A method for faulted line detection of four parallel lines on the same tower is presented, based on foursumming and doubledifferential sequences of one terminal current. Foursumming and doubledifferential sequences of fault current can be calculated using a certain transformation matrix for parameter decoupling of four parallel transmission lines. According to fault boundary conditions, the amplitude and phase characteristics of foursumming and doubledifferential sequences of fault current is studied under conditions of different types of fault. Through the analysis of the relationship of terminal current and fault current, a novel methodology for fault line detection of four parallel transmission line on the same tower is put forward, which can pick out the fault lines no matter the fault occurs in single line or cross double lines. Simulation results validate that the methodology is correct and reliable under conditions of different load currents, transient resistances and fault locations.
1. Introduction
With the advantage of large transmission capacity within small passageway, the constructions of four parallel transmission lines on the same tower become an inevitable trend for the main network of power system
[1

3]
. Due to the short distance of the four lines on the same tower, there are not only mutual inductances and capacitances between two phases in single line, but also complex mutual inductances and capacitances between every two phases in different lines
[4]
. Because faults cross lines probably occur besides faults within one line, there are sum to 8184 types of fault on four parallel lines on the same tower
[5

7]
. To minimize system insecurity and instability, fault lines must be accurately detected and tripped swiftly to make the sound lines back to normal operation when fault occurs. Therefore, accurate and quick fault line detection is very important for the selective operation of protection on four parallel lines on the same tower
[8

9]
.
Due to the complexity of line parameters of four parallel transmission lines on the same tower, there are few methods for fault line detection have been proposed recently. Based on an transformer matrix for decoupling ideal but not actual parameter of four parallel line, which supposes the mutual impedances and capacitances between every two lines only share the same value respectively, a method for fault line detection of four parallel lines on the same tower was proposed, using the homodromous and circumfluent sequences of fault current
[10]
. Based on an different transformer matrix for decoupling ideal but not actual parameter of four parallel lines, which supposes the mutual impedances and capacitances between every two lines only share two different values respectively, a method for fault line detection of four parallel lines on the same tower is proposed, using the circumfluent sequences of fault current
[11

12]
. In fact, the mutual impedances and capacitances between every two lines are different from each other, so there are three different values respectively, which means both methods above cannot be put into implementation
[13

14]
. Moreover, they can only be used for fault line selection of fault in single line but not for fault cross lines. Indeed, though the probabilities of fault cross lines are on the lower level, it is much more serious of the damage it brings. So research on fault line detection for faults cross lines seems to be more necessary.
Based on the transformer matrix for actual parameter decoupling of four parallel transmission lines, which supposes the mutual impedances and capacitances between every two lines share different values from each other respectively, foursumming and doubledifferential sequences of fault current are calculated and the characteristics of their amplitudes and phases are studied. Through the analysis of the relationship of terminal current and fault current, an effective method for fault line detection of all fault types in four parallel transmission lines on the same tower is proposed using one terminal current.
2. Parameter Decoupling Method of Four Parallel Transmission Lines on the Same Tower
The model of four parallel transmission lines on the same tower with system at both terminals is shown in
Fig. 1
.
Model of four parallel transmission lines on the same tower
There are various tower structures of four parallel transmission lines on the same tower, such as six layers, four layers, three layers, et al. Tower structures which have been put into practical application in China are presanted in
[13]
. The parameters of four parallel transmission lines change with tower structures.
Due to the short distances, there are mutual impedances and capacitances between every two phases in different lines besides that between two phases in single line. According to Carson formula, the mutual parameters depend on the distance between two conductors. Therefore, taking the different distance between every two lines into account, the mutual impedances and capacitances between every two lines are different from each other, so there are three different values of mutual impedances and capacitances respectively. Based on a reasonable approximation of impedances and capacitances according to actual line construction, Line parameter decoupling matrix M is obtained in
[13]
, which is shown in (1):
Using matrix M to transform the 12 phasors of currents in four lines into foursumming positive, negative and zero sequences and three different doubledifferential positive, negative and zero sequences, column vector of TF
is shown as follows.
Where
is foursumming positive, negative or zero sequences of currents,
,
,
are three different doubledifferential positive, negative or zero sequences of currents.
i
is 1, 2 or 0.
Assuming that the sequences of currents in each line are shown as follow:
The relationships between components in
and
are shown in (4):
Where
i
=1, 2, 0.
can be obtained through transformation matrix P, which is used for mutual parameter decoupling between phases in single line.
Where
P
= (1 3)diag[
S S S S
] .
It can be seen that foursumming sequence is the sum of a certain sequence of current in each line, and three doubledifferential sequences are the difference between a certain sequence of currents in double lines and other double lines.
3. Characteristics of Fault Current in the Case of Fault in Single Line
Taking faults in line I as an example, foursumming and doubledifferential sequences of fault current are deduced as follows in the case of single phase to ground fault, interphase fault, double phase to ground fault and threephase fault.
When ground fault occurs on phase A in line I, boundary conditions of fault current are shown in (6):
Through (3), the relationship between foursumming and doubledifferential sequences of fault current can be obtained as follows:
Where
i
=1, 2, 0.
When ground fault occurs on phase B and C in line I, boundary conditions of fault currents are shown in (8):
The relationship between foursumming and doubledifferential sequences of fault current is the same as (7).
When interphase fault occurs on phase B and C in line I, boundary conditions of fault currents are shown in (9):
The relationship between foursumming and doubledifferential sequences of fault current is the same as (7). Because it is an interphase but not ground fault, there is no foursumming and doubledifferential zero sequences of fault current, which means i can be 1 or 2 but not 0.
When threephase fault occurs in line Ⅰ, boundary conditions of fault currents are shown in (10).
The relationship between foursumming and doubledifferential sequences of fault current is the same as (7). Because it is a symmetrical fault, there is no foursumming and doubledifferential positive and zero sequences of fault current, which means i can be 1 but not 2 or 0.
When different types of fault occur in line Ⅱ, based on boundary conditions, the relationship between foursumming and doubledifferential sequences of fault current are shown in (11):
When different types of fault occur in line Ⅲ, based on boundary conditions, the relationship between foursumming and doubledifferential sequences of fault current are shown in (12):
When different types of fault occur in line Ⅳ, based on boundary conditions, the relationship between foursumming and doubledifferential sequences of fault current are shown in (13):
Relationships of three doubledifferential sequences of fault current in the case of fault in single line.
Relationships of three doubledifferential sequences of fault current in the case of fault in single line.
The same as fault in line I, i depends on the type of fault. It can be concluded that when a fault occurs in single line, three doubledifferential sequences of fault current have the same amplitude and 0° or 180° phase difference, which can be seen in
Table 1
.
It can be seen from
Table 1
that doubledifferential sequences of fault current have different characteristics when fault occurs in different line, so it is an effective method for fault line detection in the case of fault in single line. Both doubledifferential positive and negative sequences of fault current can be used for asymmetric faults, but only doubledifferential positive sequences can be used for threephase faults.
4. Characteristics of Fault Current in the Case of Fault Cross Double Lines
There are thousands of cross line fault types in four parallel transmission lines on the same tower, but the possibility of fault cross more than two lines is extremely small, so only faults cross double lines are considered in the paper.
When fault occurs cross line Ⅰ and Ⅱ, phasors of fault current in nonfault lines are 0 no matter which fault type it is, so boundary conditions are shown in (14).
Fault current sequences of each line are shown in (15):
where
i
can be 1, 2 or 0, which depends on the type of fault.
The foursumming sequences and three doubledifferential sequences of fault current can be calculated by (4), shown in (16):
When fault occurs cross line Ⅰ and Ⅲ, foursumming sequences and three doubledifferential sequences of fault current are shown in (17).
When fault occurs cross line Ⅰ and Ⅳ, foursumming sequences and three doubledifferential sequences of fault current are shown in (18).
When fault occurs cross line Ⅱ and Ⅲ, foursumming sequences and three doubledifferential sequences of fault current are shown in (19).
When fault occurs cross line II and IV, foursumming sequences and three doubledifferential sequences of fault current are shown in (20).
When fault occurs cross line Ⅲ and Ⅳ, foursumming sequences and three doubledifferential sequences of fault current are shown in (21).
it can be concluded that when a fault cross double lines occurs, the relationships of three doubledifferential sequences of fault current is complex, which are summarized in
Table 2
.
Relationship of three doubledifferential sequences of fault current in case of fault cross double lines
Relationship of three doubledifferential sequences of fault current in case of fault cross double lines
It can be seen from
Table 2
that doubledifferential sequences of fault current have different characteristics when fault occurs cross different lines, so it is an effective method for fault line detection in the case of fault cross double lines.
In
Table 2
, at least one magnitude ratio of two doubledifferential sequences of fault current is not 1, because the sum of current sequence of two fault lines is not equal to the difference of these two. However, in
Table 1
, both magnitude ratios of two doubledifferential sequences of fault current are 1. Based on the difference of these two magnitude radios, it is an effective way to identify whether a fault in single line or cross double lines.
In exceptional cases, when phase difference of current sequences of two fault lines are 90°or 270°, the sum of these two current sequences has the same magnitude as the difference of these two current sequences, so both magnitude ratios of two doubledifferential sequences of fault current are 1. However, According to
Table 1
, both phase differences of two doubledifferential sequences of fault current are 0°or 180°, not 90°or 270°, when fault in single line occurs. In order to identify whether a fault in single line or cross double lines in these exceptional cases, phase differences of two doubledifferential sequences are used besides magnitude radios.
According to the above analysis, criterion for distinguishing faults in single line from fault cross double lines is shown in (22).
Where
l
is setting value of amplitude ratio for identifying 0,
φ
is setting value of phase difference for identifying 0°, and i depends on the type of fault.
If fault phase(s) in double lines is(are) the same, for example, fault types ofⅠABCⅡABC(G), ⅠABⅡAB(G), ⅠAⅡAG, et al, current sequences of two fault lines will be equal to each other, so the doubledifferential sequence of fault current, which correspond to the difference of current sequences of the two fault lines, is 0, as shown in
Table 3
.
Doubledifferential sequences of fault currents in the case of fault with same fault phase(s) cross double lines
Doubledifferential sequences of fault currents in the case of fault with same fault phase(s) cross double lines
These fault types must be considered separately for fault line selection. Criterion for identifying these fault types from others is shown in (23).
Where
i
can be 1 or 2.
5. Line Detection Methodology for Four Parallel Transmission Line on the Same Tower
Based on the characteristics of
,
,
,
fault lines can be detected effectively no matter what type of fault it is. Types of fault are classified by (22) and (23), and then different methods of fault line detection are used for different types of fault. The basic flowchart is shown in
Fig. 2
.
The caption for a figure must follow the figure
Foursumming and doubledifferential negative sequence of terminal current are used for asymmetric fault; Foursumming and doubledifferential positive sequences of terminal current are used for threephase fault which negative sequences of current are 0.
 5.1 Fault line detection method for fault in single line
According to
Table 1
, phase differences of
and
are used for fault line detection when fault in single line occurs. Specific method is shown in
Fig. 3
.
Fault line selection method for fault in single line
 5.2 Fault line detection method for fault cross double lines
Faults cross double lines are classified into two categories according to the characteristics of fault currents. Two methods for fault line selection are proposed respectively
 5.2.1 Fault line detection method for asymmetric fault with different fault phases cross double lines:
Not a doubledifferential sequence of current is 0 when these types of fault occur. According to
Table 2
, characteristics of
and
which used for fault line detection, are illustrated as follows: when fault occurs cross line Ⅰ&Ⅱ, amplitude of
and
are the same and both phases are identical; when a fault occurs cross line Ⅲ & Ⅳ, amplitudes of
and
are the same, but their phase difference is 180°; when a fault occurs cross line Ⅰ& Ⅲ, Ⅰ& Ⅳ, Ⅱ& Ⅲ and Ⅱ& Ⅳ, at least one amplitude of
and
is 1, and their phases is 0°or 180°. The specific method is shown in
Fig. 4
.
Fault line selection method for asymmetric fault with different fault phases cross double lines
 5.2.2 Fault line detection method for fault with same fault phase(s) cross double lines:
According to
Table 3
, only one of the three doubledifferential sequences of fault currents is not 0 when these types of fault occur, so the foursumming sequence of current needs to be used as an reference. The amplitude ratio of this doubledifferential sequence and foursumming sequence of fault currents is 1 and phase difference is 0°or 180°. The specific method is shown in
Fig. 5
.
Fault line selection method for fault with same fault phase(s) cross double lines
For asymmetric fault, foursumming and doubledifferential negative sequences of terminal currents are used in the method, and for threephase fault, foursumming and doubledifferential positive sequence of terminal current are used in the method which negative sequences of currents are 0.
 5.3 Analysis for the effect of branch coefficient
The above theoretical analysis and methods are based on fault current, but in practical application, fault line detection methods shown in
Figs. 2

5
can only use terminal currents of M or N. therefore, the effect of branch coefficient of fault current must be taken into account.
For fault in single line and asymmetric fault with different fault phases cross double lines, only doubledifferential negative or positive sequence currents are used for fault line detection. Through the analysis of network of doubledifferential sequences, it is known that
,
and
are current sequences circulating from two lines into the other two lines but not to systems on both sides. Branch coefficients of doubledifferential sequences of fault currents only depend on fault point location and have the same real number for each sequence, as
, so ratios of every two doubledifferential sequences of terminal current are equal to that of fault current, which means branch coefficients have no influence on fault line detection method when terminal current is used instead of fault current in method 1 and 2. The errors of unbalanced currents, CT measurement and other factors should be considered for setting values. l can be set as about 0.3 for identification of 0, and
ϕ
can be set as 90°at most for nonoverlapping area of fault line detection.
For fault with same fault phase(s) cross double lines, not only doubledifferential negative or positive sequences of fault current but also foursumming negative or positive sequences of fault current are used for fault line detection. Through the analysis of network of foursumming sequence, it is known that
is the sum of sequences of currents in four lines, which circulate from four lines into systems on both sides. So, branch coefficients of foursumming sequences of fault current, which are not only depend on fault location but also on system resistances, are different from that of doubledifferential sequence of fault currents. Angle errors, as foursumming sequences of terminal currents are used instead of that of fault current, must be considered as an important factor for setting of
ϕ
.
6. Simulation Studies
The simulation studies were carried out with PSCAD/ EMTDC software. The model of d four parallel transmission lines on the same tower with system at both terminals is shown in
Fig. 1
. The line parameters are based on a practical four parallel transmission lines in Shanghai city in China. The conductor type of line I and II is LGJ400/35 and that of line III and IV is LGJ630/45 with 4 bundle of each phase. The line is 100 km long. Frequency dependent (phase) model is used and geometric parameter with ideal transposition in each line is set for simulation.
System parameters of M side are as follows:
Z_{M1}=3.93+i49.34 Ω, Z_{M0}=2.52+i46.03 Ω;
System parameters of N side are as follows:
Z_{N1}=2.76+i41.34 Ω, Z_{N0}=1.57+i134.09 Ω.
Typical fault types in single line or cross double lines are simulated to verify the effectiveness of the fault line detection methodology. Transition resistance is set as 0 Ω for metal fault, 300 Ω for nonmetal grounding LG fault and 30 Ω for nonmetal interphase or interline LL fault
 6.1 Simulation results of fault in single line
Doubledifferential positive sequence currents are used for threephase fault in single line. Simulation results of this type of fault with 0 Ω and 30 Ω transition resistances are shown in
Table 4
to validate fault line detection method 1. Doubledifferential negative sequence currents are used for asymmetric faults in single line. Simulation results of these types fault with 0 Ω and 30 Ω LL and 300 Ω LG transition resistances are shown in
Table 5
. It is shown that phase differences of
and
in simulation are consistent with theoretical analysis and not influenced by transition resistance.
Simulation results of threephase fault in single line
Simulation results of threephase fault in single line
Simulation results of asymmetric fault in single line
Simulation results of asymmetric fault in single line
 6.2 Line simulation results of fault cross double lines
Doubledifferential negative sequence currents are used for asymmetric fault with different fault phases cross double lines. Simulation results of these types of fault with 0 Ω and 30 Ω LL and 300 Ω LG transition resistances is presented in
Tables 6
and
7
. It is can be known that the simulation results are consistent with theoretical analysis and not influenced by transition resistance.
Simulation results of asymmetric fault with different fault phases cross double lines with 0 Ω transition resistances
Simulation results of asymmetric fault with different fault phases cross double lines with 0 Ω transition resistances
Simulation results of asymmetric fault with different fault phases cross double lines with 30 Ω LL and 300 Ω LG transition resistances
Simulation results of asymmetric fault with different fault phases cross double lines with 30 Ω LL and 300 Ω LG transition resistances
Foursumming negative sequence current is used as a reference current for asymmetric fault with the same fault phases cross double lines. Simulation results of these types of fault with 0 Ω and 30 Ω LL and 300 Ω LG transition resistances is presented in
Tables 8
and
9
. Because the influence of its branch coefficients, errors of simulation results are larger than fault line detection method 1 but still small because the angle of system resistance are close to that of the line.
Simulation results of asymmetric fault with the same fault phases cross double lines with 0 Ω transition resistances
Simulation results of asymmetric fault with the same fault phases cross double lines with 0 Ω transition resistances
Simulation results of asymmetric fault with the same fault phases cross double lines with 30 Ω LL and 300 Ω LG transition resistances
Simulation results of asymmetric fault with the same fault phases cross double lines with 30 Ω LL and 300 Ω LG transition resistances
Foursumming positive sequence current is used as a reference current for threephase fault cross double lines. Simulation results of these fault type with 0 Ω and 30 Ω LL and 300 Ω LG transition resistances are shown in
Tables 10
and
11
. Because of the influence of its branch coefficients and load currents, errors of simulation results are up to 18.3°, which are much larger than fault line detection method 1 and 2.
Simulation results of threephase fault cross double lines with 0 Ω transition resistances
Simulation results of threephase fault cross double lines with 0 Ω transition resistances
Simulation results of threephase fault cross double lines with 30 Ω LL and 300 Ω LG transition resistances
Simulation results of threephase fault cross double lines with 30 Ω LL and 300 Ω LG transition resistances
The above simulation results shown in
Tables 4

Table 11
are all under the following scenarios: the angle between system source M and N is set as 30°for simulating a certain load current; the fault locations are all at the middle of the line。For the full verification of the fault line detection methodology, lots of simulations were carried out while angle between system source M and N is set as 0°, 15°, 30°and 45°, fault location is set as 0 km, 20 km, 40 km, 60 km and 100 km, and transition resistance is set as 0 Ω, 15 Ω, and 30 Ω for LL fault and 0 Ω, 100 Ω, 200 Ω, 300 Ω for LG fault. For different typical fault types mentioned in
Fig. 2
, simulations were carried out through multiplerunmodel in PSCAD. The variables used to compare with the setting
ϕ
and l in
Figs. 2

5
are calculated and recorded in order to identify the fault line(s). The summary of simulation result is shown in
Fig. 6
.
Variables compared with ϕ in different fault types and conditions
The Variables compared with l in different fault types and conditions
All the points shown in
Figs. 6
and
7
are the calculated variables compared with the setting
ϕ
or l. the maximum value is shown for these variables in different fault types in
Fig. 2
. It can be known from
Fig. 6
that the maximum value compared with
ϕ
is among the variables used for threephase fault cross double lines, because it is affected by branch coefficients and load currents. However, the methodology can work well with that error, for
ϕ
can be set as 90°at most for nonoverlapping area of fault line detection. Variables compared with
l
are very small in different fault types because they are not affected by branch coefficients, load currents and fault locations. Therefore,
l
can be set as a small value such as 0.1 to identify if the calculated variables are 0.
 6.3 Simulation results of fault in unbalanced four parallel transmission lines on the same tower
Conductors of four parallel transmission lines on the same tower may not be transposed because of short length in actual situation, so lots of simulations are carried out on four parallel transmission lines without uniform transposition. From the simulation results, it is known that a certain error exists between results of simulation and theoretical analysis, but the three methods can operate accurately with reasonable setting values.
7. Conclusion
Based on the characteristics of foursumming and doubledifferential sequence components of fault currents, fault line detection methodology of parallel transmission lines on the same tower is proposed. Types of fault are identified first and then different methods are presented respectively for asymmetric or threephase fault in single line, asymmetric fault with different fault phases cross double lines and asymmetric or threephase fault with same fault phase(s) cross double lines. Because terminal currents are used for the raised methodology, the effect of branch coefficient is analyzed for each different method according to fault types. Lots of Simulation is done in EMTDC and it is shown that the method can operate correctly and reliably for all the fault types in single line or cross double lines under conditions of different load current, transient resistance and fault location.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (NSFC) (No. 51207102, No. 51277130)
BIO
BongTong Li He received PH.D. degree in electrical engineering and automation from Tianjin university, Tianjin China. He is a lecturer in School of electrical and automation, Tianjin University, China. His research interests are power system protection and control.
Yongli Li was born in Hebei, China, in 1963. She received the B.S. degree and M.S. degree in Electrical Engineering from Tianjin University, China, in 1984 and 1987, respectively. In1993, she received the Ph.D degree in Electrical Engineering from Universite Libre de Bruxelles, Belgium. Dr. Li is currently a Professor in the Department of Electrical Engineering and Automation, Tianjin University, and also a member of CIGRE SCB5. Her research interests include fault analysis of power system and fault diagnosis of electrical equipment, protection and adaptive reclosing of EHV/UHV transmission system, protection and control of the microgrid and distribution network.
Chuang Yao was born in Shanxi China, in January 1989. She received her master degree in electrical engineering and automation at Tianjin University in 2013, and her research interests are power system protection and control.
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