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E L S E V I E R Electric Power Systems Research, 30 (1994) 17-23 ,,
A neuro-fuzzy hybrid power system stabilizer
A.M. SharaP, T.T. Lie b ~Department of Electrical Engineering, University of New Brunswick, Fredericton, NB, Canada
bSchool of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 2263, Singapore
Accepted 22 January 1994
Abstract
The paper presents a novel neuro-fuzzy hybrid power system stabilizer (PSS) design for damping electromechanical modes of oscillation and enhancing power system synchronous stability. The hydrid PSS comprises a front-end conventional analog PSS design, an artificial neural network (ANN) based stabilizer, and a fuzzy logic post-processor gain scheduler. The stabilizing action is controlled by the post-processor gain scheduler based on an optimized fuzzy logic excursion based criterion Jo- The two PSS stabilizers, conventional and neural network, have their damping action scaled online by the magnitude of Jo and its rate of change dJo. The ANN feedforward two-layer based PSS design is the curve fitted nonlinear mapping between the damping vector signals and the desired optimized PSS output and is trained using the benchmark analog PSS conventional design. The fuzzy logic gain scheduling post-processor ensures adequate damping for large excursions, fault conditions, and load rejections. The parallel operation of a conventional PSS and a neural network PSS provides optimal sharing of the damping action under small as well as large-scale generation-load mismatch or variations in external network topology due to fault or switching conditions.
Keywords: Power system stabilizer; Neural networks; Fuzzy logic applications
I. Introduction
Power system stabilizers (PSSs) are usually used to enhance the power system synchronous stability [1-3] and damp the electromechanical oscillatory modes such as machine local, intra-area, and inter-area modes of oscillation. These electromechanical modes of oscilla- tion are usually characterized by natural frequencies in the range 0.1 - 3 Hz.
Most PSSs used in the electric utility industry are analog type with lead-lag compensators, washout, and amplifier gains. The additional damping is introduced through the excitation system by the extra damping electric torque modulation.
Most PSSs utilize speed deviation AoJ, accelerating power APa, and actual generator active and reactive powers AP G and AQG or current Ig as effective modu- lating signals. The design and selection of the best PSS structure, transfer function, and gains is a complex, iterative process which is usually optimized for a given power system topology structure and for given loading conditions. Such a PSS design method usually results in a fixed-structure, fixed-parameter type PSS, which is limited and may lose its effective damping robustness for any network topology variations, loading condi-
0378-7796•94•$07.00 (() 1994 Elsevier Science S.A. All rights reserved SSDI 0378-7796(94)00832-0
tions, large excursions such as short-circuit faults and load rejections.
New techniques [4-8] such as self-tuning, online adaptive control, Lyapunov methods result in effective PSS designs which can be made adaptive to any changes in the network or loading conditions, These new techniques require extensive knowledge of the power system dynamics and utilize estimators which are rather slow to implement online and are less effective due to incorrect state identification under noisy mea- surements and system nonlinearities. New techniques [9-12] have started to emerge for use with expert system, rule based, fuzzy logic, and neural network PSS designs.
This paper presents a novel supplementary artificial neural network (ANN) based PSS design with a fuzzy switching criterion to allow independent, sequential, or parallel operation in conjunction with the fixed 'opti- mized' PSS structure.
2. Power system dynamic model
Fig. 1 depicts the sample single-machine infinite-bus system with the excitation and speed control blocks and
18 A.M. Sharaf T.T. Lie / Electric Power
i(..I) ref
[ ~ - - ] _ +~.~e, ~7" { Exciter ~ Upss
Vt Fig. l. Sample single-machine infinite-bus system.
l
R e ) D-
x ~ • e ~
A(O = K j (sT a ) (I + sTI) E d ~,,T~ ~ TZf, T2) (a)
Fuzzy Logic ANN ~ . Scheduling
2-Layer ~ U Hidden-Sigmoid Outer-Purelinear
~ _ _ PSS
"1~f(x(k))--x(k)-x(k- 1)
T~mad
"7 B2
(b) B1 Fig. 2. (a) ANN based PSS; (b) ANN structure.
the configurations of the conventional and ANN PSS structures with the fuzzy logic gain scheduling post- processor. Fig. 2 depicts the detailed ANN based PSS with output UNPss using the input modulating damping vector X,
X = [e,o, de,,,, e~, R,~, X,,,, dUNPsS , d(dUNpss)] T
and the ANN PSS output
y = [UNpss]
where d is the differential rate operator, dx = x ( k ) -
x ( k - 1), and
e,,, = Aco = ~o -- co o
de,,)(new) = co(new) - oJ(old)
R,,~ = (e,,~ 2 + de,,~ 2 + e,, 2) 1/2
X,., = e,o de,,,
e,,j, and de,,) define the speed error deviation A~o and its incremental rate. e~, is the incremental 'error' change in machine terminal voltage. UNPSS is the output of the ANN based PSS block. The system was simulated using the MATLAB software package with a sampling period
Systems Research 30 (1994) 17-23
of 10ms. R,,, and X~,, are the synthesized additional damping signals representing the excursion vector length and the momentum measure, respectively. These additional signals are utilized to improve the curve fitting and training of the ANN based PSS design.
2.1. T h e d y n a m i c m o d e l
The generator is adapted as three first-order differen- tial equations:
3 = o~0~o (1)
1 (J") = ~ f ( P M ~- a -~ Kd(O -- P c ) ( 2 )
j~tq = 1 0 [Efd - - (X d - - X'a)id - - E'q] (3)
where
id _ Eq -- E cos 6 X e -J- Xq
i t . Eq = Eq _L (Xq -- Xd)t d !
Pe = Eq E sin 6 t
X,, + X d
For the AVR and exciter, the following dynamic model is adapted:
KA (Vref-- Vt + Upss) Era El'd : ~ - - K ( 4 )
where
vt = (Vd2 + V2) ~/2
and
Vd = x q E sin 6 X e -~- Xq
gq p r • E q - - x d 1 d
For the governor, the following transfer function is considered:
For the conventional PSS, the following transfer function is considered:
K A 1 + S T Q , ] \ I + s T e J (6)
3. A N N based P S S design
The best conventional analog PSS benchmark stabi- lizer data were utilized to train and validate the ANN PSS. The feedforward neural network [13] structure was utilized with one hidden layer of 17 neurons and
A.M. Sharaf, T.T. Lie / Electric Power Systems Research 30 (1994) 17-23 19
Table l A N N weighting matrices
- -0 .4362 0.5069
-0 .4346 -0 .0295
0.4778 0.3279
--0.1555 -0 .5595
W I = 0.4711 -0 .3463
0.0628 -0 .1810 -0 .4381
0.4055 -0 .5172
0.2299 -0 .3575
Columns 1-7 of W 2
0.0294 0.1892 0.5598 0.0871 -0 .4871 -0 .5537 - -0 .0757 -0 .1572 -0 .6269 0.0873 -0 .4562 -0 .4549
0.2138 -0 .4362 0.0898 -0.4631 -0 .1235 -0 .6545 0.2717 -0 .2574 -0.6941 -0 .6224 -0 .2388 -0 .1866
-0 .5561 -0 .2359 0.0659 -0 .3935 0.5742 -0 .1458 0.5151 0.3070 -0 .1883 -0 .2351 0.6328 -0 .3802
-0 .1894 0.7028 -0 .0288 0.4760 0.2317 -0 .5177 0.0705 0.4509 -0 .1834 -0 .3953 -0 .1027 0.6016 0.5497 0.4643 -0 .1066 -0 .3609 0.3530 -0 .2980 0.2942 -0 .5741 0.0707 -0 .1022 0.5760 0.4699 0.3805 0.6216 0.5720 0.1922 -0 .2579 0.3777 0.3680 -0 .6395 0.2832 0.0422 0.1090 0.6603
-0 .0865 -0 .1460 0.1652 -0 .7902 0.0960 -0 .4611 -0 .1233 -0 .4488 -0 .6462 -0 .0397 -0 .4717 -0 .2855 -0 .5008 0.5719 -0 .4250 -0 .2370 0.0128 0.0975
0.2926 -0 .7069 -0 .1628 -0 .5802 -0 .0002 0.3166 -0 .6224 0.2376 0.5341 -0 .0147 0.4994 0.0333
0.9480 0.3077 -0 .4429 Columns 8 -14 of W 2 0.9962 1.1379 0.1402 Columns 15-17 of W 2
0.3774 0.0587 -0 .5768 B 2 = -0 .3833
B~=
0.0972 0.5015 -0 .2688 -0 .2349
0.3169 0.1816 -0.8891 0.4623
0.1758 -0 .2436
0.6865 -0 .1610 -0 .9499
0.8323 0.9314 0.3520 1.1352
-0 .1769 0.1142 0.6557
-0 .3672 -0 .9007
0.1787 -0 .7602
0.0322
tansigmoid activation functions. The output layer has one neuron with a pure linear activation function. The ANN network was trained using offline simulation data correlating the input vector X and the output UNPSS for different disturbances such as external network switch- ing, generation-load mismatch, and fault conditions. The ANN block behaves as a nonlinear function ap- proximator correlating the selected input vector X with the output UNpss. The error goal was 0.001 while the learning rate was selected as 10 -5 . To ensure better function approximation, the speed error difference and PSS output rates (first and second changes) were uti- lized as extra input variables. Additional signals R,~ and X,o were added to enhance the function approximation.
The ANN feedforward structure is shown in Fig. 2(b). Table 1 depicts the weighting matrices W~, B~, W2, and B2, establishing the nonlinear mapping from the multivariable input vector X to the ANN PSS output UNpss.
4. Fuzzy logic gain scheduler
Fig. 3 depicts the fuzzy logic gain scheduling struc- ture comprising the fuzzification, decision table, and defuzzification stages. The two fuzzy variables I]J0 I] and its rate of change dllJo] I are defined as fuzzy input variables as follows:
[IJo(k) [I = [e,f(k) + deo~2(k) + el(k)] ,/2 (7)
dllJo(k) tl--IIJo(k) I I - I I J o ( k - 1)tl (8)
Jo
dJo
Stage 1 Stage 2 Stage 3
IS M L VL
0 0.1 O~ 0.5 0.7 0.9 I.o 1,2 ~0
I dJol
Fig. 3. Post-processor gain scheduler (/7 varies from 0 to 1.0 p.u.).
Table 2 Assignment table (decision rules)
S M L VL
1 5 9 13 S S S M L
2 6 10 14
M S M M L
3 7 11 15 L M M L L
4 8 12 16 VL M L L VL
20 A.M. Sharaf, T.T. Lie/Electric Power Systems Research 30 (1994) 17-23
NO PSS • d 2.5
".~ 1.5
~, 0.5 ½ ,1 6 8 10
Time (see.)
NO PSS ~" 0.02 5 = 0.01
'-0.01
m~"0"02 ½ 4 6 8 10
(a) Time (see.)
NO PSS
"" 300
i~ 2oo
lOO o
~' 00 2 4 6 8 10
Time (see.)
NO PSS ~" 0.2-
W m 01 2 4 6 8 10
(b) Time (see-)
Fig. 4. Response to (a) a 0.5 p.u. step-change increase in PM and (b) a 0.6 p.u. step-change increase in x~ without the conventional PSS.
~" 2.5
= 2
.~ 1.5
1
0.5
~" 0.02 5 .~ ~o1
,~ o
~-0.01
CONV. PSS
~ ~ 8 lO Time (see.)
CONV. PSS
CONV. PSS
t~ 100 ~J
• ~ 0 2 4 6 8 10
~" 0.15 5 = 0.1
-~ 0.05
o
~-0.05 2 4 6 8 10
Time (see.)
= 2 x l0~ . . . . x10"3 .o
¢/2
121 0.015 ~ . 5
2 x l0a
i -2
Speed Deviation
~ 6 h lO (a) Time (see..) (b)
Time (see.)
CONV. PSS
i ~ ; 8 lO Time (sec.)
CONV. PSS
e ~
0.05 011 0.15
Speed Deviation
0 xlO'3 CONV. PSS
-80 ~ ~ ~ 8 lO Time (see.)
Fig. 5. Response to (a) a 0.5 p.u. step-change increase in PM and (b) a 0.6 p.u. step-change increase in xe with the conventional PSS.
A.M. Sharaf, T.T. Lie~Electric Power Systems Research 30 (1994) 17-23 21
Each fuzzy variable is assigned four membership classes, namely {small, medium, large, very large}. The fuzzy output scaling fl is also assigned the same four membership classes {small, medium, large, very large}. The membership functions are also shown in Fig. 3.
The decision table (rule base) is given in Table 2 as follows. The online gain scaling fl is assigned the fol- lowing numerical weights corresponding to the four classes {small (0.1), medium (0.6), large (0.8), very large (1.0)}. So, gain vector flj has the following dis- crete weights [0.1,0.6,0.8, 1.0] depicting the possible relative sharing levels between the conventional and ANN based PSSs.
The equivalent weighting /~ is obtained using the modified center of area (COA) criterion as follows:
i f - J - - k = 0 . 8 - 1 . 8 ( 9 )
J
and
coj = min{#Jo,/tdJ0} (10)
The modified COA criterion on k around 1.5 results in better damping and less overshoot/undershoot under power system fault and excursion conditions.
5. Sample simulation results
To validate the hybrid neuro-fuzzy PSS damping effectiveness, two large excursions FI and F2 were intro- duced. A 0.5 step-change increase in Pm was applied at time t = 1.5 s and a 0.6 step-change increase in the external system equivalent impedance x e was also ap- plied at time 1.5 s. Figs. 4 and 5 depict the system response without and with the conventional PSS. Figs. 6 and 7 depict the performance with the ANN PSS and the hybrid neuro-fuzzy PSS designs, respectively.
~" 1.4
"~ 1.2
0,8
• ~ 0.6
ANN PSS
2 4 6 8 10
Time (see.)
ANN PSS
~.~r0.005
{ ,o Time (see.)
xl0 "s ANN PSS
. 1 -o.~5 6 0.6o5 O.Ol Speed Deviation
ANN PSS 4
ANN PSS
°
~, 0.5 ' ' ' 0 2 4 6 8 10
e~ O
O
2
0
Time (see.)
,oX'°-' , ~ .
. . . . 10
Time (see,)
2 x10-3 ANN PSS
-11 , -2 0 i ~ 6
Speed Deviation xl0 3
ANN PSS 10
-10
"20 2 4 6 8 10 2 4 6 8 10
(a) Time (sec.) (b) Time (sec.)
Fig. 6. Response to (a) a 0.5 p.u. step-change increase in PM and (b) a 0.6 p.u. step-change increase in x e with the A N N PSS.
22 A.M. Sharaf, T.T. Lie~Electric Power Systems Research 30 (1994) 17 23
~" 1.4 PROPOSED PSS
1.2 ..~
0.8 o
$ 0.6 ½ ,~
Time (sec.)
i~ m~--~ 11 ~ xI!10s~ROPOSED m ~
" 2 4 6 Time (see.)
PROPOSED PSS
10
xlO-3 o
10
PROPOSED PSS 1.4
~1.
~ 0.8 •
06 i i i i o
Time (sect)
10 xl0"a , PROPOSED PSS .
,
o Time (se~)
• ~." 2 x10-3 PROPOSED PSS
0
• ~ -1' 0 5 10
Speed Deviation xl0 -3
PROPOSED PSS
°it t ~ o I r~
-10 (
-20 "20 2 4 6 8 10 Time (see.)
Speed Deviation xl0 "3
PROPOSED PSS
2 4 6 8 10 (a) (b) Time (see.)
Fig. 7. Response to (a) a 0.5 p.u. step-change increase in PM and (b) a 0.6 p.u. step-change increase in x e with the hybrid neuro-fuzzy PSS.
6. C o n c l u s i o n s
The paper presents a novel hybrid PSS design for damping electromechanical modes of oscillation and enhancing power system stability. The design utilizes a hybrid convent ional PSS, an A N N based PSS, and a gain scheduling fuzzy logic post-processor .
The p roposed hybrid PSS is a robust and effective PSS with on-line scaled damping under small-scale and large-scale disturbances.
A c k n o w l e d g e m e n t s
The authors wish to acknowledge the suppor t o f N a n y a n g Technological University, Singapore, and The Universi ty of New Brunswick, Canada .
03o = 314.16 Xd = 1.24 Xe = 0.6 T~0 = 5.0
A p p e n d i x
Synchronous machine parameters (in p.u.)
M = 6.92 Kd = -- 5.0 Xq = 0.743 x~ = 0.32 PM = 0.75 E = 1.0
A VR and excitation system parameters (in p.u.)
K A = 400 T A = 0.02 Vre f = 1.0
Governor system parameters (in p.u.)
a = --0.001238 b = - 0 . 1 7 Tg = 0 . 2 5
A.M. Sharaf, T.T. Lie~Electric Power Systems Research 30 (1994) 17 23 23
Conventional PSS parameters (in p.u.) References
Kj=40 TQ=2.5 T~=0.1 T2=0.03
Disturbances
F~ a 0.5 p.u. step-change increase in PM applied at time t = l . 5 s
F2 a 0.6 p.u. step-change increase in xe applied at time t = l . 5 s
Nomenclature
E E~ Eq X~
M /'e PM T~ T~o Upss Vt Xd x~ Xq
amplitude of infinite-bus voltage transient excitation voltage transient q-axis voltage amplifier gain constant equivalent damping factor effective inertia (2H) electrical power mechanical power amplifier time constant equivalent transient rotor time constant PSS control signal terminal-bus voltage d-axis reactance d-axis transient reactance q-axis reactance
Greek letters
~5 09
rotor angle corresponding to infinite-bus system deviation of rotor speed from synchronous speed 090
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