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Time-domain fault location algorithm for parallel transmission lines using unsynchronized currents
Electrical Power and Energy Systems 28 (2006) 253–260 www.elsevier.com/locate/ijepes
Time-domain fault location algorithm for parallel transmission lines using unsynchronized currents Suonan Jiale a,*, Song Guobing a, Xu Qingqiang a, Chao Qin b,1 a
Department of Electric Power system and Automation, School of Electric Engineering, Xi’an Jiaotong University, 28 Xianning West Road, Xi’an 710049, People’s Republic of China b Department of Electric Power system and Automation, School of Electric Engineering, Xin Jiang University, 21 Youhao North Road, Urumchi 830008, People’s Republic of China Received 16 February 2004; received in revised form 26 September 2005; accepted 10 November 2005
Abstract Parallel transmission lines under fault can be decoupled into common component net and differential component net. For the differential component net is only composed by parallel lines part and its bus voltages are zero, the voltage distributions along differential component net can be calculated by each terminal current. The novel time-domain fault location algorithm is founded by voltage distributions from both terminals have the least difference at the fault point. The synchronization is achieved by searching the minimal value of locating criterion function, which is calculated by fixing one terminal data and moving another terminal data within a setting time. The proposed algorithm needs a very short datawindow, any segment of fault data from transient to steady state can be used to locate faults, and no voltage data is needed. With distributed parameter model, simulations show that the locating errors are less than 0.1 km, and the fault types and fault resistances have no effect on locating accuracy. q 2005 Elsevier Ltd. All rights reserved. Keywords: Parallel transmission lines; Differential component net; Fault location
1. Introduction The application of high-speed protective relay device leading the fault clearing time is shortened greatly, many faults have no obvious arcing marks, it is always hard to avoid crossing intricate terrain in long-range transmission, and faults always take place in the bad weather. For the reasons above mentioned, the locating reliability and accuracy are targets for protective relaying researchers. A super fault location algorithm will do good to find fault point and save line inspecting time, manpower, material resource and financial power. And it is helpful to restore the service and decrease the loss of power failure. With the narrow corridor, parallel transmission lines are in favor of saving soil, investment and construction time, so it will * Corresponding author. Tel.: C86 29 82668782; fax: C86 29 82665489. E-mail addresses: [email protected] (S. Jiale), [email protected] (S. Guobing), [email protected] (X. Qingqiang), [email protected] (C. Qin). 1 Tel.: C86 991 4546680; fax: C86 991 4546680.
0142-0615/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2005.11.013
evolve as primary mode of electric energy transmission. In many countries, high voltage transmission lines and city power grids mainly adopt parallel lines. Though the structure has an increasing application in power system, it is hard to find an accurate fault location algorithm due to the mutual induction between circuits. Varieties of algorithms developed for locating faults on parallel transmission lines [1–10], differing in many aspects, have been proposed so far. Ref. [1] introduces a new concept ‘distance factor’, which is the quotient of the pure-fault currents of both circuits at the local end. The fault location is actualized by comparing this ‘distance factor’ calculated by pure-fault currents with that of calculated by the system parameter when fault occurs. So, this method needs the source impedances involved. Ref. [2] constructs voltage equations both from fault point via faulty circuit to local end and from fault point via healthy circuit to local end, and eliminates remote infeed by bringing one of the equations into the other. According to boundary condition, fault distance can be computed. Refs. [3,4] are aimed at the single phase-to-earth and phase-to-phase faults on one circuit of parallel transmission lines, respectively. Similar to Ref. [2], the main idea is eliminating remote infeed by voltage equations
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of healthy circuit. Ref. [5] eliminates the remote infeed by computing the positive-sequence and zero-sequence current distribution factors according KVL equations around the parallel lines loops. Ref. [6] locates faults by calculating the zero-sequence current distribution of faulty circuit. According to the imaginary part of fault current is zero, and then iterative technique is used. Ref. [7] considers that the uncertainty of zero-sequence impedance have an adverse influence on locating accuracy, and reestablishes the weighting coefficients of sequence components for calculating voltage drop across the fault path. Comparing with Refs. [5,6], Ref. [7] is an improved method. The single-terminal algorithms presented by Refs. [1–7] are lumped parameter model based. Though the methods adopted by Refs. [2–7] are different, the intentions of those methods are eliminating the remote infeed by KVL equations around the parallel lines loops. A distributed parameter model based single-terminal algorithms proposed by Ref. [8] uses differential component nets of parallel lines as the locating networks. It makes fault location on parallel transmission lines easier than ever. Though algorithm presented by Ref. [8] has high accuracy by engaging distributed parameter line model, like above-mentioned algorithms it needs a data window longer than one cycle. The need of phasor estimation introduces additional difficulty in high-speed tripping situations where the algorithms may not be fast enough in determining fault location accurately before the current signals disappear due to the relay operation and breaker opening [11]. In this case, the fault location algorithms with short data-window will be a solution. In fact, the phasor estimation needs a data window longer than one cycle if high estimation accuracy is pursued. Especially in fault initial stages, because the harmonic is plentiful, it is hard for a shortwindow algorithm for phasor estimation with a higher accuracy. Another solution is travelling wave method. Travelling wave based fault location, representative by Gale et al. [12], needs a short data-window, but this method locates fault by distinguishing wave-heads from transient data and it is hard to be realized by computer automatically. Additionally, the fault location will be abortive if there is no wave process. Comparing with single-terminal algorithms, there are few two/multi terminals methods presented for parallel lines. Ref. [9] presents a lumped parallel model based algorithm for multiterminal parallel transmission lines. Ref. [10] provides two terminals algorithm based on distributed parameter model, but the locating process is complicated for needing six phases’ voltage distributions calculated. For differential component net based algorithm makes fault location on parallel lines is just like single circuit one, the single circuit line algorithms [13,14] are referencable. Ref. [13] needs a data window just longer than 1/4 cycle, and the principle is searching the minimal value of criterion function calculated by voltage distributions along the transmission line. So this algorithm can fit the situations of high-speed tripping [11]. Ref. [14] divides the location process into two parts, the approximate fault location and the refining fault location. The approximate fault location engages the time-domain method, and the maximum error is 7.5 km for 20 kHz sampling rate
used. The refining process is using frequency domain (phasor) method to search fault point within the two segments beside the approximate fault point provided by former process. For frequency domain method is used, the later process needs a longer data window. As the time-domain method of fault location needs a short data window, and it can use samples directly without filter algorithms involved [15]. Considering high-speed tripping has an adverse influence on phasor based fault location algorithm, it is easy to select distributed parameter line model based time-domain algorithms. For locating parallel lines faults, a novel time-domain algorithm only using two terminals currents is present in this paper. Based on differential component net, the novel algorithm is easier than the traditional ones for needless to consider the coupling between circuits. Additionally, the voltage at the buses of differential component net is zero, and no voltage is needed in the new algorithm. The cut-off frequency of capacitance voltage transformer (CVT) is too low to transfer transient signal and that of current transformer (CT) can reach 100 kHz or more [16], so the proposed algorithm only using current data will be practical. Considering the unsynchronized data, this paper presents a new time-domain synchronization method, which is different from Ref. [17]. With the development of electronic technology, high-speed sampling devices are widely available for making digital fault recorder. Using 500 kHz sampling rate, the fault location can be implemented in time-domain and the refining process in Ref. [14] is omitted. Additionally, the spline interpolation is used to improve accuracy by adding 10 points into between samples. Comparing with Refs. [13–15], the proposed method needs a shorter data window. 2. Differential component net of parallel transmission lines The parallel transmission lines is a kind of parallel operating state. With the identical line parameter, the currents of both circuits are same in normal operating conditions and different in fault conditions. A segment of parallel transmission lines and its coupling relations are shown as Fig. 1. The telegraph equations of parallel lines can be written as 8 vu vi > > ZKRiKL > > < vx vt (1) vi vu > > > ZK GuKC > : vx vt
Fig. 1. Coupling relations of parallel transmission lines.
S. Jiale et al. / Electrical Power and Energy Systems 28 (2006) 253–260
255
where 2
3
2
i1a
2
3
Rs
u1a 6 6 7 6R 7 6 i1b 7 6 6 m 6 7 6 u1b 7 6 7 6 7 6 6 7 6i 7 6 6 Rm u 6 1c 7 6 1c 7 7 i Z6 7 R Z6 uZ6 6 7 6 7 6 6 R0 6 i2a 7 6 u2a 7 6 m 7 6 7 6 6 6 7 6u 7 6 0 6 i2b 7 4 2b 5 6 Rm 4 5 4 u2c i2c Rm0 2 0 Gs Gm Gm Gm Gm0 6 0 0 6G 6 m Gs Gm Gm Gm 6 6 6 Gm Gm Gs Gm0 Gm0 6 G Z6 6 G0 G0 G0 G G 6 m m m s m 6 6 0 0 0 6 Gm Gm Gm Gm Gs 4 Gm0
Gm0
Gm0
Gm
Rm
Rm
Rm0
Rm0
Rs
Rm
Rm0
Rm0
Rm
Rs
Rm0
Rm0
Rm0
Rm0
Rs
Rm
Rm0
Rm0
Rm
Rs
Rm0
3
Cm0
Gs
Cm0
Cm0
And Cs Z C0 K2Cm K3Cm0 ; Gs Z G0 K2Gm K3Gm0 ; C0 is capacitance of phase to earth; G0 is conductance of phase to earth. Now introducing the definitions of common components and differential components of parallel transmission lines are as follows ( uCj Z 0:5ðu1j C u2j Þ (2) uDj Z 0:5ðu1j Ku2j Þ (
Lm
Lm
Lm0
Lm0
Ls
Lm
Lm0
Lm0
Lm
Ls
Lm0
Lm0
Lm0
Lm0
Ls
Lm
Lm0
Lm0
Lm
Ls
Lm0 Lm0 Lm0 3 Cm0 Cm0 7 Cm0 Cm0 7 7 7 7 Cm0 Cm0 7 7 7 Cm Cm 7 7 7 7 Cs Cm 7 5
Lm
Lm
Ls
7 6 6L Rm0 7 7 6 m 7 6 7 6 6 Lm Rm0 7 7 6 7 L Z6 7 6 L0 Rm 7 6 m 7 6 7 6 0 7 6 Lm Rm 5 4
Rm0 Rm0 Rm Rm Rs 3 2 Cs Cm Cm Gm0 7 6 6C Gm0 7 7 6 m Cs Cm 7 6 7 6 6 Cm Cm Cs Gm0 7 7 6 7 C Z6 7 6 C0 C0 C0 Gm 7 6 m m m 7 6 7 6 0 0 6 Cm Cm Cm0 Gm 7 5 4
Gm
2
iCj Z 0:5ði1j C i2j Þ
Cm0 Cm0 Cm0 Cs Cm Cm
where C is common component label; D is differential component label; jZa, b, c; 1 is the label of circuit 1; 2 is the label of circuit 2. Then the vu/vxZKRiKL(vi/vt) in Eq. (1) can be written as 2 3 2 3 vuD viD " # 6 vx 7 6 vx 7 6 7 6 7 i 6 7 ZKM$R$MK1 D KM$L$MK1 6 7 (4) 6 vuC 7 6 7 vi iC 4 4 C5 5 vx vx
Rs KRm0
6 6 Rm KRm0 6 6 6 Rm KRm0 6 K1 M$R$M Z 6 6 0 6 6 6 0 4 0
7 Lm0 7 7 7 7 Lm0 7 7 7 Lm 7 7 7 7 Lm 7 5 Ls
Cs
2
3 2 3 2 3 iDa uDa uCa 6 7 6 7 6 7 uD Z 4 uDb 5 iD Z 4 iDb 5 uC Z 4 uCb 5 uDc uCc iDc
2
1 0
6 60 6 6 0 16 MZ 6 6 261 6 60 4 0
0 K1
Rm KRm0
Rm KRm0
0
0
Rs KRm0
Rm KRm0
0
0
Rm KRm0
Rs KRm0
0
0
0
0
Rs C Rm0
Rm C Rm0
0
0
Rm C Rm0
Rs C Rm0
0
0
Rm C Rm0
Rm C Rm0
0
0
0
K1
0
1
0
0
0
0
1
0
1
0
0
1
0
1
0
0
3
7 7 7 7 7 0 7 7 0 7 R m C Rm 7 7 Rm C Rm0 7 5 0 Rs C Rm 0
0
1
Because 2
3
where
(3)
iDj Z 0:5ði1j Ki2j Þ
Cm
Lm0
0
2
3
6 7 iC Z 4 iCb 5
3
7 0 7 7 7 K1 7 7; 0 7 7 7 0 7 5 1
iCa
MK1 Z 2M 0
iCc
256
S. Jiale et al. / Electrical Power and Energy Systems 28 (2006) 253–260
2
Ls KLm0
6 6 Lm KLm0 6 6 6 Lm KLm0 6 K1 M$L$M Z 6 6 0 6 6 6 0 4 0
Lm KLm0
Lm KLm0
0
0
Ls KLm0
Lm KLm0
0
0
Lm KLm0
Ls KLm0
0
0
0
0
Ls C Lm0
Lm C Lm0
0
0
Lm C Lm0
Ls C Lm0
0
0
Lm C Lm0
Lm C Lm0
7 7 7 7 7 0 7 7 0 7 Lm C Lm 7 7 Lm C Lm0 7 5 0 Ls C Lm 0
2
Eq. (4) can be written as vuD vi ZKRD iD KLD D vx vt
(5)
(6)
Gs KGm0
6 0 GD Z 6 4 Gm KGm 2
vuC vi ZKRC iC KLC C vx vt
3
0
Gm KGm0 Gs KGm0
Gm KGm0
Gm KGm0
Gs C Gm0
Gm C Gm0
6 0 GC Z 6 4 Gm C Gm Gm C Gm0
Gs C Gm0 Gm C Gm0
2
where 2
Rs KRm0
6 0 RD Z 6 4 Rm KRm 2
Rm KRm0 Rs KRm0
Rm KRm0
Rm KRm0
Rs C Rm0
Rm C Rm0
6 0 RC Z 6 4 Rm C Rm
Rs C Rm0
Rm C Rm0
Rm C Rm0
2
Ls KLm0 Lm KLm0 6 0 0 LD Z 6 4 Lm KLm Ls KLm Lm KLm0 Lm KLm0 2 Ls C Lm0 Lm C Lm0 6 0 0 LC Z 6 4 Lm C Lm Ls C Lm Lm C Lm0
Lm C Lm0
Rm KRm0
Cs KGm0 Cm KGm0 6 0 0 CD Z 6 4 Cm KGm Cs KGm Cm KGm0 Cm KGm0 2 Cs C Gm0 Cm C Gm0 6 0 0 CC Z 6 4 Cm C Gm Cs C Gm Cm C Gm0 Cm C Gm0
3
7 Rm KRm0 7 5 Rs KRm0
Rm C Rm0
3
7 Rm C Rm0 7 5
Rs C Rm0 3 Lm KLm0 7 Lm KLm0 7 5 0 Ls KLm 3 Lm C Lm0 7 Lm C Lm0 7 5 0 Ls C Lm
Accordingly, the vi/vxZKGuKC(vu/vt) in Eq. (1) can be written as viD vu ZKGD uD KCD D vx vt
(7)
Gm KGm0
3
7 Gm KGm0 7 5 Gs KGm0
Gm C Gm0
3
7 Gm C Gm0 7 5
Gs C Gm0 3 Cm KGm0 7 Cm KGm0 7 5 0 Cs KGm Cm C Gm0
3
7 Cm C Gm0 7 5 Cs C Gm0
Above process shows that coupling parallel transmission lines can be decoupled into common component net and differential component net, and the line parameters of both decoupling nets are shown in above matrices. According to the process above, the parallel transmission lines system shown in Fig. 2 can be decoupled into common component net and differential component net, and they are shown in Fig. 3(a) and (b), respectively. As is shown in Fig. 3(a), there still exists source impedance in common component net. So the common component net is similar to single circuit transmission line system. In Fig. 3(b), there is no source impedance included, so the differential component net is free from the influence of source impedances. Because the differential components only appears in fault state and differential component currents only flow in the interior of parallel transmission lines, the differential (a) M
viC vu ZKGC uC KCC C vx vt
(8)
RC/LC/GC/CC
SM ZMC
RC/LC/GC/CC uCf
N ZNC S N
where (b) N
M SM ZM
ZN f
SN
Fig. 2. Parallel transmission lines system.
M
RD/LD/GD/CD
RD/LD/GD/CD
N
uDf
Fig. 3. (a) Common component net. (b) Differential component net.
S. Jiale et al. / Electrical Power and Energy Systems 28 (2006) 253–260
components of parallel transmission lines have following characteristics: (1) it can distinguish between fault and nonfault; (2) it is no relation to system parameters outside the parallel transmission lines; (3) its voltage distributions along parallel transmission lines are the highest at fault point and zero at the terminals. Above characteristics show that differential component is the optimal electric quantities for fault recognizing and fault locating in parallel transmission lines. So, differential component net based fault location algorithm needs only currents and is free from the influence of sources impedances.
3. Line voltage distribution computation A single-phase line model is shown in Fig. 4, where the resistance, inductance, conductance and capacitance per unit length are R0, L0, G0 and C0, respectively. According to Ref. [18], the voltage distributions along the line can be calculated by the voltages and currents of bus M or bus N, respectively. um ðx; tÞ Z 0:5 um ðtKx=vÞ C zc im ðtKx=vÞ eKR0 x=zc C 0:5 um ðt C x=vÞKzc im ðt C x=vÞ eR0 x=zc
(9)
un ðlKx; tÞ Z 0:5 un ðtKx=vÞ C zc in ðtKx=vÞ eKR0 x=zc C 0:5 un ðt C x=vÞKzc in ðt C x=vÞ eR0 x=zc (10) pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi where ZC Z L0 =C0 ; vZ 1= L0 C0 ; l is the length of line; x is the distance from fault point to terminal M, x2(0, l); t is a certain point of time. In three-phase line, the phase voltage distribution can be obtained by a complicated calculating process: firstly, implementing phase-mode transform to phase voltage for eliminating mutual coupling between phases; secondly, calculating mode voltage distribution in decoupling mode nets; thirdly, implementing phase-mode inverse transform to obtain phase voltage distribution from mode voltage distribution. The different mode nets have different parameters, so they have different propagation velocities and attenuations [18].
257
umj ðx; tÞ Z 0:5Zcj imj tKx=vj eKRj x=Zcj K0:5Zcj imj t C x=vj eRj x=Zcj (11) unj ðlKx;tÞ Z 0:5Zcj inj tKx=vj eKRj x=Zcj K0:5Zcj inj t Cx=vj eRj x=Zcj (12)
where j is mode label; Rj, Zcj, vj are differential component nets’ resistance, characteristic impedance and wave velocity in mode j, respectively; umj(x, t), unj(x, t) are j mode voltage distributions of differential component net, which are calculated by currents at terminal M and N, respectively; imj(t), inj(t) are j mode currents of differential component net at terminal M and N, respectively. Thus, given two terminals currents, the voltage distributions of differential component net can be calculated by (11) and (12), respectively. Considering the voltage distributions calculated by two terminal currents are equal at the fault point, the fault location function can be written as fj ðx; tÞ Z jumj ðx; tÞKunj ðx; tÞj
(13)
Eq. (13) is locating function based on mode voltage, and the phase voltage based function can be obtained by exerting phase-mode inverse transform to mode voltage. In differential component net, the voltages calculated by two terminals data are equal at fault point during the fault. In fault transient state, for eliminating the influence that very few non-fault points occasionally makes (13) equal to zero, summing up a period data calculated by Eq. (13) will be a practical method. So the improved criterion function should be written as ðt2
gj ðxÞ Z jumj ðx; tÞKunj ðx; tÞjdt
(14)
t1
where the time from t1 to t2 is data redundancy time. In synchronized state, Eq. (14) is equal to zero at fault point and greater than zero at non-fault points. So the fault location is a process searching x where (14) being the minimum. In unsynchronized state, the synchronization can be obtained by fixing one terminal data and moving another terminal data within the setting time. Then the fault location criterion function will be improved as ðt2
gj ðx; xÞ Z jumj ðx; tÞKunj ðx; t C xÞjdt
4. Fault location and data synchronization
(15)
t1
In above section, the distributed parameter line model and its calculating formulae of voltage distributions are given. As is shown in Fig. 3(b), the buses voltages in differential component net are invariably zero, so the (9) and (10) can be simplified as M um(t) im(t)
f uf (t) uf (t) ifm(t) ifn (t) Fig. 4. Single-phase transmission line.
N un(t) in(t)
where x2(0, t), t is setting unsynchronized time; x2(0, l), l is full length of the parallel lines; t1 and t2 have the same definition as Eq. (14). The unsynchronized time between both terminals data can be expressed as t Z ts C tc C td
(16)
where ts is unsynchronized sampling time; tc is the propagation time; td the delay of both sending and receiving devices.
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Eq. (15) is a function of fault distance and unsynchronized time. For searching fault distance, assume the step length of fault distance is Dx and lZm$Dx, and assume the step length of unsynchronized time is Dx and tZn$Dx. Then a matrix with n rows and m columns is obtained. In practical situations, considering the factor of rounding error and calculation precision, the minimal element of the matrix will determine the fault point. Its row will determine the unsynchronized time, and its column will determine the fault distance. Furthermore, the minimal element of fault location matrix should be equal to zero if the rounding error does not exist. From Eqs. (9) and (10), it is known that calculating a set of voltage distribution needs double propagation time of transmission line. Considering the redundancy time, the data window of fault location algorithm using synchronized data can be written as follows Fig. 6. Locating figures of transient currents.
tw Z 2l=vj C ðt2 Kt1 Þ
(17)
where tw is the length of data window; l is the length of the parallel lines; vj is the wave velocity in mode j; t1 and t2 have the same definition as Eq. (14). Considering the setting unsynchronized time t, the data window of locating algorithm using unsynchronized data can be written as follows tw Z 2l=vj C ðt2 Kt1 Þ C t
(18)
In practice, 1 ms is enough for the data redundancy time of Eqs. (14). According to Eqs. (17) and (18), a 3 ms data window is capable to locate a 300 km transmission line if the fault data are in synchronized state, and a longer data window is needed if the two terminal’s data are unsynchronized. 5. Simulations The simulation model is shown in Fig. 5. This is a 500 kV two-terminal system, the length from terminal M to terminal N is 300 km, and the Parallel Transmission Lines is 240 km. ATP is used for power system simulation and MATLAB is used for algorithm simulation. System equivalent parameter as follows: Terminal M: Positive sequence impedance Zm1Zj60.00 U; Zero sequence impedance Zm0Zj46.80 U. Terminal N: Positive sequence impedance Zn1Zj45.20 U; Zero sequence impedance Zn0Zj22.01 U. Single circuit line parameter: Positive sequence parameter: R1Z0.027 U/km, uL1Z 0.303 U/km, uC1Z4.27!10K6 S/km Zero sequence parameter: R 0Z0.196 U/km, uL 0Z 0.695 U/km, uC0Z2.88!10K6 S/km. Coupling parameter between circuits: Rm0 Z 0:064 U=km, uLm0 Z 0:131 U=km, uCm0 Z 0:467 !10K6 S=km. SM
M 30km
f 240km
N
In the following simulations, the sample frequency is 500 kHz, the unsynchronized time t is 1.5 ms, and the data window length tw is 3 ms. The figures are fault phase locating results of differential component net, and the origination of fault distance is terminal M. In this paper, only single circuit faults are simulated. In order to improve locating accuracy, introduce nine spline interpolation points into between samples. Locating figures of single-phase A–G faults are shown in Figs. 6 and 7, where Fig. 6 is the locating result of transient currents and Fig. 7 is that of steady currents. The simulating conditions are source M leading source N by 308, phase angle of source M is 908 when fault occurs, and fault resistance is 100 U. Figs. 6 and 7 present five fault location curves, respectively. Only locating curves calculated by synchronization data can indicate the exact fault positions and the amplitudes of fault location function equal to zero at fault point. In order to prove locating performance, plentiful simulations are given in the paper, and all the results are listed in Tables 1 and 2.
SN
30km
Fig. 5. Simulation model.
Fig. 7. Locating figures of steady currents.
S. Jiale et al. / Electrical Power and Energy Systems 28 (2006) 253–260
259
Table 1 Fault location results and errors of single-circuit fault Fault distance (km) 1.0
50.0
100.0
150.0
200.0
239.0
Fault type AG AB ABG ABC AG AB ABG ABC AG AB ABG ABC AG AB ABG ABC AG AB ABG ABC AG AB ABG ABC
Data for fault location Transient Steady Transient Steady Transient Steady Transient Steady Transient Steady Transient Steady
Fault resistance (U)
Locating result of fault phase (km)
Locating result of mode a (km)
Locating error of fault phase (err%)
Locating errors of mode a (err%)
0 100 200 1000 0 100 200 1000 0 100 200 1000 0 100 200 1000 0 100 200 1000 0 100 200 1000
0.96 0.96 0.96 0.96 49.92 49.92 49.92 49.92 100.08 100.08 100.08 100.08 150.00 150.00 150.00 150.00 199.92 199.92 199.92 199.92 239.04 239.04 239.04 239.04
0.96 0.96 0.96 0.96 49.92 49.92 49.92 49.92 100.08 100.08 100.08 100.08 150.00 150.00 150.00 150.00 199.92 199.92 199.92 199.92 239.04 239.04 239.04 239.04
0.017 0.017 0.017 0.017 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0 0 0 0 0.033 0.033 0.033 0.033 0.017 0.017 0.017 0.017
0.017 0.017 0.017 0.017 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0 0 0 0 0.033 0.033 0.033 0.033 0.017 0.017 0.017 0.017
Table 2 Locating results and errors at different separation angle of two sources and different fault inception angle Fault inception angles of two sources (degree)
Fault distance (km)
Locating results of fault phase (km)
Locating results of mode a (km)
Locating errors of fault phase (err%)
Locating errors of mode a (err%)
0–30 45–75 90–120 30–0 75–45 120–90 0–0 45–45 90–90
100 100 100 100 100 100 100 100 100
100.08 100.08 100.08 100.08 100.08 100.08 100.08 100.08 100.08
100.08 100.08 100.08 100.08 100.08 100.08 100.08 100.08 100.08
0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033
0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033
The locating results of single circuit A–G are shown in Table 1, and the simulation conditions are source M leading source N by 308. The phase angle of fault phase in source M is 908 when fault occurs. Fault resistances are 0, 100, 200 and 1000 U, respectively, and fault positions are from terminal M to terminal N. The locating results and errors of both fault phase and mode a are given in the table. The results in Table 1 show that both transient state and steady state samples can be used to locate faults, and have the same accuracy. Fault types and fault resistances also have no influence on locating accuracy. The proposed algorithm could locate faults from measuring point to the end of the line, and the locating accuracy is perfect. The largest locating error is less than 0.1 km, and the locating error percentage is less than 0.04% in proposed simulation length.
The locating results that A–G fault occurs at different separation angle of two sources and different fault inception angle are shown in Table 2. The simulation conditions are fault resistance 300 U and fault distance 100 km. The locating results and errors of both fault phase and mode a are presented. The locating results in Table 2 show that the algorithm in different fault inception angles and source separation angles all can afford to locate fault, and largest locating error is less than 0.1 km. Tables 1 and 2 show that, the algorithm could accurately locate faults from the local end to the remote end of the line, and locating errors are less than 0.1 km. From transient state to steady state, any segment of fault currents can be used to locate faults. The locating results are independent of the fault types, fault resistances, fault inception angles and source separation angles.
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6. Conclusions Differential component net based fault location algorithm can afford to locate faults of parallel transmission lines by two terminals unsynchronized currents. Comparing with travelling wave fault location, any segment of fault current can be used to locate faults, and there is no confinement from fault inception angle. Comparing with phasor-based algorithm, the novel algorithm requires a very short data window, and there is no filtering process involved. Acknowledgements The authors wish to acknowledge the financial support by the National Natural Science Foundation of China for this project, under the Research Program Grant No. 50377032. References [1] Mazo´n AJ, Min˜ambres JF, Zorrozua MA, Zamora I, Alvarez-Isasi R. New method of fault location on double-circuit two-terminal transmission lines. Electr Power Syst Res 1995;35(3):213–9. [2] Liao Y, Elangovan S. Digital distance relaying algorithm for first-zone protection for parallel transmission lines. IEE Proc Gener Transm Distrib 1998;145(5):531–6. [3] Zhang Q, Zhang Y, Song W, Yu Y. Transmission line fault location for phase-to-earth using one-terminal data. IEE Proc Gener Transm Distrib 1999;146(2):121–5. [4] Qingchao Zhang, Yao Zhang, Wennan Song, Yixin Yu, Zhigang Wang. Fault location of two-parallel transmission line for nonearth fault using one-terminal data. IEEE Trans Power Deliv 1999;14(3):863–7. [5] Ahn Yong-Jin, Choi Myeon-Song, Kang Sang-Hee, Lee Seung-Jae. An accurate fault location algorithm for double-circuit transmission systems. IEEE Power Eng Soc Summer Meet; 2000. [6] Sheng Lin Bo, Elangovan S. A fault location method for parallel transmission lines. Electr Power Energy Syst 1999;21(4):253–9. [7] Saha MM, Wikstrom K, Izykowski J, Rosolowski E. New accurate fault location algorithm for parallel lines. IEE seventh international conference, DPSP; 2001. [8] Jiale Suonan, Yaping Wu, Guobing Song. New accurate fault location algorithm for parallel lines on the same tower based on distributed parameter. Proc CSEE 2003;23(5):39–43.
[9] Funabashi Toshihisa, Otoguro Hitomi, Mizuma Yoshishige, et al. Digital fault location for parallel double-circuit multi-terminal transmission lines. IEEE Trans Power Deliv 2000;15(2):531–7. [10] Johns AT, Jamali S. Accurate fault location technique for power transmission lines. IEE Proc Pt C 1990;137(6):395–402. [11] Kezunovic M, Perunicic B. Synchronized sampling improves fault location. IEEE Comput Appl Power 1995;8(2):30–3. [12] Gale PF, Crossley PA, Bingyin X, Yaozhong G, Cory BJ, Baker JRG. Fault location based on travelling wave. IEE fifth international conference, DPSP; 1993. [13] Ranjbar AM, Shirani AR, Fathi AF. A new approach for fault location problem on power lines. IEEE Trans Power Deliv 1992;7(1): 146–51. [14] Gopalakrishnan A, Kezunovic M, McKenna SM, Hamai DM. Fault location using the distributed parameter transmission line model. IEEE Trans Power Deliv 2000;15(4):1169–74. [15] Lian Bao, Salama MMA, Chikhani AY. A time-domain differential equation approach using distributed parameter line model for transmission line fault location algorithm. Electr Power Syst Res 1998;46(1): 1–10. [16] Tziouvaras Demetrios A, McLaren Peter, Alexander George, Dawson D, Ezstergalyos J, Fromen C, et al. Mathematical models for current, voltage, and coupling capacitor voltage transformers. IEEE Trans Power Deliv 2000;15(1):62–72. [17] Novosel D, Hart DG, Udren E, Garitty J. Unsynchronized two-terminal fault location estimation. IEEE Trans Power Deliv 1996;11(1):130–8. [18] Marti JR, Marti L, Dommel HW. Transmission line models for steadyState and transients analysis. IEEE/NTUA Athens power tech conference: planning, operation and control of today’s electric power systems, Athens, Greece; September 1993.
Suonan Jiale received the MS and the PhD degrees in electrical engineering from Xi’an Jiaotong University in 1988 and 1991, respectively. He joined Xi’an Jiaotong University as a Professor of Electrical Engineering in 1998, where he is now a graduate teacher. He is a member of the council of electric power systems automation Journal. His research is mainly in EHV/UHV transmission line protection. He has developed many transmission-line protection devices which are running in power systems of china.
Song Guobing received the MS degree in electrical engineering from Xinjiang University, Urumchi, China, in 2002, and received the PhD degree at the Xi’an Jiaotong University, Xi’an, China, in 2005. His research interests in transmission line protection and fault location.
Time-domain fault location algorithm for parallel transmission lines using unsynchronized currents Suonan Jiale a,*, Song Guobing a, Xu Qingqiang a, Chao Qin b,1 a
Department of Electric Power system and Automation, School of Electric Engineering, Xi’an Jiaotong University, 28 Xianning West Road, Xi’an 710049, People’s Republic of China b Department of Electric Power system and Automation, School of Electric Engineering, Xin Jiang University, 21 Youhao North Road, Urumchi 830008, People’s Republic of China Received 16 February 2004; received in revised form 26 September 2005; accepted 10 November 2005
Abstract Parallel transmission lines under fault can be decoupled into common component net and differential component net. For the differential component net is only composed by parallel lines part and its bus voltages are zero, the voltage distributions along differential component net can be calculated by each terminal current. The novel time-domain fault location algorithm is founded by voltage distributions from both terminals have the least difference at the fault point. The synchronization is achieved by searching the minimal value of locating criterion function, which is calculated by fixing one terminal data and moving another terminal data within a setting time. The proposed algorithm needs a very short datawindow, any segment of fault data from transient to steady state can be used to locate faults, and no voltage data is needed. With distributed parameter model, simulations show that the locating errors are less than 0.1 km, and the fault types and fault resistances have no effect on locating accuracy. q 2005 Elsevier Ltd. All rights reserved. Keywords: Parallel transmission lines; Differential component net; Fault location
1. Introduction The application of high-speed protective relay device leading the fault clearing time is shortened greatly, many faults have no obvious arcing marks, it is always hard to avoid crossing intricate terrain in long-range transmission, and faults always take place in the bad weather. For the reasons above mentioned, the locating reliability and accuracy are targets for protective relaying researchers. A super fault location algorithm will do good to find fault point and save line inspecting time, manpower, material resource and financial power. And it is helpful to restore the service and decrease the loss of power failure. With the narrow corridor, parallel transmission lines are in favor of saving soil, investment and construction time, so it will * Corresponding author. Tel.: C86 29 82668782; fax: C86 29 82665489. E-mail addresses: [email protected] (S. Jiale), [email protected] (S. Guobing), [email protected] (X. Qingqiang), [email protected] (C. Qin). 1 Tel.: C86 991 4546680; fax: C86 991 4546680.
0142-0615/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2005.11.013
evolve as primary mode of electric energy transmission. In many countries, high voltage transmission lines and city power grids mainly adopt parallel lines. Though the structure has an increasing application in power system, it is hard to find an accurate fault location algorithm due to the mutual induction between circuits. Varieties of algorithms developed for locating faults on parallel transmission lines [1–10], differing in many aspects, have been proposed so far. Ref. [1] introduces a new concept ‘distance factor’, which is the quotient of the pure-fault currents of both circuits at the local end. The fault location is actualized by comparing this ‘distance factor’ calculated by pure-fault currents with that of calculated by the system parameter when fault occurs. So, this method needs the source impedances involved. Ref. [2] constructs voltage equations both from fault point via faulty circuit to local end and from fault point via healthy circuit to local end, and eliminates remote infeed by bringing one of the equations into the other. According to boundary condition, fault distance can be computed. Refs. [3,4] are aimed at the single phase-to-earth and phase-to-phase faults on one circuit of parallel transmission lines, respectively. Similar to Ref. [2], the main idea is eliminating remote infeed by voltage equations
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of healthy circuit. Ref. [5] eliminates the remote infeed by computing the positive-sequence and zero-sequence current distribution factors according KVL equations around the parallel lines loops. Ref. [6] locates faults by calculating the zero-sequence current distribution of faulty circuit. According to the imaginary part of fault current is zero, and then iterative technique is used. Ref. [7] considers that the uncertainty of zero-sequence impedance have an adverse influence on locating accuracy, and reestablishes the weighting coefficients of sequence components for calculating voltage drop across the fault path. Comparing with Refs. [5,6], Ref. [7] is an improved method. The single-terminal algorithms presented by Refs. [1–7] are lumped parameter model based. Though the methods adopted by Refs. [2–7] are different, the intentions of those methods are eliminating the remote infeed by KVL equations around the parallel lines loops. A distributed parameter model based single-terminal algorithms proposed by Ref. [8] uses differential component nets of parallel lines as the locating networks. It makes fault location on parallel transmission lines easier than ever. Though algorithm presented by Ref. [8] has high accuracy by engaging distributed parameter line model, like above-mentioned algorithms it needs a data window longer than one cycle. The need of phasor estimation introduces additional difficulty in high-speed tripping situations where the algorithms may not be fast enough in determining fault location accurately before the current signals disappear due to the relay operation and breaker opening [11]. In this case, the fault location algorithms with short data-window will be a solution. In fact, the phasor estimation needs a data window longer than one cycle if high estimation accuracy is pursued. Especially in fault initial stages, because the harmonic is plentiful, it is hard for a shortwindow algorithm for phasor estimation with a higher accuracy. Another solution is travelling wave method. Travelling wave based fault location, representative by Gale et al. [12], needs a short data-window, but this method locates fault by distinguishing wave-heads from transient data and it is hard to be realized by computer automatically. Additionally, the fault location will be abortive if there is no wave process. Comparing with single-terminal algorithms, there are few two/multi terminals methods presented for parallel lines. Ref. [9] presents a lumped parallel model based algorithm for multiterminal parallel transmission lines. Ref. [10] provides two terminals algorithm based on distributed parameter model, but the locating process is complicated for needing six phases’ voltage distributions calculated. For differential component net based algorithm makes fault location on parallel lines is just like single circuit one, the single circuit line algorithms [13,14] are referencable. Ref. [13] needs a data window just longer than 1/4 cycle, and the principle is searching the minimal value of criterion function calculated by voltage distributions along the transmission line. So this algorithm can fit the situations of high-speed tripping [11]. Ref. [14] divides the location process into two parts, the approximate fault location and the refining fault location. The approximate fault location engages the time-domain method, and the maximum error is 7.5 km for 20 kHz sampling rate
used. The refining process is using frequency domain (phasor) method to search fault point within the two segments beside the approximate fault point provided by former process. For frequency domain method is used, the later process needs a longer data window. As the time-domain method of fault location needs a short data window, and it can use samples directly without filter algorithms involved [15]. Considering high-speed tripping has an adverse influence on phasor based fault location algorithm, it is easy to select distributed parameter line model based time-domain algorithms. For locating parallel lines faults, a novel time-domain algorithm only using two terminals currents is present in this paper. Based on differential component net, the novel algorithm is easier than the traditional ones for needless to consider the coupling between circuits. Additionally, the voltage at the buses of differential component net is zero, and no voltage is needed in the new algorithm. The cut-off frequency of capacitance voltage transformer (CVT) is too low to transfer transient signal and that of current transformer (CT) can reach 100 kHz or more [16], so the proposed algorithm only using current data will be practical. Considering the unsynchronized data, this paper presents a new time-domain synchronization method, which is different from Ref. [17]. With the development of electronic technology, high-speed sampling devices are widely available for making digital fault recorder. Using 500 kHz sampling rate, the fault location can be implemented in time-domain and the refining process in Ref. [14] is omitted. Additionally, the spline interpolation is used to improve accuracy by adding 10 points into between samples. Comparing with Refs. [13–15], the proposed method needs a shorter data window. 2. Differential component net of parallel transmission lines The parallel transmission lines is a kind of parallel operating state. With the identical line parameter, the currents of both circuits are same in normal operating conditions and different in fault conditions. A segment of parallel transmission lines and its coupling relations are shown as Fig. 1. The telegraph equations of parallel lines can be written as 8 vu vi > > ZKRiKL > > < vx vt (1) vi vu > > > ZK GuKC > : vx vt
Fig. 1. Coupling relations of parallel transmission lines.
S. Jiale et al. / Electrical Power and Energy Systems 28 (2006) 253–260
255
where 2
3
2
i1a
2
3
Rs
u1a 6 6 7 6R 7 6 i1b 7 6 6 m 6 7 6 u1b 7 6 7 6 7 6 6 7 6i 7 6 6 Rm u 6 1c 7 6 1c 7 7 i Z6 7 R Z6 uZ6 6 7 6 7 6 6 R0 6 i2a 7 6 u2a 7 6 m 7 6 7 6 6 6 7 6u 7 6 0 6 i2b 7 4 2b 5 6 Rm 4 5 4 u2c i2c Rm0 2 0 Gs Gm Gm Gm Gm0 6 0 0 6G 6 m Gs Gm Gm Gm 6 6 6 Gm Gm Gs Gm0 Gm0 6 G Z6 6 G0 G0 G0 G G 6 m m m s m 6 6 0 0 0 6 Gm Gm Gm Gm Gs 4 Gm0
Gm0
Gm0
Gm
Rm
Rm
Rm0
Rm0
Rs
Rm
Rm0
Rm0
Rm
Rs
Rm0
Rm0
Rm0
Rm0
Rs
Rm
Rm0
Rm0
Rm
Rs
Rm0
3
Cm0
Gs
Cm0
Cm0
And Cs Z C0 K2Cm K3Cm0 ; Gs Z G0 K2Gm K3Gm0 ; C0 is capacitance of phase to earth; G0 is conductance of phase to earth. Now introducing the definitions of common components and differential components of parallel transmission lines are as follows ( uCj Z 0:5ðu1j C u2j Þ (2) uDj Z 0:5ðu1j Ku2j Þ (
Lm
Lm
Lm0
Lm0
Ls
Lm
Lm0
Lm0
Lm
Ls
Lm0
Lm0
Lm0
Lm0
Ls
Lm
Lm0
Lm0
Lm
Ls
Lm0 Lm0 Lm0 3 Cm0 Cm0 7 Cm0 Cm0 7 7 7 7 Cm0 Cm0 7 7 7 Cm Cm 7 7 7 7 Cs Cm 7 5
Lm
Lm
Ls
7 6 6L Rm0 7 7 6 m 7 6 7 6 6 Lm Rm0 7 7 6 7 L Z6 7 6 L0 Rm 7 6 m 7 6 7 6 0 7 6 Lm Rm 5 4
Rm0 Rm0 Rm Rm Rs 3 2 Cs Cm Cm Gm0 7 6 6C Gm0 7 7 6 m Cs Cm 7 6 7 6 6 Cm Cm Cs Gm0 7 7 6 7 C Z6 7 6 C0 C0 C0 Gm 7 6 m m m 7 6 7 6 0 0 6 Cm Cm Cm0 Gm 7 5 4
Gm
2
iCj Z 0:5ði1j C i2j Þ
Cm0 Cm0 Cm0 Cs Cm Cm
where C is common component label; D is differential component label; jZa, b, c; 1 is the label of circuit 1; 2 is the label of circuit 2. Then the vu/vxZKRiKL(vi/vt) in Eq. (1) can be written as 2 3 2 3 vuD viD " # 6 vx 7 6 vx 7 6 7 6 7 i 6 7 ZKM$R$MK1 D KM$L$MK1 6 7 (4) 6 vuC 7 6 7 vi iC 4 4 C5 5 vx vx
Rs KRm0
6 6 Rm KRm0 6 6 6 Rm KRm0 6 K1 M$R$M Z 6 6 0 6 6 6 0 4 0
7 Lm0 7 7 7 7 Lm0 7 7 7 Lm 7 7 7 7 Lm 7 5 Ls
Cs
2
3 2 3 2 3 iDa uDa uCa 6 7 6 7 6 7 uD Z 4 uDb 5 iD Z 4 iDb 5 uC Z 4 uCb 5 uDc uCc iDc
2
1 0
6 60 6 6 0 16 MZ 6 6 261 6 60 4 0
0 K1
Rm KRm0
Rm KRm0
0
0
Rs KRm0
Rm KRm0
0
0
Rm KRm0
Rs KRm0
0
0
0
0
Rs C Rm0
Rm C Rm0
0
0
Rm C Rm0
Rs C Rm0
0
0
Rm C Rm0
Rm C Rm0
0
0
0
K1
0
1
0
0
0
0
1
0
1
0
0
1
0
1
0
0
3
7 7 7 7 7 0 7 7 0 7 R m C Rm 7 7 Rm C Rm0 7 5 0 Rs C Rm 0
0
1
Because 2
3
where
(3)
iDj Z 0:5ði1j Ki2j Þ
Cm
Lm0
0
2
3
6 7 iC Z 4 iCb 5
3
7 0 7 7 7 K1 7 7; 0 7 7 7 0 7 5 1
iCa
MK1 Z 2M 0
iCc
256
S. Jiale et al. / Electrical Power and Energy Systems 28 (2006) 253–260
2
Ls KLm0
6 6 Lm KLm0 6 6 6 Lm KLm0 6 K1 M$L$M Z 6 6 0 6 6 6 0 4 0
Lm KLm0
Lm KLm0
0
0
Ls KLm0
Lm KLm0
0
0
Lm KLm0
Ls KLm0
0
0
0
0
Ls C Lm0
Lm C Lm0
0
0
Lm C Lm0
Ls C Lm0
0
0
Lm C Lm0
Lm C Lm0
7 7 7 7 7 0 7 7 0 7 Lm C Lm 7 7 Lm C Lm0 7 5 0 Ls C Lm 0
2
Eq. (4) can be written as vuD vi ZKRD iD KLD D vx vt
(5)
(6)
Gs KGm0
6 0 GD Z 6 4 Gm KGm 2
vuC vi ZKRC iC KLC C vx vt
3
0
Gm KGm0 Gs KGm0
Gm KGm0
Gm KGm0
Gs C Gm0
Gm C Gm0
6 0 GC Z 6 4 Gm C Gm Gm C Gm0
Gs C Gm0 Gm C Gm0
2
where 2
Rs KRm0
6 0 RD Z 6 4 Rm KRm 2
Rm KRm0 Rs KRm0
Rm KRm0
Rm KRm0
Rs C Rm0
Rm C Rm0
6 0 RC Z 6 4 Rm C Rm
Rs C Rm0
Rm C Rm0
Rm C Rm0
2
Ls KLm0 Lm KLm0 6 0 0 LD Z 6 4 Lm KLm Ls KLm Lm KLm0 Lm KLm0 2 Ls C Lm0 Lm C Lm0 6 0 0 LC Z 6 4 Lm C Lm Ls C Lm Lm C Lm0
Lm C Lm0
Rm KRm0
Cs KGm0 Cm KGm0 6 0 0 CD Z 6 4 Cm KGm Cs KGm Cm KGm0 Cm KGm0 2 Cs C Gm0 Cm C Gm0 6 0 0 CC Z 6 4 Cm C Gm Cs C Gm Cm C Gm0 Cm C Gm0
3
7 Rm KRm0 7 5 Rs KRm0
Rm C Rm0
3
7 Rm C Rm0 7 5
Rs C Rm0 3 Lm KLm0 7 Lm KLm0 7 5 0 Ls KLm 3 Lm C Lm0 7 Lm C Lm0 7 5 0 Ls C Lm
Accordingly, the vi/vxZKGuKC(vu/vt) in Eq. (1) can be written as viD vu ZKGD uD KCD D vx vt
(7)
Gm KGm0
3
7 Gm KGm0 7 5 Gs KGm0
Gm C Gm0
3
7 Gm C Gm0 7 5
Gs C Gm0 3 Cm KGm0 7 Cm KGm0 7 5 0 Cs KGm Cm C Gm0
3
7 Cm C Gm0 7 5 Cs C Gm0
Above process shows that coupling parallel transmission lines can be decoupled into common component net and differential component net, and the line parameters of both decoupling nets are shown in above matrices. According to the process above, the parallel transmission lines system shown in Fig. 2 can be decoupled into common component net and differential component net, and they are shown in Fig. 3(a) and (b), respectively. As is shown in Fig. 3(a), there still exists source impedance in common component net. So the common component net is similar to single circuit transmission line system. In Fig. 3(b), there is no source impedance included, so the differential component net is free from the influence of source impedances. Because the differential components only appears in fault state and differential component currents only flow in the interior of parallel transmission lines, the differential (a) M
viC vu ZKGC uC KCC C vx vt
(8)
RC/LC/GC/CC
SM ZMC
RC/LC/GC/CC uCf
N ZNC S N
where (b) N
M SM ZM
ZN f
SN
Fig. 2. Parallel transmission lines system.
M
RD/LD/GD/CD
RD/LD/GD/CD
N
uDf
Fig. 3. (a) Common component net. (b) Differential component net.
S. Jiale et al. / Electrical Power and Energy Systems 28 (2006) 253–260
components of parallel transmission lines have following characteristics: (1) it can distinguish between fault and nonfault; (2) it is no relation to system parameters outside the parallel transmission lines; (3) its voltage distributions along parallel transmission lines are the highest at fault point and zero at the terminals. Above characteristics show that differential component is the optimal electric quantities for fault recognizing and fault locating in parallel transmission lines. So, differential component net based fault location algorithm needs only currents and is free from the influence of sources impedances.
3. Line voltage distribution computation A single-phase line model is shown in Fig. 4, where the resistance, inductance, conductance and capacitance per unit length are R0, L0, G0 and C0, respectively. According to Ref. [18], the voltage distributions along the line can be calculated by the voltages and currents of bus M or bus N, respectively. um ðx; tÞ Z 0:5 um ðtKx=vÞ C zc im ðtKx=vÞ eKR0 x=zc C 0:5 um ðt C x=vÞKzc im ðt C x=vÞ eR0 x=zc
(9)
un ðlKx; tÞ Z 0:5 un ðtKx=vÞ C zc in ðtKx=vÞ eKR0 x=zc C 0:5 un ðt C x=vÞKzc in ðt C x=vÞ eR0 x=zc (10) pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi where ZC Z L0 =C0 ; vZ 1= L0 C0 ; l is the length of line; x is the distance from fault point to terminal M, x2(0, l); t is a certain point of time. In three-phase line, the phase voltage distribution can be obtained by a complicated calculating process: firstly, implementing phase-mode transform to phase voltage for eliminating mutual coupling between phases; secondly, calculating mode voltage distribution in decoupling mode nets; thirdly, implementing phase-mode inverse transform to obtain phase voltage distribution from mode voltage distribution. The different mode nets have different parameters, so they have different propagation velocities and attenuations [18].
257
umj ðx; tÞ Z 0:5Zcj imj tKx=vj eKRj x=Zcj K0:5Zcj imj t C x=vj eRj x=Zcj (11) unj ðlKx;tÞ Z 0:5Zcj inj tKx=vj eKRj x=Zcj K0:5Zcj inj t Cx=vj eRj x=Zcj (12)
where j is mode label; Rj, Zcj, vj are differential component nets’ resistance, characteristic impedance and wave velocity in mode j, respectively; umj(x, t), unj(x, t) are j mode voltage distributions of differential component net, which are calculated by currents at terminal M and N, respectively; imj(t), inj(t) are j mode currents of differential component net at terminal M and N, respectively. Thus, given two terminals currents, the voltage distributions of differential component net can be calculated by (11) and (12), respectively. Considering the voltage distributions calculated by two terminal currents are equal at the fault point, the fault location function can be written as fj ðx; tÞ Z jumj ðx; tÞKunj ðx; tÞj
(13)
Eq. (13) is locating function based on mode voltage, and the phase voltage based function can be obtained by exerting phase-mode inverse transform to mode voltage. In differential component net, the voltages calculated by two terminals data are equal at fault point during the fault. In fault transient state, for eliminating the influence that very few non-fault points occasionally makes (13) equal to zero, summing up a period data calculated by Eq. (13) will be a practical method. So the improved criterion function should be written as ðt2
gj ðxÞ Z jumj ðx; tÞKunj ðx; tÞjdt
(14)
t1
where the time from t1 to t2 is data redundancy time. In synchronized state, Eq. (14) is equal to zero at fault point and greater than zero at non-fault points. So the fault location is a process searching x where (14) being the minimum. In unsynchronized state, the synchronization can be obtained by fixing one terminal data and moving another terminal data within the setting time. Then the fault location criterion function will be improved as ðt2
gj ðx; xÞ Z jumj ðx; tÞKunj ðx; t C xÞjdt
4. Fault location and data synchronization
(15)
t1
In above section, the distributed parameter line model and its calculating formulae of voltage distributions are given. As is shown in Fig. 3(b), the buses voltages in differential component net are invariably zero, so the (9) and (10) can be simplified as M um(t) im(t)
f uf (t) uf (t) ifm(t) ifn (t) Fig. 4. Single-phase transmission line.
N un(t) in(t)
where x2(0, t), t is setting unsynchronized time; x2(0, l), l is full length of the parallel lines; t1 and t2 have the same definition as Eq. (14). The unsynchronized time between both terminals data can be expressed as t Z ts C tc C td
(16)
where ts is unsynchronized sampling time; tc is the propagation time; td the delay of both sending and receiving devices.
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Eq. (15) is a function of fault distance and unsynchronized time. For searching fault distance, assume the step length of fault distance is Dx and lZm$Dx, and assume the step length of unsynchronized time is Dx and tZn$Dx. Then a matrix with n rows and m columns is obtained. In practical situations, considering the factor of rounding error and calculation precision, the minimal element of the matrix will determine the fault point. Its row will determine the unsynchronized time, and its column will determine the fault distance. Furthermore, the minimal element of fault location matrix should be equal to zero if the rounding error does not exist. From Eqs. (9) and (10), it is known that calculating a set of voltage distribution needs double propagation time of transmission line. Considering the redundancy time, the data window of fault location algorithm using synchronized data can be written as follows Fig. 6. Locating figures of transient currents.
tw Z 2l=vj C ðt2 Kt1 Þ
(17)
where tw is the length of data window; l is the length of the parallel lines; vj is the wave velocity in mode j; t1 and t2 have the same definition as Eq. (14). Considering the setting unsynchronized time t, the data window of locating algorithm using unsynchronized data can be written as follows tw Z 2l=vj C ðt2 Kt1 Þ C t
(18)
In practice, 1 ms is enough for the data redundancy time of Eqs. (14). According to Eqs. (17) and (18), a 3 ms data window is capable to locate a 300 km transmission line if the fault data are in synchronized state, and a longer data window is needed if the two terminal’s data are unsynchronized. 5. Simulations The simulation model is shown in Fig. 5. This is a 500 kV two-terminal system, the length from terminal M to terminal N is 300 km, and the Parallel Transmission Lines is 240 km. ATP is used for power system simulation and MATLAB is used for algorithm simulation. System equivalent parameter as follows: Terminal M: Positive sequence impedance Zm1Zj60.00 U; Zero sequence impedance Zm0Zj46.80 U. Terminal N: Positive sequence impedance Zn1Zj45.20 U; Zero sequence impedance Zn0Zj22.01 U. Single circuit line parameter: Positive sequence parameter: R1Z0.027 U/km, uL1Z 0.303 U/km, uC1Z4.27!10K6 S/km Zero sequence parameter: R 0Z0.196 U/km, uL 0Z 0.695 U/km, uC0Z2.88!10K6 S/km. Coupling parameter between circuits: Rm0 Z 0:064 U=km, uLm0 Z 0:131 U=km, uCm0 Z 0:467 !10K6 S=km. SM
M 30km
f 240km
N
In the following simulations, the sample frequency is 500 kHz, the unsynchronized time t is 1.5 ms, and the data window length tw is 3 ms. The figures are fault phase locating results of differential component net, and the origination of fault distance is terminal M. In this paper, only single circuit faults are simulated. In order to improve locating accuracy, introduce nine spline interpolation points into between samples. Locating figures of single-phase A–G faults are shown in Figs. 6 and 7, where Fig. 6 is the locating result of transient currents and Fig. 7 is that of steady currents. The simulating conditions are source M leading source N by 308, phase angle of source M is 908 when fault occurs, and fault resistance is 100 U. Figs. 6 and 7 present five fault location curves, respectively. Only locating curves calculated by synchronization data can indicate the exact fault positions and the amplitudes of fault location function equal to zero at fault point. In order to prove locating performance, plentiful simulations are given in the paper, and all the results are listed in Tables 1 and 2.
SN
30km
Fig. 5. Simulation model.
Fig. 7. Locating figures of steady currents.
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Table 1 Fault location results and errors of single-circuit fault Fault distance (km) 1.0
50.0
100.0
150.0
200.0
239.0
Fault type AG AB ABG ABC AG AB ABG ABC AG AB ABG ABC AG AB ABG ABC AG AB ABG ABC AG AB ABG ABC
Data for fault location Transient Steady Transient Steady Transient Steady Transient Steady Transient Steady Transient Steady
Fault resistance (U)
Locating result of fault phase (km)
Locating result of mode a (km)
Locating error of fault phase (err%)
Locating errors of mode a (err%)
0 100 200 1000 0 100 200 1000 0 100 200 1000 0 100 200 1000 0 100 200 1000 0 100 200 1000
0.96 0.96 0.96 0.96 49.92 49.92 49.92 49.92 100.08 100.08 100.08 100.08 150.00 150.00 150.00 150.00 199.92 199.92 199.92 199.92 239.04 239.04 239.04 239.04
0.96 0.96 0.96 0.96 49.92 49.92 49.92 49.92 100.08 100.08 100.08 100.08 150.00 150.00 150.00 150.00 199.92 199.92 199.92 199.92 239.04 239.04 239.04 239.04
0.017 0.017 0.017 0.017 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0 0 0 0 0.033 0.033 0.033 0.033 0.017 0.017 0.017 0.017
0.017 0.017 0.017 0.017 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0 0 0 0 0.033 0.033 0.033 0.033 0.017 0.017 0.017 0.017
Table 2 Locating results and errors at different separation angle of two sources and different fault inception angle Fault inception angles of two sources (degree)
Fault distance (km)
Locating results of fault phase (km)
Locating results of mode a (km)
Locating errors of fault phase (err%)
Locating errors of mode a (err%)
0–30 45–75 90–120 30–0 75–45 120–90 0–0 45–45 90–90
100 100 100 100 100 100 100 100 100
100.08 100.08 100.08 100.08 100.08 100.08 100.08 100.08 100.08
100.08 100.08 100.08 100.08 100.08 100.08 100.08 100.08 100.08
0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033
0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033
The locating results of single circuit A–G are shown in Table 1, and the simulation conditions are source M leading source N by 308. The phase angle of fault phase in source M is 908 when fault occurs. Fault resistances are 0, 100, 200 and 1000 U, respectively, and fault positions are from terminal M to terminal N. The locating results and errors of both fault phase and mode a are given in the table. The results in Table 1 show that both transient state and steady state samples can be used to locate faults, and have the same accuracy. Fault types and fault resistances also have no influence on locating accuracy. The proposed algorithm could locate faults from measuring point to the end of the line, and the locating accuracy is perfect. The largest locating error is less than 0.1 km, and the locating error percentage is less than 0.04% in proposed simulation length.
The locating results that A–G fault occurs at different separation angle of two sources and different fault inception angle are shown in Table 2. The simulation conditions are fault resistance 300 U and fault distance 100 km. The locating results and errors of both fault phase and mode a are presented. The locating results in Table 2 show that the algorithm in different fault inception angles and source separation angles all can afford to locate fault, and largest locating error is less than 0.1 km. Tables 1 and 2 show that, the algorithm could accurately locate faults from the local end to the remote end of the line, and locating errors are less than 0.1 km. From transient state to steady state, any segment of fault currents can be used to locate faults. The locating results are independent of the fault types, fault resistances, fault inception angles and source separation angles.
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6. Conclusions Differential component net based fault location algorithm can afford to locate faults of parallel transmission lines by two terminals unsynchronized currents. Comparing with travelling wave fault location, any segment of fault current can be used to locate faults, and there is no confinement from fault inception angle. Comparing with phasor-based algorithm, the novel algorithm requires a very short data window, and there is no filtering process involved. Acknowledgements The authors wish to acknowledge the financial support by the National Natural Science Foundation of China for this project, under the Research Program Grant No. 50377032. References [1] Mazo´n AJ, Min˜ambres JF, Zorrozua MA, Zamora I, Alvarez-Isasi R. New method of fault location on double-circuit two-terminal transmission lines. Electr Power Syst Res 1995;35(3):213–9. [2] Liao Y, Elangovan S. Digital distance relaying algorithm for first-zone protection for parallel transmission lines. IEE Proc Gener Transm Distrib 1998;145(5):531–6. [3] Zhang Q, Zhang Y, Song W, Yu Y. Transmission line fault location for phase-to-earth using one-terminal data. IEE Proc Gener Transm Distrib 1999;146(2):121–5. [4] Qingchao Zhang, Yao Zhang, Wennan Song, Yixin Yu, Zhigang Wang. Fault location of two-parallel transmission line for nonearth fault using one-terminal data. IEEE Trans Power Deliv 1999;14(3):863–7. [5] Ahn Yong-Jin, Choi Myeon-Song, Kang Sang-Hee, Lee Seung-Jae. An accurate fault location algorithm for double-circuit transmission systems. IEEE Power Eng Soc Summer Meet; 2000. [6] Sheng Lin Bo, Elangovan S. A fault location method for parallel transmission lines. Electr Power Energy Syst 1999;21(4):253–9. [7] Saha MM, Wikstrom K, Izykowski J, Rosolowski E. New accurate fault location algorithm for parallel lines. IEE seventh international conference, DPSP; 2001. [8] Jiale Suonan, Yaping Wu, Guobing Song. New accurate fault location algorithm for parallel lines on the same tower based on distributed parameter. Proc CSEE 2003;23(5):39–43.
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Suonan Jiale received the MS and the PhD degrees in electrical engineering from Xi’an Jiaotong University in 1988 and 1991, respectively. He joined Xi’an Jiaotong University as a Professor of Electrical Engineering in 1998, where he is now a graduate teacher. He is a member of the council of electric power systems automation Journal. His research is mainly in EHV/UHV transmission line protection. He has developed many transmission-line protection devices which are running in power systems of china.
Song Guobing received the MS degree in electrical engineering from Xinjiang University, Urumchi, China, in 2002, and received the PhD degree at the Xi’an Jiaotong University, Xi’an, China, in 2005. His research interests in transmission line protection and fault location.