Design of Power System Stabilizer using Power Rate Reaching Law based Sliding

Mode Control Technique

Vitthal Bandal, Student Member, IEEE, B. Bandyopadhyay, Member, IEEE, and A. M. Kulkarni,

LIST OF SYMBOLS
Gp (s) generic power system stabilizer transfer func- I. I NTRODUCTION
tion
Ks stabilizer gain Power system stabilizer (PSS) units have long been re-
ωB system base frequency ( 377 rad/sec at 60 Hz) garded as an effective way to enhance the damping of elec-
Tw washout time constant
T1 , T2 , T3 , T4 lead/lag time constant tromechanical oscillations in power system [1]. The action of
Gf (s) filtering in stabilizer PSS is to extend the angular stability limits of a power system
E
q voltage proportional to the field flux linkages by providing supplemental damping to the oscillation of
of machine
Td
d-axis transient open circuit time constant synchronous machine rotors through the generator excitation
xd d-axis synchronous reactance of machine [2]. This damping is provided by an electric torque applied
x
d d-axis transient reactance of machine to the rotor that is in phase with the speed variation. Once
Ef d generator field voltage
δ machine shaft angular displacement (degree) the oscillations are damped, the thermal limits of the tie-lines
ω rotor speed ( rad./sec.) in the system may then be approached. This supplementary
Sm machine slip signal is very useful during large power transfers and line
Sm0 nominal slip of the machine
id direct axis armature current (pu) outages [3].
H inertia constant(sec.) Over the past four decades, various control methods have
D damping coefficient
KE AVR gain
been proposed for PSS design to improve overall system
TE AVR time constant(sec.) performance. Among these, conventional PSS of the lead-
xe line reactance (pu) lag compensation type [1], [4], [5] have been adopted by
Pg0 mechanical power on the shaft of machine
Pe electrical power output of machine
most utility companies because of their simple structure,
xq q-axis synchronous reactance of machine flexibility and ease of implementation. However, the per-
Vref reference input voltage formance of these stabilizers can be considerably degraded
Vs correction voltage
A state (plant) matrix of the system
with the changes in the operating condition during nor-
B control input matrix mal operation. Since power systems are highly nonlinear,
C output matrix conventional fixed-parameter PSSs cannot cope with great
x state vectors
y output vectors
changes in the operating conditions. There are two main
t time approaches to stabilizing a power system over a wide range
u stabilizing signal of operating conditions, namely adaptive control and ro-
T transpose
s switching function
bust control [6]. Adaptive control is based on the idea of
continuously updating the controller parameters according
to recent measurements. However, adaptive controllers have
generally poor performance during the learning phase, unless
Abstract— The paper presents a new method for design they are properly initialized. Successful operating of adap-
of power system stabilizer (PSS) using discrete time power tive controllers requires the measurements to satisfy strict
rate reaching law based sliding mode control technique. The persistent excitation conditions. Otherwise the adjustment of
control objective is to enhance the stability and to improve the controller’s parameters fails. Robust control provides an
the dynamic response of a single machine infinite bus (SMIB)
system, operating in different conditions. The control rules are
effective approach to dealing with uncertainties introduced
constructed using discrete time power rate reaching law based by variations of operating conditions.
sliding mode control. We apply this controller to design power Among many techniques available in the control literature,
system stabilizer for demonstrating the efficacy of the proposed H∞ and variable structure have received considerable atten-
approach. tion in the design of PSSs. The H∞ approach is applied
by Chen [6] to PSS design for a single machine infinite
Vitthal Bandal is a Research Scholar with Systems and Control Engi- bus system. The basic idea is to carry out a search over
neering, Indian Institute of Technology Bombay, Mumbai-400076, INDIA all possible operating points to obtain a frequency bound on
(e-mail:vsbandal@ee.iitb.ac.in)
Prof. B. Bandyopadhyay is with Systems and Control Engineering, the system transfer function. Then a controller is designed
Indian Institute of Technology Bombay, Mumbai-400076, INDIA (e- so that the worst-case frequency response of the closed
mail:bijnan@ee.iitb.ac.in)(corresponding author) loop system lies within prespecified frequency bounds. It
Prof. A. M. Kulkarni is with Electrical Engineering department,
Indian Institute of Technology Bombay, Mumbai-400076, INDIA (e- is noted that the H∞ design requires an exhaustive search
mail:anil@ee.iitb.ac.in) and results in a high order controller. On the other hand the

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 variable structure control is designed to drive the system to a and T4 . A torsional filter may not be necessary with signals
sliding surface on which the error decays to zero [7]. Perfect like power or delta-P-omega signal [9].
performance is achieved even if parameter uncertainties are A power system stabilizer can be most effectively applied
present. However, such performance is obtained at the cost if it is tuned with an understanding of the associated power
of high control activities (chattering) [8]. characteristics and the function to be performed by the sta-
In this paper a PSS design for SMIB system using discrete bilizer. Knowledge of the modes of power system oscillation
time power rate reaching law based sliding mode control to which the stabilizer is to provide damping establishes the
technique is proposed. In the sliding mode controller a range of frequencies over which the stabilizer must operate.
switching surface is designed. When the sliding mode occurs, Simple analytical models, such as that of a single machine
the system dynamic behaves as a robust state feedback infinite bus (SMIB) systems, can be useful in determining the
control system. A discrete time power rate reaching law frequencies of local mode oscillations during the planning
based sliding mode controller is investigated, which is used stage of a new plant. It is also desirable to establish the
to minimize the chattering. Simulations results for single weak power system conditions and associated loading for
machine infinite bus (SMIB) system are presented to show which stable operation is expected, as the adequacy of the
the effectiveness of the proposed control strategy in damping power system stabilizer application will be determined under
the oscillation modes. these performance conditions. Since the limiting gain of the
The paper is organized as follows. Section II presents some stabilizers, viz., those having input signal from speed
basics of power system stabilizer and power system analysis. or power, occurs with a strong transmission system, it is
Section III presents the review on multirate output feedback. necessary to establish the strongest credible system as the
Section IV presents discrete time power rate reaching law “tuning condition” for these stabilizers. Experience suggest
based sliding mode controller design; the same is used that designing a stabilizer for satisfactory operation with an
for PSS design of SMIB system as discussed in section external system reactance ranging from 20% to 80% on the
V. Conclusions are drawn in Section VI. The controller is unit rating will ensure robust performance [10].
validated using non-linear model simulation.
C. Power System Analysis
II. P OWER S YSTEM S TABILIZER
Analysis of practical power system involves the simul-
A. Basic concept
taneous solution of equations consisting of synchronous
The basic function of a power system stabilizer is to extend machines ,associated excitation system , prime movers, in-
stability limits by modulating generator excitation to provide terconnecting transmission network, static and dynamic (
damping to the oscillation of synchronous machine rotors motor ) loads, and other devices such as HVDC converters,
relative to one another. The oscillations of concern typically static var compensator. The dynamics of the machine rotor
occur in the frequency range of approximately 0.2 to 3.0 circuits, excitation systems, prime mover and other devices
Hz, and insufficient damping of these oscillations may limit are represented by differential equations. This results in
ability to transmit power. To provide damping, the stabilizer the complete system model consisting of large number of
must produce a component of electrical torque, which is in ordinary differential and algebraic equations [9].
phase with the speed changes. The implementation details 1) Generator Equations: The machine equations ( for
differ, depending upon the stabilizer input signal employed. jth machine ) are
However, for any input signal, the transfer function of
the stabilizer must compensate for the gain and phase of

excitation system, the generator and the power system, dEqj −1


=
[Eqj − (xdj − xdj )idj − Ef dj ], (1)
which collectively determines the transfer function from the dt Td0j
stabilizer output to the component of electrical torque which dδj
can be modulated via excitation system [4]. = ωB (Smj − Smj0 ), (2)
dt
dSmj −1
B. Classical Stabilizer implementation procedure = [Dj (Smj − Smj0 ) − Pmj + Pej ]. (3)
dt 2H
Implementation of a power system stabilizer implies ad-
justment of its frequency characteristic and gain to produce Model 1.0 is assumed for synchronous machines by ne-
the desired damping of the system oscillations in the fre- glecting the damper windings. In addition, the following
quency range of 0.2 to 3.0 Hz. The transfer function of a assumptions are made for simplicity [11].
generic power system stabilizer may be expressed as 1. The loads are represented by constant impedances.


2. Transients saliency is ignored by considering xq = xd .
Tw s (1 + sT1 ) (1 + sT3 )
Gp (s) = Ks Gf (s) 3. Mechanical power is assumed to be constant.
(1 + Tw s) (1 + sT2 ) (1 + sT4 ) 4. Ef d is single time constant AVR.
where Ks represents stabilizer gain and Gf (s) represents
combined transfer function of torsional filter (if required) and 2) State space model of power system (Machine model
input signal transducer. The stabilizer frequency characteris- 1.0): The state space model of a SMIB power system, the
tic is adjusted by varying the time constant Tw , T1 , T2 , T3 block diagram of which is shown in Fig. 1 can be obtained

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 K1 IV. D ISCRETE T IME P OWER R ATE R EACHING L AW
∆Τm ∆ Te1 BASED S LIDING M ODE C ONTROL
− ∆Sm
1 ωB ∆δ Consider a SISO plant described by a continuous time
Σ 2Hs s
+
linear model
∆ Te2
K2 K4
K5 ẋ = Ax + Bu, (6)
∆Εq
_ _ ∆ vref y = Cx.
K5 + KE
Σ Σ Where x ∈ Rn , u ∈ R, y ∈ R and the matrices A, B and
1 + s Td0 K3 1 + s TE +
∆Εfd C are of appropriate dimensions.
Let ( Φτ , Γτ , C) be the system given by Eqn.(6) sampled
K6 at sampling interval τ seconds and is represented as,
Fig. 1. Block diagram of a Single Machine Infinite Bus (SMIB) system
x(k + 1) = Φτ x(k) + Γτ u(k), (7)
y(k) = Cx(k). (8)
using generator, transformer, network and loadflow data as
given below [11], In [20] a power rate reaching law approach for continuous
time systems had been proposed. The discrete power rate
.
x= Ax + B (
Vref +
Vs ) , (4) reaching law can be directly obtained from the continuous
power rate reaching law as,
y = Cx, (5)
where s(k + 1) − s(k) = −kτ |s(k)|α sgn(s(k)) (9)
x denotes the states of the machine and are given as
x = [Sm , δ, Ef d , Eq ]. Similarly, y = Sm denotes the output where τ > 0, is the sampling period, 0 < kτ < 1 and
equation of the machine and C is the output matrix.( C = 0 < α < 1. s(k) is the switching function defined as a
[1, 0, 0, 0] ). function of system states as,
Where Sm is machine slip and is given by,
(ω − ωB ) s(k) = cT x(k) (10)
Sm = ,
(ωB ) Hence,
δ is machine shaft angular displacement in degrees, Ef d is
generator field voltage in pu and Eq is voltage proportional
s(k + 1) = cT x(k + 1) (11)
to field flux linkages of machine in p.u.
The elements of matrix A are dependent on the operating So, from Eqns. (7), (9) and (11)
condition.
III. R EVIEW O N M ULTIRATE O UTPUT F EEDBACK s(k + 1) − s(k) = cT [Φτ − I]x(k) + cT Γτ u(k) (12)
In the following, multirate output feedback is briefly Comparing the Eqns.(9) and (12), the control law is
reviewed. obtained as follows [21],
Multirate Output Feedback is the concept of sampling the
control input and sensor output of a system at different rates. u(k) = −(cT Γτ )−1 [cT Φδ x(k) + ρ|s(k)|α sgn(s(k))] (13)
It was found that multirate output feedback can guarantee where, Φδ = Φτ − I and ρ = kτ . Thus, a discrete time
closed loop stability, a feature not assured by static output sliding mode control based on power rate reaching law is
feedback [12] while retaining the structural simplicity of obtained. This control law is designed using the states of the
static output feedback. Much research has been performed in system. Switching gain c can be obtained using the procedure
this field [13]–[17]. In multirate output feedback, the control given in [22]
input [15], [17] or the sensor output [16] is sampled at a As, in practice all states of the system are not available
faster rate than the other. In this paper, the term multirate for measurement and therefore control derived with the help
output feedback is used to refer the situation wherein the of only output information of the system will be more useful
system output is sampled at a faster rate as compared to the from practical point of view. A generalized expression for the
control input. switching surface and the control using output information
It was found that state feedback based control laws of only has been derived and is given as [21],
any structure may be realized by the use of multirate output
feedback, by representing the system states in terms of the
past control inputs and multirate sampled system output [18], x(k) = Φτ C0−1 yk + [Γτ − Φτ C0−1 D0 ]u(k − 1), (14)
[19]. s(k) = cT (Φτ C0−1 yk + [Γτ − Φτ C0−1 D0 ]u(k − 1)) (15)

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 Single Machine Infinte Bus System
1.3

= −(cT Γτ )−1 [cT Φδ Φτ C0−1 yk

POWER RATE REACHING LAW BASED SMC PSS

u(k) 1.28
CLASSICAL PSS

−(cT Γτ )−1 [cT Φδ (Γτ − Φτ C0−1 D0 )u(k − 1) 1.26

−(cT Γτ )−1 ρ|s(k)|α sgn(s(k))] (16) 1.24

Delta (radians)
1.22

Thus, it can be seen from the Eqns. (15) and (16) that the 1.2

states of the system are needed neither for switching function 1.18

evaluation nor for the feedback purpose. 1.16

1.14
V. C ASE S TUDY
1.12
0 1 2 3 4 5 6 7 8 9 10
A single machine infinite bus power system is considered Time in sec.

here for PSS design using power rate reaching law based
sliding mode control technique. (a) Delta responses

A. Linearization of power system x 10

−3 Single Machine Infinte Bus System
4
POWER RATE REACHING LAW BASED SMC PSS
The Nonlinear differential equations governing the be- CLASSICAL PSS

havior power system can be linearized about a particular 3

operating point to obtain a linear model which represents

2
the small signal oscillatory response of a power system. A
SIMULINK based block diagram including all the nonlinear
Slip

blocks can also be used to generate the linear state space

model of the system. 0

−1
B. Classical power system stabilizer design for a power
system −2
0 1 2 3 4 5 6 7 8 9 10
Time in sec.
The classical power system stabilizer (PSS) is designed in
the following way.
(b) Slip responses
The eigenvalue analysis of a power system is carried out
and the participation ratio of the machine towards instability
in the network, is estimated. Power system stabilizer using Fig. 2. Delta and slip responses with classical PSS and PSS using power
rate reaching law based sliding mode control technique for Pg0=0.5 pu,
phase compensation technique is designed according to the Vref=1.0 pu and Xe=0.25 pu
participation ratio of the machine towards instability, till
satisfactory closed loop performance of the power system
is achieved. The above design of classical power system As discussed in the previous section, the SISO linearized
stabilizer (PSS) is iterative in nature and optimal tuning of model of entire system obtained at nominal operating condi-
parameters is based on the experience. If the power charac- tion is obtained, which is represented by Eqn. (6). The power
teristics of the system changes, then the whole procedure of rate reaching law based sliding mode control given by Eqn.
PSS design has to be repeated. So, design of classical power (16) is then applied to the actual nonlinear system to carry
system stabilizers cannot be considered robust in nature for out simulations.
all operating points. The proposed power rate reaching law
based sliding mode control technique used for power system D. Simulation with Non-linear model
stabilizer design is robust in nature for all the models and The slip of the machine is taken as output. This output
is not iterative . signal of the controller and a limiter is added to Vref signal.
This is used to damp out the small signal disturbances
C. Design of PSS using discrete time power rate reaching
via modulating the generator excitation. The disturbance
law based sliding mode control for Single Machine Infinite
considered here is a self clearing fault which is cleared after
Bus (SMIB) system
0.1 second. The limits of PSS output are taken as ±0.1.
The single machine infinite bus power system data is Simulation results for SMIB system for various operating
considered for designing PSS using power rate reaching law conditions, with power rate reaching law based sliding mode
based sliding mode control. The block diagram of the system controller and classical controller are shown in Fig. 2 to Fig.
is shown in Fig. 1. 5.
The following parameters are used for simulation of the As shown in plots, the proposed controller is able to damp
single machine infinite bus system model [11]: out the oscillations in 1 to 2 seconds after clearing the fault.


H = 5 sec., D = 0, Tdo = 6 sec., KE = 100, TE = 0.02 Even in some cases where, classical PSS cannot damp out
sec., xe = 0.2 p.u. the oscillations, the proposed controller is able to damp out

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 Single Machine Infinte Bus System Single Machine Infinte Bus System
1.5 1.5
POWER RATE REACHING LAW BASED SMC PSS POWER RATE REACHING LAW BASED SMC PSS
CLASSICAL PSS CLASSICAL PSS

1.4 1.4

1.3 1.3
Delta (radians)

Delta (radians)
1.2 1.2

1.1 1.1

1 1

0.9 0.9
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
Time in sec. Time in sec.

(a) Delta responses (a) Delta responses

−3 Single Machine Infinte Bus System −3 Single Machine Infinte Bus System
x 10 x 10
8 10
POWER RATE REACHING LAW BASED SMC PSS POWER RATE REACHING LAW BASED SMC PSS
CLASSICAL PSS CLASSICAL PSS
8
6

6
4

4
2
Slip

Slip

0
0

−2
−2

−4
−4

−6 −6
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
Time in sec. Time in sec.

(b) Slip responses (b) Slip responses

Fig. 3. Delta and slip responses with classical PSS and PSS using power Fig. 4. Delta and slip responses with classical PSS and PSS using power
rate reaching law based sliding mode control technique for Pg0=1.0 pu, rate reaching law based sliding mode control technique for Pg0=1.25 pu,
Vref=1.0 pu and Xe=0.25 pu Vref=1.0 pu and Xe=0.25 pu

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 Single Machine Infinte Bus System
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1

L IST OF F IGURES
0.9
0 1 2 3 4 5
Time in sec.
6 7 8 9 10
1 Block diagram of a Single Machine Infinite Bus
(SMIB) system . . . . . . . . . . . . . . . . . . 3
(a) Delta responses 2 Delta and slip responses with classical PSS
and PSS using power rate reaching law based
−3 Single Machine Infinte Bus System
sliding mode control technique for Pg0=0.5 pu,
x 10
10
POWER RATE REACHING LAW BASED SMC PSS Vref=1.0 pu and Xe=0.25 pu . . . . . . . . . . 4
CLASSICAL PSS
8
3 Delta and slip responses with classical PSS
6 and PSS using power rate reaching law based
4 sliding mode control technique for Pg0=1.0 pu,
2 Vref=1.0 pu and Xe=0.25 pu . . . . . . . . . . 5
Slip

0
4 Delta and slip responses with classical PSS
−2
and PSS using power rate reaching law based
−4
sliding mode control technique for Pg0=1.25
pu, Vref=1.0 pu and Xe=0.25 pu . . . . . . . . 5
−6
5 Delta and slip responses with classical PSS
−8
0 1 2 3 4 5
Time in sec.
6 7 8 9 10 and PSS using power rate reaching law based
sliding mode control technique for Pg0=1.35
(b) Slip responses pu, Vref=1.0 pu and Xe=0.25 pu . . . . . . . . 6

Fig. 5. Delta and slip responses with classical PSS and PSS using power
rate reaching law based sliding mode control technique for Pg0=1.35 pu,
Vref=1.0 pu and Xe=0.25 pu

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