Indian Journal of Science and Technology, Vol 7(S6), 40–47, October 2014
ISSN (Print) : 0974-6846
ISSN (Online) : 0974-5645
Multi-Machine Small Signal Stability Analysis For
Large Scale Power System
C. Chennakesavan* and P. Nalandha
Department of Electrical and Electronics Engineering, AMET University, Chennai;
cckesavan@gmail.com; nalanpsel@gmail.com
Abstract
The main objective of this paper is a fundamental study of the nature of inter-area oscillations in power systems. ­Small-signal
stability refers to the ability of a power system to maintain synchronism under small disturbances Instability may arise
in two forms increase of rotor angle due to lack of sufficient synchronization torque and rotor oscillations of increasing
amplitude due to lack of sufficient damping torque. To study small signal stability analysis synchronous machine model,
transmission line model and two area system model and the dynamic state matrix eigenvalues and eigen vectors are constructed and the small signal stability analysis done with the developed algorithm. The 11 bus systems are considered here
for study of oscillations. The effects of the system structure, generator modelling, excitation type, and system loads are discussed in detail. In the study, only small signal stability analyses are used to determine the characteristics of the system. In
the power system leads to the development of many oscillations at low frequency in the power system. This paper presents
the effect of the load model it was easier to identify unstable modes of oscillation.
Keywords: Power System Modeling, Eigen Value Analysis, Participation Factors, Small Signal Stability, TYPE-2B MODEL
1. Introduction
With the increasing electric power demand, power
systems can reach stressed conditions, resulting in undesirable voltage and frequency conditions1. The eigen value
analysis was done for small signal stability of 11 bus 2 area
system. The drawback of eigenvalue analysis is that the
higher order terms neglected from the liniearised set of
equation, Here particularly in a system is described by
strongly non-linear equations.
Modern electrical power systems have grown to a
large complexity due to increasing interconnections,
installation of large generating units and extra-high voltage tie-lines etc.
The system response to such disturbances involves
large excursions of generator rotor angles, power flows,
bus voltages, and other system variables2. It is ­important
that, while steady-state stability is a function only of
­operating conditions.
*Author for correspondence
Small signal stability is the ability of the system to
maintain synchronism under small disturbances which
occur continually on the system due to small variations
in loads and generation or other small disturbances on
the system4. A disturbance is considered to be small if
the equations that represent the dynamic performance
of the system can be linearized for the purpose of analysis function of both the operating conditions and the
disturbance.
In this paper a new method for calculation of the two
area four generator system is investigated. Stability studies for power system planning, operation and control
rely immensely on computer based power system simulation tools. Simulation tools use mathematical models
that predict the dynamic performance of the system7.
It is crucial that these power system models be modeled accurately to predict the actual performance of the
­system.
C. Chennakesavan and P. Nalandha
2. System Description and
Modelling
are also connected to bus 7 and 9 as shown in the figure 1
below. The system has the fundamental frequency 60 Hz.
The system comprises two similar areas connected
by a weak tie3. Each area consists of two generators, each
having a rating of 900 MVA and 20 kV. We will perform:
A standard IEEE 11 bus system1 as shown in Figure 1.
2.1 Synchronous Machine Model
The synchronous generators are modelled as classical
machines with δ and ω as state variables. The generator
is represented by the Norton equivalent for network solution. The admittance of the generator is included in the
main diagonals of Y matrix.
J
dw
= Tm - Te dt
(1)
dd
= w - w0 dt
(2)
2.2 Transmission Line Model (Short Line)
A transmission line is modeled as a short line. It consists
of a series impedance comprised of a resistance R and
reactance X between the terminal nodes.
2.3 Load Model
The loads are modeled as a constant admittance in
­transient stability studies.
2.4 Introduction of Two-Area System
The machines are considered to be TYPE -2B Model. The
system contains eleven buses and two areas, connected
by a weak tie between bus 7 and 9. Totally two loads are
applied to the system at bus 7 and 9. Two shunt capacitors
∗ Power-flow calculation
∗ Linear Analysis and Modal Analysis
∗ Time-domain simulation.
The left half of the system is identified as area 1 and the
right half is identified as area 2. The saturation of the synchronous machines are not identical.
This chapter describes the methodology for incorporation of the model for small-signal analysis of single
machine-infinite bus and multi-machine configurations.
This is followed by relevant results obtained for these
­configurations.
3. Small Signal Stability
Small Signal Stability is the ability of the power system to
maintain Synchronism under small perturbations. Such
perturbations occur continuously on the system because
of small variations in load and generation where the system continuously adjusts itself to changing conditions.
Restoring forces acting on the machine help them
to maintain stable conditions. The system must be able
to operate satisfactory under these conditions and successfully supply the maximum load. The “Small Signal”
disturbances are considered suffiently small for linearization of system equations to be permissible for the purpose
of analysis.
The most common form of synchronism, monotonically, in the first few seconds following the fault due to lack
of synchronizing torque and damping torque. The stability of the following types of oscillations is of concern.
Local modes are associated with the oscillations of
generating units at a particular station with respect to the
rest of system. These oscillations are localized in a small
part of the power system.
Inter area modes are associated with the oscillations
of many machines in one part of the system against
machines in the other parts7.
3.1 State Equation
Figure 1. IEEE two area 11 Bus System
Vol 7 (S6) | October 2014 | www.indjst.org
For a classical model of the synchronous machine the
states are incremental changes in rotor speed and angle.
Indian Journal of Science and Technology
41
Multi-Machine Small Signal Stability Analysis For Large Scale Power System
In our analysis the super bar notation on the per unit
quantities are dropped out. The state equations for the
SMIB system shown in the Figure 2 are given by
È D.w ˘ È - K D
Í ˙ = Í 2H
Í . ˙ Í
ÍÎ Dd ˙˚ Î w 0
-K S ˘
È 1 ˘
È Dw ˘ Í
˙
2H Í ˙ + 2H ˙ DTm ˙ Dd
0 ˚ Î ˚ ÍÎ 0 ˙˚
(3)
This is the form of X = Ax + Bu. The elements of the
state matrix A are seen to be dependent on the system
parameters KD, H, XT and the initial operating condition
represented by the values of E1 and δ0.
Where,
∆ω is in per unit
∆δ is in radians
The synchronizing torque coefficient is given by
KS =
E ¢ V•
( Xd¢ + Xtr + Xline )
cos d (4)
3.2 Multimachine Model
The base system is symmetric; it consists of two identical
areas connected through a relatively weak tie. Each area
including two generating units with equal outputs. The
full symmetry of the base system clarifies the effect that
various factors have on the inter-area mode.
Dynamic data for the generator and excitation systems
used in the study are given in Appendix-A. In setting up
the various power flows used in the studies, capacitors
were added as necessary to ensure that the systems voltage profile was satisfactory.
The electro-mechanical modes of oscillation are present in this system; two inter-area modes, one in each area,
and one inter area low frequency mode, in which the generating units in one area oscillate against those in other
area.
Our experience with large inter connected ­systems
confirms some of the results of our studies using the
small system, and we are confident that the general
conclusions drawn from our work, will apply to large
systems. The ­following equations pertain to the fourmachine system.
3.3 State Space Model
Our aim in this section is to derive the equations for
multi machine system that represent the dynamics of
the machine and the controller in the state space form.
42
Vol 7 (S6) | October 2014 | www.indjst.org
The
state variables for the machine are
 Dw Dd DEq′ DEd′ DEd′ DEd′′ . The right hand side of the


differential equations for the machine and the ­controllers
will contain algebraic variables such as current, voltage or
power and incremental changes in them should be eliminated to obtain the equations in state variables canonical
form. Once the equations in the state variables canonical
form are obtained, we can apply eigen value technique to
access stability.
 − K D ∆T 0 0 0 0 
 ∆T m 
e

 ∆.w   2H


0
 


ws 0 0 0 0 0 

∆w   1
 .  


∆EFD 
 ∆d   0 1 − 1 0 0 0   ∆d  
 .   Td′0 Td′0

  Td′0

 ∆Eq′  

 1
E
∆

′

q

1
1

∆
E
=
+

 (5)



FD
−
0
0
0
0
0
.
 ∆Ed′
Tq′0
 ∆E ′   Tq′0


Tq′0



 d 




E
∆
′
1
 .   0 1 0 0 − 1 0   d  
∆EFD 
 ∆Eq′′  Td′0

Tq′0   ∆Ed′′  Td′0
 .  

 1

∆EFD 

 ∆Ed′′  0 1 0 0 0 − 1 

 Tq′0
Tq′0 
 Td′0
Assumptions:

The machines are considered to be ­classical (no
­controllers)

Damping ignored
Loads are assumed as constant a­ dmittances

Preparation:
The initial conditions for δ, ω and voltages are obtained
from load flow and the past history terms for δ and ω are
obtained from the initial conditions.
4. Multi-Machine Small Signal
Stability Algorithm for
Type-2b Model
1. Taken IEEE 11 bus 2 area 4 machine system
2. Formulate the admittance matrix
YV = I
(6)
3. After that reduced the network equations
-1
red
YGG
= YGG - YG , NG * YNG
, NG * YNG , G (7)
4. Initialize the power, current, voltage in transient
­condition & sub-transient condition etc.
Indian Journal of Science and Technology
C. Chennakesavan and P. Nalandha
5. Find out the transformation matrix & inverse
­transformation matrix
È e jq1
Í
Í 0
T=Í
Í 0
Í
ÍÎ 0
0
e
0
jq 2
0
e
0
0
jq 3
0
È e - jq1
0 ˘
˙
Í
Í 0
0 ˙
˙ inv T = Í
Í 0
0 ˙
˙
Í
jq 4
e ˙˚
ÍÎ 0
e
0
0
- jq 2
0
0
e
0
- jq 3
0
0 ˘
˙
0 ˙
˙
0 ˙
˙
- jq 4
e
˙˚
(8)
6. Then the reduced equations into individual machine
rotor coordinates.
M = inv(T) ∗ YGGred ∗ T
(9)
12. Finally our state variables are
 ∆ω ∆δ ∆Eq′ ∆Ed′ ∆Ed′ ∆Ed′′ 

(21)
13. Find out the eigen values from above state variables
5. Eigen Analysis
With the linearized power system model in matrix form
shown in equation the system can now be analyzed through
the use of eigenvalues and eigenvectors. To examine the
free response of the system the inputs are put to zero.
Δx = [A] x
(22)
I = I0 + DI (10)
E ¢ = E0¢ + DE ¢ (11)
where x is a state vector and A is the state matrix of size
n x n;
The state equation given by is further analyzed by taking the Laplace transform. The new equation derived in
the s domain is given by
M = M0 + DM (12)
T = T0 + DT (13)
7. After applying the linearization of network equations
in individual machine rotor coordinates.
8. Small change in current is
DI = [M0 DE ¢ - j(DdM0 - M0 Dd)E 0¢ ] (14)
9. Linearization of differential equations:
The mechanical side differential equations are given by
dDw i DTmi - DTei - K D Dw i
=
2H i
dt
(15)
10. Linearization of swing equation is
dDd i
= w s Dw i dt
(16)
11. We illustrate how to include the damper windings in
the formulation of state variables2.
1
[DEq¢ + (X d - X d¢ )DId - DEFD ] Tq¢0
(17)
1
[DEd¢ + (X q - X q¢ )DIq ]
Tq¢0
(18)
.
1
[DEq¢¢ + (X q¢ - X q¢¢)DId - DEq¢ ] Tq¢¢0
(19)
.
1
[DEd¢¢ + (X d¢ - X d¢¢)DId - DEd¢ ] Td¢¢0
(20)
.
DE q¢ = .
DE d¢ = -
DE q¢¢ = -
DE d¢¢ = -
Vol 7 (S6) | October 2014 | www.indjst.org
det (s[I] – [A]) = 0
(23)
The values of s that satisfy (poles of system) are known
as eigenvalues of the matrix A. The eigenvalues may be
real or complex. For an n x n matrix they are n eigenvalues. If A is real eigenvalues occur in complex conjugates
of the form [6]
λ = σ ± jω.
(24)
5.1 Eigenvalues and Stability
The power system is stable if all of the eigenvalues are on
the left-hand side of the imaginary axis of the complex
plane; otherwise it is unstable. If any of the eigenvalues
comes over into the right hand side of the imaginary axis,
the corresponding modes are said to be unstable, and so
is the system.
The desired aim then is to have all eigenvalues in the
left hand plane. This stability is confirmed by looking at
the time dependent characteristic of the oscillatory modes
corresponding to each eigenvalue λ given by e λ t.
A real eigenvalue corresponds to a non oscillatory
mode. A negative real eigenvalue indicates a mode that
decays in time (the larger the magnitude of the eigenvalue the quicker the decay)9. A positive real eigen value
indicates a mode that grows with time and is system will
experience aperiodic instability.
A conjugate pair complex eigenvalues indicate oscillatory modes of response.
λ = σ ± jω
(25)
Indian Journal of Science and Technology
43
Multi-Machine Small Signal Stability Analysis For Large Scale Power System
1. If a conjugate of pair of complex eigenvalues has negative real parts σ this corresponds to an oscillatory
mode that decays with time and the system is said to
be globally stable.
2. If a pair has positive real parts, the corresponding
oscillatory mode grows exponentially with time and
eventually dominates the system behavior. Such a system is said to be unstable.
3. If any one of the eigenvalues has a real part, the system
will have an undamped oscillatory response.
Eigenvalues associated with an unstable or poorly damped
oscillatory mode are called dominant modes since
their contribution dominates the time response of the
system.
The real component of an eigen value gives the
damping, and the imaginary part gives the frequency of
oscillation.
Frequency of oscillation (Hz):
f =
w
2p
(26)
The damping ratio:
z=
-s
s 2 +w2
(27)
Eigen analysis also includes the computation of eigenvectors. For any eigen value λi, the column vector φi that
satisfies is called the right eigenvector for λi
[A]φi = λ φi i = 1, 2, … n
Vol 7 (S6) | October 2014 | www.indjst.org
ÈY i1 ˘
ÍY ˙
Í i2 ˙
Yi = Í  ˙
Í ˙
Í  ˙
ÍY in ˙
Î ˚
(29)
The left eigenvectors, together with the initial conditions of the system state vector x, determine the
magnitudes of the modes.
5.3 Mode Shape
The right eigenvector gives the mode shape.
The mode shape is the relative activity of the state variables when a particular mode is excited. Thus the degrees
of activity of the Kth state variable xk in the Ith mode is
given by the element φki if the right eigenvector φi.
The magnitude of the elements if φi gives the extent
of the activities of the n state variables in the Ith mode,
and the angles of the elements give phase displacement of
the state variables with regard to the mode. Thus we can
use the mode shape to analyze the magnitude and phase
displacement of the speed and rotor angle state variables
in an oscillatory mode.
The participation factor P is useful in identifying those
states which have the most influence on any mode. The
participation factor is non-dimensional.
(28)
The right eigenvector show the distribution of the
modes of response (eigenvalues) through the power system state variables. Correspondingly, there is a row vector
ψi that satisfies (24) and is called the left eigenvector of
[A] associated with the eigen value λi.
44
ψi [A] =λ ψi
6. Participation Factors
5.2 Eigen Vectors
Èj1i ˘
Íj ˙
Í 2i ˙
ji = Í  ˙ Í ˙
Í  ˙
ÍÎj 2i ˙˚
È P1i ˘ È j1iY i1 ˘
Í P ˙ Íj Y ˙
2i
2i i 2 ˙
Pi = Í ˙ Í
Í  ˙Í  ˙
˙
Í ˙Í
ÎPni ˚ ÎjniY in ˚
(30)
The element pki = ψki φik is called a participation
f­ actor. It is a measure of the relative participation of the
kth state variable in the ith mode, and vice versa5. In effect
participation factors are useful in identifying those state
variables which have the most influence on any mode.
The higher the value of participation factor of a state
for a corresponding mode, the more active that state is in
that mode when compared with the other states.
Thus the values of participation factors can reveal
which generators are involved in a particular mode. It can
Indian Journal of Science and Technology
C. Chennakesavan and P. Nalandha
reveal which machine or machines could go out of step
for any known mode or modes that might cause problem
in the power system in the advent of load variation10.
The participation factors are used to identify the areas
in the power system where any mode or (oscillation) has
most of its effect.
6.1 Eigen Value Sensitivity to Load Changes
The sensitivity of an eigen value λi to an element αkj of the
state matrix is equal to the product of the left eigenvector
element ψki and right eigenvector element φji.
∂li
= Y i k ji j ∂a k j
(31)
In this analysis the load power at a bus in the power
system was varied and the system state matrix computed.
The elements αkjC of the state matrix that showed the
greatest changes due to the load change were identified.
Thus the sensitivity of the eigenvalues to these elements
of the state matrix A sys can be computed using equation.
In computing the eigen value sensitivity the elements
that show a large variation with a change in load give
an indication of the sensitivity of the eigen value to load
changes.
Eigen-analysis of the system can then be performed by
obtaining the eigenvalues of the state matrix. The eigenvalues are then check for stability.
6.3 Load Model Procedure
1. The type of load at various buses was selected i.e.
changing parameters np and nq
2. The system matrix A sys was computed
3. The eigenvalues are computed and check for stability
analysis.
7. Simulation Results
A standard 11-bus 4 machine 2 area as shown in system.
The initial relative rotor angle obtained from the load flow
solution or initial condition.
Table 1.
EIGEN VALUES
NO
Real
Imaginary
1
−0.76e – 3
0.22e – 2
2
–
−0.22e – 2
Frequency (HZ)
Damping
Ratio
0.0003
0.331
–
–
0.545
0.032
3
−0.096
–
4
−0.111
3.43
5
−0.111
−3.43
6
−0.117
–
–
–
7
−0.265
–
–
–
8
−0.276
–
–
–
9
−0.492
6.82
10
−0.492
−6.82
11
−0.506
7.02
12
−0.506
−7.02
∗ Loads system data
∗ Performs load flow of system, calculates initial
­condition, linearize system equation and construct the
matrices and finally the system state matrix A sys.
∗ Calculate system’s eigenvalues a check for unstable
mode and gets the participation and mode shape for
critical Eigenvalues.
13
−3.428
–
–
–
14
−4.139
–
–
–
15
−5.287
–
–
–
16
−5.303
–
–
–
17
−31.03
–
–
–
18
−32.45
–
–
–
19
−34.07
–
–
–
The effect of loading was analyzed by increasing the load
at a particular bus or buses. At each change in loading
the initial conditions of the state variables were calculated
after running a load flow [9]. Next, linearization of the
system equation was done and the state matrix A sys is
formed.
20
−35.53
–
–
–
21
−37.89
0.142
22
−37.89
−0.142
23
−38.01
0.038
24
−38.01
−0.038
6.2 Simulation
The test system used in this investigation is the IEEE
– 11 bus test system. Matlab 7.5 was used to analyze this
system. The system is a consists of 4 generators. Software
was written to perform the small signal analysis on the
system. This program follows three main algorithms:
Vol 7 (S6) | October 2014 | www.indjst.org
1.087
0.072
1.117
0.072
0.023
1.0
0.006
1.0
Indian Journal of Science and Technology
45
Multi-Machine Small Signal Stability Analysis For Large Scale Power System
Figure 2. (a,b,c) Mode shapes of rotor angle
From this table 1 all eigen values are coming negative.
So we see that system is Stable. There are three rotor angle
modes of oscillation. Their mode shapes are shown in
Figure 2(a, b, c)
8. Conclusion
In this investigation a dynamic model for the power system was developed to investigate the development of low
frequency oscillations in a power system. The test system
was used the IEEE 11 bus test system. A power system
model was enveloped in Matlab and testing was done to
determine the power system stability.
Software developed performed the following function:
load flow of power system, calculation of the initial values
of the power system model and the construction of the
system state matrix. The software then applied the linear
analysis tools to find Eigenvalues of A sys, Participation
factors, and mode shape.
The Eigenvalues of A sys, Participation factors, and
Mode shape can be used to determine if system could
develop oscillations and where they occur in the power
system if excited.
The purpose of this research was to investigate the
Contribution of load to low frequency oscillations in
Power System8. As a result the system was analysis by first
examining the effect of loading on the development of
system oscillations. Secondly the effect of load model on
low frequency oscillation was determined.
In the power system leads to the development of
many oscillations at low frequency in the power system.
Finally the effect of the load model was that it was easier
to ­identify unstable modes of oscillation.
46
Vol 7 (S6) | October 2014 | www.indjst.org
Bus 7
PL = 967MW
QL = 100 MWAR
QC = 200 MWAR
Bus 9
PL = 1,767MW
QL = 100 MWAR
QC = 350 MWAR
9. Appendix
The system consists of two similar areas connected by a
weak tie. Each area consists of two coupled units, each
having a 900MVA & 20 KV. The generator parameters in
per unit on the rated MVA and KV base are as follows.
Xd = 1.8
Xq = 1.7
Xl = 0.2
Xd¢ = 0.25
Xq¢ = 0.25
Td0
¢ = 8.0 S
Tq0
¢ = 0.4 SS
Td0
¢ = 0.03 S
Tq0
¢ = 0.05 S
Asat = 0.015
Bsat = 9.6
jT1 = 0.9
kD = 0
Xq¢ = 0.55
Ra = 0.0025
H = 6.5 (for G! & G2)
Xd¢ = 0.3
H = 6.175 (for G3 & G4)
Each Step-up transformer has an impedance of
0 + j0.15 per unit on 900 MVA and 20/230 KV base, and
has an off-nominal ratio of 1.0
The parameters lines in per unit on 900 MVA &
230KV
R = 0.0001 pu/km
XL = 0.001 pu/km
Bc = 0.00175 pu/km
Then the generating units loaded as given below.
G1
G2
G3
G4
P = 700 MW
P = 700 MW
P = 719 MW
P = 700 MW
Q=185MW
Q = 235MW
Et = 1.03 – 20.2
G1

G2
Et = 1.01 – 10.5

Q = 176MW
Et = 1.03 – - 6.8
Q = 202MW
Et = 1.01 – - 17.0
G3
G4
The load and reactive power (Qc) by the shunt ­capacitors
at b uses 7 and 9 as follows
Indian Journal of Science and Technology
C. Chennakesavan and P. Nalandha
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Indian Journal of Science and Technology
47