20
CHAPTER 2
DYNAMIC STABILITY MODEL OF THE POWER SYSTEM
2.1
GENERAL
Dynamic stability of a power system is concerned with the dynamic
behavior of the system under small perturbations around an operating
condition and more specifically it is a phenomena of slow and poorly damped
or sustained or even diverging power oscillations which are essentially due to
varying system loads and ill controlled controllers of the system. Computer
analysis of this problem requires mathematical models which simulate as
accurately as the behavior of physical system but at the same time not very
complex to handle.
This chapter presents the development of the mathematical model
for the dynamic stability analysis of Single Machine Infinite Bus System
(SMIB) and Multi- machine power system.
2.2
SIMPLIFIED LINEAR MODEL FOR SMIB SYSTEM
In stability analysis, the mathematical model is used for dynamic
analysis of power systems.
2.2.1
Assumptions
The following assumptions are made for the development of
simplified linear model of SMIB system:
21
1. Damper windings both in the d and q axes are neglected.
2. Armature resistance of the machine is neglected.
3. Excitation system is represented by a single time constant.
4. Balanced conditions are assumed and saturation effects are
neglected.
2.2 2
Classical Machine Model
In the classical methods of analysis, the simplified model or
classical model of the generator is used (Kundur 1994). Here, the machine is
modeled by an equivalent voltage source behind impedance connected to an
infinite-bus as shown in Figure 2.1.
Infinite Bus
An infinite bus is a source of invariable frequency and voltage
(both in magnitude and angle). A major bus of a power system of very large
capacity compared to the rating of the machine under consideration is
approximately an infinite bus.
xe
Figure 2.1 One-line diagram of SMIB
The state space classical machine model is shown in Figure 2.2.
22
Ks
Figure 2.2 Classical Machine Model
The state equations of the classical model are given in equation (2.1):
p( ∆ω) =
1 
∆T − KS ∆δ − D∆ω

2H  m
p(∆δ) = ω 0 ∆ω
(2.1)
State vector x T = ( ∆ω ∆δ)
(2.2)
And when the effect of flux linkage is included, three states are used to model
the generator: ∆ω, ∆δ and ∆Eq'. The state equations are given in equation (2.3)
as follows:
p∆ω =
∆Tm K1
K
D
- ∆δ- 2 ∆E 'q - ∆ω
τj τj
τj
τj
p( ∆δ) = ω 0 ∆ω
p∆E 'q =
where
-1
1
K
∆E 'q + ' ∆E FD - ' 4 ∆δ
'
K 3 τ do
τ do
τ do
τj = 2H, H is the inertia constant.
(2.3)
23
State Vector of the SMIB system including the effect of flux
linkage is given by equation (2.4)
x t = [∆ω ∆δ ∆E 'q ]
(2.4)
where the variables are
Eq' - quadrature axis component of voltage behind transient
reactance
ω
- angular velocity of rotor
δ
- rotor angle in radians
K1 to K6 is the Heffron Philips constants (Padiyar 2002).
2.3
EXCITATION SYSTEM REPRESENTATION
The excitation system model considered is the simplified form of
ST1A model shown in Figure 2.3. A high exciter gain, without derivative
feedback, is used. By inspection of Figure 2.3, the state space equations can
be written as,
∆V1
∆Vt
KA
1 + sTA
∆EFD
Figure 2.3 Excitation System Representation
p∆EFD = -1/TA (KA ∆Vt +∆ EFD)
with TR is neglected, Vref =constant.
And ∆Vt
=
K5 ∆δ + K6 ∆Eq'
(2.5)
24
where
EFD - Equivalent stator emf proportional to field voltage
KA - Gain of the Exciter
TA - Time constant of the exciter
TR - Terminal Voltage Transuder Time Constant
Vt
- Terminal voltage of the Synchronous machine
Vref - Reference voltage of the Synchronous machine
Combining the Equations (2.3) with the exciter equation (2.5), the
complete state space description of SMIB system including exciter is given in
equation (2.6).
p∆ω =
∆Tm K1
K
D
- ∆δ- 2 ∆E 'q - ∆ω
τj τj
τj
τj
p∆δ = ω0 ∆ω
p∆E 'q =
-1
1
K
∆E 'q + ' ∆E FD - ' 4 ∆δ
'
K 3 τ do
τ do
τ do
pEFD = -1/TA (KA K5 ∆δ + KA K6 ∆Eq' + EFD)
(2.6)
The state vector is thus defined by Equation (2.7):
x t = [∆ω ∆δ ∆E 'q ∆E FD ]
2.4
(2.7)
SMIB SYSTEM REPRESENTATION WITH CPSS
The block diagram of the CPSS is shown in Figure 2.4. The state
equations for the same can be written as follows.
p∆V2 = KPSS p∆ω – (1/Tw) ∆V2
(2.8)
p∆Vs = (T1/T2) p∆V2 + (1/T2) ∆V2 – (1/T2) ∆ Vs
(2.9)
25
where
KPSS
-
CPSS gain
T1, T2
-
Phase compensator time constants
Tw
-
Wash out time constant
CPSS
∆Vs
∆Vt
Figure 2.4 CPSS Representation
State vector of the synchronous machine model including PSS is
given by equation (2.10):
x T = [∆ω ∆δ ∆E q ' ∆E FD ∆V2 ∆Vs ]
(2.10)
The block diagram of simplified linear model of a synchronous
machine connected to an infinite bus with exciter and PSS is shown in
Figure 2.5.
∆Vt
Conventional Power System Stabilizer
Figure 2.5 State Space Model of SMIB system representation with CPSS
26
2.5
DYNAMIC STABILITY MODEL OF MULTI MACHINE
POWER SYSTEM
In stability analysis of a multi-machine system, modelling of all the
machines in a more detailed manner is exceedingly complex in view of the
large number of synchronous machines to be simulated. Therefore simplifying
assumptions and approximations are usually made in modelling the system.
In this thesis two axis model is used for all machines in the sample system
taken for investigation.
2.5.1
Assumptions Made
In this work the synchronous machine is modeled using the twoaxis model (Anderson and Fouad 2003). In the two-axis model the transient
effects are accounted for, while the sub transient effects are neglected. The
transient effects are dominated by the rotor circuits, which are the field circuit
in d-axis and an equivalent circuit in the q-axis formed by the solid rotor. The
amortisseur winding effects are neglected. An additional assumption made in
this model is that in stator voltage equations the terms pλ and pλ q are
d
negligible compared to the speed voltage terms and that ω ≅ ω =1p.u. The
R
block diagram representation of the synchronous machine in two-axis model
is shown in Figure 2.6.
Figure 2.6 Block diagram representation of two axis model for synchronous machine
27
28
2.5.2
Synchronous Machine Representation
Using the block diagram reduction technique and with the
simplifying assumptions the state equations for the two-axis model in p.u.
form
pEd'
=
{-Ed' - (xq-xq') Iq} / τqo'
pEq'
=
{EFD - Eq' - ( xd - xd' ) Id } / τdo'
pω
=
{Tm - Dω - Te } / τj
pδ
=
ω-1
(2.11)
where the state variables are
Ed' -
direct axis component of voltage behind transient reactance
Eq' -
quadrature axis component of voltage behind transient
reactance
ω
-
angular velocity of rotor
δ
-
rotor angle in radians
and
Te =
Ed'Id + Eq'Iq – (xq' – xd' ) Id Iq
τj
4πfH
=
xd
-
direct axis synchronous reactance
xq
-
quadrature axis synchronous reactance
xd '
-
direct axis transient reactance
xq '
-
quadrature axis transient reactance
τdo' -
direct axis open circuit time constant
29
τqo' -
quadrature axis open circuit time constant
Te
-
electrical torque of synchronous machine
Tm
-
mechanical torque of synchronous machine
D
-
damping coefficient of synchronous machine
EFD -
Equivalent stator emf corresponding to field voltage
Iq
-
quadrature axis armature current
Id
-
direct axis armature current
H
-
inertia constant of synchronous machine in sec
f
-
frequency in Hz
A multi-machine power system is shown in Figure 2.7 and the
network has n machines and r loads. The active source nodal voltages in
Figure 2.7 are taken as the terminal voltages Vi , i = 1.2….n instead of the
internal EMF`s. The loads are represented by constant impedances and the
network has n active sources representing the synchronous machines.
Figure 2.7 Multi-machine with constant impedance loads
30
This network is reduced to a n-node network shown in
Figure 2.8 in which the current and voltage phases of each node are expressed
in terms of the respective machine reference frame.
Figure 2.8 Reduced n-port network
The objective here is to derive relations between vdi and
vqi, i=1,2,….n, and the state variables. This will be obtained in the form of a
relation between these voltages, the machine currents iqi and idi , and the
angles δi , i=1,2,….n. For convenience we will use a complex notation as
follows.
For a machine i we define the phasors Vi and Ii as
V = V + jV
; I = I + jI
i
qi
di
i qi
di
(2.12)
31
where
V =v / 3 : V =v / 3
qi
qi
di
di
I
qi
=i
qi
/ 3 :I
di
=i
di
/ 3
and where the axis qi is taken as the phasor reference in each case. Then we
define the complex vectors V and I by
 V   Vq1 + jVd1 

 

 V2   Vq 2 + jVd 2 
V= 
=

...

 ............. 
 V   Vqn + jVdn 

 n 
 I1   I q1 + jI d1 
  

 I 2   I q 2 + jI d 2 
I = =

 ...  ............. 
 I   I qn + jI dn 

 n 
(2.13)
(2.14)
The voltage Vi and the current Ii are referred to the q and d axes of
machine i. In the other words the different voltages and currents are expressed
in terms of different reference. To obtain general network relationships, it is
desirable to express the various branch quantities to the same reference which
is given by equation (2.15):
ˆ and ˆI ,
The node voltages and currents are expressed as V
i
i
i = 1,2,…..n, and
ˆI = YV
ˆ
where
Y
is the short circuit admittance of the network.
(2.15)
32
2.5.3
Converting to Common Reference Frame
Let us assume that we want to convert the phasor V i = Vqi + jVdi to
the common reference frame (moving at synchronous speed). Let the same
voltage, expressed in new notation, be V̂i = VQi + jVDi as shown in Figure 2.9.
where,
V i = Vqi + jVdi
and
V̂i = VQi + jVDi
(2.16)
Dref
VDi
V i = Vˆi
di
qi
Vqi
Vdi
δi
δi
Qref
VQi
Figure 2.9 Two frames of reference for phasor quantities
From the Figure 2.9
VQi = [Vqi cos δ i − Vdi cos δ i ]
(2.17)
VDi = [Vdi cos δ i − Vqi cos δ i ]
(2.18)
VQi + VDi = (Vqi cos δi − Vdi cos δi ) + (Vqi sin δi + Vdi cos δi )
(2.19)
V̂i = Vi e jδi
(2.20)
33
The equation (2.20) can be written in generalized matrix form as below
 jδ1
0
 V + jV
 e
Q1
D1

 
jδ 2
e
 V + jV   ⋅
D2  =
 Q2
 ⋅
⋅
.............
 

  ⋅
⋅
 V + jV  
Dn 
 Qn
0
 0
e jδ1
0

jδ 2

e
 ⋅
T = ⋅
⋅

⋅
 ⋅

0
 0
0   V + jV 
 q1
d1 

⋅ ⋅
0   V + jV 
q2
d2 
0 
⋅ ⋅

 .............
0 
⋅ ⋅

V
jV
+


jδn   qn
dn 
0 0 e

⋅
0 


⋅ ⋅
0 
⋅ ⋅
0 

⋅ ⋅
0 
jδn 
0 0 e

⋅
⋅
(2.21)
⋅
(2.22)
The equation (2.20) can be written as
V̂ = TV
(2.23)
Thus T is a transformation that transforms the d and q quantities of all
machines to the system frame, which a common frame is moving at
synchronous speed. The transformation matrix T contains elements only at the
leading diagonal and hence we can show that T is orthogonal, i.e. T-1 = T*.
Now the equation (2.23) can be rewritten as
ˆ
V=T*V
(2.24)
Similarly for node current
Î = TI
(2.25)
34
I = T* ˆI
(2.26)
Substituting equation (2.25) and equation (2.23) in equation (2.15),
we get
where
I = MV
(2.27)
M = T-1YT
(2.28)
Linearizing equation (2.27) and making necessary substitutions
(Anderson and Fouad 2003), the following equations are obtained.
∆Iqi = Gii ∆Vqi – Bii ∆Vdi +
-
n
∑ [ Yij sin
j=1
≠i
n
∑ [ Yii
j=1
≠i
(θij - δij0 ) ∆Vdj] +
cos(θij - δij0 ) ∆Vqj]
n
∑ [ Yij
j=1
≠i
{sin(θij - δij0) Vqj0 + cos(θij - δij0)Vdj0]∆δij
; i = 1, ...n
∆Idi = Bii ∆Vqi + Gii ∆Vdi +
+
n
∑ [ Yij sin
j=1
≠i
n
∑ [ Yij
j=1
≠i
(θij - δij0 ) ∆Vqj] +
(2.29)
cos (θij - δij0 ) ∆Vdj]
n
∑ [ Yij
j=1
≠i
{sin (θij - δij0 )Vdj0 - cos(θij - δij0)Vqj0]∆δij
; i = 1, ...n
(2.30)
The state space model for linearized system is obtained by
linearizing the differential and algebraic equations at an operating point.
While doing this linearization process, additional terms involving terminal
voltage components (which are not state variables) remain in the differential
35
equations. To express the voltage components in terms of state variables, the
machine currents are also linearized and expressed in terms of state variables
and voltage components. Finally the current components are eliminated using
the interconnecting network algebraic equations. From the initial conditions,
Ed'i0, Eq'i0, Iqi0, Idi0, EFDi0 and δi0 are determined.
Linearizing equation (2.11) we get
p∆Ed'i = {- ∆Ed'i - (xqi - xq'i) ∆Iqi } / τqo'i ; i = 1,...n
p∆Eq' i = {∆EFDi - Eq'i + ( xdi - xd'i ) ∆Idi } / τdo'i ; i = 1,...n
p∆ωi
= {∆Tmi –(Idi0 ∆Ed'i + Iqi0 ∆Eq'i + Ed'i0 ∆Idi +Eq'i0 ∆Iqi)Diωi } / τj ; i = 1,...n
p∆δi
= ∆ωi ; i = 1, …n
(2.31)
Substituting equations (2.29) and (2.30) in equation (2.31).
(replacing V by E'):
p∆Ed’i =
1
τqo '
i
{[(xqi - xq'i) Bii –1] ∆Ed'i
+ (xqi - xq'i)
- (xqi - xq'i)
- (xqi - xq'i)
n
∑ [ Yik
k=1
≠i
n
∑ [ Yik
k=1
≠i
n
∑ [ Yik
k=1
≠i
{sin (θik - δik0 ) ∆Ed'k -(xqi - xq'i) Gii ∆Eq'i
cos (θik - δik0 ) ] ∆Eq'k
cos(θik-δik0)] Ed'k0 +Yik sin ((θik - δik0) Ed'k0] ∆δik}
i =1,2 ….n
(2.32)
36
p∆Eq' i =
1
τ
{[(xdi – xd'i) Bii –1] ∆Eq'i
'
do i
+ (xdi – xd’i)
+ (xdi – xd'i)
- (xdi–xd'i)
n
∑ [ Yik
k=1
≠i
n
∑ [ Yik
k=1
≠i
n
∑ [ Yik
k=1
≠i
{sin (θik - δik0 )] ∆Ed'k + (xdi – xd'i) Gii ∆Ed'i
sin (θik - δik0 ) ] ∆Eq'k
cos(θik-δik0) Eq'k0-Yik sin((θik-δik0)Ed'k0]∆δik+ ∆EFDi)}
i =1,2 ….n
p∆ωi =
(2.33)
1
{[∆Tmi - Di∆ωi -[Idi0 + GiiEd'i0 - Bii Eq'i0] ∆Ed'i
τ
ji
- [ Iqi0 + Bii Ed'i0 + Gii Eq'i0 ] ∆Eq'i
-
-
-
n
∑ [ Yik
k=1
≠i
n
∑ [ Yik
k=1
≠i
n
∑ [ Yik
k=1
≠i
cos (θik - δik0 ) Ed'i0 - Yik sin (θik - δik0 ) Eq'i0 ] ∆Ed'k
sin (θik - δik0 ) Ed'i0 + Yik cos (θik - δik0 ) Eq'i0 ] ∆Eq'k
cos (θik - δik0 ) (-Eq'k0Ed'i0 +Ed'k0 Eq'i0) + Yik sin ((θik - δik0)
(-Ed'k0Ed'i0 +Eq'k0 Eq'i0)∆δik}
i =1,2 ….n
p∆δ1i = ω1 - ωi
i = 2,3 ….n taking machine 1 as reference.
(2.34)
(2.35)
37
The above set of equations (2.32 to 2.35) gives the state space
model of n-machine system.
2.6
EXCITER REPRESENTATION
The state space equation of the exciter can be derived from the
block diagram of the exciter shown in the Figure 2.3.
From the Figure 2.3, we get
∆E FD =
−K A
∆Vt
1 + STA
(2.36)
For n, number of exciters, the state equations is as follows:
p ∆E fdi =
-K Ai
1
-∆VRefi +∆Vi ∆E fdi ; i=1,…n
TAi
TAi
(
)
(2.37)
Now the state vector of the n machine state model including exciter
equation is as follows.
XTi = [∆Ed'i
2.7
∆Eq' i ∆ω i ∆δ i ∆EFD i] ; i=1,…n
CONVENTIONAL
POWER
SYSTEM
(2.38)
STABILIZER
REPRESENTATION
The Conventional Power System Stabilizer (PSS) adds damping to
the generator rotor oscillations by controlling its excitation using auxiliary
stabilizing signals. To provide damping, the stabilizer must produce a
component of electrical torque in phase with the rotor speed deviations.
38
The important blocks in a power system stabilizer are:
• Washout circuit.
• Phase compensator.
• Stabilizer gain.
The state space equation for the power system stabilizer (PSS) can be
obtained from the block diagram shown in Figure 2.10.
Figure 2.10 Conventional Power System Stabilizer Structure (CPSS)
From the wash out block, we get
∆V2 =
sTw
(K PSS ∆ω)
1 + sTw
p∆V2i = KPSSi p∆ωi – (1/Twi) ∆V2i
(2.39)
; i = 1,...n
(2.40)
From the phase compensator block we get
 1 + sT 
∆V = ∆V 

1 + sT 
1
s
2
(2.41)
2
From equation (2.41) we get
p∆Vsi = (T1i/T2i) p∆V2i +(1/T2i) ∆V2i–(1/T2i) ∆Vsi ; i = 1,...n (2.42)
39
The state vector of the complete system after the inclusion of power
system stabilizer is as follows:
xTi = [∆E'di ∆E'qi ∆ωi ∆δi ∆EFDi ∆V2i ∆Vsi] ; i=1,…n
2.8
(2.43)
FUZZY LOGIC BASED POWER SYSTEM STABILIZER
(FPSS)
Figure 2.11 shows the schematic block diagram of the system with
FPSS.
∆ω
FPSS
d
dt
∆Vs
+
∆Vt
-
+∆Vref
∆ω
Power System
∆ω
Generator and
Exciter
Figure 2.11 Structure of the Power system with FPSS
Speed Deviation of the synchronous machine (∆ω) and its deviation
•
(∆ ω) are chosen as inputs to the FPSS. Simulation of the sample SMIB
system without PSS is carried out for several operating conditions and
different disturbances and the inputs are normalized using their estimated
peak values. Seven labels are taken for both the inputs and output. The labels
are LP (large positive), MP (medium positive), SP (small positive), VS (very
small), SN (small negative), MN (medium negative) and LN (Large negative).
Linear triangular membership function is used in the design of FPSS. In our
design of FPSS, the fuzzy sets with triangular membership function for ∆ω
•
are shown in Figure 2.12. The membership function for ∆ ω and Vs are
similar to the above Figure 2.12.
40
MN
LN
-1
-0.66
SN
VS
SP
MP
LP
-0.33
0
0.33
0.66
1
Figure 2.12 Triangular membership function of ∆ω
Table 2.1 shows the rules of fuzzy logic based PSS (Lakshmi and
Khan 2000).
Table 2.1 Rule Table of fuzzy logic PSS
•
∆ω
∆ω
LP
MP
SP
VS
SN
MN
LN
LP
LP
LP
LP
LP
MP
SP
VS
MP
LP
LP
MP
MP
SP
VS
SN
SP
LP
MP
SP
SP
VS
SN
MN
VS
MP
MP
SP
VS
SN
MN
N
SN
MP
SP
VS
SN
SN
MN
LN
MN
SP
VS
SN
MN
MN
LN
LN
LN
VS
SN
MN
LN
LN
LN
LN
41
2.9
CONCLUSION
Mathematical model of SMIB system for dynamic stability analysis
is presented in this chapter. Various state variables with PSS, system matrix
including static exciter and CPSS are included in this chapter. Block diagram
of simplified linear model of SMIB including exciter and CPSS is also neatly
presented in this chapter. Non –linear mathematical model representing the
dynamics of the multi machine power system combining the synchronous
machine model, excitation system (IEEE Type ST 1A), with conventional
power system stabilizers are described in this chapter. The fuzzy logic based
PSS model is also described.