Journal of Power and Energy Engineering, 2014, 2, 616-623
Published Online April 2014 in SciRes. http://www.scirp.org/journal/jpee
How to cite this paper: Liu, X.M. and He, Z.Y. (2014) Transmission Lines Distance Protection Using Differential Equation
Algorithm and Hilbert-Huang Transform. Journal of Power and Energy Engineering, 2, 616-623.
Transmission Lines Distance Protection
Using Differential Equation Algorithm and
Xingmao Liu, Zhengyou He
School of Electrical Engineering, Southwest Jiaotong University, Chengdu, China
Received December 2013
This paper proposed the scheme of transmission lines distance protection based on differential
equation algorithms (DEA) and Hilbert-Huang transform (HHT). The measured impedance based
on EDA is affected by various factors, such as the distributed capacitance, the transient response
chara cte ris tics of current transformer and voltage transformer, etc. In order to overcome this
problem, the proposed scheme applies HHT to improve the apparent impedance estimated by DEA.
Empirical mode decompos ition ( E MD) is used to decompose the data set from DEA into the intrin-
sic mode functions (IMF ) and the residue. This residue has monotonic trend and is used to eva-
luate the impedance of faulty line. Simulation results show that the proposed scheme improves
significantly the accuracy of the estimated impedance.
Hilbert-Huang T ra nsf orm ; Differential Equation Algorithm; Distance Prot ecti on ;
Transmission lines are responsible for delivering a mass of energy from generator plants to load centers. Due to
covering long distance outdoor, those lines have the highest fault rate in the power network. After faults occur-
ring, they must be cleared immediately to decrease the disturbance the faults will impose on the power system.
Distance protection is one of the most used protective principles for transmission lines. The traditional dis-
tance relays which have been widely applied to transmission line calculate the impedance to the fault using the
fundamental frequency component of the local voltage and local current  . However, the operation time of
these relays is at least one cycle and their performance can be affected by current transformer saturation, expo-
nentially decaying current component (dc offset), etc. The travelling wave distance protections can achieve ultra
high speed operation time , while their reliability needs to be improved further. Unlike above protections, the
differential equation algorithm (DEA) based distance protection can operate within 6 - 8 ms after fault occurring
. This algorithm is not affected by dc offset and can make tripping decision before current transformer satura-
X. M. Liu, Z. Y. He
tion . Therefore, DEA is a promising distance relaying algorithm for transmission lines where faster operation
time is required.
However, affected by various factors, such as the transient response characteristics of voltage transformer and
current transformer, the distributed capacitance, etc, the result of DEA fluctuates up and down around the actual
value frequently after a fault occurring -. The direct using of the result calculated from DEA might lead to
mal -opera tio n of relay. To overcome above problems, several schemes have been suggested. In  , an ar-
tificial neural network (ANN) based methods we r e presented to estimate the impedance of the faulty transmis-
sion line. In  , several filtering algorithms have been applied to DEA based distance relaying system.
However, ANN based methods are difficult to be trained and tested in engineering application and filtering al-
gorithms cause the time delay for tripping decision of relay.
This paper presents a Hilbert-Huang transform (HHT) based scheme to extract high-frequency components of
the results estimated by DEA. The data set obtained from DEA was decomposed into two parts: a collection of
intrinsic mode functions (IMF) and a residue which has the characteristic of monotonic. Then, the residue was
used to evaluate the impedance of the faulty line.
2. The Hilbert-Huang Transform
The Hilbert-Huang transform (HHT), presented by an American Scientist Norden E Huang and others in 1998
 , is a time-freque nc y analysis method. In contrast to other methods like the wavelet transform which
has been introduced to analyze linear but nonstationary data, The HHT works well for data that are nonlinea r
and nonstationar y.
The Hilbert-Huang transform (HHT) consists of the empirical mode decomposition (EMD) and the Hilbert
spectral analysis (HSA). The EMD method is the fundamental part of the HHT. Using the EMD method, any
given signal can be decomposed into a collection of IMF. An IMF is defined as a function which meets the fol-
1) In the whole data set, the number of extrema and the number of zero-crossings must either be equal or dif-
fer at most by one.
2) At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the
local minima is zero.
The EMD method is implemented to extract IMF through a sifting process. The procedure of the sifting is
sum mar ized as follows:
1) For a given signal, x(t), the EMD starts by identify all the local extrema.
2) Separately connect all the local maxima and the local minima with cubic splines interpolation to form the
upper envelope, u(t), and the lower envelope, l(t).
3) The mean of the two envelopes is calculated as m1(t) = [u(t) + l(t)]/2. Take the difference between the sig-
nal and m1(t) as the proto-IMF, h1(t) = x(t) − m1(t).
4) Check h1(t) against the IMF criteria to determine if it is an IMF.
5) If h1(t) does not satisfy the IMF criteria, repeat step 1 to 4 on h1(t) as k times until the first IMF is realized.
That is h1(k−1)(t) − m1k(t) = h1k(t). Then, it is assigned as an IMF component, c1(t) = h1k(t).
6) Repeat the step 1 to 5 on the residue, r1(t) = x(t) − c1(t).
7) The procedure ends when the residue, rn(t), becomes a monotonic function from which no more IMF can
be extracted .
After completion of the above procedure, the signal can be written as follow:
( )( )( )
where ci(t) is an IMF component and rn(t) is the residual.
3. The Differential Equation Algorithm
In order to derive the differential equation algorithm (DEA), the following presumptions are required:
1) The voltage transformer is ideal in the frequency range from 50 to 500 Hz.
2) The transmission line is perfectly transposed.
3) The shunt capacitances of the line are neglected.
X. M. Liu, Z. Y. He
A briefly derivation of DEA will be mentioned subsequently and details of DEA is described in . The
protected transmission line is modeled as follows:
where ua, ub, uc and ia, ib, ic are the phase quantities of voltages and currents, respectively. The zero sequence
current is calculated as i0 = (ia + ib + ic)/3. R1, and L1 are the positive sequence resistance and inductance, respec-
tively, and R0 and L0 are the zero sequence resistance and inductance, respectively. The parameters kr and kl are
the relative line length for the positive sequence resistive part and the positive sequence inductive part, respec-
tively. For a faulty transmission line, both kr and kl should be bounded between 0 and 1. In an ideal case, kr
should be equal to kl. According to Equation (2), the equations for each specific fault type can be derived.
3.1. Single Phase-to-Ground Fau lt
Assume a single phase-to-ground fault occurs on phase A at an unknown distance of the protected line length,
the voltage of the phase A at the relay position is given by
For phase-b-ground or phase-c-ground faults, similar equations can be derived based on Equation (3).
3.2. Double Phas e-to-Gr ound Fault
For a double phase-to-ground fault, there is no zero sequence current. Consider a fault between phase-a and
phase -b, the equation can be written as:
()( )( )
3.3. Equation s
In the case of a three-phase fault, the zero sequence current is null and Equation (2) becomes:
Since it is a symmetrical fault, any phase can be used to determine the values kr and kl.
4. HHT Based Distance Relaying Scheme
Affected by various factors, such as the transient response characteristics of voltage transformer and current
transformer, the distributed capacitance, etc, the parameters kr and kl obtained from DEA fluctuate up and down
around the actual value frequently after a fault occurring, which might cause the relay isn’t able to operate cor-
On the fault condition, direct use of DEA does not give acceptable estimates of the parameters kr and kl.
Therefore, the paper introduces a HHT based distance relaying algorithm which adopts HHT to process the re-
sult from DEA. The procedure of the suggested algorithm is presented below for input data windows which
contains the latest K + 1 samples.
1) Input the latest K + 1 samples to first in first out (FIFO) queue.
2) Applying DEA to obtain kr(n − K), ···, kr(n), and kl(n − K), ···, kl(n).
3) Input kr(n − K), ···, kr(n), and kl(n − K), ···, kl(n) to FIFO queue. Two data sets in the FIFO queue are fol-
X. M. Liu, Z. Y. He
4) Using EMD to decompose Kr and Kl, respectively. According to Equation (1 ) , the residual obtained from
EMD can be expressed as follows
are monotonic functions from which no more IMFs can be extracted.
be denoted as:
from normalized values to ohms (resistance and
reactance) and determining whether to trip or not. The protection zone can be either Mho or Quadrilateral cha-
racteristic which can be found in any conventional distance protection relay.
5. Power System Simulation
In order to verify the validity of the proposed scheme in this paper, the following simulation has been conducted
using PSCAD/EMTDC. The simulated model which is a 500 kV power system shown in F igure 1. The para-
meters of the system model are listed below.
The transmission line is 300 km, the sequence data of which are the following:
R1 = 0.0201 Ω/km, L1 = 0.9045 mH/km, R0 = 0.1016 Ω/km, L0 = 2.4048 mH/km
The parameters of the equivalent source reactance at bus A are:
R1 = 1.0518 Ω, L1=0.1371 H, R0 = 0.608 Ω, L0 = 0.0922 H
The parameters of the equivalent source reactance at bus B are:
R1 = 25.6 Ω, L1 = 0.1426 H, R0 = 21 Ω, L0 = 0.1175 H
The voltages of source generators are:
Mho relay was selected to decide whether to trip or not and the relay settings for zone 1 was set to reach 80%
of the line length.
To evaluate the performance of proposed algorithm, the following algorithms were compared:
1) The differential equation algorithm (DEA). It is direct use of algorithm described by Equations (3)-(5).
2) The least-square (LS) estimate based algorithm. This algorithm was presented in . The data window is
a quarter-cycle of fundamental signal.
3) The HHT based algorithm. It is the algorithm presented in this paper.
Figures 2 and 3 show the simulation results of the above algorithms. After a fault occurring, the results cal-
culated by DEA fluctuate up and down around the actual value frequently, which might lead to mal-operation of
the relay. Therefore, it is not acceptable estimates. The LS estimate based algorithm is not very accurate and its
operation time is half a period. The HHT based algorithm seems the most precise algorithm and its operation
time is about 6 - 8 ms.
Figure 1. Diagram of the simulation system.
X. M. Liu, Z. Y. He
Figure 2. Resistance and reactance of single phase to ground
fault on phase-a at 72% of line length.
Figure 3. Resistance and reactance of double phase fault be-
tween phase-a and phase-b at 72% of line length.
024681012 14 16 18 20
0246810 12 14 16 1820
0246810 12 14 16 1820
024 6 810 12 14 16 18 20
X. M. Liu, Z. Y. He
The operation time of HHT based algorithm has been simulated for single phase to ground fault and double
phase fault at the location 10%, 50%, 90% of the protected zone. The operation time consists of impedance cal-
culation and tripping decision. The fault resistance was set to zero ohm for all cases. Tables 1 and 2 show the
simulation results. The proposed algorithm improves the convergence time of DEA algorithm.
To analysis the effect of varying the fault resistance, the proposed algorithm has been simulated for single
phase to ground fault and double phase fault considering different fault resistances (0, 3, 6, 9, 12 and 15 Ω).
Two cases are presented here. In the first case, the faults occurred at the location 95% of the protected zone. The
phase difference between the source generator ES and the source generator ER varied from −30˚ to 30˚. In the
second case, the faults occurred at the location 105% of the protected zone. The phase difference between the
source generator ES and the source generator ER varied from −45˚ to 45˚. F igure 4 shows the estimated imped-
ance for different fault resistances.
Figure 4. Affect of fault resistance on the
evaluated impedance. (a) Single phase to
ground fault on phase-a; (b) Double phase
fault between phase-a and phase-b.
Table 1. Operation time for single phase to ground on phase-a.
Fault distance/% Fault angle/˚ Operation time/ms Distance error/%
0 7 2.4
45 7 2.3
90 8 2.5
0 7 2.0
45 7 2.1
90 8 2.3
0 8 1.8
45 8 1.9
90 8 1.9
510 15 20 25 30 35 40
10 2030 40 50 60 70
X. M. Liu, Z. Y. He
Table 2. Operation time for double phase fault between phase-a and phase-b .
Fault distance/% Fault angle/° Operation time/ms Distance error/%
0 6 2.5
45 6 2.4
90 7 2.4
0 7 2.2
45 8 2.3
90 8 2.3
0 8 1.7
45 8 1.8
90 8 1.9
The paper has suggested the HHT based scheme to improve the performance of EDA. The scheme employs
Hilbert-Huang transform (HHT) to extract high-frequency components of the results of DEA. Through empirical
mode decomposition (EMD), the intrinsic mode functions (IMF) of high frequency harmonics are separated out,
and the residue which has monotonic trend is used to evaluate the impedance of faulted line. The proposed
scheme improves the convergence time of DEA and accuracy in impedance estimating. Simulation results show
that the operation time is around 6 - 8 ms.
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