Abstract
Addressed to the cascading outages, it is computationally burdensome for the reliable calculation of active and reactive power flows. This paper builds a comprehensive framework with three algorithms, including the distribution factor (DF), the Newton-Raphson (NR), and the first iteration of NR algorithm (termed as 1J). Classifiers are designed to determine whether the NR algorithm should be employed for accuracy. Classifier features are extracted upon the analytical error of 1J. As reactive power is partially considered in the 1J but neglected in the DF algorithm, the deviation between the solutions is taken as one crucial feature. The support vector machine (SVM) is then utilized for classifier training. As the deep integration of the causal inference and the statistical paradigm, this framework calculates active and reactive power flows rapidly, reliably, and robustly. The effectiveness and robustness are fully validated in three typical IEEE systems.
AS one of the major causes of large-scale blackouts, the cascading outage has aroused great concerns [
Considering the power flow (PF) nonlinearity and a number of the cascading outages, it is computationally infeasible to seek precise PF solutions in each stage. The PF equation can be described as:
(1) |
where P and Q are the vectors of active and reactive injections, respectively; θ and V are the vectors of voltage angle and magnitude, respectively; and fP and fQ are the functions of active and reactive power injections, respectively [
Serving as the primary PF calculation tool, the model-driven method can be divided into three categories, including nonlinear iterative algorithms (NIAs) such as the Newton Raphson (NR) and the fast decoupled (FD) algorithms, nonlinear noniterative algorithms (NNAs) like the first iteration of the NR algorithm (termed as 1J) and the first iteration of the fast decoupled algorithm (termed as 1P1Q), and linear algorithms (LAs) like the direct current (DC) algorithm and the line outage distribution factor (DF) algorithm.
Though NIAs can offer precise PF solutions [
There is always one NNA that corresponds to an NIA. Algorithms like 1J and 1P1Q can be described in an incremental form as (2) [
(2) |
where and are the vectors of initial active and reactive injections, respectively.
Unlike the NR algorithm fully modeling the PF issue, the FD algorithm assumes to neglect the coupling between active and reactive power, which is termed as PQ coupling in this paper. Accordingly, the system is conceptually divided into the P and Q subsystems. In each subsystem, and in (2) are approximated to two zero matrices. By repetitive iterations between active and reactive PF functions, errors of angles and magnitudes originating from neglecting the PQ coupling can be eliminated [
The DC algorithm is formulated in an incremental form as follows [
(3) |
where B is the constant admittance matrix.
As the voltage is simplified as 1.0 p.u. and the reactive power is ignored, a tiny amount of computation is required here. With the application of matrix inverse lemma, the DF and the generalized DF algorithms are proposed to model the single and multiple line outage scenarios, respectively [
Apart from the model-driven methods, the data-driven methods, such as back-propagation neural networks (BPNNs), radial basis function neural networks (RBFNNs), and convolutional neural networks (CNNs), are also applied to rapid PF estimation. Features are usually directly extracted from raw data like topologies and operation conditions (OCs) [
From above, the physical mechanism is retained in the model-driven method for robustness enhancement, which is termed as causal inference. In contrast, statistical regularities are intelligently learned in the data-driven method for efficiency promotion, termed as statistic paradigm [
Addressed to the rapid, reliable, and robust PF calculation of the cascading outages, the DF, 1J, and NR algorithms are involved in a comprehensive framework, where two newly designed classifiers determine the employment of NR algorithm. The classifier features are extracted upon the analytical errors of 1J, and support vector machine (SVM) is employed for classifier training. The contributions are twofold.
Firstly, a comprehensive framework is constructed involving DF, 1J, and NR algorithms, where two binary classifiers intelligently determine the switch for the NR algorithm. Unlike conventional model-driven and data-driven methods, causal inference and statistical paradigms are deeply integrated for robustness and efficiency.
Secondly, both active and reactive PF errors of 1J are deduced in analytical forms, and classifier features are extracted accordingly, ensuring high generalization capability. Specifically, the influence of PQ coupling on the error of 1J is quantitatively evaluated and characterized by the deviation between the 1J and DF solutions, which is proved to be indispensable and practical.
The remainder of this paper is organized as follows. Section II builds a comprehensive framework. In Section III, the error of 1J is analyzed in analytical forms, and classifier features are extracted accordingly. Section IV employs SVM for classifier training. Numerical simulations are offered in Section V, and the conclusion is given in Section VI.
As to the PF calculations in the cascading outages, the rapidity and accuracy requirements can barely be met simultaneously by only one algorithm. NIAs can acquire precise active and reactive PF solutions, yet with substantial computational costs. By contrast, little computation amount is involved in LAs and NNAs in the sacrifice of accuracy. Besides causal algorithms supported by the physical mechanism, machine learning (ML) is applied to accelerate the computation further, but with limited generalization capability. This paper proposes a comprehensive framework covering the model analysis, the statistical analysis, and the theoretical analysis. Both robustness and efficiency are ensured in calculating the PF of the cascading outages.

Fig. 1 Comprehensive framework integrating statistical analysis, model analysis, and theoretical analysis.
The cascading outages involve massive scenarios to be analyzed within a short time. Requirements on rapidity and accuracy are contradictory. Since the extremely accurate PF solution is not always required, approximate algorithms like LAs and NNAs can be used to speed up the calculation. In fact, the sum of computations of an LA and an NNA is far smaller than that of an NIA alone. Compared with all samples calculated by an NIA, the computation amount can be significantly reduced if a particular portion is analyzed by an LA and an NNA instead. As to the approximate solutions acquired by the LA and the NNA, cases with small potential errors may directly deliver the approximate PF solutions for rapidity, while cases with large potential errors should switch to an NIA for accuracy. Therefore, it is necessary to pre-estimate the potential errors of approximate solutions before an NIA is employed, and case classifiers should be designed to classify cases upon the potential errors.
Correspondingly, a comprehensive framework is built to tackle the problem mentioned above, and three involved analyses are listed as follows.
1) The model analysis is supported by three mathematical algorithms, including an LA with the least computation amount, an NIA with accurate solutions in all cases, and an NNA with accuracy and computation amount between the former two. Though they perform distinctly in speed and precision, physical mechanisms are always retained under individual assumptions, ensuring robustness.
2) The theoretical analysis involves causalities in extracted features so that the generalization capability of the classifier can be ensured.
3) The statistical analysis employs a supervised learning technique for classifier training. For cases with large potential errors, an NIA should be applied to enhance accuracy. For the rest, NNA solutions can be delivered as the final results to save computation time.
The model analysis and the statistical analysis ensure robustness and efficiency, while the theoretical analysis supports the coordination among different algorithms.
According to
Afterward, the framework can be applied online, and the corresponding algorithm is designed as
This subsection designates the DF, 1J, and NR algorithms as the LA, NIA, and NNA, respectively, for the framework for the following reasons.
Like DC algorithm, DF algorithm neglects voltage and reactive power, simplifies as , and avoids iterations. Besides, the DF calculates faster than the DC by one order of magnitude in outages [
Therefore, the DF algorithm is designated as the LA to rapidly offer the essential information for the feature extraction in Section III.
Given the convergence rate, we prefer the NR to the FD algorithms as the NIA. With the PQ coupling fully considered, the NR iteration is quadratically convergent while acquiring precise PF solutions, yet with a large computation amount.
The NR algorithm is thereby selected to offer precise PF solutions for unreliable cases to ensure framework accuracy. As PF is scarcely divergent, the solution error can be ignored here.
As two classical NNAs, the 1J provides the first iteration of NR algorithm, and 1P1Q offers the first iteration of the FD algorithm. Comparing with 1P1Q, we prefer the 1J for two reasons. One is that the features of truncation errors in different classifiers can be unified by the 1J, which saves the feature computation. More details will be given in Section III-B. The other one is that the NR algorithm has already been designated as the NIA. Subsequent iterations can be directly loaded if the NR solutions are needed.
The 1J remarkably saves the computation time by delivering the first iteration. As PQ coupling is partially considered, 1J solutions include active and reactive PFs, but they are unreliable in strong PQ coupling cases.
Hence, the 1J is selected to facilitate feature calculation and fast PF estimation for the framework.
It is computationally infeasible for classifiers to deal with all raw data in large-scale power systems. Conventional methods select key features among raw data in statistical manners, which inevitably misses information and limits the generalization capability.
To enhance the robustness, we theoretically analyze the 1J errors and extract classifier features. Instead of statistical selection from raw data, features are extracted upon causalities to enhance the generalization capability.
Taking NR solutions as references, we analyze the 1J errors in analytical forms. It is discovered that the PF nonlinearity determines the error, and the PQ coupling is proved to be a significant indicator. As is partially considered in the 1J but neglected in the DF algorithm, the PQ coupling can be indicated by the deviation between the two solutions, which will be thoroughly discussed in Section III.
Classifiers CP and CQ are designed to categorize cases upon the potential active and reactive PF errors of 1J. As to the classifier training, ML is employed to learn the mapping relationship between the extracted features and the labels of the training samples. Statistical analysis is deeply integrated into the causal algorithms for efficiency enhancement.
Class labels are designed as
Features are delivered as the classifier inputs, while the reliable and unreliable labels are tagged as outputs, determining whether to switch to NR algorithm. If both and label the reliable tags on a case, the 1J solution will be directly delivered for rapidity. Otherwise, the framework will switch to the NR algorithm for accurate PF solutions.
According to the PQ coupling involvement, Section III-A analytically divides the 1J error into the truncation and decoupling errors. In each subsystem, a feature of truncation error and a feature of decoupling error are extracted accordingly. However, the total error is not merely their sum. Section III-D offers an auxiliary feature to determine the dominant error so that the feature of the dominant error can indicate the total error. The truncation error feature, the decoupling error feature, and the auxiliary feature are served as inputs for both CP and CQ.
The NR solutions are denoted as and and can be expanded into the Taylor series at the pre-fault equilibrium point . Notably, point corresponds to the pre-fault PFs of the last outage stage. For example, point of contingencies corresponds to the PFs of the initial state of power systems, while of an contingency corresponds to the PFs of the corresponding state.
The 1J solutions and are formulated in (2) and neglect the nonlinear items in (4). Taking and as references, we can quantify the active power error and the reactive power error of the 1J as (5).
(4) |
(5) |
Depending on the involvement of the PQ coupling, (5) can be divided into two parts. Error consists of the truncation error of the P subsystem and the PV decoupling error , while error comprises the truncation error of the Q subsystem and the Qθ decoupling error , given by follows.
(6) |
(7) |
In (6) and (7), only first- and second-order partial derivatives with respect to voltage are nonzero, as active power and reactive power are quadratic functions of voltage.
The feature of truncation error should reflect the PQ coupling. Here, we compare the branch active PF solution of the 1J () with that of the DF algorithm (), and utilize the 2-norm of their deviation as the feature of truncation error , as described in (8). The 2-norm is defined as the Euclidean distance of two points and equals the square root of a vector inner product with itself.
(8) |
Since the is designed to demonstrate and , their correlations are thoroughly discussed as follows. To better explain the correlation between and , we rewrite in (6) as (9). The derivation is listed in Appendix A.
(9) |
As is indicated in (9), two terms compose . The item is the deviation of active power function between (2) and (3). If the difference between and B is ignored, the active power injection deviation between the 1J and the DF algorithms can be utilized to estimate this item, coinciding with the definition of . The item describes the influence of voltage variation on active power when the angle keeps constant. As the deviation between the 1J and the DF algorithms accords with the item description, can also quantify this term. Considering the similar and the different [
Then, the feasibility of applying to represent is discussed. Similarly, error in (6) is rewritten as (10). The detailed deviation is given in Appendix B.
(10) |
Also, the error consists of two components. The latter item describes the influence of angle variation on reactive power and is neglected here, because the element of this item is always smaller than , where and are the real and imaginary parts of the branch admittance connecting nodes i and j, respectively; and N is the node number. The value limitation is the consequence of the trigonometric function . The term is thereby dominant in , and a feature is extracted accordingly. Except for the diagonal elements, other elements in are opposite to the corresponding elements in [
Notably, the identity between the features of and is by no means accidental. It is the consequence of selecting the 1J as the NNA in the framework. Unlike in the 1P1Q, the angle and the magnitude variations are simultaneously taken into account in the 1J. Since only the angle variation is considered in the DF algorithm, the PQ coupling degree can be quantified by the deviation between the 1J and the DF solutions.
Simulation results in Appendix C have supported the monotonic relation between and 1J errors.
Unlike the feature of truncation error, the features of decoupling error are different in and , which will be discussed separately. For sake of description, we assume a power system with N buses and L transmission lines, and an arbitrary line connecting nodes i and j.
As to in (7), given that the third- and higher-order terms are relatively small and ignorable, the second-order term of the Taylor series can be employed for estimation. When the perturbation method is applied to evaluate the Hessian matrix , the error can be estimated by:
(11) |
We denote the active PF variation of outage line as , a sparse vector where elements corresponding to the outage line numbers equal to their pre-fault active PF. In the DF algorithm, the relationship can be described as (12). As the DF angle is similar to the 1J angle, (12) can be roughly held in the 1J [
(12) |
where the node-branch incidence matrix is a sparse matrix of , and its and elements are 1 and -1, respectively; identity matrix is of ; is the power transfer distribution factor (PTDF) matrix of ; and the operator is the Hadamard product, where two matrices of the same dimensions are element-wisely multiplied. Given the linear correlation between and , we define the 2-norm of as the feature of PV decoupling error , as shown in (13).
(13) |
Since reactive power is the quadratic function of voltage, the term of in (7) is a constant matrix, which depends on topology only, leaving as the dominant part of . We designate the quadratic sum of all elements in as the feature of the Qθ decoupling error , which can also be described as the square of a 2-norm, as shown in (14).
(14) |
where the vector is the 1J voltage variation. In the vector , each element equals the reciprocal value of the corresponding reference voltage.
The sum of the features of truncation error and the decoupling errors is unequal to the actual error due to the information loss in the feature extraction. An auxiliary feature is thereby designed to determine the dominance between the two features, and is proved to be useful for the primary error identification.
As to , when is small, , and can be ignored, leaving as the dominant error. At this point, eP can be indicated by . When is large, , and a contrary conclusion can be drawn that becomes the leading feature.
When it comes to the analysis, two scenarios are discussed. As to a small value of , , and is too trivial to be considered. Under such circumstances, can be directly applied to demonstrate . When takes a large value, is jointly determined by and . Correspondingly, both and are the key features.
We take the 2-norm of as the auxiliary feature by:
(15) |
As is served to determine the leading feature between and , a breaking point can be learned offline. Considering that SVM is applied to learn the mapping relationship, the breaking point of is not particularly studied.
The features of include , , and , while the features of include , , and .
Mapped by SVM, the nonlinear relationships between features in Section III and the targeted tags can be visualized in feature spaces, which takes each feature as a dimension. Both CP and satisfy classification performances. For description purposes, we give an example on CP and trained under all and contingencies of the standard OC in the IEEE 30-bus system. As to SVM setting, we take k as 5 for the k-folder cross-validation and employ the library for support vector machine (LIBSVM) package [

Fig. 2 CP and CQ learned by SVM. (a) Hyperplanes of CP based on training set. (b) Hyperplanes of CQ based on training set. (c) Classification of testing set based on CP. (d) Classification of testing set based on CQ.
As both and involve three features, their corresponding feature spaces are three-dimensional, and one point in the feature space represents one contingency.
As to the classifier training, hyperplanes of and can be plotted in their individual feature spaces, once the relationships between features and tags are well established by SVM based on the training data. Cases locating inside the hyperplane will be predicted as reliable ones, whereas the rest will be predicted as unreliable ones.
As to the classifier testing,
The correct classification ensures the accuracy and rapidity of the PF calculation. Under the following two circumstances, the actual cases are just as predicted. One is red points locating outside the envelope (the NR algorithm is applied to the large error cases), the other one is blue points locating inside the envelope (the 1J is employed for the small error cases).
Contrary to the correct classification, misclassification leads to either low efficiency or large errors. If blue points locate outside the envelope, the small error cases are predicted to be unreliable, and the NR algorithm will be employed, resulting in unnecessary computation with little accuracy improvement. If red points are inside the envelope, the large error cases are predicted to be reliable, and their 1J solutions will be directly given as the final results. With the application of these unreliable results, tremendous risks may be incurred to static security assessment (SSA) and system adjustment decision (SAD), which is practically unacceptable. To improve accuracy, we may adjust or apply more advanced ML techniques, but with the sacrifice of calculating time.
In summary, the hyperplanes in
For the evaluation of effectiveness, relevant indices are given in Section V-A. Afterward, two particular scenarios of the cascading outages are given. Section V-B involves the contingency considering the multi-element failure in each outage stage. Section V-C designs the contingency simulation, where the time sequence is considered. All cases are studied in MATLAB on a PC with an Intel-Core I5-7500 CPU, and system data are provided in MATPOWER 7.0 [
Notably, for robustness validation, samples are significantly different in outage line numbers, outage stages, and OCs. Moreover, the total number of testing cases is much higher than that of training cases.
In order to demonstrate the advantages of causal features, CNN is utilized for comparison. Considering that CNN only acquires voltages and angles, we calculate corresponding active and reactive PF solutions based on the branch PF equation listed in Appendix D.
The confusion matrix is used to demonstrate classification accuracy. When a specific classifier and instances are given, four outcomes are listed in
Both TPR and FPR are irrelevant to the proportion of the actual cases, facilitating their applications even in the unbalanced dataset. Two relevant indicators are proposed.
Receiver operating characteristic (ROC) curve visualizes the diagnostic ability of a binary classifier [
Area under the ROC curve (AUC) is defined as the area of the ROC curve in the normalized unit [
The single case error e can be the error of branch PF listed in (16) or the error of node state variable given in (17), where the NR solutions , , , and are taken as references. The error of branch PF is relative, where , , and are the active PF solution, the reactive PF solution, and the apparent flow limit of branch l, respectively. Since relative errors of low-power branches may submerge that of other branches, we rule out the branches whose power is less than 10% of for better verification. We denote the sum of branches involved as . Different from the relative error of branch PF, the error of node state variable widely takes an absolute value.
(16) |
(17) |
All relevant indices are defined in
The simultaneous outage ( contingency) is a particular case of the cascading outage. With k2 taking 0, the simultaneous outage involves multiple tripping elements in the first outage stage and can be extended to an arbitrary stage.
To verify the effectiveness of the classifier and the framework, we evaluate the performance of the IEEE 30-bus system under the standard OC. All , , and contingencies are sampled. In the contingency set, 38 and 677 contingencies are sampled for training, while 7504 contingencies are for testing. Unbalanced datasets are also considered to verify the generalization capability of the classifier, which is detailly introduced in Appendix E.

Fig. 3 Comparison of ROC curves with varying thresholds. (a) ROC curves of CP. (b) ROC curves of CQ.
Admittedly, the AUC of the proposed classifiers is not as outstanding as other classifiers with advanced classification learning algorithms. We need to point out that the total number of testing cases is ten times that of training cases here, whereas the situation usually turns the opposite for the traditional classifiers. With the increasing proportion of training cases, the AUC of our classifiers will reach an excellent level. Rather than advancing the classification learning algorithm for a perfect AUC, we aim to obtain reliable active and reactive PF solutions, which can be achieved by adjusting and .

Fig. 4 Comparison of error distributions. (a) CNN. (b) Proposed framework.
In

Fig. 5 Comparison of calculation time T.
Identifications of large error cases are compared in
The values of and dramatically increase in the sequence of the proposed framework, 1J, and CNN. The comparison between 1J and the proposed framework is underlined since the CNN accuracy here can hardly meet the industrial demands. Comparatively, the proposed framework limits under 5.5% and declines by at least one order of magnitude. The value of of the proposed framework reduces from 1.67% to 0.10% in testing cases, while declines from 13.29% to 1.39%.
In addition, the accuracy of the proposed framework is verified under a varying number of outage lines. Though the error generally increases with the growing number of outage lines, the decrease in is more prominent, and the values of and are controlled under 0.1% and 1.4%, respectively. Besides, more accurate solutions can be acquired in the testing cases if we take larger values of and .
The successive outage like contingency is another particular case for the cascading outages. Different outage stages are involved in order to consider the time sequence. For simplification purposes, only three outage stages are taken, and we compare the performances of different methods in the IEEE 57-bus, IEEE 118-bus, and 2000-bus Texas systems.
In the IEEE 57-bus system, the proposed framework is evaluated in multiple outage stages under the standard OC. In the training set, there are 2000 contingencies randomly selected. Given that each outage stage in contingencies is learned in the training set, 76 , 2810 , and 10000 contingencies are included in the testing set for performance verifications.
Also, the accuracy of the proposed framework is compared at multiple outage stages, and the robustness of the proposed framework is validated by stable accuracy in different scenarios. For example, of 1J in the testing , , and contingencies are 0.68%, 0.97%, and 1.29%, respectively. For the proposed framework, the in , , and contingencies are 0.25%, 0.25%, and 0.27%, respectively. The EP variation of the proposed framework is much flatter than that of 1J, demonstrating a better generalization capability among different outage stages.
In the IEEE 118-bus system, we randomly sample 2000 contingencies under the standard OC as the training set and 10000 contingencies each under different OCs as the testing set. The maximum OC is defined as 200% of the generation and load under the standard OC. The minimum OC takes the percentage as 50%.
Like
Compared with the causal method, a higher accuracy is achieved in the proposed framework, where is decreased by at least one order of magnitude when the 1J solutions are taken as a reference. Besides, and can be controlled under 2.5% and 4%, respectively. Comparing with the NR algorithm, we increase the speed by nearly ten times. As to the statistical method, the and can be decreased by three orders of magnitude in the sacrifice of a little computation time. Both speed and accuracy are ensured in the proposed framework.
Besides, the performances of the proposed framework are verified under multiple OCs. Though 1J errors are apparently different between the standard OC and the minimum OC, of the framework can be intelligently adjusted. As the 1J solutions of most cases are reliable under the minimum OC, should adjust to a small value to avoid unnecessary NR computations and accelerate the PF calculation. The adaptive compromise between the speed and accuracy under multiple OCs further verifies the intelligence of the proposed framework. It should be noted that the decreases of and under the minimum OC are not as remarkable as those under the standard OC. This can be explained by the larger 1J errors indicated under the standard OC, leaving a larger proportion of cases under the minimum OC under the threshold. Robustness is thereby indicated by the similar results between the training and testing sets.
The MATPOWER offers the data of the 2000-bus system. In the system under the standard OC, contingencies, containing 2000 training cases and 10000 testing cases, are randomly selected by the Monte Carlo method. Considering the CNN training difficulty of the 2000-bus system,
To estimate the active and reactive PFs of the cascading outages, we construct a comprehensive framework constituting the DF, 1J, and NR algorithms. The deep integration of the causal inference and the statistical paradigm facilitates a robust and efficient performance. As indicated in abundant simulations, the speed of PF calculation is increased by five to ten times after introducing ML into the causal algorithms, comparing with the NR algorithm. Besides, the proportion of large error cases is reduced by one or two orders of magnitude, compared with 1J. The rapidity, accuracy, and generalization capability are all ensured in the proposed framework.
Considering the enormous scenarios of cascading outages, we propose a novel framework to calculate the PF of each outage stage rapidly with satisfying accuracy. After that, SSA and SAD are accelerated, providing reliable results for independent system operators (ISOs) to ensure the secure and stable operation of power systems.
Since we aim to study the steady-state PFs, the duration between the two outage stages is assumed to be long enough that the PF results are independent of the transient process here. Also, the NR algorithm is supposed to be convergent so that the solution is precise. Further research may be conducted to investigate the impacts of the transient process and the ill-conditioned PFs.
Appendix
Error in (6) can be rewritten as:
(A1) |
We may simultaneously add and minus in the multiplier of , as presented in (A2).
(A2) |
As both and are differentiable for n times at the equilibrium point , they can be expanded into the Taylor series, as formulated by (A3) and (A4), respectively.
(A3) |
(A4) |
Substituting (A3) and (A4) into (A2), we get
(A5) |
Since active power is a quadratic function of voltage, the third- and the higher-order partial derivatives of voltage are equal to 0. We expand active power into the Taylor series at a new equilibrium point .
(A6) |
Substituting (A6) into (A5), we may calculate the truncation error of the P subsystem as presented in (9).
(A7) |
Still, we rewrite in (6) as (B1).
(B1) |
We may add and minus in the term without the multiplier , as presented in (B2).
(B2) |
All Q, , and are differentiable at the equilibrium point . We expand the three into the Taylor series, as shown in (B3), (B4), and (B5), respectively.
(B3) |
(B4) |
(B5) |
Substituting (B3), (B4), and (B5) into (B2), we obtain
(B6) |
As reactive power is a quadratic function of voltage, the third- and the higher-order partial derivatives of voltage are equal to 0. Taking and as equilibrium points, we expand the reactive power function into the Taylor series at the two new points, respectively.
(B7) |
(B8) |
Substituting (B7) and (B8) into (B6), we may obtain the truncation error of Q subsystem as (10).
(B9) |
In order to clarify the relationship between and the 1J error, we sample all , , and contingencies under the standard OC in the IEEE 30-bus system. We calculate the 1J errors and defined in (16) and scatter their relationship with in Fig. C1.

Fig. C1 Scatter plot of error versus . (a) plane. (b) plane.
In the error-feature plane, a point corresponds to an outage. When ideally designed, will be linearly related to the error. Under such circumstances, it can be utilized to adjust errors and acquire accurate solutions. However, instead of correcting errors, is designed to identify cases with large potential errors. Its effectiveness can be demonstrated by the monotonic relationship with the 1J errors.
Active and reactive PF solutions can be calculated by
(D1) |
(D2) |
where and are the active and reactive branch PFs on the branch connecting nodes i and j, respectively; is the voltage at the node ; is the voltage angle difference; and are the real and imaginary parts of the element in the bus admittance matrix, respectively; and and are the transformer ratio and half the capacity, respectively.
After employing both 1J and NR algorithms to the training cases, we get and , which represent the percentage of cases with and of 1J solutions under the threshold . For example, when takes 0.2, is the percentage of cases with eP smaller than 0.2%. Notably, the (threshold proportion) and the (large error proportion) in Table III are different. Indicator refers to the proportion of different categories in the training set only, whereas is applied to represent the large error proportions of the final PF results. Besides, of and can be different, corresponding to and in Table I.
The values of and increase with the growth of . Table EI offers four classifiers with different , and both balanced and unbalanced scenarios are considered. The performances of the four classifiers are depicted in Fig. 3.
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