2.3.1 K-Nearest neighbors (KNN).

It is well known that the K-Nearest Neighbors algorithm (KNN) for machine learning (ML) has been introduced as the non-parametric approach employed to classify and regress areas [43]. Within these two areas, the input data includes k-closest training samples in the feature space. Therefore, output based on if the KNN is utilized to classify or regress is categorized into two parts as below:

1. In k-NN classification, the output is a class membership. An object is classified by a plurality vote of its neighbors, with the object being assigned to the class most common among its k nearest neighbors (k is a positive integer, typically small). If k = 1, then the object is simply assigned to that single nearest neighbor’s class.
2. In k-NN regression, the output is the property value for the object. This value is the average of the values of k nearest neighbors.

Moreover, according to the KNN classification, the output is a class membership. An object is categorized by a plurality vote of its neighbors so that the object has been allocated to the class that is the commonest among its k-nearest neighbors (k represents a positive integer, usually small). Therefore, if k = 1, then the object is readily allocated to the class of that single-nearest neighbor. Besides, KNN has been presented as one of the types of sample-based learning or lazy learning wherein the function is merely estimated locally, and each computation would be deferred till classification" [43].

2.3.2 Support vector machine (SVM).

Vapnik and Cortes (1995) introduced SVM theory in which a hyperplane or a series of hyperplanes is constructed and utilized for both classification and regression [44,45]. By considering the labeled training set S = (x_l, y_l); l = 1;…; L of size L, and y_l ∈ 1,-1. The SVM can be obtained by solving: (1) subject to (2) (3) where ϕ(x_l) represents a nonlinear transformation mapping x_l in a high dimensional space that is called kernel function. The slack variable ξ_l represents nonlinearly separable training sets, and C denotes the parameter of a tunable positive regularization. To achieve a distributed SVM, (1) can be rewritten as follows: (4) subject to (5) (6) where N denotes the number of groups working together to train the SVM and w_i represents the parameter of the local optimization for each group. By introducing a global variable z, the (4) can be reformulated as: (7) (8) (9) (10)

To solve (7) distributively, variables {z, w_i} i = 1,…, N can be partitioned into two sets represented by {z} and {w_i}, i = 1,… N, and the Alternating Direction Method of Multipliers (ADMM) can be applied to solve the problem. Specifically, the scaled augmented Lagrangian function can be expressed as follows: (11) where ρ denotes the step size and μ_i represents the scaled dual variable. At each iteration k, {w_i},{z}and μ_i can be updated as follows: (12) (13) (14)

Note that the process of updating the w_i can be done locally in the i_th group. Moreover, it involves the fitting of a SVM to the local data using an offset in the quadratic regularization term. The vector {z} is expressed as: (15) which can be solved analytically as: (16) in which and . Finally, the scaled dual variable μ_i can be updated by: (17)

In constructing historical data, since the number of classes in topology samples depends on the number of switches, so the number of classes can be more than two groups, therefore a multi-class SVM is used [46,47], and for each nonlinear classifier, Gaussian kernel is applied in which δ is a mutable parameter. The main factors affecting the performance of SVM comprise kernel function and its parameters as well as the soft margin parameter C. The optimal Gaussian kernel parameter and soft margin (C) can be used for improving the efficiency of nonlinear SVM. Gaussian kernel, which has a single parameter (γ), is a typical choice for SVM [48]. The common practice for finding the best values of C and γ is to conduct a grid search, i.e., repeat the calculations with different C and γ combinations and determine the values yielding the best accuracy through cross-validation [49]. In Step 1, historical data are prepared by the independent system operator (ISO). In Step 2, the local and global optimization parameters are initialized. Step 3 comprises two parts. First, the local optimization parameter w_i is obtained by solving the optimization problem defined in (12) under constraints (13) and (14). Then, the global optimization parameter z is optimized in (16), where and are obtained by in-network processing through messages that were only exchanged among neighboring groups. Following the optimization solution and the achievement of convergence of w_i by each group, the local and global optimization parameters, i.e., w_i and z, respectively, are returned in Step 4.

2.3.3 Decision Tree (DT).

A Decision Tree (DT) is a predictive model where supervised learning and non-parametric methods with a hierarchical structure are used to classify different data types, and results are delivered in a flowchart with a tree-like structure. According to the dependent variable type, this algorithm is divided into two categories: regression trees for continuous variables and classification trees for discrete variables. In the DT algorithm, fragmentation of data is implemented using the features as a tree, and for better understanding, it is written using if-then rules [50]. Depending on the training data, a feature for the data is selected at each stage of this method, and the data set is decomposed into a further class grouping based on chosen features. This process continues until all the data in a category has a single label.

2.3.4 Ensemble Learning (EL).

Ensemble Learning (EL) methods represent a machine learning approach in which multiple learning algorithms are used in a parallel fashion to achieve more acceptable predictive functions in comparison to the situation that may be obtainable from each constituent learning algorithm [51,52]. While in statistical mechanics, a statistical ensemble is generally infinite, an ML ensemble contains a finite set of alternative models. This, in turn, leads to a structure with higher flexibility than the individual alternatives. Ensemble methods are meta-algorithms that combine several machine learning techniques into one predictive model to decrease variance (bagging), bias (boosting), or correlation (random subspace).

2.3.5 Bagging algorithm.

Bagging, also known as bootstrap aggregation, is a machine learning ensemble meta-algorithm designed to improve machine learning algorithms’ stability and accuracy in statistical regression and classification. Moreover, the algorithm can decrease the variance and prevent overfitting. Leo Breiman (1994) proposed bagging to improve the classification by combining classifications of randomly produced training sets. The bagging method is modeled based on the instability of base learners, which can be utilized to modify such unstable base learners’ predictive performance. The main idea is that, given a training set S of size n and a learner L, which commonly is a decision tree, bagging creates m new training sets S_i with replacement. Then, the bagging algorithm applies L to each S_i to build m models. The final output of bagging is based on simple averaging [53].

Even though it is commonly used in combination with DT methods, bagging can be applied with any kind of technique. In fact, bagging is a special case for the model averaging approach [52].

2.3.6 Boosting algorithm.

In ML, boosting is an ensemble meta-algorithm designed primarily to reduce bias [54], and variance in supervised learning includes a family of ML algorithms that convert weak learners into strong ones [55]. Boosting is based on the question first posed by [56,57]: "Can a set of weak learners create a single strong learner?" A weak learner is defined to be a classifier that is only slightly correlated with the proper classification (although it can label examples better than random guessing). In contrast, a strong learner is a classifier that is arbitrarily well-correlated with the true classification. While boosting is not algorithmically constrained, most boosting algorithms consist of learning weak classifiers concerning distribution and adding a final strong classifier. After the addition, the weak learners are weighted in a way related to the weak learners’ accuracy, and the data weights are readjusted, a process is known as “re-weighting.” Misclassified input data gain a higher weight while examples that are classified correctly lose weight. Thus, future weak learners focus more on the examples than on previous misclassified weak learners.

2.3.7 Random subspace algorithm.

In ML, the random subspace method, also called attribute bagging or feature bagging, is an ensemble learning method that attempts to reduce the correlation between estimators in an ensemble by training them on random samples of features instead of the entire feature set. The random subspace method is similar to bagging, except that the features are randomly sampled with the replacement for each learner. Informally, this prevents individual learners from over-focusing on features that appear highly predictive/descriptive in the training set but fail to be as predictive for points outside that set. For this reason, random subspace algorithms are an attractive choice for problems where the number of features is much larger than the number of training points. The random subspace method has been used with DT. When combined with "ordinary" bagging of decision trees, the resulting models are called random forests. The method has also been applied to linear classifiers, support vector machines, k-nearest neighbors, and other types of classifiers [58]. Fig 1 summarizes the main classification of SML.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Classification of SML algorithms.

These methods are in a supervised paradigm.

https://doi.org/10.1371/journal.pone.0252436.g001

2.4.1 Description of the dataset used to test the method.

We consider 7 switch devices in simulation, implying 7 states for possible events, and correspondingly, 7 voltage curves in the related time domain for 15 seconds (the simulation time duration). We set the voltage changes per second as a first feature. Maximum and minimum of the instantaneous voltage are considered as second and third features. Hence, the first column of Excel files in the supplementary material describes when the event occurs. Voltage changes, its maximum, and minimum are featured in the second, third, and fourth columns. Each excel file indicates one state of event or fault, including transient voltage and type of fault. The excel file is available at: https://doi.org/10.6084/m9.figshare.14123018.v2.

2.4.2 Rationale for choosing a dataset.

The power system analysis is generally done by monitoring of current status and computing several distinct parameters. One of the most important parameters in the power system is the transient voltage. As is well known, any change in the network, such as line and generation outages, affects the network operation status and its parameters, such as voltage in transient and steady-state mode. Thus, the tracking of the status of the current power system requires the analysis of the transient voltage. To simulate the fault in the network, we used the switch, so that when a switch is disconnected, a line or generation unit is disconnected from the network. This in turn implies that the analysis of the power system where the fault occurred requires tracking the changes in the network transient voltage. The changes and faults that occurred in the network are exactly proportional to the voltage changes, which means that the machine learning process can learn all the voltage changes in all possible fault scenarios and, after the training process, can detect the type of fault and the fault scenario that occurred.

2.4.3 The source of the data and how it was obtained.

We simulated a real standard power distribution system in ETAP software. To simulate the fault, we used 7 power switches, and, consequently, any switch disconnecting is considered a network fault. We simulated the fault in the network and recorded the transient voltage after the fault. For each fault, we obtained a transient voltage from the power grid. Hence, the labels define the type of fault, while the output is represented by the transient voltage. We recorded a series of transient voltages for all possible faults until we could train the machine learning process using these transient voltages and types of faults.