Fault location for transmission lines compensated with MOV‐protected SCs using voltage phasors
2018; Institution of Engineering and Technology; Volume: 13; Issue: 3 Linguagem: Inglês
10.1049/iet-smt.2018.5458
ISSN1751-8830
Autores Tópico(s)HVDC Systems and Fault Protection
ResumoIET Science, Measurement & TechnologyVolume 13, Issue 3 p. 392-402 Research ArticleFree Access Fault location for transmission lines compensated with MOV-protected SCs using voltage phasors Khalil Gorgani Firouzjah, Corresponding Author Khalil Gorgani Firouzjah k.gorgani@umz.ac.ir Faculty of Engineering & Technology, University of Mazandaran, Babolsar, IranSearch for more papers by this author Khalil Gorgani Firouzjah, Corresponding Author Khalil Gorgani Firouzjah k.gorgani@umz.ac.ir Faculty of Engineering & Technology, University of Mazandaran, Babolsar, IranSearch for more papers by this author First published: 01 May 2019 https://doi.org/10.1049/iet-smt.2018.5458Citations: 4AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study presents a fault location algorithm (FLA) for series compensated transmission lines (SCLs). It is established based on synchronised voltages sampling and Thevenin impedance estimation from both ends of SCL. The method is based on the use of phasor measurement units technology which is aimed to be independent of current measurements. This method includes two subroutines for the faults located on the right-hand and left-hand sides of series capacitor (SC). Lumped modelling is considered for SCL with SC equipped with metal–oxide varistor (MOV) arrester. The non-linear behaviour of SC-MOV system is investigated in the analysis. The proposed current independent FLA has been thoroughly tested using signals taken from simulations. According to the results, the percentage errors for the fault distances estimation are in proper ranges. 1 Introduction Series compensated lines (SCLs) with series capacitors (SCs) brought several advantages to power system. In spite of its capabilities, SCLs have serious challenges regarding fault location scheme. During-fault period, the system is affected by the non-linear operation of parallel metal–oxide varistor (MOV) as a protective unit of the SC [1]. Therefore, the fault location process has to be adapted. Various fault location algorithms (FLAs) for SCLs have been analysed in the literature. In some of them, the process is established based on one end [2, 3] and two ends [4-8] impedance measurement. These techniques have to be adjusted to deal with the non-linear characteristics of the MOV. These solutions include the use of fundamental frequency model of the SCL and single-ended measurements [2], distributed time-domain modelling of the line and two-ended synchronous sampling of voltages/currents [3]. An FLA is proposed in [4] for SCL using unsynchronised two-ended measurements. Another FLA is proposed in [5] for single- and double-circuit SCLs. This method is performed by taking the differences of the voltages at both sides of the fault point. An FLA is carried out in [6] by lumped model of SCLs. However, this technique is sensitive to the distance between fault and SC. Apostolopoulos and Korres [7] presented an FLA using phasor measurement units (PMUs). The main drawback of conventional two-ended method is synchronous sampling. Fulczyk et al. [8] solved this issue by generalised fault loop model at two-terminal lines. The fault loop contains the line section from the sending or receiving ends up to the fault point and also the fault path resistance. Besides impedance-based FLA, there are various methods based on transient approaches such as wavelet and travelling wave theory [9, 10]. The travelling-wave-based methods require high-frequency measurements. Along with the transient methods, artificial intelligence-based methods have been proposed [11, 12]. These methods need no assumptions about the line conditions. The intelligent methods extract the relation between faulted system signals and the fault location. Although the main drawback of this method is parameter setting and algorithm training, one of the biggest drawbacks is lack of well-defined guide for choosing the ideal number of hidden layers and neurons in the design of such systems. The majority of FLAs uses the voltage and current samples to develop the algorithm. Current and voltage transformers (CTs and VTs) and their undesired dynamic behaviour under short circuits lead to certain construction limitations. Using only the voltage measurements is beneficial by elimination of some inherent errors such as CTs saturation (at higher fault currents lead to maloperation of FLA). The solution methods are aimed to make FLAs independent of distorted secondary currents due to CT saturation, and consequently avoid its malfunction. The first remedies for assuring high accuracy under CT saturation can be performed by limiting the use of fault current phasors. The other remedies, except an intentional use of voltage signals, require identifying the CT saturation [13-15]. Among them, Galijasevic and Abur [13] developed a technique based on voltage measurement at multi-terminal network. Also, in two-terminal lines, some algorithms are designed for the faults location with the use of three-phase voltage measured at one line terminal [14]. However, these algorithms are not independent of fault type and resistance. Generally, the two-terminal FLAs are more accurate than one-terminal methods [15]. Many FLAs are presented based on these technologies due to the recent development in PMU technology. There are a number of references that suggest using synchronised phasors for two-terminal fault location. Several methods based on PMU are developed in [16-18] for two-terminal lines. Owing to CT inherent problems during-fault interval, some new techniques based on voltage measurement have been presented in [19, 20]. Although proposed methods in [19, 20] pass over the current dependency in the FLA, it is not independent of fault type and resistance. Firouzjah and Sheikholeslami [21] presented an FLA independent of fault type, resistance and inception angle. This paper presents an FLA based on synchronised voltage measurement for SCLs. The aim is to present a method independent during-fault current. The algorithm is performed into two subroutines. One of them is for faults behind the SC and another one for faults in front of the SC. Simulation results are presented to show the accuracy of the proposed technique. Also, different fault resistances are applied at different points. 2 Proposed FLA for two-terminal SCL 2.1 One SC placed in the middle of the line Fig. 1a shows a two-terminal SCL. EThA, ZThA, EThB and ZThB show Thevenin model from ends. Here, lRight and lLeft are the distances of SC (with nominal capacitor impedance XCS) point from A and B, respectively. As network theory, two-end transmission line can be modelled with ABCD matrix called transfer matrix of the network (1) where VA and VB are the end voltages. IA and IB are the send and receive currents from buses A and B, respectively. Fig 1Open in figure viewerPowerPoint Single line diagram of typical two-terminal SCL (a) Typical model in normal mode, (b) π Model of typical line in a symmetrical form, (c) Thevenin model of the faulted system (fault on the right-hand side of SC) For a medium line, the ABCD parameters are (2) where z and y are series impedance and shunt admittance per unit length. Applying differential equation of the system in the distributed modelling, transmission matrix is as shown below: (3) where γ and Zc are propagation constant and characteristic impedance of the line, respectively. These parameters can be calculated as (4) Similar to medium line model, π model can be presented for long line. So, the equivalent π model of long line is as follows: (5) The compensation is performed with three-phase SC banks at the distance lRight from B. The per unit distance is kc = lRight/lline. Symmetrical components theory is used in order to transform three-phase components, where T is transformation matrix (6) The superscript '012' denotes the zero, positive and negative sequences, respectively. EThA012 and EThB012 are Thevenin voltage sources of A and B, respectively. Note the Thevenin equivalent models at A and B are assumed to be known. Regarding Fig. 1b, the procedure is proposed below: (7) (8) (9) (10) Equations (7)–(10) yield the pre-fault SC current (IC) as (11) 2.1.1 During-fault interval on the right-hand side of SC In this stage, the algorithm is performed for unknown fault on right-hand side of the SC. According to Fig. 1c, per unit fault distance from B can be expressed as kF. According to superposition theory, voltages and current have been considered as the difference between pre-fault (V°,I°) and during-fault (Vf,If) data (12) Regarding Fig. 1c(13) (14) (15) According to the SC current (IC) and its protective current level (Ipr), the equivalent impedance is defined by [1] (16) (17) (18) (19) SC-MOV model is formulated using the compensator current during-fault interval (Icf). Regarding the superposition theory, this current can be obtained using (11) and (12) (20) The capacitor current (ΔIC) is determined based on the difference between during-fault (ICf) and pre-fault capacitor currents (ICo). Then, it is transferred to symmetrical form (21) Using this equivalent model, following are performed: (22) (23) (24) The analytical process aims to calculate the fault point voltage (VF) using system parameters as follows: (25) (26) where ΔVFL and ΔVFR are fault point voltages obtained using bus data located on its left-hand side (bus A) and right-hand side (bus B), respectively. Therefore, the difference is defined as (27) kF value obtained using this equation is named as kFR. 2.1.2 During-fault interval on the left-hand side of SC To analyse the faults on the left-hand side of the capacitor, a hypothetical substitution has been used as follows. Thus, similar to (25)–(27), relation is presented as follows: (28) (29) (30) kF value obtained using this equation is named as kFL. 2.1.3 Selecting the fault region with respect to SC Determination of fault side is depending on the values of kFR and kFL from (27) and (30). As recogniser algorithm, if only one of these variables is in the range of [0, 1], the fault is in its associated side. In the case of both kFR and kFL values are obtained in the [0, 1] interval, the fault side is determined by comparison of and in such a way that the fault occurs in the side which its objective function is lower (31) (32) (33) 2.2 Two SCs placed at each end of the line In this section, compensation is studied for the case which two SCs are located at both ends of the line. Fig. 2a shows the line with two SCs at both ends at pre-fault conditions. Flowing current of each SC for pre-fault conditions can be calculated as (34) Fig 2Open in figure viewerPowerPoint Two-terminal SCL with two SCs (pre-fault) (a) Typical model in normal mode, (b) Thevenin equivalent model of the faulted system Line side voltage of capacitors can be achieved using their flowing currents and the voltage of Thevenin model as (35) Under unknown fault, the new network can be modelled as Fig. 2b. Here, l1 and l2 are the distances from B and A to fault point, respectively. Per unit fault distance from B is kF = ll/lline. The difference between capacitors currents at pre-fault and during-fault condition are as follows: (36) SC fault currents can be calculated using (34) and (20). Then, equivalent SC impedance is estimated using (19) (37) (38) (39) Using Fig. 2b, following relations are performed: (40) (41) The process aims to calculate the fault point voltage as (42) where ΔVFA and ΔVFB are fault point voltages obtained using bus data located on its left-hand side and right-hand side, respectively. The difference between these values must be minimised to zero (43) 2.3 Thevenin impedance estimation In the proposed FLA, it is assumed that the Thevenin model of the network is available from both sides of the line. In this section, a Thevenin impedance estimation method is developed. It considered a Thevenin bus with Eth, Zth as (44) (45) Kirchhoff's Voltage Law (KVL) at bus A can be written as (46) The voltage difference of the branches is expressed in the following way from the continuous to the discrete form: (47) [n], [n + 1] indicate value at current and next samples, respectively. ΔT is the sampling interval. As (45) and (47) (48) This equation can be rewritten for N + 1 samples as (49) where X and V are known and unknown vectors, respectively. Two measurement points with the same power factor are considered [22]. H is the measurement matrix at point 1. Measured current and voltage are placed in the H and V matrices as (49). At point 2, loading is changed and the equations will be written (50) It is an overdetermined equation where the number of equations is more than the number of variables. Solution of this equation is based on least-squares method and singular value decomposition (SVD). X vectors are searched in order to minimise . 2.4 Flowchart of the proposed FLA Different steps of presented fault location method are mentioned at Fig. 3. As mentioned earlier, the main step of the proposed method is calculation of fault point voltage through left and right subroutines. Fig 3Open in figure viewerPowerPoint Flowchart of the proposed algorithm 3 Simulation study and results analysis Simulation for a 300 km, 400 kV two-terminal transmission line with 50% compensation was carried. Tables 1 and 2 summarise the required analysis and considered parameters. Performing the FLA evaluation, different factors are taken into account. The commonly main factors are: fault type (LG, LLG, LLLG, LL, LLL), fault resistance (RF), line loading angle (δ), fault inception angle (α), line length (lline), synchronisation error (ɛTsync), magnitude error in voltage (ɛMag), noise [signal-to-noise ratio (SNR)], capacitor voltage transformer (CVT) transients, uncertainty of line parameters (ɛzy), harmonic, sampling rate (fs), Thevenin impedance variation (ΔZth), Thevenin impedance estimation error (ɛZth), SC-MOV model, arcing fault and SC point. Table 1. System data used for the transmission line model [21] Parameter Transmission line Thevenin impedance positive/negative sequence R+,− 0.249168 Ω/km ZSA+,−ZSA0 17.177 + j45.5285 Ω2.5904 + j14.7328 Ω L+,− 1.556277 mH/km C+,− 19.469 nF/km zero sequence R0 0.60241 Ω/km ZSB+,−ZSB0 15.31 + j45.9245 Ω0.7229 + j15.1288 Ω L0 4.8303 mH/km C0 12.06678 nF/km MOV data→reference current: 44 kA; exponent: 23; reference voltage: 330 kV Table 2. FLA accuracy evaluation factors used in simulation Factors Fault type RF, Ω α, deg δ, deg fs, kHz lline, km SC point Other parameters RF, α, δ all 5 1 m, 10, 50 0, 30, 60, 90 5, 30 100 300 mid Table 1 lline all 5 1 m, 10, 100 90 35 100 200, 250, 300, 350, 400 mid Table 1 ɛTsync all 5 1 m, 10, 50 0, 30, 60, 90 45 100 300 mid Table 1 ɛTsync = 8, 62 mRad ɛMag all 5 1 m, 10, 100 0, 30, 60, 90 45 100 300 mid Table 1 ɛMag = 2% SNR all 5 1 m, 10, 50 90 30 100 300 mid Table 1 SNR = 10, 20, 40, 50 dB CVT transient all 5 1 m, 10, 100 0, 30, 60, 90 45 100 300 mid Table 1 CVT model as [23] ɛzy all 5 1 m, 10, 100 90 35 100 300 mid ±10% uncertainty of line data in Table 1 harmonic all 5 1 m, 10, 100 0, 30, 60, 90 45 100 300 mid I5 = 2.04%, I7 = 2.77%, I11 = 1.1%, I13 = 2.2% fs all 5 1 m, 10, 100 90 35 10, 20, 50, 100 300 mid Table 1 ΔZth all 5 1 m, 10, 100 90 35 100 300 mid ΔZth = ±50% ɛZth all 5 1 m, 10, 100 90 35 100 300 mid ±10% uncertainty of ɛZth SC-MOV model all 5 1 m, 10,100 90 35 100 300 mid ±10% uncertainty of and arcing fault LG Emanuel model 0, 30, 60, 90 45 100 300 mid Table 1 arc model [24, 25] SC point all 5 1 m, 10, 50 45 30 100 300 each end Table 1 3.1 One SC placed in the middle of the line The results for five fault types on the right-hand and left-hand sides of SC are shown in Table 3. The right subroutine results kR value <1 and the left subroutine results kL value higher than 1 for the faults near B. However, for the faults near SC, both right and left subroutines result kR and kL values <1. For example, for a fault located on the 125 km from A, kR and kL are 0.992 and 0.833, respectively. Residual values of (28) and (29) are 9 × 103 and 8 × 102, respectively. So, the left-hand side of SC is detected as faulted section. The fault location is 0.833 × 150 km from A. To evaluate the proposed FLA, it should be useful to consider following factors which affects the accuracy. Table 3. Performance of right and left subroutines Type Side Distance, km from Estimated Residual of kFR<1 kFR<1 Residual of (27)≪(30) Estimated B A kFR kFL (27) (30) Side Point LG R 5 — 0.034 1.488 4 × 10 2 × 104 Y N Y R 5.04 L — 5 1.488 0.034 2 × 104 4 × 10 N Y N L 5.04 R 45 — 0.300 1.203 3 × 102 1 × 104 Y N Y R 44.99 L — 45 1.203 0.300 1 × 104 3 × 102 N Y N L 44.99 R 85 — 0.567 0.926 6 × 102 1 × 104 Y Y Y R 85.03 L — 85 0.927 0.566 1 × 104 6 × 102 Y Y N L 84.97 R 125 — 0.833 0.992 8 × 102 9 × 103 Y Y Y R 125.00 L — 125 0.992 0.833 9 × 103 8 × 102 Y Y N L 125.01 R 145 — 0.967 0.858 6 × 102 1 × 104 Y Y Y R 145.01 L — 145 0.858 0.967 1 × 104 6 × 102 Y Y N L 145.03 LLG R 5 — 0.033 1.488 7 × 10 3 × 104 Y N Y R 4.98 L — 5 1.489 0.033 3 × 104 7 × 10 N Y N L 4.99 R 45 — 0.300 1.203 7 × 102 3 × 104 Y N Y R 44.99 L — 45 1.203 0.300 3 × 104 7 × 102 N Y N L 45.00 R 85 — 0.567 0.926 1 × 103 2 × 104 Y Y Y R 85.01 L — 85 0.927 0.567 2 × 104 1 × 103 Y Y N L 84.98 R 125 — 0.833 0.992 2 × 103 2 × 104 Y Y Y R 125.00 L — 125 0.992 0.833 2 × 104 2 × 103 Y Y N L 124.98 R 145 — 0.967 0.859 1 × 103 2 × 104 Y Y Y R 145.01 L — 145 0.859 0.967 2 × 104 1 × 103 Y Y N L 144.98 LLLG R 5 — 0.033 1.488 1 × 102 5 × 104 Y N Y R 5.00 L — 5 1.238 0.033 5 × 104 1 × 102 N Y N L 5.00 R 45 — 0.300 1.203 1 × 103 4 × 104 Y N Y R 45.00 L — 45 1.203 0.300 4 × 104 1 × 103 N Y N L 45.00 R 85 — 0.567 0.926 2 × 103 3 × 104 Y Y Y R 84.98 L — 85 0.927 0.567 3 × 104 2 × 103 Y Y N L 84.98 R 125 — 0.834 0.992 2 × 103 3 × 104 Y Y Y R 125.03 L — 125 0.992 0.833 3 × 104 2 × 103 Y Y N L 124.97 R 145 — 0.967 0.859 2 × 103 3 × 104 Y Y Y R 144.98 L — 145 0.859 0.967 3 × 104 2 × 103 Y Y N L 144.98 LL R 5 — 0.034 1.488 1 × 102 4 × 104 Y N Y R 5.05 L — 5 1.488 0.033 9 × 104 2 × 102 N Y N L 5.02 R 45 — 0.300 1.329 7 × 103 3 × 105 Y N Y R 45.01 L — 45 1.398 0.300 4 × 105 8 × 103 N Y N L 45.01 R 85 — 0.562 1.077 1 × 104 3 × 105 Y N Y R 84.25 L — 85 1.080 0.563 3 × 105 1 × 104 N Y N L 84.52 R 125 — 0.832 0.791 2 × 104 2 × 105 Y Y Y R 124.79 L — 125 0.788 0.832 2 × 105 2 × 104 Y Y N L 124.86 R 145 — 0.965 0.660 2 × 104 2 × 105 Y Y Y R 144.82 L — 145 0.655 0.966 2 × 105 2 × 104 Y Y N L 144.86 LLL R 5 — 0.034 1.488 2 × 102 9 × 104 Y N Y R 5.07 L — 5 1.488 0.034 2 × 105 4 × 102 N Y N L 5.04 R 45 — 0.300 1.373 1 × 104 6 × 105 Y N Y R 44.97 L — 45 −0.56 0.298 7 × 105 2 × 104 Y Y N L 44.69 R 85 — 0.565 1.003 3 × 104 6 × 105 Y N Y R 84.73 L — 85 1.144 0.565 6 × 105 3 × 104 N Y N L 84.82 R 125 — 0.832 0.849 4 × 104 4 × 105 Y Y Y R 124.79 L — 125 0.850 0.832 4 × 105 4 × 104 Y Y N L 124.82 R 145 — 0.965 0.713 4 × 104 5 × 105 Y Y Y R 144.70 L — 145 0.714 0.965 4 × 105 4 × 104 Y Y N L 144.75 3.1.1 Fault type, RF, α and δ Transmission lines are subject to the faults with various types and resistances. As shown in Fig. 4, the proposed FLA is tested for the fault resistances in the range of 1 mΩ to 100 Ω for five fault types, fault inception angle of 0–90° and loading angle of 5° and 30°. The error in all cases is below 0.8%. Fig 4Open in figure viewerPowerPoint Percentage errors for various faults versus α (a) With δ = 5°, (b) With δ = 30° 3.1.2 Transmission line length Fig. 5a shows the estimation error for various faults versus line lengths. It can be observed that for different line lengths and applying fault in all line locations (as a percentage of line length), estimated error is <1% in all cases. Fig 5Open in figure viewerPowerPoint Percentage errors for various faults versus variation of system parameter (a) Transmission line length, (b) Thevenin impedance variation ΔZth 3.1.3 Thevenin impedance variation In this section, sensitivity analysis of the FLA to variations of the Thevenin impedance is simulated. The Thevenin impedance is varied around the typical values (as Table 1). The impedance used in the simulation model and algorithm is the same. Also, the calculation error of the ZTh is not applied. The Thevenin impedance rate is defined as (51) Fig. 5b shows the percentage error for various faults and Thevenin impedance rates. Results are presented for various faults. According to the results, applying a ±50% of Thevenin impedance variation (ΔZth), estimation error is limited to 0.8%. It is also observed that the FLA error is relatively similar in terms of the range of the ZTh. 3.1.4 Loss of sampling synchronisation Synchronisation is a very important issue in the PMU. In fact, a 1 µs error in this synchronisation can result in an error of 0.022° at 60 Hz [26]. P-class PMUs have maximum phase errors (ɛTsync) of 8 and 62 mRad in steady and transient conditions, respectively [27]. As shown in Figs. 6a and b, the FLA is tested with the mentioned values of ɛTsync. As a result, error is <0.8%. Fig 6Open in figure viewerPowerPoint Percentage errors for faults in uncertain voltage measurement (a) Errors versus α with ɛTsync = 8 mRad, (b) Errors versus α with ɛTsync = 62 mRad, (c) Errors versus α with 2% voltage magnitude error, (d) Error versus SNR 3.1.5 Magnitude error in voltage measurements Magnitude error of the signal is also considered as distortion in the measurement. As seen from Fig. 6c, the max error does not exceed 1% for various faults with a 2% magnitude error (ɛMag). It verifies the high accuracy of the proposed algorithm. 3.1.6 Noise in the measurements In PMU measurement noise modelling, a white Gaussian noise has been investigated with SNR of 10–50 dB. Results for different faults are shown in Fig. 6d. The error is ultimately limited to 2% by applying noise with low SNR (10 dB). The error for SNR = 20 dB and above is <1%. 3.1.7 Uncertainty of line parameters Fig. 7a shows the percentage estimation error for various faults (1 mΩ–100 Ω) considering the uncertainty of line parameters. According to the results, applying a ±10% of line parameters error (ɛzy), estimation error is limited to 2.2% (52) Fig 7Open in figure viewerPowerPoint Percentage errors for various faults versus uncertainty of system (a) Line parameters uncertainty, (b) Thevenin impedance estimation error rate (ɛZth), (c) SC-MOV parameters error 3.1.8 Thevenin impedance estimation error In this paper, the typical Thevenin impedance value is used in the simulation, but the value used in FLA is varied. In fact, the FLA is tested by Thevenin impedance estimation error as (53) The estimation error of the proposed algorithm for faults located on the right-hand and left-hand sides of SC versus Thevenin impedance estimation error rate (ɛZth) is shown in Fig. 7b. As Fig. 8, max error of Thevenin impedance estimation is 3%. However, to consider a worst-case scenario, a ±10% error is considered where leads to a maximum 2.5% fault location error. Fig 8Open in figure viewerPowerPoint Percentage errors for Thevenin impedance estimation 3.1.9 SC-MOV equivalent parameter error As (17) and (19), the SC-MOV equivalent model is defined by capacitive–resistive impedance and normalised compensator current rate. The estimation error of the proposed algorithm for faults on the right-hand and left-hand sides of SC versus variations of the SC-MOV parameters is shown in Fig. 7c. Note that a ±20% parameters variation is considered for and . As shown in this figure, the error is kept below 3% in all cases. 3.1.10 CVT transients Owing to fundamental structural constraints, CVT has its own dynamic behaviour. To model CVT behaviour, the model of [23] is used. Fig. 9a shows the estimation error for different faults considering CVT transient model. As shown in this figure, the error does not exceed 2%. Fig 9Open in figure viewerPowerPoint Percentage errors for various faults considering transient and frequency effect (a) Considering CVT transient, (b) Load harmonic, (c) Sampling frequency 3.1.11 Presence of harmonic in system The effects of the harmonic load (total harmonic distortion = 4.23%) is investigated in the analysis which is shown in Fig. 9b with a maximum estimation error of 1.2%. Harmonic contents of the load current waveform are 2.04, 2.77, 1.10 and 2.20% of fundamental order for 5th, 7th, 11th and 13th orders, respectively. 3.1.12 Effect of sampling rate Variation of sampling rate (from 10 to 100 kHz) on the estimation error is investigated in Fig. 9c for different fault parameters. As this figure, the error rates are limited to 1.5%. 3.1.13 Arcing fault or high-impedance fault In this section, the effect of the arc fault on the accuracy of the algorithm is evaluated. The impedance error will be notable when there is an ineffective connection to the ground. These types of low-current faults are known as high-impedance fault (HIF). Consequently, the compensator's behaviour is assumed linear but the fault behaviour is non-linear due to the nature of the arc. To evaluate the proposed FLA in HIF, the arc fault model is carried out using the Emanuel arc model as Fig. 10a with the parameters of Table 4 [24, 25]. To study the effect of the parameters of the model, the values of the voltage sources Vp and Vn are multiplied by a coefficient called HIF factor. Also, the connection time (0.06 s) is added with Δt to consider the effect of the fault inception angle. The simulation results for LG fault are presented in Fig. 10b for different fault points and the parameters Δt and HIF factor. By increasing the number of resources used in the arc model, we can see a slight increase in the estimation error. Overall, it is observed that the estimated error for the fault location is <0.5%. Fig 10Open in figure viewerPowerPoint Fault location considering HIF model (a) HIF based on the Emanuel arc model [25], (b) Percentage errors for faults versus the HIF model parameters Table 4. Arcs parameters of HIF model [24] Breaker Switching time (on/off) Rp Rn Vp, k Vn t1 0.06 1000 1050 4 4.5 t2 0.07/0.11 2900 3000 8 8.1 t3 0.09 3500 3550 7.5 7.6 t4 0.1 3700 3750 10 10.5 t5 0.08 4000 4010 1 1.3 t6 0.13 2800 2850 3 3.5 3.2 Two SCs placed at each end of the line Fig. 11 shows the results for the case which two SCs are placed at both ends with various faults. As shown in this figure, the errors for studied cases are below 0.03%. Fig 11Open in figure viewerPowerPoint Percentage errors for various faults for two SC placed at each end 3.3 Assessment of Thevenin impedance estimation To evaluate the proposed Thevenin impedance estimator, simulations are implemented for random execution, where the Thevenin model and line loading angles are also generated randomly. The voltage and current are measured at two different operating conditions. Then, according to the least-squares method, unknown values are estimated. As shown in Fig. 8, the estimation error is <3%. 3.4 Comparative study The proposed FLA is compared with different references [5-7, 19]. Table 5 summarises parameters and considered analysis of references which should be considered in a fair comparative study. Table 5. Summarised information about proposed methods Title Parameter Method Proposed [5] [6] [7] [19] test system number of buses 2 2 2 2 2 number of circuits 1 2 1 2 1 voltage, kV 400 440 380 400 500 f, Hz 60 60 50 50 60 network model Thevenin Thevenin generator Thevenin Thevenin line model π equal π equal π equal π equal π section line length, km 300 220 300 300 320 SC parameter number of SCs 1, 2 4 1 2 0 loading angle variable fixed fixed fixed fixed SC point mid, ends ends mid mid — SC model SC-MOV SC-MOV SC-MOV SC-MOV — fault parameter type LG, LLG, LL,LLL, LLLG LG, LLG,LL, LLL LG, LLG,LL, LLL LG, LLG, LL,LLL, LLLG LG, LLG,LL, LLL resistance, Ω 1 m, 10,50, 100 0, 1, 5, 10,50, 100 1, 10,50 0.25, 0.5, 1, 5, 10,50, 100 10,50 inception angle, deg 0, 30, 60, 90 0, 90 90 0, 45, 90, 135, 180 — measuring fs, kHz 10, 20, 50, 100 2 10 1 — voltage yes yes yes yes yes current no yes yes yes no sensitivity analysis of method by: fault location * * * * * fault type * * * * * fault resistance * * * * * inception angle * * — * — compensation — — — * — CVT transients * — — * — Zth variation * — — * — Zth estimation * — — — — loading angle * — — — — line length * — — — — line parameters * * — — — phasor mag * * — — — synchronisation * * — — — noise * — — — — harmonic * — — — — sampling rate * — — — — SC mod
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