State Space Model Generation For Stability Analysis Of Power System - repository civitas UGM
Proceedings of EEE FWS 2007.
PW-8 November 22-23, 2007, Montien Hotel, Bangkok, Thailand.
State Space Model Generation
for Stability Analysis of Power Systems
Avrin Nur Widiastuti
Electrical Dept., Engineering Fac.Gadjah Mada University, Jl Grafika 2 Yogyakarta 55281, Indonesia
Tel./Fax.: 62-274 - 5552305 HP: 62-85228698528 e-mail:[email protected], [email protected]
- – This study develops a program to
Abstract
- Synchronous generator and the associated
generate linearized state space representation of the
excitation systems
power systems. The two-axis model of synchronous
- AC transmission network including static
generator and the IEEE type I model of exciter are
fixed-impedance, loads
used to represent dynamic behaviors of the
For system stability studies concerning
generating units. The developed program can
electromechanical oscillation, it is appropriate to
accommodate an actual structure of the power
neglect the transmission network and the machine
systems with some numbers of generating units and
stator transients (2). The dynamics of synchronous the AC buses. generator rotor circuits and excitation systems are
The popular Western System Coordinating
represented by the sets of differential equations. The
Council (WSCC) 3-machine, 9-bus is used for
result is that the complete system model consists of
validation of the obtained eigenvalues of the
a large number of non-linear ordinary differential
linearized system. It is found that the program can and algebraic equations. function well. The results can then be used for
We can describe complex power system as our
analyzing small signal stability of the system, and
plant which consists of generators, transmission for control design, thereafter. network as depicted in Fig. 1. It is noted that there can be a number of generators in the system.
Keywords: linearization, state space ,modeling, Network including load power system generator generator
I. INTRODUCTION
2.1 Synchronous Generator Model
The analysis of dynamic stability can be We will use a simplified generator model performed by deriving the linearized state space with two axes adopted from [8] as show in Fig. 2 π model of the system in the following form : j ( − ) δ
j ′
X d R ( s d q D Q
I jI + + x = Ax Bu (2) 2
- I jI ) e =
where the matrices A and B also depend on the given operating conditions. The eigenvalues of the ( − ) π V jV e Ve 2 θ j V jV
- j δ
- ( ) = = d q D Q
state matrix A determine the small-signal stability E X X I j E e j ( ) δ − π 2 around the given operating point. The eigenvalues [ ′ ( ′ − ′ ) ′ ] d q d q q + + analysis can be used not only for the determination of the stability regions, but also for the design of the
Figure 2 Two-axis synchronous machine model. controllers [1].
Defines : In this study we develop the program to
X ′ is the d-axis transient synchronous reactance d
generate linearized state space model of
X ′ is the q-axis transient synchronous reactance q
multimachine power systems that is necessary for the study of both small signal stability analysis, R is the armature resistance
s voltage stability and for control design, thereafter.
I is the d-axis current refer to the mach. ref. frame d
I Is the q-axis current refer to the mach. ref. frame q
II. POWER SYSTEM DYNAMIC MODEL
The overall power system representation in this
I is the d-axis current refer to the netw. ref. frame D
thesis includes models for the following components
I is the q-axis current refer to the netw. ref. frame
Q
′ ′ − ′ − = ′
E R fd E
R
E K i Rfi
fdi Fi Fi F fi
τ τ τ τ τ (13)
=
A A R R − + − +
−
A fd F A F A f A
V V ref A
V V K E K K R K
) (
(12)
τ ) (
V E E S K E +
E fd E fd T
Figure 3 Block diagram of the IEEE type I excitation control. From the block diagram above, we have three following equations :
τ
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + F F s sK τ 1 V ref V − +
−
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + A A s K τ 1 R V + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + E E s K τ E S 1 ) ( E fd − fd E- =
- − =
- ′ + +
In order to improve the damping characteristics excitation controls have been widely employed. In this study, we use the IEEE type I exciter as shown in Fig. 3 [8,9].
D s D T ω ω − = (11)
′
I E T ) ( ′ −
I E
X X
I I
( ) q d d q d d q q e
2 Δ = (9)
ω
s H M
(8) where
2 ) ( τ
fd E EX B
EX E
(14) where
Proceedings of EEE FWS 2007. November 22-23, 2007, Montien Hotel, Bangkok, Thailand.
A e S
× ⋅ = (15)
Defines :
fd E
is the exciter voltage
R
V F R
is the stabilizer voltage
PW-8
- − − − ′
A
is the rotating exctr saturation at ceiling voltg
A K
is the voltage regulator gain
The network characteristics in this study described in the form of power balance. The network can consist of many electrical buses and branches depending on the system of interest. Some of the buses connect to the generators and called generator buses. The rest of the buses which are not connected to generating units called as load bus.
2.3 AC Transmission Network
is the saturation const for rotating exciters
EX B
is the saturation const for rotating exciters
EX A
EX S
τ is the voltage regulator time constant
τ is the regulator stabilizing circ time constant
F
is the regulator stabilizing circuit gain
F K
τ is the exciter time constant
τ
is the exciter gain
E K
E
X E
Multiplying eq. (2) by
H is the inertia constant
I X
j q q d q d E e j
π δ
= ′ + ′ − ′ + ′ − −
2 (
] ) ( [ )
I X j R Ve
j q d d s j e jI
−
δ θ
) )( ( π
2 (
From the above figure, the stator algebraic equation can be written as : )
T is the damping torque component D is the damping factor
I X
T is the electromagnetic torque developed D
M T is the mechanical torque applied e
ω is the rated value of an angular velocity of rotor
E is the DC field voltage s
E′ is the d-axis transient voltage fd
E′ is the q-axis transient voltage d
is the angular velocity of rotor q
δ is the angular position of the rotor ω
X is the q-axis synchronous reactance
X is the d-axis synchronous reactance q
′ is the d-axis transient open circuit time constant d
τ
′ is the q-axis transient open circuit time constant d
X E (2)
)
q
(5)
E
( ) ) ( q d q q q d
= ′ (7)
τ ′ ′ − ′ − −
X E E
I X
E
d q d d d fd q
) (
( )
= ω (6)
) ( − −
M T T T D e M
ω ω δ − =
2 (
s
In addition, the dynamic behaviors of the rotor motion and the rotor circuits can be describing using the following differential equations :
V E θ δ (4)
I R
I X
d d q s q
′
V E θ δ (3) ) cos( =
I R
I X
) sin( = + − − − q q d s d
real and the imaginary parts, we obtain :
e and equating the
π δ − − j
τ
- ′ = (10) ) (
- ′
2.2 Excitation System
PW-8
∂ ∂ ∂ = r n n r u f u f u f u f
A , which reveal the stability of system.
After the linearized state space representation of the system is obtained such as eq.21, we can see the eigenvalues of the system by inspection of matrix
3.2 Eigenvalue and Stability
(22)
...
1 ... ... ...
1
1
B
∂ ∂ ∂ ∂
For a complex pair of eigenvalues ω σ λ j
⎢ ⎢ ⎢ ⎢ ⎣ ⎡
⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢
...
1 ... ... ...
1
1
1
A
= n n n n x f x f x f x f
∂ ∂ ∂ ∂
A negative real eigenvalue represents a decaying mode. The larger its magnitude, the faster the decay (system stable), while a positive real eigenvalue represent instability
± =
⎢ ⎢ ⎢ ⎢ ⎣ ⎡
1 (25)
November 22-23, 2007, Montien Hotel, Bangkok, Thailand.
The formulation of the state equations involves the development of linearized equation about an operating point and elimination of all non state and non input variables. However, the need to allow for the representation of extensive transmission network, loads and excitation system, makes the process very complex [2]. Therefore, the Proceedings of EEE FWS 2007.
3.4 Linearized State Space Model of Power Systems
eigenvalue A further normalization can be done by making the largest of the participation factors equal to unity.
th i
and right eigenvector associated with the
w and ki v are the th k entries in the left
p is the participation factor relating the th k state variable to the th i eigenvalue
Where
∑ = = n k ik ki ik ki ki w v w v p
the frequency of oscillation in Hz is given by : π
ki p is defined as
Participation factor analysis aids in the identification of how each state variable is reflected on a given mode or eigenvalue. Participation factor
3.3 Participation Factor
= (24)
σ ζ
2 ω σ
2
In addition, the damping ratio can be computed as :
2 = f (23)
ω
∂ ∂ ∂ ∂
⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢
1 = − − −
) sin( ) cos( i i i qi i i i di
) cos( ) (
,..., m i 2 , 1 =
for
1 ) sin( α θ θ (17)
V V
Y
= = − − − n k ik k i ik k i
V I θ δ θ δ − + − ∑
V I
For reactive power balance,
∑ = ik k n k Li i ik k i i
1 ) cos( α θ θ (16)
V V
Y
= − − − n k ik k i ik k i
∑ =
V I θ δ θ δ − + −
V I
) cos( ) sin( i i i qi i i i di
The two equations representing real and reactive power balance for each generator bus are given below. For real power balance,
- Load buses For the load buses, we have equations for both real and reactive power also. For real power balance,
Y
- −
- =
- ki
Li P and
- ki
) , ( u x f x = (20)
The behavior of a dynamic system, such as a power system, may be described by a set of n first order nonlinear ordinary differential equations of the following form :
3.1 Linearized State Space Model
III. LINEARIZED STATE SPACE MODEL OF POWER SYSTEMS
This calculation will be used for linearization process which will be the topic of discussion in the next chapter.
It is necessary to compute the initial values of all dynamic states under the given inputs. In power system dynamic analysis, the fixed inputs and initial conditions are normally found from a base case load flow solution, assuming that such load-flow solution exists.
2.4 Initial Condition
this study, loads are assumed as constant power type.
Li Q and are voltage dependence loads. In
If equations (20) is linearized around the given operating states and inputs ) , ( u x , then the linearized state equation can be written as follows : where
V V
,... n m i
For
V Q α θ θ (19)
V V
Y
∑ = ik k n k Li i ik k i i
1 = − − −
) sin( ) (
(18) For reactive power balance,
V P α θ θ
1
1
X X
V E E E K K K K E f M
I R
X X
I M E
X X
X X M E
I I
I M D R
V K
V T K
Ai refi Mi Ai i i Ai Ai qi di i q qi i q i d di i d i qi qi i d di i di qi di qi Fi Ri fdi di qi i i Fi Ei Fi Ai Ai Ai Fi Ai Fi Ai Ei fdi si i q i d i d i di i qi i i Fi Ri fdi di qi i i
Δ Δ
Δ Δ Δ
Δ Δ
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡
I M
V E E E
− =
τ τ
For the m-machine system, eq. 26 can be written as
(26) For m ., 1, i … =
τ τ ω δ
τ τ
τ τ τ τ τ τ
ω δ
θ τ
) ( ) ( ) (
1 τ
I B x A x Δ + Δ + Δ + Δ = Δ
1
1
1 ) (
1
1
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤
′ −
- ⎥ ⎥ ⎥ ⎥ ⎥
- ⎥ ⎦ ⎤
- ⎥ ⎦ ⎤
V B
2. Linearized the stator algebraic equations from generating unit yields: ( )
⎢ ⎣ ⎡
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤
Δ Δ
⎢ ⎣ ⎡
(27)
⎥ ⎥ ⎥ ⎥ ⎦ ⎤
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤
Δ Δ
θ δ
November 22-23, 2007, Montien Hotel, Bangkok, Thailand.
Figure 4 An algorithm to generate linearized state space model. Fig. 4 shows the steps to get the linearized state space model of power systems. These are those steps :
systematic procedure for treating the wide range of devices.
PW-8
Δ Δ
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡
′ −
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡
′ −
− − − −
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤
Δ Δ
Δ Δ Δ
Δ Δ
⎥ ⎥ ⎥ ⎥ ⎦ ⎤
⎢ ⎣ ⎡
′ − ′ − ′ ′ − ′ − ′
′ − −
− ′
′ ′
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤
Δ Δ
E u
- Collect differential and algebraic equations
− − −
a. Generating Unit 1.
Δ Δ
Δ Δ Δ
⎥ ⎦ ⎤
⎢ ⎣ ⎡
Four differential equations for generator
Fi i i i i i i i i i i i i i i qi di si di qi si Ri fdi di qi i i i i i i i i
V V
V V
V I
I R
X X R R
V E E E
V V θ θ δ θ δ
θ δ θ δ ω δ θ δ
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡
−
(28) For
⎥ ⎦ ⎤
( ) ( ) ( )
( ) ( ) cos sin sin cos
1 sin 1 cos =
- ⎥ ⎥ ⎥ ⎥ ⎥
- ⎥ ⎦ ⎤
⎥ ⎦ ⎤
⎢ ⎣ ⎡
Δ Δ
⎢ ⎣ ⎡
− ′
− − − − − −
⎢ ⎣ ⎡
Δ Δ
⎥ ⎦ ⎤
⎢ ⎣ ⎡
− ′
and three differential equations for exciter : seven differential equations for each generating unit, 7m.
2. Two real stator algebraic equations for each generating unit.
b. Network
., m 1, i … =
For the m-machine system, eq. 27 can be written as
g g
V D
I D x C Δ + Δ + Δ =
2
1
1 (29)
3. Linearized network algebraic equations yields:
a. Generator Bus Proceedings of EEE FWS 2007.
1. The linearization of the differential equations from generating unit yields : ⎥ ⎦ ⎤
1. Two real network equations for each generator bus.
2. Two real network equations for each load bus.
- Linearize all equations with respect to all
variables and construct equations in matrix form
- ⎥ ⎥ ⎥ ⎦ ⎤
- n l m l n m m m n m gm g m m m m gm g m m m
⎥
⎥
⎥
⎦
⎤
- ⎥ ⎥ ⎥ ⎦ ⎤
- ln >
- ⎥ ⎥ ⎥ ⎦ ⎤
- Eliminate all variables other than state or
Δ Δ
4 D D B B K − − =
Then we can arrange eq. 34, 35, 36 as follow :
u E
V V x D D D K K
K K x g l g
Δ ⎥ ⎥ ⎥ ⎦ ⎤
⎢ ⎢ ⎢ ⎣ ⎡
⎢ ⎢ ⎢ ⎣ ⎡
Δ ⎥ ⎥ ⎥ ⎦ ⎤
1
⎢ ⎢ ⎢ ⎣ ⎡
= ⎥ ⎥ ⎥ ⎦ ⎤
⎢ ⎢ ⎢ ⎣ ⎡Δ
7
6
5
1
2
2
1
3
1
2
1
1
3
4
1 D D D D K − − =
2
1
3
1
2
2 C D D C K − − =
1
1
1
1
1
3 C D B A K − − =
2
4
hj (37)
6
B (41)
Δ + Δ = Δ
(39) Where :
3
1
4
2 1 sys A M M M M
− − = (40)
E = sys
This completes the calculation to get linearized state space model of power systems.
model :
IV. STATE SPACE MODEL GENERATION
4.1 Input Data Preparation
The developed program can handle power system with many numbers of generators and buses. The input which is needed by the program are parameter datas of generators (including exciter) and network. The same network data for load flow studies is used for the calculation.
4.2 Flow Chart of the Program
The first step of the program is read input data from all components. After that the program will run load flow to calculate the operating point of the network. From operating point, program will continue to calculate initial condition all variables in generator. Then program will calculate the matrix for each equation and do many calculations based on the algorithm to develop state model. The flow chart is in Fig. 5 below.
Calculate initial values for all state and input variables needed at the obtained operating conditions start end Run load flow To obtain steady-state initial operating conditions Calculate elements of matrix from developed state model Using eq 39 Get the linearized matrices A and B Read all the required input data
Figure 4 Flow chart of the program to generate Proceedings of EEE FWS 2007.
November 22-23, 2007, Montien Hotel, Bangkok, Thailand.
E u x A x sys
V V , so that we get this
We can write as :
= ⎥ ⎦ ⎤
u E V x
M M M M x
Δ ⎥ ⎦ ⎤
⎢ ⎣ ⎡
⎢ ⎣ ⎡
Δ Δ
⎥ ⎦ ⎤
⎢ ⎣ ⎡
⎢ ⎣ ⎡Δ
l g
4
3
2
1 (38) with
⎥ ⎦ ⎤ ⎢
⎣ ⎡ = l g
V V
V
then we need to eliminate
(36) Where
7
PW-8
l g g
41
1
3
31
1
2
21
(30) We can write eq. 30 :
V D
1 , 4 1 ,
V D
I D x C Δ + Δ + Δ + Δ =
5
4
3
2
(31)
b. Load Bus
4 , 41 1 ,
51
⎢ ⎢ ⎣ ⎡
Δ ⎥ ⎥ ⎥ ⎦ ⎤
⎥ ⎥ ⎥ ⎦ ⎤
⎢
⎢
⎢
⎣
⎡
Δ Δ
⎥
⎥
⎥
⎦
⎤
⎢⎢ ⎢ ⎣ ⎡
⎢ ⎢ ⎢ ⎣ ⎡ Δ
Δ ⎥ ⎥ ⎥ ⎦ ⎤
⎢ ⎢ ⎢ ⎣ ⎡
⎢ ⎢ ⎢ ⎣ ⎡ Δ
⎢ ⎢ ⎢ ⎣ ⎡
5 , 51 1 ,
⎢ ⎢ ⎢ ⎣ ⎡ Δ
Δ ⎥ ⎥ ⎥ ⎦ ⎤
⎢ ⎢ ⎢ ⎣ ⎡ =
V V D D D D
V V D D D D
I I D D x x C
C ... ... ... ...
... ... ... ...
1 , 5 1 ,
⎥ ⎥ ⎥ ⎦ ⎤ ⎢
Δ Δ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢
V D Δ + Δ =
Δ + Δ + Δ = Δ
6
(33)
input variables
From eq. 27, 29, 31, 33 we need to eliminate
n g g
V V I , , . Firstly, we eliminate g
I , so that we get E u
V K x K x g
1
V D Δ + Δ =
4
3
(34)
V D
V K x K Δ + Δ + Δ =
(35)
l g
V D
7
V D
⎢ ⎢ ⎣ ⎡
7
⎢ ⎢ ⎢ ⎣ ⎡ Δ
Δ ⎥ ⎥ ⎥ ⎦ ⎤
⎢ ⎢ ⎢ ⎣ ⎡ =
1 , 7 1 ,
7 ,
1
7 1 ,
1
1 , 6 1 ,
l g
6 ,
1
6 1 ,
1
5 ... ...
... ...
V V D D D D
V V D D D D lm n n m n n m m m gm g m n n m m m
(32) We can write eq. 32 :
- ⎥ ⎥ ⎥ ⎦ ⎤
- ⎥ ⎦ ⎤
Proceedings of EEE FWS 2007.
PW-8 November 22-23, 2007, Montien Hotel, Bangkok, Thailand.
4.3 Outputs of the Program Table 2. Computed initial conditions
The developed program will generate state Variables Gen 1 Gen 2 Gen 3 space representation of the power systems with δ 3.586 61.098 54.137
HVDC link. It produces matrices A and B of the 0.303 1.290 0.561
I d
systems. After having these matrices, the program 0.671 0.932 0.619
I q
will calculate eigenvalues, frequency of oscillation, damping ratio, participation factor of the system.
0.065 0.806 0.779
V d
The developed program is useful to analyze the 1.038 0.634 0.666
V q
small signal stability of the system. We can get the 0 0.622 0.624 eigenvalues of the system thus we can check the E′
d
stability of it. We also calculate the frequency of 1.056 0.788 0.768
E′ d
oscillation and damping ratio, thus we can 1.082 1.789 1.403
E fd investigate what kind of problem in the system.
Moreover, we can see the participation factor to see 0.195 0.322 0.253
R f the participation of states to each mode.
1.105 1.902 1.451
V R
V. TEST RESULTS
1.095 1.120 1.098
V ref
The developed program is applied for the 0.716 1.630 0.850
T M
Western System Coordinating Council (WSCC) 3- machine, 9-bus system shown in Fig. 6. This system is used to validate the developed program. In
In this system we have three generators and literature [8] there are some discussions about this each generator has 7 states. Its mean, we will have system. They calculate the operating condition from 21 states. After we get operating point of network the nominal loading, initial condition of all and initial condition of all variables, we can variables, eigenvalues and participation factor of the compute the matrices A and B. After that the system. Those calculation will be compared with the program will give eigenvalues, frequency and calculation from developed program so that we sure damping ratio of the system (see Table 3). Fig. 7 that the developed program is correct. lists the comparison of eigenvalues obtained from
With the data that we have, we can calculate developed program and literature [8] for the WSCC operating condition of the network by running load system. flow program. We can see the results in Table 1.
Table 3. Eigenvalue, Frequency and Damping Table 1 Load flow solutions of the WSCC
Ratio from the program
Bus Type of Voltage Voltage PG QG -PL -QL bus Magnitude Angle (pu) (pu) (pu) (angle)
- 1 (swing) 1.04 0.000 0.716 0.271 - 2 (PV) 1.025 9.280
1.63 0.067 - - 3 (PV) 1.025 4.665 0.85 - - - 0.109 4 (PQ) 1.026 -2.217 - - - -
5 (PQ) 0.996 -3.989 - - 1.25
0.5 6 (PQ) 1.013 -3.687 - - 0.9 0.3 7 (PQ) 1.026 3.720 - - - -
8 (PQ) 1.016 0.728 - - 1.00
0.35 9 (PQ) 1.032 1.967 - - - - After we get operating point from running load flow program, we will calculate initial variables in the system. The computed initial conditions are given in Table 2. Proceedings of EEE FWS 2007.
PW-8 November 22-23, 2007, Montien Hotel, Bangkok, Thailand.
Figure 6 A single line diagram of the Western System Coordinating Council (WSCC)
15 eigenvalues obtained from developed program eigenvalues obtained from literature [14]
10
5 is ax y nar agi Im
- 5
- 10
1 Real axis
Figure 7. Comparison of eigenvalues obtained from developed program and literature [8] for the WSCC system. For the participation factor, we get : Table 4. Participation factors for the WSCC (mode 1-11)
Eigenvalue
State G 1 2 3 4 5 6 7 8 9 10
fd E
E ′
2 0.001733 0.67086 0.67086 0.77789 0.77789 0.42542 0.42542 3.19E-10 3.19E-10
dω
2 0.001953 0.002853 0.002853 0.002876 0.002876 0.001697 0.001697 0.25761 0.25761
qδ
2 0.001953 0.002853 0.002853 0.002876 0.002876 0.001697 0.001697 0.25761 0.25761
F R
1 0.025508 0.73996 0.73996 0.77701 0.77701 0.00651 0.00651 7.45E-11 7.45E-11
1.64E-06 0.10405 0.10405 0.091477 0.091477 0.000729 0.000729 7.47E-13 7.47E-13
1
V
R
7.34E-05 0.15518 0.15518 0.1401 0.1401 0.001126 0.001126 4.18E-11 4.18E-11
1
1
E
1 0 0 0 0 0 0 0 0 0
d E ′
2.12E-10
2.12E-10
1 0.003037 1 1 1 1 0.00822 0.00822
q E ′
1
1
ω 1 0.001217 0.000884 0.000884 0.001012 0.001012 3.01E-06 3.01E-06
1
E′
2 0.89458 0.046364 0.046364 0.038942 0.038942 0.02802 0.02802 3.15E-13 3.15E-13
fd2
δ 1 0.001217 0.000884 0.000884 0.001012 0.001012 3.01E-06 3.01E-06
d E′
November 22-23, 2007, Montien Hotel, Bangkok, Thailand.
4.86E-06 0.041734 0.041734 0.022258 0.022258 0.098534 0.098534 4.67E-11 4.67E-11 Proceedings of EEE FWS 2007.
3
V
R
5.75E-05 0.059269 0.059269 0.032313 0.032313 0.144 0.144 4.13E-11 4.13E-11
3
fd E
1 0.026088 0.026088 0.021011 0.021011 0.11337 0.11337 6.53E-13 6.53E-13
3
1 1 8.30E-11 8.30E-11
5.08E-05 0.10175 0.10175 0.10838 0.10838 0.057586 0.057586 4.91E-12 4.91E-12
3 0.006602 0.38428 0.38428 0.22343 0.22343
q E ′
0.009039 0.001701 0.001701 0.000743 0.000743 0.003754 0.003754 0.12233 0.12233
3
ω
δ
3 0.009039 0.001701 0.001701 0.000743 0.000743 0.003754 0.003754 0.12233 0.12233
F R
2 0.017652 0.48516 0.48516 0.60108 0.60108 0.33293 0.33293 2.74E-11 2.74E-11
1.73E-05 0.079927 0.079927 0.084018 0.084018 0.044466 0.044466 1.25E-10 1.25E-10
2
V
R
1
Eigenvalue
State G 12 13 14 15 16 17 18 19 20 21
11 δ
V
q E ′
1 1 0.002605 0.002605 0.000123 0.000123 0.000408 0.000408 0.013933
ω 2 0.22394 0.22394
1 1 0.002605 0.002605 0.000123 0.000123 0.000408 0.000408 0.013933
δ 2 0.22394 0.22394
1.11E-06 1.11E-06 2.33E-05 2.33E-05 0.004952 0.004952 0.0537 0.0537 0.30573 0.30573 1.50E-05
1
F R
1.48E-05 1.48E-05 9.82E-05 9.82E-05 0.017438 0.017438 0.1842 0.1842 1 1 8.94E-07
1
R
E ′ 2 0.013862 0.013862 0.019714 0.019714 0.004372 0.004372 0.001584 0.001584 0.003313 0.003313
1.52E-05 1.52E-05 0.000106 0.000106 0.016878 0.016878 0.17896 0.17896 0.97329 0.97329 8.76E-08
1
E
1 0 0 0 0 0 0 0 0 0 0 0 fd
d E′
2.30E-05 2.30E-05 0.000118 0.000118 0.000567 0.000567 0.003139 0.003139 0.050991 0.050991 5.69E-06
1
q E ′
0.012077 0.012077 0.41915 0.41915 0.000537 0.000537 0.000538 0.000538 0.00043 0.00043 4.90E-05
1
1 0.012077 0.012077 0.41915 0.41915 0.000537 0.000537 0.000538 0.000538 0.00043 0.00043 4.90E-05 ω
2 0.01285 0.01285 0.042875 0.042875 0.035023 0.035023 0.004633 0.004633 0.006454 0.006454 0.015974 d
1
Table 5 Participation factors for the WSCC (mode 12-21)
q E ′
F R 3 0.000117 0.000117 0.000233 0.000233 0.026931 0.026931 0.29295 0.29295 0.049877 0.049877 0.010403
0.001562 0.001562 0.00099 0.00099 0.094347 0.094347 1 1 0.16237 0.16237 0.000562
3
V
3 0.001609 0.001609 0.00106 0.00106 0.091795 0.091795 0.97626 0.97626 0.15878 0.15878 6.08E-05 R
fd E
0.080829 0.080829 0.001141 0.001141 0.001313 0.001313 0.007865 0.007865 0.001295 0.001295 0.91959
3
E ′
3 0.031014 0.031014 0.005812 0.005812 0.001848 0.001848 0.031142 0.031142 0.010844 0.010844 0.030766 d
1 1 0.15921 0.15921 0.000755 0.000755 0.003355 0.003355 0.000921 0.000921 0.021408
fd E
3
ω
1 1 0.15921 0.15921 0.000755 0.000755 0.003355 0.003355 0.000921 0.000921 0.021408
3
δ
3.16E-05 3.16E-05 0.000847 0.000847 0.28855 0.28855 0.018931 0.018931 0.018299 0.018299 0.009019
2
F R
1 1 0.063984 0.063984 0.059003 0.059003 0.000366
V 2 0.000424 0.000424 0.003658 0.003658
2 0.000434 0.000434 0.003849 0.003849 0.9835 0.9835 0.063088 0.063088 0.058253 0.058253 5.27E-05 R
PW-8 Proceedings of EEE FWS 2007.
PW-8 November 22-23, 2007, Montien Hotel, Bangkok, Thailand.
VI. CONCLUSION
11. William L. Brogan, Modern Control The developed program can handle Theory , Prentice Hall International calculating state space representation of actual Edition, 1991. power systems which consist of generating units
12. Dr. David Banjerdpongchai, Lecture and network.
note of Control System Theory : Control
Eigenvalues, frequency, damping ratio and System Theory , Control System participation factor can be computed. Research Laboratory, 2002. From the result we see that the obtained eigenvalues and participation factor of WSCC are correct.
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