Mrinal K Pal. Power system stability
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NUMERICAL SOLUTION OF THE TRANSIENT STABILITY PROBLEM
4-19
distinctly different from the sequence impedances that would be used for unbalanced short
circuit at the same point.)
The component voltages and currents at the point of open conductor can therefore be expressed
as
3/
210 aaaaaaaa
VVVV
′
=
′
=
′
=
′
0
210
=
+
+
aaa
III
and
02
02
1
1
ZZ
ZZ
Z
E
I
a
a
+
+
= (4.51)
Therefore, the simulation of an open conductor in a transmission line is accomplished by
connecting the parallel combination of the zero- and negative-sequence impedances looking into
the network at the point of open conductor in series with the transmission line.
References
1. B. Stott, "Power System Dynamic Response Calculations," Proc. IEEE, Vol. 67, No. 2, pp.219-241, February
1979.
2. B. Stott, "Power System Step-by-Step Stability Calculations," IEEE Int. Symp. Circuits and Systems,
Phoenix, AZ, April 1977.
3. "Power System Dynamic Analysis Phase I," Boeing Computer Services, EPRI EL-484, Project 670-1, Final
Report, 1977.
4. H.W. Dommel and N. Sato, "Fast Transient Stability Solutions," IEEE Trans., PAS - 91, pp.1643-1650, 1972.
5. W.D. Humpage, K.P. Wong and Y.W. Lee, "Numerical Integration Algorithms in Power System Dynamic
Analysis," Proc. IEE, Vol. 121, pp.467-473, June 1974.
6. W.D. Humpage and B. Stott, "Predictor-Corrector Methods of Numerical Integration in Digital Computer
Analysis of Power System Transient Stability," Proc. IEE, Vol. 112, pp.1557-1565, August 1965.
7. J.B. Scarborough, Numerical Mathematical Analysis, The Johns Hopkins Press, Baltimore, 4th Ed. 1958.
8. E.W. Kimbark, Power System Stability, Vol. 1, Wiley, N.Y. 1948.
9. H.W. Dommel, W.F. Tinney and W.L. Powell, "Further Developments in Newton's Method for Power System
Applications," IEEE PES Winter Meeting, New York, Jan. 1970, 70 CP 161-PWR.
10. H.E. Brown, Solution of Large Networks by Matrix Methods, 2nd Ed. Wiley, N.Y. 1985.
11. N. Sato and W.F. Tinney, "Techniques for Exploiting the Sparsity of the Network Admittance Matrix," IEEE
Trans., PAS-82, pp.944-990, 1963.
12. W.F. Tinney and J.W. Walker, "Direct Solutions of Sparse Network Equations by Optimally Ordered
Triangular Factorization," Proc. IEEE, Vol. 55, pp.1801-1809, November 1967.
13. W.D. Stevenson, Elements of Power System Analysis, 4th Ed. Mc Graw-Hill Book Co. 1982.
14. E. Clarke, Circuit Analysis of A-C Power System, Vol. 1, Wiley, N.Y. 1943.
5-1
CHAPTER 5
SYNCHRONOUS MACHINES
Synchronous machines play a vital role in power system stability, and a knowledge of the theory
and performance of these machines is essential for a proper understanding of the subject. So far,
the simplest synchronous machine representation -- the so-called classical model -- has been
used to explain some of the fundamentals of power system stability. In this chapter we will
review the basic theory of synchronous machines and derive models suitable for a more detailed
analysis of the stability problem, including the effects of control.
A synchronous machine consists of two major components, the stator and the rotor, that are in
relative motion and that are different in structure. In the usual machine the exciting magnetic
field is produced by a set of coils (the field winding) on the rotor which rotates within the stator
which supports and provides a magnetic flux path for the armature windings. There are
ordinarily three armature coils placed around the stator at 120° electrical degrees apart so that,
with uniform rotation of the magnetic field, voltages displaced 120° in phase will be produced in
the coils. The rotor magnetic paths and all of its electrical circuits are assumed to be symmetrical
about both the pole (direct) and interpole (quadrature) axes. The field winding is separate from
the other rotor circuits and has its axis in line with the direct axis. The other rotor circuits are
formed by the amortisseur or damper bars set in pole-face slots, arranged symmetrically and
connected together at the ends in the case of salient-pole machines, and by the eddy current paths
in the iron in the case of the solid rotors of the turbo-generators. The effect of these additional
rotor circuits can be approximated by two sets of equivalent damper or amortisseur circuits with
their axes in line with the direct and quadrature axes, respectively. Usually one circuit on each
axis is sufficient to account for the various transient and subtransient effects. The symmetrical
choice of the rotor circuits has the virtue of making all mutual inductances and resistances
between direct and quadrature axis rotor circuits equal to zero.
All mutual inductances between stator and rotor circuits are periodic functions of rotor angular
position. In addition, because of the rotor saliency, the self inductances of the stator phases and
the mutual inductances between any two stator phases are also periodic functions of rotor angular
position. Therefore, the resulting circuit equations are awkward to handle. However, if certain
reasonable assumptions are made, a relatively simple transformation of variables will eliminate
the troublesome functions of angle from the equations.
The assumptions are:
1. The windings are sinusoidally distributed along the air gap as far as all mutual effects with the
rotor are concerned.
2. The stator slots cause no appreciable variation of any of the rotor inductances with change in
rotor position.
3. Saturation may be neglected. The effect of saturation can be accounted for separately.
A schematic representation of a synchronous machine is shown in Figure 5.1.
SYNCHRONOUS MACHINES
5-2
Fig. 5.1 Schematic representation of a synchronous machine
The electrical performance of a synchronous machine may now be described by the following
equations.
Voltage Relations
The voltage relationship of any of the armature windings is of the form
ri
dt
d
e −=
ψ
(5.1)
where e,
ψ
and i are the terminal voltage, total flux linkage, and current, of the winding,
respectively, and r is the winding resistance. The direction of positive current corresponds to
generator action.
Denoting the three phases by subscripts a, b, c, the armature voltage equations can be written in
matrix form as
−
=
c
b
a
c
b
a
c
b
a
i
i
i
r
dt
d
e
e
e
ψ
ψ
ψ
(5.2)
or
abcabcabc
r
dt
d
iΨe −= (5.3)
where
abcabcabc
iΨe and,, are the vectors of the voltages, flux linkages and currents, respectively,
of phases a, b and c. The three phases, a, b, c, are lettered in the direction of rotation of the rotor,
as shown in Figure 5.1.
SYNCHRONOUS MACHINES
5-3
In a similar manner, the voltage relationships of the field and amortisseur windings, assuming an
unspecified number of amortisseur windings, can be written in matrix form as
+
=
M
M
M
O
O
O
M
M
M
M
M
M
q
d
fd
q
d
fd
q
d
fdfd
i
i
i
r
r
r
dt
d
e
1
1
1
1
1
1
0
0
ψ
ψ
ψ
(5.4)
or
rrrr
dt
d
iRΨV += (5.5)
where
,
0
0
=
M
M
M
fd
r
e
V
=
=
M
M
M
M
M
M
q
d
fd
r
q
d
fd
r
i
i
i
1
1
1
1
, iΨ
ψ
ψ
ψ
and
[
]
LLL
qdfdr
rrr
11
diag=R
Flux-Linkage Relations
The armature flux-linkage relationships can be written in matrix form as
+
−=
M
M
M
LL
LL
LL
q
d
fd
qcdccfd
qbdbbfd
qadaafd
c
b
a
cccbca
bcbbba
acabaa
c
b
a
i
i
i
lll
lll
lll
i
i
i
lll
lll
lll
1
1
11
11
11
..
..
..
ψ
ψ
ψ
(5.6)
or
rsrabcssabc
ililΨ
+
= (5.7)
Similarly, the rotor flux-linkage relationships can be written as
SYNCHRONOUS MACHINES
5-4
+
−=
M
M
M
LL
L
LLLL
LLLL
LL
LL
LLL
LLL
LLL
M
M
M
q
d
fd
q
dfd
dfffd
c
b
a
cqbqaq
cdbdad
fcdfbdfad
q
d
fd
i
i
i
l
ll
ll
i
i
i
lll
ll
lll
1
1
11
111
1
111
111
1
1
ψ
ψ
ψ
(5.8)
or
rrrabcrsr
ililΨ
+
= (5.9)
Inductance Relations
Armature self-inductances
The current in phase a produces an mmf Ni
a
, where N is the effective number of turns of the
phase a winding. This mmf has a component Ni
a
cos
θ
along the direct axis, and a component
−Ni
a
sin
θ
along the quadrature axis, where
θ
is the angle of the direct axis from the axis of phase
a, measured in the direction of rotor rotation (Fig. 5.1). If P
d
and P
q
are the permeances of the
magnetic circuits in the d and q axes, respectively, the fundamental flux components along these
two axes are:
φ
d
= P
d
Ni
a
cos
θ
,
φ
q
= −P
q
Ni
a
sin
θ
The components of these fluxes along the axis of phase a are
φ
d
cos
θ
and −
φ
q
sin
θ
, respectively,
and hence the flux which links phase a is
θθ
2222
sincos
aqad
iNPiNP +
There is also some flux linking phase a that does not link the rotor -- the leakage flux. If the flux
leakage paths have a permeance P
l
, assumed to be independent of
θ
, the leakage flux is P
l
Ni
a
and
the flux linkage due to it is
allal
iPPNiPN )sincos(
2222
θθ
+= .
The total flux linkage is therefore
++=
θφ
22
cos)(
aldaa
iPPN
θ
22
sin)(
alq
iPPN +
=
a
qd
l
qd
i
PP
P
PP
N
−
+
+
+
θ
2cos
22
2
Since inductance is defined as flux linkage per ampere, the self-inductance of phase a may be
expressed as
θ
2cos
20 aaaaaa
LLl
+
= (5.10)
Self-inductances of phases b and c are obtained by noting that they are displaced by ±120° from
phase a. Therefore the self-inductances of phases b and c are:
)120(2cos
)120(2cos
20
20
o
o
++=
−+=
θ
θ
aaaacc
aaaabb
LLl
LLl
(5.11)
SYNCHRONOUS MACHINES
5-5
Armature mutual inductances
The mutual inductances can be found in a similar manner. The flux produced by phase a that
links phase b, assuming the effective number of turns of phases a, b and c are equal, is
a
qdqd
aqad
qdab
i
PPPP
N
iNPiNP
NN
−
−
+
+
−=
−+−=
−−−=
)1202cos(
24
)120sin(sin)120cos(cos
)120sin()120cos(
2
22
o
oo
oo
θ
θθθθ
θφθφφ
The mutual inductances between phase a and phase b can thus be written in the form
)1202cos(
20
o
−+−==
θ
aaabbaab
LLll (5.12)
Note that the magnitudes of the variable parts of the self and mutual inductances are the same
and that the magnitude of the constant part of the mutual inductance is half that of the constant
part of the self inductance, excluding leakage.
Similarly,
θ
2cos
20 aaabcbbc
LLll
+
−
=
=
)1202cos(
20
o
++−==
θ
aaabacca
LLll
Rotor self-inductances
Since the effects of stator slots and of saturation are being neglected, all the rotor self-
inductances
qdffd
lll
1111
,, , etc. are constants.
Rotor mutual inductances
All mutual inductances between any two circuits both in the direct axis and between any two
circuits both in the quadrature axis are constants and also,
fddf
ll
11
=
, etc. Because of the rotor
symmetry there is no mutual inductance between any direct and any quadrature axis circuit.
Mutual inductances between stator and rotor circuits
From consideration of the variable coupling between rotor circuits and armature circuits as
function of rotor position, it follows that all stator-rotor mutual inductances vary sinusoidally
with angle and that they are maximum when the two coils in question are in line. Thus
)120cos(
)120cos(
cos
)120cos(
)120cos(
cos
111
111
111
o
o
o
o
+==
−==
==
+==
−==
==
θ
θ
θ
θ
θ
θ
dacddc
dabddb
daadda
afdfcdcfd
afdfbdbfd
afdfadafd
Lll
Lll
Lll
Lll
Lll
Lll
(5.13)
etc.
SYNCHRONOUS MACHINES
5-6
θ
sin
111 qaaqqa
Lll
−
==
)120sin(
)120sin(
111
111
o
o
+−==
−−==
θ
θ
qacqqc
qabqqb
Lll
Lll
etc.
Transformation of Equations
The equations of the synchronous machine can be greatly simplified by applying a certain
transformation of variables. Referring to Figure 5.1, if the mmfs produced by the three armature
currents are resolved along the direct and quadrature axes, currents proportional to the resultant
mmfs in the direct and quadrature axes can be defined as
[]
[]
)120(sin)120(sinsin
3
2
)120(cos)120(coscos
3
2
oo
oo
++−+−=
++−+=
θθθ
θθθ
cbad
cbad
iiii
iiii
(5.14)
This is the basis of the transformation originally used by R. H. Park. In defining the above
transformation, the axis of phase a was chosen as reference for convenience. Note that the effect
of the transformation is to transform the phase quantities into variables in a reference frame
which is fixed on the rotor.
The factor 2/3 is introduced, so that, for balanced phase currents of any given (maximum)
magnitude, the maximum values of i
d
and i
q
as the phase of the currents is varied will be of the
same magnitude.
Since three phase currents -- i
a
, i
b
, i
c
-- are to be transformed, three new variables are required in
order for the transformation to be reversible. A third (stationary) current, i
o
, which is
proportional to the conventional zero sequence current of symmetrical component theory, is
therefore introduced.
The transformation can therefore be expressed in matrix form as
abcdq
iTi
=
o
(5.15)
where
=
o
o
i
i
i
q
d
dq
i
and
T
+−−−−
+−
=
2
1
2
1
2
1
)120(sin)120(sinsin
)120(cos)120(coscos
3
2
oo
oo
θθθ
θθθ
(5.16)
The inverse transformation is
o
1
dqabc
iTi
−
= (5.17)
SYNCHRONOUS MACHINES
5-7
where
+−+
−−−
−
=
−
1)120(sin)120(cos
1)120(sin)120(cos
1sincos
1
oo
oo
θθ
θθ
θθ
T
(5.18)
Similar transformation can also be defined for armature flux linkages and voltages.
abcdq
ΨTΨ
=
o
(5.19)
abcdq
eTe
=
o
(5.20)
where
=
0
o
ψ
ψ
ψ
q
d
dq
Ψ
and
=
o
o
e
e
e
q
d
dq
e
Applying the transformation (5.17) to equation (5.9), we obtain
rrrdqrsr
iliTlΨ +=
−
o
1
Using the relationships (5.13), the above reduces to
rrrdqrsr
iLiLΨ
+
=
o
(5.21)
where
−
−
−
=
0
2
3
0
00
2
3
00
2
3
1
1
qa
da
afd
rs
L
L
L
L
and
rrrr
lL =
Similarly, equation (5.7) can be transformed as
rsrdqssdq
iliTlΨT +=
−−
o
1
o
1
or
rsrdqssdq
ilTiTlTΨ +=
−
o
1
o
(5.22)
After carrying out the indicated matrix operations, using the relationships (5.10) - (5.13), the
above reduces to
rsrdqssdq
iLiLΨ
+
=
oo
(5.23)
where
[
]
o
diag LLL
qdss
−
−
−
=L
SYNCHRONOUS MACHINES
5-8
ooo
2oo
2oo
2
2
3
2
3
abaa
aaabaaq
aaabaad
LLL
LLLL
LLLL
−=
−+=
++=
(5.24)
and
=
..
....
1
1
qa
daafd
sr
L
LL
L
In equation (5.23)
ψ
d
and
ψ
q
may be regarded as the flux linkages in fictitious windings moving
with the rotor and centered over the direct and quadrature axes, respectively. L
d
and L
q
are self-
inductances of these equivalent armature circuits.
In a similar manner, the armature voltage equations become
o
1
o
1
o
1
)(
dqdqdq
r
dt
d
iTΨTeT
−−−
−=
or
oo
1
o
)(
dqdqdq
r
dt
d
iΨTTe −=
−
(5.25)
After carrying out the indicated matrix operation, the above reduces to
oooo dqdqdqdq
r
dt
d
dt
d
iΨSΨe −+=
θ
(5.26)
where
S
−
=
000
001
010
We note that equation (5.26) is similar to the original equation (5.3), but with the addition of the
generated or speed voltage terms,
odq
dt
d
ΨS
θ
.
The complete set of machine performance equations now consists of the circuit voltage equations
(5.26) and (5.5), and the flux-linkage equations (5.23) and (5.21). At constant rotor speed these
equations are linear differential equations with constant coefficients, and even with variable rotor
speed they are considerably simpler than the original set of equations.
The phase quantities may be found from the transformed variables by applying the inverse
transformation as indicated in equation (5.17).
Per Unit System
In order for the equations describing the synchronous machine to be expressible in terms of an
electrical equivalent circuit the mutual inductance coefficients must be reciprocal. Inspection of
equations (5.21) and (5.23) shows that this condition is not completely satisfied in the present
form of the equations. This difficulty arises because of the particular transformation used.