Design and analysis of an adaptive fuzzy power system stabilizer

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					IEEE Transactions on Energy Conversion, Vol. 11, No. 2, June 1996

455

DESIGN

AND ANALYSIS POWER

OF AN ADAPTIVE STABILIZER
K. Tomsovic Science

FUZZY

SYSTEM
P. Hoang

School of Electrical Engineering and Computer Washington State University 99164-2752 Pullman. WA

Abstract Power system stabilizers (PSS) must be capable of providing range of PSS rely

system stabilizer

(PSS) provides

a positive

damping

torque

[1,2] in phase with the speed signal to cancel the effect of a broad appropriate stabilization signals over operating conditions and disturbances. Traditional the system negative damping torque. Because the gains of this controller are determined for a particular operating 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 rule-based controllers Control implemented has shown promising results [6,7]. algorithms based on fuzzy logic have been in many processes [8,9]. The application of has been motivated (1) improved by the desire for robustness over

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. around The emphasis in fizzy control design centers uncertainties in system parameters 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 industry control deregulation. Fuzzy logic

such control techniques that obtained (2) simplified

one or more of the following: controllers are based on empirical to fuzzy logic for a specific criteria. This in controller

rules. In this paper, a systematic approach control design is proposed. Implementation machine reqtures specification criteria which of performance into off-line performance parameters real-time translates

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 the control numerical has been undergoing of FACTS solutions devices field [8] for a small [11] and such test in Japan. Other of these designs

three

can be calculated

or computed

in response to system changes. The robustness of and transient is directed at methods

systems have been developed for voltage regulators [12]. Most require have been tuned to a specific generally effort. In this paper, a methodology is proposed.

the controller is emphasized. Small signal analysis methods are discussed. This work developing appropriate robust stabilizer design when fuzzy logic is applied. 1. Introduction Power system stabilizers appropriate operating stabilization conditions

system. Unfortunately,

and analysis

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 parameters to obtain suitable root location Although [13]. a broad range of fuzzy logic methods have both a well-founded

(PSS) must be capable of providing signals over A traditional power

and disturbances.

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 stability measures for fuzzy logic controllers [10, 13]. In the power system, performance concerns are particularly acute with the high reliability of instabilities. models may requirements and the costly effects Yet, analysis using precise mathematical be infeasible due to the power system

96 W

paperrecommended andapprovedby the IEEE Energy Development and Power Generation Committee of the IEEE Power
039-8 EC
A Engineering Society for presentation at the 1996 IEEE/PES Winter Meeting, January 21-25, 1996, Baltimore, MD. Manuscript submitted July 27, 1995; made available for printing January 4, 1996.

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 In this analysis. methods work, to address these performance a first step is taken towards studies. the non-linear concerns. systematic

For simulation

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 assessment in terms of Fig. 2. Excitation MODELS and model system models are 1. System

2. SYSTEM In this section. the

generator plant

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 lead network. is greater than that obtainable In this case. cascaded lead

presented (details can be found in the Appendix). diagram of the generator

The block

is shown in Fig.

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:

stages are used where k is the number of lead stages. 2.2 Fuzzy stabilizer The development of the fuzzy logic approach here is limited to the controller structure and design. More detailed discussions on tizzy logic controllers are widely available, e.g., [9, 12]. For the proposed FPSS. the second term of (2) is replaced with a fuzzy logic rule-base using the filtered That is. speed deviation and acceleration of the machine.

2.1 Power system stabilization The stabilizing system. signal is introduced in conjunction with the exciter

the deviation from synchronous speed and acceleration of the machine are the error, e, and error change. &, signals, respectively. stabilizing IFe
THEN

for the controller.

The control

output,

u. is the

reference voltage to obtain feedback for the regulatorIn this study, both a traditional (FPSS) are analyzed. logic based stabilizer

signal Vs. Each control rule R, is of the form:

PSS and a fuzzy The traditional is A,
u is C,

PSS is modeled by the following

transfer function:
k

AND e is B,

~(s)

.~T

.

()

I+sTl — 1+sT2

(2)

where

.-t,. B I

and

C,

are

fuzzy

sets

with

triangular -1 and

membership

functions

as shown normalized

between

The first term in (2) is a reset term that 1s used to “wash out” the compensation effect after a time lag T. The use of reset control will assure no permanent offset in the terminal voltage due to a prolonged error in frequency, which may

1 in Fig. 3. These same fuzzy sets are used for each variable of interest: only the constant of proportionality is changed. These constants are h-,, K, and k- for the error. error change and control output, respectively. The error and error change are classified according to these fuzzy membership functions Similarly. contribution modtiled by an appropriate membership signal one rule. control constant. may Rule A specific the are signal may have non-zero a specific of more than in more than one set. represent conditions

joined by using the minimum intersection operator the resulting membership fimction for a rule is: (e,

so that

UR,

e) = min(h(e)>

!-h,(e))

(3)

\

—-[

Ffiter

The suggested control output from rule I is the center of the membership function C,. Rules are then combined using the center of gravity method to determme a normalized control output U

Fig. 1 Generator Plant Model

457

@
LN LN -1
(LP=lqge positwe; negatwe; LP LP LP MP MP

MN
LP MP MP IUP SP

SN
LP A@ SP SP

ZE
AfP IUP SP ZE SN

SP
IMP SP ZE SN SN

MP I
SP ZE NV ‘AIN MN

LP
I ZE Slv I

-.65
posltwe: ZE=zero: SN=

-.3
MP=

0

.3
medium

.65

1
SP=small MN=medlum e

MN SN ZE

Fig. 3. Membership

functions

scaled from -1 to 1
posltwe: negatwe; AL%i ~ lMAT LN

SN=snlall negative)

small

SF’
(4)

IZE
I I I

I 2.3 Proposed FPSS design steps development so far is general. A
K,

I ZE I SIV

I

I ./X

I
~ Lhr

LP

IL.V , Table. 1. Rules table
outputs are italicued)

MN

MN

1 I

The fhzzy logic controller

(Control

particular control design requires specification of all control The control rules are rules and membership functions. designed from an understanding controller, of the desired effect of the For example, consider the rule:

and A“ for a particular The methodolo~

system and range of operating is described below: control output for K based on

conditions.

1. Select the maximum

IF e is SN AND e is SP THEN /.{ k ZE This rule anticipates The complete that the desired operating control point will be needed.

the physical limitations of the controller. 2. Replace the FPSS with a constant gain K. 3. Simulate a significant disturbance until oscillations exceeds the either begin to settle or the stability limit. 4. Set A-. and h-,to the maximum error e and e. respectively, penocl. If damping appears inadequate then: 5. Linearize operating 6. Using
K.

system

observed values for the simulation

reached soon and stabilization set of control

is no longer

during

rules is shown in Table 1. Each

of the 49 control rules represents a desired controller response to a particular situation. The control rules were designed to be symmetric under the assumption that if 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 literature membership for a specific and many rely on manual tuning of control rules and functions authors to establish have the desired performance e.g. [15], neural net may artificial

the system and FPSS around the nominal point (see section 4). traditional eigen value to obtain desired damping. is to manage a wide uncertainties, analysis, adjust

and K. together (i.e. maintaining

the same relative

magnitude) As an objective

of the fuzzy controller conditions

range of operating

and modelling

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 are response to

system. (There are some exceptions, proposed

methods for tuning the controller).

Such manual tuning

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 The membership the constants K,.

scenarios

intended

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

w
-1-

angle response to this disturbance for systems with the PSS and the FPSS design (see Fig. 6) Case 2: Three phase to ground a fault clearing time tc=0.2
sec

fault

at A.. (Fault is

30% of the distance along line). Line is removed with The plots show the response of the systems with design (see Fig. 7). 3.2 Multimacbine bus A system with two machines connected to an infknite bus is parameters shown in Fig. 5. The system has the following (all values in per unit): Network: line 1-2 R=.O 18, X=.11, B=.226; B=.098: both lines 2-3 the PSS and the FPSS

Fig. 4 Single machine connected inthite

,1

2

R=.008, X=.05,

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: - three bus system + Case 3: Three phase to ground fault at B. (Fault is

+
Fig, 5. Two machine step change in mechanical model of Appendix A.

power input using the non-linear The PSS is designed using a

30’% of the distance along line 1-3).

The line returns

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 intlnite bus. adapted from [17], was used in the design phase studies of a and then this controller was used for multi-machine system. adapted from [18]. 3.1 Single machine The single machine connected to an infhite bus shown in

to service with clearing time tC=O. 15 sec. (Fig. 8). Plots show response for systems with the PSS and the FPSS design. 3.3 Discussion In all cases, the FPSS shows superior the traditional controller. For improved damping is not as noticeable. or similar response to the better remains

smaller

disturbances.

For the more severe demonstrates

disturbances, performance. the controller

the FPSS controller The multi-machine robustness in that

has significantly simulations the controller

Fig. 4 has the following Generator: M=9.26s, .1”;=. 190 p.u. Voltage remdator (iVote: Network:
external equwalent The

system parameters:
D=.O1

p.u, T>O=7.76 s. A~=.973 P.U. T., = 05s.
R

effective despite significant

changes in the system dynamics. ANALYSIS

: KA=50.

lE~~l S 10
arises from modelhng V=l .05: Y= O 6 +JO.3. T=3 .0s.

4. PERFORMANCE 4.1 Small signal stability For small disturbances, system linearized of this elgenvalues stability the

negatwe

of generation.)

both lines R=-O.68, .Y=l.994; KC= 7.09, Power system stabilizer:

T,=O.6851s,

can be characterized operating point.

by the If the the

T,=O.1s ~: & =40,0,
Mechanical

about

K,

=9.25.

K=

1.0

system lie in the left hand plane,

input power’; P~=l and are shown in the figures

system is small disturbance stable. In this study. the delay caused by computation is neglected so that the tizzy logic 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.

Two cases were simulated on the following page.

@ Case 1: Step change of mechanical to P.=
1.3 p.u.

power from P.= 1 and rotor

The plots show frequency

459 A difficulty of the small signal analysis lies in the fact that the FPSS is not differentiable. This problem is managed by numerically operating point. calculating Table a linear 2 shows approximation the eigenvalues near the for the fuzzy logic preliminary critical operating controllers time but our results in this area are stall approach is pursued here. The of (CCT) is calculated for a number

A numerical

clearing

points for the PSS and FPSS systems.

The results

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 at the controller operating second point and the eigenvalues operating While point without are recalculated changing the

are shown in Table 3. In all cases, the FPSS designs improve the margin of stability as indicated by the CCT. 5. CONCLUSIONS This AND DISCUSSION for a fuzzy logic

paper proposes a general

structure

parameters.

the uncompensated

system has

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 performance with relative insensitivity point Similar results were found for system.
4.2 Transient

stabilizer. maximum

Controller design requires calculation of the ranges for frequency and frequency deviation

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 to disturbances has demonstrated the effectiveness of this design technique. Small signal and transient stability analysis give some evidence of the robustness of the controller. This methods research of design control. under is directed and at developing for tizzy systematic logic based which model

to the operating the multimachine

stability the system non-linearities to apply Lyapunov FPSS must be to

analysis

For large disturbances, considered. Operating Point Nominal ~lm=]

stabilization are effective

The ability extreme

to design controllers of dynarnlc

It is possible Pss

functions

uncertainty

parameters is felt to be of growing relevance as the number of energy suppliers connected to the network increases. 6. REFERENCES [1] F. P. deMello and Machine Control ,“
and S’stems,

-1.361 -4.290 -18.88 -0.33

+ j4.452 ~ j8.199

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~ j3.235 j13.312

C.
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316-329. [2] S. E. M. de Oliveira, “Effect of Excitation Systems and Power Systems Stabilizers on Synchronous Generator Damping and Synchronizing Torques. ” Ii7E Vol. 136, September 1989, pp. 70 Proceedings, [3] G. Honderad, M.R. Chetty and J. Heydman. “An Expert System
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No.3. 1987, pp. 343-348 [5] D.Xia and G, T. Heydt, “Self-tuning Controller for the Transactions on Generator Excitation Control, ” IE~E Vol. PAS-102, 1983. Power .4pparatus and Systems, pp. 1877-1885. [6] T. Hiyama. “Real Time Control of Micro-machine System using Micro-computer based Fuzzy Logic Power 11’mter .Ileelmg. 93 System Stabilizer,” 1993 IEEIOPES
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Table 3: Critical

clearing

times for example systems

Columbus,

OH. January 1993.

460

““’~

,. ,, ,, . .

. .

o

2

4 Time (sec.)

6

8

0

2

4 Tune (SN )

6

$

Fig 6 SteD chm?e of me-ctical ‘“0081~

lxnver from P-=l to P.=1 3,--- F’SS, . FPSS 100 I
I

/-,

J

I

1

,, ,,
0.998 0 2 4 Tu-ne (see ) 6

I I
8

I
20 0 2 4 Tune (see ) 6

1 8

Fig. 7 Faull at A with t, =0 2 sec , -- PSS, - FPSS 1.02: -1,01 3 & c w $ I ,,, ,,, ,,. ,,, ,,’! ‘,>,, ,,, ,, ;’ Machine 1
101[

Madune 2

I
> ,. ? . ‘, ,(’ ‘,’’., ,, 1

,
1

,

‘, ‘, 1 2

I
3 Tune (sez ) 4 5 6

0
20-

1

2

3

4

5

6

099— o
20

Time (SW ) Excitation \olt.age 1

Excitation voltage 2

-1

-lo~ 0

I I 1

1“ 2 3 Tme (SW) 4
Fig

I 5 6

0

1

2

3 Tune (S= )

4

5

6

$ Fault at B mTth :,=0

15sec, -- Pss, - FPss

461 [7] M. A. Hassan and 0. P. Malik, Laboratory Self-Tuned Test Results for Power “Implementation a Fuzzy Logic and Based (B-7) (B-%) (B-9) (B-IO) (B-11) (B-12)

IEEE/PES System Stabilizer.” Summer .Weeting, 92 SM 474-9 EC, Seattle, July 1992 [8] S.T Wierzchon, “Mathematical Tools for Knowledge

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Approximate

Reasorung

in

Expert

1985, pp. 61-69.
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K. to

Tanaka Backing

and up
on

M.

Sane,

“A

Robust

Stabilization Generator rotor angle Synchronous Internal Internal frequency axis voltage quadrature
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of Fuzzy Control Control
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direct axis voltage axis voltage voltage factor power input

~;
F D Pm PG pL QG QL Id ltf .Yd

Direct axis voltage Quadrature Terminal Damping Mechanical

[12]

Generator real power Load real power Generator reactive power Load reactive power Direct axis Moment of Direct axis Quadrature Quadrature current inertia reactance axis reactance reactance reactance axis transient

Oscillations

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Field voltage Number of generators Y bus matrix Angle of Y bus matrix Reference voltage Stabilizing input signal Regulator amplifier gain Regulator amplifier time constant

Fuzzy Logic
$vstems.

IEEE

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Nov.

1993.

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298-301. [16] K. Tomsovic and P. Hoang. “Design and Analysis Methodology for Fuzzy Logic Stabilization Control of Power System Disturbances.” to be submitted to LEEE
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APPENDIX

A: MODELING

The study systems are described by the following and algebraic equations: & co, – WYef (~, – (l&j]+
– (.Y~, –.~j, pm, – pG, )/Af, ) Id, + Efd, )/~dOz

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

(B-1) (B-2) (B-3) (B-4) (B-5) (B-6)

& = (–D, o
E&, = (–Ej,

PG, = ~’>,Id, i- l’~,Iql QC, = t ‘qId, – ~’d,I~, Id, = (E&–
~“~,)/.\:,