European Journal of Scientific Research - EuroJournals

928 Samuel N. Ndubisi and Marcel .U. Agu
3.4. Linear model of SMIB
Figure 3: A single machine infinite bus power system.
By linearizing the above equations about the operating point, we have the state variable model of a
single machine to infinite bus (SMIB) as follows
X � = AX
+ BU
(14)
y = CX
(15)
Where the state variable x is defined by
X = [ Δδ
, Δω,
ΔE′
q ]
In the above system matrix A, B, and C are represented by
(16)
⎡
⎢
⎢
0
1
⎢ ω0
V ′ s E q cosδ
D
A = ⎢−
−
⎢ 2H
X ′ ds 2H
⎢ ( X ′ d − X d ) Vs
sinδ
⎢−
0
⎢⎣
T ′ ′
d 0 X ds
⎤
⎥
0
⎥
ω0
V sin ⎥
s δ
−
⎥
2H
X ′ ds ⎥
1 ( X − ′ ) 1 ⎥
d X d
−
⎥
T ′ ′ ′
d 0 Td
0 X d 0 ⎥⎦
⎡ ⎤
⎢
0
⎥
⎢ ⎥
B = ⎢ 0 ⎥
⎢ ⎥
⎢
K c ⎥
⎢⎣
T ′ ⎥ d 0 ⎦
⎡ E′
q − X d I d
C = ⎢
⎢⎣
Vt
− X ′ d Vs
sinδ
+
X ′ ds
2
X ′ d I q
Vt
⎛ Vs
⎞
⎜
⎟
⎟cosδ
⎝ X ′ ds ⎠
0
E′
− ′ q X d I
Vt
where sub index 0 shows that matrices are calculated at operating point.
( )( ) ( ) ( )
3.5. Simplified voltage and reactive power control loop model (AVR)
0
d
⎛ X ′ ⎤
d ⎞
⎜
⎜1-
⎟ ⎥
⎝ X ′ ds ⎠ ⎥⎦
It is observed from [15] that in an interconnected power system, that the control is made up of load
frequency control loop (LFC) and automatic voltage control loop (AVR). The LFC loop controls
changes in real power due to changes in rotor angle while the AVR loop controls the changes in
reactive power which depends on excitation voltage magnitude. Because the excitation time constant is
much smaller than the prime mover time constant, its transient decay much faster. For this, the crosscompiling
between the LFC loop and AVR loop is negligible and hence the load frequency and
excitation voltage control can be analyzed independently (see fig 4).
0