Applying Cross-Permutation-Based Quad-Hybrid Feature Selection Algorithm on Transient Univariates to Select Optimal Features for Transient Analysis

Neglect feature selection matter for high-dimensional transient data obtained from phasor measurement units (PMUs) negatively affects the inconsistent -linked indices, namely data labeling time (DLT) and data labeling accuracy (DLA) in the transient analysis (TA). A reasonable trade-off between DLT and DLA or a win-win solution (low DLT and high DLA) necessitates feature-based mining on transient multivariate excursions (TMEs) via designing the comprehensive feature selection scheme (FSS). Hence, to achieve high-performance TA, we offer the cross-permutation-based quad-hybrid FSS (CPQHFSS) to select optimal features from TMEs. The CPQHFSS consists of four filter-wrapper blocks (FWBs) in the form of twin two-FWBs mounted on two-mechanism of the incremental wrapper, namely incremental wrapper subset selection (IWSS) and IWSS with replacement (IWSSr). The IWSS2FWBs and IWSSr2FWBs contain filter-fixed and wrapper-varied approaches (FfWv) that first block-specific FfWv of IWSS2FWBs and IWSSr2FWBs includes relevancy ratio-support vector machine (RR-SVM) and second block-specific FfWv of IWSS2FWBs and IWSSr2FWBs accompanied by relevancy ratio-twin support vector machine (RR-TWSVM). Generally, RRIWSSSVM and RRIWSSTWSVM is in IWSS2FWBs, and RRIWSSrSVM and RRIWSSrTWSVM is in IWSSr2FWBs. Besides direct relations in two-FfWvBs per incremental wrapper mechanism, by plugging different kernels into the hyperplane-based wrapper, all possible cross-permutations of hybrid FSS are applied on transient data to extract the optimal transient features (OTFs). Finally, the evaluation of the effectiveness of the CPQHFSS-based OTFs in TA is conducted based on the cross-validation technique. The obtained results show that the proposed framework has a DLA of 98.87 % and a DLT of 152.525 milliseconds for TA.


RR
IWSSr TWSVM is in IWSSr 2FWBs . Besides direct relations in two-F f W v Bs per incremental wrapper mechanism, by plugging different kernels into the hyperplane-based wrapper, all possible crosspermutations of hybrid FSS are applied on transient data to extract the optimal transient features (OTFs). Finally, the evaluation of the effectiveness of the CPQHFSS-based OTFs in TA is conducted based on the cross-validation technique. The obtained results show that the proposed framework has a DLA of 98.87 % and a DLT of 152.525 milliseconds for TA.

I. INTRODUCTION
Nowadays, information technology (IT) by integrating different data-driven systems, plays the pivot role in collecting a large amount of data in different sensitive industries. The raw data obtained by the IT paradigm provide the necessary conditions for conducting dataoriented actions instead of experience-based operations in all tasks and responsibilities of system operators [1][2][3][4][5]. Such restructuring in the decision-making process will be possible through data mining (DM) technology which triangulated machine learning (ML), statistical learning (SL), and dataset to discover useful patterns for predicting different phenomena [6,7]. Besides the importance of the type of ML and SL methods for achieving efficiency in learning procedures, the high-dimensional data (HDD) with sparse-dissimilar features is the most significant factor that negatively affects the training-testing procedures (TTP) of learning frameworks. In this regard, the concept of the curse of dimensionality is defined as a great challenge on the way of high-performance DM [8,9]. Furthermore, the importance of inconsistent-linked indices like accuracy and time (A&T) to make A&T-critical prediction in real-world problems exacerbates the necessitate of focusing on HDD concern. To fill two needs with one deed, DM engineers apply FSS on HDD for extracting the optimal features (OFs) set [10,11]. The survived OFs based on the FSS will bring two points: first, the low processing time to predict unseen cases due to the mapping HDD to low-dimensional feature space, and second, high accuracy prediction induced by selecting most discriminative features (MDFs). One of the HDD-oriented real-world problems is transient stability assessment (TSA) related to secure power supply [12][13]. The IT-based grid infrastructure equipped with phasor measurement units (PMUs) gathers highdimensional transient features (HDTF) for transient analysis (TA) [14]. In HDTF, the presence of irrelevant and redundant features is problematic for TTP of predictive approaches, which cause low-accuracy transient stability prediction (TSP). By applying the FSS scheme on HDTF, optimal transient features (OTFs) are selected to achieve high data labeling accuracy (DLA) in TSP. Also, the severe-sudden essence of transient stability necessitates using the FSS for compacting the HDTF to decrease data labeling time (DLT) in TSP, including observed time and prediction time [15]. In terms of the low DLT, low dimensional optimal transient features (LDOTF) caused fast learning in TTP scenarios leads to low prediction time, and existing the most relevant features in LDOTF allow picking up small-optimal observations. Consequently, by applying the FSS scheme on HDTF, system operators will be able to take timely-accurate corrective control actions to provide secure-adequate exploitation of the power grid. Hence, to achieve a win-win trade-off (high DLA and low DLT), designing the comprehensive FSS has been widely considered by DM researchers for TA.

II. RELATED WORKS
Reviewing the FSS-based transient studies shows that optimal transient features are selected by filter and filterwrapper (hybrid) methods. In term of filter-oriented FSS, in Reference [16,17], mutual information theory applied on transient characteristics related to power and angle to select optimal features regarding two principles: selected transient features have maximum relevance to the target class and have minimum relevance to one another, which is called minimum-redundancy and maximum-relevance (mRMR) FSS. Reference [18] introduce the ReliefF algorithm for calculating relevancy of rotor faults features to predict the health state of induction motor. To calculate total transfer capability regarding transient stability limitations, designing the feature pre-screening scenario based on the fast correlation-based filter (FCBF) is considered in [19]. Based on FCBF, optimal features of active and reactive load power, phase angles of bus voltages, and the induced electromotive force of generators are selected to achieve training-testing efficiency. In Reference [20], for constructing the dynamic security assessment (DSA) model for predicting the transient stability margin, applying the FSS algorithm based on partial mutual information (PMI) and the Pearson correlation coefficient (PCC) is considered as the main step of the DSA model. In terms of hybrid FSSs, in Reference [21], filter-wrapper FSS includes feature weight ranking by Relief (filter as the preliminary) step, and five-fold cross-validation SVM model (wrapper as the complementary step) are applied on trajectory cluster features for selecting optimal feature set. Based on the proposed hybrid FSS in [22], first, normalized mutual information (NMI) ranks the initial features in the form of strongly relevant feature subset (SRFS) and the weakly relevant feature subset (WRFS). Next, the obtained knowledge of the filter phase is fed to the wrapper phase equipped with an easy-implementing search algorithm called binary particle swarm optimization (BPSO) to improve the effectiveness of FSS results. Considering highdimensional multivariate time series data obtained by transient simulations, Reference [23] designed hybrid FSS in bi-mode, including trajectory-based filter-wrapper method (TFWM) and point-based filter-wrapper method (PFWM). In TFWM, mutual information-entropy-based (MIE) calculations (filter) and fuzzy imperialist competitive algorithm (FICA)-IWSS-based trihedral kernel-SVM (wrapper) find the optimal transient series. Next, the PFWM, including MIE calculations (filter) and the Gaussian kernel-SVM (wrapper) utilized to find optimal point features per optimal time series.
Regardless of the precise mining on transient feature space by the abovementioned FSSs, which have led to the acceptable performance in TSA, designing the comprehensive hybrid framework to extract maskedrelevant transient features is the greatest challenge to achieve timely-accurate TSA. Lack of cross-oriented learning mechanisms in the form of multifaceted hybrid FSS (MHFSS) causes some features with the discriminative character don't survive in the feature selection process. In this regard, focusing on the structure of the filter or hybrid FSSs in previous studies shows the fact that the mining of intrinsic characteristics of transient data for selecting optimal features is based on the unilateral strategy equipped with vertically learning. Such a mechanism may be applicable in selecting the optimal features to improve the TSA, but it will ignore optimal-blurred transient features. Furthermore, the characteristic of transient data is the main parameter in determining how to apply the proposed MHFSS to it. Having a glance at FSS-based studies shows that the FSSs applied on multivariate point or time series data in the whole-manner. Such a strategy tends to select the OTFs without regard to the possibility of optimal features sacrificing related to each univariate trajectory. Streaming k-variate time-series data obtained by the PMUbased synchronized measurement necessitate extracting univariate-specific OTFs in the form of univariate-oriented learning in both filter and wrapper phase.
The main contributions of this study to solve abovementioned challenges in FSS-based TSA are summarized as follows:  A new feature selection algorithm called crosspermutation quad-hybrid FSS (CPQHFSS) was considered to select OTFs to achieve highperformance TA. This scheme was designed via a multifaceted hybrid scenario accompanied by the entropy-based metric in the filter phase and hyperplane-based learning methods in the wrapper phase.  The CPQHFSS applied on high-dimensional TMEs based on partial-manner learning for extracting univariate-specific MDFs. The optimal features per univariates are survived by the conducting cross-permutation scenario of the proposed FSS. Such a mechanism guaranteed the optimal-blurred transient features extraction to achieve high DLA and low DLT.  The performance of CPQHFSS-specific OTFs in TA was compared with selected OTFs by other FSS based on the cross-validation technique. The rest of the paper is organized as follows: The detailed descriptions of the CPQHFSS are remarked in Section 3. Experimental results of applying CPQHFSS on univariates of TMEs for TSA are presented in Section 4. Also, the comparison results between the proposed FSS and the other FSSs are interpreted in Section 4. Finally, the conclusion is depicted in Section 5.

III. CROSS-PERMUTATION-BASED QUAD-HYBRID FEATURE SELECTION SCHEME (CPQHFSS)
The overall framework of FSS-oriented TA based on CPQHFSS is shown in Fig. 1. After transient data gathering phase, we offer CPQHFSS including twin two-filterwrapper blocks (2FWBs) mounted on the two-mechanism of the incremental wrapper namely incremental wrapper subset selection (IWSS 2FWBs ) and IWSS with replacement (IWSSr 2FWBs ). The relevancy ratio-support vector machine (RR-SVM) is the filter-fixed wrapper-varied (F f W v ) embedded in the first block of IWSS 2FWBs and IWSSr 2FWBs . The second block-specific F f W v of IWSS 2FWs and IWSSr 2FWBs is designed by relevancy ratio-twin support vector machine (RR-TWSVM). Besides direct relations in four hybrid blocks, cross-permutation-based relations in IWSS 2FWBs and IWSSr 2FWBs are defined in CPQHFSS. By plugging the different kernels into hyperplane-based wrapper methods of IWSS 2FWBs and IWSSr 2FWBs , all possible cross-permutations of two states per IWSS 2FWBs and IWSSr 2FWBs caused to applying different hybrid FSS on each univariate of transient multivariate excursions (TMEs) to extract OTFs. In the third step, transient analysis based on survived OTFs in the presence of cross-validation scenario evaluates the effectiveness rate of the OTFs in achieving high-performance TA.
The CPQHFSS by interlacing the filter-wrapper methods, incremental wrapper mechanisms, and cross-permutation scenario, selects OTFs of TMEs for high-performance TA. According to Fig. 2, each univariates of the transient multivariate trajectories dataset is entered into the filter phase as the first step of CPQHFSS (See Fig. 2, filter funnel). Then, the obtained filter-based results per univariate are used in the wrapper phase of the CPQHFSS, which consists of four hybrid blocks categorized in twin two-FWBs, which are mounted on dual incremental wrapper mechanisms (IWSS and IWSSr). Overall, in the CPQHFSS, four hybrid blocks formed as IWSS 2FWBs (the left rectangle box of Fig. 2) and IWSSr 2FWBs (the right rectangle box of Fig. 2). The direct relations in each block of IWSS 2FWBs or IWSSr 2FWBs be caused that two F f W v -states (totally four F f W v -states). In CPQHFSS, due to plugging elastic and non-elastic kernels into hyperplane-based approaches situated in the wrapper methods, different cross-permutation hybrid FSS can be considered for two filter-wrapper-states of IWSS 2FWBs and IWSSr 2FWBs (See cross-permutation box in Fig. 2). Taking into cognizance the concise explanation of CPQHFSS depicted in Fig. 2, the pseudocode of CPQHFSS is shown in Table 1. As can be seen in Table 1, the main body of CPQHFSS includes filter phase (RR analysis), incremental wrapper mechanisms (IWMs), and cross-permutation function. In the main body of CPQHFSS, symmetric uncertainty values (Line 3) of    [10][11][12][13][14]. Finally, structure arrays are entered into the crosspermutation function to extract final OTFs (fOTFs) per TU (Line 16). After conducting union-intersection operations on subsets of optimal features based on cross-permutation scenario, the union of fOTFs (UfOTFs) is obtained (Line 17). To better understand the details of the pseudocode of CPQHFSS, we elaborate on the triple components of it (filter, IWMs, and cross-permutation) in Sections III-A to III-C. Besides the above-mentioned describing the main body of pseudocode of CPQHFSS, we present the complexity of CPQHFSS for readers at a glance. The complexity of CPQHFSS is related to IWMs accompanied by hyperplanebased learning methods. By analyzing these main functions, we can approximate the complexity of CPQHFSS. In the worst case, the complexity of IWSS and IWSSr is O(n) and O(n 2 ), respectively [24]. Also, the complexity of SVM and TWSVM is O(n 3 ) and O(2×(n/2) 3 ), respectively [25]. Hence, the complexity of SVM IWSS TWSVM is O(max{(n×n 3 ), (n×2×(n/2) 3 )}) and SVM IWSSr TWSVM has O(max{(n 2 ×n 3 ), (n 2 ×2×(n/2) 3 )}) complexity. Since the complexity of the SVM is 4 times larger than of the TWSVM, the complexity of SVM IWSS TWSVM and SVM IWSSr TWSVM will be equal to O(n×n 3 ) and O(n 2 ×n 3 ), respectively. On the other hand, the experiment results of Reference [24] show the fact that the complexity of IWSSr is near to IWSS when the number of variables to be selected is a very small number of wrapper evaluations. Consequently, the CPQHFSS has O(2×n 4 ) complexity.

1) IWSS
The IWSS mechanism [26] is utilized in the first set of twin 2FWBs (IWSS 2FWBs ) in CPQHFSS as IWM. How to navigate in the incremental process of IWSS to select optimal features depends on the results of the embedded filter and wrapper method. As the preliminary step of IWSS, by conducting the filter method on feature set based on relevancy ratio (RR), the features are sorted based on // Permutations of IWSS-based learning models (9 permutations (LP1: LP9)). (6); // Permutations of IWSSr-based learning models (9 permutations (RP1: RP9)). (7) RP1= IWSSr mat (1) ∪ IWSSr mat (2); RP2= IWSSr mat (1) ∪ IWSSr mat (4); RP3= IWSSr mat (1) ∪ IWSSr mat (6) RR values in descending manner. Then, for completing the first increment of IWSS, the feature inserted into the first position of the sorted array (fh1: feature with highest RR) is fed to the classification learner (CL), and then fh1 with the prediction accuracy ( Acc (fh1)) is recorded in the candidate features subset (CFS) based on the TTP. In the next increment, the feature with second-highest RR (fh2) is added to the CFS, and the updated CFS-based learning model reports the Acc (fh1, fh2). If the classification performance of CFS including fh1 and fh2 is higher than the performance of fh1, the third increase (adding fh3) is accompanied by fh1 & fh2; otherwise, fh2 is deleted from CFS, and fh3 is added to the CFS and placed next to fh1. Fig. 3 shows how to select OTFs by the IWSS in the form of a numerical example.

2) IWSSr
In the second set of twin 2FWBs, filter-wrapper blocks are mounted on IWSSr [24] algorithm (IWSSr 2FWBs ) as IWM. In IWSSr, similar to the dependence of the IWSS on filter and wrapper method results, based on sorted RR values of features, in the first increment, fh1 is added to CFS, then CL trained by fh1 and Acc (fh1) is recorded. In the second increment, fh2 is added to the preceding CFS in two modes. In the first mode, fh1 is replaced by fh2 (only fh2 added to CFS) and in the second mode, fh1 and fh2 are added to CL together. Now, in the second increment, Acc (fh1) and Acc (fh1, fh2) are obtained. Fig. 4, shows the process of IWSSr which the third increment starts from node 3 (create node 4 to 6).

B. TWIN TWO FILTER-WRAPPER BLOCKS (2FWBs)
Twin 2FWBs in CPQHFSS refer to applying two sets of 2FWBs in the presence of IWMs which are called IWSS 2FWBs and IWSSr 2FWBs . The 2FWBs include frozen information theory concept (filter-fixed) and unfixed machine learning classifier (wrapper-varied). Filter-related model of 2FWBs including the relevancy ratio (RR) [27]

1) Filter-Fixed methods in 2FWBs
Relevancy ratio (RR): The symmetric uncertainty (SU) is considered as the HB 1-2 -specific filter index in IWSS 2FWBs and IWSSr 2FWBs , which triangulated the entropy, conditional entropy, and mutual information (MI) to measure the relevancy rate between features of transient univariates and class label. The SU index is calculated as: Where k TU i f represents i th features of k th transient univariate, and C is the class label of transient samples. In (1), the entropy H(Z) is defined as: (1) is defined as: (3) is called conditional entropy as follow:

2) Wrapper methods in 2FWBs
(a) Support vector machine (SVM) SVM in [28] introduced as a supervised learning model that draws hyperplane in feature space for classifying binary or multi-class data in the form of the linear SVM (hard margin or soft margin approach) and nonlinear SVM (kernel-based approach). Regardless of the different factors that affected the SVM formula, SVM aims a low structural risk without overfitting data to achieve high accuracy in train-test procedures. For example, achieving such a goal in the presence of data that has a nonlinear decision boundary (e.g., PMU-gathered HDTF), necessitates plugging the kernel trick into SVM computations as follows: (5) is employed for mapping the data from the main space to the new space (high dimensional space) so that in the new space the data are linearly separable. The maximum-margin separating hyperplane in feature space is solved by (6): (b) Twin support vector machine (TWSVM) The standard SVM is formulated based on finding the middle boundary (maximum margin) between two parallel planes with the maximum distance from each of the existing classes. SVM geometry space can be reshaped by cross planes in which each plane could nearest distance to the samples of one class and farthest from the samples of the other class. This idea was raised as the generalized proximal eigenvalue support vector machine (GEPSVM) [28]. In another effort by [29], the spirit of the GEPSVM was kept into a new skeleton (new formulation) was termed TWSVM. In TWSVM, cross planes are obtained by solving the following optimization problems: Where c1, c2, e1, and e2 are vectors with a value of 1 and a proper dimension. By calculating the Lagrangian function for (7) and (8), the Karush-Kuhn-Tucker (KKT) equations are formed. By placing the KKT terms and relations in the Lagrangian function for each of the relations (7) and (8), the dual optimal relations are obtained according to the following relations: Based on the dual optimization problems, the value  and  is obtained via quadratic programming and by placing these values in the KKT relations, the values [w (1) , b (1) ] and [w (2) , b (2) ] related to hyperplanes of the binary-class task are obtained: Finally, the class label of the unseen point n x   is determined from the plane that is close to this point: Based on the above-mentioned principle of TWSVM, the nonlinear classification of HDTF can be considered via kernel-based cross planes [29]: ( , ) 0 ( , ) 0 are obtained by solving the following optimization problem: (2) (2)

C. CROSS-PERMUTATION SCENARIO BASED ON WRAPPER KERNELS
The cross-permutation scenario in CPQHFSS is conducted based on plugged triple kernel into wrapper approaches of twin 2FWBs. The kernels situated in wrapper approaches of IWSS 2FWBs and IWSSr 2FWBs are categorized into two types: 1) non-elastic kernel: linear kernel (LinKer) [31], polynomial kernel (PolKer) [31], and standard Gaussian radial basis function ( S GRBF) [28], and 2) elastic kernels: dynamic time warping in GRBF ( DTW GRBF) [32] and recursive edit distance kernel (REDK) [33]. The definitions per kernel are summarized below: (a) LinKer: LineKer is calculated based on the inner product plus constant c, which is considered as the simplest kernel function: The degree-based variant of LinKer (the d value more than 1) is known as PolKer, which is defined as follow: (c) S GRBF: S GRBF is known as a non-elastic kernel due to linear alignment (point to point) in pattern matching in feature space. The S GRBF for using as ( , ) K x x in (5) and (13) is defined as follows: According to (19), ( , ) K x x can be defined as the DTWbased elastic kernel as follow: (e) REDK: In [33], constructing the kernel based on the aggregation of scores recursively caused that the REDK is introduced as an efficient elastic similarity measure against classical elastic distances. According to (21), if for any pair of trajectory, such following equation is satisfied, the function , :U U R     termed as REDK: , , , is the trajectory with a discrete index varying between 1 and p (or q). Also, ( ) h  is the cost function for edit operation.
After applying different kernels in the hyperplane classifiers of wrapper phase ( S GRBF, DTW  IWSSr in IWSSr 2FWBs ), first, by regarding possible kernel-based permutation between two-state of IWSS 2FWBs (9 permutations), for each permutation of IWSS 2FWBs , the union of the selected preliminary OTFs (pOTFs) per the state of IWSS 2FWBs is recorded (See Fig. 2, left panel (LLP and RLP) of the cross-permutation box). In terms of IWSSr 2FWBs , the mentioned scenario is conducted to record union results per permutation in IWSSr 2FWBs , which is shown in the right panel (LRP and RRP) of the permutation box. Next, based on the cross-manner scenario, each left union is linked with right unions (9 links are established; e.g., 9 black links, 9 blue links, and so on). Then, the intersection of both sides of the link in each colored 9 links (e.g., the intersection of both sides of the first black link) is calculated. After calculating the intersection of both sides of all colored links (e.g., both sides of the second black link to both sides of the ninth black link), the union of sets is recorded (See Fig. 2), union sign in the polygonal black side). Such mechanism (intersection and union) is conducted on other colored 9 links (9 blue links, 9 yellow links, and so on). Finally, the intersection of the obtained sets in polygonal sides is considered as univariate-specific final OTFs (intersection of polygonal black side, polygonal brown side, and so on).

A. TRANSIENT DATASET CONSTRUCTION
As can be seen in Fig. 1, transient dataset construction containing multivariate trajectories is considered as the first step of the proposed framework for FSS-based TSA. In this step, we executed contingency simulation based on the twostep transient dataset generation workflow (TDGW) proposed in [34], which is shown in Fig. 5. In TDGW, first, transient responses are extracted from output channels of basic features (OCBF-X; X indicated basic features namely bus voltages (VOLT), voltage phase angle (VANGLE), machine active power (PELEC), machine reactive power (QELEC), and reactive power consumption (QLOAD)). The application program interface (API) functions of the SIEMENS power system simulator for engineering (PSS/E) [35] in the form of Python-based automation file are used for exerting contingency simulation on the New England test system-New York power system (NETS-NYPS) (See Fig. 6) [36]. The generated transient samples are derived by substation outages, generator outages, and line outages where the fault duration time is set to 0.23 seconds (the time step is 0.0167 seconds). Also, the fault clearing time is set after the end of fault duration time. As an important uncertainty parameter in generating transient samples, the convert load (CONL) API is added to the PSS/E-Python automation file to regard different load characteristics which the percent of active and reactive power load to be converted to the constant current and constant admittance load characteristics [34,36]. In the second step of TDGW, a set of Matlab-based commands cause adding factors to OCBF-X which leads to the defined 28 univariates trajectory features listed in Table 2 Fig. 7.

B. FINAL OTFs (fOTFs) SET PER TRANSIENT UNIVARIATE (TU)
Extracting the transient univariate-specific final OTFs set called fOTFs TU# (namely fOTFs TU1 to fOTFs TU28 ) by applying CPQHFSS on each univariate (TU 1 to TU 28 ) of transient multivariate trajectory dataset (TMTD) is elaborated in this section. According of Fig. 2, in the first step of CPQHFSS, each univariate of TMTD is entered to the filter-fixed phase of twin 2FWBs. According to (1), RR of point features per TU of TMTD is calculated based on SU measure. According to was what mentioned about the filter method in Section 2.B.1, for example, the obtained RR of pfs of TU 6 ( pfs TU 6 ; 9 cycles) based on SU is shown in Fig. 8. As can be seen in Fig. 8, based on SU values of pfs TU 6 , {pf8}is ranked as a high SU feature, and {pf5}is considered as the low SU pfs. Next, by sorting pfs TU 6 based on SU values in descending manner ( Spfs TU 6 ), the order in which the pfs TU 6 enter the wrapper phase in twin 2FWBs is specified (See Table 3; sixth row). For more clarity, Table 3 show the Spfs TU 1 to Spfs TU 28 based on SU values.   After conducting the wrapper phase of CPQHFSS in the form IWSS 2FWBs and IWSSr 2FWBs , the obtained results are shown in Table 4 to Table 7. Table 4 show the obtained preliminary OTFs (pOTFs) of TU x ( pOTFs TU x ) based on IWSS-SVM regarding SVM-specific kernels in the form of  Table 5 show pOTFs TU x based on IWSS-TWSVM regarding TWSVM-specific kernels in the form of pOTFs IWSS-TWSVM LinKer , pOTFs IWSS-TWSVM PolKer , and pOTFs IWSS-TWSVM SGRBF columns. In terms of IWSSr-based results, Table 6 and Table 7, show the pOTFs TU x based on IWSSr-SVM ( pOTFs IWSSr-SVM SGRBF , pOTFs IWSSr-SVM DTW-GRBF , pOTFs IWSSr-SVM REDK ) and IWSSr-TWSVM ( pOTFs IWSSr-TWSVM LinKer , pOTFs IWSSr-TWSVM PolKer , pOTFs IWSSr-TWSVM SGRBF ) respectively. For example, after applying the wrapper phase of CPQHFSS on Spfs TU 3 , the IWSS-based pOTFs TU 3 (See third row of Table 4 and Table 5) and IWSSr-based pOTFs TU 3 (See third row of Table 6 and Table 7 Fig. 9 show how IWSS-SVM SGRBF -based pOTFs TU 3 are selected in IWSS iterations. Also, the Acc metric (23) measured the performance of SVM IWSS TWSVM and SVM IWSSr TWSVM . Furthermore, the finetuning on learning parameters (C and σ) in each iteration of IWMs is considered for TTP which for the SVM and TWSVM classifiers and their plugged kernels as (24). In the case of IWSS-SVM SGRBF (See Fig. 9), in each iteration, the maximum value of the Acc (retrieved by optimal pair of learning parameters) is recorded. As can be seen in Fig. 9, the pOTFs TU 3 based on IWSS-SVM SGRBF is {pf1, pf5, pf6}, which Acc variations related to optimal iteration (node 5: green-face) is depicted in 3-D plot.

SVM
in the presence of UfOTFs TU1:TU28 is measured by classification metrics which are shown in Table 9.
By regarding the above-mentioned points in TTP for TSP, the value of triple indices in TSP (Acc, TPR, and TNR) per fold is shown in Table 10. Based on setting the different value for learning parameters, the fluctuations of the Acc index per fold is recorded, and these values are inserted into Table 10. For example, the Acc variation of fold 2 , fold 5 , fold 7 , and fold 10 are depicted in Fig. 10. Furthermore, the    average of the Acc, TPR, and TNR is regarded in Table 10.
The results of  Table 11), indicating a low DLT to address corrective control actions.

D. Comparison of Experimental Methods: CPQHFSS Vs. Vertically Unilateral FSSs (VUFSSs) and MHFSS
For further evaluation on the efficacy of CPQHFSS-based UfOTFs TU1:TU28 in TSP, the comparison between with three VUFSSs OTFs (3VUFSSs OTFs ) is considered in this section. The 3VUFSSs includes mRMR [16,17], ReliefF [18] and fast correlation-based filter (FCBF) [19]. Also, we compared the CPQHFSS with the bi-mode hybrid feature selection scheme (BMHFSS) [23] and partial-injective trilateral hybrid (filter-wrapper) scheme called PITHS [34] as the MHFSS. By applying the 28-variate time-series data to 3VUFSSs and 2MHFSS, the 3VUFSSs-based OTFs and 2MHFSS-based OTFs are extracted. Next, the 3VUFSSs OTFs and 2MHFSS OTFs are entered into the  s GRBF SVM learning model in the same learning conditions expressed in subsection C of section 3 (10-fold crossvalidation technique and the fine-tuning of the learning parameters).
As can be seen in Table 12, CPQHFSS OTFs have better performance in TSP than 3VUFSSs OTFs and 2MHFSS OTFs (ignoring only 0.25% less than TPR than PITHS). The obtained results of Table 12 show that CPQHFSS in the presence of 48-cycles of 28-variate trajectory (See Table 8) has better performance (Acc, TPR, and TNR) than mRMR (9-cycle of 4-variate trajectory features), FCBF, ReliefF, and BMHFSS (9-cycle of 3-variate trajectory features [23]. In terms of the PITHS, method, the selected cycles via CPQHFSS leading to high Acc and TPR than PITHS (24cycle of 18-variate trajectory features [34] Table 13). Generally, the CPQHFSS DLT causes the system operator to have enough time to take corrective actions. The detail information about CPQHFSS DLT , 3VUFSSs DLT , and 2MHFSS DLT are shown in Table 11 and Table 13.

V. CONCLUSION
The presence of tightly correlated indices in the TSA problem, namely, accuracy and time of transient prediction, extracting optimal transient features (OTFs) from HDTF space via multifaceted FSS to meet low DLT and high DLA, was defined as the main agenda of this paper. To this end, we proposed cross-permutation-based quad-hybrid FSS (CPQHFSS) designed by integrating four filterwrapper blocks (FWBs) in the form of twin two-FWBs mounted on dual incremental wrapper mechanisms (IWMs) called IWSS 2FWBs and IWSSr 2FWBs . Regarding filter-fixed and wrapper-varied approaches (F f W v ) embedded into IWMs 2FWBs of CPQHFSS, the relevancy ratio-support vector machine (RR-SVM) raised as F f W v in the first block of IWSS 2FWBs and IWSSr 2FWBs , and relevancy ratio-twin support vector machine (RR-TWSVM) situated in the second block of IWSS 2FWBs and IWSSr 2FWBs . Furthermore, the nonlinear nature of transient feature space makes the plugging elastic and non-elastic kernels into the wrapper phase an integral part of CPQHFSS.