455
IEEE Transactions on Energy Conversion, Vol. 11, No. 2, June 1996
DESIGN
AND ANALYSIS
POWER
OF AN ADAPTIVE
SYSTEM
FUZZY
STABILIZER
K. Tomsovic
P. Hoang
School of Electrical Engineering and Computer
Washington State University
99164-2752
Pullman. WA
system stabilizer
Abstract
Science
(PSS) provides
a positive
damping
torque
[1,2] in phase with the speed signal to cancel the effect of
Power system stabilizers
(PSS) must be capable of providing
a broad
appropriate
stabilization
signals over
operating conditions and disturbances. Traditional
range
of
PSS rely
on robust linear design methods. In an attempt to cover a
wider range of operating conditions,
expert or rule-based
controllers have also been proposed. Recently, fuzzy logic as
a novel robust control design method has shown promising
results.
The
around
emphasis
uncertainties
in
fizzy
control
design
in system parameters
centers
and operating
conditions. Such an emphasis is of particular relevance as
the difficulty
of accurately
modelling
the connected
generation
is expected
to increase
under
power
controllers
are based on empirical
rules. In this paper, a systematic approach
control design is proposed.
Implementation
reqtures specification
performance
parameters
real-time
criteria
which
of performance
translates
into
can be calculated
control
to fuzzy logic
for a specific
criteria.
three
off-line
This
controller
or computed
in
in response to system changes. The robustness of
the controller
is emphasized.
Small signal
analysis methods are discussed. This work
developing
robust
appropriate
when fuzzy logic is applied.
stabilizer
design
and transient
is directed at
and analysis
methods
Power system stabilizers
operating
stabilization
conditions
039-8 EC
Control
implemented
has shown promising
(PSS) must be capable of providing
signals
over
and disturbances.
results [6,7].
algorithms
based on fuzzy logic have been
in many processes [8,9]. The application of
such control techniques
has been motivated
that obtained
(2) simplified
by the desire for
(1) improved
robustness over
using conventional
linear control algorithms.
control design for difficult to model systems.
e.g., the truck backer-upper problem [10], and (3) simplified
implementation.
In power systems, several controllers have
been developed for PSS. One such controller
hdyro unit
has been undergoing
field
[8] for a small
test in Japan. Other
systems have been developed for voltage regulators
the control
of FACTS
devices
have been tuned to a specific
numerical
solutions
generally
[12]. Most
[11] and
of these designs
system. Unfortunately,
require
such
a large computational
general
design and analysis
The proposed method also
ptirsues small signal stability analysis which provides the
opportunity
to design a system with adjustable controller
a broad
range
A traditional
of
power
paperrecommendedandapprovedby the IEEE Energy
Development and Power Generation Committee of the IEEE Power
96 W
rule-based controllers
effort.
In this paper, a
methodology
is proposed.
1. Introduction
appropriate
condition,
they may not be valid for a wide range of
operating conditions. Considerable efforts have been directed
towards developing adaptive PSS. e.g. [3,4]. In an attempt to
cover a wide range of operating
conditions,
expert or
rule-based controllers
have been proposed for PSS [5].
Recently,
the introduction
of fuzzy
logic
into
these
one or more of the following:
deregulation.
Fuzzy logic
machine
industry
the system negative damping torque. Because the gains of
this controller
are determined
for a particular
operating
A
parameters to obtain suitable root location
Although
[13].
fuzzy logic methods have both a well-founded
theoretical basis and numerous successful implementations,
controversy has surrounded the developed systems. This is
due in part to the lack of satisfying performance measures.
Recently, there have been efforts directed at appropriate
Engineering
Society for presentation at the 1996 IEEE/PES Winter Meeting,
January 21-25,
1996, Baltimore, MD. Manuscript submitted July 27, 1995;
stability measures for fuzzy logic controllers [10, 13]. In the
power system, performance concerns are particularly
acute
made available
with the high reliability
for printing
January 4, 1996.
of instabilities.
models
may
requirements
and the costly effects
Yet, analysis using precise mathematical
be infeasible
due to the power
system
complexity
(i.e.,
large
dimension,
non-linearities,
uncertainties in load fluctuations, disturbances and generator
dynamics, and so on). The viewpoint
offered here is that
0885-8969/96/$05.00 @ 1996 IEEE
456
fuzzy Ioglc has been introduced
because of the above
ditllculties
and thus, the approach should be better equipped
than
conventional
methods
concerns.
In
this
work,
systematic
analysis.
to address these performance
a first
For simulation
step
is
taken
towards
studies. the non-linear
power system and controller are linearized and small signal
stability analysis is performed,
It is proposed to rethink the
traditional
methods
greater uncertainties.
of
stability
this
section.
in
of
the
MODELS
generator
and
system
models
presented (details can be found in the Appendix).
diagram
terms
Fig. 2. Excitation
2. SYSTEM
In
assessment
of the generator
plant
model
are
The block
is shown in Fig.
1.
Specifically,
the plant is modeled based on a generator
model incorporating
single-axis field flux variation and the
simple excitation system shown in Fig. 2. The PSS used for
comparison
studies is a lead compensator
The following
H=.(s)
models the excitation
continuous
transfer function
system and regulator:
occur in an overload or islanding
condition.
The second
term of (2) is a lead compensator to account for the phase
lag through the electrical system [14].
In many practical
cases. the phase lead required
from
a single
The development
of the fuzzy logic approach here is
limited to the controller structure and design. More detailed
with
a fuzzy
signal is introduced
in conjunction
In this study, both a traditional
logic based stabilizer
PSS is modeled by the following
with the
exciter
respectively.
stabilizing
()
using
the filtered
of the machine.
That is.
for the controller.
The control
output,
u. is the
signal Vs. Each control rule R, is of the form:
The traditional
transfer function:
IFe
is A,
AND e is B,
THEN
.
rule-base
PSS and a fuzzy
(FPSS) are analyzed.
I+sTl
—
1+sT2
logic
and acceleration
the deviation from synchronous speed and acceleration of the
machine are the error, e, and error change. &, signals,
reference voltage to obtain feedback for the regulator-
.~T
case. cascaded lead
discussions on tizzy logic controllers are widely available,
e.g., [9, 12]. For the proposed FPSS. the second term of (2)
2.1 Power system stabilization
~(s)
In this
2.2 Fuzzy stabilizer
speed deviation
system.
is greater than that obtainable
lead network.
stages are used where k is the number of lead stages.
is replaced
The stabilizing
System
u is C,
k
(2)
where
.-t,. B I
membership
and
C,
functions
are
fuzzy
sets
with
as shown normalized
triangular
between
-1 and
The first term in (2) is a reset term that 1s used to “wash
1 in Fig. 3. These same fuzzy sets are used for each variable
out” the compensation effect after a time lag T. The use of
reset control will assure no permanent offset in the terminal
of interest: only the constant of proportionality
is changed.
These constants are h-,, K, and k- for the error. error
voltage
change and control output, respectively. The error and error
change are classified according to these fuzzy membership
due to a prolonged
error in frequency,
which
may
functions
modtiled
by an appropriate
signal may have non-zero
Similarly.
a specific
contribution
membership
control
of more than
constant.
A specific
in more than one set.
signal
one rule.
may
represent
the
Rule
conditions
are
joined by using the minimum
intersection operator
the resulting membership fimction for a rule is:
UR,
\
—-[
Ffiter
Fig. 1 Generator Plant Model
(e,
e) = min(h(e)>
!-h,(e))
so that
(3)
The suggested control output from rule I is the center of
the membership function C,. Rules are then combined using
the center
of gravity
control output U
method
to determme
a normalized
457
@
MN
LN
SN
SP
ZE
MP
LP
I
-1
-.65
-.3
0
Fig. 3. Membership
(LP=lqge
posltwe:
positwe;
negatwe;
MP=
ZE=zero:
SN=
1
LP
LP
LP
AfP
IMP
SP
I ZE
MN
LP
MP
A@
IUP
SP
ZE
Slv
SN
LP
MP
SP
SP
ZE
NV
AL%i ~
ZE
MP
IUP
SP
ZE
SN
‘AIN
lMAT
SF’
MP
SP
IZE
SN
SN
MN
LN
I
scaled from -1 to 1
medium
SN=snlall
small
.65
.3
functions
LN
SP=small
posltwe:
negatwe;
MN=medlum
e
negative)
I
(4)
I
2.3 Proposed
LP
FPSS design steps
The fhzzy logic controller
development
so far is general.
particular control design requires specification of all control
The control rules are
rules and membership
functions.
designed from an understanding
controller,
K,
to be symmetric
control
and
many
manual
functions
needed.
rules is shown in Table 1. Each
under
the
assumption
tuning
to establish
of
control
that
have
proposed
methods for tuning the controller).
rules
artificial
and
e.g. [15],
neural
net
Such manual tuning
may
be very time consuming and perhaps more importantly
sheds
some doubt on the claims for robustness of the fuzzy logic
approach. In this work. a systematic tuning methodology is
proposed.
It is assumed that the fimdamental
control laws
change quantitatively
not qualitatively
with the operating
In this vein. control rules and membership
condition.
functions
functions
are designed once as above.
are modified by scaling through
error
is described below:
control
output for K
based on
The membership
the constants K,.
a significant
disturbance
e and e.
respectively,
until
system
oscillations
exceeds
the
observed values for
during
the
simulation
penocl.
If damping appears inadequate then:
5. Linearize
if
the desired performance
system. (There are some exceptions,
authors
are italicued)
either begin to settle or the
stability limit.
4. Set A-. and h-,to the maximum
point will be
is no longer
necessary any asymmetries could be best handled through
scaling. In addition, adjacent regions in the rule table allow
only nearest neighbor changes in the control ouput (LN to
MN. MN to SN and so on).
This ensures
that small
changes in e and e result in small changes in u.
Many of the fuzzy logic controllers
proposed in the
membership
1
I
system and range of operating
1. Select the maximum
of the 49 control
rules represents a desired controller
response to a particular
situation.
The control rules were
for a specific
outputs
~Lhr
./X
of the desired effect of the
that the desired operating
set of control
on
MN
The methodolo~
3. Simulate
This rule anticipates
rely
MN
I
I
the physical limitations of the controller.
2. Replace the FPSS with a constant gain K.
reached soon and stabilization
literature
I SIV
and A“ for a particular
conditions.
IF e is SN
AND e is SP
THEN /.{ k ZE
designed
I
(Control
For example, consider the rule:
The complete
I
IL.V
,
Table. 1. Rules table
ZE
A
I
I
the system and FPSS around the nominal
operating
6. Using
K.
point (see section 4).
traditional
eigen value
and K. together (i.e. maintaining
magnitude)
As an objective
adjust
to obtain desired damping.
of the fuzzy controller
range of operating
analysis,
the same relative
conditions
is to manage a wide
and modelling
uncertainties,
the simulation
in step 3 for method 1 may need to be
repeated under a set of parameter variations.
The controller
is adaptive in the sense of the varying of these gains but not
in terms of varying the control rules. Further discussion on
these design steps can be found in [16].
3. NUMERICAL
In this section, simulations
to
several
disturbances.
exercise the controller
SIMULATION
illustrate
The
the controller
scenarios
are
response
intended
to
rather than to represent any specific
system scenario. Two simple systems are presented but
higher order systems have been simulated
with similar
results. The FPSS constants are found by simulations of a
458
angle response to this disturbance for systems with
the PSS and the FPSS design (see Fig. 6)
w
Case 2: Three phase to ground
fault
at A.. (Fault is
30% of the distance along line). Line is removed with
a fault
clearing
time tc=0.2
The plots show the
sec
response of the systems with
the PSS and the FPSS
design (see Fig. 7).
-1-
3.2 Multimacbine
Fig. 4 Single machine connected inthite
bus
A system with two machines
2
,1
connected to an infknite bus is
shown in Fig. 5. The system has the following
parameters
(all values in per unit):
Network:
line 1-2 R=.O 18, X=.11, B=.226;
R=.008, X=.05,
B=.098:
both lines 2-3
line 1-3 R=.007, X= 04. B=.082;
1~1=0.6-j0.3, Y,2=0.4-J0.2
Mechanical Dower: Pml=l .2, P~2=l
Generator parameters are the same as in the single machine
case. One scenario is presented here:
+
Fig, 5. Two machine
step change in mechanical
model of Appendix
A.
- three bus system
+
power input using the non-linear
The PSS is designed using a
Case 3: Three phase to ground
intlnite
bus. adapted from [17], was used in the design phase
studies
of
a
connected
Fig. 4 has the following
Generator: M=9.26s,
.1”;=. 190 p.u.
external
In all cases, the FPSS shows superior
to an infhite
D=.O1
The
T., = 05s.
negatwe
R
arises
=9.25.
K=
from
modelhng
V=l .05:
T,=O.6851s,
on the following
to P.=
robustness
effective despite significant
in
response to
disturbances.
the
For the more severe
has significantly
simulations
that
the
better
demonstrates
controller
remains
changes in the system dynamics.
ANALYSIS
4.1 Small signal stability
For small disturbances,
linearized
of this
stability
about
the
can be characterized
operating
point.
by the
If
system lie in the left hand plane,
the
the
system is small disturbance stable. In this study. the delay
caused by computation
is neglected so that the tizzy logic
and are shown in the figures
page.
1.3 p.u.
the FPSS controller
elgenvalues
1.0
@ Case 1: Step change of mechanical
is not as noticeable.
The multi-machine
controller
or similar
smaller
disturbances,
system
input power’; P~=l
Two cases were simulated
For
4. PERFORMANCE
Y= O 6 +JO.3.
T=3 .0s.
controller.
damping
lE~~l S 10
both lines R=-O.68, .Y=l.994;
KC= 7.09,
Power system stabilizer:
Mechanical
traditional
performance.
the
p.u, T>O=7.76 s. A~=.973 P.U.
of generation.)
K,
bus shown in
system parameters:
: KA=50.
equwalent
T,=O.1s
~:
& =40,0,
3.3 Discussion
improved
The single machine
Voltage remdator
(iVote:
Network:
FPSS design.
the
3.1 Single machine
is
The line returns
to service with clearing time tC=O. 15 sec. (Fig. 8).
Plots show response for systems with the PSS and the
conventional
phase lead technique to precisely compensate
for the phase lag of the electrical loop. Two systems are used
for the simulations.
A single machine connected to an
and then this controller
was used for
multi-machine
system. adapted from [18].
at B. (Fault
fault
30’% of the distance along line 1-3).
power from P.= 1
The plots show frequency
and rotor
controller can be modeled as a zero memory non-linearity.
This is a reasonable assumption as the rotor oscillations of
interest are orders of magnitude
slower than the time
required for the FPSS computations.
The FPSS does not
introduce new poles but acts to shift the eigen values of the
uncompensated
system.
459
A difficulty
of the small signal analysis lies in the fact
that the FPSS is not differentiable.
This problem is managed
fuzzy logic
by numerically
critical
operating
calculating
point.
Table
a linear
2 shows
approximation
the
near the
eigenvalues
for
the
system with the traditional PSS and the FPSS design. As the
FPSS should provide proper stabilization control over a wide
range of operating conditions, the eigenvalues are found at
The system is designed for the
two operating
points.
nominal
operating
at
second
the
controller
point and the eigenvalues
operating
parameters.
point
While
controllers
preliminary
but our results in this area are stall
A numerical
clearing
operating
time
approach
(CCT)
is pursued
is calculated
here.
The
for a number
points for the PSS and FPSS systems.
of
The results
are shown in Table 3.
In all cases, the FPSS designs
improve the margin of stability as indicated by the CCT.
5. CONCLUSIONS
AND
DISCUSSION
are recalculated
without
changing
the uncompensated
the
system has
This
paper proposes a general
stabilizer.
maximum
structure
for a fuzzy
logic
Controller
design requires calculation
of the
ranges for frequency and frequency deviation
two eigenvalues in the right hand plane, both the traditional
PSS and FPSS act to move the eigenvalues into left hand
plane and establish
small
disturbance
stability.
It is
interesting to note that the I?PSS shows good small signal
during some specified disturbance. The advantage of th~s
design approach is that the controller is insensitive to the
precise dynamics of the system. Simulation
of the response
performance
with relative
insensitivity
point
Similar results were found for
system.
to disturbances has demonstrated the effectiveness of this
design technique.
Small
signal
and transient
stability
analysis give some evidence of the robustness of the
to the operating
the multimachine
controller.
4.2 Transient
stability
This
research
methods
For large disturbances,
considered.
the system non-linearities
It is possible
to apply Lyapunov
Operating
Point
Pss
Nominal
-1.361
+ j4.452
-4.290
~ j8.199
~lm=]
must be
functions
-2.072
~ j8.703
-19.59
-0.33
- 8.027~
j13.312
under
[1] F.
-1.790
P.
-8.309
.4pparatus
~ j14.412
316-329.
for the single machine
systems
FPSS
machine
P,”=l.30
0.08 sec.
0.84 sec.
0.14 sec.
System
0.25 sec.
Pm,=l.2,
Pm,=l.2,
Pm,=l.o
Pm,=l.2
Table 3: Critical
0.23 sec.
0.21
sec.
clearing
O. 31 sec.
0.28 sec.
systematic
logic
based
to design controllers
which
of dynarnlc
model
sec.
times for example systems
C.
Concordia,
Stability
,“
IEEE
and S’stems,
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as
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PAS-88.
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on Po~er
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for
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of
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to Power
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of
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July 1989. pp. 465-568.
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deMello
~ j2.634
J
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The ability
Synchronous
-0.33
Table 2: Eigenvalues
Single
are effective
Excitation
t j3.727
-4.814
Operating
control.
at
analysis
6. REFERENCES
-0.33
Pm=l.30
stabilization
directed
and
~ j3.235
-0.33
-0.818
is
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of energy suppliers connected to the network increases.
FPSS
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460
““’~
,.
,,
,,
. .
. .
o
6
4
2
0
8
4
2
6
$
Tune (SN )
Time (sec.)
Fig 6 SteD chm?e of me-ctical
lxnver from P-=l to P.=1 3,--- F’SS, . FPSS
100
‘“0081~
I
I
/-,
J
I
1
,,
,,
II
0.998
0
4
2
6
I
1
20
0
8
2
4
Tu-ne (see )
6
8
Tune (see )
Fig. 7 Faull at A with t, =0 2 sec , -- PSS, - FPSS
Machine 1
1.02:
,,,
I
-1,01
3
&
I
,,,
,,.
,,,
,,’!
>
‘,>,,
c
w
$
,,,
Madune 2
101[
;’
,,
,.
‘, ,(’ ‘,’’.,
,,
?
.
-
1
,
0
1
2
3
4
5
6
Time (SW )
Excitation \olt.age 1
20-
‘,
‘,
,
1
099—
o
1
I
2
3
4
5
6
5
6
Tune (sez )
Excitation voltage 2
20
-1
-lo~
0
1“
I I
1
I
2
3
Tme (SW)
4
Fig
5
6
$ Fault at B mTth :,=0
0
1
15sec, -- Pss, - FPss
2
3
Tune (S= )
4
461
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PI-Type
on
Sets
and
Soft
Power
Fuzzy Logic
Fuzzy
System
1994.
Methods
and
for Improving
on Fuzzy
Controllers,”
J.J. Grainger
flna[vsis,
and
McGraw-Hill,
APPENDIX
1, No.
IEEE
4,
&stenz
E&, = (–Ej,
Generator real power
pL
Load real power
QG
Generator reactive power
QL
Load reactive power
Id
Direct axis
Moment of
Direct axis
Quadrature
– (.Y~, –.~j,
reactance
axis transient
Number of generators
Y
Y bus matrix
Y
V;ef
Reference voltage
pp.
reactance
Field voltage
1993.
Angle of Y bus matrix
K.
Stabilizing input signal
Regulator amplifier gain
T.
Regulator
F,
amplifier
time constant
Academic
Stevenson,
Power
Systems
1994.
DETAIL
differential
(B-1)
pm, – pG, )/Af,
(B-2)
) Id, + Efd, )/~dOz
(B-3)
PG, = ~’>,Id, i- l’~,Iql
(B-4)
QC, = t ‘qId, – ~’d,I~,
(B-5)
Id, = (E&–
(B-6)
~“~,)/.\:,
current
inertia
reactance
axis reactance
BIOGRAPHIES
W.D.
A: MODELING
(~, – (l&j]+
power input
n
Dynanucs,
co, – WYef
& = (–D,
o
voltage
factor
Transactions
Nov.
The study systems are described by the following
and algebraic equations:
&
axis voltage
Mechanical
of
~vstems.
Yu, Electrlc
Power
Press, 1983, pp.8(J-81
Y.N.
[18]
Quadrature
Direct axis transient
K. Tomsovic and P. Hoang. “Design and Analysis
Methodology
for Fuzzy Logic Stabilization
Control of
Power System Disturbances.”
to be submitted to LEEE
Transactions
[17]
Direct axis voltage
PG
Control,
Performance
axis voltage
Pm
.Yd
298-301.
[16]
direct axis voltage
Quadrature
Vol.
$vstems.
Stablhty
Internal
D
ltf
Computing
quadrature
Terminal
Damping
F
San Jose. CA, Nov. 1994, pp. 262-269.
Svmposlum.
[14]
Rough
~;
frequency
Internal
Patrick
Hoang
Received the B.S. and M. S. degrees from
Washington
State University.
Pullman. in 1992 and 1994.
respectively, all in Electrical Engineering.
He is currently
employed at DISC Inc. His research interests include expert
systems and fuzzy set application to power system and plant
controls.
the
B. S.E.E. from
Kevin
Tomsovic
(M’87)
received
Michigan
Tech. University.
Houghton,
in 1982, and the
M. S.E.E. and Ph.D. degrees from University of Washington,
Seattle, in 1984 and 1987. respectively.
He has held visiting
professorships at National Cheng Kung University. National
Sun Yat-Sen
University
and the Royal
Institute
of
Technology
in Stockholm.
His research interests include
expert systems and fuzzy set applications to power systems