Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Intelligent Assessment of Active and Reactive Power Flow with Satisfying Accuracy for N-k1-k2 Cascading Outages  PDF

  • Yusheng Xue
  • Yun Liu
NARI Group Corporation (State Grid Electric Power Research Institute), Nanjing, China; School of Electrical Engineering, Shandong University, Jinan, China

Updated:2021-09-27

DOI:10.35833/MPCE.2020.000312

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Abstract

Addressed to the N-k1-k2 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.

I. Introduction

AS one of the major causes of large-scale blackouts, the cascading outage has aroused great concerns [

1]-[3]. According to the IEEE PES Computing and Analytical Methods Subcommittee, a cascading outage is a sequence of events in which an initial disturbance, or a set of disturbances, triggers a sequence of one or more dependent component outages [4]. Clear as the definition is, two controversies remain. One is the number of initial contingencies. Some researchers believe that only one fault initiates the whole cascade [5], whereas others claim that clustered outages may also incur a cascading outage [6], [7]. The other one is the number of faults that occur in each stage. Some researchers insist that faults successively weaken the grid [8], while others hold that multiple faults may occur in each stage [9]. Consistent with the standard definition, the term “N-k1-k2 cascading outages” is proposed in this paper to emphasize that multiple outages should be considered both in the initial fault and follow-up outage stages. Particularly, if k2=0, it turns out to be the high-order contingency [10]. If k1=k2=1, it becomes the N-1-1 contingency [11].

Considering the power flow (PF) nonlinearity and a number of the N-k1-k2 cascading outages, it is computationally infeasible to seek precise PF solutions in each stage. The PF equation can be described as:

P=fPθ,VQ=fQθ,V (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 [

12].

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 [

13], [14], they fail to meet the online requirements when a number of the N-k1-k2 cascading outages are involved. The parallel computation technique is applied and demonstrates superior calculation speed, especially in large-scale systems [15]. However, it is constrained by industrial computational resources. Apart from enhancing the computation technique, efforts are made on model simplification to relieve the massive computational burden with satisfying accuracy.

There is always one NNA that corresponds to an NIA. Algorithms like 1J and 1P1Q can be described in an incremental form as (2) [

16], where matrices [Pi/θj] and [Pi/Vj] indicate the influence of active power on the voltage angle and magnitude, respectively; matrices [Qi/θj] and [Qi/Vj] indicate the influence of reactive power the on the voltage angle and magnitude, respectively.

P=P0+PiθjΔθ+PiVjΔVQ=Q0+QiθjΔθ+QiVjΔV (2)

where P0 and Q0 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, [Pi/Vj] and [Qi/θj] 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 [

14]. Hence, the PQ coupling can be indicated by two correlations. One is between active power and magnitude. The other one is between reactive power and angle. For cases with strong PQ coupling, the two pairs are usually tightly related, and both [Pi/Vj]ΔV and [Qi/θj]Δθ cannot be ignored. Contrarily, for cases with weak PQ coupling, the two pairs are barely affected by each other, so [Pi/Vj]ΔV and [Qi/θj]Δθ can be ignored. Compared with the complete involvement of the NR algorithm into the PQ coupling, NNAs are too rough to be applied to the strong PQ coupling cases, though with a little computation amount. Lacking a sufficient mechanism analysis for error evaluations, NNAs fail to identify cases with large potential errors, impeding industrial usages.

The DC algorithm is formulated in an incremental form as follows [

17].

P=P0+BΔθ (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 [

18], [19]. These LAs further simplify the PF calculation for outages, but the active PF errors of LAs are generally larger than those of NNAs and will dramatically increase with the growing number of outage lines. Also, no reactive PF solutions are available. Though some LAs involve the calculation of reactive power and voltage [20]-[22], the insufficient accuracy limits their further developments. For description purposes, the LA in this paper only considers active power.

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) [

23], [24]. Based on the training samples, statistical regularities between features and the PF are then learned. Once the relationship is statistically mapped, it is employed to unknown testing cases to obtain PF solutions. A remarkable speed-up has been witnessed. For example, CNN is reported to achieve 100x acceleration in N-1 contingency analysis [25], however, it is very case-dependent. Its generalization capability remains to be further improved in practical power systems.

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 [

26]. Neither the causal inference nor the statistical paradigm alone could accelerate PF calculations with satisfying accuracy. However, their integration makes it attainable. Currently, this has been utilized to active PF analysis in cascading outages [27]-[29], leaving the reactive PF analysis unsolved yet.

Addressed to the rapid, reliable, and robust PF calculation of the N-k1-k2 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.

II. Comprehensive Framework to Analyze Cascading Outages

As to the PF calculations in the N-k1-k2 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 N-k1-k2 cascading outages. Figure 1 depicts the proposed framework.

Fig. 1 Comprehensive framework integrating statistical analysis, model analysis, and theoretical analysis.

A. Three Involved Analyses

The N-k1-k2 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 Fig. 1, the offline preparations for the proposed framework include three steps, as shown in Algorithm 1.

Algorithm 1  : pseudo-code of offline preparations

Step 1: select an LA, an NNA, and an NIA.

Step 2: evaluate the analytical error of the selected NNA and extract features accordingly.

Step 3: employ ML for classifier training.

Afterward, the framework can be applied online, and the corresponding algorithm is designed as Algorithm 2.

Algorithm 2  : pseudo-code of online applications

Employ the selected NIA to analyze the initial state of the power system (base case).

Loop while any case is not analyzed.

Step 1: analyze a case with the selected LA and NNA.

Step 2: calculate features according to the extraction formula deduced offline.

Step 3: label a reliable or unreliable tag to the case according to the classifier trained offline.

Step 4: if it is the reliable tag, jump to Step 5, else employ the selected NIA.

Step 5: deliver the active and the reactive PF solutions.

End loop

B. Model Analysis for PF Calculation

This subsection designates the DF, 1J, and NR algorithms as the LA, NIA, and NNA, respectively, for the framework for the following reasons.

1) Selection for LA

Like DC algorithm, DF algorithm neglects voltage and reactive power, simplifies [Pi/θj] as B, and avoids iterations. Besides, the DF calculates faster than the DC by one order of magnitude in outages [

28]. It utilizes the change in active power injections to model outages by rectifying B. Actually, tiny modifications on B of the base case can save massive computations of multiple outages.

Therefore, the DF algorithm is designated as the LA to rapidly offer the essential information for the feature extraction in Section III.

2) Selection for NIA

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.

3) Selection for NNA

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.

C. Theoretical Analysis for Feature Extraction

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.

D. Statistical Analysis for Classifier Training

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 Table I, where eP and eQ are the active and reactive PF errors of 1J, respectively; and ηP and ηQ are the user-defined thresholds for active and reactive PF errors, respectively.

TABLE I Class Labels
ClassifierLabelExplanation
CP Reliable Cases with small active PF errors (eP<ηP)
Unreliable Cases with large active PF errors (ePηP)
CQ Reliable Cases with small reactive PF errors (eQ<ηQ)
Unreliable Cases with large reactive PF errors (eQηQ)

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 CP and CQ 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.

III. Error Analysis and Feature Extraction

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.

A. Error Analysis for 1J

The NR solutions are denoted as PNR and QNR and can be expanded into the Taylor series at the pre-fault equilibrium point (θ0,V0). Notably, point (θ0,V0) corresponds to the pre-fault PFs of the last outage stage. For example, point (θ0,V0) of N-2 contingencies corresponds to the PFs of the initial state of power systems, while (θ0,V0) of an N-1-1 contingency corresponds to the PFs of the corresponding N-1 state.

The 1J solutions P1J and Q1J are formulated in (2) and neglect the nonlinear items in (4). Taking PNR and QNR as references, we can quantify the active power error eP and the reactive power error eQ of the 1J as (5).

PNR=P1J+C202!2Pθ2Δθ2+C212!2PVθΔVΔθ+C222!2PV2ΔV2+QNR=Q1J+C202!2Qθ2Δθ2+C212!2QVθΔVΔθ+C222!2QV2ΔV2+ (4)
eP=-C202!2Pθ2Δθ2+C212!2PθVΔθΔV+C222!2PV2ΔV2+eQ=-C202!2Qθ2Δθ2+C212!2QθVΔθΔV+C222!2QV2ΔV2+ (5)

Depending on the involvement of the PQ coupling, (5) can be divided into two parts. Error eP consists of the truncation error of the P subsystem ePtr and the PV decoupling error ePde, while error eQ comprises the truncation error of the Q subsystem eQtr and the decoupling error eQde, given by follows.

ePtr=-C212!2PVθΔθ+ΔV-C222!2PV2+ΔV2eQtr=-C202!2Qθ2Δθ2+-C212!2QVθΔθ+ΔV-        C323!3QV2θΔθ+ΔV2 (6)
ePde=-C202!2Pθ2Δθ2+C303!3Pθ3Δθ3+eQde=-C222!2QV2ΔV2 (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.

B. Feature of Truncation Error

The feature of truncation error should reflect the PQ coupling. Here, we compare the branch active PF solution of the 1J (PL1J) with that of the DF algorithm (PLDF), and utilize the 2-norm of their deviation as the feature of truncation error λtr, 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.

λtr=PL1J-PLDF 2 (8)

Since the λtr is designed to demonstrate ePtr and eQtr, their correlations are thoroughly discussed as follows. To better explain the correlation between λtr and ePtr, we rewrite ePtr in (6) as (9). The derivation is listed in Appendix A.

ePtr=PVθ0,V0ΔV+fP(θ0+Δθ,V0)-fP(θ0+Δθ,V0+ΔV) (9)

As is indicated in (9), two terms compose ePtr. The item (P/V)|θ0,V0ΔV is the deviation of active power function between (2) and (3). If the difference between [Pi/θj] 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 λtr. The item fP(θ0+Δθ,V0)-fP(θ0+Δθ,V0+ΔV) 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, λtr can also quantify this term. Considering the similar Δθ and the different ΔV [

30], we may approximate fP(θ0+Δθ,V0) and fP(θ0+Δθ,V0+ΔV) as the active power solutions of the DF algorithm and the 1J, respectively. Therefore, the feature λtr is applicable to indicate ePtr.

Then, the feasibility of applying λtr to represent eQtr is discussed. Similarly, error eQtr in (6) is rewritten as (10). The detailed deviation is given in Appendix B.

eQtr=Qθθ0,V0Δθ+fQ(θ0,V0+ΔV)-fQ(θ0+Δθ,V0+ΔV) (10)

Also, the error eQtr consists of two components. The latter item fQ(θ0,V0+ΔV)-fQ(θ0+Δθ,V0+ΔV) describes the influence of angle variation on reactive power and is neglected here, because the ith element of this item is always smaller than 2Vij=1NVjGij2+Bij2, where Gij and Bij 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 fQ. The term (Q/θ)|θ0,V0Δθ is thereby dominant in eQtr, and a feature is extracted accordingly. Except for the diagonal elements, other elements in Pi/Vj are opposite to the corresponding elements in Q/θj [

12], which means (Q/θ)|θ0,V0 is nearly equal to -(P/V)|θ0,V0. As is indicated in abundant experiments, the absolute values of Δθ and ΔV are often monotonically related and increase with the growing number of outage lines [28]. Therefore, eQtr can be represented by (P/V)|θ0,V0ΔV and further indicated by λtr.

Notably, the identity between the features of ePtr and eQtr 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 λtr and 1J errors.

C. Feature of Decoupling Error

Unlike the feature of truncation error, the features of decoupling error are different in CP and CQ, 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 l connecting nodes i and j.

1) Feature of PV Decoupling Error

As to ePde 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 2P/θ2, the error ePde can be estimated by:

ePde-C202!2Pθ2Δθ212Pθθ0,V0-Pθθ0+Δθ,V0Δθ (11)

Formula (11) approximates ePde as the multiplication of (P/V)|θ0,V0-(P/V)|θ0+Δθ,V0 and Δθ. All elements in (P/V)|θ0,V0-(P/V)|θ0+Δθ,V0 are smaller than 2Gij2+Bij2, because P/V is the trigonometric function of θ. Comparatively, the absolute value of Δθ often increases with the propagation of cascading failure. Accordingly, Δθ is a critical variable in determining ePde.

We denote the active PF variation of outage line as ΔP˜L, a sparse L×1 vector where elements corresponding to the outage line numbers equal to their pre-fault active PF. In the DF algorithm, the Δθ-ΔP˜L relationship can be described as (12). As the DF angle is similar to the 1J angle, (12) can be roughly held in the 1J [

29].

Δθ=B-1CI-I(ΨC)-1ΔP˜L (12)

where the node-branch incidence matrix C is a sparse matrix of N×L, and its (i,l)th and (j,l)th elements are 1 and -1, respectively; identity matrix I is of L×L; Ψ is the power transfer distribution factor (PTDF) matrix of L×N; and the operator is the Hadamard product, where two matrices of the same dimensions are element-wisely multiplied. Given the linear correlation between ΔP˜L and Δθ, we define the 2-norm of ΔP˜L as the feature of PV decoupling error λPde, as shown in (13).

λPde=ΔP˜L 2 (13)

2) Feature of Qθ Decoupling Error

Since reactive power is the quadratic function of voltage, the term 2Q/V2 of eQde in (7) is a constant matrix, which depends on topology only, leaving ΔV2 as the dominant part of eQde. We designate the quadratic sum of all elements in ΔV as the feature of the decoupling error λQde, which can also be described as the square of a 2-norm, as shown in (14).

λQde=ΔV1JV*22 (14)

where the N×1 vector ΔV1J is the 1J voltage variation. In the N×1 vector V*, each element equals the reciprocal value of the corresponding reference voltage.

D. Auxiliary Feature

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 ΔV is proved to be useful for the primary error identification.

As to eP, when ΔV is small, ePtrePde, and ePtr can be ignored, leaving ePde as the dominant error. At this point, eP can be indicated by λPde. When ΔV is large, ePtrePde, and a contrary conclusion can be drawn that λtr becomes the leading feature.

When it comes to the eQ analysis, two scenarios are discussed. As to a small value of ΔV, eQtreQde, and eQde is too trivial to be considered. Under such circumstances, λtr can be directly applied to demonstrate eQ. When ΔV takes a large value, eQ is jointly determined by eQtr and eQde. Correspondingly, both λtr and λQde are the key features.

We take the 2-norm of ΔV as the auxiliary feature λau by:

λau=ΔV1JV* 2 (15)

As λau is served to determine the leading feature between λtr and λde, a breaking point can be learned offline. Considering that SVM is applied to learn the mapping relationship, the breaking point of λau is not particularly studied.

The features of CP include λtr, λPde, and λau, while the features of CQ include λtr, λQde, and λau.

IV. Case Classification Based on SVM

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 CQ satisfy classification performances. For description purposes, we give an example on CP and CQ trained under all N-1 and N-2 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 [

31]. The testing set covers 100000 randomly sampled cases of N-1, N-2, and N-3 contingencies under multiple OCs in the IEEE 30-bus system. Figure 2 presents the training and testing of CP and CQ.

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 CP and CQ 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 CP and CQ 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, Fig. 2(c) and (d) offers four sections of distributed testing samples when λau takes 0.001, 0.1, 0.2, and 0.3. Points in the sections are the testing samples, and points of different colors demonstrate testing samples of different categories. The actual reliable cases with mean errors smaller than 1.2% are marked in blue, while the actual unreliable ones with mean errors no smaller than 1.2% are marked in red. As it is practically unacceptable to leave actual unreliable cases unrecognized, we always cover the red on the blue if they overlap.

Figure 2(c) and (d) also demonstrates the sections of hyperplanes in Fig. 2(a) and (b), respectively. The relationship between the envelopes and the points illuminates the performances of classifiers.

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 η or apply more advanced ML techniques, but with the sacrifice of calculating time.

In summary, the hyperplanes in Fig. 2(a) and (b) demonstrate the nonlinear relationships between features and tags learned by SVM. Actual cases are distributed in the feature space, and their positions relative to the hyperplanes visualize the classification accuracy. For a better visualization, Fig. 2(c) and (d) offers sections of hyperplanes. In the section, small error cases are mostly located inside the envelope, while most large ones stay outside, indicating the consistency between the actual situations and the predictions.

V. Case Study

For the evaluation of effectiveness, relevant indices are given in Section V-A. Afterward, two particular scenarios of the N-k1-k2 cascading outages are given. Section V-B involves the N-k contingency considering the multi-element failure in each outage stage. Section V-C designs the N-1-1-1 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 [

32]. Both training and testing cases are randomly sampled from the contingency set by the Monte Carlo method. As PF unsolvability is not the main issue, we rule out cases where the NR solutions are not convergent.

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.

A. Evaluation Indices

1) Classifier Evaluation

The confusion matrix is used to demonstrate classification accuracy. When a specific classifier and instances are given, four outcomes are listed in Table II. Two indices are proposed accordingly: one is true positive rate (TPR), namely the true positive (TP) percentage in all actual reliable cases, and the other one is false positive rate (FPR), namely the false positive (FP) percentage in all actual unreliable cases.

TABLE II Confusion Matrix of Classifier
PredictedActual Output
ReliableUnreliable
Reliable TP FP
Unreliable False negative (FN) True negative (TN)

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 [

33]. It presents a curve in ROC space consisting of multiple TPRs and FPRs under varying thresholds. A point in the ROC space represents a classifier. One is better than another if it is northwest to the other (higher TPR and lower FPR). Specifically, the point (0,1) achieves perfect classification, while the diagonal line connecting (0,0) and (1,1) indicates random guessing. Generally, the classification performs better when the curve is closer to the northwest of the ROC space.

Area under the ROC curve (AUC) is defined as the area of the ROC curve in the normalized unit [

33]. It statistically equates to the probability that a classifier will rank a random choice of reliable cases higher than that of unreliable ones. The AUC value is between 0 and 1. For the two particular classifiers mentioned above, the AUC values of the perfect classifier and the random guesser are equal to 1 and 0.5, respectively. Generally, the classifier obtains a higher accuracy when AUC is closer to 1.

2) Evaluation for PF Calculation

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 PlNR, QlNR, VNR, and θNR are taken as references. The error of branch PF is relative, where Pl, Ql, and Sl,lim 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 Sl,lim for better verification. We denote the sum of branches involved as L'. Different from the relative error of branch PF, the error of node state variable widely takes an absolute value.

eP=1L'PlNR>0.1Sl,limPl-PlNRPlNR×100%eQ=1L'QlNR>0.1Sl,limQl-QlNRQlNR×100% (16)
eθ=1Nθ-θNReV=1NV-VNR (17)

All relevant indices are defined in Table III. Since it is unfair to compare T for two sets involving different numbers of cases, we also employ σNR.

TABLE III Symbols of Indicators
IndicatorSymbolQuantityExplanation
Accuracy e Single case error Error of one case, including eP, eQ, eV, and eθ, as formulated in (16) and (17)
E Mean error Mean value of e, including EP, EQ
emax Maximum error Maximum error of e, including emax,P and emax,Q
στ Proportion of cases with large errors Percentages of cases with errors larger than τ, including σP,τ and σQ,τ. We take τ as 1%, 3%, and 5% here (σP,3 is the percentage of cases of which eP exceeds 3%)
Speed T Time The total time in seconds
σNR Proportion of cases with NR analysis Case percentage analyzed by NR algorithm

B. Simultaneous Outage

The simultaneous outage (N-k contingency) is a particular case of the N-k1-k2 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 N-1, N-2, and N-3 contingencies are sampled. In the contingency set, 38 N-1 and 677 N-2 contingencies are sampled for training, while 7504 N-3 contingencies are for testing. Unbalanced datasets are also considered to verify the generalization capability of the classifier, which is detailly introduced in Appendix E.

Figure 3 illustrates the ROC curves and AUCs of the classifier with varying values of ηP and ηQ. The ROC curves of CP and CQ concentrate in the northwest corner of the ROC space, which is little influenced by the imbalanced dataset. The AUC values remain stable around 0.89. Therefore, both accuracy and generalization capability are ensured even in highly imbalanced scenarios.

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 ηP and ηQ.

Figure 4 compares the error distributions of CNN and the proposed framework. In the eV-eθ plane, one point corresponds to an outage. It should be noted that the CNN errors in Fig. 4(a) are close to the published results [

25].

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

In Fig. 4(a), the maximal eV and eθ in training cases only occupy 35% and 18% of those in testing cases, respectively. As to the proposed framework, the two proportions are 73% and 55%, respectively. Therefore, the training and testing samples of the framework are more concentratedly distributed than those of CNN. The consistent error distribution between the training and the testing data indicates a better generalization capability. As the causal involvement in features contributes to the most significant distinction between CNN and our framework, we may conclude that the causality can significantly enhance the generalization capability.

Figure 5 compares the calculation time of the NR algorithm and the proposed framework. The simulation results show that the comprehensive framework accelerates the computation by five to ten times, compared with the traditional NR algorithm. Considering the ML employment in the framework, involving the statistical paradigm can intensify the causal inference efficiency.

Fig. 5 Comparison of calculation time T.

Identifications of large error cases are compared in Table IV.

TABLE IV Accuracy Analyses of Simultaneous Outages in IEEE 30-bus System
DatasetScenario (total number)Methodemax,P (%)σP,3 (%)emax,Q (%)σQ,3 (%)

Training

set

N-1(38) 1J 1.14 0 8.20 2.63
CNN 24.76 100.00 22.55 100.00
Proposed framework 1.14 0 2.78 0
N-2(677) 1J 10.69 0.30 14.32 6.94
CNN 87.53 100.00 66.23 97.22
Proposed framework 2.34 0 4.57 0.74

Testing

set

N-3

(7504)

1J 15.38 1.67 17.27 13.29
CNN 176.64 100.00 139.09 100.00
Proposed framework 5.46 0.10 5.10 1.39

The values of emax and σ3 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 emax under 5.5% and declines σ3 by at least one order of magnitude. The value of σP,3 of the proposed framework reduces from 1.67% to 0.10% in testing cases, while σQ,3 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 emax is more prominent, and the values of σP,3 and σQ,3 are controlled under 0.1% and 1.4%, respectively. Besides, more accurate solutions can be acquired in the N-3 testing cases if we take larger values of ηP and ηQ.

C. Successive Outage

The successive outage like N-1-1--1 contingency is another particular case for the N-k1-k2 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.

1) IEEE 57-bus System

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 N-1-1-1 contingencies randomly selected. Given that each outage stage in N-1-1-1 contingencies is learned in the training set, 76 N-1, 2810 N-1-1, and 10000 N-1-1-1 contingencies are included in the testing set for performance verifications.

Table V compares the accuracy and speed of 1J, NR, CNN, and the proposed framework. Compared with 1J, the proposed framework remarkably enhances the accuracy, decreases σ5 by at least two orders of magnitude, and controls emax,P and emax,Q under 4% and 8.5%, respectively. Compared with the NR algorithm, the framework accelerates the computation by nearly five times. Compared with CNN, the framework presents remarkable advantages in accuracy and robustness.

TABLE V Analysis on Accuracy and Speed of Successive Outage in IEEE 57-bus System
DatasetScenario (total number)MethodAccuracy of active PFAccuracy of reactive PFSpeed
ParadigmAlgorithmEP (%)emax,P (%)σP,5 (%)EQ (%)emax,Q (%)σQ,5 (%)σNR (%)T (s)

Training

set

N-1-1-1(2000) Causal 1J 1.27 15.84 6.35 2.96 33.76 18.33 0 7.34
NR 0 0 0 0 0 0 100.00 44.06
Statistical CNN 12.50 94.68 100.00 10.78 88.99 99.00 3.22
Comprehensive Proposed framework 0.27 3.88 0 0.61 8.12 0.23 13.80 13.97

Testing

set

N-1(76) Causal 1J 0.68 9.49 2.63 1.68 19.28 9.21 0 0.12
NR 0 0 0 0 0 0 100.00 0.72
Statistical CNN 10.95 18.22 100.00 8.24 15.48 100.00 1.51
Comprehensive Proposed framework 0.25 1.95 0 0.54 3.70 0 13.16 0.22
N-1-1(2810) Causal 1J 0.97 15.53 4.18 2.34 30.24 14.29 0 7.07
NR 0 0 0 0 0 0 100.00 45.68
Statistical CNN 11.52 78.32 100.00 9.33 49.97 99.14 4.05
Comprehensive Proposed framework 0.25 2.37 0 0.55 5.13 0.14 14.56 13.50

N-1-1-1

(10000)

Causal 1J 1.29 29.32 6.11 3.06 38.83 19.56 0 37.66
NR 0 0 0 0 0 0 100.00 217.23
Statistical CNN 13.08 125.56 99.96 11.23 93.15 99.25 14.17
Comprehensive Proposed framework 0.27 3.57 0 0.62 8.23 0.24 14.21 71.37

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, EP of 1J in the testing N-1, N-1-1, and N-1-1-1 contingencies are 0.68%, 0.97%, and 1.29%, respectively. For the proposed framework, the EP in N-1, N-1-1, and N-1-1-1 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.

2) IEEE 118-bus System

In the IEEE 118-bus system, we randomly sample 2000 N-1-1-1 contingencies under the standard OC as the training set and 10000 N-1-1-1 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 Table V, Table VI compares the causal method, the statistical method, and the comprehensive framework.

TABLE VI Analysis on Accuracy and Speed of Successive Outage in IEEE 118-bus System
Dataset

Scenario

(total number)

MethodAccuracy of active PFAccuracy of reactive PFSpeed
ParadigmAlgorithmEP (%)emax,P (%)σP,1 (%)EQ (%)emax,Q (%)σQ,1 (%)σNR (%)T (s)

Training

set

Standard OC (2000) Causal 1J 0.32 8.53 6.13 0.55 13.24 9.58 0 10.60
NR 0 0 0 0 0 0 100.00 161.28
Statistical CNN 3.16 39.97 100.00 2.18 23.35 99.50 4.25
Comprehensive Proposed framework 0.08 1.07 0.12 0.11 1.09 0.12 4.53 14.66

Testing

set

Maximum OC (10000) Causal 1J 0.30 12.13 3.91 0.53 18.10 10.61 0 50.39
NR 0 0 0 0 0 0 100.00 644.01
Statistical CNN 41.46 74.17 100.00 19.82 46.85 100.00 20.08
Comprehensive Proposed framework 0.08 2.33 0.11 0.11 3.59 0.13 4.32 72.14
Minimum OC (10000) Causal 1J 0.10 3.43 1.75 0.14 4.78 3.09 0 48.23
NR 0 0 0 0 0 0 100.00 685.29
Statistical CNN 67.07 123.50 100.00 28.79 56.60 100.00 19.95
Comprehensive Proposed framework 0.04 1.26 0.05 0.06 1.61 0.08 0.01 48.99

Compared with the causal method, a higher accuracy is achieved in the proposed framework, where σ1 is decreased by at least one order of magnitude when the 1J solutions are taken as a reference. Besides, emax,P and emax,Q 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 EP and EQ 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, σNR of the framework can be intelligently adjusted. As the 1J solutions of most cases are reliable under the minimum OC, σNR 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 EP and EQ 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.

3) The 2000-bus Texas System

The MATPOWER offers the data of the 2000-bus system. In the system under the standard OC, N-1-1-1 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, Table VII compares 1J, NR algorithm, and the proposed framework of the testing cases. Compared with 1J, the proposed framework enhances the accuracy by decreasing the large error proportion to one-fourth of 1J. Compared with the NR algorithm, the framework speeds up the calculation by nearly four times. The acceleration is not as obvious as the previous simulation because only about four iterations are required in each NR analysis. However, the NR iterations tend to be larger than 10 in real systems, indicating a significant PF acceleration of the framework. Therefore, the framework can rapidly calculate both active and reactive PFs with a guaranteed accuracy.

TABLE VII Analysis on Accuracy and Speed of Successive Outage in 2000-bus Texas System
MethodAccuracy of active PFAccuracy of reactive PFSpeed
EP (%)emax,P (%)σP, 0.1 (%)EQ (%)emax,Q (%)σQ, 0.5 (%)σNR (%)T (s)
1J 0.02 0.38 3.42 0.15 3.90 6.84 0 426
NR 0 0 0 0 0 0 100.00 1908
Proposed framework 0.01 0.30 0.69 0.06 1.83 1.75 1.80 538

VI. Conclusion and Future Work

To estimate the active and reactive PFs of the N-k1-k2 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

Appendix A

Error ePtr in (6) can be rewritten as:

ePtr=-n=2Cn1n!nPVθn-1θ0,V0Δθn-1ΔV-n=1Cn+12n+1!n+1PV2θn-1θ0,V0Δθn-1ΔV2 (A1)

We may simultaneously add and minus (P/V)|θ0,V0ΔV in the multiplier of ΔV, as presented in (A2).

ePtr=-n=1Cn1n!nPVθn-1θ0,V0Δθn-1-PVθ0,V0ΔV-12n=12Cn+12n+1!n+1PV2θn-1θ0,V0Δθn-1ΔV2=-n=11n-1!nPVθn-1θ0,V0Δθn-1-PVθ0,V0ΔV-12n=11n-1!n+1PV2θn-1θ0,V0Δθn-1ΔV2 (A2)

As both P/V and 2P/V2 are differentiable for n times at the equilibrium point (θ0,V0), they can be expanded into the Taylor series, as formulated by (A3) and (A4), respectively.

PVθ0+Δθ,V0=n=11n-1!nPVθn-1θ0,V0Δθn-1 (A3)
2PV2θ0+Δθ,V0=n=11n-1!n+1PV2θn-1θ0,V0Δθn-1 (A4)

Substituting (A3) and (A4) into (A2), we get

ePtr=-PVθ0+Δθ,V0-PVθ0,V0ΔV-122PV2θ0+Δθ,V0ΔV2=PVθ0,V0ΔV-PVθ0+Δθ,V0ΔV+122PV2θ0+Δθ,V0ΔV2 (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 (θ0+Δθ,V0).

fP(θ0+Δθ,V0+ΔV)=fP(θ0+Δθ,V0)+PVθ0+Δθ,V0ΔV+122PV2θ0+Δθ,V0ΔV2 (A6)

Substituting (A6) into (A5), we may calculate the truncation error of the P subsystem as presented in (9).

ePtr=PVθ0,V0ΔV+fP(θ0+Δθ,V0)-fP(θ0+Δθ,V0+ΔV) (A7)

Appendix B

Still, we rewrite eQtr in (6) as (B1).

eQtr=-n=2Cn0n!nQθnθ0,V0Δθn-n=1Cn+11n+1!n+1QVθnθ0,V0ΔθnΔV-n=1Cn+22n+2!n+2PV2θnθ0,V0ΔθnΔV2 (B1)

We may add and minus (Q/θ)|θ0,V0Δθ in the term without the multiplier ΔV, as presented in (B2).

eQtr=-n=1Cn0n!nQθnθ0,V0Δθn-Qθθ0,V0Δθ-n=1Cn+11n+1!n+1QVθnθ0,V0ΔθnΔV-n=1Cn+22n+2!n+2QV2θnθ0,V0ΔθnΔV2=-n=11n!nQθnθ0,V0Δθn-Qθθ0,V0Δθ-n=11n!n+1QVθnθ0,V0ΔθnΔV-12n=11n!n+2QV2θnθ0,V0ΔθnΔV2 (B2)

All Q, Q/V, and 2Q/V2 are differentiable at the equilibrium point (θ0,V0). We expand the three into the Taylor series, as shown in (B3), (B4), and (B5), respectively.

fQθ0+Δθ,V0=fQθ0,V0+n=11n!nQθnθ0,V0Δθn (B3)
QVθ0+Δθ,V0=QVθ0,V0+n=11n!n+1QVθnθ0,V0Δθn (B4)
2QV2θ0+Δθ,V0=2QV2θ0,V0+n=11n!n+2QV2θnθ0,V0Δθn (B5)

Substituting (B3), (B4), and (B5) into (B2), we obtain

eQtr=-fQθ0+Δθ,V0-fQθ0,V0-Qθθ0,V0Δθ-QVθ0+Δθ,V0-QVθ0,V0ΔV-122QV2θ0+Δθ,V0-2QV2θ0,V0ΔV2=Qθθ0,V0Δθ+fQθ0,V0+QVθ0,V0ΔV+122QV2θ0,V0ΔV2-fQθ0+Δθ,V0+QVθ0+Δθ,V0ΔV+122QV2θ0+Δθ,V0ΔV2 (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 (θ0,V0) and (θ0+Δθ,V0) as equilibrium points, we expand the reactive power function into the Taylor series at the two new points, respectively.

fQθ0,V0+ΔV=fQθ0,V0+QVθ0,V0ΔV+122QV2θ0,V0ΔV2 (B7)
fQ(θ0+Δθ,V0+ΔV)=fQ(θ0+Δθ,V0)+QVθ0+Δθ,V0ΔV+122QV2θ0+Δθ,V0ΔV2 (B8)

Substituting (B7) and (B8) into (B6), we may obtain the truncation error of Q subsystem as (10).

eQtr=Qθθ0,V0Δθ+fQ(θ0,V0+ΔV)-fQ(θ0+Δθ,V0+ΔV) (B9)

Appendix C

In order to clarify the relationship between λtr and the 1J error, we sample all N-1, N-2, and N-3 contingencies under the standard OC in the IEEE 30-bus system. We calculate the 1J errors eP and eQ defined in (16) and scatter their relationship with λtr in Fig. C1.

Fig. C1 Scatter plot of error versus λtr. (a) eP-λtr plane. (b) eQ-λtr plane.

In the error-feature plane, a point corresponds to an outage. When ideally designed, λtr 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, λtr is designed to identify cases with large potential errors. Its effectiveness can be demonstrated by the monotonic relationship with the 1J errors.

Appendix D

Active and reactive PF solutions can be calculated by

Pij=ViVjGijcosθij+Bijsinθij-tijGijVi2 (D1)
Qij=ViVjGijsinθij-Bijcosθij+tijBij-bij0Vi2 (D2)

where Pij and Qij are the active and reactive branch PFs on the branch connecting nodes i and j, respectively; Vi is the voltage at the node i; θij is the voltage angle difference; Gij and Bij are the real and imaginary parts of the element in the bus admittance matrix, respectively; and tij and bij0 are the transformer ratio and half the capacity, respectively.

Appendix E

After employing both 1J and NR algorithms to the training cases, we get σP,η and σQ,η, which represent the percentage of cases with eP and eQ of 1J solutions under the threshold η. For example, when η takes 0.2, σP,η is the percentage of cases with eP smaller than 0.2%. Notably, the σP,η (threshold proportion) and the σP,τ (large error proportion) in Table III are different. Indicator σP,η refers to the proportion of different categories in the training set only, whereas σP,τ is applied to represent the large error proportions of the final PF results. Besides, η of CP and CQ can be different, corresponding to ηP and ηQ in Table I.

The values of σP,η and σQ,η increase with the growth of η. Table EI offers four classifiers with different η, and both balanced and unbalanced scenarios are considered. The performances of the four classifiers are depicted in Fig. 3.

TABLE EI Accuracy Analyses of Simultaneous Outages in IEEE 30-bus System Classifiers with Different Values of η
η (%)σP,η (%)σQ,η (%)Description
0.2 25 22 Unbalanced for CP and CQ
0.4 50 36 Balanced for CP, unbalanced for CQ
0.5 61 52 Unbalanced for CP, balanced for CQ
0.8 75 76 Unbalanced for CP and CQ

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