Model based PI power system stabilizer design for damping low frequency oscillations in power systems

Research article
Model based PI power system stabilizer design for damping low
frequency oscillations in power systems
Aprajita Salgotra*
, Somnath Pan
Department of Electrical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India
a r t i c l e i n f o
Article history:
Received 7 December 2017
Received in revised form
22 February 2018
Accepted 18 March 2018
Available online 26 March 2018
Keywords:
Frequency response matching
Reference model
Direct synthesis
Low frequency oscillation
One machine infinite bus (OMIB) system
Proportional-integral power system
stabilizer (PI-PSS)
a b s t r a c t
This paper explores a two-level control strategy by blending local controller with centralized controller
for the low frequency oscillations in a power system. The proposed control scheme provides stabilization
of local modes using a local controller and minimizes the effect of inter-connection of sub-systems
performance through a centralized control. For designing the local controllers in the form of
proportional-integral power system stabilizer (PI-PSS), a simple and straight forward frequency domain
direct synthesis method is considered that works on use of a suitable reference model which is based on
the desired requirements. Several examples both on one machine infinite bus and multi-machine sys-
tems taken from the literature are illustrated to show the efficacy of the proposed PI-PSS. The effective
damping of the systems is found to be increased remarkably which is reflected in the time-responses;
even unstable operation has been stabilized with improved damping after applying the proposed
controller. The proposed controllers give remarkable improvement in damping the oscillations in all the
illustrations considered here and as for example, the value of damping factor has been increased from
0.0217 to 0.666 in Example 1. The simulation results obtained by the proposed control strategy are
favourably compared with some controllers prevalent in the literature.
© 2018 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
The low frequency oscillations in power systems had been
noticeably observed in 1920s in the form of spontaneous oscilla-
tions and became more prominent when synchronous generators
are interconnected to build-up more capacity. These oscillations are
undesirable even at low frequencies (i.e. 0.1e2 Hz) as they reduce
the power transfer capability of the transmission line [1]. The ef-
forts to transfer bulk power across weak transmission line may lead
to the low frequency oscillations [2]. It may be due to any distur-
bance such as sudden change in transmission line parameters or
fluctuations in the turbine output power. In order to improve the
transient stability, the gain is kept high for fast excitation that
further aggravates the problem. To address this problem, a viable
solution has been widely accepted by the power industries in the
form of power system stabilizer (PSS). It adds a stabilizing signal to
the excitation system to compensate the phase lag resulting from
the voltage regulator, exciter and synchronous generator. In effect
of this, the overall damping of the system is improved.
The two-level scheme has been observed in Ref. [3] where the
overall system is decomposed into different sub-systems and an
optimal controller is designed for each sub-system. This method
involves tedious computation for obtaining the optimal gain of
order n as nðn þ 1Þ=2 Riccatti equations are to be solved.
Proportional-integral (PI) controllers (analog and digital) for one
machine infinite bus (OMIB) and multi-machine systems have been
designed in Ref. [4], however, interactions among the machines are
not considered. Design of two-level PSS has been discussed in
Ref. [5] where the effects of one machine over the other machines
following a change in the mechanical torque in one or both the
machines are simulated. A dynamic pole assignment technique has
been proposed in Ref. [6] that uses pre-specified eigen-structure. In
Ref. [7] a power system stabilizer is designed using the output
feedback control based on the reduced-order model derived by the
balanced truncation method. A two-level control strategy com-
bined with order-reduction has been addressed in Ref. [8] to ensure
fast convergence of the designed results. However, an approximate
model matching technique for obtaining the parameters of PID
controller may be observed in Ref. [9]. This technique may be
applied without model-reduction, even for high-order systems.
Moreover, model free control has been employed for differential
* Corresponding author.
E-mail addresses: aprajita.salgotra3@gmail.com (A. Salgotra), somnathpan123@
gmail.com (S. Pan).
Contents lists available at ScienceDirect
ISA Transactions
journal homepage: www.elsevier.com/locate/isatrans
https://doi.org/10.1016/j.isatra.2018.03.013
0019-0578/© 2018 ISA. Published by Elsevier Ltd. All rights reserved.
ISA Transactions 76 (2018) 110e121

algebra based intelligent PID control in Ref. [10], time-delay esti-
mation based intelligent PI control in Refs. [11,12], sliding mode
control based PI control in Ref. [13]. In Ref. [14], a model matching
approach is used to solve a H∞ problem and the parameters are
determined by closed-loop shaping. Recently, Yaghooti et al. [14]
has designed a coordinated PSS using model reference adaptive
technique. For multi-machine system, the mathematical modeling
and block diagram showing the interactions among various sub-
systems has been explained elaborately in Ref. [15]. Among the
efforts to address the low frequency oscillations problem, energy
storage devices offer a viable solution to maintain the power sys-
tem stability and its role has been elaborated in Ref. [16].
Apart from these methods, some authors have used the soft
computing techniques that do not require a mathematical model of
the system. These methods include the artificial intelligence tech-
niques such as artificial neural networks [17], self-tuned fuzzy logic
[18], bacteria foraging optimization [19] and the heuristic searching
algorithm such as genetic algorithm [20], differential evolution
[21], particle swarm optimization [22], harmony search algorithm
[23], etc. These methods rely on iterative procedure and most of
these techniques have high computation burden.
Furthermore, many methods prevalent in the literature require
reduction of the system model to design the controllers. With all
these in view, always there is a need for finding a simple method
which will result in a simple but effective controller for operation
and performance.
In this paper, a PI-PSS is proposed owing to its simplicity in
structure, robustness, ease of implementation and maintenance. A
suitable phase-lead may be obtained by employing a PI controller
to compensate the phase-lag introduced between the exciter input
and the electrical torque [4]. Here, the aim is to propose a design
procedure that is simple in mathematics, less involved in compu-
tation and independent of order (without requiring system order
reduction) and structure of the system. These objectives are ach-
ieved by blending two controllers, local and centralized e that re-
sults in effective achievement for the damping of the system. The
local controller is designed using the concept of a method as in
Ref. [9] which is involved in finding only two parameters of the PI
controller against finding six parameters of the conventional lead-
lag PSS. Unlike in Ref. [9], proposed method investigates in finding a
suitable reference model considering oscillatory/unstable dy-
namics of the nominal system along with the consideration of
desired response. This local controller ensures a minimum system
performance even when the centralized controller becomes inef-
fective in the event of any contingency. The centralized controller is
proposed to design using the method as in Refs. [5,8], for reduction
of the interactions among the sub-systems. Examples are taken
from the literature on one machine infinite bus (with IEEE type-
DC1 and IEEE type-ST1 exciters) and multi-machine power
systems.
The rest of the paper is organized as follows. The design
methodology is shown in Section 2. In Section 3 simulation of four
examples taken from the literature is illustrated. Finally, conclusion
is drawn in Section 4.
2. Design methodology
The small perturbation based block diagram of the one machine
infinite bus (OMIB) system with IEEE type DC excitation as shown in
Fig. 1 is considered here [24] for design of the PI-PSS as the local
controller.
2.1. Design of the local PI controller
The design method presented in Ref. [9] is followed here for
design of the PI controller for mitigating the low frequency oscil-
lations. The overall transfer function of the control system with the
unknown controller has been derived analytically considering the
mechanical torque deviation ðDTmÞ as input and the speed devia-
tion ðDuÞ as output as given by:
where, HðsÞ is the controller to be designed. GAðsÞ; GUðsÞ and GSðsÞ
represents the transfer function of the AVR, the saturation
compensation and the AVR stabilization circuit, respectively, while
G1ðsÞ consisting of all these three transfer functions represents the
IEEE Type-DC1 exciter system [25].
Nomenclature
xd; xq Direct and quadrature axis synchronous reactance, in
order
x0
d Direct axis transient reactance
xe; re Transmission line reactance and resistance, in order
vd; id; fd Direct axis voltage, current and flux linkage, in order
vq; iq; fq Quadrature axis voltage, current and flux linkage, in
order
ifd Field current
E0
q Q-axis generator internal voltage
EFD Field voltage
V∞, Vt Infinite bus and terminal voltage, in order
u Angular velocity
KA, TA Exciter gain and time constant, in order
T0
d0 Open-circuit generator field time constant
M Inertia constant
D Prime mover damping
Tm, Te Prime mover and Electrical torque, in order
q Angular position of direct axis w.r.t. stator
D Small excursions about an initial operating point
d Angle between quadrature axis and infinite bus
voltage
KE, TE Exciter gain and time constant, in order
KF ;TF Regulating stabilizing circuit gain and time constant,
in order
VF Stabilizing feedback signal of IEEE Type-DC1
excitation system
TCðsÞ ¼
G3ðsÞ½1 þ G1ðsÞG2ðsÞK6ðsÞ Š
1 þ G3ðsÞG4ðsÞK1 À G1ðsÞG2ðsÞG3ðsÞG4ðsÞK2K5 À G2ðsÞG3ðsÞG4ðsÞK2K4
þG1ðsÞG2ðsÞK6 þ G1ðsÞG2ðsÞG3ðsÞK2HðsÞ þ G1ðsÞG2ðsÞG3ðsÞG4ðsÞK1K6
(1)
A. Salgotra, S. Pan / ISA Transactions 76 (2018) 110e121 111

Considering, nðsÞ ¼ G3ðsÞ½1 þ G1ðsÞG2ðsÞK6ðsÞ Š;
dðsÞ ¼ 1 þ G3ðsÞ G4ðsÞK1 À G1ðsÞ G2ðsÞ G3ðsÞ G4ðsÞK2K5 À
G2ðsÞ G3ðsÞ G4ðsÞK2K4 þ G1ðsÞG2ðsÞK6 þ G1ðsÞ G2ðsÞ G3ðsÞ G4ðsÞK1K6
and gpðsÞ ¼ G1ðsÞG2ðsÞG3ðsÞK2 the closed-loop transfer function
along with the controller HðsÞ may be written as:
TcðsÞ ¼
nðsÞ
dðsÞ þ gpðsÞHðsÞ
(2)
The procedure for designing the local controller may be stated in
the following steps.
Step (a) From the step response of the open-loop system the
peak overshoot ðypÞ, the settling time ðtsÞ, damping factor ðzÞ, un-
damped natural frequency ðunÞ, etc. are determined.
Step (b) An achievable reference model, Mref is chosen based on
the desired requirements and the plant behavior.
Step (c) The transfer function of the closed-loop system along
with the unknown controller is derived as given by eqn. (2).
Step (d) Here, the closed-loop control system TcðsÞ is considered
to be following the performance ofMref ðsÞ. Hence, it may be written
that TcðsÞyMref ðsÞ wherefrom, the expression of the controller is
analytically derived as:
HðsÞ ¼
nðsÞ À Mref ðsÞ Â dðsÞ
Mref ðsÞ Â gpðsÞ
(3)
Step (e) The controller HðsÞ is now approximated in the form of a
PI controller, CðsÞ in terms of the frequency response using the
divided difference calculus as in Ref. [26] to obtained the following
relations:
CðsÞyHðsÞ where C ðsÞ ¼ KP þ
KI
s
(4)
CRðuÞju¼uk
yHRðuÞju¼uk
; k2½0; N À 1Š
CIðuÞju¼uk
yHIðuÞju¼uk
; k2½0; N À 1Š

(5)
where, CRðuÞ; CIðuÞ; HRðuÞ and HIðuÞ are the real and the imaginary
parts of CðsÞ and HðsÞ, respectively, and are all real functions of u.
Here, uk are sufficiently small frequency points around u ¼ 0. In
this case, for designing a PI controller with two unknowns only one
low frequency point is required.
Step (f) A suitable frequency point of matching u0 is chosen from
the frequency response of the open-loop system for the purpose of
matching.
It is evident from eqn. (5) that a value of u0 resulting in two
linear algebraic real equations with the unknown parameters
ðKp and KiÞ of the controller. The following expressions are
obtained:
aK ¼ b
a ¼
2
6
4
1 0
0 À
1
u0
3
7
5; K ¼ ½ KP KI ŠT
b ¼ ½ HRðu0Þ HIðu0Þ Š
9
=
;
(6)
2.2. Criteria for the selection of a suitable reference model
As the nominal system has low damping factor, the step
response of the system is highly oscillatory. In such case, designing
a control system aiming to have enormously high damping factor
without any oscillation would be difficult and may lead to degra-
dation of other performances of the system such as the controller
output, etc. This is more stringent in case of unstable system that
has negative damping factor. Therefore, a pair of complex conjugate
poles (allowing some oscillations in step response) is taken for the
reference model which would lead to an easily achievable perfor-
mance for the control system to be designed. Obviously, the
damping factor of the reference model is chosen a higher value
than that of the nominal system. In order to have zero steady-state
speed deviation, one zero at origin in the s-plane is considered. In
the sequel, good overall responses have been obtained that are
illustrated through the examples taken from the literature.
2.3. Design of the centralized controller
For multi-machine systems, the overall power system is
Fig. 1. Control configuration of OMIB with IEEE Type DC1 excitation system and PI-PSS.
A. Salgotra, S. Pan / ISA Transactions 76 (2018) 110e121112

assumed to be divided into separate sub-systems. When the sub-
systems are interconnected, the interactions act as perturbations
[27]. In order to reduce the effects of interactions, a centralized
controller is utilized and the strategy shown in Refs. [5,8], has been
adopted here. For the purpose of minimizing the effect of in-
teractions among various sub-systems in a multi-machine power
system (with the state-space matrices A; B; C), the global gain
matrix G is obtained as follows:
G ¼ ÀB0
H
where Hij ¼ Aij; isj
¼ 0; i ¼ j
(7)
where, B0 is the pseudo-inverse of B and Aij represents the off-
diagonal elements of state matrix. The elements of matrix H are
obtained by replacing the diagonal elements of the state matrix A
by zero. Now, the appropriate entries in the centralized controller
matrix are replaced by the local controller parameters to ﬁnally
obtain the overall controller that incorporate the effect of both the
controllers in the interconnected system.
3. Simulation results
3.1. Example 1: an OMIB with IEEE type-DC1 excitation system
An OMIB with IEEE Type-DC1 excitation system (Fig. 1) is
considered here for which the linearized state-space model is
available in Refs. [4,6]. As the proposed design procedure works
with the transfer function model, the transfer function of the
nominal system (with DTm, the mechanical torque deviation as
input and Du, the speed deviation as output) has been derived as:
From (8), the open-loop poles are at
À8:132±j8:985; À0:234±j10:773; À3:0854; and À1:55: The damp-
ing factor ðzÞ and natural frequency ðunÞ corresponding to the
dominant poles ðÀ0:234±j10:773Þ are 0.0217 and 10.8 rad/s,
respectively. The step response of the open-loop system with 0.05
pu input and the bode plot are shown in Fig. 2. It is observed that
the step response is highly oscillatory. According to the criteria for
selection of the achievable reference model (given in Section 2.2), a
pair of conjugate complex poles with z ¼ 0:53 and un ¼ 10:8 rad/s
are considered along with a zero at origin of the s-plane to ensure
zero steady-state speed deviation. Hence, the reference model,
Mref ðsÞ is chosen as:
Mref ðsÞ ¼
0:0193s
½1 þ sð0:052 þ 0:84jÞŠ ½1 þ sð0:052 þ 0:84jÞŠ
¼
0:0193s
0:0099s2 þ 0:1049s þ 1
(9)
For the chosen reference model, the settling time ðtsÞ is 0.859 s,
the peak overshoot ðypÞ is 5:163 Â 10À4s and time for peak over-
shoot ðtpÞ is 0.1212 s. According to the design procedure, a fre-
quency point of u0 ¼ 0:05 rad=s is chosen and a PI controller has
been obtained as:
CðsÞ ¼ À21:068 À
0:163
s
(10)
Then, the closed-loop control system becomes:
TcðsÞ
¼
0:211s5 þ4:509s4 þ50:26s3 þ183:9s2 þ157:9s
s6 þ21:37s5 þ353:4s4 þ3301s3 þ19160s2 þ56000sþ57520
(11)
The closed-loop poles are À4:66 ± j5:22; À 3:08±j11:8; À3.80
and À 2:09. The z and un corresponding to the dominant pole are
0.666 and 7 rad/s, respectively which shows a signiﬁcant
improvement on the damping factor. For the purpose of
Fig. 2. Open-loop bode plot and step response (Example 1).
ToðsÞ ¼
0:211s5 þ 4:509s4 þ 50:26s3 þ 183:9s2 þ 157:9s
s6 þ 21:37s5 þ 353:4s4 þ 3301s3 þ 27520s2 þ 88740s þ 81900
(8)

performance comparison 0.05 p. u. step change in DTm has been
considered here. Performance comparison of the closed-loop con-
trol system with that of the reference model is shown in Fig. 3
which shows that the designed control system is closely
following the response of the reference model chosen.
The performance of the proposed control system has been
compared with that of the control systems proposed by
Refs. [4,6,25,28e30], in Table 1 where as some of these controllers
([6,25,28 and 29]]) are further compared graphically in Figs. 4 and
5.
It is observed from the tables and the ﬁgures that the proposed
method gives the highest damping factor and is comparable with
Fig. 3. Performance comparison of bode plots and step responses of the reference model and the design system (Example 1).
Table 1
Performance comparison among various control systems for 5% change in DTm (Example 1).
Method Controller tSðsÞ ypð Â 10À4Þ tpðsÞ x TVCO ISE ð Â 10À7Þ
Huang Chen [28] À 23:67 À 0:1109
s
1.391 À10.25 0.1581 0.282 2.221 2.041
Huang et al. [6] À 16:305 À 0:3608
s
3.901 À10.17 0.1558 0.164 2.786 6.348
Lee Wu [29] À 11:37 À 0:2288
s
2.671 À10.08 0.1530 0.141 2.079 2.096
Bhattacharya [25] À 18:08 À 0:2483
s
2.168 10.18 0.1562 0.159 0.979 1.876
Feliachi et al. [30] À 23:63 À 0:1113
s
1.388 À10.24 0.1585 0.281 2.411 2.085
Hsu et al. [34] 20ð1þ0:017sþ0:02s2
Þ
ð1þ0:05sÞ ð1þ0:05sÞ
2.692 À9.192 0.1336 0.111 8.603 2.017
Hsu Hsu (Optimal control) [4]
K ¼
½12:61 0:157 0:0039
À0:935 4:66 À304:3Š
0.7682 ¡8.182 0.1280 0.361 8.604 1.061
Hsu Hsu (Sub-optimal control) [4] À 19:24 À 0:219
s
2.033 À10.19 0.1558 0.222 2.210 1.871
Hsu Hsu (Root-locus) [4] À 29 À 0:23
s
1.562 À10.36 0.1618 0.259 2.345 2.119
Proposed À 26 À 0:163
s
1.148 À10.22 0.1580 0.666 0.579 2.038
Fig. 4. Comparison of step responses due to 0.05 p. u. change in DTm (Example 1). Fig. 5. Comparison of controller outputs due to 0.05 p. u. change in DTm (Example 1).

the other methods in terms of settling time ðtsÞ, time for peak
overshoot ðtpÞ, peak overshoot ðypÞ, total variance of controller
output (TVCO) which is a measure of smoothness of the controller
output and integral square error (ISE).
3.2. Example 2: synchronous generator with IEEE type-ST1
excitation system
A synchronous generator with IEEE-Type ST1 exciter connected
to an infinite bus through transmission line is considered here. The
relevant system parameters, system matrices and the linearized
state-space representation of this test system are available in
Ref. [31]. In this system, G1 ¼ KA=ð1 þ sTAÞ. The following two cases
with different operating conditions have been considered as
follows:
3.2.1. Case a: nominal operating condition with active
power ¼ 136 MW and reactive power ¼ 83.2 MVAr
The transfer function of the nominal system with DTm as input
and Du as output is computed as:
ToðsÞ ¼
0:211s3 þ 4:336s2 þ 21:5s
s4 þ 20:55s3 þ 192s2 þ 1828s þ 7985
(12)
The open-loop poles are À0:3349±j9:422; À12:946 and À 6:936.
The z and un corresponding to the dominant poles
ðÀ0:3349±j9:422Þ are 0.0355 and 9.43 rad/s, respectively. For the
design of achievable reference model, a pair of conjugate complex
poles with z ¼ 0:295 and un ¼ 8:45 rad=s is considered with a zero
fixed at origin of the s-plane. With this, the transfer function of the
reference model is chosen as:
Mref ðsÞ ¼
0:00269s
½1 þ sð0:035 þ 0:1128jÞŠ½1 þ sð0:035 À 0:1128jÞŠ
¼
0:00269s
0:0139s2 þ 0:0697s þ 1
(13)
For the chosen reference model, the settling time ðtsÞis 1.739 s,
shoot ðtpÞ is 0.1658 s. With a frequency point of matching at u0 ¼
0:08 rad=s, the PI controller is designed as:
CðsÞ ¼ À7:8604 À
0:332
s
(14)
With this controller, the transfer function of the closed-loop is
obtained as:
TcðsÞ ¼
0:211s3 þ 4:336s2 þ 21:51s
s4 þ 20:55s3 þ 192s2 þ 1481s þ 3392
(15)
The z and un corresponding to the dominant pole ðÀ2:14±j8:68Þ
are 0.24 and 8.94 rad/s, respectively, which shows a significant
improvement in the damping factor.
3.2.2. Case b: unstable operating point with active
power ¼ 112 MW and reactive power ¼ À32MVAr
The open-loop transfer function of the nominal system with
same input and output is as follows:
ToðsÞ ¼
2:11s3 þ 4:336s2 þ 10:91s
s4 þ 20:55s3 þ 125:3s2 þ 1472s þ 4631
(16)
In this case, z and un corresponding to the dominant poles
are À0.0308 and 8.45 rad/s, respectively. A pair of complex conju-
gate poles having z ¼ 0:250 and un ¼ 7:25 rad=s with zero at origin
is considered for the reference model as given by:
Mref ðsÞ ¼
0:00235s
½1 þ sð0:035 þ 0:134jÞŠ½1 þ sð0:035 À 0:134jÞŠ
¼
0:00235s
0:0191s2 þ 0:0689s þ 1
(17)
For the chosen reference model, the settling time ðtsÞis 2.142 s,
the peak overshoot ðypÞis 1:028 Â 10À4s and time for peak over-
shoot ðtpÞ is 0.1791 s. A frequency point of matching
u0 ¼ 0:14 rad=s is chosen to obtain the PI controller represented as:
CðsÞ ¼ À9:11 À
0:186
s
(18)
The transfer function of the close-loop control system becomes:
TcðsÞ ¼
0:211s3 þ 4:336s2 þ 10:91s
s4 þ 20:55s3 þ 125:3s2 þ 953:7s þ 1305
(19)
For the designed system, the z and un corresponding to the
dominant pole are 0.197 and 7.04 rad/s, respectively, that shows a
noteworthy improvement in the damping factor.
The performance comparison with the controller in Ref. [32] for
nominal and unstable operating conditions for 0.01 p. u. step
change in DTm is shown in Table 2 and Fig. 6. It may be observed
from this table that the proposed controller shows the better re-
sults in terms of settling time and comparable results for other
performance indices for both the cases.
3.3. Example 3: two machine infinite bus system
A two bus two hydro plants (each having 10 machines) is
considered as the test system [6]. The state vector and output
vector of the system are considered as
h
Du1 Dd1 Deq
0
1 DeFD1 Du2 Dd2 Deq
0
2 DeFD2
i
and
½ Du1 Dd1 Du2 Dd2 ŠT
.
3.3.1. Design of the local controllers
The system is decomposed into two sub-systems as given below
[3]:
Machine 1: With X1 ¼
h
Du1 Dd1 De0
q1 DeFD1
i
as state vec-
tor, the open-loop transfer function of the nominal system with
DTm as input and Du as output is determined as:
ToðsÞ ¼
0:122s3 þ 6:153s2 þ 583s
s4 þ 50:7s3 þ 4821s2 þ 2585s þ 1:293 Â 105
(20)
The z and un corresponding to the dominant poles are 0.0245
and 5.20 rad/s, respectively. The reference model with a pair of
complex conjugate poles (with z ¼ 0:825 and un ¼ 4:60rad=s) and
a zero at origin is chosen as:
Mref ðsÞ ¼
0:0038s
½1 þ sð0:18 þ 0:1236jÞŠ½1 þ sð0:18 À 0:1236jÞŠ
¼
0:0038s
0:047s2 þ 0:36s þ 1
(21)
Table 2
Performance comparison for 0.01 p.u. change in DTm (Example 2).
Case Method tSðsÞ ypð Â 10À4Þ tPðsÞ
Case a Ellithy et al., 2014 [31] 5.261 2.152 1.161
Proposed 1.793 2.445 0.191
Case b Ellithy et al., 2014 [31] 10.61 1.955 1.345
Proposed 2.996 2.848 0.224

shoot ðtpÞ is 0.2317 s. The obtained PI controller with frequency
point of matching of u0 ¼ 3 rad=s is:
CðsÞ ¼ À91:593 À
1:11
s
(22)
The transfer function of the closed-loop system becomes:
TcðsÞ ¼
0:122s3 þ 6:153s2 þ 583s
s4 þ 50:7s3 þ 4821s2 þ 82380s þ 5:434 Â 105
(23)
The z and un corresponding to the dominant poles are 0.830 and
11.5 rad/s, respectively that show appreciable improvement in
damping factor.
h
Dd2 Du2 De0
q2 DeFD2
i
as state vec-
tor, the open-loop transfer function of the nominal system with
DTm as input and Du as output is computed as:
ToðsÞ ¼
0:1237s3 þ 6:223s2 þ 248:2s
s4 þ 50:55s3 þ 2086s2 þ 3869s þ 1:401 Â 105
(24)
and 8.35 rad/s, respectively. For designing the reference model, a
pair of complex conjugate poles with z ¼ 0:7 and un ¼ 11:5 rad=s is
chosen with a zero at origin as given by:
Mref ðsÞ ¼
0:0018s
½1 þ sð0:061 þ 0:062jÞŠ½1 þ sð0:061 À 0:062jÞŠ
¼
0:0018s
0:0076s2 þ 0:122s þ 1
(25)
shoot ðtpÞ is 0.097 s. Considering, the frequency point of matching
at u ¼ 2:12 rad=s, the designed PI controller is obtained is obtained
as:
CðsÞ ¼ À65:683 À
1:2586
s
(26)
With this, the closed-loop control system becomes:
TcðsÞ ¼
0:122s3 þ 6:153s2 þ 583s
s4 þ 50:7s3 þ 4821s2 þ 39590s þ 3:981 Â 105
(27)
19 rad/s, respectively that show a signiﬁcant improvement in
damping factor.
3.3.2. Design of the centralized controller
For the design of the centralized controller, a reduced order
power system, as stated in Ref. [8], is considered here to achieve
approximate optimum performance. For this purpose, a reduced
order model is obtained by eliminating the states that have the
least contribution to input/output behavior by using the Hankel
singular values. The reduced order state model ðAr; Br; CrÞ is ob-
tained as:
Ar ¼
2
6
6
4
À0:244 À0:073 0 0:0731
377 0 0 0
0 0:1843 À0:2473 À0:1847
0 0 377 0
3
7
7
5
Br ¼

À0:2212 0 0:1017 0
0:07233 0 À0:3008 0

; Cr ¼

1 1 0 0
0 0 1 1

9
=
;
(28)
The gain matrix for global control using eqn. (28) for minimizing
the interaction between the sub-systems as in Ref. [8] is computed
as:
G ¼

0 0:1458 0 À0:1459
0 À0:5634 0 0:5647

(29)
It is to note, the bold faced diagonal blocks of the gain matrix G
are to be replaced by the local controller gains that are already
designed to arrive at the overall gain matrix. Accordingly, the overall
gain matrix, including local and global controllers, is obtained as:
K ¼

À91:593 À1:11 0 À0:1459
0 À0:5634 À65:683 À1:2586

(30)
The simulation of the complete system has been shown in Fig. 7.
The performance comparison of the proposed controller with the
controllers stated in Refs. [3,5,7 and 30]] are shown in Fig. 7. From
Table 3 and Fig. 7, it is observed that the proposed controllers give
the better overall performances in all cases of this example than the
other controllers.
3.4. Example 4: three bus inﬁnite bus system
A three bus two hydro plants is considered here with
h
Dd1 Du1 De
0
q1 DeFD1 Dd2 Du2 De
0
q2 DeFD2
i
as state
vector and ½ Dd1 Du1 Dd2 Du2 Dd3 Du3 Š as output vector
[32].
Fig. 6. Comparison of step responses for 0.01 p. u. change in DTm (Example 2): (a) for case a, (b) for case b.

Fig. 7. Comparison of transient responses in one machine due to 0.05 p. u. change in DTm in other or both machines (Example 3).

3.4.1. Design of the local controllers
The system is decomposed into three machines as discussed
below:
Machine 1: With
h
Dd1 Du1 De0
q1 DeFD1
i
as state vector, the
transfer function of the open-loop system with DTm1
as input and D
u1 as output is:
T0ðsÞ ¼
0:2165s3 þ 4:529s2 þ 17:18s
s4 þ 21:6s3 þ 137:7s2 þ 1164s þ 4226
(31)
and 7.42 rad/s, respectively. For the above transfer function model,
a pair of conjugate complex poles is considered with z ¼ 0:325 and
un ¼ 7:15 rad=s with a zero at origin. The chosen reference model
is given as:
Mref ðsÞ ¼
0:0031s
½1 þ sð0:049 þ 0:132jÞŠ½1 þ sð0:049 À 0:132jÞŠ
¼
0:0031s
0:00195s2 þ 0:098s þ 1
(32)
shoot ðtpÞ is 0.1761 s. Choosing the frequency point of matching at
u0 ¼ 6:4 rad=s, the designed PI controller obtained as:
CðsÞ ¼ À29:733 þ
0:5953
s
(33)
The transfer function of the closed-loop is obtained as.
TCðsÞ ¼
0:2165s3 þ 4:529s2 þ 17:18s
s4 þ 21:06s3 þ 137:7s2 þ 1475s þ 1891
(34)
8.64 rad/s, respectively that shows the increase in damping factor.
h
Dd2 Du2 De0
q2 DeFD2
i
as state vec-
tor, the transfer function of the open-loop system with DTm2 as
input and Du2 as output is:
ToðsÞ ¼
0:1471s3 þ 2:972s2 þ 3804s
s4 þ 20:06s3 þ 81:38s2 þ 1125s þ 1677
(35)
The z and un corresponding to the dominant poles are À0.0142
and 7.45 rad/s, respectively. A pair of conjugate complex poles is
considered with z ¼ 0:295 and un ¼ 4:75 rad=s for selection of the
reference model as given by:
Mref ðsÞ ¼
0:0055s
½1 þ sð0:062 þ 0:201jÞŠ½1 þ sð0:062 À 0:201jÞŠ
¼
0:0055s
0:0443s2 þ 0:129s þ 1
(36)
shoot ðtpÞ is 0.296 s. A frequency point of matching u0 ¼ 4:7 rad/s is
chosen for obtaining the PI controller and as is computed as:
CðsÞ ¼ À22:436 þ
0:4699
s
(37)
The closed-loop control system becomes:
TcðsÞ ¼
0:1471s3 þ 2:972s2 þ 3:804s
s4 þ 20:18s3 þ 81:38s2 þ 1286s þ 401:2
(38)
and 8.08 rad/s, respectively that clearly indicate the improvement
of damping factor.
h
Dd3 Du3 De
0
q3 DeFD3
i
as state vector, the
transfer function, taking DTm3 as input and Du3 as output is
determined for the system considered.
TOðsÞ ¼
0:108s3 þ 2:181s2 þ 6:3s
s4 þ 20:21s3 þ 79:79s2 þ 415:1s þ 1388
(39)
The z and un corresponding to the dominant poles are À0.0358
and 4.61 rad/s, respectively. For selection of the reference model, a
Table 3
Performance comparison (Example 3).
Method Abdel-Magid and Aly [3] Flechai et al. [30] T. Huang et al. [5] C. Huang et al. [7] Proposed
Transient Response in Du1 due to 0.05 p.u. change in DTm1
tSðsÞ 1.332 2.688 0.9229 0.5868 0.6113
ypð Â 10À4Þ 8.790 10.44 7.036 5.744 4.672
tpðsÞ 0.1194 0.1910 0.1403 0.1170 0.1082
tSðsÞ 1.466 3.128 0.6884 0.7087 0.5857
ypð Â 10À4Þ 1.754 12.86 6.348 6.216 4.138
tpðsÞ 0.2288 0.1910 0.1079 4.138 0.0812
tSðsÞ 1.488 3.121 1.179 0.9961 0.6450
ypð Â 10À4Þ 6.525 6.379 1.897 1.215 1.3053
tpðsÞ 0.2286 0.4456 0.2266 0.2797 0.1677
tSðsÞ 1.339 2.995 1.1817 0.5868 0.7201
ypð Â 10À4Þ 2.026 11.58 1.006 5.744 1.8232
tpðsÞ 0.2785 0.2266 0.2267 0.1177 0.1407
Transient Response in Du1 due to 0.05 p.u. change in both DTm1 and DTm2
tSðsÞ 1.322 2.793 0.7708 1.1792 0.5947
ypð Â 10À4Þ 8.796 13.46 6.7524 1.8973 5.072
tpðsÞ 0.1193 0.1910 0.1325 0.2266 0.0974
Transient Response in Du2 due to 0.05 p.u. change in both DTm1 and DTm2
tSðsÞ 1.128 2.078 1.159 0.8701 0.5057
ypð Â 10À4Þ 9.251 20.40 10.57 7.071 6.432
tpðsÞ 0.1293 0.3356 0.1835 0.1472 0.1244

pair of conjugate complex poles are considered with z ¼ 0:215 and
un ¼ 4:484 rad/s and a zero at origin as given by:
Mref ðsÞ ¼
0:0046s
½1 þ sð0:044 þ 0:2017jÞŠ½1 þ sð0:044 À 0:2017jÞŠ
¼
0:0046s
0:0426s2 þ 0:088s þ 1
(40)
shoot ðtpÞ is 0.2676 s. With frequency point of matching at u0 ¼
1 rad=s, the PI controller is obtained as:
C3ðsÞ ¼ À28:649 þ
0:0433
s
(41)
The closed loop system is obtained as:
TCðsÞ ¼
0:108s3 þ 2:181s2 þ 6:3s
s4 þ 20:21s3 þ 79:79s2 þ 673:1s þ 1237
(42)
5.76 rad/s, respectively that shows the effective improvement in
the damping of the system. It is to note that the damping factor of
the nominal system is negative as the system is unstable.
3.4.2. Design of the centralized controller
For the design of the centralized controller, ﬁrst the reduced
order power system model is obtained as:
Ar ¼
2
6
6
6
6
6
6
4
0 377 0 0 0 0
À0:1418 0:0225 0:0196 0:0056 0:0399 0:2342
0 0 0 377 0 0
0:0098 À0:0019 À0:1724 0:2034 0:0714 0:0312
0 0 0 0 0 377
0:0031 À0:0025 0:0165 À0:0087 À0:0642 0:165
3
7
7
7
7
7
7
5
Br ¼
2
6
6
6
6
6
6
4
0 0 0
À0:1379 0:0132 0:0747
0 0 0
À0:0318 À0:2781 0:0601
0 0 0
À0:0214 0:0128 À0:1527
3
7
7
7
7
7
7
5
;Cr ¼

0 1 0 1 0 1
1 0 1 0 1 0

9
=
;
(43)
Following the procedure illustrated in the earlier example the
gain matrix for centralized controller using Eqn. (43) is obtained as:
Now, the bold faced diagonal blocks of the gain matrix G are
replaced by the local controller gains that are already designed in
Section 3.4.1 to obtain the overall gain matrix:
The performance comparison of the proposed controller with
some of the controllers stated in Refs. [5,22,33] under the various
operating conditions has been shown in Table 4. The simulation
results employing all these controllers have been shown in Fig. 8.
By comparing Fig. 8 and Table 4 simultaneously, it is clear that the
proposed controller provides the better performances than the
others controllers taken from the literature.
Table 4
Performance comparison (Example 4).
Method Huang et al. [5] Solimon et al. [22] Optimal [33] Proposed
Transient Response in Du1due to 0.05 p.u. change in DTm1
tSðsÞ 3.261 7.511 1.834 1.959
ypð Â 10À4Þ 12.688 14.317 12.59 13.933
tpðsÞ 0.1848 0.2080 0.1888 0.2747
tSðsÞ 3.910 8.266 3.369 2.498
ypð Â 10À4Þ 2.554 11.05 0.8050 0.7055
tpðsÞ 0.4681 1.491 0.6421 0.5219
tSðsÞ 2.864 7.907 3.3212 2.9262
ypð Â 10À4Þ 5.171 8.021 0.2485 4.086
tpðsÞ 0.4434 0.4161 0.3273 0.4945
tSðsÞ 4.067 8.296 3.356 2.487
ypð Â 10À4Þ 4.353 3.359 1.255 3.709
tpðsÞ 0.6651 2.080 0.6295 0.5219
tSðsÞ 2.566 7.813 2.186 1.601
ypð Â 10À4Þ 9.239 9.874 8.824 8.093
tpðsÞ 0.2093 0.2080 1.889 0.2671
tSðsÞ 2.618 7.588 2.535 2.717
ypð Â 10À4Þ 6.339 5.183 1.308 2.624
tpðsÞ 0.4680 0.4854 0.4406 0.4121
tSðsÞ 3.534 7.807 4.029 2.923
ypð Â 10À4Þ 2.352 1.985 0.1520 2.243
tpðsÞ 0.7152 0.9014 0.9064 0.4669
tSðsÞ 3.051 7.759 2.838 2.647
ypð Â 10À4Þ 1.863 4.317 1.108 0.3653
tpðsÞ 0.4081 0.5162 0.4532 0.3846
tSðsÞ 1.944 5.966 1.257 2.397
ypð Â 10À4Þ 7.554 6.289 7.197 6.5024
tpðsÞ 0.2340 0.1734 0.2140 0.2472
G ¼
2
4
0:92212 0:00971 À0:10465 À0:007508 À0:07846 À1:5628
À0:17662 0:009278 0:62303 0:01369 À0:15743 0:11626
À0:16465 0:016288 À0:04113 0:059415 0:41826 0:22959
3
5 (44)
K ¼
2
4
0:5953 29:733 À0:10465 À0:007508 À0:07846 À1:5628
À0:17662 0:009278 0:4699 22:436 À0:15743 0:11626
À0:16465 0:016288 À0:04113 0:059415 0:0433 28:649
3
5 (45)

Fig. 8. Comparison of transient responses in one machine due to 0.05 p. u. change in DTm in other machines (Example 4).

4. Conclusion
An enhancement of the small-signal stability of the power sys-
tem in terms of improving the damping factor using a two-level
control scheme has been proposed in this paper. In the local level
the proposed design method is based on frequency domain model
matching that avoids elaborate frequency response analysis and
outperforms extensive mathematical calculations. This matching
method proposed here is independent of the order and structure of
the system and, hence, does not require reduction of the system
before designing the controllers. Suitable reference models for the
purpose of model matching are developed that are easily achiev-
able by the system dynamics fulfilling the desired response. Close
matching of the designed systems with the reference models is
obtained following the design process. In the global level, the
proposed controller works on minimizing the effect of inter-
connection of the sub-systems leading to improvement in overall
responses.
The proposed design method has been applied on four examples
taken from the literature with four different configurations of po-
wer system such as OMIB with IEEE Type-DC1 excitation system,
OMIB with IEEE Type-ST1 excitation system, two-machine infinite
bus system and three-machine infinite bus system. The simulation
results are elaborately illustrated through figures and tables, which
evidently show the efficacy of the proposed scheme in improving
the damping of the low frequency oscillations appreciably. The local
controllers or its blending with the centralized (global) controllers
(as the case may be) shows favorable performance when compared
with some controllers prevalent in the literature. The extension of
the proposed control strategy for large power system and consid-
eration of time-delay in communication channels along with
practical implementation of the proposed controllers may be
considered as relevant future work.
References
[1] Pai MA, Gupta DPS, Padiyar KR. Small signal analysis of power systems, vol. 6.
New Delhi: Alpha Science Int'l Ltd.; 2004. p. 33e41.
[2] Paserba J, Kundur P, Sanchez-Gasca J, Larsen E. Small signal stability and po-
wer system oscillations. In: Power system stability and control. Boca Raton:
CRC press; 2012. p. 11e20.
[3] Abdel-Magid YL, Aly GM. Two-level optimal stabilization in multi-machine
power systems. Elec Power Syst Res 1983;6(1):33e41.
[4] Hsu YY, Hsu CY. Design of a proportional-integral power system stabilizer.
IEEE Trans Power Syst 1986;1(2):46e52.
[5] Huang T, Awang T, Yang W. Two-level optimal output feedback stabilizer
design. IEEE Trans Power Syst 1991;6(3):1042e8.
[6] Huang T-L, Lee K-T, Hwang T-Y, Chao C-C. System time-varying controller
design via E.A. method. J Inf Optim Sci 2008;29(2):253e70.
[7] Huang C, Zhang K, Dai X, Wang W. A design method of area controller for
power systems using reduced model. Energy Procedia 2012;14:1053e9.
[8] Dotta D, Silva AS, Decker IC. Wide-area measurements-based two-level con-
trol design considering signal transmission delay. IEEE Trans Power Syst
2009;24(1):208e16.
[9] Anwar MN, Pan S. A new PID load frequency controller design method in
frequency domain through direct synthesis approach. Int J Electr Power En-
ergy Syst 2015;67:560e9.
[10] Fliess M, Join C. Model-free control and intelligent PID controllers: towards a
possible trivialization of nonlinear control? IFAC Proc 2009;42(10):1531e50.
[11] Wang HP, Zheng D, Tian Y. High pressure common rail injection system
modeling and control. ISA Trans 2016;63:265e73.
[12] Zhang X, Wang H, Tian Y, Peyrodie L, Wang X. Model-free based neural
network control with time-delay estimation for lower extremity exoskeleton.
Neurocomputing 2018;272:178e88.
[13] Li S, Wang H, Tian Y, Aitouch A, Klein J. Direct power control of DFIG wind
turbine systems based on an intelligent proportional-integral sliding mode
control. ISA Trans 2016;64:431e9.
[14] Yaghooti A, Buygi M, Shanechi M. Designing coordinated power system sta-
bilizers: a reference model based controller design. IEEE Trans Power Syst
2016;31(4):2914e24.
[15] Ugalde-Loo CE, Acha E, Liceaga-Castro E. Multi-machine power system state-
space modelling for small-signal stability assessments. Appl Math Model
2013;37:10141e61.
[16] Hu X, Zou C, Zhang C, Li Y. Technological developments in batteries: a survey
of principal roles, types, and management needs. IEEE Power Energy Mag
2017;15(5):20e31.
[17] Segal R, Sharma A, Kothari M. A self-tuning power system stabilizer based on
artificial neural network. Int J Electr Power Energy Syst 2004;26(6):423e30.
[18] Ramirez-Gonzalez M, Malik OP. Self-tuned power system stabilizer based on a
simple fuzzy logic controller. Elec Power Compon Syst 2010;38(4):407e23.
[19] Ghoshal SP, Chatterjee A, Mukherjee V. Bio-inspired fuzzy logic based tuning
of power system stabilizer. Expert Syst Appl 2009;36(5):9281e92.
[20] Alkhatib H, Duveau J. Dynamic genetic algorithms for robust design of multi
machine power system stabilizers. Int J Electr Power Energy Syst 2013;45(1):
242e51.
[21] Wang Z, Chung CY, Wong KP, Tse CT. Robust power system stabiliser design
under multi-operating conditions using differential evolution. IET Gener,
Transm Distrib 2008;2(5):690e700.
[22] Soliman HM, Morsi MF, Hassan MF, Awadallah MA. Power system reliable
stabilization with actuator failure. Elec Power Compon Syst 2009;37:61e77.
[23] Sambariya D, Prasad R. Optimal tuning of fuzzy logic power system stabilizer
using harmony search algorithm. Int J Fuzzy Syst 2015;17(3):457e70.
[24] Abido MA. Pole placement technique for PSS and TCSC-based stabilizer design
using simulated annealing. Electr Power Energy Syst 2000;22:543e54.
[25] Bhattacharya K, Nanda J, Kothari ML. Optimization and performance analysis
of conventional power system stabilizers. Electr Power Energy Syst
1997;19(7):449e58.
[26] Pan S, Pal J. Reduced order modeling of discreet-time systems. Appl Math
Model 1995;19(March):133e8.
[27] Siljak DD, Sundareshan MK. A multi-level optimization of large scale systems.
IEEE Trans Automat Contr 1976;21(1):79e84.
[28] Huang TL, Chen SC. Power system PI controller design via optimal sub-
eigenstructure assignment. Elec Mach Power Syst 1993;21(4):437e48.
[29] Lee Y-C, Wu C-J. Damping of power system oscillations with output feedback
and strip eigenvalue assignment. IEEE Trans Power Syst 1995;10(3):1620e6.
[30] Feliachi A, Zhang X, Sims CS. Power system stabilizers design using optimal
reduced order models Part II: design. IEEE Trans Power Syst 1988;3(4):
1676e84.
[31] Ellithy K, Said S, Kahlout O. Design of power system stabilizers based on m-
controller for power system stability enhancement. Int J Electr Power Energy
Syst 2014;63:933e9.
[32] Esmaili MR, Khodabakhshian A, Panah PG, Azizkhanid S. A new robust multi-
machine power system stabilizer design using quantitative feedback theory.
Procedia Technol 2013;11:75e85.
[33] Arnautovic D, Medanic J. Design of decentralized multivariable excitation
controllers in multimachine power systems by projective controls. IEEE Trans
Energy Convers 1987;EC-2(4):598e604.
[34] Hsu C-Y, Chen L-M, Liou K-L, Hsu Y-Y. Effect oF power system stabilizers and
load frequency controllers on power system dynamics stability. J Chin Inst Eng
1985;8(2):135e42.

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