Contents lists available atScienceDirect
Electrical Power and Energy Systems
journal homepage:www.elsevier.com/locate/ijepesMathematical development of the sampling frequency e
ffects for improving
the two-terminal traveling wave-based fault location
F.B. Costa
a,⁎, F.V. Lopes
b, K.M. Silva
b, K.M.C. Dantas
c, R.L.S. França
a, M.M. Leal
a, R.L.A. Ribeiro
a aFederal University of Rio Grande do Norte, BrazilbBrasília University, Brazil
cFederal University of Campina Grande, Brazil
A R T I C L E I N F O Keywords: Fault location Traveling waves Transmission line Sampling frequency A B S T R A C T
This paper presents a deep analysis of sampling frequency effects on the classical two-terminal traveling wave-based transmission line fault location method performance, demonstrating that these effects can be represented by well-defined regions (in the format of lozenges) in a time-space plane. Thereby, it is demonstrated that the classical method estimates the fault location in predefined points due to sampling frequency-associated time resolution. Furthermore, faults at the same location can be randomly estimated in different predefined locations depending on the fault inception time, resulting in location uncertainties. Nevertheless, the proposed time-space lozenges support a probabilistic analysis in order to consider the sampling frequency effects by adding a probabilistic searchfield component to the classical fault location formulation which indicates the region in which the fault took place with 100% of certainty regarding the sampling frequency effects. Therefore, this paper also proposes an improvement on the classical method by considering the effects of the sampling frequency. The proposed approach was evaluated by means of digital simulations and validated experimentally in laboratory by the application of faults along a 1 km long cable.
1. Introduction
When a fault takes place on overhead transmission lines traveling waves propagate away toward the line terminals with velocity close to the speed of light, becoming the first fault evidence at the line term-inals. Information regarding the wavefront arrival times has supported a number of solutions for fault location on transmission lines[1–7]. The use of modern digital signal processors operating under high sampling frequency allows the accurate extraction of the wavefront arrival times, which successfully integrates the traveling wave principle with in-telligent electronic devices (IEDs) for fast and accurate fault location estimation[8,9].
Classical traveling wave-based fault location methods can be di-vided into single- and two-terminals categories according to the number of terminals at which the fault-induced transients are analyzed[10,11]. Line measurements on a single terminal provide a fault location esti-mation based on the comparison of thefirst wavefront arrival time and those reflected from the fault point (secondary wavefront arrival time) that reach the monitored terminal[10,12,13]. However, the detection methods require a significant amount of signal processing in order to properly detect the secondary wavefront, distinguishing it from the
traveling waves reflected at the remote terminal or adjacent line terminals[14]. In the case of earth faults, single-terminal fault location methods can also use the arriving instant of the earth mode component in order to avoid the requirement of the secondary wavefront detection [15,16]. Conversely, two-terminal traveling wave-based methods are the simplest because only the first wavefront arrival time at both terminals is required[17–20]. However, the main concerns are the required communication system and data synchronization [11]. Nevertheless, communication system and data synchronization through GPS (Global Positioning System) are usually available in overhead transmission lines, allowing the application of two-terminal algorithms without loss of reliability and accuracy[8,21–23]. The developments presented in this paper are based on the classical two-terminal fault location formulation, which has been widely applied in traveling wave-based fault location schemes.
The classical two-terminal traveling wave-based fault location method is enabled by the detection method just when thefirst wave-front arrival time of the traveling waves is detected in both line term-inals and a communication is established to provide this information to the fault location method successfully[11]. The main concerns which affect directly the performance of traveling wave-based detection
https://doi.org/10.1016/j.ijepes.2019.105502
Received 11 September 2018; Received in revised form 1 July 2019; Accepted 19 August 2019 ⁎Corresponding author.
E-mail address:flaviocosta@ect.ufrn.br(F.B. Costa).
Available online 03 September 2019
0142-0615/ © 2019 Elsevier Ltd. All rights reserved.
methods are: 1) the fault inception angle[24,25]; 2) the fault resistance [26,27]; 3) high-level noise[24]; 4) voltage and current transducers with low-frequency response [28,29]; and 5) failure in the commu-nication system[30]. However, relevant solutions have been reported in the literature to overcome these problems. For instance, currently there are accurate traveling wave-based detection methods available commercially [31,32], and consistent solutions for the first three aforementioned issues are also well reported in the literature [27,24,20,33]. Optical transducers with broad frequency response and negligible noise level are also commercially available today[34–36]. In addition, modern communication systems are commonly used in transmission lines[14]. Therefore, considering that the main focus of this paper is related to the fault location rather than the aforementioned issues, which are closely related to the performance of detection methods, they are not evaluated in this paper.
It is well-known that the accuracy of the classical two-terminal traveling wave-based fault location method is susceptive to a variety of errors, such as: (1) the exact traveling wave velocity, whose un-certainties are mainly due to the line electrical parameters[16]; (2) the estimated line length may not be accurate[37,38]; (3) the wavefront arrival time detection methods can provide errors due to low amplitude traveling waves[24,29]; (4) time synchronization through GPS, which can provide a time displacement up to 1μs[16]. However, important solutions have been proposed to overcome these problems. For in-stance, unsynchronized-based fault location methods, fault location solutions with no need of traveling wave velocity estimation, and powerful techniques for wavefront arrival time detection have been proposed[5,16,39,24,25].
Another source of error to traveling wave-based fault location methods is the sampling frequency, which is usually mitigated by the use of high sampling rates [8]. Currently, for instance, there are tra-veling wave-based devices running for sampling frequencies from 1 to 5 MHz[40–42]. Nevertheless, the effects of the sampling frequency in fault location methods must still be properly evaluated. In[30], a two-terminal traveling wave-based protection was proposed by considering the effects of the sampling frequency. However, such a protection system is entirely independent of the fault location estimation. In practice, even the protection being independent of the fault location estimation, such information is essential to be quickly and accurately transferred to the maintenance crew, especially in the case of perma-nent faults to speed up the line restoration[14]. However, there is an uncertainty in the fault point estimations due to the sampling frequency effects that are typically neglected.
This paper provides a thorough explanation of the sampling fre-quency effects in the classical two-terminal traveling wave-based fault location method, which yields an additional error component to the classical fault location equation. This error is represented by time-space lozenges where: (1) the fault location is estimated at predefined points; (2) faults at the same location can be randomly estimated in different preset locations depending on the fault inception time. The time-space lozenges support a probabilistic analysis in order to take advantage of the effects of the sampling frequency. As a consequence, this paper proposes a probabilistic searchfield component to the classical fault location estimation equation tofind faults with 100% of certainty re-garding the sampling frequency effects, i.e., disregarding other un-certainties such as the propagation velocity estimation.
This paper considered both simulation and experimental results for validating the proposed mathematical development as well as for de-monstrating the effectiveness of the proposed fault location method and its feasibility to real applications. For the experimental setup, a 1 km long cable was used to emulate faults (and its induced traveling waves) along a transmission line and a sampling rate of 500 kHz was used to assess both classical and proposed traveling wave-based fault location methods.
2. The ideal two-terminal traveling wave-based fault location
The lattice diagram inFig. 1illustrates the traveling wave propa-gation on a transmission line with length l for a fault into the protected line fardFikm from bus i (anddFj km from the bus j). From the fault inception time (tF), the traveling waves propagate along the line in both directions away fromdFiat a velocity v. Therefore, thefirst wavefront arrival time at buses i and j are tFi> tF and tFj> tF, respectively. Since only the classical two-terminal traveling wave-based fault location method is studied in this paper, all considered time parameters and propagation velocity are related to the aerial modes of the traveling waves.
The distances propagated by the traveling waves to reach the line terminals (actual fault distances) are given by:
⎜ ⎟ = ⎛ ⎝ − ⎞ ⎠ ⇒ = − d v t t t t d v , Fi Fi F F Fi Fi (1) ⎜ ⎟ = ⎛ ⎝ − ⎞ ⎠ ⇒ = − d v t t t t d v , Fj Fj F F Fj Fj (2) being the line length given by:
= +
l dFi d .Fj (3)
Therefore, considering the fault inception timetFas the reference in(1)
and (2), the fault location estimation from bus i considering information from bus j is ideally given by:
⎜ ⎟ = + − = + ⎛ ⎝ − ⎞ ⎠ d l (t t )v l v t t 2 2 2 , Fij Fi Fj Fi Fj (4) where tFiandtFj∈(continuous time). Therefore, from bus i, the fault location estimationdFijis equal to the actual fault distance dFi (i.e.,
=
dFij dFi) admitting no line parameter estimation errors (l and v) and no errors in thefirst wavefront arrival time detection (tFiand tFj). 3. Practical two-terminal traveling wave-based fault location
Most two-terminal traveling wave-based fault location methods consider(4)or a similar equation for estimating the fault location in transmission lines[4,5,23,43–45]. However, in practice, IEDs take into account discrete-time voltages and currents sampled at a specific sampling frequency ( fs). In addition, simulations consider afinite time step to provide discrete-time voltages and currents. Therefore, the continuous times tFiand tFjin practice are literally used as discrete times in the fault locator methods, i.e., a simple change from continuous to discrete time is accomplished.
Since the effects of the sampling frequency are addressed in this paper, the discrete times associated to the continuousfirst wavefront arrival times (tFiand tFj) are denoted bykFi/fsandkFj/fs, wherekFiandkFj are the kth samples associated to the discretefirst wavefront arrival times kFi/fs and kFj/fs, respectively, which can be obtained by
Internal fault Time (s) Time (s) t k /f Traveling waves F Fj S k /fFi S tF Traveling waves Effective wavefront al arriv time k /fF S k /fF S Sampling time
Bus i dFi Transmission line with kmdFj= -l ldFi Bus j
tFi tFj Wavefront al arriv time ȟtF ȟtFi ȟtFj ȟtF
normalizing the fault record time vector by the sampling period1/fs. Therefore, following the notation considered in this paper, existing fault locators would be defined as:
⎜ ⎟ = + − = + ⎛ ⎝ − ⎞ ⎠ d l k f k f v l v f k k ( / / ) 2 2 2 , Fij Fi s Fj s s Fi Fj (5) which just replaces continuous (tFiand tFj) by discrete (kFi/fsandkFj/fs) times.
Just the simple replacement of the continuous wavefront arrival time by the related discrete time disregards the effects of the sampling frequency as addressed in the remainder of this paper.
4. The sampling frequency effects
Considering discrete-time voltages and currents sampled at fs, the first wavefront arrival times are effectively kFi/fs⩾tFi andkFj/fs⩾tFj instead of the actual times tFiand tFj(Fig. 1), respectively, which must impact the accuracy of fault location estimation methods.
4.1. Time errors associated to the sampling frequency
The fault location estimation from(4)considers the traveling wave propagation time from the fault point to buses i and j (tFi−tF and
−
tFj tF, respectively), where the fault inception timetFis taken as time reference, nullified in such equation ([tFi−tF]−[tFj−tF]=tFi−tFj). Similarly the reference timetFis essential in order to develop thefinal fault location equations in the continuous time, an equivalent fault inception time in the discrete-time domain (k fF/s) is also necessary to be taken as reference.
The actual fault inception timetFis a continuous value, and at bus i with voltages and currents sampled at fsthe related discrete fault in-ception time could be either ⌊t fF s⌋/fsor ⌊(t fF s⌋ +1)/fs, where⌊ ⌋* is the largest integer value not greater than * (floor operator). However,
⌊t f⌋ + f
( F s 1)/sis higher than the actual valuetF. Therefore, the discrete fault inception time taken as reference (k fF/sinFig. 1) is given by[30]:
=⌊ ⌋ k f t f f , F s F s s (6)
wherek fF/s⩽tF<(kF+1)/fs. This procedure results in a fault incep-tion time error associated to the sampling frequency (Fig. 1), given by [30]: = − = − ⌊ ⌋ ξ t k f t t f f , t F F s F F s s F (7) where0⩽ξtF <1/fs.
Thefirst wavefront arrival time at bus i in the discrete-time domain (kFi/fsinFig. 1) is given by[30]: =⌊ ⌋ + k f t f f 1 , Fi s Fi s s (8)
which is the first sampling time soon after the first wavefront of the traveling waves had arrived at bus i. Therefore,
− < ⩽
k f t k f
( Fi 1)/s Fi Fi/s. This procedure yields in an error. Thereafter, the discretefirst wavefront arrival time at bus i is also given by:
= + k f t ξ , Fi s Fi tFi (9) where0<ξtFi<1/fsis thefirst wavefront arrival time error tokFi/fs, at bus i, associated to the sampling time.
From(8) and (9), the errorξtFiis given by:
= ⌊ ⌋ + − ξ t f f f t 1 . t Fi s s s Fi Fi (10) Therefore, from(1), wheretFi=tF+dFi/v, the errorξtFiis given by:
= ⎢ ⎣ ⎢ + ⎥ ⎦ ⎥ + − − ξ f d f v t f f d v t 1 1 . t s Fi s F s s Fi F Fi (11) The fault location estimation through(4)considers the combination of thefirst wavefront arrival time at both line terminals ( −tFi tFj), which yields the total time error considering the individual errors of the wa-vefront arrival time at both buses as follows[30]:
⎜ ⎟ = − = − − ⎛ ⎝ − ⎞ ⎠ ⇒ ξ ξ ξ k k f t t t t t Fi Fj s Fi Fj Fij Fi Fj (12) ⎜ ⎟ = ⎛ ⎝ ⎢ ⎣ ⎢ + ⎥ ⎦ ⎥ − ⎢ ⎣ ⎢ + ⎥ ⎦ ⎥⎞ ⎠ + − ξ f d f v t f d f v t f d d v 1 , t s Fi s F s Fj s F s Fj Fi Fij (13) where −1/fs <ξtFij <1/fs.
4.2. The two-terminal fault location estimation
Substituting(8)in(4), the fault location estimation in the discrete-time domain is composed by two terms:
= + − + − d l (k k ) /v f ξ v 2 2 , Fij Fi Fj s D t ξ Fij Fij DFij (14)
whereDFijis a quantized (discrete) fault location estimation and ξDFij, derived from the time errors associated to the sampling frequency, is a fault location estimation error, respectively defined as:
⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + − D k l v f k k Δ 2 2 ( ) , Fij Fij s Fi Fj k ΔFij (15) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥= − + ⎢ ⎣ + ⎥⎦− ⎢⎣ + ⎥⎦ −
(
)
ξ t d d t f t f , . D F Fi Fi l v f l d f v F s d f v F s 2 2 ( ) Fij s Fi s Fi s (16) Conventional two-terminal traveling wave-based fault location methods consider an analog equation defined in(15)for the fault location esti-mation, and disregard the fault location estimation error ξDFij, which is equivalent to the ideal fault location estimation equation in(4)by using the time difference (kFi−kFj)/fs instead of tFi−tFj, or just (5), as aforementioned. However, one feature of DFij, which has not been properly addressed in the literature, is that its value is quantized (kFi−kFj∈) and predictable (there is a limited number of fault dis-tances) such as addressed in the remainder of this paper.The fault location estimation error ξDFij, which is usually disregarded in the literature, depends on two unknown and random variables: the actual faut inception timetFand the actual fault positiondFi. In spite of
⎡ ⎣ ⎢ ⎤ ⎦ ⎥
ξDFij tF,dFi be unknown, it impacts the fault location estimation. Therefore, this paper presents a probabilistic analysis in order to re-place the term ⎡
⎣ ⎢
⎤ ⎦ ⎥
ξDFij tF,dFi in(14)by a searchfield where faults can be found with 100% of certainty regarding the sampling frequency effects, which is a valuable information to be considered in the fault location and line inspection procedures.
4.3. The predictable fault location estimation values
From(15), the discrete fault location estimation from bus i is given by
⏟
⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = + − = + D k l v f k k l l k Δ 2 2 ( ) 2 Δ Δ , Fij Fij s l Fi Fj k Fij Δ ΔFij (17)whereΔ is the line length portion propagated by the traveling wavesl
as half-sampling length, given by = l v f Δ 2s. (18)
According to(17), the following statements are verified (seeFig. 2): 1. WhenkFi−kFj=0, the fault location is estimated on the middle of
the line (DFij[0]=l/2).
2. Since l and lΔ are constants andkFi−kFj∈, the fault is estimated in predefined points located at multiple distances of lΔ km from the middle of the line.
The half-length of the line (l/2) can be divided inNΔl∈portions of the half-sampling lengthΔl, as follows[30]:
= ⎢ ⎣ ⎢ ⎥⎦⎥= ⎢ ⎣ ⎢ ⎥ ⎦ ⎥ N l l lf v 2Δ . l s Δ (19) If the line length is not a multiple ofΔ , there is an additional portionl
from each line end defined as follows[30]:
= − = −⎢ ⎣ ⎢ ⎥ ⎦ ⎥ l l N l l lf v v f Δ 2 lΔ 2 2 , s s 2 Δ (20) where0⩽Δl2<v/2fs.
The time differencekFi−kFj/fsis limited to a transit time (τ=l v/ ). Therefore, from(19),NΔlis defined as the maximum value ofkFi−kFjas follows[30]:
⏟
−⎢ ⎣ ⎢ ⎥ ⎦ ⎥ < − < ⎢ ⎣ ⎢ ⎥ ⎦ ⎥ − lf v k k lf v . s N Fi Fj s N l l Δ Δ (21)Thereby, the possible values ofkFi−kFjare
− = − + − + … − … −
kFi kFj { NΔl 1, NΔl 2, , 1, 0, 1, ,NΔl 1}. (22) From(17), the sampling differencekFi−kFjis given by:
− = − k k D l f v (2 ) . Fi Fj Fij s (23) Therefore,(20) and (21)yield:
− < < +
l
N l D l N l
2 ΔlΔ Fij 2 ΔlΔ . (24)
Thereby,DFijis a quantized distance with2NΔl−1possible values given by:
… −
D D D N
{ Fij[1], Fij[2], , Fij[2 Δl 1]}, (25)
where DFij[ ]n is the nth possible value of the fault distance estimation
given by: ⎜ ⎟ = + ⎛ ⎝ − ⎞ ⎠ = + ⇒ D [ ]n l n N l l n l 2 Δ Δ Δ Fij Δl 1 2 (26) ⎜ ⎟ = + ⎛ ⎝ −⎢ ⎣ ⎢ ⎥ ⎦ ⎥⎞ ⎠ D n l n lf v v f [ ] 2 2 , Fij s s (27)
where0<n<2NΔl. The fault location estimationsD [0]Fij and DFij[2NΔl] are not considered for two-terminal traveling wave-based fault location methods which consider only the time displacement of thefirst wave-front arrival time (kFi−kFj) such as addressed in the remainder of this
paper.
All possible values for the fault location estimation, which consist on fault distances estimated from Δl2+Δl tol−Δl2−Δl from the buses, with distance intervals ofΔ km (half-sampling distance) arel
shown inFig. 2.
4.4. The time-space regions
The total time-space region is delimited at one sampling period, from k fF/sto(kF+1)/fsand it covers the entire transmission line length (Fig. 3). All possible faults on the transmission line take place inside the total time–space region with random values of fault distancedFi and fault inception timetF.
As aforementioned, the fault location estimationdFijin(14)results in a quantized fault location estimation DFij with an unknown fault distance error ξDFij, where ξDFijis a function of unknown variables: fault distancedFi and fault inception timetF. Therefore, it is necessary to define a time-space area where all faults inside it will be estimated with a specific single distance from bus i in order to consider the effects of the error ξDFij.
The key point to identify the time-space areas where all faults will be estimated at one of the quantized distance DFij[ ]n is to identify the
time-space points where the actual fault locationdFiresults in nullified errors ξtF,ξtFi, and/orξtFj, i.e., the fault situations where the fault in-ception time as well as thefirst wavefront arrival time at bus i and/or j coincide with a sampling time (borders between time-space areas).
Thefirst wavefront arrival time at bus i coincides with a sampling time when = ⇒ = = + ξ t k f k m f 0 ( ), t Fi Fi s F s Fi (28) where0<m⩽NΔl. For instance, the lines ABC and GEHI inFig. 3 at-tend(28). Therefore, thefirst wavefront arrival time of all faults on the line ABC will be at the same sampling time at bus i, whereas thefirst wavefront arrival time of all faults on the line GEHI will be at the next sampling time at bus i.
WhenξtFi=0andξtF=0(e.g., fault at point A inFig. 3), the time propagated by the traveling waves from the fault point to the bus i is
ǻl D [ ]1 D [ ]2 D [N -1] D [N ] D [N +1] D [2 -2N ] D [2 -1N ] D [ ]0 D [ N ]2 l/2 ǻl ǻl2 ǻl ǻl Fij Fij Fij Fij Fij Fij Fij Fij Fij ǻl ǻl ǻl ǻl ǻl ǻl
Fig. 2. Possible fault location estimations.
given by = − = + − = τ t t k m f k f m f , Fi Fi F F s F s s (29) yielding = = d m f v m2Δ ,l Fi s (30) and = − dFi DFij[2 ]m Δ ,l2 (31)
which means that the fault at point A inFig. 3is far m l2 Δ km from the bus i (an integer multiple of the distance propagated by the traveling waves in one sampling time2Δ ) and farl Δl2km from the discrete fault position DFij[2 ]m.
WhenξtFi=0 andξtF→1/fs (e.g., fault at point C inFig. 3), the distance from the fault point to the bus i is given by
= − − dFi DFij[2 ]m Δl2 2Δ ,l (32) yielding ⎜ ⎟ = − = ⎛ ⎝ − ⎞ ⎠ d m f v m l 1 2 1 Δ , Fi s (33)
which means that the fault at point C inFig. 3is far2(m−1)Δ kml
from bus i and farΔl2+2Δlkm from DFij[2 ]m.
Since distances from points A and C to the discrete fault location
DFij[2 ]m are known, the distance of a fault located at any point of the
line ABC (on the border between time–space areas) can be estimated by using geometry (Fig. 3) as follows:
− − = ⇒ D m l d l ξ f [2 ] Δ 2Δ 1/ Fij Fi t s 2 F (34) = − − − ⌊ ⌋ dFi[ ]tF DFij[2 ]m Δl2 t vF 2t fF s Δ .l (35)
The same mathematical procedure used to define(35)for faults on the line ABC can be used to identify all boundary lines whose the first wavefront arrival time will reach a sampling time at bus i or j (ξtFi=0or
=
ξtFj 0). Therefore, the actual fault distance from bus i as a function of tFand DFij[2 ]m resulting in thefirst wavefront arrival time at bus i or j
equal to a sampling time (nullified error ofξtFiorξtFj) is given by: = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥+ + − ⌊ ⌋ + dFi[ ]tF DFij 2m (Δl t vF 2 t fF s Δ )l p (2Δ )l q , d 2 ΔFi (36)
where 0<m⩽NΔl,q∈,p= −1or 1, andΔdFiis the distance between the actual fault distancedFiand the respective quantized location es-timation DFij. For instance, Fig. 3 highlights the nearest lines from
DFij[2 ]m by using(36): (1) line ABC obtained withp= −1andq=0;
(2) line GEHI obtained with p= −1andq=1; (3) line DEF obtained withp=1andq=0; (4) line JBHL obtained withp=1andq= −1.
The boundaries for the fault to be estimated in a specific quantized location can be obtained through the comparison between the esti-mated fault location in(14)and the actual fault location in(36), where the last terms of both equations can be compared (ξDFij[ ]tF =ΔdFi[ ]tF) considering the limits −Δl<ξDFij <Δl. Therefore, the borders for faults to be estimated at the predicted distance DFij[2 ]m is given by
⎜ ⎟ ⎜ ⎟ = ⎛ ⎝ + − ⌊ ⌋ ⎞ ⎠ + ⎛ ⎝ ⎞ ⎠ ξDFij[ ]tF Δl2 t vF 2 t fF s Δl p 2Δl q. (37) For instance, since −Δl<ξDFij <Δl and selecting the appropriated
∈
n andp= −1or 1, all the faults located inside the polygons AB-HEDA and IHLI in Fig. 3 will be estimated on the single position
DFij[2 ]m.
Each discrete fault location estimation DFij[ ]n is associated to a
time–space region with the same area. For instance,Fig. 4depicts the time–space regions for all possible faults inside the transmission line
considering distinct sampling frequencies. In this example, the line in Fig. 4(a) is subdivided in6Δl+2Δl2subsections, whereΔl2=Δ /3l for the sake of illustration simplicity. This could be a situation where L = 100 km, v = 2.8798 km/s, and fs= 9.6 kHz ( lΔ = 15 km and lΔ2= 5 km).
According to(19), the number of lozenges (the fault location re-solution) is proportional to the sampling frequency. Therefore, the highest possible sampling frequency is desired for the two-terminal traveling wave-based fault location as addressed in next sections. For instance, by doubling the sampling frequency inFig. 4(b) the number of lozenges increased from 5 to 11 (estimation on 11 possible locations), whereas by tripling the sampling frequency inFig. 4(c) the number of lozenges increased to 19 (estimation on 19 possible locations).
4.5. Probabilistic searchfields
Since both the actual fault distancedFiand the fault inception time tFare random variables, a specific fault can present fault location es-timationdFij in two pre-defined positions D nFij[ ] or DFij[n+1] ran-domly. For instance, the faults F1 and F2 on the same location, but with different inception times, betweenD [2]Fij orD [3]Fij , are estimated on D [2]Fij or D [3]Fij , respectively, in Fig. 4. Therefore, a probabilistic searchfield is necessary to account for these variations.
Fig. 5depicts the n-th time–space region with a total areaA1+A2 (left-hand plot) where all faults will be estimated on DFij[ ]n. For the
sake of geometry simplicity, A1+A2 constitutes a lozenge (unique Fig. 4. The effects of the sampling rate (time-space regions): (a) fs; (b) 2 fs; (c)
region) with area given by: + = = A A l f l f (2Δ )(1/ ) 2 Δ . s s 1 2 (38) The faults in the distance rangeDFij[ ]n ±ΔdF, with0⩽ΔdF<Δl, will be estimated on DFij[ ]n if the combination of the actual fault locationdFi and the fault inception timetFis inside the area A1or A2(left-hand plot in Fig. 5) delimited from DFij( )n −ΔdF to DFij( )n +ΔdF, which is equivalent to be inside the area A3 in the right-hand plot inFig. 5.
The rectangle triangles with areas A4andA5are similar, yielding
= A d lf Δ 4Δ . F s 4 2 (39) In addition, four rectangle triangles with area A4 complement the re-gion with areaA3in order to form a rectangle with sides2ΔdFand1/fs. Therefore, the areaA3is given by
= − = − A d f A l d d lf 2Δ 1 4 (2Δ Δ )Δ Δ . F s F F s 3 4 (40) The probability of a fault with actual position into the distance range
∈ ±
dFi DFij[ ]n ΔdFibe estimated correctly on DFij[ ]n is given by ⎡ ⎣ ⎢ ⎤ ⎦ ⎥= + = − P d A A A l d d l Δ 100% (2Δ Δ )Δ Δ 100%, F 3 F F 1 2 2 (41) where0⩽ΔdF<Δl.
The distance rangeΔdFis a percentage of the distance propagated by the half-sampling distanceΔ as followsl
= = d η l η v f Δ Δ 2 , F s (42) and = η f v d 2 Δ , s F (43) where0⩽η<1 is a percentage of the searchfield.
Based on(42) and (41), the probability of the fault be inside the searchfield is given by
= −
P η d[ Δ ]1 η(2 η)100%, (44)
where0⩽η<1.
According to(44), the searchfield with fault location probability of 100% is achieved when
−η η= ⇒η − η+ = ⇒
(2 ) 1 2 2 1 0 (45)
=
η 1, (46)
whereas the search field with fault location probability of 90% is achieved when
−η η= ⇒ η − η+ = ⇒
(2 ) 0.9 10 2 20 9 0 (47)
=
η 0.68. (48)
The searchfield with fault location probability of about 0% is achieved when
−η η= ⇒η=
(2 ) 0 0. (49)
As a benchmark,Table 1summarizes values of η which delimit search fields for specific fault location probabilities.
5. The proposed two-terminal fault location estimation
Despite of the conventional discrete fault location estimationDFij defined in(15), this paper proposes a searchfield (distance interval of
d
Δ Fijkm) centered onDFijwith a specific probability to find the fault in order to increase the efficacy of the actual fault location search as fol-lows: ⎜ ⎟ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥= + ⎛ ⎝ − ⎞ ⎠ ± d k l v f k k η v f Δ Δ 2 2 2 , Fij Fij s Fi Fj s (50) where − ⌊lf vs/ ⌋ <kFi−kFj< ⌊lf vs/ ⌋and0⩽η<1.
For instance, the fault is found with 100% of certainty (η=1in Table 1) considering the effects of the sampling frequency in the fol-lowing searchfield
⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = + − ± d k l v f k k v f Δ Δ 2 2 ( ) 2 , Fij Fij s Fi Fj k D k s Δ [Δ ] Fij Fij Fij (51)
whereas the fault is found with 75% of certainty ( =η 0.50 inTable 1) in half of this searchfield as follows
⎜ ⎟ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥= + ⎛ ⎝ − ⎞ ⎠ ± d k l v f k k v f Δ Δ 2 2 4 . Fij Fij s Fi Fj s (52)
Based on(51) and (18), it is possible to set the sampling frequency to get a specific search field with 100% of certainty regarding the effects of the sampling frequency as follows:
= ⎡ ⎢ ⎢ ⎤⎥⎥ f v l 2Δ , s (53)
For instance, if a searchfield of about 300 m is desired, which is about one tower span[31], and the propagation velocity of the line is taken as 98% of the speed of light, such as considered in most ac transmission line fault location methods, the desired sampling frequency would be
=
fs 980kHz, which is feasible since there are traveling wave-based fault location methods running atfs =1MHz currently[40].
6. Performance assessment
Experimental and simulated faults demonstrated the performances of the classical fault location estimation method(5)and the proposed method(51). Both tests demonstrated that the proposed method con-templates the effects of the sampling frequency following the mathe-matical theory performed in this paper.
Fig. 5. Geometry of an elementary time-space region.
Table 1
Probability for specific search fields.
P η l[ Δ ](%) 100 90 75 50 0
6.1. Considerations about thefirst wavefront arrival time detection The classical two-terminal traveling wave-based fault location methods are enabled by the previous fault detection method, which must provide an accurate detection of thefirst wavefront arrival time. Both fault detection and fault location methods present their own issues as highlighted inTable 2.
Regarding the main issues related to the detection method, it is well-known that the fault-induced transients are affected by some para-meters such as the fault inception angle, the fault resistance, electrical noise, and the frequency response of instrument transformers. For in-stance, a single-line-to-ground fault at phase A (AG fault) with fault inception angle around the phase-A voltage zero (e.g., 10 degrees) will result in low amplitude traveling waves, and thefirst wavefront arrival times will probably be not detected[24,25]. Nevertheless, the effects of such parameters in fault detection methods based on the traveling waves are well reported in the literature[24,25]. Therefore, the effects of the parameters mentioned above must be dealt with by the detection method and are out of scope of this paper.
Regarding the simulations, severe transients were produced aiming to avoid errors in the fault detection in order to evaluate the sampling frequency effects in the fault location method. This was accomplished by considering solid AG faults (rF= 0 Ω) with fault inception angleθF≈
°
90 [46], minimizing the fault parameter effects in the fault detection. As a consequence, the first wavefront arrived on the line terminals presented relevant magnitude in order to be easily detected. Besides, the transducers employed have an ideal model without noise for re-ducing the data acquisition errors and errors in the fault detection. Therefore, the effects of the fault resistance, fault inception angle, electric noise, and transducers could be disregarded for the detection method (Table 2).
Regarding the experimental test, solid faults (resistance of rF= 0 Ω) were not used due to its harmful effects on the laboratory platform. Moreover, the experiments employ real voltage sensors with noise and finite frequency bandwidth. To provide a reliable detection of the first wavefront arrival times in the experimental data, manual inspection was performed. However, some mechanisms such asθF≈90°could be adjusted to generate the highest fault-induced transients as possible in order to facilitate the identification of the correct wavefront arrival times. In summary, the parameters which affect the fault detection method could not be disregarded in the experimental tests (Table 2), but most faults could be generated with detectable transients.
Besides the effects of the fault parameters and instrument trans-formers on the traveling waves, the performance evaluation of a spe-cific wavefront arrival time detection method is also beyond the scope of this paper. Therefore, the wavefront arrival time is detected using the cycle-by-cycle method, which is one of the most straightforward method for this purpose[47,48]. Once the simulation tests considered all the faults with relevant fault-induced transients, this method de-tected thefirst wavefront arrival time accurately. Regarding the ex-perimental database, the detection of thefirst wavefront arrival time
presented uncertainties in few cases, but the results obtained validated the proposed fault location method experimentally.
Thefirst wavefront arrival time detection method used a third-order Butterworth anti-aliasing low-passfilter with cutoff frequency at 90% of the Nyquist frequency, i.e., fc=0.9 /2fs in both simulated and ex-perimental data.
6.2. Experimental platform
Fig. 6 depicts the experimental setup composed by 1 km long polypropylene-typeflexible cable (1 km PP cable) with four isolated copper wires with a transverse area of 1 mm2to emulate a transmission
line. The isolated wires present no individual metallic shield to enhance the coupling among wires. Nevertheless, the cable presents an elec-tromagnetic shielding enveloping the wires, and connected to the ground at both ends in order to reduce the effects of noise, avoid electromagnetic interference among cables, and facilitate the detection of traveling waves. The local line terminal, at bus 1, was connected to the single-phase main grid at 127 V (peak voltage of ≈180 V) and grounding, whereas the remote line terminal, at bus 2, was connected to a single-phase linear load of ZL=100 Ω with no connection to ground.
According toFig. 6, two isolated parallel copper wires were used to feed the load: the first wire (phase wire) connects the single-phase source V to one end of the load impedanceZL(i.e., it connects bus 1 to bus 2); the second wire (return wire) is grounded at the source side and is connected to the ungrounded load at the other end. This connection was necessary in the experimental test bench due to some reasons:
•
To guarantee that the waves travel along the line without the grounding interference. Without the return wire (i.e., just con-necting the load and source at the common ground of theTable 2
Parameter uncertainties in the analysis.
Method Main issues Sim. Exp. Real-life
Noise ✓ ✓
Fault Fault resistance ✓ ✓
Detection Fault inception angle ✓ ✓
Transducers ✓ ✓
Sampling frequency ✓ ✓ ✓
Fault Velocity estimation ✓ ✓
Location Time synchronization ✓
Line length ✓
Sim: simulations; Exp: Experimental.
Fig. 6. Experimental platform for traveling wave propagation in a transmission line.
experimental bench), the first wavefront of the traveling waves which reaches the nearest bus interferes immediately in the farthest bus, and transients would be always detected at the same time in both buses.
•
The currents at a specific point in both phase and return wires are exactly the same and with opposite directions, even when thefirst wavefronts (desired to be detected) due to the fault travel through the wires. Therefore, the magneticfield is stronger in between the parallel phase and return wires than in other regions inside the metallic shield due to the opposite direction of currents in the par-allel wires. Furthermore, the magneticfield on the metallic shield, resultant from the parallel phase and return wires, is substantially reduced if compared to the situation without the return wire. The metallic shield is also grounded at both ends as aforementioned, which in addition to reduce the magnetic interference it ideally nullifies the electric field among cables. Therefore, the electro-magnetic interference among cables is irrelevant, and the cable could be wrapped on two reels with the fault point between them. In fact, no unusual interaction between traveling waves was verified in the experimental tests with the PP cables wrapped on reels.The selected PP cable couldfit all of the aforementioned technical re-quirements to enable the detection of thefirst wavefront arrival time of the traveling wave. However, other cable types can be used in other experimental setups for traveling wave-based analysis since questions such as electromagnetic interference can be verified.
The voltage transducers are Agilent N2894A passive probes with 10:1 attenuation ratio and 700 MHz bandwidth. Voltages are measured at buses 1 and 2 in order to detect thefirst wavefront arrival time at the line terminals, whereas a voltage at the fault location is also measured for manual validating of the traveling wave propagation time. Currents were not considered due to the limited frequency response of the available current sensors. The measurements of voltages at line ends and at the point of the fault employ a single device (MSO-X 4024A Agilent oscilloscope) to avoid communication and synchronization de-vices. This choice disregards additional errors in the fault location es-timation as addressed in the literature[5], and the effects of the sam-pling frequency are highlighted.
The oscilloscope presents a 200 MHz bandwidth, a maximum sam-pling frequency of 5 GSa/s, a maximum memory depth of 4 Mpts, and four analog channels. Therefore, based on the features of voltage transducers and oscilloscope, voltage measurements could be sampled at fsexp=5 MHz. The channel-to-channel isolation of the oscilloscope is equal or greater than 40 dB from DC to 200 MHz, and manual inspec-tions of the oscillographic data verified that there was no interference between the oscilloscope channels when the traveling waves arrived at the monitored points by using fsexp= 5 MHz. In addition, the oscillo-scope provides a standard USB 2.0 connectivity for exporting the ex-perimental data for off-line analysis.
The goal of this paper is not to emulate a transmission line with realistic parameters compatible with a high-voltage overhead ac transmission line, but rather validate the effects of the sampling fre-quency in a traveling wave-based fault location method. Therefore, for accomplishing the experiment, a 1 km PP cable could be wrapped on the reels and the traveling waves could be measured successfully at 5 MHz.
As aforementioned, the effects of the sampling frequency would be ideally highlighted if the uncertainties of all of other parameters would be nullified, which is not possible in an experimental test. Therefore, parameters related directly to the fault detection method could not be totally handled to get the most severe fault transients and no errors in the fault detection. For instance, the minimum fault resistance which could be used in order to guarantee the safety of the experiment wasRF = 33.33 Ω. The fault inception angle could not be automatically set to
°
90. Therefore, after manual inspections only faults with fault inception angle around90°were selected in order to enhance the fault detection
performance. In addition, a high noise level was present in the voltage signals, and the voltage transducers presented limited frequency re-sponse. Therefore, issues related to the fault detection method could not be completely eliminated in the experimental data (Table 2).
Ideally, the actual traveling wave velocity value should be taken into account in order to properly prove the effects of the sampling frequency with no interference of other parameters. However, in a real transmission line or through experimental tests the actual traveling wave velocity estimation is not possible since it depends on complex factors such as environmental temperature and irregular grounding along the line. In this experimental test, the traveling wave velocity was statistically estimated asv=1.4286km/s orv=0.4762cby considering several switchings and measuring the cable transit time for the tra-veling waves. Therefore, errors related to the velocity estimation are automatically included in the performed experimental analysis (Table 2) as further addressed in the remainder of this section. Never-theless, uncertainties related to the time synchronization and line length could be disregarded in the experimental tests (Table 2).
Faults were applied on the cable from 10 to 990 m, with a step of 10 m, starting from the bus 1. Therefore, 99 actual faults were applied on the experimental transmission line. Although the line is so small (1 km), the detection of thefirst wavefront arrival time at each cable end was possible due to a combination of a high sampling frequency (5 MHz), a relatively low propagation velocity ( =v 0.4762c), and fault situations with the traveling waves with the highest amplitude in ac-cordance with limitations of the experiment. Since the detection method was not the focus of the paper, a manual inspection was ac-complished in each actual data to reduce errors due to the detection method.
In order to prove that faults in the same position can result in dif-ferent fault location estimations in accordance with the fault inception time, the fault location methods (classical and proposed) were eval-uated with a sampling frequency offs=500kHz (ten times lower than fsexp). Therefore, the fault inception times provided by the detection method atfsexp=5 MHz is considered as continuous times, whereas the related fault inception times used by the fault location methods are considered as discrete times.
Fig. 7(a) depicts the voltages at the line ends (buses 1 and 2) and at the point of the fault for a fault 590 m far from bus 1. After the ex-perimental data acquisition at a sampling frequency offsexp=5 MHz, the following continuous times were provided by the detection method and manual inspection for this case: the fault inception time astF=17/fs
exp, and thefirst wavefront arrival time at buses 1 and 2 astF =37/fs
exp
1 and
=
tF2 31/fsexp, respectively. Therefore, the continuous fault inception
times could be identified with a time resolution of f1/sexp=0.2μs. The discrete wavefront arrival times (used in the fault locators at 500 kHz) are automatically identified askF1=kF 2=38 in this case.
6.3. Fault location performance assessment through experimental data
By usingfs=500kHz, =l 1000 m, andv=0.4762c, the parameters ≈
l
Δ 143m, Δl2≈71m, and NΔl=3 are obtained according to(18),
(20), and(19), respectively. Therefore, the line is divided in six regions of 143 m and two regions of 71 m, where the predictable discrete fault
locations defined in (17) or (27) are
≈
DFij {71, 214, 357, 500, 643, 786, 929}m such as depicted inFig. 8. As aforementioned, a fault takes place at a continuous time, i.e., at infinite possible values between two sampling times
⩽ < +
k fF/s tF (kF 1)/fs, and the fault can be estimated in either the two closest discrete fault points DFij[ ]n defined in(27). To verify this
fea-ture, fsexpwas set to be ten times higher than f
s (fs =5
exp MHz) and faults were applied with random fault inception time between k fF/sand
+
k f
(F 1)/s. The location results of these faults (conventional and pro-posed) are shown inFig. 8.
Regarding the conventional method, according toFig. 8(a), faults with actual position of DFij[ ]n ⩽dFi⩽DFij[n+1] were estimated
randomly in either DFij[ ]n or DFij[n+1]km, demonstrating the un-certainties of the fault location estimation due to the sampling fre-quency. For instance, faults at 360 and 640 m from the bus 1 were both estimated on the middle of the line (500 m). Another example, faults on dF1=400, 410, and 420 m were randomly estimated ondF12=500, 357, and 500 m, respectively. In a practical application, where the length of the transmission line can reach several hundreds of km, the identifi-cation of the actual position of the fault would be a critical task by the maintenance crew since both the direction and distance to follow from the fault location estimation are unknown.
Considering the proposed probabilistic searchfield with 100% of certainty regarding the sampling frequency effects defined in(51), all faults estimated at a discrete fault point DF12[ ]n, with0<n<6, were
correctly identified from DF12[ ]n −Δl to DF12[ ]n +Δl with 100% of certainty in this set of experimental faults, i.e., all actual fault points were identified inside the search fields in Fig. 8(b), even with un-certainties in the propagation velocity estimation. In a practical appli-cation, this searchfield could speed up the line restoration.
6.4. Theory verification through experimental data
Since the fault inception timetFcan be placed in ten different times between two sampling times of the fault locators (ξt ={0, 1,…, 9}/fsexp
F ),
each actual fault was evaluated 10 times in an off-line analysis.
Therefore, the fault locators could be evaluated 990 times in order to prove the time-space lozenges experimentally. For instance,Figs. 7(a) and (b) refer to the same fault. However, the fault locators are eval-uated with different sampling times:k fF/s =8/fs
exp (ξ =9/f t s exp F ) and = = kF/fs kF/fs 38/fs exp
1 2 in Fig. 7(a), whereas k fF/s =15/fs exp (ξtF=2/fsexp) andkF/fs=45/fs exp 1 , andkF/fs=35/fs exp 2 inFig. 7(b). As a
consequence, the same fault at 590 m from the bus 1 would be esti-mated in two different places according to the fault inception time.
Considering the 990 evaluated experimental data,Fig. 9also depicts the performance of the conventional fault locator defined in(5)which is equal to the first two terms of the proposed method in (51) (DFij[ΔkFij]). As expected by the mathematical development proposed in this paper, a fault in a specific location can present two fault location estimations depending on the fault inception time. In addition, there are time-space regions in the format of lozenges where the faults inside them have their location estimated to the related discrete fault location DFij(Fig. 9).
Some faults near the borders of the lozenges inFig. 9had the fault estimated with the opposite location by the classical method due to uncertainties to estimate the exact traveling wave velocity experimen-tally and issues related to fault detection method such as high-level noise and low amplitude traveling waves, which could not be overcome in the experimental setup (Table 2). By using the proposed mathema-tical approach (all terms in(51)), the faults were located in the pro-vided searchfield successfully, as shown inFig. 8, even considering the
Fig. 7. Voltages in an actual fault far 590 m from the bus 1: (a)ξtF=9/fsexp; (b)
=
ξtF 2/fsexp.
Fig. 8. Fault location estimation considering experimental data: (a) classical method; (b) proposed method with searchfield.
uncertainties in the lozenge borders.
6.5. Theory verification through simulations
The 500 kV/60 Hz transmission power system inFig. 10 [19]was simulated in Matlab Simulink with the line modeled with distributed parameters in order to consider the traveling wave phenomena in ac-cordance with the Bergeron model. According to[49], the propagation velocity of aerial modes slight changes with the frequency, so that this line model could be properly used since only thefirst wavefront arrival time of the aerial modes is required to be detected by the two-terminal fault location methods studied in this paper. Therefore, the considered
traveling wave velocity is given by v=
= × − × −
L C
1/ 1 1 1/ 873.65 10 13.806 109 = 2.8798 × 105 km/s, i.e., there are no uncertainties in the propagation velocity in the simulation data (Table 2) as required to validate the effects of the sampling fre-quency. In addition, this propagation velocity is usually recommended for traveling wave-based fault locators in overhead ac lines [5]. Nevertheless, for a practical application in actual transmission lines or when a frequency dependent model of the line is used, the propagation velocity can be obtained by means of an energization condition[8].
Two traveling wave-based IEDs were implemented and connected on buses 1 and 2. The performance of the classical fault location esti-mation defined in (5)and the proposed method defined in(51) was assessed with two databases summarized inTable 3. The line length is
=
l 100 km, and faults were simulated on the line (Fig. 10) with dis-tances from bus 1 varying from 5 to 95 km with steps of one kilometer. It is well-known that traveling wave-based fault location estimation methods are desired to be used with high-sampling frequency. However, initially, this paper considered traveling wave-based IEDs with a low-sampling frequency offs =9600Hz to get results similar to the experimental setup. In this fashion, the effects of the sampling frequency could be better highlighted by comparing experimental and simulated results. In the remainder of this paper, more realistic sam-pling frequency will be evaluated. The time step used in the simulations ( tΔsim) was 10 times less than the IED sampling period 1/f
s, i.e., =
fssim 10fs, wherefssim=1/Δtsimis defined in this paper as the simu-lation sampling frequency.
By usingfs =9600Hz, =l 100 km, and v = 2.8798×105km/s, the parametersΔl=15km,Δl2=5km, andNΔl=3are obtained according to(18), (20), and(19), respectively. As a consequence, the line is di-vided in six regions of 15 km and two regions of 5 km, where the fault location will be estimated on the discrete fault points {5, 20, 35, 50, 65, 80, or 95} km according to(27). Therefore, both experimental and si-mulations resulted in a line subsection of sixΔ and twol Δl2, but with different dimensions.
Fig. 11depicts the performance of the conventional fault location estimation method defined in(5), which is also considered in thefirst two terms on the proposed equation in (51), where the simulation sampling frequency was ten times higher than the relay sampling fre-quency, and all fault inception times from k fF/sto(kF+1)/fsin intervals ofΔtsim=1/f
s
simwere considered (database 1 inTable 3).
According toFigs. 9 and 11, the time-space areas addressed in Section4.4and shown inFig. 4were validated through experimental and simulation results, proving that:
•
There are distinct time-space regions in the lattice diagram where all faults are estimated on the same position.•
Faults on the same position can randomly present different location estimation depending on the random fault inception time.•
Spite of distinct time-space regions, a conclusive fault location es-timation is not possible due to the randomness of the actual fault distance and fault inception time. Therefore, probabilistic search fields are necessary to overcome the effects of the sampling fre-quency, such as proposed in this paper.As mentioned before, the fault location was previously estimated, considering low sampling frequency in simulations, just for comparison with the experimental results. In practical applications, the use of a high sampling frequency minimizes the errors in the fault location es-timation. Therefore, a new set of faults was simulated considering the IED sampling frequency of fs=192 kHz and fs =10f
sim
s(database 2 in
Fig. 9. Fault location estimation considering experimental data: time-space lozenges.
Fig. 10. The power system model.
Table 3
Fault inception time and distance parameters.
Database
Parameters 1 2
fssim(kHz) 96 1920
fs(kHz) 9.6 192
All steps of All steps of
⩽ t kF fs F(s)< + kF fs 1 fssim 1 fssim 1 dF(km) from Bus 1 5:1:95 19.25:0.05:20.75 Total 910 310
Table 3).
The sampling frequency of fs=192 kHz, =l 100 km, and the velo-city of v = 2.8798×105km/s result inΔl≈750m,Δl ≈125
2 m, and
=
NΔl 133according to(18), (20), and (19), respectively. Therefore, the line is divided in 133 regions of 750 m and two regions of 125 m. Faults were simulated from 19.25 to 20.75 km from bus 1 by changing 50 m at a time (i.e., faults at 31 locations). For each fault location it was con-sidered fault inception times at all the ten time subdivisions between two sampling times (i.e., ten faults for each fault location). Therefore, 310 faults were evaluated (Table 3).
Fig. 12depicts the fault location estimations for this set of faults, where each graphical point represents the results of a fault (310 gra-phical points). As predicted in(51), all faults estimated on 20 km were inside a lozenge time-space area with searchfield of 20±0.75 km with 100% of certainty.
7. Additional remarks
As aforementioned, the effects of the sampling frequency as well as uncertainties in the traveling wave velocity estimation, line length, and time synchronization through GPS affect directly the fault location es-timation, whereas noise, fault resistance, fault inception angle, and the frequency resolution of transducers are the main issues of fault detec-tion methods such as summarized inTable 2. Since the effects of the sampling frequency in the fault location method is the main goal of this paper, simulations were essential because all other issues and un-certainties such as unun-certainties in the propagation velocity, line length, and synchronization time were disregarded in simulations (known parameters). In these cases, the fault location methods were only af-fected by the sampling frequency (Table 2searchfield with 100% of certainty in the fault location estimation could be obtained.
It was not possible to verify the influence of only the sampling frequency without the interference of other parameters in the experi-mental tests because there were uncertainties in the propagation velo-city estimation. In addition, issues related to the detection method such as noise andfinite frequency response of transducers were present in the experimental tests. Fault resistance and inception angle presented minimum interference since they could be adjusted to generate strong transients. However, even with the interference of the propagation velocity, the proposed method could be validated successfully, and a search field with 100% of certainty in the fault location estimation could be obtained in these scenarios.
In a practical application where all kinds of parameter uncertainties are present such as shown inTable 2the searchfield for the fault lo-cation estimation must be expanded in order to cover about 100% of certainty in the fault location estimation. Although the inclusion of other effects besides the sampling frequency in the fault location esti-mation is out of scope of this paper, a brief addressing about them was taken into account as case studies in order to support further research in the direction of obtaining a searchfield with about 100% of certainty in the fault location estimation.
7.1. Uncertainties in the traveling wave velocity estimation
Considering an uncertainty in the propagation velocity estimation, the unknown actual velocity v is given by the estimated velocity (v)̃ plus an error (εv), i.e., v =ṽ ±εv. However, the velocity deviation is a percentage of the actual one, which is also unknown. Therefore, a maximum percentage (Pv) can be used in order to cover the un-certainties in the velocity estimation, i.e., P vv ̃is used instead of εv. Therefore, substituting v byṽ±P vv ̃in the classical two-terminal tra-veling wave fault location defined in(5)yields:
̃ ̃ ⎜ ⎟ = + ⎛ ⎝ − ⎞ ⎠ ± − d l v f k k P v f k k 2 2 2 . Fij s Fi Fj v s Fi Fj (54) Therefore, including this logic in(51), the effects of both the sampling frequency and uncertainties in the velocity estimation can be overcome by ̃ ̃ ⎜ ⎟ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥= + ⎛ ⎝ − ⎞ ⎠ ± + − d k l v f k k v P k k f Δ Δ 2 2 (1 ) 2 . Fij Fij s Fi Fj v Fi Fj s (55)
For instance, considering that the used transmission line presents a maximal velocity deviation of 3% due to external factors such as tem-perature, the search field in Fig. 12 would be extended from [19.25 20.75] km to [18.35 21.63] km in order to locate properly all faults from 19.25 to 2.75 km with 100% of certainty considering un-certainties of both sampling frequency and traveling wave velocity.
Following this preliminary analysis the maximal velocity deviation identification is a key point which must be properly evaluated in further studies. In[50], for instance, a method for the transmission line para-meter estimation is proposed based on fault record data, and errors were less than 2%. In addition, the modal traveling wave velocity is properly given by the transmission line parameters (series inductance L and shunt capacitance C). Therefore, based on[50], errors of 2% in the propagation velocity estimation would be expected, and Pv=3% as used in this case study would be enough to cover uncertainties in the velocity estimation.
To minimize this kind of uncertainty, propagation velocity has been successfully estimated during the line commissioning by analyzing transients generated during line switching maneuver[40].
7.2. Uncertainties in the time synchronization process
According to[16], a GPS provides time-of-day information with an accuracy better than 1μs anywhere in the world. In the conventional fault location estimation defined in(5), the effects of the GPS can be inserted in the time particle(kFi−kFj)/fsby adding a time error (εt) as follows: ⎜ ⎟ = + ⎛ ⎝ − ⎞ ⎠ ± d l v f k k ε v 2 2 2. Fij s Fi Fj t (56) Therefore, including this logic in(51), the effects of both the sampling frequency and uncertainties in the time synchronization can be over-come by ⎜ ⎟ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + ⎛ ⎝ ⎜ − ⎞ ⎠ ⎟ ±⎛ ⎝ + ⎞ ⎠ d k l v f k k v f ε v Δ Δ 2 2 2 2 . Fij Fij s Fi Fj s t (57) For instance,εtcan be set to 2μs, which is two times higher than the expected maximum GPS time error of 1μs. In this fashion, an extension ofε v/2t =287.98 m (by considering v = 2.8798×105 km/s) in the pro-posed searchfield would identify faults with 100% of certainty con-sidering both the effects of the sampling frequency and time synchro-nization errors.
The faults of database 2 inTable 3were again simulated considering a GPS time error of 1μs in the IED at bus 2, and the fault location results for these 310 new cases considering both the effects of the
