Evolutionary Large-Scale Multiobjective Optimization for Ratio Error Estimation of Voltage Transformers

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEVC.2020.2967501, IEEE Transactions on Evolutionary Computation IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. , NO. , MONTH YEAR 1 Evolutionary Large-Scale Multiobjective Optimization for Ratio Error Estimation of Voltage Transformers Cheng He IEEE Member, Ran Cheng IEEE Member, Chuanji Zhang, Ye Tian, Qin Chen, and Xin Yao IEEE Fellow Abstract—Ratio error estimation of the voltage transformers plays an important role in modern power delivery systems. Existing ratio error estimation methods mainly focus on periodical calibration but ignore the time-varying property. Consequently, it is difficult to efficiently estimate the state of the voltage transformers in real time. To address this issue, we formulate a time-varying ratio error estimation (TREE) problem into a largescale multiobjective optimization problem, where the multiple objectives and inequality constraints are formulated by statistical and physical rules extracted from the power delivery systems. Furthermore, a set of TREE problems from different substations are systematically formulated into a benchmark test suite for characterizing their different properties. The formulation of these TREE problems not only transfers an expensive ratio error estimation task to a relatively cheaper optimization problem, but also promotes the research in large-scale multiobjective optimization by providing a real-world benchmark test suite with complex variable interactions and correlations to different objectives. To the best of our knowledge, this is the first time to formulate a real-world problem into a benchmark test suite for large-scale multiobjective optimization, and it is also the first work proposing to solve TREE problems via evolutionary multiobjective optimization. Index Terms—Time-varying ratio error estimation, voltage transformer, large-scale multiobjective optimization, benchmark test suite, inequality constraint voltage value. They can be used to measure the voltage values and as transforming devices. By the end of 2011, more than 120, 000 VTs have been installed in the high voltage grid in the State Grid of China. According to the statistics in Guangzhou, China, the failure rate of VT has reached 0.8% per year during the last decade [1]. The ratio error drift fault is one of the main failures for VT, which will mislead the downstream applications due to the wrong data in the grid, such as those in the relaying, metering, and controlling tasks [2]. For example, the entire electricity consumption of China was 6.3077×1013 kilowatt-hour during 2017 [3], and a metering error of 6×108 kilowatt-hour electricity would occur if the ratio errors of the VTs were 0.001%. Hence, the estimation of the timevarying ratio error (TREE) of VTs plays a crucial role in power delivery systems [4]. Primary Side ‫ܣ‬଴ , ‫ܤ‬଴ , ‫ܥ‬଴ VT-1 Measured Voltage ‫ܣ‬ଵ , ‫ܤ‬ଵ , ‫ܥ‬ଵ C. He, R. Cheng, and Xin Yao are with the University Key Laboratory of Evolving Intelligent Systems of Guangdong Province, Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China. X. Yao is and also with CERCIA, School of Computer Science, University of Birmingham, Birmingham B15 2TT, U.K. E-mail: chenghehust@gmail.com, ranchengcn@gmail.com, xiny@sustech.edu.cn. (Corresponding author: Ran Cheng) Y. Tian is with the Institutes of Physical Science and Information Technology, Anhui University, Hefei 230601, China. Email: field910921@gmail.com. C. Zhang and Q. Chen are with the State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China. E-mail: chuanjizhang@hust.edu.cn, wfr101@163.com. This work was supported by the National Natural Science Foundation of China (No. 61903178 and 61906081), the Program for Guangdong Introducing Innovative and Entrepreneurial Teams grant (No. 2017ZT07X386), the Shenzhen Peacock Plan grant (No. KQTD2016112514355531), and the Program for University Key Laboratory of Guangdong Province grant (No. 2017KSYS008). 1 The proposed method in this work applies to general voltage transformers, including the capacitor ones (i.e., CVTs). VT-2 Measured Voltage ‫ܣ‬ଶ , ‫ܤ‬ଶ , ‫ܥ‬ଶ Secondary Side ܽ଴ , ܾ଴ , ܿ଴ I. I NTRODUCTION In the area of industrial power delivery, a voltage transformer (VT)1 is an elementary device widely used in the substation of the power delivery system for measuring the Power Transformer VT-3 Measured Voltage ܽଵ , ܾଵ , ܿଵ VT-4 Measured Voltage ܽଶ , ܾଶ , ܿଶ Fig. 1. The diagram of the VTs in a representative power delivery system, where A0 , B0 , C0 and a0 , b0 , c0 are the true primary and secondary threephase voltage values, respectively. Note that, the primary side and the secondary side are connected by the power transformer. The ratio error (RE) of a VT can be mathematically formulated as [5]: Vm − Vt , (1) RE = Vt where Vm is the primary voltage value measured by the VT, and Vt (always unknown) is the corresponding true primary voltage value. Fig. 1 presents the diagram of a representative 110kV substation with four sets of three-phase VTs in the power delivery system. To be specific, VT-1 and VT-2 measure the same three-phase primary voltage values A0 , B0 , C0 in 1089-778X (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEVC.2020.2967501, IEEE Transactions on Evolutionary Computation IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. , NO. , MONTH YEAR parallel, while VT-3 and VT-4 measure the same three-phase secondary voltage values a0 , b0 , c0 in parallel. Thus, the four VTs are interacting by the true voltage values. For instance, VT-1 and VT-2 measure the same primary voltage value, while VT-3 and VT-4 measure the same secondary voltage value. Since the primary side voltage value is proportional to the secondary side voltage value, VT-1 and VT-2 are related to VT3 and VT-4. Meanwhile, we denote the primary and secondary voltage values obtained by the ith set of VTs as Ai , Bi , Ci and ai , bi , ci , respectively. In this substation, the RE can be calculated according to one of the primary voltage values, e.g., (Ai − A0 )/A0 , (Bi − B0 )/B0 , or (Ci − C0 )/C0 . To obtain the REs of VTs, there are mainly two kinds of methods, i.e., calibration methods and data-driven estimation methods. The calibration is conducted by de-energizing the power and compare the uncalibrated VT with the standard one manually [6]. Typical methods include the off-line and on-site calibrations [7], [8], [9]. By contrast, the data-driven estimation methods build specific models of the VTs, and estimate the REs based on the constraints derived from the electrical relationship and data collected from VTs, e.g., the adaptive detection method [10] and the line parameter estimation method [11]. However, both calibration and data-driven estimation methods suffer from some practical deficiencies. As for the calibration methods, despite that the REs are accurate under the standard work condition. They suffer from the expensive economy/time consumption and have to be used periodically [6]. For example, most substations conduct calibration every four years or even longer, and the calibration will cause economic losses (e.g., over 0.3 million CNY) and take a long time (e.g., more than one day) [12]. As for the datadriven estimation methods, in spite of the cheaper cost, their accuracy and versatility are unsatisfying due to the limitations of the VT-specific models [13], [14]. To alleviate the issues in the existing calibration and datadriven methods, in this work, we propose to formulate the TREE problem into a multiobjective optimization problem (MOP) by considering the difference between the true voltage values and those measured by the VTs. As is done in calibration, the optimization target of the MOP is also to estimate the true voltage values. Compared to the calibration and the datadriven error estimation methods, the proposed method has the major advantages of lower cost and better versatility. The premise of the formulation is that most of the VTs in a substation should work in a good condition and only several VTs suffer from relatively high REs. Consequently, the combination of many VTs for formulating an optimization problem will lead to a solution that can reveal the true REs of most VTs, and statistically, the variance of the REs can be reduced due to the average of many uniformly distributed solutions. As for the versatility, three major types of substations are included in our proposed benchmark problems, and the formulation of these benchmark problems does not require any specific VT models/types. Thus, our proposed method can be applicable to different VTs, which makes it more practical for real-world applications. Besides, in our formulated benchmarks, the true voltage values over time are the optimization target, and the REs of different VTs can be calculated according to (1). In 2 this way, different behaviors of different VTs can also be captured without the requirement of similar behaviors. The on-site calibration results of VTs in a real-world substation of the power delivery system have verified the validity of our proposed evolutionary multiobjective optimization method (refer to Section IV-D). Moreover, by collecting three different types of data (i.e., the primary voltage values, the secondary voltage values, and the phase angle values) measured by VTs from different substations, we systematically formulate six TREE problems into a benchmark test suite for large-scale multiobjective optimization. In this way, an expensive error estimation task is transferred to a relatively cheaper optimization problem. To the best of our knowledge, this is the first test suite for large-scale multiobjective optimization which is formulated by using a real-world application problem, and it is also the first work proposing to solve TREE problems via evolutionary multiobjective optimization. To be specific, the contributions of this work are as follows: 1) We formulate the economically expensive and timeconsuming RE estimation task for VTs into a relatively cheaper multiobjective optimization problem. The proposed method can be used to provide real-time estimations as early warnings of potential faulty VTs. 2) We propose a large-scale multiobjective optimization test suite constructed by different types of measured voltage/phasor data from different substations, which involve complex variable interactions, correlations to different objectives, and scalable number of decision variables. The proposed test suite would promote the research in large-scale multiobjective optimization by providing a real-world benchmark test suite. 3) We adopt several evolutionary multiobjective optimization methods to analyze the formulated problems, which helps better understand the RE estimation task by providing the function separability and variable interactions of each problem. 4) We use eight representative multiobjective evolutionary algorithms (MOEAs) to optimize 30 test instances selected from our proposed test suite. Furthermore, we validate the proposed formulation method for the TREE problems and the corresponding performance of evolutionary multiobjective optimization using a real-world power delivery system. The rest of this paper is organized as follows. In Section II, we briefly review some background of RE estimation approaches and evolutionary multiobjective optimization. The details of the formulations and analyses of the real-world TREE problems are given in Section III. Evolutionary optimization of the proposed test suite by using eight representative MOEAs is presented in Section IV. Conclusions are drawn in Section V. II. BACKGROUND A. Ratio Error Estimation Approaches The periodical calibration techniques and the data-driven RE estimation methods are two main kinds of RE estimation 1089-778X (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEVC.2020.2967501, IEEE Transactions on Evolutionary Computation IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. , NO. , MONTH YEAR approaches in modern power delivery systems, among which the on-site calibration technique is strongly recommended by IEEE [6]. The detailed procedures of the on-site calibration technique are given as follows: first, the substation should be de-energized by disconnecting the uncalibrated VTs from the high-voltage line; second, a high voltage power source with a low voltage generator and a step up transformer (SUT) should be set up; third, the uncalibrated VTs and the standard VT should be connected to the SUT in parallel; finally, the RE can be obtained by a comparator after turning on the generator. The visualized mechanism of the on-site calibration is displayed in Fig. 2, where Vm is measured by the uncalibrated VT and Vt is measured by the standard VT (refer to (1), where the standard VT measured voltage is assumed to be the ground truth). S Voltage Generator SUT Standard VT (ܸ௧ ) Uncalibrated VT (ܸ௠ ) Load Load ȟܸ Comparator Fig. 2. The mechanism of the on-site calibration technique, where the standard VT and comparator are two additional devices for obtaining the reference voltage value and the difference between Vt and Vm , respectively. Despite the wide adoption of the on-site calibration technique, it suffers from several practical problems. First, the on-site calibration technique requires that the standard VT should be far more accurate than the uncalibrated VT. For instance, the RE of the standard VT shall not exceed 0.075% for calibrating a 0.3 class VT. Second, the on-site calibration techniques can only obtain the static RE but fail to characterize the time-varying property. As a result, this calibration should be repeated many times to obtain a discontinuous time-varying RE. Finally, there are usually dozens of VTs in a substation, and the entire operating time/economic cost for the on-site calibration would be unbearable. The data-driven RE estimation methods also require some pre-calibrated information as a reference, while they estimate the RE based on the constraints derived from the electrical relationship and data collected from VTs [12]. For example, the adaptive detection method estimates the true primary voltage based on the transmission line equivalent circuit model, where the adopted model is assumed to be accurate [10]. Similarly, a line parameter estimation method has been proposed to estimate the true primary voltage with some pre-calibrated data [11]. These data-driven error estimation methods aim to address the drawbacks of the periodical calibration techniques, but they only consider a single objective about the equivalent circuit model and require additional pre-calibrated information, resulting in the unsatisfying estimation accuracy. 3 Moreover, all these approaches have only been validated via simulations. B. Evolutionary Multiobjective Optimization The formulation of multiobjective optimization problems (MOPs) with inequality constraints is mathematically presented as follows: Minimize F (x) = (f1 (x), f2 (x), . . . , fM (x)) (2) subject to gi (x) ≤ 0, i ∈ {1, 2, . . . , J}, and x ∈ RD , where M is the number of objectives, D is the number of decision variables, J is the number of constraints, RD is the feasible space, and g is the constraint function [15]. The concept of Pareto efficiency, also known as Pareto optimality, in the area of economics is introduced to characterize the optimal result of the MOPs, which indicates the situation that it is impossible to improve one decision variable without degenerating others in multiobjective optimization [16]. Meanwhile, it is also used to define the so-called Pareto dominance relationship to distinguish the qualities of two solutions of an MOP [17]. Solution x1 is said to Pareto dominate another solution x2 (denoted as x1 ≺x2 ) iff the following equations are satisfied [18], [19], [20]: { ∀i ∈ 1, 2, . . . , M, fi (x1 ) ≤ fi (x2 ), (3) ∃j ∈ 1, 2, . . . , M, fj (x1 ) < fj (x2 ). If x ∈ RD is not Pareto dominated by any other solutions in RD , x is called the Pareto optimal solution, {x∈RD | ∄ y ∈RD , y≺x} is called the Pareto optimal set (PS), and {F (x) | x∈P S} is called the Pareto optimal front (PF). To solve MOPs, many MOEAs have been proposed since 1984 [21], which can be roughly classified into three categories [22], i.e., the dominance based MOEAs, the decomposition based MOEAs, and the performance indicator based MOEAs [23]. Typical MOEAs include the elitist non-dominated sorting genetic algorithm (NSGA-II) [24], the improved strength Pareto EA (SPEA2) [25], the decomposition-based MOEAs (MOEA/D) [26], the indicator based EA (IBEA) [27], and the S-metric selection based MOEA (SMS-EMOA) [28], etc. 1) Multiobjective Optimization Benchmarks: The earliest multiobjective benchmark problems can date back to 1984 when Schaffer proposed two single-variable test problems [29]. Later in 1990, Kursawe proposed the KUR test problems with scalable decision variables [30]. Then in the following years, many scalable test suites with different characters were proposed, e.g., the Zitzler-Deb-Thiele test suite (ZDT) [31], the unconstrained function problem test suite (UF) [32], the Deb-Thiele-Laumanns-Zitzler test suite (DTLZ) [33], the DTLZ with hardly-dominated boundaries (mDTLZ) [34], the walking fish group toolkit (WFG) [35], the Paretobox problems [36], the multi-line distance minimization problem (MLDMP) [37], and the large-scale multiobjective and many-objective test suite [38]. Most recently, a generator for multiobjective benchmark problems with difficult-toapproximate Pareto front boundaries has been proposed [39]. In these test suites, some specific properties of MOPs, e.g., 1089-778X (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEVC.2020.2967501, IEEE Transactions on Evolutionary Computation IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. , NO. , MONTH YEAR the variable linkages, the shapes of the Pareto fronts or Pareto sets, and the fitness landscapes, are well considered. Those existing benchmarks have characterized some properties of MOPs well in many perspectives, e.g., separability, modality, geometry, bias, many-to-one mappings, etc [35]. Usually, these scalable benchmark problems follow the generic multiplication/addition-based formulations as f (x) = f (x) = (1 + g(x)) · h(x), g(x) + h(x), (4) (5) where g is the landscape function related to the convergence, h is the shape function characterizing the shape of the PF. Meanwhile, the PS is related to both the g function and the h function [33], [40]. Sequentially, these test suites are scalable in many aspects by using these mathematical and systematical construction strategies. 2) Large-Scale Multiobjective Optimization: The largescale multiobjective optimization problems (LSMOPs) often refer to the MOPs with large-scale decision variables [41]. As the number of decision variables increases, the volume of search space and the complexity of fitness landscape grow exponentially, which poses stiff challenges to existing MOEAs [42]. Furthermore, the interactions between the decision variables, a.k.a. the nonseparability, make LSMOPs even more challenging [43]. Note that, there are many different techniques, e.g., perturbation [44], interaction adaption [45], modeling [46], or randomization [47], can be used to detect the interdependence between the pairwise decision variables. Specifically, if an objective function contains several groups of nonseparable decision variables, it is known as partially separable. Accordingly, the objective function can be decomposed into different subproblems if it is partially separable, and then the subproblems can be optimized independently in a divide-and-conquer manner [48]. Considering the partial separability and variable interaction properties in the objective functions, some researchers have also attempted to analyze the variable interactions in MOPs [49]. Recently, there are also some other works proposed for solving LSMOPs, e.g., the third-generation cooperative coevolutionary differential evolution algorithm (CCGDE3) [50], the weighted optimization framework (WOF) [42], and the largescale multiobjective optimization framework based on problem reformulation (LSMOF) [51]. III. M ULTIOBJECTIVE F ORMULATION In this work, we formulate the TREE problems into six LSMOPs with inequality constraints, termed TREE1 to TREE6. To be specific, these TREE problems can be classified into three types according to the types of involved data, i.e., data with primary voltage values only (Type 1), data with both primary and secondary voltage values (Type 2), and data with both voltage and phase angle values (Type 3). As summarized in Table I, TREE1, TREE2, and TREE3 include two objectives and three constraints; TREE4 and TREE5 include two objectives and four constraints; TREE6 includes three objectives and six constraints. To better visualize the results of each optimization problem, the primary/secondary 4 voltages values or the phase angle values measured by the uncalibrated VTs are presented in Fig. 3, where each subfigure displays one type of measured data. In the following formulations, three symbols, i.e., K, P , and T , are used to formulate the TREE problems, where the detailed settings are summarized in Table I. • K denotes the total number of the primary and/or secondary voltage phases; • P denotes the number of VT sets involved in the TREE problem; • T denotes the length of the measured sequential data. The value of K is 3 in TREE1 and TREE2, 6 in TREE3 to TREE5, and 12 in TREE6, respectively. Meanwhile, the value of P is 2 in TREE3, TREE4, and TREE6, 3 in TREE1 and TREE5, and 12 in TREE2, respectively. Note that M , J, and K are specified parameters determined by the topology structure of the VTs in the substation, while T is determined by how many samples have been collected and controls the expected accuracy level of the estimated voltage value. TABLE I S ETTINGS OF M , J , K, P , T , AND THE T YPE OF I NVOLVED DATA FOR E ACH TREE P ROBLEM , W HERE M AND J D ENOTE THE N UMBERS OF O BJECTIVES AND C ONSTRAINTS , R ESPECTIVELY. Problem M J K P T TREE1 TREE2 TREE3 TREE4 TREE5 TREE6 2 2 2 2 2 3 3 3 3 4 4 6 3 3 6 6 6 12 3 12 2 2 3 2 14918 21950 8000 45156 52029 8000 Data Type Type Type Type Type Type Type 1 1 1 2 2 3 In our formulation2 , the decision variables of the TREE problems are given as x = (x1,1 , · · · , x1,T , · · · , xK,1 , · · · , xK,T ), (6) where xi,j denotes the true ith phase voltage at time j (refer to Vt in (1)), and the number of decision variables is D=K·T . Similarly, the data collected from the ith set of VTs can be formulated as(7). di = (di1,1 , · · · , di1,T , · · · , diK,1 , · · · , diK,T ), (7) where i ∈ [1, 2, · · · , P ] and dp,q denotes the pth measured data at time q. Then, the REs of these VTs can be calculated according to (8) on the basis of (1). ei =( di1,1 −x1,1 ,··· x1,1 diK,1 −xK,1 ,··· xK,1 , di1,T −x1,T x1,T , diK,T −xK,T xK,T ,··· , (8) ). For simplicity, the REs are denoted as ei = (ei1,1 , · · · , ei1,T , · · · , eiK,1 , · · · , eiK,T ). (9) Moreover, the variations of REs over time is ∆ei = (ei1,2 − ei1,1 , · · · , ei1,T − ei1,T −1 , · · · , eiK,2 − eiK,1 , · · · , eiK,T − (10) eiK,T −1 ). 2 In this paper, | ∗ |denotes the absolute value of ∗, ∗j denote the complex number of ∗, and std(∗) denotes the standard deviation of ∗. 1089-778X (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEVC.2020.2967501, IEEE Transactions on Evolutionary Computation IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. , NO. , MONTH YEAR 5 Fig. 3. The primary/secondary voltage (termed PV and SV respectively) values or the phase angle values of TREE1 to TREE6 problems respectively, which are measured by the real-world uncalibrated VTs. Note that the dimension number also denotes the sample index of the measurement. For simplicity, the RE variation over time is denoted as ∆ei = (∆ei1,1 , · · · , ∆ei1,T , · · · , ∆eiK,2 , · · · , ∆eiK,T ). (11) For TREE6, the phase angle information measured from the primary and secondary sides of the power transformer is denoted as: Φ = (∆ϕ12 , ∆ϕ23 , ∆ϕ13 ) (12) The first objective is the total time-varying REs of all the uncalibrated VTs, which reveals the matching degree of the true voltage values and the measured ones. Note that the timevarying REs over the VTs are meant to be minimized, and the detailed formulation of the first objective is f1 (x, e1 , · · · , eP ) = P K ∑ T ∑ ∑ eik,j , (16) j=1 k=1 i=1 with ∆ϕij = [(xi,1 − xi+3,1 ) − (xj,1 − xj+3,1 ), · · · , (xi,T − xi+3,T ) − (xj,T − xj+3,T )]. (13) Besides, the complex values of the measured voltage and phase angle can be calculated by ci,j = xi,j · exp (xi+⌊K/2⌋,j · 1j), (14) ci = (ci,1 , ci,2 , · · · , ci,T ). (15) where ei is given in (9). The second objective is the sum of the RE variation over time, which reveals the time-varying relationship between the true voltage values and the measured ones. Note that the variance of the RE variations for different VTs is meant to be minimized, and the detailed formulation of the second objective is and thus In the following, we demonstrate the details of the objective functions, constraints, and problem analysis, respectively. A. Objective Functions Three objective functions of the TREE problems are formulated based on the relationship between the true voltage values and those measured by the VTs. For each objective function, a smaller objective value indicates a better estimation accuracy. f2 (x, ∆e1 , · · · , ∆eP ) = P √ ∑ std((∆ei1,1 , · · · , ∆eiL,T )), i=1 (17) where ∆ei is given in (11), and L is K for TREE1 to TREE5 and ⌊K/2⌋ for TREE6 respectively. In contrast to the first two objectives which consider the voltage values only, the third objective reveals the phase angle relationship among the true voltage values, the measured phase angle values, and the measured voltage values. Note that the variance of the phase angle RE variations for different VTs is 1089-778X (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEVC.2020.2967501, IEEE Transactions on Evolutionary Computation IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. , NO. , MONTH YEAR meant to be minimized, and the detailed formulation of this objective is 6 where i ∈ [1, 2, 3] and j ∈ [1, 2 · · · , T ]. The corresponding constraint can be derived as T −1 ∑ P √ ∑ f3 (x, ∆e , · · · , ∆e ) = std((∆ei⌊K/2⌋,1 , · · · , ∆eiK,T )). 1 P j=1 i=1 (18) Despite that the formulations of the second and third objectives are similar, we do not merge them due to the existence of different types of data. K ∑ (τi − C1 ) < 0. (19) i=1 Furthermore, since the primary side voltage value is proportional to the secondary side voltage value, all the proportions for three-phase voltages should be the same. Thus, we can derive the second constraint from its fluctuation by K−1 ∑ [std(τ2 − τ1 , · · · , τK − τK−1 ) − C2 ] < 0. (20) i=1 The detailed formulation of VUF is given as τ = (τ1 , τ2 , · · · , τK ), (21) where τi = µj = max j∈[1,2,··· ,T ] 1 3 3 ∑ |xi,j − µj |/µj , (22) xi,j . (24) T −1 ∑ j=1 [ ] std(tj ) − C4 < 0, (25) where tj = ( x1,j · x4,j x2,j · x6,j x3,j · x4,j , , ). x3,j · x2,j x4,j · x3,j x6,j · x1,j (26) 3) Phase constraint: VT is an inductive sensor in the power delivery system, and the primary voltage phasors will be lagged after being transformed to the secondary side. Besides, the phase angle difference between every two phasors mainly depends on the equivalent impedance of the power transformer and the load current through it. The basic assumption of the phase constraint is that the difference between threephase angles in a power transformer system is held constant. Therefore, the first phase constraint indicates the tolerance of phase angle imbalance, and its formulation is std(Φ) − C5 < 0, (27) where Φ is given in (12). Furthermore, we can obtain the following matrix according to the phase and voltage balance in the power delivery system.   1    4π 1 exp ( 2π c c4 IA Ia 3 j) exp ( 3 j) 1  IB Ib  = × 1 exp ( 4π j) exp ( 2π j)   c2 c5 , 3 3 3 c3 c 6 IC Ic 1 1 1 (28) where ci is given in (15). Then we can obtain ψ 1 = IB/IA = (ψ1,1 , ψ1,2 , · · · , ψ1,T ), ψ 2 = Ib/Ia = (ψ2,1 , ψ2,2 , · · · , ψ2,T ). (29) The variation of ψ over time can be derived as ∆ψ 1 = (ψ1,2 − ψ1,1 , · · · , ψ1,T − ψ1,T −1 ), ∆ψ 2 = (ψ2,2 − ψ2,1 , · · · , ψ2,T − ψ2,T −1 ). (30) Thus, the final constraint, which reveals the tolerance of phase angle variation, can be derived as max {std(∆ψ 1 ), std(∆ψ 2 )} − C6 < 0. (31) i=1 2) Series constraints: As for the series constraints, they mainly describe the time-varying relationships of the same decision variable. The basic assumption is that the decision variable varies slowly and continuously rather than abruptly, and the variation of decision vector x at time j is defined as ∆xj = ( ] std(∆xj ) − C3 < 0. Furthermore, another constraint for the primary and secondary voltages (i.e., TREE3 to TREE6) is derived. Note that this constraint reveals the imbalance between the primary and secondary sides, and its formulation is B. Constraints In a power delivery system, VTs in the same substation are connected through conductors, and the measurement of voltage and phase angle values is a long-term procedure. Hence, three main constraints, i.e, the topology constraint, the time series constraint, and the phase constraint, can be formulated for restricting the feasible area of the problem. 1) Topology constraints: Topology constraints are derived from the three-phase balance principle, where the true threephase primary voltages are kept in balance if the grid is under control [52]. This balance is described by a factor, termed voltage unbalance factor (VUF), which indicates the balance between three-phase impedance. Since the three-phase impedance is kept in balance and varies slowly (especially in the high voltage grid) [53], the VUF value and its fluctuation can be restricted in a range as [ x1,j+1 −x1,j x2,j+1 −x2,j xK,j+1 −xi,j , ,··· , ), x1,j x2,j xK,j (23) C. Problem Analysis Here, we investigate the characters of the TREE problems by using three different methods, i.e., the differential grouping method in [48] and the variable interaction analysis methods in MOEA/DVA [44] and LMEA [49] respectively. This investigation aims to reveal the relationships among the decision variables, the objectives, and the constraints in each TREE problem. In these analyses, T is set to 200 for each problem, 1089-778X (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEVC.2020.2967501, IEEE Transactions on Evolutionary Computation IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. , NO. , MONTH YEAR 7 TABLE II T HE D ECISION VARIABLE I NTERACTION IN A SSOCIATION WITH D IFFERENT O BJECTIVE AND C ONSTRAINT F UNCTIONS . Problem Part f1 f2 f3 g1 g2 g3 g4 g5 g6 TREE1 Part 1 Part 2 0 600 200:200:200 0 – – 600 0 600 0 600 0 – – – – – – TREE2 Part 1 Part 2 0 1200 400:400:400 0 – – 1200 0 1200 0 1200 0 – – – – – – TREE3 Part 1 Part 2 0 600 200:200:200 0 – – 600 0 600 0 600 0 – – – – – – TREE4 Part 1 Part 2 0 1200 400:400:400 0 – – 1200 0 1200 0 1200 0 1200 0 – – – – TREE5 Part 1 Part 2 0 1200 400:400:400 0 – – 1200 0 1200 0 1200 0 1200 0 – – – – TREE6 Part 1 Part 2 0 1200 300:150:150 600 300:150:150 600 1200 0 1200 0 1200 0 1200 0 600 600 600:600 0 ‘Part 1’ and ‘Part 2’ present the numbers of interacting decision variables in the same group and the number of independent decision variables, respectively. and the number of decision variables is 600, 600, 1200, 1200, 1200, and 2400 for TREE1 to TREE6, respectively. The numbers of interacting and independent decision variables associated with each objective/constraint of each TREE problem are given in Table II, while the numbers of different types of decision variables obtained by MOEA/DVA and LMEA are given in Table III. It can be observed from Table II that the first objectives of TREE1 to TREE5 are fully separable, while the second ones can be partially divided into three equal-sized groups without any separable variables. Additionally, all the constraints of TREE1 to TREE5, except for TREE6, are fully nonseparable. To sum up, the proposed test suite involves both separable and nonseparable parts. As summarized in Table III, in terms of the relationships between the decision variables and the objectives, the analysis results obtained by both of the methods in MOEA/DVA and LMEA indicate that almost all the decision variables of TREE1 to TREE5 are position variables, and around half of the decision variables of TREE6 are distance variables. However, the results are significantly different when applying the methods in MOEA/DVA and LMEA to analyze the relationships between the decision variables and the constraints. This can be attributed to the fact that the two methods were proposed for analyzing the relationships between the decision variables and the objectives, but may not work properly when dealing with constraints. The decision variable analysis results indicate that the TREE problems are challenging in variable interactions and function separabilities. Besides, their rich characteristics, especially the complex variable correlations to different objectives, also indicate the good potentials as benchmark LSMOPs. IV. E VOLUTIONARY O PTIMIZATION In this work, we have transformed the RE estimation task, which is always handled by manual calibration techniques, into a relatively cheap large-scale multiobjective optimization problem. To solve our formulated TREE problems and investigate their properties, we conduct experiments by using eight representative MOEAs, namely, NSGA-II [24], TABLE III T HE N UMBERS OF P OSITION VARIABLES AND D ISTANCE VARIABLES OF E ACH TREE P ROBLEM O BTAINED BY MOEA/DVA AND LMEA. MOEA/DVA Problem TREE1 TREE2 TREE3 TREE4 TREE5 TREE6 LMEA Objectives Constraints Objectives Constraints 600:0 1199:1 597:3 1197:3 599:1 1193:1207 598:2 1200:0 600:0 1200:0 1200:0 2380:20 600:0 1200:0 600:0 1200:0 1200:0 1200:1200 337:263 341:859 14:586 126:1074 199:1001 2346:54 MOEA/D [26], GDE3 [54], CMOPSO [17], MOPSO [55], IBEA [27], MOEA/DVA [44], and WOF [42]. We select these MOEAs since they have used different kinds of genetic operators and/or represented different kinds of MOEAs. For each test problem, the number of decision variables and the maximum number of function evaluations are given in Table IV.3 In the following, we first present a brief introduction to the adopted performance indicators, and then give the parameter settings of the adopted algorithms. Afterwards, each algorithm is run for 20 times to optimize each test problem independently to obtain the statistical results. Finally, we validate the performance of evolutionary multiobjective optimization on the TREE problem using a real-world power delivery system. A. Optimization Setting 1) Performance Indicators. We adopt two widely used performance indicators, i.e., the hypervolume (HV) indicator [57] and the inverted generational distance (IGD) indicator [58], to evaluate the performance of the adopted algorithms. Both HV and IGD can assess the convergence as well as the distribution of the obtained solution set. The two indicators require a reference point/set based on the true PFs, which is usually a set 3 In this work, all the adopted algorithms and the formulated TREE problems are implemented on PlatEMO [56]. 1089-778X (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEVC.2020.2967501, IEEE Transactions on Evolutionary Computation IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. , NO. , MONTH YEAR of uniformly distributed solutions on the true PF. Since we do not have the true PFs of the six test problems, we collect all the non-dominated solutions obtained by the adopted algorithms over 20 runs into a set, termed P ′ , as the reference points. Then, the reference point is set to 1.01∗n (n is the nadir point of set P ′ ) for HV calculation, and the reference point set is set to 0.99∗P ′ for IGD calculation. 2) Constraint Handling. The Ray-Tai-Seow’s constrainthandling approach by using a non-domination check of constraint violations is adopted [59]. 3) Reproduction Operators. The simulated binary crossover (SBX) and polynomial mutation (PM) are adopted for offspring generation in NSGA-II, MOEA/D, and IBEA. To be specific, the distribution indices of crossover and mutation are set to nc =20 and nm =20, respectively; the crossover and mutation probabilities are set to pc =1.0 and pm =1/D respectively, where D is the number of decision variables [60]. In GDE3 and MOEA/DVA, the differential evolution (DE) operator [61] and PM are used for offspring generation, where the control parameters are set to CR=1, F =0.5, pm =1/D, and η=20 as recommended in [62]. In CMOPSO, the competitive mechanism based particle swarm optimizer (PSO) is adopted without additional parameters; the particle swarm update strategy in SMPSO [63] is adopted during the second phase of WOF; the standard PSO is adopted in MOPSO [55]. 4) Population Sizing. The population is set to 100 for biobjective test instances and 105 for tri-objective test instances. (5) Specific Parameter Settings. The number of neighborhoods is set to T =10 in MOEA/D; the number of divisions in each objective is set to div=10 in MOPSO; the fitness scaling factor is set to κ=0.05 in IBEA [27]. In MOEA/DVA [44], the number of sampling solutions in control variable analysis is set to N CA=20, and the maximum number of tries required to judge the interaction is set to N IA=6. In WOF, the embedded MOEA is SMPSO [63] as suggested, the number of groups is set to γ=4, the grouping method is random grouping, the transformation function ψ is set to the interval function, the number of evaluations for original problem is set to t1 =1000, the number of evaluations for transformed problem is set to t2 =500, the number of chosen solutions to do weight optimization is set to q=M + 1, and the fraction of function evaluations to use for the alternating weight-optimization phase is set to delta=0.5 as recommended in [42], where M is the number of objectives in the LSMOP. No other specific parameters are involved in NSGA-II, GDE3, or CMOPSO. (6) Termination Condition. For all the test instances, the total number of function evaluations (FEs) is set to be proportional to the number of decision variables, and the detailed values are given in Table IV. B. Results The statistical results of the IGD and HV values achieved by the eight MOEAs on TREE1 to TREE6 are summarized in Table V and Table VI, respectively. It can be observed from these two tables that WOF-SMPSO has achieved most of the best results, especially on TREE1 and TREE2, which are relatively simple in terms of the variable interactions and 8 TABLE IV S ETTINGS OF THE N UMBERS OF D ECISION VARIABLES AND F UNCTION E VALUATIONS FOR E ACH T EST I NSTANCES , R ESPECTIVELY. TREE1 TREE2 TREE3 TREE4 TREE5 TREE6 D FEs D FEs D FEs D FEs D FEs D FEs 6.00E+03 2.00E+04 6.00E+03 4.00E+04 6.00E+03 2.00E+05 1.20E+04 1.00E+06 6.00E+04 6.00E+04 1.20E+04 4.00E+05 1.20E+04 4.00E+04 1.20E+04 8.00E+04 1.20E+04 4.00E+05 2.40E+04 2.00E+06 1.20E+05 8.00E+04 2.40E+04 8.00E+05 1.80E+04 6.00E+04 1.80E+04 1.20E+05 2.40E+04 6.00E+05 3.60E+04 4.00E+06 1.80E+05 1.00E+05 4.80E+04 1.20E+06 2.40E+04 8.00E+04 2.40E+04 1.60E+05 3.60E+04 8.00E+05 4.80E+04 6.00E+06 2.40E+05 1.20E+05 7.20E+04 1.60E+06 3.00E+04 1.00E+05 3.00E+04 2.00E+05 4.80E+04 1.00E+06 6.00E+04 8.00E+06 3.00E+05 1.50E+05 9.60E+04 2.00E+06 the numbers of constraints; however, it has failed to obtain any feasible solution on high-dimensional TREE4 to TREE6. Meanwhile, NSGA-II, MOEA/D, GDE3, CMOPSO, MOPSO, and IBEA have achieved similar optimization results and suffered the same failures on TREE4 and TREE6. Nevertheless, due to the complex variable interactions and correlations to different objectives in TREE problems, MOEA/DVA has obtained worse results in comparison with other algorithms. To be specific, TREE1 problem is the simplest problem among the six TREE problems, which involves two objectives and three constraints. The non-dominated solutions obtained by the eight MOEAs on TREE1 with 6000 decision variables in the run associated with the medium IGD value are displayed in Fig. 4, where the windows with approximate PF indicate the convergence of the obtained solutions, and the zoomin windows are used to show the distribution of obtained solutions. It is obvious that only WOF-SMPSO can obtain a set of diverse solutions, while other MOEAs fail to converge or spread the solutions over the approximate PF. Note that the algorithms with PSO operators have performed slightly better than those with SBX operators, which may be attributed to their ability in tracking the global best particle(s) and thus perform faster convergence on problems with relatively simple fitness landscapes. By contrast, TREE6 is difficult as it involves three objectives, six constraints, and complex variable interactions. The non-dominated solutions obtained by the selected algorithms on TREE6 with 3000 decision variables are given in Fig. 5. It can be observed that WOF-SMPSO has obtained the best converged and diverse solutions again. However, it fails to find solutions around the tails of the approximate PF, and thus its diversity maintenance strategy should be further enhanced for solving TREE problems. Besides, the PSO/DE based algorithms only find a small number of feasible solutions clustered to several regions, and their poor performance could be attributed to their poor capabilities in diversity maintenance, especially on problems with complex fitness landscapes such as TREE6. In summary, the eight MOEAs exhibit slightly different capabilities in dealing with the challenges in our proposed test suite. On one hand, it indicates the promising potentials of evolutionary multiobjective optimization, especially for those MOEAs with PSO operators, in solving the TREE problems; on the other hand, however, it calls for the development and 1089-778X (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEVC.2020.2967501, IEEE Transactions on Evolutionary Computation IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. , NO. , MONTH YEAR 9 TABLE V T HE IGD R ESULTS O BTAINED BY NSGA-II, MOEA/D, GDE3, CMOPSO, MOPSO, IBEA, MOEA/DVA, I NSTANCES . T HE B EST R ESULT IN E ACH ROW IS H IGHLIGHTED . Problem TREE1 TREE2 TREE3 TREE4 TREE5 TREE6 WOF-SMPSO AND ON 30 TREE T EST Dim NSGA-II MOEA/D GDE3 CMOPSO MOPSO IBEA MOEA/DVA WOF-SMPSO 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1.99E+01(5.61E-02) 2.58E+01(6.28E-02) 3.85E+01(2.70E-01) 5.26E+01(3.46E-01) 6.64E+01(2.85E-01) 5.06E+01(4.65E-01) 1.00E+02(6.52E-01) 1.56E+02(9.61E-01) 1.96E+02(2.52E-01) 2.53E+02(0.00E+00) 2.20E+02(1.93E-01) 4.34E+02(3.77E-01) 8.04E+02(4.10E-01) 1.09E+03(3.17E-01) 1.33E+03(4.23E-01) 2.31E+01(1.03E-02) 5.32E+01(1.50E-14) – 8.42E+00(1.77E+01) – 1.50E+01(4.76E+01) 1.12E+02(1.50E-14) 1.93E+02(6.37E-01) 2.49E+02(1.14E+00) 9.60E+01(1.55E+02) 2.07E+03(3.91E-01) 6.78E+07(8.24E-02) – – 1.59E+01(2.55E+01) 2.00E+01(1.82E-02) 2.58E+01(8.64E-02) 3.87E+01(1.34E-01) 5.25E+01(2.33E-01) 6.60E+01(2.59E-01) 5.05E+01(2.72E-01) 9.90E+01(4.81E-01) 1.53E+02(5.84E-01) 1.92E+02(1.52E-02) 2.47E+02(0.00E+00) 3.58E+02(3.44E+00) 7.43E+02(2.77E+00) 1.27E+03(8.98E+00) 1.69E+03(6.43E+00) 2.24E+03(8.61E+00) 2.09E+01(0.00E+00) 4.27E+01(7.49E-15) – 4.13E+01(8.39E-05) – 1.53E+01(4.83E+01) 1.19E+02(1.50E-14) 1.99E+02(5.38E-01) 2.54E+02(1.03E+00) 3.26E+02(1.25E+00) 2.07E+03(7.90E-02) 6.78E+07(4.83E-02) – – 3.96E+01(1.15E-01) 2.30E+01(4.28E-01) 2.88E+01(2.78E-01) 4.28E+01(5.00E-01) 6.04E+01(4.71E-01) 7.45E+01(4.77E-01) 5.40E+01(7.52E-01) 1.09E+02(6.22E-01) 1.69E+02(2.16E-01) 2.12E+02(1.34E-01) 2.81E+02(5.99E-14) 2.20E+02(5.57E-02) 4.33E+02(9.86E-02) 8.00E+02(1.21E-01) 1.08E+03(5.38E-02) 1.32E+03(2.33E-01) 2.18E+01(0.00E+00) 4.45E+01(7.49E-15) – – – 2.16E+01(6.84E+01) 2.60E+02(5.99E-14) 4.01E+02(4.41E+00) 5.57E+02(5.07E+00) 1.44E+02(3.03E+02) 2.07E+03(7.15E-02) 6.78E+07(4.58E-03) – – – 2.21E+01(1.73E-01) 2.68E+01(2.17E-01) 3.90E+01(6.36E-01) 5.36E+01(1.23E+00) 6.64E+01(7.95E-01) 5.16E+01(2.75E-01) 9.81E+01(2.42E+00) 1.54E+02(3.10E+00) 1.97E+02(2.95E-01) 2.48E+02(0.00E+00) 1.13E+02(3.36E+00) 2.56E+02(5.72E+00) 5.44E+02(3.40E+00) 7.47E+02(6.34E+00) 9.59E+02(4.15E+00) 1.96E+01(3.74E-15) 4.07E+01(7.49E-15) – – – 1.80E+01(5.70E+01) 1.37E+02(3.00E-14) 2.48E+02(1.46E+00) 3.27E+02(1.78E+00) 9.02E+01(1.90E+02) 2.07E+03(1.86E+00) 6.78E+07(1.57E-08) – – – 2.47E+01(9.22E-02) 3.05E+01(1.55E-01) 4.44E+01(1.89E-01) 6.16E+01(1.96E-01) 7.48E+01(2.87E-01) 5.63E+01(1.34E-01) 1.10E+02(2.21E-01) 1.70E+02(2.73E-01) 2.12E+02(7.87E-02) 2.79E+02(5.99E-14) 2.15E+02(1.33E+00) 4.27E+02(7.88E-01) 7.96E+02(1.49E+00) 1.08E+03(6.31E-01) 1.32E+03(2.04E+00) 1.99E+01(0.00E+00) 4.07E+01(7.49E-15) – – – 2.33E+01(7.37E+01) – 4.14E+02(1.78E-01) 1.12E+02(2.37E+02) 1.36E+02(2.86E+02) 2.07E+03(2.19E-02) 6.78E+07(0.00E+00) – – – 2.02E+01(8.83E-02) 2.52E+01(5.01E-02) 3.73E+01(5.16E-02) 5.09E+01(7.49E-02) 6.39E+01(1.50E-01) 4.90E+01(6.07E-02) 9.57E+01(6.25E-02) 1.49E+02(3.36E-01) 1.88E+02(6.09E-02) 2.44E+02(0.00E+00) 2.02E+02(1.07E+00) 4.04E+02(1.56E+00) 7.60E+02(1.24E+00) 1.03E+03(1.68E+00) 1.27E+03(2.61E+00) 2.07E+01(3.74E-15) 4.64E+01(7.49E-15) – – – 1.49E+01(4.73E+01) – 1.88E+02(2.37E-01) 2.43E+02(1.77E-01) – 2.07E+03(3.10E-02) 6.78E+07(1.57E-08) – – – 5.91E+01(2.38E-01) 1.30E+02(3.68E-01) 1.96E+02(2.95E-01) 2.60E+02(5.99E-01) 3.27E+02(4.74E-01) 2.57E+02(5.69E-01) 5.19E+02(9.28E-01) 7.76E+02(9.57E-01) 1.05E+03(6.07E-01) 1.30E+03(0.00E+00) 2.92E+02(6.34E+00) 6.05E+02(4.48E+00) 1.17E+03(6.12E+00) 1.69E+03(4.87E+00) 2.10E+03(1.44E+01) 9.32E+01(0.00E+00) 1.88E+02(0.00E+00) – – – 5.97E+01(1.89E+02) – – – – 2.10E+03(9.98E-01) 6.78E+07(0.00E+00) – – – 1.39E+00(4.01E-01) 2.48E+01(1.17E+00) 3.56E+01(7.90E-01) 4.49E+01(1.22E+00) 5.59E+01(9.68E-01) 3.49E+01(1.53E+00) 6.95E+01(2.08E+00) 1.11E+02(1.56E+01) 1.25E+02(2.40E+00) 1.75E+02(3.00E-14) 6.78E+02(4.00E-03) 1.36E+03(2.91E-03) 2.80E+03(2.08E-02) 4.28E+03(9.48E-03) 5.89E+03(1.87E+00) 2.28E+01(0.00E+00) 4.38E+01(7.49E-15) – – – 2.39E+01(7.57E+01) – – – – 1.83E+03(6.33E+02) 3.41E+05(6.14E-11) – – – ‘–’ indicates that the compared algorithm fail to obtain any feasible solution. TABLE VI T HE HV R ESULTS O BTAINED BY NSGA-II, MOEA/D, GDE3, CMOPSO, MOPSO, IBEA, MOEA/DVA, I NSTANCES . T HE B EST R ESULT IN E ACH ROW IS H IGHLIGHTED . Problem TREE1 TREE2 TREE3 TREE4 TREE5 TREE6 AND WOF-SMPSO ON 30 TREE T EST Dim NSGA-II MOEA/D GDE3 CMOPSO MOPSO IBEA MOEA/DVA WOF-SMPSO 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 6.59E+00(2.13E-02) 6.77E+00(1.98E-02) 1.00E+01(6.98E-02) 1.31E+01(7.46E-02) 1.60E+01(5.72E-02) 5.60E+01(3.85E-01) 1.10E+02(6.47E-01) 1.61E+02(9.05E-01) 2.10E+02(4.67E-01) 2.61E+02(0.00E+00) 2.48E+03(1.14E+00) 4.77E+03(2.16E+00) 9.24E+03(3.15E+00) 1.38E+04(1.50E+00) 1.81E+04(3.40E+00) 4.65E+00(9.29E-03) 8.75E+00(1.87E-15) 0.00E+00(0.00E+00) 2.32E-03(4.93E-03) 0.00E+00(0.00E+00) 4.41E+00(1.40E+01) 4.93E-01(1.17E-16) 6.85E-01(8.66E-03) 8.43E-01(7.83E-03) 2.84E-01(4.57E-01) 8.88E+07(8.17E+05) 4.85E+11(1.54E+09) 1.61E+15(2.64E-01) 0.00E+00(0.00E+00) 2.50E-02(4.90E-02) 6.22E+00(2.58E-02) 6.28E+00(1.26E-02) 9.26E+00(2.74E-02) 1.21E+01(3.03E-02) 1.49E+01(3.93E-02) 5.23E+01(2.80E-01) 1.03E+02(4.63E-01) 1.53E+02(3.59E-01) 2.02E+02(8.38E-02) 2.52E+02(3.00E-14) 3.09E+03(1.56E+01) 5.93E+03(1.05E+01) 1.06E+04(4.13E+01) 1.55E+04(2.40E+01) 2.01E+04(5.72E+01) 6.63E+00(9.36E-16) 1.30E+01(1.87E-15) 0.00E+00(0.00E+00) 7.89E-01(9.93E-04) 0.00E+00(0.00E+00) 4.32E+00(1.37E+01) 3.95E-01(5.85E-17) 5.78E-01(4.64E-03) 7.62E-01(6.95E-03) 8.55E-01(8.51E-03) 1.28E+08(1.03E+06) 8.45E+11(4.39E+09) 2.65E+15(0.00E+00) 0.00E+00(0.00E+00) 2.72E+00(6.20E-02) 8.08E+00(2.12E-01) 8.88E+00(1.29E-01) 1.36E+01(2.71E-01) 1.84E+01(2.65E-01) 2.33E+01(3.15E-01) 7.29E+01(2.02E+00) 1.51E+02(2.47E+00) 2.37E+02(1.53E+00) 3.22E+02(1.19E+00) 4.05E+02(5.99E-14) 2.49E+03(5.55E-01) 4.79E+03(4.10E-01) 9.30E+03(7.37E-01) 1.39E+04(4.12E-01) 1.82E+04(1.10E+00) 7.06E+00(0.00E+00) 1.39E+01(1.87E-15) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 6.90E+00(2.18E+01) 1.09E+01(1.87E-15) 1.88E+01(4.61E-01) 2.62E+01(5.55E-01) 7.17E+00(1.51E+01) 1.26E+08(4.18E+06) 7.64E+11(1.13E+09) 2.75E+15(5.27E-01) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 7.65E+00(9.60E-02) 8.02E+00(1.17E-01) 1.17E+01(5.13E-01) 1.52E+01(7.57E-01) 1.91E+01(7.09E-01) 6.69E+01(6.84E-01) 1.22E+02(6.39E+00) 1.88E+02(1.09E+01) 2.63E+02(1.32E+00) 2.89E+02(5.99E-14) 3.47E+03(3.60E+01) 6.35E+03(6.53E+01) 1.14E+04(3.26E+01) 1.65E+04(6.68E+01) 2.12E+04(4.65E+01) 6.29E+00(9.36E-16) 1.22E+01(1.87E-15) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 5.82E+00(1.84E+01) 4.02E+00(9.36E-16) 6.89E+00(1.34E-01) 8.35E+00(1.26E-01) 2.05E+00(4.32E+00) 1.32E+08(1.59E+06) 8.12E+11(0.00E+00) 2.56E+15(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 9.25E+00(6.50E-02) 9.82E+00(9.48E-02) 1.45E+01(1.27E-01) 1.91E+01(1.41E-01) 2.36E+01(2.19E-01) 8.09E+01(5.85E-01) 1.59E+02(9.48E-01) 2.38E+02(1.32E+00) 3.18E+02(5.22E-01) 3.91E+02(5.99E-14) 2.51E+03(4.60E+00) 4.81E+03(3.94E+00) 9.30E+03(4.86E+00) 1.38E+04(2.53E+00) 1.82E+04(6.26E+00) 6.40E+00(9.36E-16) 1.23E+01(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 7.52E+00(2.38E+01) 0.00E+00(0.00E+00) 2.03E+01(1.29E-02) 5.34E+00(1.13E+01) 6.24E+00(1.32E+01) 1.56E+08(5.27E+05) 1.00E+12(0.00E+00) 3.30E+15(5.27E-01) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 6.79E+00(1.79E-02) 7.09E+00(3.26E-02) 1.05E+01(6.01E-02) 1.37E+01(8.40E-02) 1.67E+01(1.42E-01) 5.96E+01(2.05E-01) 1.17E+02(5.28E-01) 1.70E+02(1.19E+00) 2.21E+02(2.16E-01) 2.67E+02(0.00E+00) 2.56E+03(6.65E+00) 4.90E+03(6.34E+00) 9.42E+03(5.69E+00) 1.40E+04(8.05E+00) 1.83E+04(1.22E+01) 5.53E+00(9.36E-16) 1.05E+01(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 4.57E+00(1.45E+01) 0.00E+00(0.00E+00) 8.80E-01(1.31E-02) 1.05E+00(5.98E-03) 0.00E+00(0.00E+00) 1.38E+08(6.37E+05) 8.37E+11(1.29E-04) 2.55E+15(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 7.21E-01(1.67E-02) 2.10E-01(2.53E-03) 2.97E-01(4.41E-03) 3.60E-01(4.02E-03) 4.31E-01(4.05E-03) 1.78E+00(3.89E-02) 3.13E+00(6.17E-02) 4.38E+00(3.79E-02) 5.69E+00(2.22E-02) 6.60E+00(9.36E-16) 3.42E+02(6.86E+00) 5.39E+02(3.79E+00) 9.11E+02(6.19E+00) 1.31E+03(1.28E+01) 1.62E+03(2.39E+01) 1.13E-01(1.46E-17) 1.77E-01(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 1.09E-01(3.45E-01) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 1.38E+07(2.36E+05) 1.93E+10(0.00E+00) 5.43E+13(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 1.03E+01(2.96E-03) 1.11E+01(2.98E-03) 1.65E+01(4.27E-03) 2.18E+01(4.53E-03) 2.70E+01(6.03E-03) 9.35E+01(5.41E-02) 1.85E+02(1.23E-01) 2.76E+02(1.24E-01) 3.67E+02(7.07E-02) 4.56E+02(1.20E-13) 6.44E+03(2.15E-01) 1.26E+04(2.94E-02) 2.48E+04(1.72E-01) 3.70E+04(1.51E-01) 4.90E+04(2.03E-01) 7.40E+00(9.36E-16) 1.45E+01(1.87E-15) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 8.87E+00(2.81E+01) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 1.72E+08(2.33E+02) 1.12E+12(2.57E-04) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) 0.00E+00(0.00E+00) application of tailored algorithms for TREE problems as none of the tested MOEAs can solve them perfectly; moreover, the deterministic grouping based algorithms may not perform well due to the complex variable interactions and correlations to different objectives in TREE problems. C. Challenges Generally, the main challenges of our proposed TREE problems can be attributed to three aspects. First, the TREE problems are intrinsically scalable to the number of decision variables. Since the complexities of the proposed TREE problems are substantially increased as the number of decision variables grows, it poses more challenges than generic multiplication/addition-based large-scale benchmark MOPs. Moreover, since the true voltage values are determined by the load of the power system, the optimal values of decision variables in the TREE problems are irregularly distributed. Consequently, the PSs of TREE problems are more complex than those in conventional benchmark problems, thus posing more challenges to effective offspring generation [62]. Second, the large number of constraints is another challenge to existing MOEAs. For example, results in Table V indicate that most algorithms fail to find any feasible solutions on 1089-778X (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEVC.2020.2967501, IEEE Transactions on Evolutionary Computation IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. , NO. , MONTH YEAR 0.025 Non-dominated solutions Approximate PF MOEA/D 0.02155 0.015 f2 94.9 80 85 90 0 70 95 8 10-3 Non-dominated solutions Approximate PF 0.02 0.015194 f2 80 80 85 0.002 90 0.01519 84.18 84.19 84.2 f1 0.002 0 0 95 0.015192 0.004 81 f1 74 76 78 80 f1 MOPSO 7 -1 f1 75 0.015196 0.01 0.006 0.004 94.82 0.005 f1 0.012 0.008 0 f2 0.021544 94.8 94.95 f1 75 f2 0.01 0.01878 94.85 0 70 0.006 0.021546 0.01879 0.005 0.021548 f2 f2 f2 0.0188 0.01 Non-dominated solutions Approximate PF 0.014 1 0.008 0.01881 CMOPSO 0.016 Non-dominated solutions Approximate PF 0.01 0.02 0.015 GDE3 0.012 Non-dominated solutions Approximate PF f2 NSGA-II 0.02 10 82 84 74 86 76 78 IBEA 0.06 Non-dominated solutions Approximate PF MOEA/DVA 1.8 Non-dominated solutions Approximate PF 10 -3 1.6 0.015 2 77.25335 77.2534 1 0.005 76 78 80 f1 82 84 86 0 70 75 80 1.3 f2 f1 85 236 95 0 50 238 84 86 74 76 78 80 82 f1 1.1 f1 0.01 90 10 9.5 1.2 0.0512 91.1 91.15 91.2 f1 0 74 0.02 0.0181 f1 1.4 0.0514 3 7.237 10-4 1.5 0.0516 0.03 0.0182 10.5 f2 0.01 86 f2 7.238 f2 f2 4 0.0518 0.04 0.0183 f2 5 84 WOF-SMPSO f2 10-3 f2 7.239 82 Non-dominated solutions Approximate PF 1.7 0.05 6 80 f1 f1 1 0.9 100 f1 150 200 250 74 76 78 80 82 Fig. 4. The non-dominated solutions obtained by the compared algorithms on TREE1 in the run associated with the median IGD value. Fig. 5. The non-dominated solutions obtained by the compared algorithms on TREE6 in the run associated with the median IGD value. high-dimensional TREE4 to TREE6. Since the number of constraints is even bigger than the number of objectives, constraint handling strategies for mating selection, environmental selection, and offspring generation should be specially enhanced. Moreover, since the nonlinear properties of the constraints are even more complex than those of the objectives, the complex mutual dependency between the objectives and the constraints can be a challenge. Third, the interactions among the neighborhood decision variables in the proposed TREE problems lead to further challenges in terms of the nonseparability of the objective functions. Note that, due to the requirement of smoothness and continuity, the neighborhood decision variable interactions widely exist in the real-world problems, e.g., the automatic systems [64] or the engineering design models [65]. Consequently, the PSO based MOEAs perform better than the generic operator-based ones, which is mainly attributed to the fact that PSO is more effective in tracking the neighborhood variable interactions. Nevertheless, due to a large number of 1089-778X (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEVC.2020.2967501, IEEE Transactions on Evolutionary Computation IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. , NO. , MONTH YEAR local optima in LSMOPs, the diversity maintenance in the PSO based offspring generation should be enhanced to alleviate premature convergence. D. Real-World Validation We validate the proposed formulation method for the TREE problems and the corresponding performance of evolutionary multiobjective optimization using a real-world power delivery system, where two state-of-the-art RE estimation approaches (as introduced in Section II), i.e., the adaptive detection method [10] and the line parameter estimation method [11], are adopted in our validation test. Furthermore, a state-of-theart calibration method, i.e., the on-site calibration method [6], is adopted to obtain the REs of the VTs as the ground truth. Note that a smaller difference between the estimated result and the calibration result indicates a better performance. The flowchart of the proposed method in estimating the REs of uncalibrated VTs is presented in Fig. 6. First, a precise sampling device (±0.05% maximum permissible error) and a workstation are installed in the central control room of the selected substation. Then, all analog signals of VTs in this substation are sampled by this device and saved in the workstation as historical data. If the sampled data is not enough, i.e., n≤T (T is set to 8000 in this experiment), more data should be sampled; otherwise, the collected data is used to formulate a TREE problem. Next, an MOEA is adopted to optimize the formulated problem to obtain the estimated true voltage values by averaging all feasible solutions. Afterwards, the RE of each VT is calculated and the faulty one(s) will be reported to staff; otherwise, the above process will be repeated. Start Data Collection (Voltage Values by VTs) History Dataset of n Samples No n>T Yes TREE Problem Formulation MOEA Optimization Average of the Feasible Solutions RE Calculation No Faulty VTs Yes Report End Fig. 6. The flowchart of our proposed method in handling the real-world TREE problem. As the formulation, we use the first problem in the proposed benchmark set (i.e., TREE1), and then we conduct optimization using the algorithm having the most stable performance as tested above, i.e., the MOPSO [55]. Note that the parameter 11 settings of the MOPSO are the same as that in Section IV-A, except that the maximum number of FEs is set to 300,000. TABLE VII S TATIC RE S OF T HREE -P HASE VT S O BTAINED BY MOPSO AND O N -S ITE C ALIBRATION , W HERE REA , REB , AND REC ARE THE S TATIC RE S OF THE P RIMARY S IDE P HASE A, P HASE B, AND P HASE C VT S , R ESPECTIVELY. Phase Optimization (%) Calibration (%) Difference(%) REA REB REC 0.195 0.267 0.237 0.124 0.227 0.126 0.071 0.040 0.111 In the static scenario, i.e., the simplified TREE problem, the on-site calibration method obtains a single RE value for each VT as the ground truth. In contrast, the time-varying REs of each VT obtained by the MOPSO are averaged and then used as the static RE. The detailed static RE values obtained by the on-site calibration and MOPSO are displayed in Table VII, where REA , REB , REC correspond to the threephase REs, respectively. The RE differences between our method and the calibration method are only 0.071%, 0.040%, and 0.111% respectively, indicating its estimation accuracy. As for the results achieved by two compared estimation methods, i.e., the adaptive detection method and the line parameter estimation method, the maximum RE differences are 0.31% and 0.88%, respectively. Generally, MOPSO significantly outperforms these two data-driven RE estimation methods. E. Discussions Based on the above, we can conclude that the proposed formulation method is generally effective. In spite of the slight differences between the REs obtained by the proposed formulation method and the calibration method (i.e., ground truth), the performance is significantly better than the compared data-driven RE estimation methods. More specifically, the proposed formulation method has the following advantages. 1) Cheap Cost: Our proposed method is much cheaper than any on-site/off-line calibration. Besides, this method does not need to de-energize the substation or conduct additional labor operations. 2) Real-Time: Once the voltage sensors are installed, the voltage values from each VT can be obtained easily. Consequently, the data can be used to estimate the REs whenever it is needed. Instead of the time-consuming calibration process, this method provides real-time estimations as early warnings of potential faulty VTs. It is worth noting that, in practice, the method does not have to be in full-time operation; alternatively, it could also be deployed periodically. 3) Versatility: In our proposed benchmark problems, three major types of substations are included, where the formulation does not require any specific VT types/models. Besides, the true voltage values over time are the optimization target of our proposed method, and the REs of different VTs can be calculated according to the RE calculation method (refer to (1)). Thus, our proposed method can be applicable to VTs in different substations, which makes it more practical for realworld application problems. 1089-778X (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEVC.2020.2967501, IEEE Transactions on Evolutionary Computation IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. , NO. , MONTH YEAR Hence, in practice, the proposed formulation method can provide the decision-makers with early warnings of potential faulty VTs, thus enabling them to decide whether to conduct expensive on-site/off-line calibration or not. It is worth noting that, however, the proposed method is merely tailored for estimating the REs of VTs, since there are mainly three difficulties in applying this method to current transformers (CTs). First, the properties of the current are totally different from those of the voltages, and thus it will be difficult to formulate the objectives/constraints. Second, the measuring range of the CTs is larger than that of the VTs, resulting in a larger search space. Thus, the optimization of the problem would be too difficult to obtain feasible results. Third, since the sampling devices for VTs are installed in parallel while those for CTs are installed in series, the fault of sampling devices can lead to serious damages to the CT systems. V. C ONCLUSION In this work, we have formulated the RE estimation tasks in power delivery systems into benchmark large-scale multiobjective optimization problems. Then, we have used eight representative MOEAs to optimize them. Furthermore, we have validated the proposed multiobjective formulation method as well as the corresponding performance of evolutionary multiobjective optimization using a real-world substation of the power delivery system. The formulated TREE problems are suitable for examining the performance of large-scale MOEAs in solving real-world problems due to the following reasons. Firstly, the number of decision variables in those TREE problems can vary from hundreds to even millions, which indicates good scalability. Secondly, the decision variable interactions in the TREE problems cover different scenarios, i.e., being fully separable, fully nonseparable, and partially separable. Thirdly, the TREE problems are irregular in terms of variable interactions, correlations to different objectives, and shapes of PFs/PSs, which can reflect the complexity of real-world problems. Fourthly, we have considered many constraints for the TREE problems, thus making them more practical. We have optimized the formulated TREE problems using eight representative MOEAs, i.e., NSGA-II, MOEA/D, GDE3, CMOPSO, MOPSO, IBEA, MOEA/DVA, and WOF-SMPSO. Empirical results indicate that WOF-SMPSO has achieved the best general performance while the other algorithms have exhibited slightly different capabilities. To be specific, the deterministic grouping based algorithms may fail in solving TREE problems due to their complex variable interactions and correlations to different objectives; by contrast, MOEAs with PSO operators have performed slightly better than algorithms with crossover operators. Furthermore, we have validated the proposed multiobjective formulation method and the corresponding performance of evolutionary multiobjective optimization using a real-world substation of the power delivery system, where three state-of-the-art RE estimation approaches are compared with our proposed multiobjective formulation and evolutionary optimization method. In conclusion, the contributions of this work are two-fold. On one hand, the multiobjective formulation of the TREE 12 problems can promote the research in evolutionary large-scale multiobjective optimization. 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Lagoudas, “Modeling and analysis of origami structures with smooth folds,” Computer-Aided Design, vol. 78, pp. 93–106, 2016. 1089-778X (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TEVC.2020.2967501, IEEE Transactions on Evolutionary Computation IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. , NO. , MONTH YEAR Cheng He (M’2019) received the B.Eng. degree from the Wuhan University of Science and Technology, Wuhan, China, in 2012, and the Ph.D. degree from the Huazhong University of Science and Technology, Wuhan, China, in 2018. Currently, he is a postdoctoral research fellow with the Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen, China. His current research interests include model-based evolutionary algorithms, multiobjective/many-objective optimization, largescale optimization, deep learning, and their applications. Ran Cheng (M’2016) received the B.Sc. degree from the Northeastern University, Shenyang, China, in 2010, and the Ph.D. degree from the University of Surrey, Guildford, U.K., in 2016. He is currently an Assistant Professor with the Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen, China. His current research interests include evolutionary multiobjective optimization, model-based evolutionary algorithms, large-scale optimization, swarm intelligence, and deep learning. He is the founding chair of IEEE Symposium on Model-Based Evolutionary Algorithms (IEEE MBEA). He is the recipient of the 2018 IEEE Transactions on Evolutionary Computation Outstanding Paper Award, the 2019 IEEE Computational Intelligence Society (CIS) Outstanding Ph.D. Dissertation Award, and the 2020 IEEE Computational Intelligence Magazine Outstanding Paper Award. 14 Xin Yao (M’91-SM’96-F’03) received the B.Sc. and Ph.D. degrees from the University of Science and Technology of China, Hefei, China, in 1982 and 1990, respectively. He is a Chair Professor of Computer Science with the Southern University of Science and Technology, Shenzhen, China, and a part-time Professor of Computer Science with the University of Birmingham, U.K. He has been researching multi-objective optimization since 2003, when he published a well-cited EMO’03 paper on many-objective optimization. His current research interests include evolutionary computation, ensemble learning, and their applications in software engineering. Dr. Yao was a recipient of the 2001 IEEE Donald G. Fink Prize Paper Award, the 2010, 2016, and 2017 IEEE Transactions on Evolutionary Computation Outstanding Paper Awards, the 2010 BT Gordon Radley Award for Best Author of Innovation (Finalist), the 2011 IEEE Transactions on Neural Networks Outstanding Paper Award, the Prestigious Royal Society Wolfson Research Merit Award in 2012, the IEEE Computational Intelligence Society (CIS) Evolutionary Computation Pioneer Award in 2013, and many other best paper awards. He was the President of IEEE CIS from 2014 to 2015, and from 2003 to 2008 he was Editor-in-Chief of the IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION. He is a Distinguished Lecturer of IEEE CIS. Chuanji Zhang received the B.S. degree in 2014 from the Huazhong University of Science and Technology. He is currently a Ph.D. candidate with the School of Electric and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China. His main research interests include electrical measurement, measurement error estimation. Ye Tian received the B.Sc., M.Sc., and Ph.D.degrees from the Anhui University, Hefei, China, in 2012, 2015, and 2018, respectively. He is currently a Lecturer with the Institutes of Physical Science and Information Technology, Anhui University, Hefei, China. His current research interests include multiobjective optimization methods and their application. He is the recipient of the 2018 IEEE Transactions on Evolutionary Computation Outstanding Paper Award. Qing Chen received the B.S., M.S., and Ph.D degrees from the Huazhong University of Science and Technology, in 2002, 2004 and 2008, respectively. He is currently an Associate Professor with the School of Electric and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China. His main research interests include electrical measurement, condition monitoring, and Oilfield electrical engineering. 1089-778X (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.