14th PSCC, Sevilla, 24-28 June 2002
Session 14, Paper 2, Page 1
Damping oscillations improvement by fuzzy power system stabilizers
tuned by Genetic Algorithms
D. Menniti, A. Burgio, A. Pinnarelli, V. Principe, N. Scordino, N. Sorrentino
Department of Electronic, Computer and Systems Science
University of Calabria - ITALY
Fax (+39) 984-494713 Phone (+39) 984-494707 e-mail sorrentino@deis.unical.it
Abstract – A very important matter of discussion in power
system operation is the oscillations damping problem. As an
issue, new more effective power system stabilizers (PSSs) have
been implemented, based on non linear control laws and/ or
non conventional control techniques such as those provided by
use of fuzzy logic and neural networks. Because of
simultaneous presence of several PSSs in the power system, the
problem of their coordination arises. For this reason, in this
paper the authors utilize Genetic Algorithms to coordinate
power system stabilizers based on fuzzy logic previously
proposed by the authors.
The co-ordination technique has been tested on the wellknown New England power system and its effectiveness has
been confirmed by the obtained results.
Keywords: Decentralized control, Power System
stabilizer, Fuzzy control, Multimachine power system,
Feedback Linearization.
1. INTRODUCTION
The severe restrictions on the expansion of the
transmission network of modern power systems, due to
economic constraints, environmental impact and right-ofway limitations, lead to the operation of power systems
under conditions increasingly stressed, implying more
tight transient and steady-state stability margins.
Therefore, a major effort has to be made to improve
power system stabilizers (PSSs) performance and
characteristics. PSSs are usually designed once a time, by
conventional control methods, which restrict the system
model to low order single-input-single-output linear
models, whereas the power system oscillatory instability
is actually a large-scale multivariable problem.
In order to improve the PSSs performance, generally,
the explicit and systematic formulation of two major
issues in their design has to be taken into account: the
ability of the control scheme to provide satisfactory
performance for a large variety of operating conditions the robustness issue- and the interactions among the
various controllers -the co-ordination issue [1].
In [2], to meet the robustness issue, the authors have
proposed a fuzzy logic power system stabilizer (FPSS).
In this case, the power system was modelled using the
direct feedback linearization technique (DFL). The DFL
is able to represent a n machine power system as an
interconnected system of n linear systems by non-linear
local feedback for each machine [3]. After the
linearization, the dynamic of each subsystem, controlled
as it was isolated, is independent of the actual operating
point. The DFL compensating law has the ability to
alleviate but not to eliminate the non-linearities effects in
the n machine power system. The model still so contains
non-linearities and interconnections effects.
The proposed FPSS is able to compensate these nonlinearities and interconnections effects improving power
system dynamic performance. The FPSS is a fuzzy
controller based on a double decision table approach. In
the former table, the antecedent of each rule conjoins the
speed deviation and the generated power deviation in
fuzzy set values. In latter one, the antecedent of each rule
conjoins the previous fuzzy output value and the angle
deviation in new fuzzy set values.
A very important task to assure a good performance of
the controllers is to choose the FPSSs membership
functions ranges (MFRs). Due to the complexity of both,
the power system and the FPSS, the MFRs are determined
by a hand made trial and error procedure considering, for
sake of simplicity, a FPSS once a time.
Moreover, in order to assure the power system stability
and to improve power system dynamic performance the
co-ordination of the FPSS actions is necessary.
For this aim, in this paper, it is proposed a genetic
algorithm (GA) to tune FPSSs membership function
ranges previously determined to achieve the better coordination when the FPSSs simultaneously work in the
same power system.
GAs are search procedures based on the mechanics of
natural selection and natural genetics. They were
developed to allow computers to evolve solutions to a
problem, to select the best solutions for recombination of
the others and to use their offspring to replace poorer
solutions. Usual concerns in optimization problems, such
as non-differentiability, non-linearity and non-convexity,
do not limit the use of this search method. The greater
flexibility provided by GAs regarding controller structure
and objective function specifications, is the main
advantage of the method.
In the first section of the paper, the FPSS controller is
briefly recalled, then the used GA is described in details.
Finally, the numerical experiments performed on the New
England power system are summarized.
2. THE FUZZY LOGIC POWER SYSTEM
STABILIZER: FPSS
In the last years, the interest in modelling power system
by feedback linearization techniques has increased more
and more. The main advantage of this technique is based
on the fact that the linearization does not depend on the
actual operating point. Then, the controller designed on
the basis of the so linearized model is more robust than
14th PSCC, Sevilla, 24-28 June 2002
Session 14, Paper 2, Page 2
the conventional one. Unfortunately, by feedback
linearization, residual non-linearities exist: it is so
necessary to adopt an adaptive control technique such as
fuzzy logic in order to assure effective control and to
enhance the system dynamic performance [2, 4].
In this section the combination of feedback linearization
and fuzzy control, proposed in [2], is briefly reported.
A. Direct feedback linearization
As well known, in an n machines power system, only
the m critical ones are equipped with PSS. To design
effective controllers and to co-ordinate them, an
important step is to evaluate their dynamic interactions.
The following equations for the n machines power system
can be written:
δDi (t ) = ω i (t )
(1)
Di
1
ωD i (t ) = −
ω i (t ) +
( Pmi − Pei (t ))
(2)
Mi
Mi
)
(
1 
'
'
'
ED qi
u − E qi
I di (t )
(t ) =
(t ) + x di − x di
 i

'
Tdoi
(3)
where i = 1,..., n and
'
'
Pei (t ) = ∑ E qi
(t ) E qj
(t ) β ij (t )
j =1
n
'
I di (t ) = ∑ E qj
(t )α ij (t )
j =1
with
β ij (t ) = Bij sin δ i (t ) − δ j (t ) + Gij cos δ i (t ) − δ j (t )
[ (
)
(
)]
α ij (t ) = [Bij cos(δ i (t ) − δ j (t ) ). − Gij sin (δ i (t ) − δ j (t ) )]
and the other variables of obvious meaning.
It is useful to recall also the two following expressions:
n
'
'
(t ) E qj
(t )α ij (t )
∑ E qi
(4)
j =1
n
'
I qi (t ) = ∑ E qj
(t ) β ij (t )
j =1
(5)
In order to eliminate the non-linearities in the electric
equations (1)-(3), let us firstly consider the differentiation
of the active electric power Pei(t). Hence the new variable
is ∆Pei(t)=Pei(t)-Pmi:
1
∆Pei (t ) = −
∆Pei (t )
'
Tdoi
+
1
'
Tdoi
' I (t ) I (t ) + P
[u i (t ) I qi (t ) + x d − x di
mi
qi
di
'
+ Tdoi
Qei (t )ω i (t )] +
−
)
(
n
'
'
(t ) E qi
(t ) β ij (t )
∑ E qj
(
)
' Q (t )ω (t )]
− Tdoi
ei
i
(6) so becomes
∆PDei (t ) = −
1
'
Tdoi
∆Pei (t ) +
1
'
Tdoi
(6)
j =1
n
'
'
(t ) E qj
(t )α ij (t )ω j (t )
∑ E qi
j =1
Being in operative regions of synchronous machines
n
'
'
vi (t ) + ∑ ED qj
(t ) Eqi
(t )β ij (t )
j =1
(8)
n
'
'
− ∑ Eqi
(t ) Eqj
(t )α ij (t )ω j (t )
j =1
Considering for each machine equipped with a PSS
x i (t ) = [δ i (t ) ω i (t ) ∆Pei (t )] as state vector, and vi(t)
as input vector, the set of m machines equipped with PSS
can be seen as the aggregation of m subsystems described
by:
xD i = Ai x i (t ) + Bi v i (t ) + Bi f i (t , x(t ))
where
i=1,2..m




0 
 0 
1 


Bi =  0 
−

Mi 
 1 
1 
T ' 
−
 doi 
' 
Tdoi

 n '

' (t ) β (t )
 ∑ ED qj (t ) E qi

ij
 j =1

' 

f i (t , x(t )) = Tdo
n


'
'
 − ∑ E qi (t ) E qj (t )α ij (t )ω j (t ) 
 j =1





0
1

D
Ai = 0 − i
Mi

0
0


n
Qei (t ) =
Iqi(t)≠0, for the m machines the following non-linear
feedback law can be chosen:
1
'
u i (t ) =
[vi (t ) − x d − x di
I di (t ) I qi (t ) − Pmi
I qi (t )
(7)
(9)
(10)
(11)
B. Fuzzy logic controller
From equations (9)-(11), it is clear that the dynamic of
each machine is linear when it is considered decoupled
from the other ones, while the residual non-linearities are
due to the interconnections. Adopting an opportune
decentralised adaptive control law it is possible to face
the aforesaid non-linearities thus improving system
performance [4]. In [2], the authors have already
demonstrated the effectiveness of the proposed fuzzy
power system stabilizer (FPSS) to achieve such a goal.
The basic configuration of the FPSS can be simply
represented in four parts: the fuzzifier, the knowledge
base, the inference engine and the defuzzifier. The
fuzzifier maps the FPSS input crisp values into fuzzy
variables using normalized membership functions and
input gains. The fuzzy logic inference engine then infers
the proper control action based on the available fuzzy
rules base. The fuzzy control action is in turn translated to
the proper crisp value through the defuzzifier using
normalized membership functions and output gains.
In the proposed FPSS, the model expressed by (9)-(11)
14th PSCC, Sevilla, 24-28 June 2002
Session 14, Paper 2, Page 3
is utilized and consequently the state variables, angle
deviation, speed deviation, and active power deviation,
are selected as FPSS inputs.
Although the angle deviation (∆ δ) cannot be directly
measured, it can be easily estimated by local
measurements as demonstrated in [5].
Each of the FPSS input and output signals are
represented by a given number of linguistic variables. A
reasonable number is seven, because increasing the
number of linguistic variables results in a corresponding
increase in the number of rules. Each linguistic variable
has its fuzzy membership function. The membership
function maps the crisp values into fuzzy variables. The
chosen set of membership are NB, NM, NS, Z, PS, PM,
PB which stand for negative big, negative medium,
negative small, zero, positive small, positive medium and
positive big, respectively. For sake of simplicity, it is
assumed that the membership functions are triangular,
symmetrical and each one overlaps the adjacent functions
by 50%.
As in FPSS there are three control variables, the
authors propose an original solution based on a double
decision table approach, as shown in Fig. 1.
In Table 1, the antecedent of each rule conjoins the
speed deviation, ∆ω, and the generated active power
deviation, ∆Pe, in fuzzy set values (OutFLC1). In Table 2,
the antecedent conjoins OutFLC1 and the angle deviation
∆δ in new fuzzy set values (OutFLC2).
FPSS
∆Pe
∆ω
∆δ
FLC1
Outflc1
FLC2
νi
time, for which complex optimization algorithms have
been devised to obtain the tuning sequence [8].
In this paper, to simultaneously tune all the FPSSs
parameters, a genetic algorithm is used and an opportune
performance index, calculated by the dynamic simulation
of the overall non linear system, is adopted.
In this section, firstly, the peculiarities of GA are
recalled, then the formulation of the problem as an
optimization problem and its translation in genetic code
are illustrated.
A. Genetic Algorithm
GAs are search procedures inspired by the mechanism
of evolution and natural genetics. They combine the
survival of the fittest principle with information exchange
among individuals to computationally produce simple yet
powerful tools for system optimization and other
applications.
The first step in the solution of an optimization problem
using GA is the encoding of the optimization problem’s
variables. The most usual approach is to represent these
variables as strings of 0s and 1s. These strings are often
referred to as chromosomes, by analogy to the natural
genetic process. Firstly, the relationship between these
strings and the actual value of the variables may be
established in different ways, and the best way may differ
from one problem to another. Secondly, the objective
function is converted into a fitness function, which is
used to evaluate each string.
An initial population of alternative solutions is usually
chosen at random. Potentially good candidate solutions
beforehand known can be included in the initial
population to speed up computation and increase the
chances of finding the global optimum.
Table 1 Rules Table of FLC1
Fig. 1 FPSS structure
Then the by the centroid defuzzyfication rule is applied
on OutFLC2 to evaluate vi. The fuzzy rules for FLC1 and
FLC2 are reported in Tables 1 and 2, respectively.
The membership function range, i.e. the FPSS
parameters, are determined by an hand made trial and
error procedure considering in the power system only a
FPSS once a time (CFPSS).
3. FPSS TUNING PROCEDURE
In a multi-machine power system with several poorly
damped modes of oscillation, several PSSs have to be
used. For large-scale power systems, comprising many
interconnected machines, the problem of tuning the PSS
parameters is not a straightforward exercise and in some
cases could become too complex to resolve [6,7].
Researchers have addressed the PSS parameters tuning
problem in a multi-machine power system in two basic
ways: one concerns algorithms to simultaneously tune the
parameters of all the machines of the system; the other
one to sequentially tune the PSS parameters, one PSS at
∆ω
∆ Pe
NB
NM
NS
ZE
NB
NB
NB
NB
NM
NM
NB
NB
NM
NM
NS
NB
NM
NM
NS
ZE
NB
NM
NS
ZE
PS PM PB
NM NS ZE
NS ZE PS
ZE PS PM
PS PM PM
PS
PM
PB
NM
NS
ZE
NS
ZE
PS
ZE
PS
PM
PS
PM
PB
PM
PM
PB
PM
PB
PB
PB
PB
PB
Table 2 Rules Table of FLC2
Outflc1
NB
NM
NS
ZE
PS
NB
NB
NB
NB
NB
NM NM NS
PM PB
NM
NB
NM
NM
NM
NS
NS
∆δ NS
NM
NM
NS
NS
ZE
ZE
PS
ZE
NM
NS
NS
ZE
PS
PS
PM
ZE
PS
NS
ZE
ZE
PS
PS
PM PM
PM
ZE
PS
PS
PM
PM
PM
PB
PB
PS
PM
PM
PB
PB
PB
PB
After this step, successive populations are generated
14th PSCC, Sevilla, 24-28 June 2002
Session 14, Paper 2, Page 4
using the following genetic operators:
• Selection: this is a process in which strings are coupled
to a mating pool according to their fitness value, i.e. the
ones with a higher value have a higher probability of
contributing with one or more offspring in the next
generation;
• Crossover: this operator proceeds in three steps: firstly,
pairs of strings are picked at random from the mating
pool; secondly, if a number generated randomly in the
range 0 to 1 is greater then pc (the crossover rate), this
pair is selected for crossover, otherwise it remains
unaltered; thirdly, the crossover process itself consists
of interchanging the portions of the strings beyond a
position, which is randomly selected.
• Mutation: this is the random flipping of bits in the
population of strings, according to a mutation rate pm.
Selection is responsible for the implementation of the
survival of the fittest principle. Crossover implements
information exchange among individuals of a population
with the attempt to generate new better fitted individuals.
Mutation has the role of restoring good genetic material
that may have been lost by selection and crossover. There
are different ways of performing selection, crossover and
mutation [9].
B. Formulation of the optimization problem
To apply GA, it is firstly necessary to formulate the
problem as an optimization problem.
At this scope, it is opportune to remember that the main
goal of the proposed approach is to choose the
membership function ranges of the m FPSSs so as to take
into account the interactions among the different
controllers and to enhance the dynamic performance of
the power system.
Obviously, it is necessary to determine an index able to
quantify the dynamic performance of the power system
for a chosen set of parameters. The adopted dynamic
performance index, which quantifies the deviation of the
machine speed from its nominal value, is the following
[10]:
J =
are the parameter bounds.
Of course, this optimization problem cannot be solved
with a conventional technique, being characterized by a
non-differentiable objective function and non-linear
constraints.
In this case, genetic algorithms (GA)can provide
advantage.
C. Definition of GA elements
The formulation (13) can be translated into genetic
algorithm code. The most important step is to code the
potential solution of the problem as an individual.
The variables to be determined are the triangular fuzzy
set ranges. Each FPSS is made up of four input and two
output fuzzy sets; maintaining the ranges bounds as
further variables then the total amount of variables for the
FPSS is twelve. Obviously, these variables are the same
for each of the m FPSSs.
The variables are floating numbers that can be
converted in binary strings of appropriate dimension in a
very easy way, assuring a fixed precision.
A generic solution of the problem can be so coded as in
Fig. 2, where LbIj, UbIj, LbOk and UbOk are respectively
the lower and upper bounds of the j-th input and k-th
output fuzzy set ranges.
For each individual it is possible to evaluate its fitness
calculating the associated performance index J simulating
the non-linear power system model under a test fault.
Having in mind to maximize the powersystem dynamic
performance the fitness is computed as the inverse of J.
m
T
∑ ∫0 ∆ω i dt
i _1
(12)
Then, the optimization problem can be formulated as
follows:
θ∗=Min J(θ,x)
θi i=1,2..m
s.t.
x i = Ai xi (t ) + Bi vi (t ) + Bi f i (t , x(t )) i=1,2..m
(13)
νi=fuzzy_controller(θi,xi)
f(t,x,θ)=0 ( machines without PSS)
θi,min <θ i <θi,max
where f is the non-linear power system model, θi is the
set of the membership function parameters and θi,min, θi,max
Fig. 2 Coding the generic problem solution
D. Structure of GA
The scheme of GA is conventional: the roulette rule for
the selection, the simplest crossover for the generation
evolution and a binary technique for the mutation [11].
The population so evolves from an initial one, chosen
randomly, to a final issue, for a fixed number of
generations. Indeed, the more is the number of requested
generations better is the solution, but this would worsen
the computational burden. By fixing the number of
generations, the most convenient trade off between the
improvement of the solution and computational effort is
14th PSCC, Sevilla, 24-28 June 2002
Session 14, Paper 2, Page 5
reached.
Then, the best element of the final population is
considered as the optimal configuration for FPSS
(GFPSS).
4. NUMERICAL EXPERIMENTS
The proposed approach has been tested on the New
England test system constituted by ten machines and
thirty-five buses. For this power system the most critical
machines to be equipped with PSS are the 6-th and 9-th.
For each machine, the proper FPSS membership
functions have been designed by a trial and error
procedure. Starting from this knowledge, 10 possible
FPSSs configurations have been randomly generated. The
GA starts from this initial population and uses as test case
the line 3-9 in fault condition for 0.10 sec.. The GA
evolves for 10 generations and then stops having reached
an acceptable level of fitness. In Fig. 3, the trend of the
fitness in the population during the evolution in the
generation is depicted.
It is possible to see that the best solution improves
generation after generation as well as the population mean
fitness.
FPSSs parameters before and after the tuning procedure
are reported in Table 3-4 respectively.
In order to test the effectiveness of the so obtained
controller, the results of the GFPSS are compared with
CFPSS.
Several fault conditions have been considered and the
performance of the system is quantified by (12).
The most significant results of the performed tests are
reported in Table 5. The first column reports the line in
fault (fault duration 0.15 sec), the second and third ones,
respectively, report the value of the index (12) when
CFPSS and GFPSS are working calculated using the
results of the non linear dynamic system simulations.
10.8
x 10
-3
10.6
10.4
10.2
10
1/J
Best fitness
Mean fitness
9.8
9.6
9.4
9.2
9
8.8
1
2
3
4
5
6
7
generations
Fig. 3 Genetic algorithm solutions
Table 3 FPSS on machine 6 parameters before and after
10
optimization
Membership
Function range
∆ω
∆P
∆δ
V
CFPSS
GFPSS
-2÷2
-10÷10
-1.5÷1.5
-300÷300
-2.60÷1.40
-12.77÷6.99
-2.07÷1.06
405÷201
Table 4 FPSS on machine 9 parameters before and after
optimization
Membership
Function range
∆ω
∆P
∆δ
V
CFPSS
GFPSS
-1÷1
-10÷10
-1.5÷1.5
-500÷500
-1.34÷1.31
12.82÷7.17
-2.00÷1.00
-698÷333
Table 5 Results of the most relevant tests
Line in fault CFPSS GFPSS Improvement [%]
4-19
1396
1149
17.7%
8-9
2446
2047
16.2%
3-4
2434
2223
8.7%
16-17
2445
2047
16.3%
In order to show the effectiveness of the proposed
approach, the last column reports the improvement
allowed by the GFPSS with respect to CFPSS. It can be
noticed that an average improvement of 14.6% can be
reached after the tuning procedure.
As an example, in Fig. 4, the speed of machines 2, 6
and 9 are reported in comparison when CFPSS and
GFPSS are working, respectively, and a fault on line 4-19
is considered.
5. CONCLUSIONS
In the paper a co-ordinated design of the fuzzy power
system stabilizers (FPSS) proposed by the authors in [2]
is described.
The problem of the co-ordinated selection of FPSSs
parameters formulated as an optimization problem to
minimize a dynamic performance index is very hard to
solve by conventional optimization methods, owing to
system large dimension and to the non-differentiable
nature of the problem. For this reason, in the paper, a
genetic algorithm has been proposed. Then, the
optimization problem has been translated into a genetic
formulation, in terms of fitness and chromosomes, so that
a standard genetic algorithm can find an “optimal”
solution.
The numerical results obtained on the New England
power system confirm the goodness of the proposed
approach comparing the dynamic power system
performance before and after the tuning procedure.
14th PSCC, Sevilla, 24-28 June 2002
Session 14, Paper 2, Page 6
Machine 2
0.8
GFPSSs
CFPSSs
0.6
0.4
rad/sec
0.2
0
-0.2
-0.4
-0.6
-0.8
0
1
2
3
4
5
sec
6
7
8
9
10
Machine 9
1
GFPSSs
CFPSSs
0.8
0.6
rad/sec
0.4
0.2
0
-0.2
-0.4
0
1
2
3
4
5
sec
6
7
8
9
10
Machine 6
0.7
GFPSSs
CFPSSs
0.6
0.5
0.4
rad/sec
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
0
1
2
3
4
5
sec
6
7
8
9
10
Fig. 4.: Speed machines when a fault on line 4-19 occurs
VI. REFERENCES
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[3] Y. Wang, G. GuoD.J. Hill “Robust decentralized
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power
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