regularization coefficient is set to be .
To test the validity of the boundary in a more general way, 20000 samples (not boundary samples) are generated as the test set based on the proposed sampling method. The accuracy , precision , and recall rate are also introduced to evaluate the model performance, which are calculated as (23).
(23) |
where is the number of samples labeled as secure and judged as secure; is the number of samples labeled as insecure and judged as insecure; is the number of samples labeled as secure and judged as insecure; is the number of samples labeled as insecure and judged as secure; is the proportion of correctly classified samples in the dataset and reflects the overall performance of the proposed sampling method; is the proportion of samples that are truly secure among those classified as secure by the model, reflecting the security of the fitted boundary; and is the proportion of secure samples that are correctly classified in the dataset, reflecting the conservativeness of the boundary to some extent. In this paper, it is crucial to minimize the number of insecure samples classified as secure samples, i.e., minimize , thus a conservative translation of the fitted boundary is required.
Model | Depth | Training set | Validation set | Test set | ||||||
---|---|---|---|---|---|---|---|---|---|---|
UDT | 1 | 0.9907 | 0.9876 | 0.9938 | 0.9881 | 0.9843 | 0.9921 | 0.9957 | 0.9967 | 0.9983 |
2 | 0.9924 | 0.9869 | 0.9982 | 0.9898 | 0.9830 | 0.9967 | 0.9959 | 0.9966 | 0.9987 | |
3 | 0.9943 | 0.9933 | 0.9953 | 0.9929 | 0.9923 | 0.9936 | 0.9959 | 0.9981 | 0.9972 | |
4 | 0.9946 | 0.9947 | 0.9945 | 0.9922 | 0.9929 | 0.9914 | 0.9956 | 0.9986 | 0.9963 | |
UDT without FNC | 1 | 0.5637 | 0.5340 | 1.0000 | 0.5648 | 0.5347 | 0.9998 | 0.8717 | 0.8719 | 0.9998 |
2 | 0.6049 | 0.5586 | 1.0000 | 0.6058 | 0.5592 | 1.0000 | 0.8715 | 0.8719 | 0.9994 | |
3 | 0.6412 | 0.5822 | 1.0000 | 0.6438 | 0.5840 | 1.0000 | 0.8715 | 0.8718 | 0.9991 | |
4 | 0.6412 | 0.5822 | 1.0000 | 0.6438 | 0.5840 | 1.0000 | 0.8712 | 0.8718 | 0.9991 | |
IGR-WODT | 1 | 0.9944 | 0.9936 | 0.9953 | 0.9929 | 0.9927 | 0.9932 | 0.9957 | 0.9985 | 0.9965 |
2 | 0.9944 | 0.9937 | 0.9951 | 0.9931 | 0.9927 | 0.9934 | 0.9957 | 0.9985 | 0.9965 | |
3 | 0.9944 | 0.9924 | 0.9964 | 0.9923 | 0.9938 | 0.9938 | 0.9957 | 0.9985 | 0.9965 | |
4 | 0.9946 | 0.9943 | 0.9949 | 0.9926 | 0.9923 | 0.9929 | 0.9957 | 0.9985 | 0.9965 | |
IGR-WODT without FNC | 1 | 0.5637 | 0.5340 | 1.0000 | 0.5648 | 0.5347 | 1.0000 | 0.8717 | 0.8719 | 0.9998 |
2 | 0.6049 | 0.5586 | 1.0000 | 0.6058 | 0.5592 | 1.0000 | 0.8715 | 0.8719 | 0.9994 | |
3 | 0.6411 | 0.5821 | 1.0000 | 0.6438 | 0.5840 | 1.0000 | 0.8713 | 0.8718 | 0.9992 | |
4 | 0.6638 | 0.5980 | 1.0000 | 0.6642 | 0.5983 | 1.0000 | 0.8713 | 0.8718 | 0.9992 | |
Translated boundary | 0.9563 | 0.9998 | 0.9129 | 0.9559 | 0.9993 | 0.9124 | 0.9865 | 0.9999 | 0.9846 |
As shown in
IGR-WODT achieves high accuracy across all indicators at the depth of 1, with an accuracy exceeding 99.29% in the training, validation, and test sets. Therefore, IGR-WODT with the depth of 1 is selected as the boundary approximation model, and its security region boundary under the feature space is shown in (24), which is constructed using Algorithm SAII of Supplementary Material A.
(24) |
To enhance the ability of the model to distinguish insecure samples more accurately, the boundary is conservatively translated. This means that the insecured samples, currently designated as security by the existing boundary, are reclassified as outside the security region.
Specifically, the distance between insecure samples within the current security region of the training set and the current SSRB is calculated. There are 117 insecure samples within this region, and their distances from the current boundary are presented, as shown in

Fig. 9 Distance distribution of insecure samples within security region from boundary for WESS 3-machine 9-bus system.
As shown in Fig.
(25) |
The performance on datasets under the new boundary is shown in
To further evaluate the computational efficiency of the proposed sampling method, we compare it with the pointwise method by calculating the time required for each method to classify the state of 20000 samples in the test set. The results shown in
Method | Time (s) |
---|---|
Proposed sampling method | 0.20 |
Pointwise method | 524.77 |
In order to further verify the effectiveness and generality of the proposed sampling method in the high-dimensional SSRB approximation of large-scale power systems, the proposed sampling method is verified on the IEEE 118-bus system. The IEEE 118-bus system is configured with 54 generators and 99 load nodes, with bus 69 set as the slacking bus, whose active power injection space is 152-dimensional.
Similar to the experiments on the WECC 3-machine 9-bus system, the proposed sampling method is compared with the LHS, MCS, IS, and MCMCS methods. The experiments are performed with five different random seeds. The results are investigated, as shown in
Method | Number of secure samples | Proportion of secure samples (%) |
---|---|---|
Proposed sampling method | ||
LHS | ||
MCS | ||
IS | ||
MCMCS |
After completing the search for boundary samples, 87268 pairs of boundary sample pairs are obtained. A 1D-CNN based feature non-linear converter is designed for the IEEE 118-bus system, containing one layer of 1D-CNN and three layers of FCN with the number of neurons being 128, 64, and 3, respectively. The aforementioned boundary sample pairs are divided into training and validation sets according to 8:2. Additionally, 100000 samples, which are not boundary samples, are generated as the test set.
After 1000 rounds of training, we select the optimal model as the feature non-linear converter. The performance of the SSRB fitted using IGR-WODT in the feature space is presented in
Model | Depth | Training set | Validation set | Test set | ||||||
---|---|---|---|---|---|---|---|---|---|---|
UDT | 1 | 0.9673 | 0.9705 | 0.9639 | 0.9429 | 0.9438 | 0.9418 | 0.9483 | 0.9770 | 0.9634 |
2 | 0.9747 | 0.9776 | 0.9615 | 0.9469 | 0.9504 | 0.9323 | 0.9483 | 0.9770 | 0.9634 | |
3 | 0.9791 | 0.9766 | 0.9819 | 0.9497 | 0.9492 | 0.9503 | 0.9481 | 0.9759 | 0.9643 | |
4 | 0.9803 | 0.9717 | 0.9894 | 0.9502 | 0.9426 | 0.9589 | 0.9503 | 0.9702 | 0.9729 | |
IGR-WODT | 1 | 0.9807 | 0.9800 | 0.9815 | 0.9511 | 0.9491 | 0.9533 | 0.9490 | 0.9731 | 0.9683 |
2 | 0.9800 | 0.9755 | 0.9847 | 0.9512 | 0.945 | 0.9581 | 0.9495 | 0.9724 | 0.9696 | |
3 | 0.9805 | 0.9820 | 0.9792 | 0.9507 | 0.9511 | 0.9503 | 0.9485 | 0.9740 | 0.9667 | |
4 | 0.9807 | 0.9803 | 0.9812 | 0.9504 | 0.9507 | 0.9500 | 0.9483 | 0.9742 | 0.9664 | |
Translated boundary | 0.9385 | 0.9991 | 0.8771 | 0.9205 | 0.9846 | 0.8543 | 0.9223 | 0.9882 | 0.9219 |
As shown in Tables
Given the comparable performance, the model with lower complexity is chosen as the boundary approximation model, which is the IGR-WODT model with the depth of 1. Its boundary expression is shown in (26).
(26) |
Similar to Case 1, a conservative translation of the aforementioned boundary is performed. One thousand three hundred and ninety nine insecure samples are located within the current security region. The distances between these samples and the current SSRB are distributed. As shown in
(27) |

Fig. 10 Distance distribution of insecure samples within security region from boundary for IEEE 118-bus system.
We conduct a further comparison of the computational efficiency between the proposed sampling method and the pointwise method in Case 2. Specifically, we analyze the time required to assess the states of 100000 operating points for both methods. As illustrated in
Method | Time (s) |
---|---|
Proposed sampling method | 10.32 |
Pointwise method | 2677.71 |
Further, considering the noise interference and data missing during data measurement and transmission process in real power systems, we conduct experiments to test the proposed sampling method. Specifically, we add Gaussian random noise with a range from -1 to 1 to a randomly selected 10% of the dataset and simultaneously nullified 0.1% of the data to simulate the aforementioned issues. The performance of the proposed sampling method on the dataset is subjected to noise interference and data missing, as shown in

Fig. 11 Model performance on dataset subjected to noise interference and data missing.
As shown in
This paper proposes a novel method to approximate the high-dimensional SSRB via the feature non-linear converter and improved oblique DT. The proposed sampling method is evaluated on the WECC 3-machine 9-bus system and IEEE 118-bus system. The experimental results demonstrate that it outperforms previous methods by effectively approximating the high-dimensional boundary and reducing the error between the approximated and real boundaries. As shown in Tabels
Future research work will mainly focus on: ① integrating renewable energy and load forecasting tools to assess the system state in real-time within time steps based on the proposed sampling method, and guiding the formulation of real-time dispatch and control strategies for power systems; and ② data-driven methods for predicting the steady-state security margin.
References
A. Moreira, D. Pozo, A. Street et al., “Reliable renewable generation and transmission expansion planning: co-optimizing system’s resources for meeting renewable targets,” IEEE Transactions on Power Systems, vol. 32, no. 4, pp. 3246-3257, Jul. 2017. [Baidu Scholar]
J. Xu, H. Gao, R. Wang et al., “Real-time operation optimization in active distribution networks based on multi-agent deep reinforcement learning,” Journal of Modern Power Systems and Clean Energy, vol. 12, no. 3, pp. 886-899, May 2020. [Baidu Scholar]
S. Li, D. Cao, W. Hu et al., “Multi-energy management of interconnected multi-microgrid system using multi-agent deep reinforcement learning,” Journal of Modern Power Systems and Clean Energy, vol. 11, no. 5, pp. 1606-1617, Sept. 2023. [Baidu Scholar]
Y. Yu, Y. Liu, C. Qin et al., “Theory and method of power system integrated security region irrelevant to operation states: an introduction,” Engineering, vol. 6, no. 7, pp. 754-777, Jul. 2020. [Baidu Scholar]
H.-D. Chiang and C.-Y. Jiang, “Feasible region of optimal power flow: characterization and applications,” IEEE Transactions on Power Systems, vol. 33, no. 1, pp. 236-244, Jan. 2018. [Baidu Scholar]
J. Xiao, G. Zu, X. Gong et al., “Observation of security region boundary for smart distribution grid,” IEEE Transactions on Smart Grid, vol. 8, no. 4, pp. 1731-1738, Jul. 2017. [Baidu Scholar]
S. Chen, Z. Wei, G. Sun et al., “Convex hull based robust security region for electricity-gas integrated energy systems,” IEEE Transactions on Power Systems, vol. 34, no. 3, pp. 1740-1748, May 2019. [Baidu Scholar]
T. Ding, R. Bo, H. Sun et al., “A robust two-level coordinated static voltage security region for centrally integrated wind farms,” IEEE Transactions on Smart Grid, vol. 7, no. 1, pp. 460-470, Jan. 2016. [Baidu Scholar]
S. Chen, Q. Chen, Q. Xia et al., “Steady-state security assessment method based on distance to security region boundaries,” IET Generation, Transmission & Distribution, vol. 7, no. 3, pp. 288-297, Mar. 2013. [Baidu Scholar]
F. Wu and S. Kumagai, “Steady-state security regions of power systems,” IEEE Transactions on Circuits and Systems, vol. 29, no. 11, pp. 703-711, Nov. 1982. [Baidu Scholar]
H. D. Nguyen, K. Dvijotham, and K. Turitsyn, “Constructing convex inner approximations of steady-state security regions,” IEEE Transactions on Power Systems, vol. 34, no. 1, pp. 257-267, Jan. 2019. [Baidu Scholar]
D. Lee, H. D. Nguyen, K. Dvijotham et al., “Convex restriction of power flow feasibility sets,” IEEE Transactions on Control of Network Systems, vol. 6, no. 3, pp. 1235-1245, Sept. 2019. [Baidu Scholar]
X. Li, T. Jiang, L. Bai et al., “Orbiting optimization model for tracking voltage security region boundary in bulk power grids,” CSEE Journal of Power and Energy Systems, vol. 8, no. 2, pp. 476-487, Mar. 2022. [Baidu Scholar]
T. Jiang, R. Zhang, X. Li et al., “Integrated energy system security region: concepts, methods, and implementations,” Applied Energy, vol. 283, p. 116124, Feb. 2021. [Baidu Scholar]
X. Li, G. Tian, Q. Shi et al., “Security region of natural gas network in electricity-gas integrated energy system,” International Journal of Electrical Power & Energy Systems, vol. 117, p. 105601, May 2020. [Baidu Scholar]
X. Li, T. Jiang, G. Liu et al., “Bootstrap-based confidence interval estimation for thermal security region of bulk power grid,” International Journal of Electrical Power & Energy Systems, vol. 115, p. 105498, Feb. 2020. [Baidu Scholar]
T. Yang and Y. Yu, “Static voltage security region-based coordinated voltage control in smart distribution grids,” IEEE Transactions on Smart Grid, vol. 9, no. 6, pp. 5494-5502, Nov. 2018. [Baidu Scholar]
R. Yan, G. Geng, and Q. Jiang, “Data-driven transient stability boundary generation for online security monitoring,” IEEE Transactions on Power Systems, vol. 36, no. 4, pp. 3042-3052, Jul. 2020. [Baidu Scholar]
J. An, J. Yu, Z. Li et al., “A data-driven method for transient stability margin prediction based on security region,” Journal of Modern Power Systems and Clean Energy, vol. 8, no. 6, pp. 1060-1069, Nov. 2020. [Baidu Scholar]
P. Yong, Y. Wang, T. Capuder et al., “Steady-state security region of energy hub: modeling, calculation, and applications,” International Journal of Electrical Power & Energy Systems, vol. 125, p. 106551, Feb. 2021. [Baidu Scholar]
O. F. Avila, J. A. P. Filho, and W. Peres, “Steady-state security assessment in distribution systems with high penetration of distributed energy resources,” Electric Power Systems Research, vol. 201, p. 107500, Dec. 2021. [Baidu Scholar]
Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature, vol. 521, no. 7553, pp. 436-444, May 2015. [Baidu Scholar]
S. Wu, L. Zheng, W. Hu et al., “Improved deep belief network and model interpretation method for power system transient stability assessment,” Journal of Modern Power Systems and Clean Energy, vol. 8, no. 1, pp. 27-37, Jan. 2020. [Baidu Scholar]
Z. Shi, W. Yao, L. Zeng et al., “Convolutional neural network-based power system transient stability assessment and instability mode prediction,” Applied Energy, vol. 263, p. 114586, Apr. 2020. [Baidu Scholar]
X. Liu, X. Miao, H. Jiang et al., “Box-point detector: a diagnosis method for insulator faults in power lines using aerial images and convolutional neural networks,” IEEE Transactions on Power Delivery, vol. 36, no. 6, pp. 3765-3773, Dec. 2021. [Baidu Scholar]
D. Yang, Y. Pang, B. Zhou et al., “Fault diagnosis for energy internet using correlation processing-based convolutional neural networks,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 49, no. 8, pp. 1739-1748, Aug. 2019. [Baidu Scholar]
J. van Gompel, D. Spina, and C. Develder, “Satellite based fault diagnosis of photovoltaic systems using recurrent neural networks,” Applied Energy, vol. 305, p. 117874, Jan. 2022. [Baidu Scholar]
Y. Zhao, P. Liu, Z. Wang et al., “Fault and defect diagnosis of battery for electric vehicles based on big data analysis methods,” Applied Energy, vol. 207, pp. 354-362, Dec. 2017. [Baidu Scholar]
H. Shi, M. Xu, and R. Li, “Deep learning for household load forecasting – a novel pooling deep RNN,” IEEE Transactions on Smart Grid, vol. 9, no. 5, pp. 5271-5280, Sept. 2018. [Baidu Scholar]
C. Hu, J. Zhang, H. Yuan et al., “Black swan event small-sample transfer learning (BEST-L) and its case study on electrical power prediction in covid-19,” Applied Energy, vol. 309, p. 118458, Mar. 2022. [Baidu Scholar]
Y. Jiang, T. Gao, Y. Dai et al., “Very short-term residential load forecasting based on deep-autoformer,” Applied Energy, vol. 328, p. 120120, Dec. 2022. [Baidu Scholar]
Q. Hou, N. Zhang, D. S. Kirschen et al., “Sparse oblique decision tree for power system security rules extraction and embedding,” IEEE Transactions on Power Systems, vol. 36, no. 2, pp. 1605-1615, Mar. 2021. [Baidu Scholar]
T. Behdadnia, Y. Yaslan, and I. Genc, “A new method of decision tree based transient stability assessment using hybrid simulation for real-time pmu measurements,” IET Generation, Transmission & Distribution, vol. 15, no. 4, pp. 678-693, Dec. 2020. [Baidu Scholar]
R. Yan, G. Geng, and Q. Jiang, “Data-driven transient stability boundary generation for online security monitoring,” IEEE Transactions on Power Systems, vol. 36, no. 4, pp. 3042-3052, Jul. 2021. [Baidu Scholar]
A. Vaswani, N. Shazeer, N. Parmar et al., “Attention is all you need,” Advances in Neural Information Processing Systems, Long Beach, USA, Jun. 2017, pp. 1-10. [Baidu Scholar]
B. Yang, S. Shen, and W. Gao, “Weighted oblique decision trees,” in Proceedings of the AAAI Conference on Artificial Intelligence, Hawaii, USA, Sept. 2019, pp. 5621-5627. [Baidu Scholar]
Y. Dai, Q. Chen, J. Zhang et al., “Enhanced oblique decision tree enabled policy extraction for deep reinforcement learning in power system emergency control,” Electric Power Systems Research, vol. 209, p. 107932, Aug. 2022. [Baidu Scholar]
R. D. Zimmerman, C. E. Murillo-Snchez, and R. J. Thomas, “Matpower: steady-state operations, planning, and analysis tools for power systems research and education,” IEEE Transactions on Power Systems, vol. 26, no. 1, pp. 12-19, Feb. 2011. [Baidu Scholar]