chapter 4 effective stabilizing system for multi-machine...
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CHAPTER 4
EFFECTIVE STABILIZING SYSTEM FOR
MULTI-MACHINE POWER SYSTEM USING
OPTIMAL TUNING OF PSS PARAMETERS
4.1 INTRODUCTION
One of the major issues in the PSS application in the multi-machine
power system for the small signal stability enhancement is tuning of the PSS
parameters. There are so many approaches for finding solution to the problem
of optimal tuning of power system stabilizers in multi-machine power system.
In the past two decades, there has been considerable research in the area of
optimal tuning of power system stabilizers in multi-machine power system. In
multi-machine system with poorly damped modes of oscillations, several
stabilizers have to be used and the problem of tuning of PSS parameters
becomes relatively complicated. This chapter addresses the optimal tuning of
the PSS parameters in the multi-machine power system.
4.2 PROPOSED METHOD FOR THIS WORK
The objective is to find an effective stabilizing system for damping
different critical modes under different operating conditions of a multi-
machine power system using optimal tuning of PSS parameters.
The proposed method has the following steps:
1. Determining the critical modes of power system for different
operating conditions from the corresponding linearized
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models and then selecting a set of dominant critical modes and
ranking the same based on the damping ratio.
2. Identification of the best location of the PSSs for damping the
selected critical modes.
3. Optimal Tuning of the parameters of the PSS for the selected
location using the parameter constrained non-linear
optimization technique.
4. Checking the dynamic responses of the multi-machine power
system after installing the optimally tuned PSSs in the
identified locations for the adequacy of the stabilization.
These steps are explained in detail in the following sections.
4.3 IDENTIFYING AND RANKING THE DOMINANT
CRITICAL MODES OF THE POWER SYSTEM
The multi-machine test system is modeled with each generator has
two axis with four states, '
d∆E , '
q∆E , ∆ω , ∆δ . The IEEE type ST1A model
excitation system has been included for all the generators.
The linearized state equations for the two-axis, fourth order model
generator are as follows:
X AX•
= (4.1)
where A is a state matrix.
As explained in Chapter 2, the linearized state equations for
generator and exciter are given in Equation (4.2) and Equation (4.3)
respectively. The complete state vector of the multi-machine power system
with exciter is given by the Equation (4.4):
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They are as follows:
p∆Ed' i = { - ∆Ed'i - (xqi - x'i) ∆Iqi } / τqo'i ; i = 1,...n
p∆Eq'i = { ∆EFDi - Eq'i + ( xdi - x'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 (4.2)
The state space equation for the exciter,
( )i
Aifdi Ref i fdi
Ai Ai
-K 1p∆E = -∆V +∆V - ∆E ;
T T i=1,…n (4.3)
xT
i = [∆E'di ∆E'qi ∆ωi ∆δi ∆EFDi ] ; i=1,…n (4.4)
From the A matrix, the critical modes are identified for each one of
the critical operating conditions. From all the critical modes corresponding to
different operating conditions, a set of dominant critical modes are identified
and ranked based on the ascending order of the damping ratio.
4.4 IDENTIFICATION OF BEST LOCATION OF THE PSSs IN
A MULTI-MACHINE POWER SYSTEM
For the selected set of dominant critical modes, the best location of
the PSS for each one of the dominant critical modes is identified using RGA
method, In frequency domain, the RGA matrix can be calculated for the
frequency corresponding to the critical mode. Then the rows and columns of
the RGA matrix are rearranged in such a way that the RGA matrix is closer to
identity matrix. This is done by the application of Genetic Algorithms.
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The selection of control loops from the RGA analysis is explained
in such a way that the RGA matrix is used to find out the best input – output
signals which give the good control of PSS which has the speed input signal
and voltage reference signal as output and also that because of the selection of
this input – output pair, the interaction effects by the other PSS control loops
if any over this feedback loop (PSS) should be minimized.
4.5 OPTIMAL TUNING OF THE PARAMETERS OF THE PSS
USING PARAMETER CONSTRAINED NON-LINEAR
OPTIMIZATION TECHNIQUE
In order to determine the tuned parameters of PSS (to damp a dominant
critical mode) connected to the identified machine, the multi-machine power
system is reduced to an equivalent SMIB system retaining that particular machine
alone. This step is repeated for each one of the selected dominant critical modes.
This problem is posed as a non-linear optimization problem.
4.5.1 Problem Formulation for the Optimal Tuning of the PSS
The objective of the optimization problem is to maximize the damping
ratio ζ of the dominant critical mode of that machine which is equivalent to the
minimization of non-linear function (i.e) Damping Index (DI).
DI = (1 )− ζ (4.5)
The optimization programming problem is stated as follows:
To determine KPSS and T1 which will minimize the following function:
K Gc(j ) GEP (j )n nPSSMin: (1- ) = 1 -
2 Mn
ω ω ζ
ω (4.6)
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Subject to KPSS min
≤ KPSS ≤ KPSS max
(4.7)
T1min
≤ T1 ≤ T1max
(4.8)
The PSS parameters used for the optimization are KPSS and T1. Parameter T2 is
fixed.
The expression for damping ratio ‘ζ’ in terms of machine and PSS
parameters is derived and obtained as follows:
Electric torque ∆Te and mechanical torque ∆Tm are assumed to be
on the ∆δ- ∆ω plane .To derive an extra damping ∆TE through the
supplementary excitation, ∆Te must be in phase with the ∆ω. Similarly extra
damping through the governor control must be in phase with the -∆ω. In the
general case of supplementary excitation and governor control of low
frequency oscillations, the extra electric damping ∆TE be included in the ∆Te
and the extra mechanical damping ∆TM be included in ∆Tm (Yu 1983).
The derivation of objective function is done based on the SMIB model of the
system.
Electric torque, ∆Te = K1∆δ + DE∆ω (4.9)
where, DE damping coefficient due to the extra damping
K1 is the Phillips Heffron constant
Assume that the original ∆Tm from the regulator governor control is still
negligible,
So Mechanical torque, ∆Tm = - DM ∆ω (4.10)
where, DM is the mechanical damping coefficient due to extra damping.
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The mechanical damping torque ∆TD synchronous generator model can be
expressed
∆TD = D ∆ω (4.11)
The characteristics equation of the system mechanical loop is given as
(Ms2+ (Dm+DE+D) s+ωb K1) ∆δ = 0 (4.12)
where ∆δ = (ωb ∆ω) / s,
ωb is the system frequency
Normalization yields the equation (4.12) as,
(s2 + 2 ζ ωn s + ωn
2 ) ∆δ = 0 (4.13)
Equating equation (4.12) and equation (4.13) yields
ζ = (Dm+DE+D)/2ωnM (4.14)
where, ωn - undamped mechanical mode oscillating frequency in radian
per second
ζ - damping coefficient in per unit
M = 2H where H is the inertia constant.
So undamped mechanical mode oscillating frequency can be directly
calculated as per the equation, neglecting all damping in the characteristics
equation,
ωn = (ωb K1/ M )1/2
(4.15)
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From the study of SMIB system model, the AVR loop can be
represented as shown in Figure 4.1.
Figure 4.1 AVR loop for Plant Transfer Function (GEP(s))
The transfer function of this loop is given by:
A 2 3s
A 3 do' A 3 6
K K KGEP(s) = Te(s)/ V (s) =
(1+ sT ) (1+sK T ) + K K K∆ ∆ (4.16)
where GEP(s) is the plant tranfer function
K1 to K6 are the Heffron Philips Constants (Padiyar 2002).
The PSS transfer function can be expressed from Figure 2.10 from the
Chapter 2 is as follows:
∆Vs(s)/∆ω(s) = KPSS Gc(s)Gw(s) (4.17)
where, KPSS - PSS gain
Gw(s) - washout transfer function transfer function sTw
(1 sTw)=
+
∆Vt
∆Te(s) A
A
K
1 s T+3
3 do
K
1 sK T '+K2
K6
∆Vs(s)
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Gc(s) - phase compensator transfer function = 1 sT
1
1 sT2
+
+
From equation (4.17):
{∆Vs(s)/∆Te(s)} x {∆Te(s) / ∆ω(s)} = Kpss Gc(s) Gw(s) (4.18)
Since from equation (4.9), ∆Te = K1∆δ+DE∆ω
Te(s)
DE0(s)
∆=
∆δ =∆ω (4.19)
From Equation (4.16),
∆Vs(s)/ ∆Te(s) = 1 / GEP(s) (4.20)
From Equation (4.19),
∆Te(s)/∆ω(s) = DE (4.21)
Using Equation (4.20) and Equation (4.21), the Equation (4.18) can
be written as follows:
[1/GEP(s)] * [DE] = KPSS Gc(s) Gw(s) (4.22)
From the Equation (4.14) neglecting mechanical damping,
DE = 2 ζ ωnM (4.23)
So equation (4.22) by substituting DE,
[1 / GEP(s)]*[2 ζ ωn M] = KPSSGc(s) Gw(s) (4.24)
Assuming, Gw(s) = 1 because Tw >> 1.
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From the equation (4.24), the damping ratio in terms of PSS parameters is
given by
K Gc (s) GEP (s)
PSS = 2 Mn
ζω
(4.25)
where, s = jωn
4.5.2 Parameter Constrained Non-Linear Optimization
The main objective of this method can be very clear with the help
of the Figure 4.2. Among dominant swing modes only those have damping
ratio less than cr
ξ are considered in the optimization. In Figure 4.2, '+' sign
indicates eigen values before optimization and '∗ ' sign indicates eigen values
after optimization. The optimization is done by using the Sequential
Coordinated Programming.
Figure 4.2 Objective of optimization
Cos-1 ζ
+ Inter area modes
+ Local modes *
*
σ
jω
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4.6 PLACEMENT OF OPTIMALLY TUNED PSS IN THE
IDENTIFIED MACHINES
After obtaining the optimized parameters of the PSSs for the
identified individual machines, the PSSs with the optimal parameters are
installed on the identified machines. Then the simulation is carried out in
order to check the adequacy of the damping.
4.7 SYSTEM INVESTIGATED AND SIMULATION RESULTS
In order to verify the performance of the proposed tuning method,
10-machine 39-bus New England test System is used. The system data and
the parameters of all the generating units, transmission lines and loads are
given in Appendix 2.
For the nominal operation condition of the test system, there are
eight critical modes (Table 4.1).
Table 4.1 Critical Modes of New England Test System (Nominal
operating condition)
Mode No. Eigen Values Damping Ratio
1 -1.0361 ± j9.7046 0.1062
2 -0.8359 ± j8.7394 0.0952
3 -1.0388 ± j7.8218 0.1317
4 -0.5390 ± j6.6460 0.0808
5 -0.5013 ± j6.4141 0.0779
6 -1.0606 ± j6.4005 0.1635
7 -0.6701± j5.4632 0.1217
8 -0.7063 ± j3.5853 0.1933
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From the damping factors of the eigen values, it is observed that the
damping of all the critical modes is unsatisfactory. Robustness of the test
system is checked by choosing the following different operating conditions:
a) Line outage (21-22) in the system
b) Line outage (21-22) and 25% load increase in the 16th
and 21st
bus
c) 25% generation increase in generator 7.
The critical swing modes obtained for the above three operating
conditions are given in Table 4.2, Table 4.3 and Table 4.4.
Table 4.2 Critical Modes of Test System (operating condition (a))
Mode No. Eigen Values Damping Ratio
1 -0.9652 ± j9.7412 0.0986
2 -0.08668±j8.6973 0.0992
3 -0.9455±j7.9622 0.1179
4 -0.4930±j6.6234 0.0742
5 0.4986±j6.3299 0.0785
6 -0.8394 ±j5.6291 0.1475
7 -0.7620±j4.9294 0.1528
8 -1.1606 ± j8.4005 0.1839
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Table 4.3 Critical modes of Test System (operating condition (b))
Mode No. Eigen Values Damping Ratio
1 -0.9568 ± j9.7649 0.0975
2 -0.8588 ± j8.7064 0.0982
3 -0.9319 ± j7.9825 0.1160
4 -0.9607 ± j7.4006 0.1815
5 -0.4976 ± j6.5604 0.0756
6 -0.5281 ± j6.3384 0.0830
7 -0.8243 ± j5.6254 0.1450
8 -0.7550 ± j4.9210 0.1516
Table 4.4 Critical Modes of Test System (operating condition (c))
Mode No. Eigen Values Damping Ratio
1 -0.7684 ±j 11.051 0.0694
2 -0.6112 ±j10.1157 0.0603
3 -0.8154 ±j 9.0787 0.0895
4 -0.5308 ±j 8.2943 0.0639
5 -0.3265 ±j 7.5605 0.0431
6 -0.2409 ±j 7.1092 0.0339
7 -0.3061 ±j 6.8213 0.0448
8 -0.2928 ±j 4.3861 0.0666
All the above four different operating conditions (including the
nominal operating condition) are taken and a set of dominant critical modes
(say nine modes) are selected .The damping ratios corresponding to all the
critical modes for the four different critical operating conditions are tabulated
as shown in Table 4.5. Then the dominant critical modes are selected and
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ranked based on ascending order of the damping ratios as shown in the
Table 4.6.
Table 4.5 Selection of Critical Modes
Damping ratio of
the critical modes
of base case
condition
Damping ratio of
the critical modes
of op condition
(a)
Damping ratio of
the critical modes
of op condition
(b)
Damping ratio of
the critical
modes of op
condition(c)
0.1062 0.0986 0.0975 0.0694
0.0952 0.0992 0.0982 0.0603
0.1317 0.1179 0.1160 0.0895
0.0808 0.0742 0.1815 0.0639
0.0779 0.0785 0.0756 0.0431
0.1635 0.1475 0.0830 0.0339
0.1217 0.1528 0.1450 0.0448
0.1933 0.1839 0.1516 0.0666
Table 4.6 Ranking of critical dominant modes
Damping ratio of
dominant critical
modes
0.0339
0.0431
0.0448
0.0603
0.0639
0.0666
0.0694
0.0742
0.0756
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The operating conditions corresponding to all the dominant critical
modes are tabulated in Table 4.7.
Table 4.7 Operating condition corresponding to the selected dominant
critical modes
Critical modes Damping ratio Operating Condition
-0.2409 ±j7.1092 0. 0339 Case (c)
-0.3265±j7.5605 0. 0431. Case (c)
-0.3061±j6.8213 0.0448 Case (c)
-0.6112±j10.1157 0.0603 Case (c)
-0.5308±j8.2943 0.0639 Case (c)
-0.2928± j4.3861 0.0666 Case (c)
-0.7684±j11.0517 0.0694 Case (c)
-0.4930±j6.6234 0.0742 Case (a)
-0.4976 ± j6.5604 0.0756 Case (b)
In Table 4.7, for the first most critical mode (-0.2409 ±7.1092i)
with the damping ratio is 0.0339 which corresponds to the operating
condition- case (c). The optimum locations of PSSs to damp out this dominant
critical mode is found out using RGA method. The RGA matrix is calculated
for the frequency of this critical mode. Then the modified RGA matrix is
found out using Genetic Algorithm (Table 4.8).
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Table 4.8 Modified RGA Matrix for the first mode dominant critical mode
∆Vs
∆ω ∆Vs1 ∆Vs2 ∆Vs3 ∆Vs4 ∆Vs5 ∆Vs6 ∆Vs7 ∆Vs8 ∆Vs9 ∆Vs10
∆ω1 1.0168 -0.001098 0.000687 -0.000765 0.000001 0.0000154 0.0000991 0.0000122 -0.004987 0.000085
∆ω2 0.00089 1.0198 -0.004235 0.000987 0.00067 0.0000234 -0.0000772 -0.000987 0.0002897 0.0000076
∆ω3 -0.00138 2.987E-18 0.98765 1.0987E-18 0.000987 0.000009 3.88E-18 -0.0000002 1.07E-14 0.00001
∆ω4 -0.00548 0.00987 0.000685 1.0190 -0.00097 -0.00000112 5.687E-25 7.87E-18 -1.0497E-15 0.000064
∆ω5 0.00025 -0.00023 -3.82E-18 -1.087E-19 1.0177 -0.000087 -3.87E-18 5.678E-12 0.00006879 7.89E-21
∆ω6 9.04E-17 -0.00045 0.000986 -0.0000099 0.000067 0.93465 0.0000112 0.0000677 0.0002354 0.00009
∆ω71 -0.0009 9.04E-17 -0.004926 -0.0000987 -0.000055 0.0000123 0.93918 -0.000076 -0.0009987 0.00012
∆ω8 -0.00025 0.00067 0.000987 -0.000076 0.000078 0.000998 -1.987E-25 0.92350 -0.00000632 -1.176EE-25
∆ω9 0.00125 0.000099 -1.687E-19 0.000004 -0.000076 -1.468E-18 0.000987 0.000987 1.0021 6.58E-18
∆ω10 -0.00617 0.00187 0.000987 -1.4687E-18 0.000056 3.224E-18 5.687E-14 0.0000076 2.956E-18 1.0035
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From Table 4.8, the machine location order is found out as per
RGA method and shown below.
1. Machine No. 9
2. Machine No. 10
3. Machine No. 1
4. Machine No. 5
5. Machine No. 4
6. Machine No. 2
After identified the best locations of PSS using RGA method, as
per Table 4.8, the optimal parameters are found out using the detail explained
in section 4.4.
Non-linear programming problem for the first machine can be
formulated like this:
n nK GEP (j ) Gc(j )
PSSMin: (1- ) = 1 - 2 Mn
ω ωζ
ω
Subject to 10 ≤ KPSS ≤ 90
0.001 ≤ T1 ≤ 1.6
T2 is fixed as 0.06 seconds. For the above problem, optimized
controller parameters for the first machine are obtained as KPSS = 12.0432 and
T1 = 0.0010 secs.
Like above the optimal PSS parameters are found out for all the
machines to damp out its critical mode and the optimized values are shown in
Table 4.9.
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Table 4.9 Optimized controller parameters
Gen.no KPSS T1 in secs
1 12.0432 0.0010
2 12.099 0.0011
3 11.2791 0.0016
4 11.2792 0.0016
5 12.0056 0.0015
6 11.2792 0.0016
7 10.6075 0.00165
8 12.0432 0.0016
9 11.9235 0.0018
10 12.9567 0.0016
For the first most critical mode (-0.2409 ±7.1092i), the first PSS
with the optimized parameters is connected to the most dominant RGA
element with input signal of ∆ω9 and ∆Vs9 as the output signal to place PSS
on the Machine No.9. The simulation result i.e., dynamic response of the rotor
angle deviation (∆δ13) of the test system, when only one PSS with the
optimized parameters is connected to the 9th
machine is shown in Figure 4.3.
The time axis is taken as time in per unit. The actual time
t actual = time in per unit / ωbase
Figure 4.3 System response of rotor angle deviation ∆δ13 when PSS
connected to the 9th
machine
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From Figure 4.3, it is observed that the damping is not adequate.
The dynamic response is full of oscillations. So one more PSS machine with
its optimized parameters is connected to 10th
which is the machine
corresponding to the next dominant element in the RGA matrix corresponding
to the first critical mode. Simulation result is shown in Figure 4.4.
Figure 4.4 System response of rotor angle deviation ∆δ13 when PSS
connected to the 10th
machine
From Figure 4.4, it is observed that the damping is not adequate.
The dynamic response is full of oscillations. So one more PSS is connected to
first machine which is the machine corresponding to the next dominant
element in the RGA matrix corresponding to the first critical mode.
Simulation result is shown in Figure 4.5.
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Figure 4.5 System response of rotor angle deviation ∆δ13 when PSS
connected to the 1st machine
Simulation result of Figure 4.5 reveals that damping of rotor
oscillations is not still adequate. So more PSSs are connected to the system in
machine no.5, machine no.4 and machine no.2 sequentially as per the locating
order of machines. The PSSs are installed in the identified machines with the
optimized controller parameters as shown in Table 4.9.
The system responses when PSSs located in machine no.5, machine
no.4 and machine no.2 sequentially are shown in Figure 4.6 to Figure 4.8.
Finally well damped condition is obtained after connecting six PSS
parameters in the above locating order.
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Figure 4.6 System response of rotor angle deviation ∆δ13 when PSS
connected to the 5th machine
Figure 4.7 System response of rotor angle deviation ∆δ13 when PSS
connected to the 4th
machine
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Figure 4.8 System response of rotor angle deviation ∆δ13 when PSS
connected to the 2nd machine
Hence finally PSSs are connected to the machines 9, 10, 1, 5, 4 and
2 of the system sequentially. In this case, simulation results demonstrate that
the oscillations are well damped for each mode separately in the system.
Totally six PSSs are connected sequentially in the test system in order to
damp out the first most critical mode. The above steps are repeated for all the
selected modes in the Table 4.7 to get the better damping performance in the
system. After installation, the improvement in damping ratios of all the
selected dominant critical modes is checked and is given in Table 4.10.
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Table 4.10 Comparison between the damping ratios of the dominant
critical modes before and after the placement of PSS
Critical modes Damping ratio
Before PSS
Damping ratio
After PSS
-0.2409 ±j7.1092 0.0339 0.3271
-0.3265±j7.5605 0.0431 0. 4050
-0.3061±j6.8213 0.0448 0.3140
-0.6112±j10.1157 0.0603 0.2981
-0.5308±j8.2943 0.0639 0.3362
-0.2928±j 4.3861 0.0666 0.2980
-0.7684±j11.0517 0.0694 0.3458
-0.4930±j6.6234 0.0742 0.3745
-0.4976 ± j6.5604 0.0756 0.4041
Table 4.10 reveals the enhancement of stability after placing the
PSSs for the selected critical modes. The same procedure is carried out for all
the remaining critical modes to improve the dynamic performance. Figure 4.9
to 4.17 shows the simulation results for the rotor angle deviation for the
operating condition (c) with the step change in 5° rotor angle perturbation.
Figure 4.9 System response of the test system for the rotor angle
deviation (∆δ12) for operating condition (c)
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Figure 4.10 System response of the test system for the rotor angle
deviation (∆δ13) for operating condition (c)
Figure 4.11 System response of the test system for the rotor angle
deviation (∆δ14) for operating condition (c)
87
Figure 4.12 System response of the test system for the rotor angle
deviation (∆δ15) for operating condition (c)
Figure 4.13 System response of the test system for the rotor angle
deviation (∆δ16) for operating condition (c)
88
Figure 4.14 System response of the test system for the rotor angle
deviation (∆δ17) for operating condition (c)
Figure 4.15 System response of the test system for the rotor angle
deviation (∆δ18) for operating condition (c)
89
Figure 4.16 System response of the test system for the rotor angle
deviation (∆δ19) for operating condition (c)
Figure 4.17 System response of the test system for the rotor angle
deviation (∆δ110) for operating condition (c)
90
4.8 CONCLUSION
This chapter discussed the design of an effective stabilizing system
for a multi-machine power system. The proposed approach is applied to the
10 machines, 39 bus New England test system. For the test system, the critical
modes are identified for different operating conditions from the linearized
model and a set of dominant critical modes among them are chosen and
ranked based on the damping ratio. For each one of the selected dominant
critical modes, the best location of PSS is identified based on RGA method.
For the identified PSSs as per the RGA method, the parameters are optimally
tuned by non-linear optimization technique using Sequential Quadratic
Programming Method. Then the optimal parameters are installed on the
respective machines. Simulation is carried out by taking the 5% step change
in rotor angle as the disturbance and the results demonstrate the effectiveness
in improving the small signal stability of the system.