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Fault location method applied to transmission lines of general configuration
Electrical Power and Energy Systems 69 (2015) 287–294
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Fault location method applied to transmission lines of general configuration Silvio Giuseppe Di Santo ⇑, Carlos Eduardo de Morais Pereira Department of Electrical Energy and Automation Engineering, University of Sao Paulo, Av. Prof. Luciano Gualberto, trav. 3 n° 158, 05508-900 SP, Brazil
a r t i c l e
i n f o
Article history: Received 27 January 2014 Received in revised form 4 January 2015 Accepted 11 January 2015
Keywords: Fault location Transmission lines Optimization
a b s t r a c t Power quality is an important concern once automation is present in almost all industrial process. Since fault occurrences affects the power quality considerably, in this paper is proposed a new fault location method applied to transmission lines constituted of any configuration, as example double circuit, untransposed sections, and multiple derivations. In order to locate the fault, the method uses voltage and current phasors gathered from terminals with measures, however the method does not need these measures from all terminals. The proposed method is composed of three blocks to locate the fault, which are: Algorithm’s Main Control, Grid Scanning Process, and Objective Function’s Minimization Process. A large number of simulations were conducted and the results show the accuracy and efficiency of the method, even in cases of high impedance faults. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction The demand for power quality is increasingly becoming a rule in almost all industrial processes, where automation is highly present. These processes demand for an adequate voltage level and power frequency. Hence, locating faults in transmission lines with accuracy is an important need, since the repair time, in the case of permanent faults, affects considerably the reliability of the system as well as causes economic damages by the industrial processes stopping. In the literature there are many fault location methods, where it may be cited those that use data from one terminal [1–5], from two terminals [6–11], and those that use data from more than one terminal in transmission lines with derivations [12–17]. In the followings paragraphs will be briefly discussed some fault location methods. The fault location method proposed in [2] is applied to parallel transmission lines. It is based on transmission line’s distributed parameters model and require voltage and current measures from only one terminal. The method determine the fault point by means of the sequence networks (positive, negative and zero sequences), where are calculated the sequence voltages at the fault point in terms of the voltage and current measures and of the unknowns point and resistance of the fault. A concern about this method is ⇑ Corresponding author. Tel.: +55 11 97503 5310; fax: +55 11 3091 5719. E-mail address: [email protected] (S.G. Di Santo). http://dx.doi.org/10.1016/j.ijepes.2015.01.014 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
that an increase in the fault location errors may occur when untransposed transmission lines and unknown tapped loads is present. In [6] is proposed an adaptive fault location method that depends on synchronized measures gathered by Phasor Measurements Units (PMUs), being independent of measures that shall be provided by electric utilities. This method is applied only to single circuit transmission lines transmission without tapped loads. The method proposed in [12] is based on traveling waves. Using the waves’ arrival time in the terminals it is built a matrix whose coefficients are the ratio between the fault distance and the length of the branch formed by the pair of terminals considered. With this matrix the author determines the local of the fault and after the fault distance. The main concern about this method is that measurements from all terminals may not be available and as in [6] the method is applied only to single circuit transmission lines. In [13] is proposed a two-stage fault location optimization model, along with defining a matching degree index. The method could be used in large transmission networks. It is also proposed the corresponding PMU placement strategy. Once the method uses only voltages in the calculations, the results obtained it may be wrong when faults with high impedance occur. The method proposed by [14] locates faults on transmission lines constituted of single circuit. In the fault location, the method uses the current and voltage phasors measured at local and remote terminals of the transmission line. With these phasors, are calculated the current and voltage phasors at the tap points and so it
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is used some one terminal fault location method to determine the fault point. As in [6,12,13], the method is only applicable to single circuit transmission lines and the impedance of the loads have to be known, once the method is not capable to estimate them. In [15] is proposed a fault location method to locate faults on single and double circuit transmission lines, with loads connected in taps. The method uses current and voltage phasors to locate the fault point. In order to locate the fault the method needs to use two different algorithms, one for data available in two terminals and another for data available from one terminal. In order to locate the fault, the method proposed in this paper uses the current and voltage phasors gathered from terminals with measures, i.e., it is not necessary to use measures from all terminals. In this paper was considered that there are measures at two main terminals only. The main characteristic of this new method is the possibility to locate faults on transmission lines with any topology, as tapped loads, double circuits, and untransposed sections. In addition, no further examinations on fault data or the use of more than one fault location algorithm are necessary. A large number of simulations were conducted aiming to show the efficiency and accuracy of the proposed method in locating the fault. The simulations were performed in ATP program [18].
Description of the proposed fault location method The proposed method uses the following data to locate the fault: transmission line’s parameters and length; transformer’s rated data; pre-fault and post-fault current and voltage’s phasors, gathered from terminals with measures; and terminal’s equivalents, gathered from short-circuit programs. It is also necessary to perform a correction in the angles of the current and voltage’s phasors at the terminals, once the signals corresponding to these quantities may be unsynchronized among the terminals. This is done by taking one terminal as reference. It is also necessary to perform the load’s impedance estimation. In ‘Algorithm for estimation of loads’ impedance and phasors’ correction angle’ is described the algorithm to accomplish these tasks. In the proposed method all the quantities and models were developed in phase sequence (abc sequence), which allowed to address the fault location’s issues in untransposed double-circuit
transmission lines more appropriately than in symmetrical components. The proposed fault location method is composed of three blocks, which are: Algorithm’s Main Control, Grid Scanning Process, and Objective Function Minimization Process. These blocks are described in ‘Proposed fault location algorithm’. The grid depicted in Fig. 1 is used for description of the method. The notations for the quantities are summarized as follows: t, /, c, and n: respectively, number of terminals, phases, circuits, and line’s sections; pn: sections’ distance variables (or distance pointers); lengn: sections’ length; [zl] and [yl]: line’s series impedance and shunt admittance by length unit matrices. mea
calc
mea
calc
ðV ij Þpre ; ðV ij Þpre ; ðV ij Þpos , and ðV ij Þpos : measured and calculated prefault and post-fault voltages at terminal i and phase j; mea
calc
mea
calc
ðIijk Þpre ; ðIijk Þpre ; ðIijk Þpos , and ðIijk Þpos : measured and calculated prefault and post-fault currents at terminal i, circuit j, and phase k; ½V bus pre ; ½V bus pos ; ½Ibus pre , and ½Ibus pos : vectors of pre-fault and post-fault voltages and currents at the grid’s nodes; [Ybus] and [Zbus]: grid’s admittance and impedance matrices; [Yf]: fault admittance matrix; f : vector of three phases’ fault currents; ½Iabc Rf: Fault resistance; h i h i h i ð1Þ ð1Þ ð1Þ Z Lp1 ; Y L , and Y p1 : respectively, quadrupole’s series
impedance matrix between terminal L and point p1, shunt admittance matrices of terminal L side and p1 side of the section 1’s first part; h i h i h i ð1Þ ð1Þ ð1Þ ALp1 ; BLp1 , and DLp1 : quadrupole’s constants of the first part of the line’s section 1; 1 1
½Z p p : grid’s impedance matrix viewed from the point p1; h 1 ipre V pabc : vector of three phases’ pre-fault voltages at the point p1; h i h i L2 R1 R2 IL1 abc ; Iabc ; ½Iabc , and ½I abc : vectors of three phases’ post-fault currents injected into circuits 1 and 2 of the local (L) and remote (R) terminals;
Fig. 1. Grid constituted of double circuit, m derivations and n P 8 sections used for description of the method.
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h i V Labc and ½V Rabc : vector of three phases’ post-fault voltages at the local (L) and remote (R) terminals; [.]D: diagonal matrix. Algorithm for estimation of loads’ impedance and phasors’ correction angle The algorithm described in this section starts providing an initial ‘‘kick’’ (based on the grid’s loading forecast) for the load’s impedance and phasors’ correction angle to the optimization algorithm (Direct Search Algorithm [19]). Hence, the DS algorithm minimizes the objective function, whose flowchart is depicted in Fig. 2, by taking these quantities as unknowns. In (1) is showed the function to be minimized, where are calculated the differences among voltages and currents at the terminals, similarly to [15]:
mea mea i calc ij calc / ½ðV i Þ / ½ðI ij Þ t X t X c X X j pre ðV j Þpre X k pre ðIk Þpre þ e¼ mea mea i¼1 k¼1 j¼1 ðV ij Þpre ðIijk Þpre i¼1 j¼1
ð1Þ
At the terminals, in the voltage errors’ computation, was not taken into account the number of circuits once the voltages are the same for all of them (refer to Fig. 1). The voltages and currents at the terminals are calculated by means of the grid’s admittance matrix using the relationship (2) and the terminal’s equivalent impedance.
½V bus pre ¼ ½Y bus 1 ½Ibus pre
ð2Þ
The error function (1) is minimized by means of the loads’ impedance variation inside the grid’s impedance matrix and by variation of the voltage and current phasors’ angles gathered from the terminals by taking one of them as reference. The search for the minimum stops when some of stopping criteria of the optimization algorithms is satisfied (as example: number of function evaluations).
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As observed in (1), the voltages and currents measured at the terminals, whose phasors’ angles are being corrected, also are compared to the calculated ones. By using this approach, the convergence to the global minimum showed to be more efficient than using only the voltage and current phasors measured at the reference terminal. As in [15], in the cases where the loads’ feeding circuit is unknown, the optimization process is repeated for each combination of the circuits and the errors obtained for all of this combination are compared. This is an uncommon scenario and makes the process of locate the fault slower, but it is an adequate alternative for these cases. Proposed fault location algorithm The proposed algorithm is composed of three blocks. The first block, that is the Algorithm’s Main Control, supplies the data used in the fault location, performs the fault type variation (in the cases this information is not available) and outputs the fault location information to the user. The second block controls the grid’s scanning process. The third block performs the minimization of the objective function that contains the grid model, accordingly to the fault type, returning the following fault information: error, distance, and resistance. To explain the blocks functionality will be used the grid depicted in Fig. 1, where the transmission lines were divided into sections being each section’s length with begin at zero and end at lengn. Therefore, the distance pointers, pn, will vary from zero to lengn (only the pointer belonging to the section under analysis will vary while the others are fixed at the middle of their respective line’s section). Block 1 – Algorithm’s Main Control The steps performed by this block are: 1. Supplies the data used in the fault location: pre-fault and postfault voltage and current phasors, measured at local and remote terminals (L and R terminals); line’s parameters and length; impedances of the equivalents, loads, and transformers; and faulty circuit (determined by the line’s relay protection or by Block 2). 2. Verifies whether the fault type were supplied or not. 3. If the fault type is known, this information is sent to Block 2, which returns the fault information, then step 4 is skipped. But instead, if the fault type was not supplied, then Block 2 is performed for all fault types and it is storage the minimization error as well as the fault information for each fault type. 4. Compares the minimization errors obtained for each fault type and get the case with minimal error. 5. Outputs to the user the fault information, that are: distance, resistance, occurrence section, and circuit. Block 2 – Grid’s Scanning Control There are two ways for scanning the grid, which depends on whether the faulty circuit is known or not. The first way (faulty circuit known) is described as follows:
Fig. 2. Objective Function Minimization Process of the algorithm for estimation of loads’ impedance and phasors’ correction angle.
1. The scanning process takes the first transmission line’s section and the known faulty circuit as a possible container of the fault and follows to Block 3 which returns the minimization error that is associated to this section. 2. The scanning process follows to the next line’s section, where the same procedure is performed, and the minimization’s error of both line’s sections are compared, being the one with higher error dismissed. This process continues until the analysis of all line’s sections.
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The second way (faulty circuit unknown) is described as follows: 1. The scanning process takes the first line’s section and one of the circuits as a possible container of the fault and follows to Block 3 which returns the minimization error that is associated to this section’s circuit. 2. The scanning process follows to the next section’s circuit, where the same procedure is performed, and the minimization’s error of both section’s circuits are compared, being the one with higher error dismissed and then the minimization error is associated to the line’s section. 3. The scanning process follows to the next line’s section and the procedures described in the steps 1 and 2 are performed. 4. The error associated to both line’s section are then compared, being the one with higher error dismissed. This process continues until the analysis of all line’s sections. Block 3 – Objective Function Minimization Process This block describes how the objective function is minimized, where was used the Direct Search optimization algorithm [19] to perform this task. Direct Search algorithm minimizes the objective function by varying the fault’s resistance and distance. Depending on the fault type one objective function is chosen to be minimized. It is important to highlight that all quantities (currents, voltages, impedances, admittances, etc.) are in phase sequence. Aiming to describe the minimization process, is supposed that the grid’s scanning process (Block 2) is searching for faults in the line’s section 1 – circuit 1 and that the fault is of phase A-to-ground type. The distance pointers p1 to pn change the transmission line’s impedance and susceptance inside the grid’s admittance matrix, once these quantities are functions of hyperbolic cosine and sine of the distance [20]. The fault resistance (4) is also inserted into the grid’s admittance matrix. The grid’s admittance matrix is built inserting into it the admittances obtained from the series impedance and shunt admittance matrices between each grid’s nodes according to Eqs. (3)–(5). The mentioned equations were extracted from [20]. For instance, will be showed only the equations belonging to the first part of the section 1, i.e., the section between terminal L and point p1, as depicted by Fig. 3.
h h
ð1Þ
Z Lp1 ð1Þ
YL
i
i1
¼
h
ð1Þ
¼ BLp1
nh
i1
i oh i1 ð1Þ ð1Þ DLp1 ½I BLp1
ð3Þ ð4Þ
h i h i1 nh i o ð1Þ ð1Þ ð1Þ Y p1 ¼ BLp1 ALp1 ½I
where [I] is the identity matrix and (6)–(8) are the phase sequence quadrupole’s constants matrices: ð1Þ
ð6Þ
ð1Þ
ð7Þ
ð1Þ
ð8Þ
½ALp1 ¼ ½yl 1 ½M½coshðcj p1 ÞD ½M1 ½yl ½BLp1 ¼ ½yl 1 ½M½cj sinhðcj p1 ÞD ½M1 ½DLp1 ¼ ½M½coshðcj p1 ÞD ½M1
The quantity [M] and cj are, respectively, the eigenvector and eigenvalue matrices of the matrix (9):
½P ¼ ½yl ½zl
ð9Þ
Once the grid depicted in Fig. 3 have two three-phase circuits, the matrices of Eqs. (3)–(9) are of order six. The following are the steps of the objective function’s minimization process: 1. The minimization process starts setting up an initial ‘‘kick’’ for the fault resistance and for point p1 values (it will be varied from zero to leng1). The point p1 is taken as the fault point. 2. The others points are fixed at the middle of the line’s section which they belong to, once the scanning process is in the line’s section 1. 3. The pre-fault voltages on the nodes (10) are calculated by means of the grid’s impedance matrix, which it is constituted, in addition to the terminals’ bar, by the points p1 to pn as nodes, and by the vector of currents injected into the nodes.
½V bus pre ¼ ½Z bus ½Ibus pre
ð10Þ
4. The phases’ fault current (11) is calculated by means of the fault resistance, the impedance ‘‘viewed’’ from the point p1 (extracted from the grid’s impedance matrix) and by the prefault voltage at this point, as depicted by Fig. 4. 1 p1
f ½Iabc ¼ ½Y f f½Z p
1
1
þ ½Y f g ½Z p
1 p1
1
1
½V pabc
pre
ð11Þ
The fault resistance model is fault type dependent. Eq. (12) shows the fault resistance model for a phase A-to-ground fault type in phase sequence. The model for the other fault types is set up modifying the matrix coefficients:
2
1=Rf
0 0
0
0 0
6 ½Y f ¼ 4 0
Fig. 3. First part of the line’s section 1 of the grid depicted in Fig. 1, viewed in terms of matrix equivalent p circuit, representing the two line’s three-phase circuits.
ð5Þ
3
7 0 05
ð12Þ
Fig. 4. Thevenin’s equivalent for the computation of the fault current, considering the point p1 and fault admittance (12).
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5. The nodes’ post-fault voltages (13) are calculated by means of the impedance matrix and by the vector of currents injected into the nodes including the fault current, which is injected at the point p1. In this case [If] is inserted inside [Ibus]pre, transforming it into [Ibus]pos.
½V bus pos ¼ ½Z bus ½Ibus pos
ð13Þ
6. From the post-fault voltages vector is extracted the post-fault voltages at the terminals with measures (only at L and R terminals in this example). Thus, using the nodal calculated postfault voltages and the line’s equivalent model, are calculated the post-fault currents flowing in the circuits 1 and 2. 7. The calculated post-fault voltages and currents are compared to the measured ones, resulting in an error (14).
mea i calc R X c ½ðV i Þ X j pos ðV j Þpos e¼ mea ðV ij Þpos i¼L j¼a mea ij calc R X 2 X c ½ðI ij Þ X k pos ðI k Þpos þ mea ðIijk Þpos i¼L j¼1 k¼a
ð14Þ
8. Point p1 and the fault resistance are varied (performed by Direct Search optimization algorithm). 9. Steps 3–8 are repeated until one of the stopping criteria is reached (e.g. error less than an established value, number of function evaluation, etc.). The steps previously described are shown in a flowchart depicted in Fig. 5.
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Results The performance of the fault location method proposed in this paper was analyzed face to a large number of fault simulations, where the voltage and current measurements before and during the faults were generated by means of ATP program. In the simulations were included fault distance, type, and resistance variations as well as deviations in the phasors and transmission line’s parameters. It was considered that there are voltage and current measures from local (L) and remote (R) terminals only. Other consideration is that all transformers were connected in grounded wye – delta configuration aiming to include the contribution of the zero sequence current to the fault. In the simulations presented in ‘Results’, the loads’ impedance, its feeding circuit, and phasors’ correction angle were estimated by the method described in ‘Algorithm for estimation of loads’ impedance and phasors’ correction angle’. The simulations were conducted taking the grid depicted in Fig. 6 as example, where: LT 138 kV is constituted of double circuit and untransposed sections, with parameters (in symmetrical components):
r1 ¼ 0:18432 X=km; x1 ¼ 0:483 X=km; c1 ¼ 9:075 nF=km; r0 ¼ 0:62321 X=km; x0 ¼ 1:508 X=km; c0 ¼ 5:707 nF=km r0m ¼ 0:43935 X=km and x0m ¼ 0:933 X=kmðmutualÞ Transformers (138/13.8 kV): Tr1: 15 MVA, Tr2 and Tr4: 30 MVA, Tr3: 50 MVA, Tr5: 50 MVA. Xsc = 10% for all transformers. Terminals’ equivalents (in symmetrical components): Terminal L impedances: Z Leql1 ¼ 2 þ 19:5i X and Z Leql0 ¼ 5 þ 39:5i X; Terminal R impedances: Z Reql1 ¼ 1 þ 8:5i X and Z Reql0 ¼ 2 þ 18:5i X; In order to facilitate the analysis of the results, the following considerations were taken related to the fault distances and errors: The fault distances for all line’s sections are referred to the local terminal (L), thus the reference distances, for the computation of the distance relative errors, was set up as:
Fig. 5. Objective function’s minimization process of Block 3.
Fig. 6. Simulated grid for evaluation of the proposed fault location method.
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For For For For
faults faults faults faults
in in in in
sections 1, 2, and 3: 100 km; sections 5 and 6: 75 km; section 7: 80 km; sections 8 and 9: 150 km.
Fault distance variation This section presents the simulation results for fault distance variations. Fig. 7 presents the fault distance relative error results for phase to ground (AG) and three phase (ABC) fault types, both with resistance of 5 X, considering all transmission line’s sections. The relative errors were calculated taking the reference distances, as stated in ‘Results’.
For all results the relative errors were calculated taking the reference distances, as stated in ‘Results’. Fig. 9 presents the fault distance relative error results for faults located in line’s section 1 – circuit 1 at 5 km from the local terminal. Fig. 10 presents the fault distance relative error results for faults located in line’s section 4 – circuit 2 at 95 km from the local terminal. Fig. 11 presents the fault distance relative error results for faults located in line’s section 6 – circuit 2 at 67 km from the local terminal.
Fault type variation This section presents the simulation results for fault type variations. Fig. 8 presents the fault distance relative error results for faults in line’s sections 1, 4, 6, and 7 considering phase to ground (AG), three phase (ABC), double phase (AB), and double phase to ground (ABG) fault types, all with resistance of 5 X. The relative errors were calculated taking the reference distances, as stated in ‘Results’. Fault resistance variation This section presents the simulation results for fault resistance variations.
Fig. 9. Fault distance relative error results for fault resistance variations considering faults in line’s section 1 – circuit 1 at 5 km from local terminal.
Fig. 7. Fault distance relative error results for fault distance variations, where S stands for section, C for circuit, and FD for fault distance.
Fig. 10. Fault distance relative error results for fault resistance variations considering faults line’s in section 4 – circuit 2 at 95 km from local terminal.
Fig. 8. Fault distance relative error results for fault type variations, where S stands for section, C for circuit, and FD for fault distance.
Fig. 11. Fault distance relative error results for fault resistance variations considering faults in line’s section 6 – circuit 2 at 67 km from local terminal.
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S.G. Di Santo, C.E.d.M. Pereira / Electrical Power and Energy Systems 69 (2015) 287–294 Table 4 Results for line parameters errors – line’s section 7. Fault type
Max. error (m)
Min. error (m)
Mean error (m)
Std. dev. (m)
Phase to ground Three phase
639.77 332.79
26.49 39.81
196.41 172.20
152.01 90.72
Table 5 Comparison of simulated and estimated loads and circuit switch state.
Fig. 12. Fault distance relative error results for fault resistance variations considering faults in line’s section 7 – circuit 1 at 70 km from local terminal.
Load
Sim. circ.
Sim. R (X)
Sim. X (X)
Calc. circ.
Calc. R (X)
Calc. X (X)
1 2 3 4 5
2 1 1 1 1
12.06 6.03 3.62 7.24 4.52
3.96 1.98 1.19 2.38 1.49
2 1 1 1 1
12.04 6.03 3.62 8.76 4.05
3.92 1.97 1.19 2.33 1.90
Fig. 12 presents the fault distance relative error results for faults located in line’s section 7 – circuit 1 at 70 km from the local terminal.
Table 6 Comparison of simulated and estimated phasors’s synchronization angle.
Voltage and current phasors deviations Aiming to evaluate the method against phasors deviations, a large number of errors (150 in total) with normal distribution and maximum of 3% were applied to the voltages’ module, errors with maximum of 10% were applied to the currents’ module, and errors with maximum of 10° were applied to the voltages’ and currents’ angle [7]. Table 1 presents the simulation results for a fault located at 95 km, line’s section 4, circuit 2 with resistance of 5 X. Table 2 presents the simulation results for a fault located at 70 km, line’s section 7, circuit 1 with resistance of 5 X. Line parameters deviations Aiming to evaluate the method against line parameters deviations, a large number of errors (150 in total) with normal
Table 1 Results for phasors errors – line’s section 4.
Sim. sync angle
Calc. sync angle
20.00°
20.00°
distribution and maximum of 3% were applied to the line’s parameters real and imaginary parts. The base for these errors was extracted from the simulations performed varying the ground resistivity and tower height on ATP program. Table 3 presents the simulation results for a fault located at 95 km, line’s section 4, circuit 2 with resistance of 5 X. Table 4 presents the simulation results for a fault located at 70 km, line’s section 7, circuit 1 with resistance of 5 X. Performance evaluation of the loads estimation and phasor’s correction angle algorithm Tables 5 and 6 shows the performance evaluation results for the loads’ impedance estimation (as well as their feeding circuit) and for the phasors’ correction angle estimation algorithm. Discussion
Fault type
Max. error (m)
Min. error (m)
Mean error (m)
Std. dev. (m)
Phase to ground Three phase
534.85 765.26
30.35 19.16
261.31 184.86
166.04 192.05
Table 2 Results for phasors errors – line’s section 7. Fault type
Max. error (m)
Min. error (m)
Mean error (m)
Std. dev. (m)
Phase to ground Three phase
1452.7 1096.2
32.42 9.58
795.23 482.83
696.94 347.01
Table 3 Results for line parameters errors – line’s section 4. Fault type
Max. error (m)
Min. error (m)
Mean error (m)
Std. dev. (m)
Phase to ground Three phase
126.87 121.55
3.36 13.50
43.21 48.98
34.85 31.99
Analyzing the simulation results for fault distance, type, and resistance variations was observed that the errors were small compared to the line’s section length, where the maximum relative error obtained for distance variations was 0.29%, for type variations was 0.29%, and for resistance variations was 0.39%. The errors obtained for fault resistance variations indicate a good characteristic of the proposed method that can be justified by the use of non-simplified models of the grid’s components in the objective function minimization procedures. Hence, without deviations in the phasors as well as in the line’s parameter, the error obtained for any fault conditions are very small, as observed in the simulation results. For phasors deviations, which can reach 3% in the voltage module, 10% in the current module, and 10° in the angle, a maximum error of 1.82% for a fault at 70 km, line’s section 7, circuit 1 with resistance of 5 X was obtained. For the line’s parameters deviations, which can reach 3%, a maximum error of 0.8% was obtained for the fault location and conditions as previously mentioned. In relation to phasors and line’s parameters deviations, the errors, although being larger than the errors mentioned in the previous paragraphs, continue to be small compared to the transmission line’s length.
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In the case of loads’ impedance estimation (including their feeding circuit) and phasors’ correction angle algorithm, the results obtained in ‘Performance evaluation of the loads estimation and phasor’s correction angle algorithm’ show the efficiency in estimating these quantities. The time interval to locate the fault is an important issue to be considered, however in the literature review the authors did not present the time taken by their methods. Thus, in order to open a discussion about this issue, considering that the loads’ impedance are known, the time interval taken by the proposed method to locate the fault was about 58.5 s (considering all method stages, since phasor calculation from measures until the outcome of the fault location to the user). However, in the case where the impedance of the loads is unknown, which is the case of all simulations performed in this paper, the method took about 43.5 s longer (considering known the loads’ feeding circuit, which is the most common case). Therefore, the total time to locate the fault is short compared to the time taken by the utility to fix the transmission line damage. For the simulations a PC with Core i7 – 2.0 GHz processor and a RAM memory of 6 GB was used. Conclusion A new fault location method for transmission lines is proposed in this paper. The main feature is the capability to locate faults on transmission lines with any topology, as double circuits, tapped loads, and untransposed sections. In addition, the availability of measurements from all terminals is unnecessary. Futhermore, the search for the fault location is done in each line’s section independently, and then the errors associated to each one are compared, outcoming the fault information in a direct way to the user, unlike methods that first determine the faulty section and after have to apply other algorithms to determine the fault distance. The simulation results show the efficiency and accuracy of the proposed method in obtaining the fault information, such as distance, occurrence section and circuit, type, and resistance, mainly in the cases where loads’ impedance is not available. References [1] Mamisß MS, Arkan M, Kelesß C. Transmission lines fault location using transient signal spectrum. Int J Electr Power Energy Syst 2013;53(December):714–8. [2] Chaiwan P, Kang N, Liao Y. New accurate fault location algorithm for parallel transmission lines using local measurements. Electric Power Syst Res 2014;108(March):68–73.
[3] Niazy I, Sadeh J. A new single ended fault location algorithm for combined transmission line considering fault clearing transients without using line parameters. Int J Electr Power Energy Syst 2013;44(1):816–23. [4] Johns AT, Moore PJ, Whittard R. New technique for the accurate location of earth faults on transmission systems. IEE Proc-Gener, Transm Distrib 1995;142(2):119–27. [5] Takagi T, Yamakoshi Y, Yamaura M, Kondow R, Matsushima T. Development of a new type fault locator using the one-terminal voltage and current data. IEEE Power Eng Rev 1982;PER-2(8):59–60. [6] Al-Mohammed AH, Abido MA. An adaptive fault location algorithm for power system networks based on synchrophasor measurements. Electric Power Syst Res 2014;108(March):153–63. [7] Apostolopoulos CA, Korres GN. Accurate fault location algorithm for doublecircuit series compensated lines using a limited number of two-end synchronized measurements. Int J Electr Power Energy Syst 2012;42(1):495–507. [8] Fulczyk M, Balcerek P, Izykowski J, Rosolowski E, Saha MM. Two-end unsynchronized fault location algorithm for double-circuit series compensated lines. In: Power and energy society general meeting – conversion and delivery of electrical energy in the 21st century, 2008 IEEE; 2008. p. 1–9. [9] Gopalakrishnan A, Kezunovic M, McKenna SM, Hamai DM. Fault location using the distributed parameter transmission line model. IEEE Trans Power Delivery 2000;15(4):1169–74. [10] Wang B, Dong X, Bo Z, Klimek A. Impedance phase faults location algorithm for UHV transmission lines. In: Transmission and distribution conference and exposition: Latin America, 2008 IEEE/PES; 2008. p. 1–4. [11] Faybisovich V, Khoroshev MI. Frequency domain double-ended method of fault location for transmission lines. In: Transmission and distribution conference and exposition, 2008. T&D. IEEE/PES; 2008. p. 1–6. [12] Zhu Yongli, Fan Xinqiao. Fault location scheme for a multi-terminal transmission line based on current traveling waves. Int J Electr Power Energy Syst 2013;53(December):367–74. [13] Jiang Q, Li X, Wang B, Wang H. PMU-based fault location using voltage measurements in large transmission networks. IEEE Trans Power Delivery 2012;27(3):1644–52. [14] Pereira CEM, Zanetta LC. Fault location in multitapped transmission lines using unsynchronized data and superposition theorem. IEEE Trans Power Delivery 2011;26(4):2081–9. [15] Manassero G, Senger EC, Nakagomi RM, Pellini EL, Rodrigues ECN. Faultlocation system for multiterminal transmission lines. IEEE Trans Power Delivery 2010;25(3):1418–26. [16] Liao Y. Fault location for single-circuit line based on bus-impedance matrix utilizing voltage measurements. IEEE Trans Power Delivery 2008;23(2):609–17. [17] Chen Y, Liu D, Xu B. Wide-area traveling wave fault location system based on IEC61850. IEEE Trans Smart Grid 2013;4(2):1207–15. [18] ATP: Alternative Transient Program Rule Book, Leuven EMTP Center, Leuven, K.U.; 1987. [19] Hooke R, Jeeves TA. ‘‘Direct Search’’ solution of numerical and statistical problems. J ACM (JACM) 1961;8(2):212–29. [20] Bowman WI, Mcnamee JM. Development of equivalent Pi and T matrix circuits for long untransposed transmission lines. IEEE Trans Power Apparatus Syst 1964;83(6):625–32.
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Fault location method applied to transmission lines of general configuration Silvio Giuseppe Di Santo ⇑, Carlos Eduardo de Morais Pereira Department of Electrical Energy and Automation Engineering, University of Sao Paulo, Av. Prof. Luciano Gualberto, trav. 3 n° 158, 05508-900 SP, Brazil
a r t i c l e
i n f o
Article history: Received 27 January 2014 Received in revised form 4 January 2015 Accepted 11 January 2015
Keywords: Fault location Transmission lines Optimization
a b s t r a c t Power quality is an important concern once automation is present in almost all industrial process. Since fault occurrences affects the power quality considerably, in this paper is proposed a new fault location method applied to transmission lines constituted of any configuration, as example double circuit, untransposed sections, and multiple derivations. In order to locate the fault, the method uses voltage and current phasors gathered from terminals with measures, however the method does not need these measures from all terminals. The proposed method is composed of three blocks to locate the fault, which are: Algorithm’s Main Control, Grid Scanning Process, and Objective Function’s Minimization Process. A large number of simulations were conducted and the results show the accuracy and efficiency of the method, even in cases of high impedance faults. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction The demand for power quality is increasingly becoming a rule in almost all industrial processes, where automation is highly present. These processes demand for an adequate voltage level and power frequency. Hence, locating faults in transmission lines with accuracy is an important need, since the repair time, in the case of permanent faults, affects considerably the reliability of the system as well as causes economic damages by the industrial processes stopping. In the literature there are many fault location methods, where it may be cited those that use data from one terminal [1–5], from two terminals [6–11], and those that use data from more than one terminal in transmission lines with derivations [12–17]. In the followings paragraphs will be briefly discussed some fault location methods. The fault location method proposed in [2] is applied to parallel transmission lines. It is based on transmission line’s distributed parameters model and require voltage and current measures from only one terminal. The method determine the fault point by means of the sequence networks (positive, negative and zero sequences), where are calculated the sequence voltages at the fault point in terms of the voltage and current measures and of the unknowns point and resistance of the fault. A concern about this method is ⇑ Corresponding author. Tel.: +55 11 97503 5310; fax: +55 11 3091 5719. E-mail address: [email protected] (S.G. Di Santo). http://dx.doi.org/10.1016/j.ijepes.2015.01.014 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
that an increase in the fault location errors may occur when untransposed transmission lines and unknown tapped loads is present. In [6] is proposed an adaptive fault location method that depends on synchronized measures gathered by Phasor Measurements Units (PMUs), being independent of measures that shall be provided by electric utilities. This method is applied only to single circuit transmission lines transmission without tapped loads. The method proposed in [12] is based on traveling waves. Using the waves’ arrival time in the terminals it is built a matrix whose coefficients are the ratio between the fault distance and the length of the branch formed by the pair of terminals considered. With this matrix the author determines the local of the fault and after the fault distance. The main concern about this method is that measurements from all terminals may not be available and as in [6] the method is applied only to single circuit transmission lines. In [13] is proposed a two-stage fault location optimization model, along with defining a matching degree index. The method could be used in large transmission networks. It is also proposed the corresponding PMU placement strategy. Once the method uses only voltages in the calculations, the results obtained it may be wrong when faults with high impedance occur. The method proposed by [14] locates faults on transmission lines constituted of single circuit. In the fault location, the method uses the current and voltage phasors measured at local and remote terminals of the transmission line. With these phasors, are calculated the current and voltage phasors at the tap points and so it
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is used some one terminal fault location method to determine the fault point. As in [6,12,13], the method is only applicable to single circuit transmission lines and the impedance of the loads have to be known, once the method is not capable to estimate them. In [15] is proposed a fault location method to locate faults on single and double circuit transmission lines, with loads connected in taps. The method uses current and voltage phasors to locate the fault point. In order to locate the fault the method needs to use two different algorithms, one for data available in two terminals and another for data available from one terminal. In order to locate the fault, the method proposed in this paper uses the current and voltage phasors gathered from terminals with measures, i.e., it is not necessary to use measures from all terminals. In this paper was considered that there are measures at two main terminals only. The main characteristic of this new method is the possibility to locate faults on transmission lines with any topology, as tapped loads, double circuits, and untransposed sections. In addition, no further examinations on fault data or the use of more than one fault location algorithm are necessary. A large number of simulations were conducted aiming to show the efficiency and accuracy of the proposed method in locating the fault. The simulations were performed in ATP program [18].
Description of the proposed fault location method The proposed method uses the following data to locate the fault: transmission line’s parameters and length; transformer’s rated data; pre-fault and post-fault current and voltage’s phasors, gathered from terminals with measures; and terminal’s equivalents, gathered from short-circuit programs. It is also necessary to perform a correction in the angles of the current and voltage’s phasors at the terminals, once the signals corresponding to these quantities may be unsynchronized among the terminals. This is done by taking one terminal as reference. It is also necessary to perform the load’s impedance estimation. In ‘Algorithm for estimation of loads’ impedance and phasors’ correction angle’ is described the algorithm to accomplish these tasks. In the proposed method all the quantities and models were developed in phase sequence (abc sequence), which allowed to address the fault location’s issues in untransposed double-circuit
transmission lines more appropriately than in symmetrical components. The proposed fault location method is composed of three blocks, which are: Algorithm’s Main Control, Grid Scanning Process, and Objective Function Minimization Process. These blocks are described in ‘Proposed fault location algorithm’. The grid depicted in Fig. 1 is used for description of the method. The notations for the quantities are summarized as follows: t, /, c, and n: respectively, number of terminals, phases, circuits, and line’s sections; pn: sections’ distance variables (or distance pointers); lengn: sections’ length; [zl] and [yl]: line’s series impedance and shunt admittance by length unit matrices. mea
calc
mea
calc
ðV ij Þpre ; ðV ij Þpre ; ðV ij Þpos , and ðV ij Þpos : measured and calculated prefault and post-fault voltages at terminal i and phase j; mea
calc
mea
calc
ðIijk Þpre ; ðIijk Þpre ; ðIijk Þpos , and ðIijk Þpos : measured and calculated prefault and post-fault currents at terminal i, circuit j, and phase k; ½V bus pre ; ½V bus pos ; ½Ibus pre , and ½Ibus pos : vectors of pre-fault and post-fault voltages and currents at the grid’s nodes; [Ybus] and [Zbus]: grid’s admittance and impedance matrices; [Yf]: fault admittance matrix; f : vector of three phases’ fault currents; ½Iabc Rf: Fault resistance; h i h i h i ð1Þ ð1Þ ð1Þ Z Lp1 ; Y L , and Y p1 : respectively, quadrupole’s series
impedance matrix between terminal L and point p1, shunt admittance matrices of terminal L side and p1 side of the section 1’s first part; h i h i h i ð1Þ ð1Þ ð1Þ ALp1 ; BLp1 , and DLp1 : quadrupole’s constants of the first part of the line’s section 1; 1 1
½Z p p : grid’s impedance matrix viewed from the point p1; h 1 ipre V pabc : vector of three phases’ pre-fault voltages at the point p1; h i h i L2 R1 R2 IL1 abc ; Iabc ; ½Iabc , and ½I abc : vectors of three phases’ post-fault currents injected into circuits 1 and 2 of the local (L) and remote (R) terminals;
Fig. 1. Grid constituted of double circuit, m derivations and n P 8 sections used for description of the method.
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h i V Labc and ½V Rabc : vector of three phases’ post-fault voltages at the local (L) and remote (R) terminals; [.]D: diagonal matrix. Algorithm for estimation of loads’ impedance and phasors’ correction angle The algorithm described in this section starts providing an initial ‘‘kick’’ (based on the grid’s loading forecast) for the load’s impedance and phasors’ correction angle to the optimization algorithm (Direct Search Algorithm [19]). Hence, the DS algorithm minimizes the objective function, whose flowchart is depicted in Fig. 2, by taking these quantities as unknowns. In (1) is showed the function to be minimized, where are calculated the differences among voltages and currents at the terminals, similarly to [15]:
mea mea i calc ij calc / ½ðV i Þ / ½ðI ij Þ t X t X c X X j pre ðV j Þpre X k pre ðIk Þpre þ e¼ mea mea i¼1 k¼1 j¼1 ðV ij Þpre ðIijk Þpre i¼1 j¼1
ð1Þ
At the terminals, in the voltage errors’ computation, was not taken into account the number of circuits once the voltages are the same for all of them (refer to Fig. 1). The voltages and currents at the terminals are calculated by means of the grid’s admittance matrix using the relationship (2) and the terminal’s equivalent impedance.
½V bus pre ¼ ½Y bus 1 ½Ibus pre
ð2Þ
The error function (1) is minimized by means of the loads’ impedance variation inside the grid’s impedance matrix and by variation of the voltage and current phasors’ angles gathered from the terminals by taking one of them as reference. The search for the minimum stops when some of stopping criteria of the optimization algorithms is satisfied (as example: number of function evaluations).
289
As observed in (1), the voltages and currents measured at the terminals, whose phasors’ angles are being corrected, also are compared to the calculated ones. By using this approach, the convergence to the global minimum showed to be more efficient than using only the voltage and current phasors measured at the reference terminal. As in [15], in the cases where the loads’ feeding circuit is unknown, the optimization process is repeated for each combination of the circuits and the errors obtained for all of this combination are compared. This is an uncommon scenario and makes the process of locate the fault slower, but it is an adequate alternative for these cases. Proposed fault location algorithm The proposed algorithm is composed of three blocks. The first block, that is the Algorithm’s Main Control, supplies the data used in the fault location, performs the fault type variation (in the cases this information is not available) and outputs the fault location information to the user. The second block controls the grid’s scanning process. The third block performs the minimization of the objective function that contains the grid model, accordingly to the fault type, returning the following fault information: error, distance, and resistance. To explain the blocks functionality will be used the grid depicted in Fig. 1, where the transmission lines were divided into sections being each section’s length with begin at zero and end at lengn. Therefore, the distance pointers, pn, will vary from zero to lengn (only the pointer belonging to the section under analysis will vary while the others are fixed at the middle of their respective line’s section). Block 1 – Algorithm’s Main Control The steps performed by this block are: 1. Supplies the data used in the fault location: pre-fault and postfault voltage and current phasors, measured at local and remote terminals (L and R terminals); line’s parameters and length; impedances of the equivalents, loads, and transformers; and faulty circuit (determined by the line’s relay protection or by Block 2). 2. Verifies whether the fault type were supplied or not. 3. If the fault type is known, this information is sent to Block 2, which returns the fault information, then step 4 is skipped. But instead, if the fault type was not supplied, then Block 2 is performed for all fault types and it is storage the minimization error as well as the fault information for each fault type. 4. Compares the minimization errors obtained for each fault type and get the case with minimal error. 5. Outputs to the user the fault information, that are: distance, resistance, occurrence section, and circuit. Block 2 – Grid’s Scanning Control There are two ways for scanning the grid, which depends on whether the faulty circuit is known or not. The first way (faulty circuit known) is described as follows:
Fig. 2. Objective Function Minimization Process of the algorithm for estimation of loads’ impedance and phasors’ correction angle.
1. The scanning process takes the first transmission line’s section and the known faulty circuit as a possible container of the fault and follows to Block 3 which returns the minimization error that is associated to this section. 2. The scanning process follows to the next line’s section, where the same procedure is performed, and the minimization’s error of both line’s sections are compared, being the one with higher error dismissed. This process continues until the analysis of all line’s sections.
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The second way (faulty circuit unknown) is described as follows: 1. The scanning process takes the first line’s section and one of the circuits as a possible container of the fault and follows to Block 3 which returns the minimization error that is associated to this section’s circuit. 2. The scanning process follows to the next section’s circuit, where the same procedure is performed, and the minimization’s error of both section’s circuits are compared, being the one with higher error dismissed and then the minimization error is associated to the line’s section. 3. The scanning process follows to the next line’s section and the procedures described in the steps 1 and 2 are performed. 4. The error associated to both line’s section are then compared, being the one with higher error dismissed. This process continues until the analysis of all line’s sections. Block 3 – Objective Function Minimization Process This block describes how the objective function is minimized, where was used the Direct Search optimization algorithm [19] to perform this task. Direct Search algorithm minimizes the objective function by varying the fault’s resistance and distance. Depending on the fault type one objective function is chosen to be minimized. It is important to highlight that all quantities (currents, voltages, impedances, admittances, etc.) are in phase sequence. Aiming to describe the minimization process, is supposed that the grid’s scanning process (Block 2) is searching for faults in the line’s section 1 – circuit 1 and that the fault is of phase A-to-ground type. The distance pointers p1 to pn change the transmission line’s impedance and susceptance inside the grid’s admittance matrix, once these quantities are functions of hyperbolic cosine and sine of the distance [20]. The fault resistance (4) is also inserted into the grid’s admittance matrix. The grid’s admittance matrix is built inserting into it the admittances obtained from the series impedance and shunt admittance matrices between each grid’s nodes according to Eqs. (3)–(5). The mentioned equations were extracted from [20]. For instance, will be showed only the equations belonging to the first part of the section 1, i.e., the section between terminal L and point p1, as depicted by Fig. 3.
h h
ð1Þ
Z Lp1 ð1Þ
YL
i
i1
¼
h
ð1Þ
¼ BLp1
nh
i1
i oh i1 ð1Þ ð1Þ DLp1 ½I BLp1
ð3Þ ð4Þ
h i h i1 nh i o ð1Þ ð1Þ ð1Þ Y p1 ¼ BLp1 ALp1 ½I
where [I] is the identity matrix and (6)–(8) are the phase sequence quadrupole’s constants matrices: ð1Þ
ð6Þ
ð1Þ
ð7Þ
ð1Þ
ð8Þ
½ALp1 ¼ ½yl 1 ½M½coshðcj p1 ÞD ½M1 ½yl ½BLp1 ¼ ½yl 1 ½M½cj sinhðcj p1 ÞD ½M1 ½DLp1 ¼ ½M½coshðcj p1 ÞD ½M1
The quantity [M] and cj are, respectively, the eigenvector and eigenvalue matrices of the matrix (9):
½P ¼ ½yl ½zl
ð9Þ
Once the grid depicted in Fig. 3 have two three-phase circuits, the matrices of Eqs. (3)–(9) are of order six. The following are the steps of the objective function’s minimization process: 1. The minimization process starts setting up an initial ‘‘kick’’ for the fault resistance and for point p1 values (it will be varied from zero to leng1). The point p1 is taken as the fault point. 2. The others points are fixed at the middle of the line’s section which they belong to, once the scanning process is in the line’s section 1. 3. The pre-fault voltages on the nodes (10) are calculated by means of the grid’s impedance matrix, which it is constituted, in addition to the terminals’ bar, by the points p1 to pn as nodes, and by the vector of currents injected into the nodes.
½V bus pre ¼ ½Z bus ½Ibus pre
ð10Þ
4. The phases’ fault current (11) is calculated by means of the fault resistance, the impedance ‘‘viewed’’ from the point p1 (extracted from the grid’s impedance matrix) and by the prefault voltage at this point, as depicted by Fig. 4. 1 p1
f ½Iabc ¼ ½Y f f½Z p
1
1
þ ½Y f g ½Z p
1 p1
1
1
½V pabc
pre
ð11Þ
The fault resistance model is fault type dependent. Eq. (12) shows the fault resistance model for a phase A-to-ground fault type in phase sequence. The model for the other fault types is set up modifying the matrix coefficients:
2
1=Rf
0 0
0
0 0
6 ½Y f ¼ 4 0
Fig. 3. First part of the line’s section 1 of the grid depicted in Fig. 1, viewed in terms of matrix equivalent p circuit, representing the two line’s three-phase circuits.
ð5Þ
3
7 0 05
ð12Þ
Fig. 4. Thevenin’s equivalent for the computation of the fault current, considering the point p1 and fault admittance (12).
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5. The nodes’ post-fault voltages (13) are calculated by means of the impedance matrix and by the vector of currents injected into the nodes including the fault current, which is injected at the point p1. In this case [If] is inserted inside [Ibus]pre, transforming it into [Ibus]pos.
½V bus pos ¼ ½Z bus ½Ibus pos
ð13Þ
6. From the post-fault voltages vector is extracted the post-fault voltages at the terminals with measures (only at L and R terminals in this example). Thus, using the nodal calculated postfault voltages and the line’s equivalent model, are calculated the post-fault currents flowing in the circuits 1 and 2. 7. The calculated post-fault voltages and currents are compared to the measured ones, resulting in an error (14).
mea i calc R X c ½ðV i Þ X j pos ðV j Þpos e¼ mea ðV ij Þpos i¼L j¼a mea ij calc R X 2 X c ½ðI ij Þ X k pos ðI k Þpos þ mea ðIijk Þpos i¼L j¼1 k¼a
ð14Þ
8. Point p1 and the fault resistance are varied (performed by Direct Search optimization algorithm). 9. Steps 3–8 are repeated until one of the stopping criteria is reached (e.g. error less than an established value, number of function evaluation, etc.). The steps previously described are shown in a flowchart depicted in Fig. 5.
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Results The performance of the fault location method proposed in this paper was analyzed face to a large number of fault simulations, where the voltage and current measurements before and during the faults were generated by means of ATP program. In the simulations were included fault distance, type, and resistance variations as well as deviations in the phasors and transmission line’s parameters. It was considered that there are voltage and current measures from local (L) and remote (R) terminals only. Other consideration is that all transformers were connected in grounded wye – delta configuration aiming to include the contribution of the zero sequence current to the fault. In the simulations presented in ‘Results’, the loads’ impedance, its feeding circuit, and phasors’ correction angle were estimated by the method described in ‘Algorithm for estimation of loads’ impedance and phasors’ correction angle’. The simulations were conducted taking the grid depicted in Fig. 6 as example, where: LT 138 kV is constituted of double circuit and untransposed sections, with parameters (in symmetrical components):
r1 ¼ 0:18432 X=km; x1 ¼ 0:483 X=km; c1 ¼ 9:075 nF=km; r0 ¼ 0:62321 X=km; x0 ¼ 1:508 X=km; c0 ¼ 5:707 nF=km r0m ¼ 0:43935 X=km and x0m ¼ 0:933 X=kmðmutualÞ Transformers (138/13.8 kV): Tr1: 15 MVA, Tr2 and Tr4: 30 MVA, Tr3: 50 MVA, Tr5: 50 MVA. Xsc = 10% for all transformers. Terminals’ equivalents (in symmetrical components): Terminal L impedances: Z Leql1 ¼ 2 þ 19:5i X and Z Leql0 ¼ 5 þ 39:5i X; Terminal R impedances: Z Reql1 ¼ 1 þ 8:5i X and Z Reql0 ¼ 2 þ 18:5i X; In order to facilitate the analysis of the results, the following considerations were taken related to the fault distances and errors: The fault distances for all line’s sections are referred to the local terminal (L), thus the reference distances, for the computation of the distance relative errors, was set up as:
Fig. 5. Objective function’s minimization process of Block 3.
Fig. 6. Simulated grid for evaluation of the proposed fault location method.
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For For For For
faults faults faults faults
in in in in
sections 1, 2, and 3: 100 km; sections 5 and 6: 75 km; section 7: 80 km; sections 8 and 9: 150 km.
Fault distance variation This section presents the simulation results for fault distance variations. Fig. 7 presents the fault distance relative error results for phase to ground (AG) and three phase (ABC) fault types, both with resistance of 5 X, considering all transmission line’s sections. The relative errors were calculated taking the reference distances, as stated in ‘Results’.
For all results the relative errors were calculated taking the reference distances, as stated in ‘Results’. Fig. 9 presents the fault distance relative error results for faults located in line’s section 1 – circuit 1 at 5 km from the local terminal. Fig. 10 presents the fault distance relative error results for faults located in line’s section 4 – circuit 2 at 95 km from the local terminal. Fig. 11 presents the fault distance relative error results for faults located in line’s section 6 – circuit 2 at 67 km from the local terminal.
Fault type variation This section presents the simulation results for fault type variations. Fig. 8 presents the fault distance relative error results for faults in line’s sections 1, 4, 6, and 7 considering phase to ground (AG), three phase (ABC), double phase (AB), and double phase to ground (ABG) fault types, all with resistance of 5 X. The relative errors were calculated taking the reference distances, as stated in ‘Results’. Fault resistance variation This section presents the simulation results for fault resistance variations.
Fig. 9. Fault distance relative error results for fault resistance variations considering faults in line’s section 1 – circuit 1 at 5 km from local terminal.
Fig. 7. Fault distance relative error results for fault distance variations, where S stands for section, C for circuit, and FD for fault distance.
Fig. 10. Fault distance relative error results for fault resistance variations considering faults line’s in section 4 – circuit 2 at 95 km from local terminal.
Fig. 8. Fault distance relative error results for fault type variations, where S stands for section, C for circuit, and FD for fault distance.
Fig. 11. Fault distance relative error results for fault resistance variations considering faults in line’s section 6 – circuit 2 at 67 km from local terminal.
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S.G. Di Santo, C.E.d.M. Pereira / Electrical Power and Energy Systems 69 (2015) 287–294 Table 4 Results for line parameters errors – line’s section 7. Fault type
Max. error (m)
Min. error (m)
Mean error (m)
Std. dev. (m)
Phase to ground Three phase
639.77 332.79
26.49 39.81
196.41 172.20
152.01 90.72
Table 5 Comparison of simulated and estimated loads and circuit switch state.
Fig. 12. Fault distance relative error results for fault resistance variations considering faults in line’s section 7 – circuit 1 at 70 km from local terminal.
Load
Sim. circ.
Sim. R (X)
Sim. X (X)
Calc. circ.
Calc. R (X)
Calc. X (X)
1 2 3 4 5
2 1 1 1 1
12.06 6.03 3.62 7.24 4.52
3.96 1.98 1.19 2.38 1.49
2 1 1 1 1
12.04 6.03 3.62 8.76 4.05
3.92 1.97 1.19 2.33 1.90
Fig. 12 presents the fault distance relative error results for faults located in line’s section 7 – circuit 1 at 70 km from the local terminal.
Table 6 Comparison of simulated and estimated phasors’s synchronization angle.
Voltage and current phasors deviations Aiming to evaluate the method against phasors deviations, a large number of errors (150 in total) with normal distribution and maximum of 3% were applied to the voltages’ module, errors with maximum of 10% were applied to the currents’ module, and errors with maximum of 10° were applied to the voltages’ and currents’ angle [7]. Table 1 presents the simulation results for a fault located at 95 km, line’s section 4, circuit 2 with resistance of 5 X. Table 2 presents the simulation results for a fault located at 70 km, line’s section 7, circuit 1 with resistance of 5 X. Line parameters deviations Aiming to evaluate the method against line parameters deviations, a large number of errors (150 in total) with normal
Table 1 Results for phasors errors – line’s section 4.
Sim. sync angle
Calc. sync angle
20.00°
20.00°
distribution and maximum of 3% were applied to the line’s parameters real and imaginary parts. The base for these errors was extracted from the simulations performed varying the ground resistivity and tower height on ATP program. Table 3 presents the simulation results for a fault located at 95 km, line’s section 4, circuit 2 with resistance of 5 X. Table 4 presents the simulation results for a fault located at 70 km, line’s section 7, circuit 1 with resistance of 5 X. Performance evaluation of the loads estimation and phasor’s correction angle algorithm Tables 5 and 6 shows the performance evaluation results for the loads’ impedance estimation (as well as their feeding circuit) and for the phasors’ correction angle estimation algorithm. Discussion
Fault type
Max. error (m)
Min. error (m)
Mean error (m)
Std. dev. (m)
Phase to ground Three phase
534.85 765.26
30.35 19.16
261.31 184.86
166.04 192.05
Table 2 Results for phasors errors – line’s section 7. Fault type
Max. error (m)
Min. error (m)
Mean error (m)
Std. dev. (m)
Phase to ground Three phase
1452.7 1096.2
32.42 9.58
795.23 482.83
696.94 347.01
Table 3 Results for line parameters errors – line’s section 4. Fault type
Max. error (m)
Min. error (m)
Mean error (m)
Std. dev. (m)
Phase to ground Three phase
126.87 121.55
3.36 13.50
43.21 48.98
34.85 31.99
Analyzing the simulation results for fault distance, type, and resistance variations was observed that the errors were small compared to the line’s section length, where the maximum relative error obtained for distance variations was 0.29%, for type variations was 0.29%, and for resistance variations was 0.39%. The errors obtained for fault resistance variations indicate a good characteristic of the proposed method that can be justified by the use of non-simplified models of the grid’s components in the objective function minimization procedures. Hence, without deviations in the phasors as well as in the line’s parameter, the error obtained for any fault conditions are very small, as observed in the simulation results. For phasors deviations, which can reach 3% in the voltage module, 10% in the current module, and 10° in the angle, a maximum error of 1.82% for a fault at 70 km, line’s section 7, circuit 1 with resistance of 5 X was obtained. For the line’s parameters deviations, which can reach 3%, a maximum error of 0.8% was obtained for the fault location and conditions as previously mentioned. In relation to phasors and line’s parameters deviations, the errors, although being larger than the errors mentioned in the previous paragraphs, continue to be small compared to the transmission line’s length.
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In the case of loads’ impedance estimation (including their feeding circuit) and phasors’ correction angle algorithm, the results obtained in ‘Performance evaluation of the loads estimation and phasor’s correction angle algorithm’ show the efficiency in estimating these quantities. The time interval to locate the fault is an important issue to be considered, however in the literature review the authors did not present the time taken by their methods. Thus, in order to open a discussion about this issue, considering that the loads’ impedance are known, the time interval taken by the proposed method to locate the fault was about 58.5 s (considering all method stages, since phasor calculation from measures until the outcome of the fault location to the user). However, in the case where the impedance of the loads is unknown, which is the case of all simulations performed in this paper, the method took about 43.5 s longer (considering known the loads’ feeding circuit, which is the most common case). Therefore, the total time to locate the fault is short compared to the time taken by the utility to fix the transmission line damage. For the simulations a PC with Core i7 – 2.0 GHz processor and a RAM memory of 6 GB was used. Conclusion A new fault location method for transmission lines is proposed in this paper. The main feature is the capability to locate faults on transmission lines with any topology, as double circuits, tapped loads, and untransposed sections. In addition, the availability of measurements from all terminals is unnecessary. Futhermore, the search for the fault location is done in each line’s section independently, and then the errors associated to each one are compared, outcoming the fault information in a direct way to the user, unlike methods that first determine the faulty section and after have to apply other algorithms to determine the fault distance. The simulation results show the efficiency and accuracy of the proposed method in obtaining the fault information, such as distance, occurrence section and circuit, type, and resistance, mainly in the cases where loads’ impedance is not available. References [1] Mamisß MS, Arkan M, Kelesß C. Transmission lines fault location using transient signal spectrum. Int J Electr Power Energy Syst 2013;53(December):714–8. [2] Chaiwan P, Kang N, Liao Y. New accurate fault location algorithm for parallel transmission lines using local measurements. Electric Power Syst Res 2014;108(March):68–73.
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