Design of Variable Structure Fuzzy Power System Stabilizer in Multi-
machine System
1TAWFIQ H. ELMENFY, 2SAMAH ABDELSALAM
1Department of Electrical and Electronics Engineering, Faculty of Engineering University of Benghazi,
LIBYA
2Higher Institute for Technology and Science, Regdalleen, LIBYA
Abstract- This paper introduces adaptive variable structure fuzzy controller as a power system stabilizer
(AFPSS) used to damp inter-area modes of oscillation following large disturbances in power systems. In
contrast to the conventional PSS, fuzzy-based stabilizers are more efficient because they cope with
oscillations at different operating points. The proposed controller is a fuzzy-logic-based PSS that has the
capability to tune its rule-base on line. The change in the fuzzy rule base is done using a variable-structure
algorithm to achieve optimum performance. The adaptive algorithm of the proposed controller
significantly reduces the rule base size due to its adaptively and improves its performance. This statement
is confirmed by simulation results of a three-machine infinite-bus system under different operating points.
Keywords: Fuzzy Logic Control, Adaptive Control, Power system stabilizer
Received: October 12, 2022. Revised: April 21, 2023. Accepted: June 3, 2023. Published: July 7, 2023.
1. Introduction
Power system stability is defined as the ability of
the system to remain in its equilibrium point that
following small and large disturbances. Power
system are becoming increasingly stressed
because of growing demand and restrictions on
building new power plants and lines. One of the
consequences of such a stressed system is the
threat of losing stability following a disturbance
due to poor power system stabilizer (PSS). So, the
increasing of power transfer capability and
improvement power system stability is necessary
to enhancement [1].
Power systems are hardly nonlinear systems that
often have low frequency oscillations due to poor
damping caused by wide range of operating
conditions which can lead to loss of synchronism
and cascade blackout [1]. Conventional Power
system stabilizers (CPSS) can provide
supplementary control signal to excitation system
to damp these oscillations [1−4]. The tune of the
fixed parameters of these conventional stabilizers
are usually based on the linearized model of the
power system around a small range of operating
points. Operating conditions change as load
variations and wide of power generations. These
wide range of operating conditions affect power
system dynamic behavior which requires fine-
designed and tuned PSS [1-4].
Adaptive controller can tune its parameters on-line
when the power system changes its operating
conditions, line switching and unpredictable fault
location in power system. Therefore, adaptive
controller is expected to give good quality of
performance under a hardly nonlinearity of power
system [5-6].
Unlike (CPSS), which requires a linear plant
model for its designing, fuzzy logic controller
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.6
Tawfiq H. Elmenfy, Samah Abdelsalam
E-ISSN: 2769-2507
41
Volume 5, 2023
(FLC) allows to design a controller with unknown
or unprecise mathematically model of the
nonlinear power system.
This paper proposes adaptive fuzzy-logic PSS
(AFPSS) that grasps the advantages of adaptive of
nonlinear fuzzy-logic techniques and overcomes
FLC drawbacks. The introduced stabilizer is
initialized using the rule-base of a standard FLPSS
to guarantee an appropriate performance during
the learning step. The rule-base is tuned in real
time so that the stabilizer can fit to wide range
operating conditions. The adaptive feature of the
introduced stabilizer results in a satisfactory
performance using a significantly small rule base
due to its tuned parameters as compared to the
standard FLPSS.
2. Adaptive Variable Structure FLPSS:
For singleton fuzzifier, , center average
defuzzifier, product inference, and Gaussian
membership function, it is easy to shown that [8]
M
l
p
ii
F
p
ii
F
M
li
x
x
xfv
l
i
l
i
11
1
1
)(
)(
)(
(1)
Where
M
is the number of rules in the FLS and
p
is the number of inputs to the FLS.
l
is the
centroids of membership functions of the output
corresponding to the
M
rules. Equation (1) can be
expressed as
)()(
1
xpxfv i
M
ii
(2)
where
)(xpi
are called fuzzy basis function (FBF)
[8] and are given by
M
l
p
ii
F
p
ii
F
ix
x
xp
l
i
l
i
11
1
)(
)(
)(
(3)
Now the FLS can be referred to as a FBF
expansion
Consider the nonlinear system
buxfy n )(
)(
(4)
Where
)(f
is unknown real continuous nonlinear
function and
b
is an unknown constant.
Ru
and
Ry
are the input and the output of
the system, respectively.
n
T
nRxxxx )1(
,,,
is the system state vector, and
)(r
y
is the rth
derivative of
y
.
The output
y
is required to follow a reference
signal
m
y
which is selected such that it is
derivatives up to the nth order exit. Define the
tracking error as
)()()(
,, nn
m
n
mm yyeyyeyye
, (5)
Hence, the tracking error state vector,
e
, can be
selected as
T
n
eeee )1(
If
bandxf )(
were known, we would choose the
feedback control law as
ekyxf
b
uT
n
m )(
1
(6)
where the design vector
k
is given by
11 kkkk nn
T
. Substituting (6) into (4)
leads to
ekyy T
n
m
n )()(
(7)
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.6
Tawfiq H. Elmenfy, Samah Abdelsalam
E-ISSN: 2769-2507
42
Volume 5, 2023
Noting the above definition of the tracking error in
(5),
e
, it is possible to rewrite (7) as
0
)1(
1
)()( ekekeeke n
nn
T
n
(8)
The characteristic equation of the error model (3.8)
is
0
1
1 n
nn ksks
(9)
The design parameters
n
kkk ,,, 21
are selected
such that the roots of (9) are in the left-hand side
of the s-plane to ensure stability. Since
bandxf )(
are unknown, the control law (3.6)
cannot be implemented. Based on the universal
approximation theorem [8,9], there is a fuzzy
system that can approximate
u
; i.e. it is possible
to write
)(xpu T
(10)
ˆ
is the vector estimated of centeroids of the
membership functions assigned to
u
and
)(xp
is
the vector of fuzzy basis functions. The control
law (10) is implemented based on an estimate
value
ˆ
of the true values
. Hence, we can write
)(
ˆ
),( xpxuu T
c
(11)
Substituting
),( xuc
in (4) leads to
),()(
)( xbuxfy c
n
(12)
Adding and subtracting
bu
to (3.12) result in
)),(()(
)( uxubbuxfy c
n
(13)
Similar to the derivative of (7), it is possible to
show that the error model corresponding to the
closed loop system (13) is
)),((
)( xuubeke c
T
n
(14)
Eq. (14) can be put in the controllable canonical
form by choosing
21 eee
,
32 eee
,
n
n
neee
)1(
1
)),((
)( xuubekee c
T
n
n
(15)
So, the state space model takes the form
)),(( xuubeAe ccc
(16)
where
121
1000
00
100
0010
kkkk
A
nn
c
,
b
bc
0
0
0
The design parameters
n
kkk ,,, 21
are selected
such that the eigenvalues of
c
A
are located in a
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.6
Tawfiq H. Elmenfy, Samah Abdelsalam
E-ISSN: 2769-2507
43
Volume 5, 2023
pre-specified region of the left-hand side of the s-
plane.
The estimate value
ˆ
is typically based on the
second method of Lyapunov to ensure the stability
of the adaptive system. To illustrate that, consider
the following Lyapunov function.
1
22
1
TT b
ePeV
(17)
where
andP
are positive definite matrixes and
ˆ
is the estimation error. The calculations
of
andP
are shown below. The designer
normally picks up the matrix
as diagonal matrix
that determine the adaptation rate as shown below.
The time derivative of
V
is
1
)(
2
1
TTT bePeePeV
(18)
Substituting for
e
from (3.16) into (3.18) leads to
𝑉
=1
2(𝑒𝑇𝑃𝐴𝑐𝑒 + 𝑒𝑇𝑃𝑏𝑐𝜑𝑇𝑝(𝑥) + 𝑒𝑇𝐴𝑐
𝑇𝑃𝑒 +
𝑏𝑇𝑃𝑒𝜑𝑇𝑝(𝑥)) + 𝑏𝜑𝑇𝛤−1𝜑 (19)
Eq. (3.19) can be simplified to
1
)()(
2
1
TT
c
T
T
cc
TbePbxpePAPAeV
(20)
The closed loop system (3.16) is stable if
V
is
negative semi-definite. Since
c
A
has stable
eigenvalues, it is true that
P
is the solution of the
algebraic Lyapunov equation
QPAPA T
cc
(21)
where
Q
is a positive semi-definite matrix that is
arbitrarily chosen by the designer. Select the
adaptation law as [10, 11]
)(
1xpePb
bn
T
c
(21)
Where
n
P
is the second column of the matrix
P
.
Hence, it is possible to rewrite (19) as
QeeV T
2
1
(22)
Eq. (22) clearly shows that the closed-loop system
(16) is stable if the adaptation law (21) is
employed. To implement (16) and calculate
ˆ
,
we assume that the variation of
is much slower
than of
ˆ
i.e.
is locally constant [61]. The
estimate value
ˆ
is given by
)(
1
ˆxpePb
bn
T
c
(23)
From 23 we have
)(
1
ˆxpePb
bn
T
c
The equivalent sigma-modification law is [11]
))0(()( iii
n
T
ii xpe
(24)
The constants
i
,
and
are design parameters.
)0(
i
is the initial estimate of
i
. By selecting
,0,1
and
)(/ xpe i
n
T
i
i
, we can
rewrite (24) as
)0()sgn( i
n
T
i
ipe
(25)
Where
i
is a constant set by the designer to
specify the possible range of variation of
i
around
)0(
i
. A smaller
i
reflect more
confidence in the corresponding initial value
)0(
i
. To chattering and ensure smooth variation
of
i
, it is common to replace the
sgn(.)
function
in (25) by the
(.)sat
function. Hence, the
estimater is implemented as
)0()( i
n
T
i
ipesat
(26)
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.6
Tawfiq H. Elmenfy, Samah Abdelsalam
E-ISSN: 2769-2507
44
Volume 5, 2023
The estimates obtained by (25) are used to
calculate
pss
U
Where
)(
ˆ
),( xpxuu T
c
(27)
3. Design Procedure Of a Direct
Variable-Structure Adaptive Fuzzy
PSS
1. Let
1
x
( speed deviation),
2
x
(speed deviation derivative ) be the inputs to the
fuzzy basis function (FBF), i.e
.][][ 21
TT
xxx
It is reasonable
to choose the generator speed deviation
and its derivative
as input signals to the
PSS controller, because the aimed is to
damping the oscillation of generator speed
(system frequency) to zero [11]. Each input is
assigned three fuzzy membership functions
PandZN ,,
that stand for the linguistic values
negative, Zero, and positive, respectively. Fig.
1 shows the membership functions
PandZN ,,
. The membership functions are
triangles and equally distributed, the author test
unequally distributed membership functions
and concentrated within a certain interval as
describe in [11], but the author found equally
distributed is more effect in our controller
specially with triangle shapes. The ranges of
membership functions are chosen according to
what is expected for maximum and minimum
of speed deviation and the derivative of speed
deviation
Figure 1: The membership functions used for
the AFPSS inputs variables
2. Develop a fuzzy basis function (FBF) rule base
with two inputs of speed deviations
and
, and one output. Set the initial value of
as initial fuzzy rule base (selected by the
designer's experience with help of the table
look-up ) as in table 1. Apply the adaptation law
(26) to compute
ˆ
online in Table 2 and
calculate the FBF from (3) and apply the results
to (27) to get the PSS output
pss
U
.
3. Use trial and error method to get the gains of
controllers
k
and the second columns of
P
which is of interest in our calculations.
Table: 1 Initial fuzzy rule base of AFPSS
P
Z
N
Z
N
N
N
P
Z
N
Z
P
P
Z
P
Table 2 Tunable fuzzy rule base of AFPSS
P
Z
N
3
ˆ
2
ˆ
1
ˆ
N
6
ˆ
5
ˆ
4
ˆ
Z
9
ˆ
8
ˆ
7
ˆ
P
4. Simulation Results
Nonlinear power system which is used in
simulation studies consists of three Generator
Multimachine power systems. The
N Z P
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.6
Tawfiq H. Elmenfy, Samah Abdelsalam
E-ISSN: 2769-2507
45
Volume 5, 2023
performance of proposed (AFPSS) and
compared with CPSS as shown in Fig. 2, is
evaluated by acting the large disturbance in
transmission line at 2 sec and cleared ant 2.133
sec.
The system performance tracking index is
characterized by ISE as:
(28)
where is the out signals deviation
Figure 2: Power system model used in study of
one Machine
Figure 3: speed deviation response for Gen.
(1)
Figure. 4: speed deviation response for Gen.
(2)
Figure 5: speed deviation response for Gen.
(3)
Table 3 Index equation (28)
Speed
Deviation
Gen. 1
Gen. 2
Gen. 3
CPSS
0.49
0.16
0.14
AFPSS
0.48
0.15
0.13
5. Conclusion
In this paper, dynamic behavior of three
machine systems installed with a conventional
power system stabilizer (CPSS) is investigated
under 3-phased fault. Proposed variable structure
Fuzzy based power are designed to improve
dtteISE )(
2
e
0 1 2 3 4 5 6 7 8 9 10
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
time (sec)
speed deviation (pu)
CPSS
AFPSS
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
time (sec)
CPSS
AFPSS
0 1 2 3 4 5 6 7 8 9 10
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
time (sec)
speed deviation (pu)
CPSS
AFPSS
Turbine
To other
Machine
s
∆ω
G
Governor
AFPSS
d/d
t
AVR
Generator
T.L
Δω
x
Exciter
CPSS
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.6
Tawfiq H. Elmenfy, Samah Abdelsalam
E-ISSN: 2769-2507
46
Volume 5, 2023
damping local and interarea mode of oscillations
following three phase faults that occurs at 1 sec.
and cleared at 1.1 sec.
The obtained results demonstrate that the proposed
AFPSS provides the smallest ISE index in three
generators. Therefore, it is recommended to be
considered in further and extended studies.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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(Attribution 4.0 International, CC BY 4.0)
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International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.6
Tawfiq H. Elmenfy, Samah Abdelsalam
E-ISSN: 2769-2507
47
Volume 5, 2023
