A new principle of traveling wave differential protection on series-capacitor-compensated transmission lines

SCIENCE CHINA Technological Sciences

© Science China Press and Springer-Verlag Berlin Heidelberg 2014 tech.scichina.com link.springer.com

*Corresponding author (email: [email protected])

• Article • April 2014 Vol.57 No.4: 754–764

doi: 10.1007/s11431-014-5503-7

A new principle of traveling wave differential protection on series-capacitor-compensated transmission lines

ZHANG YanXia, XUAN WenBo* & ZHANG HongYuan

Key Laboratory of Smart Grid of Ministry of Education, Tianjin University, Tianjin 300072, China

Received October 25, 2013; accepted December 23, 2013

Traveling wave differential protection has the ability, in theory, to entirely eliminate the effects of distributed capacitive cur-rent, but it cannot be applied on series-capacitor-compensated lines directly. In this paper, unbalanced output of conventional forward and reverse traveling wave differential currents under normal operating conditions and external faults was analyzed. A new type of traveling wave differential current was defined by combining forward and reverse traveling wave differential cur-rents. Expressions of the defined differential current when internal and external faults occur were deduced. On this basis, a new principle of traveling wave module differential protection on series-capacitor-compensated lines was proposed, and character-istics of module differential current under different faults were analyzed. The priniciple is immune to line distributed capacity, series capacitor positions, and presence or absence of MOV breakovers. The validity of this scheme was verified by PSACD simulations.

series-capacitor-compensated line, traveling wave differential protection, module differential protection

Citation: Zhang Y X, Xuan W B, Zhang H Y. A new principle of traveling wave differential protection on series-capacitor-compensated transmission lines. Sci China Tech Sci, 2014, 57: 754764, doi: 10.1007/s11431-014-5503-7

Variable definition. The variables have lowercase and italic subscript “c” is re-

lated to c-phase, for example: cfI is the fault current of c-

phase; the variables have lowercase and general subscript “c” is related to series capacitor, for example: uc(t) is the voltage drop of the series capacitor; the upper and italic subscript “C” only appers in variable “ZC”, which is wave impedance.

1 Introduction

Split phase current differential protection, simple in princi-ple and with high sensitivity, is the main protection of extra/ ultra-high voltage transmission lines. However, its selectivity and reliability are strongly influenced by large distributed

capacitive current on extra/ultra-high voltage transmission lines. The impact on performance of differential protection can be reduced, thanks to the capacitive current compensa-tion method, but it is difficult to compensate effectively in fault transient processes because of complex current fre-quency components [1–3]. Traveling wave differential pro-tection has been proposed by some researchers [4–10], uti-lizing differences between directional traveling waves from two terminals to detect faults, which can eliminate effects of distributed capacitive current theoretically.

Deploying series compensation capacitors on extra/ultra- high voltage transmission lines reduces electrical distance and thus boosts transmission capacity, optimizes power flow and voltage distribution, and improves system stabilization. Series compensation capacitors impair uniformity of impedance dis-tribution, posing difficulties to protection configuration and setting calculations [11–14]. Traditional traveling wave dif-ferential protection is also not applicable to series-capacitor-

Zhang Y X, et al. Sci China Tech Sci April (2014) Vol.57 No.4 755

compensated transmission lines directly. When a power sys-tem is operating normally and external faults occur, unbal-anced output is proportional to the voltage drops of series compensated capacitors. The longer the lines, higher the series compensation degree, and greater the through current, the larger is unbalanced output and the higher is the possibility of incorrect operation for protection. A scheme of traveling wave differential protection based on wavelet transform proposed in the literature [13] uses the header of the traveling wave to constitute the differential signal, but it needs significantly high sampling frequency and is prone to interference. A differential principle based on the Bergeron model of lossy transmission lines has also been proposed [14]. In the article, action expres-sions of each phase were deduced, proving that the principle is immune to the effects of distributed capacitive current, yet through quite complicated calculations. Another travel-ing wave differential theory for series compensated lines, which defines the sum of currents flowing into a series compensated capacitor as a differential signal, has been pro-posed [15]. However, the unbalanced output caused by line loss was not considered and analyzed, so the theoretical basis of the principle is not sufficiently sound.

In this study, a new type of traveling wave differential current has been defined by applying the characteristics of unbalanced output of conventional forward and reverse traveling wave differential currents. On the basis of the cur-rent, a new principle of module traveling wave differential protection on series-capacitor-compensated transmission lines has been proposed.

2 Problems of conventional traveling wave dif-ferential protection applied on series-capacitor- compensated lines

As shown in Figure 1, mn is a single-phase lossless line, L is the length of mn, v is wave velocity, =L/v is the propaga-tion delay of line mn, ZC is wave impedance, and the posi-tive directions of currents from two terminals are both from bus to line. Each current can be regarded as a superposition of a forward traveling wave and a reverse one. The direction of forward current traveling wave is defined as from left to right; reverse traveling wave, from right to left. The current traveling waves (double) of both terminals are

( ) ( ) ( ),

( ) ( ) ( ),

( ) ( ) ( ),

( ) ( ) ( ).

m m C m

m m C m

n n C n

n n C n

i t u t Z i t

i t u t Z i t

i t u t Z i t

i t u t Z i t

(1)

In the above equations, ( )mi t , ( )mi t , ( )ni t , and ( )ni t

are forward and reverse current traveling waves of the m and n ends. The current and voltage of the m and n ends are im(t), in(t), um(t), and un(t), respectively. The forward and

Figure 1 Traveling wave on a single-phase lossless line.

reverse traveling wave differential currents have been de-fined as [12]

1

2

( ) ( ) ( ),

( ) ( ) ( ).

D m n

D n m

i t i t i t

i t i t i t (2)

On lossless lines, the forward traveling wave of m end spreads to n end in time , becoming the forward traveling wave of n end. The reverse traveling wave of n end spreads to m end in time , becoming the reverse traveling wave of m end. So both iD1(t) and iD2(t) are zero when the power system is operating normally and an external fault occurs. When an internal fault occurs, the presence of a fault point makes the transmission line nonuniform, so both iD1(t) and iD2(t) are no longer zero and their amplitudes are equal to the amplitude of fault point current, leading to traveling wave differential protection action.

A series-capacitor-compensated transmission line is shown in Figure 2, with the propagation delay from m end to ca-pacitor and from capacitor to n end being respectively m and n, where =m+n. The voltages on both sides of the series capacitor are uc1(t) and uc2(t). The currents flowing into the series capacitor from left and right are ic1(t) and ic2(t) respec-tively, which are equal in magnitude and opposite in direc-tion. The sections between the two terminals to the series capacitor both are smooth and meet uniform transmission conditions:

c1

c2

( ) ( ),

( ) ( ).

m m

n n

i t i t

i t i t (3)

So the forward traveling wave differential current is de-duced as

1

c1 c2

( ) ( ) ( ) ( ) ( )

( ) ( ),

D m n m m n n

n n

i t i t i t i t i t

i t i t (4)

the two terminals of the capacitor satisfy:

c1 c1 c1

c2 c2 c2

( ) ( ),

( ) ( ).

C

C

i t u Z i t

i t u Z i t (5)

Figure 2 Series-capacitor-compensated transmission line.

756 Zhang Y X, et al. Sci China Tech Sci April (2014) Vol.57 No.4

Substituting eq. (5) into eq. (4):

1 c1 c2 c1 c2( ) ( ) ( ) ( ) ( )D n C n C n ni t u t Z u t Z i t i t

c1 c2( ) ( ) .n C n Cu t Z u t Z (6)

Let uc(t) be the voltage drop of the series capacitor, so uc(t)=uc1(t)uc2(t). Applying this to eq. (6):

1 c( ) ( ) ,D n Ci t u t Z (7)

eq. (7) indicates that conventional forward traveling wave differential current iD1(t) has considerable unbalanced out-put, which is equal to the ratio between the voltage drop of the capacitor and the wave impedance under normal system operation. When an external fault occurs, the unbalanced output can still be expressed by eq. (7), but the current bringing the voltage drop should be external fault current flowing through the series capacitor. Such unbalanced out-put does not exist on regular lines. Obviously, the longer the lines, the higher the series compensation degree, and the greater the through current, the larger is the unbalanced output. The unbalanced output may lead to incorrect action by con-ventional traveling wave differential protection, which is set by exceeding the largest unbalanced current (10% of the largest short-circuit current multiplied by a safety factor of 1.5–2). For the simulation system shown in Figure 8, the c-phase differential currents are under the conditions shown in Figure 3. When the system operates normally, the differ-ential currents at 40° transmission angle and zero load are

cDI and IDc, as shown in Figure 3(a). When an external

three-phase fault occurs at point f2, the differential currents under larger and smaller system operating modes, respec-tively, are cDI and IDc, as shown in Figure 3(b). As sys-

tem operating mode increases, the external fault current flowing through the series capacitor also increases, poten-tially leading to incorrect protection operation.

Two premises were assumed to deduce eq. (7): 1) The line is lossless and 2) the current transformer has no transfer error. If line loss and transfer error are considered, unbal-anced output should be higher, and conventional traveling wave differential protection more likely results in incorrect operation when an external fault occurs.

For a reverse traveling wave on series compensated transmission lines, equeations analogous to eqs. (3)–(5) can

be deduced. When the system operates normally:

2 c( ) ( ) ,D m Ci t u t Z (8)

obviously, the conventional reverse traveling wave differen-tial current iD2(t) is also not zero, which may lead to incor-rect protection operation.

3 A new principle of traveling wave differential protection on series-capacitor-compensated lines

3.1 Basic principle

In consideration that neither conventional forward traveling wave differential current iD1(t) nor reverse traveling wave differential current iD2(t) is zero under normal system opera-tion and an external fault occurs, a new traveling wave dif-ferential current is defined as

1 2( ) ( ) ( ),D D D nmi t i t i t (9)

In the above equation, nm=nm. When the system is operating normally and an external

fault occurs:

1 2

c c

c c

( ) ( ) ( )

( ) ( )

( ) ( )

0.

D D D nm

n C n m m C

n C n C

i t i t i t

u t Z u t Z

u t Z u t Z

(10)

When a fault occurs on a transmission line, the currents flowing to the fault point from left and from right, respec-tively, are if1(t) and if2(t), as shown in Figure 4. The sections from m end to capacitor, from capacitor to fault point, and from fault point to n end are smooth, and their propagation delay is m, cf, nf, respectively, n=cf+nf. The forward traveling waves on three sections satisfy:

c1

c2 c 1

2

( ) ( ),

( ) ( ),

( ) ( ).

m m

f f

f nf n

i t i t

i t i t

i t i t

(11)

The forward traveling wave currents of two terminals at fault points satisfy:

Figure 3 Differential current in traditional protection scheme. (a) Different transmission power; (b) different operating mode.


Figure 4 Internal faults that occur on a series-capacitor-compensated line.

1 1

2 2

( ) ( ) ( ),

( ) ( ) ( ).

f f C f

f f C f

i t u t Z i t

i t u t Z i t (12)

Subtracting the two eqs. in eq. (12):

1 2 1 1( ) ( ) ( ) ( ) ( ).f f f f fi t i t i t i t i t (13)

Applying eqs. (13) and (11) to iD1(t)= ( ) ( )m ni t i t ,

1

c1 c2

c

( ) ( ) ( )

( ) ( ) ( )

( ) ( ).

D m n

n n f nf

n C f nf

i t i t i t

i t i t i t

u t Z i t

(14)

By similar derivation, reverse traveling wave differential current iD2(t) should be

2

c2 c1 c

c c

( ) ( ) ( )

( ) ( ) ( )

( ) ( ).

D n m

m m f f m

m C f f m

i t i t i t

i t i t i t

u t Z i t

(15)

Therefore,

1 2

c

c c

c

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( ).

D D D nm

n C f nf

n m m C f nm f m

f nf f n f

i t i t i t

u t Z i t

u t Z i t

i t i t (16)

Both sides of eq. (16) are changed by Fourier transform:

cj j ( )

c

e e

( ).

nf n f

D f f

f nf f n f

I I I

I I

(17)

In the above equation, fI is the fault point current and

is power frequency. Obviously, when an internal fault occurs, DI is the sum of two constant-amplitude vectors,

and the amplitude of DI decreases with increasing angle

difference between the two vectors. The angle difference is related to the position of the ca-

pacitor and the location of the fault point. For existing transmission lines, the capacitor position is determinate, and the angle difference changes only with the location of the fault point. When a fault occurs at the series capacitor, n= nf, cf=0, the angle difference is minimum (i.e., zero), and

2D fI I . When a fault occurs at the n end, nf=0, n=cf,

the angle difference attains its maximum phase value, 2cf. Let the length from fault point to capacitor be 200 km and the wave velocity be equal to the speed of light. Thus 2cf =

24°, and DI =2cos(cf), fI =1.956× fI . Therefore, the

differential current value is about twice that of the fault point current when an internal fault occurs.

By the above analysis, the defined differential current

c( ),D f nf f n fI I I and the amplitude of

DI is about twice

fI when an internal fault occurs. Be-

cause If is related to transient resistance, the larger the tran-sient resistance and the lower the If , the lower is the protec-tion sensibility. However, the differential current in conven-tional traveling wave differential protection is equal to the fault point current If . By comparison, the differential pro-tection proposed in this article doubles the sensibility when internal fault occurs.

3.2 Traveling wave module differential protection for three-phase system

Karen Bauer transform is applied for phase-model decou-pling in a three-phase system. The transformation matrix is

1

1 1 1

= 1 1 0 ,

1 0 1

S (18)

currents of 0-mode, 1-mode, 2-mode are obtained by trans-form as follows:

0

1

2

( ) ( ) ( ) ( ),

( ) ( ) ( ),

( ) ( ) ( ).

a b c

a b

a c

i t i t i t i t

i t i t i t

i t i t i t

(19)

Module voltages and currents at m and n ends are given by eq. (19), and traveling waves of each module at m and n ends can be computed. So the forward and reverse traveling wave differential currents for each module are

1

2

( ) ( ) ( ),

( ) ( ) ( ).

x x x xD m n

x x x xD n m

i t i t i t

i t i t i t

(20)

In the above equation, x=0, 1, 2. Consulting eq. (9), the module differential currents are defined as follows:

0 0 0 01 2

1 1 1 11 2

2 2 2 21 2

( ) ( ) ( ),

( ) ( ) ( ),

( ) ( ) ( ).

D D D nm

D D D nm

D D D nm

i t i t i t

i t i t i t

i t i t i t

(21)

When the power system is operating normally and an external fault occurs, the module differential currents are zero. When an internal fault occurs, according to eq. (16), the module differential currents can be expressed by the


fault point currents of each module:

c( ) ( ) ( ).x x x x x xD f nf f n fi t i t i t (22)

Fault point currents for each module can be obtained by

0

1 1

2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) .

( ) ( ) ( ) ( )

a a b cf f f f f

b a bf f f f

c a cf f f f

i t i t i t i t i t

i t i t i t i t

i t i t i t i t

S (23)

All variables in eqs. (9) and (23) are instant values, but conventional traveling wave differential protection and the protection defined in this paper focus on power frequency components. So full-wave Fourier transform is applied to extract amplitude and angle from real-time sample data of two line terminals. By applying eq. (23) to eq. (22), we can acquire module differential currents from fault point cur-rents. According to boundary conditions of fault, module differential currents are transformed by full-wave Fourier transform as shown in Table 1. By analyzing Table 1, we reach the following conclusions.

1) Whatever the type of fault, there is at least one large amplitude among 0 ( )DI t , 1 ( )DI t , and 2 ( )DI t . According to

this characteristic, start-up criterion is set as follows: If one of 0 ( )DI t , 1 ( )DI t , or 2 ( )DI t is greater than its thresh-

old value, a fault is judged to occur. 2) When phase-to-phase faults occur, 0 ( )DI t =0, 1 ( ) 0,DI t

and 2 ( ) 0DI t .

3) When grounding faults occur, 0 ( )DI t 0. When and

only when a b-phase grounding fault occurs, 2 ( )DI t =0; when

and only when a c-phase grounding fault occurs, 1 ( )DI t =0.

When an a-phase grounding fault occurs, the angle difference between 1 ( )DI t and 0 ( )DI t is very small because the trave-

ling wave propagation velocities are different between zero- mode and aerial-mode. For example, in the power system shown in Figure 8, =4°. When a two-phase grounding fault happens, the angle difference between 1 ( )DI t and

0 ( )DI t is comparatively large. For a bc-grounding fault,

bfI and cfI , as shown in Figure 5(a), are the fault phase cur-

rents at the fault point, and the angle difference between them is 120° when transient resistance is zero. Because

angle difference between 0je n

and 1je n

is small, the

angle difference between 0j0 0

c2( )cos( )e nb cD f f fI I I

and 1DI 1j1c2 cos( )e nb

f fI is approximately equal to

the 120° angle difference between b cf fI I and bfI .

With increasing transient resistance Rg, bfI and cfI ro-

tate counterclockwise and clockwise respectively [1], and bfI rotates counterclockwise to be perpendicular to

b cf fI I and decreases to 90°. Therefore, 90°120°

when a bc- grounding fault occurs. For an ab-grounding fault, bfI

and

afI , as shown in Figure 5(b), are two fault

phase currents at the fault point, and the angle difference

between 0j0 0c2( )cos( )e nb b

D f f fI I I and 1 2( )a bD f fI I I

1j1

ccos( )e n

f is approximately equal to the 90° angle

difference between b af fI I and a b

f fI I . With increasing

Rg, afI and bfI rotate counterclockwise and clockwise re-

spectively [1], and remains 90°. For a ca-grounding fault, a

fI and cfI , as shown in Figure 5(c), are fault phase cur-

rents at the fault point, and the angle difference between 0j0 0

c2( ) cos( )e na cD f f fI I I and

1j1 1c2 cos( )e na

D f fI I

Table 1 Module differential currents under different faults

Fault type Phase current of fault point 0DI 1

DI 2DI

Signal-phase grounding fault

a-phase grounding ( ) ( )b cf fi t i t =0

0j0c2 cos( )e na

f fI 1j1

c2 cos( )e naf fI

1j1c2 cos( )e na

f fI

b-phase grounding ( ) ( )a cf fi t i t =0

0j0c2 cos( )e nb

f fI 1j1

c2 cos( )e nbf fI 0

c-phase grounding ( ) ( )a bf fi t i t =0

0j0c2 cos( )e nc

f fI 0 1j1

c2 cos( )e ncf fI

Two-phase grounding fault

ab-grounding fault ( )cfi t =0

0j0c2( )cos( )e na b

f f fI I 1j1

c2( )cos( )e na bf f fI I

1j1c2 cos( )e na

f fI

bc-grounding fault ( )afi t =0

0j0c2( )cos( )e nb c

f f fI I 1j1

c2 cos( )e nbf fI

1j1c2 cos( )e nc

f fI

ca-grounding fault ( )bfi t =0

0j0c2( ) cos( )e na c

f f fI I 1j1

c2 cos( )e naf fI

1j1c2( ) cos( )e na c

f f fI I

Phase-to-phase fault

ab-phase-to-phase fault

( ) ( )a bf fi t i t ( )c

fi t =0 0 1j1

c4 cos( )e naf fI

1j1c2 cos( )e na

f fI

bc-phase-to-phase fault

( ) ( )b cf fi t i t ( )a

fi t =0 0 1j1

c2 cos( )e nbf fI

1j1c2 cos( )e nc

f fI

ca-phase-to-phase fault

( ) ( )c af fi t i t ( )b

fi t =0 0 1j1

c2 cos( )e naf fI

1j1c4 cos( )e na

f fI

abc three-phase fault 0 1j1

c2( )cos( )e na bf f fI I

1j1c2( )cos( )e na c

f f fI I


is approximately equal to the 60° angle difference between a c

f fI I and afI . With increasing Rg, cfI and afI rotate

counterclockwise and clockwise respectively [1], and decreases to 90° maximum. Therefore, 60°90° when ca-grounding fault occurs. The angle is used to differen-tiate a-phase grounding faults form two-phase grounding faults.

According to the above-mentioned features, the operating principle of traveling wave module differential protection is described as follows.

1) If any one of the three inequalities shown in eq. (24) is satisfied, the start-up criterion operates and step (2) should be executed.

0

1

2

,

,

.

D G

D G

D G

I I

I I

I I

(24)

IG is the fixed threshold current, which should be set to exceed the largest unbalanced current of each module. De-tailed analysis of the principle to set IG is given in Section 4.

2) eq. (25) is used to judge whether 0-mode differential current is zero. If zero, it commits a phase-to-phase fault and trips three phases; if not zero, go to step (3).

0 .D GI I (25)

3) eq. (26) is used to judge whether 2-mode differential current is zero. If zero, a b-phase grounding fault occurs, tripping b-phase; if not zero, step (4) should be executed.

2 .D GI I (26)

4) eq. (27) is used to judge whether 1-mode differential current is zero. If zero, a c-phase grounding fault is deter-mined, tripping c-phase; otherwise, go to step (5).

1 .D GI I (27)

(5) The left side of eq. (28) expresses the angle differ-ence between 1-mode differential current and 0-mode dif-ferential current. If the inequality is satisfied, an a-phase grounding fault is judged, tripping a-phase. If not, two- phase grounding fault has been committed, tripping three phases. In eq. (28), 0

D and 1D denote the phase angle of

0-mode and 1-mode differential current respectively, and the fixed threshold value G is set to 30° according to Figure 5 and the margin of protection.

0 1 .D D G (28)

A working flow chart of the above-mentioned steps is shown in Figure 6.

The series capacitors are seen as a discontinuity point in the above derivation process. eqs. (10) and (16) show

Figure 5 Angle differences in two-phase grounding faults. (a) bc- ground; (b) ab-ground; (c) ca-ground.

Figure 6 Flow chart of traveling wave differential protection.

whether the power system is operating normally or an ex-ternal fault has occurred. In either case, unbalanced output of iD(t) has been eliminated, which means that no matter how uc(t) changes, iD(t) will not be affected. Therefore, the principle proposed is immune to location of series capaci-tors, whether MOV breakovers and nonlinear characteristics of MOV. When voltage drop of the series capacitors is zero or series capacitors are out of service, the principle is still effective.

4 Setting the principle of threshold current IG

4.1 Propagation characteristics of smooth transmission lines

Voltage and current variation on a smooth line can be de-


scribed by partial differential equations:

( , ) ( , )( , ) ,

( , ) ( , )( , ) .

u x t i x tRi x t L

x ti x t u x t

Gi x t Cx t

(29)

In eq. (29), R, L, G, and C are line parameters per unit length, respectively; u(x,t) and i(x,t) denote voltage and current at point x in moment t. For the power frequency component, the steady-state current is obtained by solving the above equation:

1 2( ) e e ,x xI x A A (30)

where ( j )( j )R L G C , and

1A and

2A are de-

termined by boundary conditions. When voltage and current at the starting point are known, as mU and mI , then

1

1

( ) (2 ) ,

( ) (2 ) ,

m C m C

m C m C

A U Z I Z

A U Z I Z

(31)

the line wave impedance ( j ) ( j )CZ R L G C .

In eq. (31), 1e

xA is the forward traveling wave compo-

nent, so let 1( ) e xI x A . Because amplitude decays and

phase shifts in the traveling wave propagation process on loss lines, given the wave travel’s unit length, the relation-ship between ( )I x and ( 1)I x can be described as

1 1( 1)

1 1

e e( ) 1e .

( 1) ee e e

x x

x x

A AI x

I x A A

(32)

Let ( ) ( ) xI x I x ,

1( 1) ( 1) xI x I x , and

j . Then

1j( )

1

( )( ) ( )e e .

( 1) ( 1) ( 1)x xx

x

I xI x I x

I x I x I x

(33)

Simplifying eq. (33):

1

( )ln j( ),

( 1) x x

I x

I x

(34)

So

1

( )ln ,

( 1)

.x x

I x

I x

(35)

In eq. (35), is amplitude decay factor per unit length, and is phase shift coefficient per unit length, both ob-

tained by solving eq. j ( j )( j )R L G C :

2 2 2 2 2 2 2

2 2 2 2 2 2 2

1 1( ) ( )( ) ,

2 2

1 1( ) ( )( ) .

2 2

RG LC R L G C

LC RG R L G C

(36)

The relationship between the change law of wave ampli-tude decay and the phase shift and line parameters is de-scribed in eq. (36). With increasing R, line energy loss in-creases, and the amplitude decay factor grows. With in-creasing L and C, electric field and magnetic field affect each other more intensely and the phase shift coefficient becomes larger. For lossless line, R=G=0, the only phase shift is in the traveling wave propagation process, and the amplitude remains unchanged.

Given that the transmission line shown in Figure 1 is lossy and smooth, the amplitudes of two terminals with forward traveling wave currents satisfy:

1 2 1

1 2 1 1

1

1 1

ln ln

ln ln ln

,

m m m m h m L n L

n m m m h m L n

m m h n L

m m h n

I I I I I I

I I I I I I

I I I

I I I

L

(37)

where m hI is the amplitude of the forward traveling wave

current at h kilometers away from m end. Simplifying eq. (37):

e .Ln mI I (38)

The angle difference between mI and

nI satisfies

1 1 1 1

.m n m m m m L m L n

L (39)

Simplifying eq. (39):

.n m L (40)

Synthesizing eqs. (38) and (40), forward traveling wave currents of two ends satisfy

je ( ) e e .L L Ln n n m m mI I I L I (41)

By similar derivation, reverse traveling wave currents of two ends satisfy:

je e .L Lm nI I (42)

4.2 Unbalanced current from different forward and reverse traveling wave currents

Let the series-capacitor-compensated line shown in Figure 2 be a loss line, and the sections from m end to capacitor and from capacitor to n end be smooth. The power frequency component of the forward traveling wave current can be


analyzed as follows. The section from m end to capacitor satisfies

c cjc1 e e .m mL L

mI I (43)

The forward traveling wave currents on each side of the section from capacitor to n end can be described as

c cjc2e e .n nL L

nI I (44)

The forward traveling wave currents on both sides of the capacitor can be described as

c c

c c

jc1 c1 c1

jc2 c2 c2

e e ,

e e .

m m

n n

L LC m

L LC n

I U Z I I

I U Z I I

(45)

Subtracting the two equations in eq. (45):

c1 c2 c1 c2 c1 c2

c1 c2 c .

C C

C C

I I U Z U Z I I

U U Z U Z

(46)

eq. (43) is equivalent to the following equation:

c c c c c c

c c c c

j j jc1

( ) j( )

j

e e e e e e

e e

e e .

n n n m m n

n m m n

L L L L L Lm

L L L Lm

L Lm

I I

I

I

(47)

Subtracting each side of eqs. (47) and (44):

c c c c

c c

j jjc2 c1

jc2 c1

e e e e e e

e ( )e .

n n n n

n n

L L L LL Ln m

L L

I I I I

I I

(48)

Applying eq. (46) to (48), and simplifying:

c c

c c

j jc2 c1

j jc

e ( )e + e e

e e + e e .

n n

n n

L L L Ln m

L L L LC m

I I I I

U Z I

(49)

Therefore, when the power system is operating normally and an external fault occurs, forward traveling wave differ-ent current can be expressed as

c c

j1

jj jc

e

e e e e e .n n

D m n

L LL Lm m C ub

I I I

I I U Z I

(50)

ubI

is the forward unbalanced current. Because L v

and v , L , the time delay in ( )mi t com-

pensates the phase shift generated by traveling wave propa-gation. Thus, eq. (50) becomes

c c

c c

jjc

jjc

e e e e e

(1 e ) e e e .

n n

n n

L Lj L Lub m m C

L LL Lm C

I I I U Z

I U Z

(51)

Defining unbalanced factor as 1 e L , then

c cjjce e e .n nL LL

ub m CI I U Z (52)

For a reverse traveling wave, similar formulas can be deduced as

c cjjce e e .m mL LL

ub n CI I U Z (53)

4.3 Setting the principle of fixed threshold

For the traveling wave different current defined in this pa-per, when a power system operates normally and an external fault occurs, unbalanced current is expressed as

c mc

c

c c c c

c c c c

j j( )

j2j

j jc c

j2 jjc

e e

e e

e e e e

e e (e e ) e .

nm n

m

n n m n

m n m n

L Lub ub ub ub ub

LLm n

L L L LC C

L L L LLm n C

I I I I I

I I

U Z U Z

I I U Z

(54)

Let c ce en mL L ; is obviously related to line length, installation locations of capacitors, and amplitude decay factor . When a series capacitor is installed at the middle of the line, Lnc=Lmc, then =0; when a series capaci-tor is installed at the end of a line, has its maximum value. Because line parameters of 0-mode and 1-mode are differ-ent, the ’s of the two modes are also different. For the simulating model used in this paper, the relationship be-tween the ’s of two modes and the installation location of the series capacitor is shown in Figure 7(a). The relationship between 0/1 and the installation location of the series ca-pacitor is shown in Figure 7(b). Wherever the capacitor is installed, 0 110 is expected to be satisfied, so 0 is far

larger than 1. When the capacitor is installed at the middle point, 0=1=0, the third item of eq. (54) is zero, and the following discussion is unaffected.

Unbalanced factor is related to line length and ampli-tude decay factor . The longer the lines and the greater the resistance per unit length, the larger is . Applying the transmission line parameters of Figure 8, 1=0.0107, 0= 0.1109, which indicates that 0 is far larger than 1.

Therefore, 0-mode unbalanced current is far larger than 1-mode and 2-mode unbalanced currents when external fault occurs. The fixed threshold IG should be set to exceed the biggest 0-mode unbalanced current.

The 0-mode unbalanced current when external grounding fault occurs can be described as

0 c 0 c 0j j2 j0 0 0 0 0 0 ce e e .m nL L L

ub m n CI I I U Z (55)

Figure 7 Value range of . (a) Value of ; (b) value of 0/1.


Applying eq. (42) to the above equation:

0 0 c c 0 c 0j j( ) j0 0 0 0 0 0 ce e e e .n m nL L L L L

ub m m CI I I U Z

(56)

According to the 0-mode network of external grounding fault,

0 0m m SI U Z , ZS is system impedance of m side,

because ZC is far larger than ZS, so 0 0m m SI U Z

0m CU Z , then

0 0 0 0m m C m mI U Z I I and 0mI

0 0m C mU Z I 0mI , which indicates that

0mI and

0mI are equal in amplitude and opposite in direction.

The third item of eq. (56) satisfies 0 c 0( )C m CU Z U Z

0mI , so this item can be neglected and eq. (56) is simpli-

fied to

c c0 0 0j j( )0 0 0 0 0e e e .n mL L L L

ub m mI I I (57)

In order to obtain maximum

0ubI , the angle between

two vectors should be minimum, thus the rotating direction of

0mI should be reversed to that of 0mI , which means

that the series capacitor should be installed at the n side; meanwhile, Lnc=0, Lmc=L, so eq. (57) is simplified to

0 0 0

0 0 0

0 0 0

j j0 0 0 0 0

j j0 0 0 0

j j0 0

e e e

e e e

(e e e ).

L L Lub m m

L L Lm m

L L Lm

I I I

I I

I

(58)

For extra/ultra-high voltage transmission lines, 0eL 1, and eq. (58) is simplified to

0 0j j

0 0 0

0 0 0

(e e )

2 i sin( ).

L Lub m

m

I I

I L

(59)

Compared with traditional traveling wave differential protection, unbalanced current as expressed in eq. (59) is smaller and also independent of degree of series compensa-tion and pre-fault transmission power. Considering the mar-gin, the fixed threshold IG of module traveling wave differ-ential protection is set as in eq. (60), where 0mI is the

maximum 0-mode current at m side when external ground-ing fault occurs.

0 02 .G mI I (60)

5 Verifying the simulation

The simulation is based on the transmission system shown in Figure 8, which was built in PSCAD. Simulation data was processed by MATLAB to verify the new protection principle proposed in this paper. The system rating voltage is 750 kV, the line length is 400 km, the series compensa-tion degree is 40%, a series capacitor is installed at the mid-dle of the line, and a voltage transformer at a bus is applied.

Figure 8 Series-capacitor-compensated line simulation model.

Fault start time is set at t=0.2 s full-wave Fourier transform is used in the filter algorithm, sampling frequency is 4 kHz. and IG is set as in eq. (60). Because 0mI is 1850 A when a

single-phase grounding fault occurs at external point f2, and

0mI is 1560 A when a two-phase grounding fault occurs at

the same point, the maximum 0mI is 1850 A. Substituting

line parameters into eq. (36), 0=2.94×104. So 0

0 1 e L 4400 2.94 101 e 0.1109 , and 0 02G mI I

2 A 0.1109 A 1850 A =410 A.

Three types of module differential currents are shown in Figure 9 when three-phase fault occurs at point f1. Because

1D GI I and 2

D GI I , the start-up criterion is satisfied and

start-up components would operate. Phase-to-phase fault is judged due to 0

D GI I .

Given that the series capacitor is installed at the n side of the line shown in Figure 8, three types of module differen-tial currents are shown in Figure 10 when three-phase fault occurs at point f1. Obviously, 1

D GI I and 2D GI I , so the

start-up criterion is satisfied and a start-up component would operate. Because 0

D GI I , this is judged to be a phase-

to-phase fault. Results of the simulation declare that the protection principle is immune to the installation locations

Figure 9 Three-phase fault internal.

Figure 10 Three-phase fault internal.


of series capacitors. When an a-phase grounding fault occurs at point f1 with

Rg=500 , amplitudes and phase angle differences occur between the three types of mode differential currents shown in Figure 11. Obviously, 1

D GI I , 2D GI I and 0

D GI I ,

so the start-up criterion is satisfied and a start-up component would operate. Because 0

D GI I , this is judged to be a

grounding fault. The b-phase ground fault is excluded ac-cording to 2

D GI I , and the c-Phase ground fault is also

excluded owing to 1D GI I . So the fault type is judged to

be a-phase grounding owing to 0 1D D G .

The module differential current of three-phase fault at external point f2 and no-load switching is shown in Fig- ure 12. These are small, much less than the fixed threshold, so protection does not actuate.

Full-wave Fourier transform is applied to extract the fundamental frequency component of the traveling waves, so calculation accuracy is not affected by sampling fre-quency. This is different from conventional traveling wave protection, which needs to accurately extract header polarity and arrival time with the help of high sampling frequency. The 0-mode differential currents under three different sam-pling frequencies are shown in Figure 13 when c-phase grounding fault occurs at point f1 with Rg=500 . As shown, current amplitudes are essentially equal under the different

Figure 11 A phase-to-ground fault. (a) Magnitude of each module cur-rent; (b) angle difference.

Figure 12 Faults at external and no-load closing. (a) Three-phase fault at external; (b) no-load closing.

Figure 13 Differential currents under different sampling frequencies.

sampling frequencies, with the difference so small that it can be neglected. Thus sampling frequency has no effect on the algorithm in this study.

6 Conclusions

In this study, module traveling wave current has been de-fined for series-capacitor-compensated transmission lines. A new principle of module traveling wave differential protec-tion has been proposed, based on the features of the defined differential current under different fault conditions. The protection has the following characteristics.

1) The protection is appropriate and not affected by the installation location of the capacitors, including at one side and at the middle of a line.

2) The protection is not affected by whether or not either MOV breakovers or capacitors are in service.

3) The unbalanced output is small and independent of load current. The protection is capable of tolerating high transient resistance.

4) Increasing the sampling frequency is unnecessary, and the protection can be realized with existing technology, so it is of exceedingly high practical value.

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A new principle of traveling wave differential protection on series-capacitor-compensated transmission lines