On travelling-wave-based protection of high-voltage networks
Citation for published version (APA):
Bollen, M. H. J. (1989). On travelling-wave-based protection of high-voltage networks. Technische Universiteit
Eindhoven. https://doi.org/10.6100/IR316599
DOI:
10.6100/IR316599
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Published: 01/01/1989
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ON TRAVELLING-WAVE-BASED PROTECTION
OF
HIGH-VOLTAGE NETWORKS
Bollen, Mathias Henricus Johannes
On travelling-wave-based protection of high-voltage networks I Mathias Henricus Johannes Bollen. -[S.l. : s.n.]. Fig., tab.
Proefschrift Eindhoven. Met lit.opg., reg. ISBN 90-9002955-9
SISO 661.55 UDC 621.316.925(043.3) NUGI 832 Trefw.: hoogspanningsnetten; beveiliging I
ON TRAVELLING-WAVE-BASED PROTECTION
OF
HIGH-VOLTAGE NETWORKS
PROEFSCHRIFf
ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus prof. ir. M. Tels, voor een commissie aangewezen door het College van Dekanen in het openbaar te verdedigen op vrijdag 15 september 1989 om 16.00 uur
door
Mathias Henricus Johannes Bollen
geboren te Stein (L)Prof. Dr. Ir. W.M. C. van den Heuvel en
If our goals are all the same We can run alone and free If we persue a different aim
Let the truth of love be lighted Let the love of truth shine clear Sensibility
Armed with sense and liberty With the heart and mind united In a single perfect sphere
1. Introduction ... .
2. Travelling waves and high-voltage lines ... 7
3. Network modelling ... 26
4. Testing of algorithms for travelling-wave-based protection ... 41
5. Directional detection ... 58
6. Differential protection ... 73
7. Other algorithms for travelling-wave-based protection ... SO 8. A protective scheme for a double-circuit line ... 102
9. Summary and conclusions ... 114
10. References ... 118
Samenvatting ... 127
cxxx ...
1301. Introduction
1.1. The protection of public supply systems
A public supply system is set up to transport and distribute electrical energy from the generators to the users. Due to the interconnection of a large number of generators (power plants) a high reliability is achieved. An important role in maintaining this high reliability is played by the power-system protection. The function of the protection is the disconnection of defective lines and apparatus from the system. This thesis concerns the
detection and disconnection of faults1 on high-voltage lines.
Before the turn of the century, this was mainly achieved by using fuses. This century showed the introduction of protective relays in combination with circuit-breakers. The first to detect the fault, the latter to disconnect it. The protective relay uses voltages or currents to detect a fault. Upon detecting, a tripping signal is sent to the circuit breaker that disconnects the fault. At first. simple relays were used, like time overcurrent relays, directional-power relays and differential relays . By using these relays it was difficult to maintain the selectivity (only disconnecting the faulted line or apparatus), especially in the growing high-voltage networks in Europe in the twenties. This led to the proposal of a number of new protection principles drawing much attention from the professional world. But all of them were overruled by the principle of distance protection. The first distance relays were used in 1925 for the 100 kV double-circuit 1 ine betw:een the power plant of Seestadl and Prague [Walter, 1967]. Nowadays the protection of high-voltage lines is largely achieved by means of distance relays. From power frequency voltages and currents measured at one line terminal an input impedance is determined for the line. If this value is below a certain preset value, a tripping signal is sent to the circuit breaker and the line is disconnected.
The main advantage of di,t}tcmce (Yl,otecUoo is the fact that no communication link is needed. Each relay can determine the position of the fault from voltages and currents measured at the relay position. Due to all kinds of errors, the reach of a distance relay cannot be determined exactly. It is therefore impossible to distinguish between a fault a 1 i ttle before the next substation and a fault a little beyond that substation. But by introducing different distance zones a highly reliable network protection
A fault (or short-circuit) is an inadvertent accidental connection between two or more phase conductors or between one or more phase conductors and ground [Blackburn, 1987].
has been achieved.
Zone 1 of each distance relay reaches up to about 90% into the line-to-be-protected. zone 2 up to 50% into the next adjacent line and zone 3 approximately 25% into the adjacent line beyond [Blackburn, 1987]. If a relay detects a fault in its zone 1 an instantaneous tripping signal will be generated (no delay introduced on purpose). In case of a fault in zone 2 or 3 a tripping signal will be generated if the fault remains during a certain time. This verification time is longer for more distant zones. A fault somewhere on a line, not too close to one of the line terminals, (position 1 in Figure 1.1) will lead to an instantaneous tripping signal of both A and B. In case of a fault at position 2 (i.e. close to one of the line terminals) only B will generate an instantaneous tripping signal. Relay A will generate a tripping signal if the fault lasts for longer than a certain time. This time is longer than the time needed to disconnect a fault at position 3 by relay C and its circuit breaker. In that way zone 2 of relay A serves as a primary protection for faults close to the other line terminal and as a remote backup for relay C in case of faults on the first half of the next adjacent line. Zone 3 serves as a remote backup for faults on the second half of the next adjacent line (e.g. position 4).
--- -
ZONE 3-·
ZONE2
Figure 1.1. Protection of a high-vottage tine by means of distance retays
When reliable communication links are available it is possible to use information from both 1 ine terminals. In that case each fault on the 1 ine wi 11 lead to an instantaneous trip. Two principles can be distinguished: differential protection and directional detection. A attte~entiai ~tection
relay uses currents from both line terminals. If the difference between corresponding currents exceeds a certain threshold a tripping signal is generated. This principle is mainly used for short lines and for the protection of busbars, as the current values must be transmitted from one line terminal to the other. A simple communication link can be used
for dL'tecti.Oitai detecti.Oit. On each line terminal a "directional detector" determines the direction to the fault (generally from the direction of the power flow). By exchanging this result each relay decides whether a tripping signal should be generated or not.
Protection needing communication links is connections in the United States [I.andoll et al.,
used for important 1981] and is already common practice in Japan [CIGRE, 1980]. In most cases distance protection is used as a remote backup. In Europe, 50-Hz based distance protection is used almost throughout the high-voltage network. In some protection schemes a signal is transmitted to the relay on the other line terminal when a zone fault is detected {a t'tan~fe't-t'ti.p ~cheme). Immediately upon receiving the signal the relay on the other line terminal is prepared to generate a tripping signal. An interruption of the communication link will not endanger the reliability of these schemes.
In most protection schemes the final backup consists of a tLme-o~e'tCU't'tent 'tel.~. In case the current exceeds a certain value during a certain time, a tripping signal will be generated.
1.2. Why fast protection?
The first public supply network was built in 1882 on the initiative of Thomas Alva Edison. An electrical power system supplied energy for a small quarter of New York by means of a relatively low D.C. voltage. Within a few years such networks arose all over the world. The protection took place by means of a large fuse at the outlet of the generator.
Developments in A.C. technology like the invention of the transformer, made it possible to connect power stations with each other. This led to an enormous growth of the electric supply networks in the twenties. With the larger transmitted power and the larger generators the need arose for faster fault clearing. Where a disconnection (or clearing) time of 10 seconds was still acceptable in 1910 a time below 2 seconds was asked for in 1925. In later years a continued growth of the supply networks is observed. This does not primarily concern the aerial extent but the generated and transmitted power as well as the network complexity.
Fault clearing times have decreased ever more. In the present state of the art fault detection times of 20 to 40 milliseconds are general practice, leading to fault clearing times of 40 to 100 milliseconds depending on the circuit breaker used [Blackburn, 1987]. In the past ten years there has been a call for even shorter fault clearing times {and detection times). Three causes for this can be given:
In case of a fault close to a generator, the disconnection of resistive load will speed up the generator. The larger the pre-fault transmitted power is, the larger the acceleration. If such a fault lasts too long, this will lead to transient instability. The increasing power exchange between networks thus calls for a decreasing fault clearing time or additional high-voltage lines. The first solution is preferable from an economic and environmental point of view. A fault clearing time of one or two cycles is needed in some situations to prevent transient instability [Hicks and Butt, 1980]. Utilizing 3/4-cycle breakers, this leaves about 5 milliseconds for fault detection. This is the main cause for research after faster fault detection in the late seventies and eighties.
The large currents during a short circuit constitute a heavy mechanical stress on the power systems apparatus. Since the electrodynamical forces are proportional to the square of the current, reducing the maximum current is an effective way of reducing the mechanical stress. Because the first current maximum will be the highest {due to exponential current terms) current limiting devices will only be effective when action is taken before the first current maximum. This calls for a fault detection within one or two milliseconds [Thuries and
Pham Van Doan, 1979].
The availability of fast and cheap microprocessors made microprocessor based protection relays possible. A few of these relays are already available, all of them using distance protection [-,1988]. But the microprocessor also enables the use of other {better, faster) protection principles. This triggers the call for fast protection. In this context Chamia [1988] speaks about "technology-driven" applications.
The development of one-cycle 500 kV air blast circuit-breakers prompted Bonneville Power Administration, Portland, Oregon to launch a special program in 1974 aiming at Ultra-High-Speed Relaying systems. The scope of this program was to dev~lop a relaying system with an operating time of 4 milliseconds or less for multi-phase faults close to the relay position. This resulted in a relay with an operating time of 4 milliseconds based on travelling wave principles [Esztergalyos, Yee, Chamia and Liberman, 1978]. It will be discussed as Chamia's algorithm in Section 5.1.
At about the same time a japanese group developed a prototype of a travelling-wave-based differential relay to solve some relaying problems encountered in EHV and UHV networks [Takagi, Baba. Uemura, Sakaguchi, 1977].
Despite promising testing results presented in a number of papers, the relay has not been applied in existing power systems. It will be discussed as Takagi's algorithm in Section 6.1.
Since 1978 more algorithms for travelling-wave-based protection of high-voltage lines have been proposed. But, except for Chamia's algorithm, none of them has reached the state of commercial availability.
Also some non travelling-wave-based algorithms for fast protection have been proposed. Most of them determine the stationary quantities from a time window of less than one cycle of the power frequency. None of them is able to determine the fault position with an acceptable accuracy within half a cycle [Glavitsch, Btirki, Ungrad, 1987].
1.3. The aim of this study
2
As already noted several algorithms for travelling-wave-based protection of high-voltage lines have been proposed. In the author's opinion none of them has been sufficiently tested to answer the question ":J!J. /]a':J.t fJ'totectioo, ba!J.ed oo, t'ta1!>eUing.-fl9(t1!>e fJ'tincipte':J. f;o!J.!J.i/..te?" On the other hand some of the proposed algorithms are based on very simple principles making it (almost) impossible to find even simpler algorithms. In view of this situation it is not appropriate to search for a completely new algorithm.
First the existing algorithms must be subject to a thorough and critical review. It must be investigated whether the algorithm will react
3
properly to all possible fault situations in the zone-to-be-protected . Further it must be studied how the algorithm reacts on disturbances not caused by faults in the zone-to-be-protected.
It is not possible to perform a large number of field experiments in high-voltage networks. Therefore we have to resort to network models for the testing of protection algorithms. ~he de1!>etopment ot netmo'tk modeL!J. j]o't the !J.tudy- ot t'ta1!>eLUng,-wa1!>e-ba!J.ed pwtecti..oo, forms the first part of this study. The models are described in Chapters 2 and 3.
An
existing model as well as a newly developed model have been used.2
A protection algorithm has been defined here as: "The way in which measured voltages and currents are used to calculate certain functions and the way how these functions are used to take decisions concerning protection".
3 For line protection, discussed here, the zone-to-be-protected consists in general of three phase-conductors (one circuit) between two substations. But for some relays the zone-to-be-protected consists of just one conductor, for others of two parallel circuits.
The second part of the study concerns the tell.Ung. ot the P"tePoll.ed aLg.o'ti.thnlll., the introduction of simplications, changes and/or extensions followed by a new round of testing. Also some new proposals have been
studied. Chapter 4 gives a list of situations that might lead to an
incorrect decision by the relay. It also describes how the network models
and the list of situations have been used for extensive testing of
protection algorithms. Chapters 5 through 7 give an overview of existing algorithms as well as the results of the testing.
The third part of the study concerns the c'teatton ot a Li.nk ~eteeen the
P'toPoll.ed aig.o'ti.thnlll.
ana
a tata'te 'teL~ that is capable of protecting a high-voltage line. The results of this part can be found in Chapter 8.The conclusions of this study may form a next ll.tep. on the ~ to a
2. Travelling waves and high-voltage lines 2.1. The line model
A high-voltage 4 line5 is a structure consisting of parallel conducting
wires, of towers to support the wires and of isolating chains to prevent
electrical contact between the wires and the towers. An example is depicted
in Figure 2.1.
Figure 2.1. A high-voltage line, consisting of 20 conductors. Six bundles of
3 conductors (6 phase conductors} are used to transport electrical energy.
Tile tlllO other conductors (shielding wires} are used to prevent a direct
lightning stroke to one of the phase conductors. Tile left set of phase
conductors can operate independently of the right set. Such a set of three
is called a circuit. The line shoun here is a "double-circuit line".
4
5
There are no internationally accepted standards to define the terms high voltage, extra high voltage and ultra high voltage. According to the IEEE standards board the term high voltage refers to voltage levels between 115 kV and 230 kV [Blackburn, 1987]. As the principles discussed in this thesis can be applied to any voltage level, the term "high voltage" is used here in a broader sense.
The term line is used here from the modelling point-of-view. All sets of electromagnetically coupled parallel conductors will be described as one
transmission line. Such a line may consist, from the protection
A detailed electronagnetic analysis of an actual high-voltage line is impossible and undesirable. As in any theoretical treatment of technical systems we need an appropriate model which is
a. simple enough to yield basic insight and to allow fast calculations; b. accurate enough to yield sufficient agreement with measured results. Some remarks concerning the last item can be found in Section 3.6.
~~~~
~ez
!. X .
•
~,'i';w-&/1
Figure 2.2. A si.x-}ilase transmission line:
Six paralleL conductors above a conducting
grCJl.llld.
The line model used in this thesis is characterized by the following assumptions:
1. The high-voltage line is viewed as a homogeneous multi -phase transmission line with constant parameters along the line, cf. Figure 2.2. Effects of towers and due to the sag of the conductors are neglected. (Calculations concerning the electrical properties of towers are given by Okumura and Kijima [1985], while Menemenlis and Zhu Tong Chun [1982] propose a model that includes the sag of the conductors); 2. Only TEM waves propagate along the line. This requires that the
characteristic transverse dimensions are much smaller than the wave length used. The TEM character implies that the total field can be represented in terms of currents through the conductors and voltages between them. The last assertion is due to the fact that the line integral of the electric field vector evaluated between two points in a transverse plane is the same for all paths in that plane;
3. Ground is considered as an additional conductor. With n aetaUtc conductors we then haven independent currents iu, u=l .. n (the current through the ground _being determined by Kirchhoff's current law) and n independent voltages Vu, u=l. .n. between the metallic conductors and
ground (all other voltages can be expressed in terms of these voltages with the aid of Kirchhoff's voltage law);
4. In due course suitable assumptions are introduced concerning the ohmic losses of the conductors. Such assumptions are required to guarantee the TEM character of the wave propagation at least in an approximate way.
Sections 2.2 and 2.3. will discuss this model for the single-phase and multi-phase line, respectively.
Assumption 2 applies if the condition
b
<
h/4. {2.1}is satisfied [Lorenz, 1971]. Here i\ is the wavelength pertaining to the
frequency of interest and b is the maximum distance between the conductors of the transmission line. Condition {2.1} results in an upper limit for the frequency for which our model is valid. To estimate this upper limit we
consider the typical example b 50 m {maximum distance between metallic
conductor and ground} leading to a frequency limit of 1.5 MHz.
In the presence of losses the rEM-character of the wave propagation is
more or less disturbed. Lorenz [1971] has shown that this effect increases
with growing frequency. However it can be neglected for frequencies below
the limit given by (2.1).
Summarizing it can be stated that the model applies to the transmission
lines under study up to some hundreds of kHz. 2.2. The single-phase line
The telegraphist's equations for a lossless single-phase transmission line
have been derived from electromagnetic theory by Heaviside [1893]. They read
as
a
i za
v z {2.2} (2.3} The constant parameters C and L denote the parallel {shunt) capacitance andthe series inductance per unit length, respectively.
If {weak) ohmic and dielectric losses have to be taken into account
(2.2) and (2.3} cannot be generalized in the time domain. Instead, we have
to confine ourselves to time-harmonic signals for which the phasors V and I of voltage and current satisfy the pair of equations
d I -YV z d
v
-ZI z where G and R conductance and y G+jc.£z
R+jwL denote the series {ohmic} {2.4} (2.5)frequency-dependent parallel {dielectric}
resistance per unit length, respectively6
6 A time-domain equivalent of (2.4} and (2.5) does not exist due to the
frequency dependence of R and G, which represent integral operators in the
time domain. It should be noted that in general also L and C can exhibit
Elimination of V or I leads to wave equations for current and voltage only d2 V 2 w="Y v {2.6} d2I 2 w="Y I (2. 7)
where "Y =
J(R
+ jwL) (G + jc.>C) "propagation coefficient" . {2.8} Equations {2.4} and (2.5) have travelling waves as solutions [e.g. King, 1955]V(z) = y{+)exp(-"YZ) + v<-)exp{+"YZ) I(z) = I(+}exp(-"Yz) + I(-)exp{+"Yz) where y(+) = Z0 I{+) ,
y(-)
= -
Z 0 I(-) Z0=
~
I
G + jc.>C "wave impedance" (2.9) {2. 10} {2.11} (2.12} (2. 13)For a qualitative description of travelling-wave phenomena the losses can be neglected, so that {2.2) and (2.3} apply. These equations have the solutions
v(z,t) = v+(t - z/c) + v-{t + z/c) (2. 14)
i(z, t} = i+{t - z/c) + i-{t + z/c) (2. 15)
where v+(t) = Z0 i+{t} (2. 16)
v-{t} = - Z0 i ( t} (2.17}
c {LC}~ "velocity of propagation" (2.18}
Zo = (UC)~ "wave impedance" (2. 19} Here v+(.} and v-(.) represent "travelling waves" in positive and negative z-direction. From {2.14} through (2.17} it is possible to derive so-called Bergeron s equations [Bergeron, 1950] for the terminal behaviour of a line with length~ {cf. Figure 2.3):
v(~.t) + Z0 i{~.t) = v{O,t- ~/c)+ Z0 i(O,t- ~/c)
v(~.t) - Z0 i{~.t) = v{O,t +~/c) - Z0 i(O,t +~/c)
i(O,t} + v{O,t} i{~. t) + v(~.t) (2.20) (2. 21) Figure 2.3. A line of finite
length~-For the study of protection algorithms we henceforth use the symmetrical notation of Figure 2.4. Bergeron's equations then transform into the form
v1(t} + Zo i1(t}
=
v2(t + ~) - Z0 i 2(t + ~}v1(t} - Z0 i1(t)
=
v2(t - ~) + Z0 i 2(t- ~)(2.22) (2.23)
The quantities vv(t) +
Zo
iv(t) and vv(t) - Z0 iv(t) (v=l.2) will bereferred to as incoming and outgoing waves at port v, respectively.
Fi.gure 2.4. A finite
i,(t)
~
Une with traveLlingtime ~.
...
+v1 ( t) v2(t)
2.3. The multi-phase line
In general a high-voltage line consists of n metallic conductors plus ground. Such a system of (n+l} conductors can be described as an n-phase transmission line through generalizing (2.4) and (2.5) [e.g. Bewley, 1933: Wedepohl, 1963; Djordjevic et al., 1987]. The result reads as
{2.24)
(2.25) The n-dimensional vectors y(P) and I(p) are formed by the phase voltages (voltages between the metallic conductors and ground) and the phase currents (currents through the metallic conductors), respectively. The nxn matrices
z(P) and y(P) are formed by the series (self or mutual) impedances and
parallel (self or mutual) admittances per unit length, respectively. Both matrices z(P) and y(P) are symmetrical. Elimination of the current vector from {2.24) and (2.25) gives rise to a second-order wave equation for the voltage vector:
d2V{p) (p) (p) (p)
~=Z Y V {2.26)
Similarly the current vector satisfies
(2.27}
In the limit of a Lo~~Le~~ line we obtain a pure TEM wave for which a close relation exists between z(P) and y{P), for the product of z(P) and y{P)
(which are both imaginary) equals the unit matrix times a scalar constant: z(P) y(P) y(P) z(P) = 72 U
7 jw/c ,
(2.28)
where c denotes the velocity of light and U denotes the unit matrix. The matrix equation (2.26} then represents a set of n ancoaPtea wave equations, leading to n different waves having equal propagation coefficients (i.e. equal velocities of propagation). the same holds for (2.27) describing the phase currents. Olrrent and voltage for each phase satisfy {2.14) through
(2. 19).
For a to!l!lff line the matrix products z(P )y(P) and y(P )z(P) are no longer diagonal so that (2.26) and (2.27) each represent n coaPtea wave equations. To uncouple Equation {2.27) "component currents" I(c) are introduced according to
(2.29)
where the. transformation matrix Q has to be chosen such that I(c) again satisfies a set of uncoupled equations. Insertion of (2.29) into (2.27),
then leads to
d21(c) -1 y(P)z(P) Q
1(c) .
w-
= Q (2.30}To achieve the above aim the transformation matrix Q has to be composed of the (suitably normalized) eigenvectors of the matrix product y(P)z(P). Equation {2.30) then transforms into
d2I(c) 2 {c)
w-
= 7 I {2.31)where 72
=
Q-1y(P)z{P)Q (2.32)is a diagonal matrix. The general solution of (2.31) is
I(c){z)
=
exp(-7z} I{c+) + exp(72) I(c-) .(2.33}
From
(2.33)
together with(2.29)
and(2.24)
the general solution of the set of equations(2.24)
and{2.25)
is found. It reads asI(p)(z)
=
Q exp{-72) I(c+) + Q exp(72) I(c-). (2.34)y(P){z) = [y(P)]-1 Q7 exp(-72) I(c+)_[y(P)]-1 Q7 exp(72) I(c-) (2.35}
The constant vectors I{c+) and I(c-) are determined by the boundary conditions. The expression exp (7z) denotes a diagonal matrix whose elements are formed by taking the exponent of the corresponding element of the diagonal matrix 7Z·
In case all elements. except one, of I(c) equal zero, the resulting current and voltage distribution travels along the multi-phase line without
changing its structure. Such a combination of voltages between and currents through the conductors is called a mode or modal wave. So the elements of the component current vector indicate which modes are present in the phase currents. The current vector belonging to such a modal wave is an eigenvector of the matrix product y(P)z(P), containing the coefficients of the right-hand members of the coupled wave equations (2.27) for the currents through the conductors. As expected the corresponding voltage vector is an eigenvector of z(P)y(P), containing the coefficients of the wave equations
(2.26)
for the voltages between the conductors.Although {2.34) and (2.35) fully describe the wave behaviour of the n-phase transmission line, some authors [e.g. Wedepohl. 1963] make use of two transformation matrices instead of one. Apart from component currents I(c), "component voltages" V(c) are introduced according to
y(P)
=
sv{c) . (2.36)Such an introduction is only attractive if it is possible to derive single-phase {frequency domain) equivalents of Bergeron's equations for the component quantities. In that case each component can be described independent of the others. From (2.34) and (2.35), together with {2.29} and (2.36) the following equation is derived for the terminal behaviour of a line of length~ (cf. Figure 2.3 and Equation 2.20):
y(c)(~)
+z~c)I(c}(~)
=exp(-~)
{v(c)(O} +~c)I(c)(O)}
, where z1c}=
s-t
[y(P)r1 Q 7 "wave impedance matrix".(2.37)
(2.38}
To obtain a set of single-phase equations, the voltage transformation matrix S must be so chosen that the matrix
~c)
becomes diagonal. It can be proved that each matrix S leading to a diagonal matrixz~c),
is a matrix of eigenvectors of the matrix product z(P)y{P). Then {2.36) will uncouple the wave equation for the voltage vector{2.26).
Within the computer program EMTP {cf. Chapter 3) a voltage trans-formation matrix S is derived from Q through the equation
{2.39)
where st denotes the transposed of
s.
Such a choice will lead to a diagonal matrix z1c) if y{P) and z{P) are symmetrical and all eigenvalues of z{P)y(P) are different. For an actual line the first condition will always be met, the second one might not be fulfilled in a very limited number of situations.As the wave impedance matrix
Z~c)
depends on the choice of the transformation matrices, their normalization deserves special attention. Within EMTP, the matrix Q is normalized such that for each column (i.e. for each eigenvector) the sum of the squares of the real parts equals unity and the sum of the squares of the imaginary parts is minimal.Recently it has been shown [Brandao Faria, 1988] that some transmission line configurations have a non-diagonizable matrix product y(P)z(P)_ In that case the above theory fails and the resulting travelling waves are no longer of the form given in (2.34) and (2.35). A more general solution of the multi-phase transmission line equations, covering this nondiagonalizable case is given by Brandao Faria and Borges da Silva [1986]. Such a degeneration is always 11mi ted to an isolated frequency for a lossy line. Whether the degeneration also appears on a physical line or is just a consequence of the approximations of the model is not yet clear [Luis Naredo, 1986; Olsen, 1986].
2.3.1. The balanced single-circuit line
The balanced single-circuit 1 ine is a hypothetical line where all three metallic conductors are considered to behave equally. Since in that case the conductors need to have equal distances to each other and to the ground plane, the balanced single-circuit line is physically impossible7 In reality the differences in distance are relatively small so that the balanced single-circuit line is useful as a first-order description. Due to the symmetrical nature of the {hypothetical) structure the resulting equations are of a simple form. These equations will be used as a basis for most of the protection algorithms discussed in forthcoming chapters. For a balanced single-circuit line the z{P) and y{P) matrices are of the following structure (s stands for "self", m for "mutual"):
z zz
l
[y yy
z{P)= z z z s m m • y{P) y y y s m m {2.40)
m s m m s m
z z z y y y
m m s m m s
The matrix products z{P)y(P) and y{P)z{P) are of the same symmetrical structure. The eigenvalues 7
1 2 (1=1,2,3) are found as 2 71 (Zs + 2Zm){Ys + 2Ym) 732 = (Zs - Zm){Ys- Ym)
7 Such a symmetry is, however, possible for a three-phase cable.
(2.41) {2.42)
while a general expression for the matrix of eigenvectors reads as
(2.43)
In this case there is a "homopolar" or "zero sequence" wave with equal currents through all three phase-conductors. closing through ground and two "aerial" waves restricted to the phase conductors. An often used transformation matrix satisfying {2.43) is
1
"]
1:.]
- l 1 Q 1 a2 a2 Q=3
a2 (2.44) 1 a a awith a=exp (j2Jr/3); Voltages and currents under this transformation are known as "symmetrical components".
Another modal transformation, used in algorithms for travelling wave-based protection is the following one:
[
1 1 2
Q
=
t
1 -2 -1 1 1 -1(2.45)
By using this transformation, component quantities are derived from (measured) phase quantities through simple calculations. Figure 2.5 shows the current distributions for this transformation.
0
®
0
0
0
0
0
0
®
X X X2.3.2 The balanced double-circuit line
Figure 2.5. Current distribu-tions corresponding
homopotar unue (left)
aerial unues on a stngte-circutt line.
to the and the balaru:ed
A balanced double-circuit line consists of six (metallic) phase conductors and one reference conductor (ground). The phase conductors are divided into two groups of three. called circuits. Each phase conductor is considered to have the same distance to ground. The distances between each pair of conductors within one circuit are equal. so are the distances between a conductor in one circuit and a conductor in the other circuit. Again this situation is physically impossible but it provides us with a simple line
model. For the balanced double-circuit line z<P) and y{P) are of the following structure {d stands for "double circuit")
z s z m z zd zd zd m z z z zd zd zd m s m z z z zd zd zd m m s z{P) zd zd zd z s z m z m {2.46) zd zd zd z m z s z m zd zd zd z m z m z s y y y yd yd yd s m m y y y yd yd yd m s m y y y yd yd yd m m s y{P) yd yd yd y s y m y m {2.47) yd yd yd y m y s y m yd yd yd y m y m y s
Three different eigenvalues are found: 2 {Z + 2 z + 3 Zd) {Y + 2 y + 3 Yd) {2.48)
"'t
s m s m 2 {Z + 2 Zm - 3 Zd) {Y + 2 y - 3 Yd) {2.49) "12 s s m 2 2 2=
'Ys2=
{Z - z ) {Y - Ym) {2.50) "!3 "14 "'s s m swith the following general expression for the matrix of eigenvectors:
Q, Qt2 Qt3 Qt4 Qts Qt6 Q, Qt2 Q23 Q24 Q25 Q26 Q, Qt2 Q33 Q34 Q35 Q36 Q Q, -Qt2 Q43 Q44 Q45 Q46 {2.51) Q, -Qt2 Q53 Q54 Q55 Q56 Q, -Qt2 Q63 Q64 Q65 Qss where Q1j + ~j +Q3j
=
0 and Q4j + ~j + ~j=
0, j=
3,4,5,6."ft is the propagation coefficient of the "homopolar wave", "{2 of the
"double-circuit wave" going "up" in one circuit and "down" in the other. "{3
through "'s correspond to the "single-circuit waves" closing inside each circuit. An example of such a transformation is found as {cf. Figure 2.6)
1 2 4 0 0 1 1 -4 -2 0 0 1 1 -1 -1 -1 Q=1/6 1 2 -2 0 0
.
Q-1_-
0 -1 1 0 0 0 (2.52) 1 -1 0 0 2 4 0 -1 0 0 0 1 -1 0 0 -4 -2 0 0 0 0 -1 1 -1 0 0 2 -2 0 0 0 1 0 -1 0 0 0 ® 0 0 0 0 0 0e
@"""
>< X :. @ 0 0 0 0 0 0 0 @ 0 0 0 0 @ 0 0 0 0 0 0 0 0 @ 0Figure 2.6. Current distributions corresponding to the homopoLar wave (top
Left), the doubLe-circuit wave (top right) and the singLe-circuit waves on a
balanced double-circuit line.
2.4. Travelling waves in high-voltage networks
2.4.1. A subdivision of transients, according to their origin
The transients occurring in a high-voltage network can be divided into three groups, according to their origin.
1. ~~~~eat~ caa~ed ~ ~~LtchLn~ oPe~tLon~ Ln the netwo~
These can be called on-purpose operations, because there is always someone or something giving a conunand to open or close the switch. Typical examples are:
energizing an unloaded line:
deenergizing a short-circuited line:
2. ~~an~Lent~ ha~Ln~ theL~ caa~e out~Lde the net~k
High-voltage lines consist of very long wires (tens of kilometers or more). These are very good antennas, so radio signals are easily picked up. Other examples are very low frequency (0.01 Hz} currents induced by ionospheric currents [e.g. Pirjola and Viljanen, 1989] and injected signals with a frequency of some kHz used for communication purposes. A more spectacular case is a lightning stroke to a high-voltage line or somewhere in the vicinity of a high voltage line.
3. ~~~Lent~ caa~ed log. taait LnLUatLoo
Faults are inadvertent, accidental connections between phase wires or between one or more phase wires and ground [Blackburn, 1987]. Causes for faults (short circuits} are lightning (induced voltage or direct strike}, switching overvoltages, wind, ice, earthquake, fire, falling trees, flying objects, physical contact by animals or human beings, digging into underground cables and so on.
These three groups will be further discussed in Section 2.4.3 through 2.4.5. A number of studies have been performed during the last years to determine the number of transient phenomena that occur in a high-voltage network. All studies make use of a transient recorder that is· triggered by a sudden change in voltage or current. The number of recorded phenomena depends largely on the kind of measurement and on the position in the network. Further the seasons turn out to have influence on the frequency of recorded phenomena.
During a three-year monitoring period in a 500 kV substation and in a 138 kV substation a total of 341 phenomena has been recorded [EPRI, 1986]; 190 of them were caused by switching operations, 114 by lightning strokes and 37 by faults. A study by EdF (Electricite de France) in the 400 kV network [Barnard et al., 1984] showed 93 transients caused by faults and 19 caused by switching operations in a period of about 2 years. Malewski, Douville and Lavallee (1987] found a total of 300 transient phenomena during a two year monitoring period in a 735 kV substation. A four-month (july-october) monitoring period in the Dutch 220 kV network yielded 39 lightning-caused transients (27 occurring within one week), 1 due to a fault and 3 due to switching operations [Koreman, 1988]. Johns and Walker [1988] found 93 transient phenomena within 39 days (December '83 till February '84), 29 of which occurred within 3 hours.
2.4.2 Superimposed quantities
A high-voltage network is fed by 50 or 60 Hz sources (in the remainder only 50 Hz will be used). During an undisturbed situation all voltages and currents in the network are sinusoidal in time. The sinusoidal voltages and
currents as existing before a certain disturbance will be called
"un.di,(l.tu'tl.ect ~titLe().". After the disturbance this term denotes their continuation, i.e. the fictitious voltages and currents that would have occurred without disturbance.
After a transient due to some disturbance a new sinusoidal state will be reached which may or may not differ from the undisturbed state. To study the transient phenomenon it is sometimes attractive not to use the actual voltages and currents but "(l.U/>e'tLIII(Jo(l.ed ~titLe().". Their values are defined as the difference between the actual and the undisturbed values.
The disturbance can be any of those described in the previous section. We illustrate the concept of superimposed quantities by means of a fault on a single-phase line. The undisturbed network is shown in Figure 2.7.
Figure 2.7. Undisturbed values.
Undisturbed voltages at line terminal A. at the future fault position F and at line terminal Bare given as
VA (t) =VA sin (~Uot + <PA) , (2.53}
VF (t) = VF sin {~Uot + <PF) •
(2.54)
VB ( t) =VB sin (~Uot + <PB)
(2.55)
respectively.
Figure 2.8 shows the network situation after the occurrence of the fault at time-zero. Actual voltages at A, F. and Bare denoted by vA(t),
vF(t), and vB(t), respectively.
t:(tl
FThe superimposed quantities, being the difference between the actual values of Figure 2.8 and the undisturbed values of Figure 2.7 are given in Figure 2.9.
A . F
Figure 2.9. Superimposed vatues:
vF
(t)=
0, t<O.Superimposed voltages at line terminal A, at the fault position F, and at line terminal B are given as
VA (t) =VA (t) - ~A (t) (2.56) {
-~F
(t) t<
0 VF (t)=
VF (t) - ~F (t) t ) 0 (2.57) VB (t) = VB (t) - \IB (t) {2.58} respectively.2.4.3. Transients caused by switching Olli!rations
Figure 2.10. Travetting waves initiated during tine energizing.
As an example Figure 2.10 shows the configuration for energizing one of the circuits of a double-circuit line. Closing the circuit breaker (1) leads to voltage jumps on both sides of it. This sets up travelling waves into the line to be energized (2) as well as into the feeding network (3). At the remote line-terminal (4) the circuit breaker in the line-to-be-energized is open. At this discontinuity a part of the wave is reflected (5) and through the coupling with the second circuit a part travels into the network behind the remote line-terminal (6). The reflected wave (5) is again reflected at the circuit breaker (1} and so gives rise to multiple reflections.
Figure 2.11 shows the voltages during 1 ina-energizing as measured on the line side of the open circuit-breaker (4).
f
1.2vr
(pu) 0 0,8f
1,~r\1
Vs ~pu) -1ms 0,4lo,4
vto
(pu) 1,22.4.4. Transients caused by lightning
Figure 2.11. Voltages at the open line terminal measured during
line energizing. Adopted from
[Kersten and Jacobs, 1988]
There are three different mechanisms for a lightning stroke to interfere with a high-voltage line; dependent on the distance between the position of
the stroke and the phase conductors.
1. ~ LLghtnLng ~t~oke ~oaemhe~e Ln the ~LcLnLtv
ot
the LLneThis can be a stroke between a cloud and earth as well as a stroke between two clouds. In most cases the waveform of the induced voltages is bipolar with substantial amplitude being attained in the first few microseconds. Theoretically amplitudes higher than 1 MV are possible for a 100 kA stroke at a distance of a few hundreds of meters from the line [Eriks~on, 1976]. Lightning strokes to earth of high current amplitude in the vicinity of a high-voltage line are rare due to the attractive effect of the line. But high objects (e.g. trees) close to the line can increase this possibility. Measurements show induced voltages up to 200 kV [Cornfield and Stringfellow,
De la Rosa et al. [1988] have installed a single-phase line in Mexico, in a region with a high ground-flash density. Their measurements show that lightning strokes tens of kilometers away from the line can induce voltages
of some kV. An overvoltage of 244 kV has been measured for a stroke at a
distance of 4.5 km. It appears that a stroke next to a line will create the
highest overvol tage some distance further down, which can be explained
electromagnetically [Cooray and De la Rosa, 1986].
l
v
(kV)
-40
-SOL---~
Figure 2.12. TypicaL voLtage shape due to a Lightning stroke dose to a high-vottage tine, adopted from [Koreman, 1988}.
A typical shape of the voltage pulse due to a nearby stroke is shown in Figure 2.12. The superimposed voltages for the three phases are shown, with
the vertical axes shifted with respect to each other.
2. d LLghtnLng ~t~oke to a ~hLeLdLng ~L~e o~ to a t~e~
To prevent dangerous lightning strokes to phase conductors, shielding wires are used. The so-called electro-geometric theory describes how a lightning
stroke develops [e.g. Eriksson, 1976]. According to this theory only
lightning strokes below a certain current amplitude can reach the phase conductors. So all high-current strokes will reach the shielding wire or a
tower. This causes intense electromagnetic fields, leading in a few cases to flashovers between phase conductors and ground.
When the lightning stroke does not lead to such a flashover the three phases are influenced in about the same manner. i.e. the induced voltages (and currents) in the three phases are of about equal shape and magnitude. Mainly a homopolar wave is initiated. The voltage amplitudes are tens of kilovolts.
3. Ell Ughtni.ng o.t'!.oke to a Pkao.e
condu.ctO'l-When the lightning strikes directly to a phase conductor. this will cause a large voltage between that conductor and the other conductors {including ground). If this voltage exceeds the breakdown voltage of the insulation a flashover wi 11 occur leading to a fault. According to electro-geometric theory such a flashover will never occur if the shielding wires are correctly positioned. A direct stroke is often called a shielding failure, although the word is sometimes reserved for direct strokes leading to a
flashover.
Figure 2.13 shows voltages due to a direct stroke leading to a fault. They have been measured in a substation about 10 km from the fault. In phase 1 a negative spike is visible followed by a transient due to the single-phase fault. At the fault position the spike must have been much higher. The amplitude reduction takes place on the 10 km line between the position of the stroke and the substation and in the substation between the line being hit and the line being monitored. The negative slope of the spike corresponds to the lightning current, the positive slope corresponds to the flashover between the phase being hit and ground.
A stroke not leading to a fault causes a high voltage spike in the phase that has been hit . The other phases show a 1 ower vo 1 tage spike. Whether the spikes in the other phases are of the same polarity or of opposite polarity is still a matter of debate. The coupling between the phases would lead to a spike of equal polarity [e.g. Anderson, 1985] but the release of the induced charge would lead to spikes of opposite polarity. Figure 2.13 shows spikes of opposite polarity.
Kinoshita et al. [1987] present measurements of the simultaneous occurrence of a positive spike in one phase and a negative in another. This is caused by a negative stroke followed, almost immediately, by a positive stroke somewhere else in the same cloud. If both strokes hit a phase conductor the above phenomenon can be observed.
Q5 - 1 , 0 t(ms}
t1
fase2 Q5 - 1 , 0 Hms) ~~---~Figure 2.13. Transient voltages caused by a. lightning stroke to a. JJw.se
conductor leading to a. Flashover. adopted from [Koreman. 1988].
2.4.5. Fault transients
A fault (short circuit) on a high-voltage line, normally implies a voltage
jump at the fault position. This voltage jump leads to travelling waves that
can be detected in a large part of the network. All protection principles in this study use these travelling waves to retrieve information from the fault.
In most studies of fault transients the resistance of the conducting path at the fault post tion is neglected. Although there is always some resistance present, the approximation is certainly acceptable for solid faults or those with only a small arc distance.
An
analysis for longer arc distances is given by Cornick. Ko and Pek [1981]. These authors also show that the fault initiation (due to the spark) takes place within one microsecond. After that it takes a few microseconds for the arc to develop. During this time the arc voltage drops from 100 V/cm down to about 10 V/cm. For arcs across wooden structures (wood-pole-lines, flashover to a tree) the arc voltage-gradient drops from 800 to SO V/cm.As long as the pre-fault voltage is large compared to the post-fault (arc) voltage, the latter can be neglected. For an arc length of 10 meter and an arc-voltage gradient of 10 V/cm, the pre-fault voltage must be much higher than 10 kV. As the systems studied have voltage amplitudes of hundreds of kilovolts, the arc voltage will not be of great influence on the fault transients. Of course this is no longer true in case of a pre-fault voltage close to zero. But then a high resistive fault (= long arc distance) will (generally) not occur.
Faults on high-voltage lines can have many causes. The main cause for faults on the high-voltage lines is lightning. According to a study of a number of networks all over the world, about 40% of all faults are
lightning-caused [Anderson. 1985]. This value shows a large deviation for different utilities. A large part of the lightning-caused faults are multi-phase or double-circuit. According to Sargent and Darveniza [1970], 40% of the lightning-caused faults on circuit lines are double-circuit faults. Figure 2.14 shows some double-double-circuit faults caused by
lightning. Other important fault causes are storm (10-30%} and galopping conductors [Mackay, Barber and Rowbottom, 1976; Leppers, 1985].
R
w
X0
0
0
X0
0
Xs
v
0
0
X X0
0
0
0
X XT
u
X X X0
X0
X0
0Figure 2.14. Examptes of doubte-circuit fautts caused by tightning [adopted from Sargent and Dnrventza, 1970]. x:fautted phase, o=non-fautted phase, the
phase order is shown on the teft.
From a number of studies the following distribution of faults among different types can be concluded [Anders, Dandeno and Neudorf, 1984, Light,
1979; Recker, Reisner and Waste, 1986; Liew and Darveniza, 1982].
Single-phase-to-ground 7o-90%
Phase-to-phase 10-20%
Two-phase-to-ground and three phase: 5-15%
single-circuit faults Double-circuit faults
75-98% 2-25%
3. Network modelling
Chapter 2 dealt with the transition from a high-voltage line (an actual physical inhomogeneous structure) to the simplified model "transmission line". In this chapter we further elaborate the concept "multi-phase transmission line" and present a number of pertinent data which are in current use in modern computer programs. Two of these programs (cf. Section 3.2 and 3.3) have been used in this thesis to determine transient voltages and currents in a high-voltage network. Such simulations are required to
test protection algorithms, as described in forthcoming chapters.
3.1. Impedance and admittance matrices
Extensive analysis is available concerning the determination of the z(P) and
y(P) matrices for configurations consisting of a set of thin metallic
conductors (cf. Figure 3.1). The basic situation studied in this context is that of a single circular conductor above a homogeneously conducting half-space (cf. Figure 3.2). Parameters for two or more metallic conductors are approximately evaluated under the assumption that the electromagnetic field can be viewed as a linear combination of the individual fields of the single-conductor configurations. Minor possible refinements reckoning with the vertical dependence of the conductivity of the ground (as discussed by Wedepohl and Wasley [1966] and by Nakagawa, Ametani and Iwamoto [1973]) will be left out of consideration.
'-
0
0
transmtsston ltne consisting of three
0
Figure 3.1. The cross-section of aconductors and a conducting half-space.
l///////J/l/77
Carson [1926] was one of the first who evaluated transmission line parameters. He found expressions for the elements of the impedance matrix
z(P) that nowadays serve as a reference under the name "Carson's
equations". These are used in most modern computer programs for the analysis of transients in high-voltage networks, including the programs used in our study. The elements of the matrix y(P) are determined from .the electrostatic fields.
Similar expressions for the transmission line parameters have been found at about the same time by Rlidenberg [1925], Mayr [1925] and Pollaczek [1926]. Whereas these authors only give integral expressions, Carson [1926]
rewrites his integral solution in the form of fast converging infinite
series. This series solution has certainly contributed much to the
widespread use of Carson's equations.
_yz
/ X
~---~---~/
h
Figure 3.2. Configuration studied by
Carson [1926].
The configuration under study is depicted in Figure 3.2. Following Carson [1926] {and in conformity with Chapter 2) we consider a time-space
dependence of the form exp(jwt-?£), where~ is the propagation coefficient
to be determined. While y(P) is assumed to have its {lossless, i.e. imaginary) electrostatic value, expressions for the {complex) elements of
z(P)
are evaluated under several low-frequency approximations.In later years other authors have found expressions for the line parameters under less simplifying approximations. Wise [1948] finds that the elements of the admittance matrix y(P) have to be corrected as a consequence of the finite conductivity of the ground. However his correction terms are small for normal configurations and frequencies below 1 MHz [Hedmann, 1965]. Wedepohl and Efthymiadis [1978] evaluate the electromagnetic field due to a
current with an exponential time-space dependence and find a "modal
equation" which is solved numerically [Efthymiadis and Wedepohl, 1978]. For practical high-voltage lines and frequencies up to a few hundreds of kHz,
their value of the propagation coefficient ~ again does not considerably
differ from that found from Carson s equations. Wait [1972] derives an even
more general modal equation that is solved by Olsen and Chang [Olsen, 1974;
Olsen and Chang, 1974; Chang and Olsen, 1975]. Contrary to Wedepohl and Efthymiadis [1978] they find more than one solution. It appears that a single wire above ground can support two discrete modes (apart from a continuous mode spectrum). One of the discrete modes {the "transmission line mode") corresponds to Carson's solution, the other (the "fast mode") is associated with a high phase velocity and a low damping. According to Olsen and Pankaskie [1983] it is safe to assume that only the transmission line
mode is present when
h ( i\/20 . (3. 1)
where h is the height of the conductor and A is the wavelength pertaining to the frequency under consideration. For a height of 50 meters a limit of 300 kHz is found. This admits the final conclusion that for a high-voltage line Carson's equations are valid up to a few hundreds of kHz.
3.2. The computer program EMTP.
3.2.1. The solution method used in EMTP
The "Electromagnetic transients program" (EMTP) is a universal computer program for the calculation of electromagnetic transients in high-voltage networks. It has been developed in the late sixties by Bonneville Power Administration, Portland, Oregon and the University of British Columbia, Vancouver, Canada. After a few years it was in widespread use through computers all over the world, where many people provided the program with extensions. This rapid growth in use as well as in size can be described to the compatibility (it only uses standard FORTRAN) and the availability (it was public domain software). Although the term EMTP is already used as a standard, there still exist many program versions. In order to create one real standard version, Leuven EMTP Center8 has started to co-ordinate all improvement efforts. As long as this standard does not exist, it is necessary to tell what version has been used. All EMTP calculations described in this thesis are performed with the M39 version on an APOLLO-domain computer.
EMTP is able to calculate voltages and currents in networks consisting of resistances, inductances, capacitances, single and multi-phase
~-circuits,
transmission lines9 and other elements. Each network element is represented by means of an equation that relates a node voltage at time t to node voltages and branch currents at times t-kAt, k=l,2, . . . By combining all element equations and the pertinent Kirchhoff's equations a matrix equation is derived for a network of n nodes:G.v(t) = j(t) hist(t) - Ge e(t)
8
Leuven EMTP Center Kard. Mercierlaan 94 3030 Heverlee Belgium.
(3.2)
9 In the EMTP-related literature the name "distributed-parameter lines" is used instead of transmission lines.
where v( t} vector of unknown voltages, j(t} vector of current sources, e(t} vector of voltage sources,
hist(t} vector of "history terms" taking into account voltages from the past,
G, Ge nxn matrices determined by the network topology and the values of the elements.
Starting from a (given} situation voltages are calculated for equidistant points in time. Finally, the currents can be determined with the aid of the element equations. More details on
EMTP
are given by Dommel[1986].
3.2.2. The LINEa:>NSTANTS routine
32.4 31.4 0 24.4 0
•
•
s
17.4w
•
R•
~T•
v
•
u
/}\0.~ \tf..--~ , __ .7 ('I) ,...:...
~~
Figure 3.3. Cross-section of a specific tine. The transmission open circLescorrespond to the shielding wires, with an outer diameter of 2.2 em and an inner df.am.eter10 of 0.8 em.
The cLosed circLes corres-pond to a bundLe of three conductors as sholm. in the insert. Each conductor has outer and inner diameters of
2.8 and 0.8 em, respectiveLy. ALL dimensions are given in meters.
From Carson's equations it is possible to determine impedance and admittance matrices for a multi-phase transmission line. This is done by EMT~ s LINECONSTANTS routine. We illustrate the routine by means of the 20-phase transmission line depicted in Figure 3.3. It is a model of the high-voltage line of Figure 2.1. The average height of the sagging conductors in the transmission line model is determined as:
h =
t
ht +~
hm ' (3.3}10
Within
EMT~
s LINEmNSTANTS routine wires are considered to be tubular conductors with certain inner and outer diameters. In reality the wires consist of a good conducting mantle and a core with a lower conductivity. Due to eddy-current effects the current will concentrate in the mantle.where ht is the height at the tower position and hm is the height midway
between two towers. The resistivity of the ground is taken to be 100 Qm,
which is an acceptable value for over 50% of the world land area. Higher
values might be needed for mountainous terrain and permafrost areas
[OCIR.19SS]. The 20 x 20 matrices y(P) and z(P) are reduced to 6 x 6
matrices by assuming:
the voltage between a shielding wire and ground is zero; - the voltage between two conductors in one bundle is zero.
From the resulting 6x6-matrices a current transformation matrix Q, a
(diagonal) wave-impedance vector zic) and propagation coefficients 7 are determined, according to the theory of Section 2.3.
Each column of Q is normalized such that the sum of the squares of the
real parts equals one and the sum of the squares of the imaginary parts is minimum. In a next step the imaginary parts are neglected. leading to the following current transformation matrix for the transmission line depicted in Figure 3.3: 0.46 -0.30 0.53 0.25 0.58 0.16 0.24 -0.20 -0.38 0.54 -Q.35 0.47 Q 0.48 -0.61 -0.27 -0.38 -0.20 -o.51 (3.4) 0.48 0.61 -Q.27 0.38 0.20 -0.51 0.46 0.30 0.53 -0.25 -Q.58 0.16 0.24 0.20 -0.38 -Q.54 0.35 0.47
This matrix considerably differs from that for a balanced double-circuit line (2.51). Obviously the transformation matrices for a balanced line cannot be applied to nonbalanced lines. Nevertheless the transformation matrix for a balanced 1 ine turns out to be sui table for the protection algorithms to be discussed in forthcoming chapters.
3.2.3. The .!MARTI SETUP routine
A number of numerical transmission-line models are available within EMTP. The model applied in this study is the ".}MARTI SETUP" proposed by Marti [1982]. It is the most detailed model available. Whether its results are close enough to reality is still a matter of debate [e.g. Empereur and Somatilake, 1986; Lima, 1987; Marti, et al., 1987; Ametani, 1988]. This question can only be answered by performing field tests (cf. Section 3.6).
The JMARTI SETUP incorporates the frequency dependence of modal wave
impedances and propagation coefficients. The transformation matrix Q is
considered to be frequency independent. (The debate concentrates on this
