14th PSCC, Sevilla, 24-28 June 2002
Session 22, Paper 6, Page 1
IDENTIFICATION OF OPTIMUM SITES FOR A POWER SYSTEM
CONTROLLER USING NORMAL FORMS OF VECTOR FIELD
Gilsoo Jang,
Insoo Lee,
Sae-Hyuk Kwon
Korea University
Seoul, Korea
[email protected]
Abstract – In stressed power system, due to the presence of increased nonlinearity and the existence of nonlinear modal interactions, there exist some limitations to the
use of conventional linear system theory to identify the
optimum sites for a controller. This paper suggests an
approach based on the method of normal forms to identify
suitable sites for controllers with incorporating the
nonlinear interaction. In this paper, nonlinear participation factors and coupling factors are proposed as measures of the nonlinear interactions, and an identification
procedure of proper sites for a controller is also proposed. The proposed procedure is applied to the 10generator New England System, and the results illustrate
its capabilities.
plied to New England 39-bus test system, and the simulation results verify the validity of the proposed method.
2
APPROACH
2.1 The Power System Model
The generators without excitation control are represented by the classical model [4]. Generators with excitation control are described by the two-axis model. The
block diagram of the static exciter model [5] is shown in
Figure 1.
Keywords: Power system dynamics, normal forms of
vector field, nonlinear participation factor, power
system stabilizer
1 INTRODUCTION
Modern power systems highly utilize available transmission and generation with large amounts of power
interchanged among electric companies and geographical
regions. The stressed nature of power networks, described above, has an impact on the system dynamic
behavior. In this stressed interconnected power systems a
small disturbance may induce a complicated dynamic
behavior that excites numerous modes of oscillation.
Only a few of these modes are of primary interest to
system designer [1]. In order to damp those electromechanical modes, it is the first task to determine some
generators which have more influence on the mode than
others. Works have been performed to determine the
most effective combination of generators for controller
application by the modal approaches [2][3]. With
stressed conditions, the interarea mode which is caused
by two or more groups of generators being interconnected by weak ties may occur, and control performance
can be altered by possible unfavorable nonlinear modal
interaction in the power system. Therefore, it is important to incorporate the nonlinear modal interactions in
the analysis and control design. Since the conventional
linear approaches neglect the possible nonlinear interaction among modes, a method to identify suitable locations of controllers for damping the oscillations with the
nonlinear information is proposed using normal forms of
vector field. An index based on the nonlinear participation factor and coupling factor is developed to quantify
the nonlinear interaction. The proposed method is ap-
Figure 1: Static Exciter Model
The network is represented classically: quasi statesteady network parameters, constant impedance loads,
etc. Assuming the generator internal reactance to be
constant, a network representation at the internal generator nodes can be obtained. The procedure described in
(Chapter 9 of [4]) yields the direct and quadrature axes
currents for the generators represented in detail. The
currents for the classically represented machines can also
be obtained. We assume that in an m-generator system
there are l generators represented by the two-axis model
and equipped with exciters, the remaining m-l generators
are presented by the classical model. Then the dynamic
equations governing the generators and the excitation
system have the general form
•
X = F(X )
(1)
where,
X = [ Eq′1 , Ed′1 , ω 1 , δ1 , EFD1 , x E1 , x E 2 , ...,
1
Eql′ , E dl′ , ω l , ..., ω m , δ m ]
T
and F is an analytic vector field.
1
14th PSCC, Sevilla, 24-28 June 2002
Session 22, Paper 6, Page 2
2.2 The Normal Form Technique
We expand (1) as a Taylor series about a stable equilibrium point XSEP and obtain using again X and xi as the
state variables.
•
1 T i
X H X + H .O.T .
2
xi = Ai x +
(2)
where,
Ai=Ith row of Jacobian A which is equal to
[∂F / ∂X ]X
SEP
3
QUANTIFICATION OF NONLINEAR
INTERACTION
3.1 Nonlinear Participation Factor
Linear participation factors are a well-known method
to find out mode-machine interactions [6]. The participation factor pki represents a measure of the participation
of the kth machine state in the trajectory of the ith mode.
It is given by
HI= [∂ Fi / ∂ x j ∂x k ] X SEP =Hessian matrix
2
pki = uki * v ik
Denote by J the (complex) Jordan form of A, and by
U the matrix of the right eigenvalues of A. Then the
transformation X=UY yields for the linear and the second
order terms of (2) the equivalent system
•
N
N
y i = λi y i + ∑∑ Ckjl y k yl
(3a)
k =1 l =1
where,
C =
j
N
∑V
1
2
p =1
T
jp
[U T H PU ] = [Ckl ]
j
Since linear participation factors are functions of
both the left and right eigenvectors, they are independent
of eigenvector scaling. Using normal forms we apply the
concept of nonlinear participation factors [7]. The normal form initial conditions, using the second order approximation of the inverse transformation in (4a), can be
used to express the solution for kth machine state variable as
N
x k ( t ) = ∑ uki ( v ik + v 2ikk )e λ t
(3b)
and V denotes the matrix of associated left eigenvectors.
If the second order non-resonance condition holds, i.e. if
λ j ≠ λ k + λ l for all three tuples of eigenvalues of A,
then the normal form transformation of (3a) is defined
by
i
N
N
+ ∑ ∑ u2 kpq ( v pk + v 2 pkk )( v qk + v 2 qkk )e ( λ
N
where,
k =1 l =1
N
j
u2 ikl = ∑ uij h2 kl
(4b)
j
j
kl
C
λk + λl − λ j
(4c)
Using the approach given in [7], one can define second order participation factors according to
N
In Z-coordinates, the system (3a) takes on the form
x k ( t ) = ∑ p2 ki e
i =1
λi t
z j (t ) = z joe λ j t
(6)
N
y j (t ) = z joe λ t + ∑ ∑ h2 klj z ko z lo e ( λ
j
k
+ λ l )t
(7)
k =1 l =1
N
∑
j =1
uij z joe
λ jt
+
N
N
N
∑ ∑∑ h2
j =1
uij [
N
p +λq ) t
p =1 q = p
(11)
(5)
Equations (2)-(5) allow us to obtain explicit second
order solutions for the system in the different coordinate
systems:
xi (t ) =
N
+ ∑ ∑ p2 kpq e ( λ
•
z j = λjz j
N
j
j =1
j =1 l =1
h2 kl =
N
v 2ipp = −∑ ∑ h2 kli v kp v lp
(4a)
h2 j ( Z ) = ∑ ∑ h2 kl z k z l
p +λq ) t
p =1 q = p
where,
N
(10)
i =1
N
Y=Z+h2(Z)
(9)
k =1 l =1
( λk + λl )t
j
]
k l z kozlo e
(8)
where,
p2 ki = uki ( v ik + v 2ikk )
p2 kpq = u2kpq ( v pk + v 2 pkk )( v qk + v 2qkk )
Note that there are two types of second order participation factors. The p2ki represents the second order participation of the kth machine state in the ith singleeigenvalue mode. In fact, the linear participation factor,
pki, is one term in the expression for p2ki which includes
the second order corrections. The p2kpq represents the
second order participation of the kth machine state in the
'mode' formed by the combination of the p and q modes.
14th PSCC, Sevilla, 24-28 June 2002
Session 22, Paper 6, Page 3
3.2 Coupling Factor
3.2.1
Linear Coupling Factor [8]
Assuming the presence of an ideal PSS which has the
static stabilizing feedback measuring angular velocity
and actuating upon accelerating torque, the eigenvalue
sensitivity with respect to stabilizer gain can be derived
in the form
∂λh
1
1
= −ui + n,h vh,i + n
= − pih
∂k s
Mi
Mi
(12)
where, k s is the stabilizer gain, Mi is the inertia constant
of the ith generator, and ui+n,h and vh,i+n are the (i+n)th
elements of right and left eigenvector. According to the
time response of linearized state equation, the value
v hi / M i measures controllability of the hth mode from
the ith input and u jh measures observability of the hth
mode on the jth machine. Therefore, the value of
u jhv hi / M i determines how much an ideal PSS applied
on the ith generator affects the jth generator motion
during the excitation of the hth oscillatory mode. Thus,
sequentially defined coupling factor between generators i
and j in the mode h will be
Cij2( h) =
u jhvhi uihv jh
Mi
Mj
=
pih p jh
M iM j
(13)
The total coupling factor which weights the influence
of stabilizers applied over generators i and j on their
dynamic performance under all mode excitation, can be
given by
Cij2
=
∑h
Cij2( h)
(14)
3.2.2
Nonlinear coupling factor
Using nonlinear participation factor the coupling factor in equation (13) can be enhanced to incorporate
nonlinear information of the system, and is given by
hancement of the total system stability in stressed conditions.
3.3 Controller Location Selection Procedure
Based on the above approach, a procedure to determine a suitable location of a controller is proposed. This
procedure ensures the enhancement of systems damping
even in stressed conditions.
1. Identify critical modes of the system using eigenanalysis.
2. Calculate participation factors of the identified critical modes, and identify participating state variables and
generators in the modes.
3. Calculate the sensitivity of all critical modes with
respect to the controller parameter of the identified
generator. If all sensitivity values have the same sign on
real part, the generators can be the possible controller
locations. Then, the selection procedure ends.
4. If we can’ t find any generator of which controller
makes the same sign on the real part of the mode sensitivity values, choose the generator which makes sensitivity values have the same sign as many as possible.
5. Identify the state variable and generator which participates in the mode with opposite sign, and calculate
the nonlinear coupling factors of the states identified at
step 2 and 5 on the mode.
6. If the nonlinear coupling factors have small values,
then the generator identified at step 2 can be the possible controller location, and the selection procedure
ends. If not, then go back to step 1 to identify new possible location.
4
CASE STUDIES
The proposed procedure is tested on the model of a
stressed power system and the numerical results are
presented here. The test system given in Figure 1 is the
NE 10-generator, 39-bus system. Nine generators are
represented by the two-axis model and one generator is
by classical model.
8
37
C 2ij2 ( h)
=
p2ih p2 jh
MiM j
2
(15)
The nonlinear coupling factor is a measure of ith and
jth generators’ nonlinear interaction, and represents how
much an ideal PSS applied on the ith generator affects
the jth generator dynamic performance during the hth
mode excitation. The difference between nonlinear and
linear coupling factors is dependent on how much the
power system is stressed. The nonlinear coupling factor
defined in equation (15) represents a suitable and reliable
base for identification of the most effective combination
of generators for PSS application, and it should be used
to find the location of a controller for maximum en-
29
26
38
25
30
9
27
17
2
28
18
24
31
23
3
21
22
15
4
39
10
14
5
6
6
20
7
11
8
35
19
12
9
33
13
10
4
34
5
1
1
36
7
16
32
3
Figure 1: New England Test System
14th PSCC, Sevilla, 24-28 June 2002
Session 22, Paper 6, Page 4
Table 1: Eigenvalues of NE 39-bus system
mode
Real Part
Imag. Part
mode
Real Part
Imag. Part
1:
3:
5:
7:
9:
11:
13:
15:
17:
19:
21:
23:
*25:
27:
29:
31:
-3.34E+00
-1.40E+00
-1.54E+00
-1.11E+00
-1.08E+00
-4.77E-01
-5.20E-01
-2.26E-01
-9.81E+00
-9.87E+00
-6.80E+00
-2.36E+00
-1.64E-03
-4.33E+00
-3.03E+00
-1.89E+00
1.60E+01
1.59E+01
1.44E+01
1.27E+01
1.22E+01
1.01E+01
7.97E+00
6.38E+00
1.68E+00
1.36E+00
1.08E+00
3.94E+00
2.92E+00
3.59E-01
5.57E-01
4.89E-01
33:
35:
37:
39:
41:
43:
45:
47:
49:
51:
53:
55:
57:
59:
61:
63:
-1.55E+00
-1.13E+00
-7.84E-01
-1.04E+00
-1.03E+00
-1.38E+00
-6.82E-01
-8.37E+00
-8.59E+00
-1.05E+01
-1.66E+01
-1.00E+02
-1.00E+02
-1.00E+02
-1.00E+02
-4.61E+01
9.56E-01
1.04E+00
7.39E-01
8.78E-01
7.48E-01
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
All eigenvalues of the system are given in Table 1.
The information of participation factors identifies 4 low
frequency inertial modes among poorly damped modes,
namely modes 13, 15, 25, and 37. The dominant one
among the critical modes is mode 25, and Table 2 shows
some large linear and nonlinear participation factors of
the mode.
Table 3: Eigenvalue sensitivity w.r.t. the controller parameter
of the 9th generator
Mode
13
15
37
25
Sign of real part
Positive
Negative
Negative
Negative
Table 4: Nonlinear Coupling Factors
h
i and j
C 2ij( h)
Mode 13
25 27
2.18E-4
Mode 15
25 27
2.94e-1
Mode 37
25 27
3.48e-9
Mode 25
25 27
2.72e-1
Figures 2 shows the relative rotor angle plots of all
generators for a 3-phase stub fault at bus #33 cleared at
0.05 second when any PSS is not installed. When a PSS
is installed at the generator 9 according to the selection
procedure, the damping of the system is enhanced as in
Figure 3.
Table 2: Participating States and Factors
Mode
25
25
25
25
25
State Var.
27
24
25
21
36
Linear pf
1.35E-0.1
8.94E-0.2
5.89E-0.2
5.35E-0.2
4.66E-0.6
Mode State Var.
25
64
25
61
25
60
25
59
25
62
Nonlinear pf
1.10E+00
1.05E+00
8.79E-0.1
7.63E-0.1
4.82E-0.1
The dominant participating states on mode 25 are the
27th state in linear case, and the 64th state which is the
exciter state of the 9th generator in nonlinear case. Also,
The states which participate heavily in mode 25 are inertial modes in linear case and exciter modes in nonlinear
case. It is important to notice nonlinear participating
factors which can incorporate nonlinear interaction only
identify the control states as dominant participating
states, and it is because control states are strongly
nonlinearly interacted with each other. In order to understand, how the controller parameter of the 9th generator
influences the critical modes, we compute their eigenvalue sensitivity with respect to the parameter, and the
results are in Table 3. The sensitivity analysis indicates
that the real parts of most of the critical modes are affected by the controller at the 9th generator. Increasing
the gain will push the eigenvalues of the critical modes
into the left half plane except mode 13.
In order to ensure the controller’ s location makes
positive damping effects on all of the critical modes, the
participating states in mode 13 are calculated. The identified state is state 25, and the nonlinear coupling factor of
the states 25 and 27 on the mode 13 is calculated in
Table 4. According to Table 4 the 9th generator can be
the suitable location for the controller since the
nonlinear participating factor for the mode 13 is small
enough to excite mode 13.
Figure 2: Rotor Angle Plot Without a PSS
Figure 3: Rotor Angle Plot With a PSS at Generator 9
14th PSCC, Sevilla, 24-28 June 2002
5
CONCLUSIONS
This work tries to improve conventional controller
location method by incorporating nonlinear information
provided by the method of normal forms. The proposed
location procedure uses the concept of nonlinear participation and coupling factors for determining the best sites
to be equipped with controllers with ensuring the controllers improve total system damping. The addition of
supplementary nonlinear information is valuable since it
can make unfavorable control interaction be considered
in the procedure.
Further research will be done to extend this approach
to an optimal method. Also, application of the proposed
approach to large stressed power system is asked to
compare the approach with the conventional linear approach.
6
ACKNOWLEDGEMENTS
This work was supported by grant No. R01-200000269 from the Basic Research Program of the Korea
Science & Engineering Foundation.
REFERENCES
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Ames, Iowa, 1997.
Session 22, Paper 6, Page 5
[2] Arcidiacono, V., Ferrari, E., Marconato, R., Dosghali, J., and Grandez, D., “Evaluation and improvement of electromechanical oscillation damping by
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results in the Central Peru power system”, IEEE
Transactions on PAS-99, 1980, pp. 769-778.
[3] Demello, F.P., Nolan, P.J., Laskowski, T.F., and
Undrill, J.M., “Coordinated application of stabilizers
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[4] Anderson, P. M., and A. A. Fouad. Power System
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[5] Lin, C. "Analysis of nonlinear modal interaction and
its effect on control performance in stressed power
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Iowa State University, Ames, Iowa, 1995.
[6] Perez-Arriaga, I. J., G. C. Vergbese, and F. C.
Schweppe. "Selective Modal Analysis with Applications to Electric Power Systems. Part 1: Heuristic
Introduction." IEEE Transactions on PAS v.101,
Sep.1982: 3117-3125.
[7] Starrett, S. K. "Application of normal forms of vector
fields to stressed power systems." Ph.D. Thesis, Iowa
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[8] Ostojic, D.R., “Identification of optimum site for
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