Estimation of Transmission Line Parameters using Fault Records Rastko Zivanovic School of Electrical & Electronic Engineering The University of Adelaide SA 5005 Australia Email:
[email protected] ABSTRACT In this paper an innovative algorithm for the transmission line parameter estimation is presented. The important factors, such as synchronisation, processing of non-stationary signals and phasor estimation that are affecting the estimation accuracy have been addressed in the paper, and the set of pre-processing tools have been proposed. The new algorithm is tested using practical fault records. Some representative results are presented in the paper.
1.
INTRODUCTION
The electrical parameters of a transmission line are generally not known with a great precision. Especially in the zero-sequence, several factors influence accuracy (for example frequency and ground resistivity). While these imprecise parameters may be sufficient for protection purposes, a higher accuracy is an important requirement for the fault location. Koglin and Schmidt [1] proposed to estimate the sequence parameters on the basis of the fault recordings obtained from digital relays at the line terminals during unsymmetrical power system conditions, such as external single-phase to ground faults and single pole auto-reclosing. They envisaged both synchronized and unsynchronized recordings. The main advantage of this technique is that it doesn’t require any special equipment or procedure. Philippot and Maun [2] proposed to use Digital Fault Recorders (DFR) instead of digital relays. This ensures better measurements thanks to a higher sampling rate, a longer duration of the recording, easier calibration, and the possibility to adjust the full scale of the current inputs. Furthermore, a DFR can be configured to record any desired event, thanks to the input expansion capabilities. The estimation procedure requires synchronized recordings to ensure accurate estimation of the aerial-mode inductances. In this procedure the DFRs at both ends should be triggered during certain system conditions, such as normal heavy load, and during intentionally caused single-pole tripping and auto-reclosing operation. The estimator yields the parameters of a distributed-parameter model of a twoterminal line. These data can be transferred to a twoended fault locator, the accuracy of which will then be essentially determined by the quality of the phasor measurements during fault condition [3]. In this paper we propose a novel parameter estimation algorithm based on unsynchronised voltage and current fault recordings obtained from DFRs located on the
opposite ends of a transmission line. The complete estimation algorithm is very complex comprising of a number of tasks. In the first task explained in section 2 we apply the abrupt change detection algorithm [4] to segment recorded signals in pre-fault and fault parts and then to synchronise those signals using the instant of fault. In the same section the phasor estimation algorithm has been described [5, 6]. The pre-fault voltage and current signals in all segments are converted to phasors and used to estimate positive sequence line parameters and synchronization error between phasors at opposite line ends. To estimate positive sequence parameters we have used the method based on measured impedances at both line ends [1]. This method is not sensitive to synchronization error. Synchronization error can be estimated using the positive sequence model. It is used to synchronise voltage and current phasors. These three steps are elaborated in section 3. Fault location and zero-sequence parameters are estimated using recordings of the ground faults. This method described in section 4 is based on the positive sequence line model and synchronised voltage and current phasors from both line sides. Section 5 concludes the paper with presentation of the representative results obtained in practical application using signals recorded during faults in the transmission network of South Africa.
2.
SEGMENTATION AND SYNCHRONIZATION
The aim is to estimate the time-instants of the abrupt changes in recorded signals and segment the signals in the pre-fault segment, fault segment, the segment when the fault is cleared by opening the circuit-breaker, the segment after breaker recloses etc. After this eventspecific segmentation of all recorded signals, the synchronization of those signals has been performed based on detection of the fault inception time. 2.1.
ABRUPT CHANGE DETECTION
Different segmentation techniques based on the abrupt change detection have been tested in a comparative manner and reviewed in [7]. These techniques are broadly categorized as simple methods, model-based approach, and model-free or nonparametric approach [7]. Among these techniques, the nonparametric approach, namely, the wavelet transform appears to be most suitable for this specific application. For detail description of the abrupt change detection based on the wavelet transform and threshold checking we refer the
reader to our previous paper [8]. Here we provide only the short summary of this technique. Multiresolution Signal Decomposition technique and the Quadrature Mirror Filter banks [9] have been used to decompose the recorded signals into localized and detailed representations in the form of wavelet coefficients. Daubechies 1 and 4 wavelets are used as the mother wavelets. After transforming the recorded signal using the Discrete Wavelet Transform, we obtain the smoothed and detailed versions. The detailed version, called the wavelet transform coefficient, is used for threshold checking to estimate the abrupt change timeinstants [8]. As wavelet coefficients are the changes of the averages, so a coefficient of large magnitude implies a large change in the original signal [8]. The change timeinstants can be estimated by the instants when the wavelet coefficients exceed a given threshold which is equal to the first order approximation of the ‘universal threshold’ of Donoho and Johnstone [10]. After the threshold checking, to indicate the change time instants as unit impulses, the specially designed smoothing filter has been used [8]. 2.2.
SYNCHRONIZATION
Not synchronised DFRs installed in sub-stations at the opposite sides of a transmission line trigger for any disturbance condition on the line. The DFRs provide short-duration voltage & current recordings as well as binary data captured during a disturbance. Those recordings differ by some time delays. To synchronize recordings we detect the instant of fault in all analogue signals using the abrupt change detection method described above. The complete synchronization algorithm is described as follows [11]: a) First we segment all recordings that are capturing the same disturbance using abrupt change detection. b) Then we estimate the individual fault inception timings of the segmented but unsynchronized recordings. c) We choose the recording with the minimum fault inception time as the reference one. d) We synchronize the rest of the recordings with the reference recording by equating their fault inception times with the reference fault inception time; i.e., we left-shift the rest of the recordings, their fault inception times equated to the reference one. e) Then we again perform the abrupt change detectionbased segmentation on these synchronized recordings to have the synchronized, segmented recordings for further analysis. The respective binary data are also synchronized and matched against the synchronized, segmented analogue signals. The matched binary plots are used as cross-check against any possible discrepancies. 2.3.
PHASOR ESTIMATION
After segmentation and synchronisation we convert signals into the fundamental frequency phasors for all segments. The phasor estimation algorithm is based on the following assumptions:
a) The fundamental frequency is constant for a signal in one segment. b) In transmission networks steady-state harmonics are usually negligible. If present they should be removed with low-pass filter prior to phasor estimation. The signal model (either voltage or current) assumed in the phasor estimation algorithm is s (k ) = A cos(θ k + ϕ ) − ∆s (k ) ,
(1)
where θ = 2πfTs ( Ts is the constant sampling interval). The model (1) parameters are constant fundamental frequency f , amplitude A and phase angle ϕ . ∆s(k )
is the residual between the measured signal s (k ) and the sinusoidal signal model. This residual is modelled as ∆s (k ) = E [∆s(k )] + e(k ) ,
(2)
where E [∆s (k )] is the trend component ( E[∗] is the mathematical expectation), representing for example DC offset or bias of the measuring instrument, and e(k ) is the noise component assumed to be independent and identically distributed (i.i.d.). The signal model (1) corresponds to the following difference equation [6]:
[s(k − 1) + ∆s(k − 1)]x + K s (k ) + ∆s (k ) + s (k − 2 ) + ∆s (k − 2) = 0.
(3)
The fundamental frequency and the coefficient x in the difference equation (3) are related through, x = −2 cos(θ ) = −2 cos(2πfT s ) .
(4)
The difference equation (3) for m+2 samples that belong to one signal segment, can be written in the matrix form ax + b + D( x)∆s = 0 ,
⎡ s (2) ⎤ ⎢ s (3) ⎥ ⎥, where a = ⎢ ⎢ M ⎥ ⎢ ⎥ ⎣ s (m + 1)⎦ ⎡1 x 1 0 ⎢0 1 x 1 D(x ) = ⎢ ⎢M O O O ⎢ ⎣0 L 0 1
(5)
⎡ s(3) + s (1) ⎤ ⎢ s(4 ) + s(2 ) ⎥ ⎥, b=⎢ ⎢ ⎥ M ⎢ ⎥ ⎣ s(m + 2) + s (m )⎦ L 0⎤ ⎡ ∆s (1) ⎤ ⎥ ⎢ ∆s (2 ) ⎥ O 0⎥ ⎥. , and ∆s = ⎢ ⎢ ⎥ O ⎥ M ⎥ ⎢ ⎥ x 1⎦ ⎣∆s(m + 2)⎦
The coefficients x and residuals ∆s can be estimated by solving the following linear least squares problem with the non-linear equality constrains,
(∆sˆ, xˆ ) = arg min ⎧⎨ 1 [∆s − E(∆s )]T [∆s − E(∆s )]⎫⎬ ,
⎩2 subject to (5), where
E(∆s )T = [E (∆s (1) ) E (∆s (2 )) K
⎭
(6) E (∆s (m + 2))] .
A smoothing filter has been used to estimate E(∆s ) and to separate offset and noise components in the residuals ∆s . Having x , the fundamental frequency is calculated using (4). Placing the frequency and s (k ) + ∆sˆ(k ) in (1), amplitudes A and phases ϕ are estimated using linear least squares technique.
3.
PARAMETER ESTIMATION USING PREFAULT RECORDS
Voltage and current phasors form both sides and for all identified segments are converted into symmetrical components for further analysis. The pre-fault segment has been used to estimate positive-sequence line parameters as well as to refine the synchronisation of phasors. The procedure is explained in this section. 3.1.
Z s1 =
Z s1 Z r1 ⋅ tanh(γ 1 l ) + Z s1 = Z r1 + Z c1 ⋅ tanh (γ 1 l ) . (11) Z c1 1 ⋅ tanh (γ 1 l ) and x 2 = Z c1 ⋅ tanh (γ 1 l ) as Z c1 the unknown linear parameters, the equation (11) gets its final form
Using x1 =
x 2 − Z s1 Z r1 x1 = Z s1 − Z r1 .
Consider a single-phase positive-sequence transmission line model [12]: V s1 = cosh (γ 1 l ) ⋅ V r1 + Z c1 ⋅ sinh (γ 1 l ) ⋅ I r1 ,
1 ⋅ sinh (γ 1 l ) ⋅ V r1 + cosh (γ 1 l ) ⋅ I r1 . Z c1
(7)
where V s1 , I s1 are positive sequence voltage and current phasors at sending-end, V r1 , I r1 are positive sequence voltage and current phasors at remote-end,
γ 1 = z1 ⋅ y1 is the positive-sequence Z c1 =
propagation constant, z1 is the positive-sequence characteristic y1
zˆ1 =
yˆ1 =
An expression for positive-sequence impedance at the sending-end can be derived using (7) as follows: V cosh (γ 1 l ) ⋅ V r1 + Z c1 ⋅ sinh (γ 1 l ) ⋅ I r1 . (8) Z s1 = s1 = 1 I s1 ⋅ sinh (γ 1 l ) ⋅ V r1 + cosh (γ 1 l ) ⋅ I r1 Z c1
After multiplying numerator and denominator with 1 / I r1 , the equation (8) becomes
(9)
Replacing V r1 / I r1 in (9) with Z r1 (positive-sequence impedance at remote-end) we obtain [1]
(12)
To achieve a better accuracy in estimating parameters x1 and x 2 , we should use at least two records. To collect enough data records DFRs can be manually triggered during various normal operating conditions. For each pre-fault or normal operating record the equation (12) is valid, where Z s1 and Z r1 are measured values, and x1 and x 2 are the unknown parameters to be estimated. For a number of data records we can write the over determined system of linear equations and apply the least squares procedure to estimate the parameters x1 and x 2 . Knowing those auxiliary parameters, the positive-sequence line parameters are estimated using the following formulas:
impedance, z1 and y1 are the positive-sequence line parameters, and l is the line length.
V I cosh (γ 1 l ) ⋅ r1 + Z c1 ⋅ sinh (γ 1 l ) ⋅ r1 I r1 I r1 Z s1 = . V r1 I r1 1 ( ) ( ) ⋅ sinh γ 1 l ⋅ + cosh γ 1 l ⋅ Z c1 I r1 I r1
(10)
After bringing the denominator in (10) to the left side and dividing with 1 / cosh (γ 1 l ) we have:
POSITIVE SEQUENCE PARAMETERS
I s1 =
cosh (γ 1 l ) ⋅ Z r1 + Z c1 ⋅ sinh (γ 1 l ) . Z r1 ⋅ sinh (γ 1 l ) + cosh (γ 1 l ) Z c1
3.2.
(
)
−1 ˆ ⋅ ˆ x 2 x1 xˆ 2 tanh , and ⋅ xˆ1 l
tanh −1
(
xˆ 2 ⋅ xˆ1
xˆ l⋅ 2 xˆ1
(13)
).
(14)
SYNCHRONISATION ERROR
The abrupt change detection algorithm has been used to synchronize signals from both line sides. Still there is a need to refine synchronisation of phasors because error with the maximum of one sampling interval is possible after the segmentation phase. For example, if we have one sampling interval shift between two signals and we use 2.5kH sampling frequency there will be 7.2° phase angle difference between two phasors obtained from those signals. The estimation algorithm is extremely sensitive to such phase angle difference and additional effort should be made to achieve better synchronisation. To solve this problem we can use the estimated positive sequence model parameters (zˆ1 , yˆ1 ) or Zˆ c1 , γˆ1 to
(
)
calculate the phase angle difference (δ v , δ i ) between two voltage and current positive-sequence phasors on opposite sides using:
V s1 exp( jδ v ) = cosh (γˆ1 l ) ⋅ V r1 + Zˆ c1 ⋅ sinh (γˆ1 l ) ⋅ I r1 , I s1 exp( jδ i ) =
bk = V r 0 ( k ) − V s 0 ( k ) ,
1 ⋅ sinh (γˆ1 l ) ⋅ V r1 + cosh (γˆ1 l ) ⋅ I r1 . ˆ Z c1
k = 1,2, K N F .
For the number of faults N F ≥ 2 we solve the system of equations (20) using the least squares method. The estimates of Zˆ e0 , Yˆe0 obtained as the solution of the least squares problem are used to calculate zero sequence parameters, either Zˆ c 0 , γˆ 0 or (zˆ 0 , yˆ 0 ) as follows:
(15)
4.
(
PARAMETER ESTIMATION USING GROUND FAULT SIGNALS
4.1.
(
FAULT LOCATION
5.
(17) suitable for calculating the distance-to-fault d .
ZERO SEQUENCE PARAMETERS
Zero-sequence fault voltage can be represented using synchronised zero-sequence voltage and current phasors from both sides and the zero-sequence equivalent π model [12],
APPLICATION EXAMPLE
1.2
abrupt change detection
1 0.8
(18)
Z e0 = Z c0 sinh (γ 0 ) and
0.6 C urrent [pu]
where the zero-sequence equivalent π parameters are
0.4 0.2 0 -0.2
(19)
-0.4 -0.6
The equation (18) can be recast into the following form: a k1 Z e0 + a k 2 Z e0Ye0 = bk ,
(21)
Important pre-processing steps in the parameter estimation algorithm are: segmentation, synchronisation and phasor estimation. In the first stage abrupt change detection has been used in identifying fault and pre-fault segments. Figure 1 shows the result of applying wavelet transform in segmentation of the current signal recorded during one single-phase earth fault in the transmission network of South Africa. The selected current samples in the pre-fault and fault segments are converted into phasors. Figure 2 shows the result of the signal decomposition procedure (described in section 2.3) applied on the fault segment. This technique removes offset and noise from the signal making the estimation of amplitude and phase angle (phasor estimation) very accurate.
V s1 − V r1 cosh (γ 1 ) + Z c1 I r1 sinh (γ 1 ) , Z c1 I s1 + Z c1 I r1 cosh (γ 1 ) − V r1 sinh (γ 1 )
Ye0 cosh (γ 0 ) − 1 = . 2 Z e0
⎠
Zˆ e0 , sinh (γˆ 0 )
γˆ zˆ 0 = γˆ 0 Zˆ c0 , yˆ 0 = 0 . Zˆ c 0
where d is a distance to fault in percentage and a line length is l = 1 . Equation (16) can be rewritten in the following form
dY ⎛ ⎞ V f 0 = V s 0 − dZ e0 ⎜⎜ I s 0 − e0 V s 0 ⎟⎟ 2 ⎝ ⎠ (1 − d )Ye0 ⎛ ⎞ = Vr 0 − (1 − d ) Z e0 ⎜⎜ I r 0 − V r 0 ⎟⎟, 2 ⎝ ⎠
2
⎝
V f 1 = V s1 cosh (γ 1d ) − Z c1 I s1 sinh (γ 1d ) (16) = V r1 cosh (γ 1 (1 − d ) ) − Z c1 I r1 sinh (γ 1 (1 − d ) ).
4.2.
)
⎛ Zˆ Yˆ ⎞ γˆ 0 = cosh −1 ⎜1 + e0 e0 ⎟ , Zˆ c0 = ⎜ ⎟
Synchronised voltage and current positive sequence phasors from both line sides are used to calculate distance-to-fault location. To obtained the formula we express the voltage at the place of fault using voltage and current phasors from both sides
tanh (γ 1d ) =
)
(20)
-0.8 0
50
100
150
200
250 300 Samples
350
400
450
500
Figure 1: Current segmentation
where a k1 = −d k (I s 0 (k ) + I r 0 (k )) + I r 0 (k ) ,
d 2 V (k ) , a k 2 = k (V s 0 (k ) − V r 0 (k ) ) + d k V r 0 (k ) − r 0 2 2
The parameter estimation algorithm described in this paper can be used to improve accuracy of transmission line parameters. To test the estimation algorithm we have compared the estimated values with the calculated
values for one transmission line in the grid of South Africa. For the purpose of estimation we were using signals recorded during two single-phase ground faults on the line. The results are shown in Table 1. The estimated parameters are very close to the calculated values. This proves that the algorithm can perform well in practical applications. Further accuracy testing of the algorithm requires comparison between estimates and the parameters obtained in the staged open and shortcircuit tests.
0.8 0.6 offset & noise C urrent [pu]
0.4
[1] H.J.Koglin and M.Schmidt, “Estimation of Transmission Line Parameters by Evaluating Fault Data Records”, 12th Power System Computation Conference, Dresden, August 19-23, 1996
[3] J.C.Maun, L.Philippot, Z.Chen, M.Mouvet, “Fault Locator based on Two-ended Measurement and Least-Squares Estimation”, Proceedings of the Southern African Power System Protection Conference, Midrand, South Africa, 8-9 November 1994
0.2 0 -0.2 -0.4 -0.6 current offset & noise removed
-0.8 -1 0
50
100
150
Samples
[4] A.Ukil, R. Živanović, “Abrupt Change Detection in Power System Fault Analysis using Adaptive Whitening Filter and Wavelet Transform”, Electric Power Systems Research, Volume 76, June 2006, pp. 815-823
Figure 2: Filtering offset & noise of the fault current segment in Figure 1
[5] J.Jordaan, R.Zivanovic, “'Frequency Estimation in Power Systems Using the Dynamic Leapfrog Method”, Measurement, Volume 39, June 2006, pp. 451-457
Table 1: Parameter estimation compared to the calculated parameters
[6] R. Živanović, “Analysis of Recorded Transients on 765kV Lines with Shunt Reactors”, Proceedings of the 2005 IEEE PowerTech Conference, St.Petersburg, Russia, June 2005
line parameters R0 [Ω/km] R1 [Ω/km] L0 [H/km] L1 [H/km] C 0 [F/km] C1 [F/km] 6.
REFERENCES
[2] L.Philippot and J.C.Maun, “An Application of Synchronous Phasor Measurement to the Estimation of the Parameters of an Overhead Transmission Line”, Fault and Distance Analysis & Precise Measurements in Power Systems Conference, Arlington, Virginia, 1995
recorded current
1
is required. The algorithm can improve settings of distance relays, fault locator settings and analysis of the DFR disturbance records.
calculated parameters 0.2173 0.01186 2.858x10-3 0.855x10-3 8.952x10-9 1.3225x10-8
estimated parameters 0.2135 0.0119 0.0029 8.553x10-4 8.97x10-9 1.3215x10-8
error [%] 1.779 0.336 1.469 0.23 0.2 0.077
CONCLUSIONS
In this paper we have presented a novel transmission line parameter estimation algorithm. The algorithm is utilising unsynchronized fault records on both line ends. Very high accuracy in signal segmentation, synchronisation and phasor estimation has been achieved with the methods presented. The accuracy of these preprocessing tools contributed significantly to the success of the proposed parameter estimation algorithm. The algorithm testing results demonstrate potential for using such algorithm to obtain more accurate line impedance and admittance values. Use of the algorithm is especially recommended when accurate zero-sequence line model
[7] A. Ukil and R. Živanović, "Detection of Abrupt Changes in Power System Fault Analysis: A Comparative Study," in Proc. Southern African Universities Power Engineering Conference (SAUPEC’05), Johannesburg, South Africa, January 2005 [8] A. Ukil and R. Živanović, "Abrupt Change Detection in Power System Fault Analysis using Wavelet Transform," in Proc. International Conf. on Power Systems Transients (IPST’05), Montreal, Canada, Jun 2005 [9] S. Mallat, “A wavelet tour of signal processing”, Academic Press, 1998 [10] D.L. Donoho and I.M. Johnstone, "Ideal Spatial Adaptation by Wavelet Shrinkage," Biometrika, vol. 81, no. 3, 1994, pp. 425-455 [11] A.Ukil and R. Živanović, “Application of Abrupt Change Detection in Relay Performance Monitoring”, 40th International Universities Power Engineering Conference (UPEC 2005), University College Cork, Cork, Ireland, September 2005 [12] J.D.Glover and M.Sarma, “Power Systems Analysis and Design”, PWS Publishing Company, Boston, 1994
