Mrinal K Pal. Power system stability
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SMALL DISTURBANCE STABILITY
8-3
expression for the solution of the linearized system (eqn. 2.43 of Chapter 2), depending on the
initial state (the system state at the end of the assumed small disturbance), instability can be
caused by any mode out of several thousands in a large system. An evaluation of the complete
set of eigenvalues (modes) is therefore needed. For this purpose a large system can be reduced to
manageable size. When the interest lies locally, distant areas can be replaced by equivalents. For
inter-area oscillation studies, individual areas can be reduced in size by combining the machines
that swing in coherent groups.
Actually, by judiciously choosing the modeling details, a complete eigenvalue analysis of fairly
large systems can be handled by today’s computing facilities. To accomplish this, machines in
the immediate vicinity of the area under investigation can be represented in full detail. Machines
outside and adjacent to the area can be represented by the classical model, and the rest of the
system can be represented by equivalents.
State Space Representation of Power System
A power system can be represented by the following sets of differential and algebraic equations:
),(
),(
yxg0
y
xfx
=
=
&
(8.1)
After linearization the above can be expressed as
∆
∆
=
∆
y
x
AA
AA
0
x
43
21
&
(8.2)
From (8.2)
[
]
xAxAAAAx ∆=∆−=∆
−
3
1
421
&
(8.3)
Single machine-infinite bus system
Considering synchronous machine model 2 (see Chapter 5), the differential equations for the
machine internal voltages are given by
dqqq
d
qo
eixx
dt
ed
T
′
−
′
−=
′
′
)( (8.4)
qdddfd
q
do
eixxE
dt
ed
T
′
−
′
−−=
′
′
)( (8.5)
The machine voltage equations:
ddqqq
qqddd
ixiree
ixiree
′
−−
′
=
′
+
−
′
=
(8.6)
The equation of motion:
)(
2
o
o
2
2
o
ωω
ω
δ
ω
−−−=
d
em
K
TT
dt
dH
(8.7)
qddqqqdde
iixxieieT )(
′
−
′
+
′
+
′
=
(8.8)
SMALL DISTURBANCE STABILITY
8-4
For the transmission system (see Fig.6.4 of Chapter 6)
deqeqb
qededb
ixireV
ixireV
−−=
+
−=
δ
δ
cos
sin
(8.9)
Linearizing equations (8.4) - (8.9)
dqqq
d
qo
eixx
dt
ed
T
′
∆−∆
′
−=
′
∆
′
)(
(8.10)
qdddfd
q
do
eixxE
dt
ed
T
′
∆−∆
′
−−∆=
′
∆
′
)( (8.11)
ddqqq
qqddd
ixiree
ixiree
∆
′
−∆−
′
∆=∆
∆
′
+
∆
−
′
∆=∆
(8.12)
ω
ω
δ
ω
∆−∆−=
∆
o
2
2
o
2
d
e
K
T
dt
dH
(8.13)
[][]
qddqqdqdqdqqdd
qdqddqqqqqdddde
iixxeiixxeieie
iiiixxieieieieT
∆
′
−
′
+
′
+∆
′
−
′
+
′
+
′
∆+
′
∆=
∆
+
∆
′
−
′
+
∆
′
+
′
∆+∆
′
+
′
∆=∆
)()(
)()(
(8.14)
We will assume constant mechanical torque -- 0
=
∆
m
T (i.e., no governor action). Inclusion of
governor control is quite straightforward.
deqeqb
qededb
ixireV
ixireV
∆−∆−∆=∆−
∆
+
∆
−
∆=∆
δδ
δ
δ
sin
cos
(8.15)
Considering the simplified excitation control system used in Chapter 6 (Fig. 6.6, IEEE type 1),
ignoring saturation for the purpose of this illustration, the linearized equations can be written in
matrix form as (see Appendix C)
t
A
A
fd
FF
AF
AF
AAF
AF
fd
V
T
K
x
E
TT
TT
KK
TTT
KK
x
E
dt
d
∆
−
+
∆
∆
−
+−
=
∆
∆
0
11
1
22
(8.16)
Since
222
qdt
eeV += , ∆V
t
can be expressed as
∆
∆
=∆
q
d
t
q
t
d
t
e
e
V
e
V
e
V
(8.17)
Note that in equations (8.14) - (8.17)
tqdqdqd
Veeeeii ,,,,,,,
δ
′
′
are all values at the initial
operating point, obtained from power flow.
Equations (8.10) - (8.17) can be arranged in the form of (8.2), where
SMALL DISTURBANCE STABILITY
8-5
[]
[]
′
∆∆∆∆=∆
′
∆∆
′
∆
′
∆∆∆=∆
qdqd
fdqd
eeii
xEee
y
x
2
ωδ
−
+−
′′
−
′
−
−−−
=
FF
AF
AF
AAF
AF
dodo
qo
qd
d
TT
TT
KK
TTT
KK
TT
T
ii
H
K
11
1
11
1
2H2H2
1
oo
1
ωω
A
[][]
−−
′
′
−
−
′
′
−
′
−
′
+
′
−
′
−
′
+
′
−
=
t
q
A
A
t
d
A
A
do
dd
qo
qq
ddqqqdqd
V
e
T
K
V
e
T
K
T
xx
T
xx
ixxe
H
ixxe
H
)(
2
)(
2
oo
2
ωω
A
−
−
−
=
δ
δ
sin
cos
1
1
3
b
b
V
V
A
,
−
−−
′
′
−
=
1
1
1
1
4
ee
ee
d
q
rx
xr
rx
xr
A
from which the state model follows.
State model of the multi-machine system
In a multi-machine system using 2-axis representation the relationship between the individual
machine terminal voltages (or currents) in terms of the machine reference frame and network
reference frame is given by equations of the form of equations (5.182) and (5.183) of Chapter 5.
For a system containing n synchronous machines the voltage relationship is
SMALL DISTURBANCE STABILITY
8-6
−
−
=
MOM
2
2
1
1
22
22
1
11
2
2
1
1
sincos
cossin
sin1cos
cossin
Q
D
Q
D
q
d
q
d
e
e
e
e
e
e
e
e
δδ
δδ
δδ
δδ
or
e = TV (8.18)
Similarly,
i = TI (8.19)
Linearizing (8.18),
δMVTVTVTe
∆
+
∆
=
∆
+∆=∆
(8.20)
where
∆
∆
=∆
−
−
=
M
O
2
1
2
2
1
1
,
δ
δ
δM
d
q
d
q
e
e
e
e
From (8.20) we can also write
δNeTδMTeTV ∆+∆=∆−∆=∆
−−− 111
(8.21)
where
−
−
=−=
−
O
2
2
1
1
1
D
Q
D
Q
e
e
e
e
MTN
Similarly, from (8.19),
δMITi ∆+∆=∆ (8.22)
and
δNiTI ∆+∆=∆
−1
(8.23)
where
,
2
2
1
1
−
−
=
O
d
q
d
q
i
i
i
i
M
−
−
=−=
−
O
2
2
1
1
1
D
Q
D
Q
i
i
i
i
MTN
SMALL DISTURBANCE STABILITY
8-7
The relationship between the machine terminal voltages and currents in network reference frame,
in linearized form, is given by
VYI
∆
=
∆
N
(8.24)
where
Y
N
is the 2n × 2n network admittance matrix written in real form after eliminating the
non-machine nodes and then separating the real and imaginary parts (see Chapter 5).
Using (8.23) and (8.21), (8.24) can be written as
[
]
δNeTYδNiT ∆+∆=∆+∆
−− 11
N
from which
[
]
δNYTMeTYTi ∆++∆=∆
−
NN
1
(8.25)
Equation (8.25) can also be written as
[
]
δNYTMiTYTe ∆++∆=∆
−−− 111
NN
(8.26)
For the multi-machine case we could write the differential equations in matrix form as
iiiii
yAxAx
∆
+
∆
=∆
21
&
i = 1, 2, … n (8.27)
the machine algebraic equations (8.12) as
iiMii
eiZE0
∆
+
∆
+
∆−= i = 1, 2, … n (8.28)
where
′
′
−
=
′
∆
′
∆
=∆
idi
qii
Mi
qi
di
i
rx
xr
e
e
ZE ,
and the network equations (8.25) as
[]
eTYTiδNYTM0 ∆+∆−∆+=
−1
NN
(8.29)
Equations (8.27) - (8.29) could be arranged in the form of (8.2), from which the state model
could be obtained as shown in (8.3).
Note that most of the submatrices involved are block diagonal and sparse except for
[]
NYTM
N
+ and
1−
TYT
N
which are full. However, the method is cumbersome for large
systems. A method that is more suitable for multi-machine systems is as follows:
For a group of machines (8.27) can be written as
eCiBxAx
∆
+
∆
+
∆
=∆
1
&
(8.30)
where
∆
∆
=∆
∆
∆
=∆
∆
∆
=∆
MMM
2
1
2
1
2
1
,, e
e
ei
i
ix
x
x
The machine algebraic equations can be grouped as
eZEZi ∆−∆=∆
−− 11
MM
SMALL DISTURBANCE STABILITY
8-8
where
=
∆
∆
=∆
−
−
−
OM
1
2
1
1
1
2
1
,
M
M
M
Z
Z
ZE
E
E
and can be expressed as
eZxDi ∆−∆=∆
−1
M
(8.31)
where
D is obtained by augmenting
1−
M
Z with zeros.
Similarly, the network equations (8.25) can be expressed as
eTYTxGi ∆+∆=∆
−1
N
(8.32)
Using (8.31), (8.30) reduces to
[]
[
]
eZBCxDBAx ∆−+∆+=∆
−1
1 M
&
(8.33)
From (8.31) and (8.32)
[
]
[
]
xGDeZTYT ∆−=∆+
−− 11
MN
from which
[]
[
]
xGDTTZTYTe ∆−+=∆
−
−
−− 1
1
11
MN
(8.34)
Substituting (8.34) in (8.33) we obtain
xAx
∆
=∆
&
where
[]
[
]
[
]
GDTTZTYTZBCDBAA −+++=
−
−
−− 1
1
111-
M1
-
MN
The formation of the coefficient matrix can be further simplified by grouping the state variables
and forming the state equations for each group, and then combining them together. The method
has the advantage of requiring the least amount of computation and storage space. It also makes
it simpler to locate the various system and control parameters in the final matrix, thus facilitating
eigenvalue sensitivity analyses.
An Efficient Method of Deriving the State Model
The method will first be illustrated for a synchronous machine model where the various flux
linkages are among the state variables. The manipulation of the equations in the intermediate
stages before the final coefficient matrix is formed is somewhat simpler for this model than for
models 1, 2, or 3 (see Chapter 5).
State model using flux linkages as state variables
The relevant equations for the synchronous machine, assuming one damper winding on each
axis, are listed below.
SMALL DISTURBANCE STABILITY
8-9
Flux linkage equations:
qqqaqq
ddfddfdadd
ddffdffddadfd
qaqqqq
dadfdadddd
ixix
ixixix
ixixix
ixix
ixixix
1111
11111
11
1
1
+−=
++−=
++−=
+−=
+
+
−
=
ψ
ψ
ψ
ψ
ψ
(8.35)
Voltage equations:
qq
q
dd
d
fdfd
fd
fd
qd
q
q
dq
d
d
ir
dt
d
ir
dt
d
ir
dt
d
e
ir
dt
d
e
ir
dt
d
e
11
1
o
11
1
o
o
oo
oo
1
0
1
0
1
1
1
+=
+=
+=
−+=
−−=
ψ
ω
ψ
ω
ψ
ω
ψ
ω
ω
ψ
ω
ψ
ω
ω
ψ
ω
(8.36)
Air-gap torque:
dqqde
iiT
ψ
ψ
−
= (8.37)
The flux linkage equations in linearized form can be written in matrix form as
∆
∆
=
∆
∆
rr
s
i
i
X
Ψ
Ψ
(8.38)
where
,
∆
∆
=∆
q
d
s
ψ
ψ
Ψ
∆
∆
∆
=∆
∆
∆
=∆
∆
∆
∆
=∆
q
d
fd
r
q
d
q
d
fd
r
i
i
i
i
i
1
1
1
1
,, iiΨ
ψ
ψ
ψ
and
−
−
−
−
−
=
qaq
ddfad
dfffdad
aqq
adadd
xx
xxx
xxx
xx
xxx
11
111
1
X
SMALL DISTURBANCE STABILITY
8-10
From (8.38)
∆
∆
=
∆
∆
r
s
r
Ψ
Ψ
Y
i
i
(8.39)
where
Y is the inverse of X.
Equation (8.39) can be written as
∆
∆
=
∆
∆
r
s
r
Ψ
Ψ
YY
YY
i
i
2221
1211
For n machines the above can be split up as
∆
∆
=
∆
∆
MOM
2
1
2
1
1
1
2
1
Ψ
Ψ
Y
Y
i
i
where
[
]
iii
12111
YYY = and
∆
∆
=∆
i
r
i
s
i
Ψ
Ψ
Ψ
i = 1, 2, …, n
or
ΨPi
∆
=
∆
(8.40)
Similarly,
ΨQi
∆
=
∆
r
(8.41)
where
=
O
2
2
1
2
Y
Y
Q and
[
]
iii
22212
YYY = i = 1, 2, …, n
The machine voltage equations (8.36), after linearization and assuming
o
ω
ω
≈ , can be written in
matrix form as
∆
∆
∆
∆
∆
−
−
−
+
∆
∆
∆
∆
∆
=
∆
∆
∆
∆
∆
q
d
fd
q
d
q
d
fd
q
d
fd
q
d
q
d
fd
q
d
i
i
i
i
i
r
r
r
r
r
1
1
1o
1o
o
o
o
1
1
o
o
1
1
-
ω
ω
ω
ω
ω
ψ
ψ
ψ
ψ
ψ
ω
ω
ψ
ψ
ψ
ψ
ψ
&
&
&
&
&
fd
doad
ffd
q
d
E
Tx
x
e
e
∆
′
+
∆
∆
+
o
o
ω
ω
SMALL DISTURBANCE STABILITY
8-11
In the above we have used the relationships
fd
fd
ad
fd
e
r
x
E = and
fd
ffd
do
r
x
T
o
ω
=
′
.
Using (8.39) the above can be written as
ex
xReWΨSΨ ∆+∆+∆=∆
&
(8.42)
where
∆
∆
=∆
′
=
2
,
00
00
0
00
00
x
E
Tx
x
fd
ex
doad
ffd
xR
For n machines,
=
=
=
OOO
2
1
2
1
2
1
,, R
R
RW
W
WS
S
S
Using (8.26) and (8.40), (8.42) becomes
[
]
exNN
xRδNYTMWΨPTYTWΨSΨ ∆+∆++∆+∆=∆
−−− 111
&
ex
xRδCΨB
∆
+∆+∆= (8.43)
where
PTYTWSB
11 −−
+=
N
,
[
]
NYTMWC
1−
+=
N
The equation of motion in linearized form, with 0
=
∆
m
T , is given by (8.13), where
qddqdqqde
iiiiT
ψ
ψ
ψ
ψ
∆
−
∆
+
∆
−∆=∆
For n machines the equation of motion can be expressed as
ωδ ∆=∆
&
(8.44)
ωKiΨΨiω
∆
+
∆
+
∆=∆
d
&
(8.45)
where
−
−
=
∆
∆
=∆
O
M
000
2H2H
000
2H2H
,
2
2
o
2
2
o
1
1
o
1
1
o
2
1
dq
dq
ii
ii
ωω
ωω
ω
ω
iω
SMALL DISTURBANCE STABILITY
8-12
−
=
O
2
2
o
2
2
o
1
1
o
1
1
o
2H2H
2H2H
dq
dq
ψ
ω
ψ
ω
ψ
ω
ψ
ω
Ψ
−−= L
2
2
1
1
22
diag
H
K
H
K
dd
d
K
Using (8.40), (8.45) becomes
ωKΨPΨΨiω
∆
+
∆
+∆=∆
d
&
ωKΨF
∆
+
∆=
d
(8.46)
where
PΨiF +=
From (8.16) and (8.17) the linearized equations for the excitation control is
eLxKx
∆
+
∆=∆
exexex
&
(8.47)
where
−
+−
=
FF
AF
AF
AAF
AF
ex
TT
TT
KK
TTT
KK
11
1
K
−−
=
00
t
q
A
A
t
d
A
A
V
e
T
K
V
e
T
K
L
For
n machine system
=
=
OO
2
1
2
1
, L
L
LK
K
K
ex
ex
ex
Using (8.26) and (8.40), (8.47) becomes
[
]
δNYTMLiTYTLxKx ∆++∆+∆=∆
−−− 111
NNexexex
&
δHΨGxK
∆
+
∆
+∆=
exex
(8.48)
where
PTYTLG
11 −−
=
N
and
[
]
NYTMLH
1−
+=
N