PSS for Multi-Machine Power System Using Adaptive Adaline-Identifier and Pole-Shift Control

Copyright @ IFAC Power Plants and Power Systems Control. Brussels. Belgium. 2000

PSS FOR MULTI-MACHINE POWER SYSTEM USING ADAPTIVE ADALINE-IDENTIFIER AND POLE-SHIFT CONTROL G. Ramakrishna,

O.P. Malik

The University of Calgary Calgary, Alberta, Canada T2N IN4

Abstract - This paper proposes a ADAptive LINEar Neuron (ADALINE) identifier and a pole-shift controller to damp out multi-mode oscillations in power system. A five machine power system is used in the simulation studies. The generating units are represented by third order discrete ARMA models. ADALINE network is modeled so that its weights have a one-to-one relationship with the ARMA parameters. The weights are updated adaptively at each sampling interval to determine the characteristics of the actual system. The pole-shift control uses the on-line updated weights to calculate the closed-loop poles of the system. The unstable poles are moved inside the unit circle in the z-plane and the control is calculated so as to achieve regulation of the system to a constant setpoint in the shortest interval of time. Behavior of the adaptive power system stabilizer (APSS) and its coordination ability with other conventional PSSs (CPSSs) in the system are demonstrated. Copyright @1!OOO [FAC. KelJWord.: Multimachine, power ill/litem IItabilizerll, ARMA mode", Identifierll, adaptive control.

1.

ating conditions, and maintain desired control ability over a wide operating range of a power system. The main limitation of self-tuning control is that it takes a large amount of computing time for on-line parameter identification.

INTRODUCTION

Multi-mode oscillations appear in multi-machine power system in which the interconnected generating units have quite different inertia constants and are weakly connected by transmission lines. These oscillations are generally analyzed in three main oscillation modes, i.e. local, inter-area and inter-machine modes. Depending upon their location in the system, some generators participate in only one oscillation mqde, while others participate in more than one mode (Larsen and Swann, 1981). The effectiveness of the proposed APSS to damp out multi-mode 0scillations in multi-machine environment needs to be verified.

To meet the above requirements, artificial neural networks (ANNs) based identification techniques have been proposed in the literature (Zhang et al., 1993; He and Malik, 1997; Shamsollahi and Malik, 1997; Shamsollahi and Malik, 1999). Once trained, the ANN can store the 'a priori' conditions and new information can be added on-line (He and Malik, 1997). However, a major drawback of the above ANN based techniques (Zhang et al., 1993; He and Malik, 1997; Shamsollahi and Malik, 1999) is the "black-box" like description between the input-output pairs. In this paper, a more precise approach of using an ANN for identifying the model parameters and an analytical technique to compute the control signal is presented. A third-order discrete ARMA model is used to describe the generating units (Chen et. al., 1993). The ADALINE network is modeled so that its weights have a one-to-one relationship with the regression coefficients of the ARMA model. This overcomes the "black-box" like description of the ADALINE and the information is analyzed at each sampling interval to track changes in the controlled system. The Pole-

Among the published literature on the APSS, selftuning regulators are the most common (Pierre, 1987; Ghosh et al., 1984; Finch et al., 1996). The structure of a self-tuning APSS has two parts: on-line parameter identification and control strategy. At each sampling .period, a mathematical model is obtained by on-line identification method to track the dynamic behavior of the plant. The control strategy calculates the control signal based on on-line identified parameters. Studies have shown that an APSS can adjust its parameters on-line according the change in oper-

159

3

6

2

where A(z-l), B(z-l) and C(z-l) are polynomials in the backward shift operator z-l and are of third order. Variables y(t), u(t) and d(t) are the system output, system input and white noise respectively. The parameter vector in this study is

8

5

O(t)

= [ol020ablb2baClc2C3].

(2)

The noise term, C(z-l )d(t), in the ARMA model of eqn. (1) plays an important role in APSS design and application in multi-machine power systems (Chen and Malik, 1995). In multi-machine power systems, the behavior of one generating unit is determined not only by itself but also by other generating units in the system which are unknown to the local APSS. It means that the output of the identified model, y(t), in eqn. (1) is not only determined by the known input u(t) but also by some other known input sources described as d(t). This term actually serves as the estimation of the coupling of this local generating unit with other generating units in the system and the local noise of the system. It is included to improve the accuracy of the dynamic description of the identified model. Rewriting eqn. (1) in the form suitable for identification:

Fig. 1. A five machine power system configuration

shift (PS) control uses the on-line updated regression coefficients to calculate the closed-loop poles of the system. The unstable poles are moved inside the unit circle in the z-plane and the control calculated so as to achieve desired performance. 2. MULTI-MACHINE POWER SYSTEM MODEL A five machine power system without infinite bus, as shown in Fig. 1, is used to evaluate the performance of the proposed APSS. Five generating units are connected through a transmission network. Generators C l l C 2 and C 4 have much larger capacities than Ca and Cs. All five generators are equipped with governors, exciters and AVRs. Parameters of all generators, governors, exciters, AVRs, transmission lines, loads and operating conditions are given in (Shamsollahi and Malik, 1999). Generators C 2 • Ca and Cs may be considered to form one area, and generators Cl and C 4 a second area. The two areas are connected through a tie-line connecting buses 6 and 7. Under normal conditions, each area serves its own load and is almost fully loaded with a small load flow over the tie-line.

y(t) = OT (t)\lf(t)

+ d(t),

(3)

where

\If(t)

[-y(t - 1) - y(t - 2) - y(t - 3) u(t - 1) u(t - 2) u(t - 3) d(t - 1) d(t - 2) d(t - 3)]. (4) 4. ADALINE-IDENTIFIER

The ADALINE model shown in Fig. 2 is used as the identifier. It consists of a single layer and has one output linear neuron. ~Pe(t) is the active power deviation and u(t) is the APSS control signal, both at time step t. The linear transformation (lin) calculates the neuron's output by simply returning the value passed to it as given in eqn. (5) below.

Due to the different sizes of the generators and system configuration, multi-mode oscillations occur when the system experiences a disturbance (Shamsollahi and Malik, 1999). The local mode occur at about 1.3 H z and the inter-area mode at about 0.65 H z. These two frequencies differ significantly due to the large difference in the inertia of the generators. The speed difference between C 2 and Ca exhibits mainly the local mode oscillations, while the speed difference between Cl and C 2 shows the inter-area mode oscillations. Both local and inter-area modes of oscillations appear in the speed difference between Cl and Ca. The effectiveness of the proposed APSS in damping local and inter-area mode oscillations is given in Section 6.

~Pe(t)

= lin(WV) = WV.

(5)

The output of the identifier is the predicted active power deviation at time step t, ~Pe(t). W is the weight vector. The input vector of the ADALINEIdentifier is V(t)

3. APSS STRUCTURE The APSS consists of a ADALINE-identifier and a PS controller. For the multi-machine model, the generating unit is identified by a discrete ARMA model of the form (Chen et al., 1995):

[~Pe(t - 1), ~Pe(t - 2), ~Pe(t - 3), u(t - 1), u(t - 2). u(t - 3), d(t - 1), d(t - 2), d(t - 3)]. (6)

The cost function for the identifier is:

160

o.2r---.,..----.----,--.,.--,--,--,---,----,---,

AP.('·.) AP.(1.2)

o

AP.(,-3) 0(.-1) 0(,-2)

..c..

0(1-3)

.- \ f--

I

"

------------------------------

d (I-I)

-O..0~~:--~-~--7--:':'0--:':'2--'-'.----.L.,,--'.L..--!20

d (.-2)

T.m•.•

d (,-3)

0.' ""'-""".,.",r:7.."" .. ", .. .,., ...".., ..".., ...'."'--,.........-..-...-..........-...-..-.....,.---..,.---... .."..'1>,..-.. --,----,

O.os ._ ..... _._._._._.- _.- ._._. _. - ._. - _. _._.- ._._._"- _. _. - ._.- ._._.

Fig. 2. ADALlNE network model

i where

~Pe(k)

is the desired active power deviation.

0

Los

III

..c.l

The Widrow-Hoff learning rule (Etxebarria, 1994) is used to train the ADALINE. The weights are updated by the following equation:

f-----------------------------------bl

-O.'50L----J'----'-----J.------'------'-'0-----'-'2--'-'.--'-'.--'-1I-.....J20

r.,.•.•

On-line variation of A and B parameters during a three-phase to ground fault

Fig. 3.

W(t+1)

=

T

W(t)+21'e(t)V (t).

(8)

where I' is the learning rate and e(t) is the identification error. e(t) is given by

erence changes and three-phase to ground fault. In eqn. (10), 21' was chosen equal to 0.999vrV.

(9)

After the off-line training, the network is further trained on-line every sampling period making it an adaptive approach. The on-line updating of weights allows the APSS to track the operating conditions of the power system and any changes in the ARMA parameters. Figure 3 shows typical curves for the online variation of the A and B parameters for generator G 1 during a three-phase to ground fault applied at 1.Os at the middle of one transmission line between buses 3 and 6. The fault is cleared 50ms later by opening the breakers at both ends, and the circuit breakers are reclosed after 11 s. The initial operating conditions are 5.11 pu active power delivered to the bus at 0.60 pI lag.

The learning algorithm adjusts the weights of the ADALINE so as to minimize this mean square error. I' decides the speed of convergence of the iterative procedure. If I' is large, learning occurs quickly, but if it is too large it leads to instability and errors even increase. In this paper, to ensure stable learning and to ensure that the weights converge to optimum W· , I' is chosen as (Hagan, et. al., 1996):

o

<

I'

<

l/Amo %,

(10)

where Amo:J: is the largest eigen value of the correlation matrix VTV of the input vectors.

5.

The proposed ADALINE-Identifier has a linear structure. From eqns. (3-6) it can be seen that the weight vector W has a one-to-one relationship with the system parameters 0 which can be used by the adaptive controller in computing the control signal. This provides insight into the operation of the ADALINE model and the Widrow-Hoff learning procedure can be used to find the optimum set of regression coefficients. The process of obtaining ARMA parameters from ADALINE is an important point, as it is possible to apply linear analysis control methods such as the PS control technique to obtain the control signal. The PS control technique is described in Section 5.

ADAPTIVE POLE-SHIFT

(PS)

CONTROL

Once the system model is identified as in eqn. (I), the control signal can be calculated based on this model. In this paper, the self-optimizing PS control algorithm described (Chen et al., 1993) is employed to generate the control signal. The PS process is briefly described next. For the system modeled by eqn. (1). assume that the feedback loop has the form (c.L Fig. 4) u(t) y( t)

G(z-l) F (z -1 ) .

(11)

From eqns. (1) and (11) the closed-loop characteristic polynomial T( z-l) can be derived as

For the studies described in this paper, active power deviation of the generating unit is sampled at the rate ~f 20Hz and is used as input to the stabilizer (y = ~Pe). For off-line training of the ADALINEIdentifier, data was collected at selected operating conditions for the generators under lagging and leading power factor conditions. The disturbances used were input torque reference disturbances, voltage ref-

A(Z-I)F(z-l)

+ B(z-l)G(z-l) = T(z-l).

(12)

Unlike the pole-assignment (PA) algorithm (Mills, et al., 1996), in which T(z-l) is prescribed, the PS control algorithm makes T(z-l) take the form of A(z-l) but the pole locations are shifted by a pole-shift factor at i.e.

161

Identified Model B(z 1)

and Malik, 1999). An IEEE Standard 421.5, Type STIA AVR and Exciter Model and an IEEE Standard 421.5, PSSIA Type Conventional PSS (CPSS) (IEEE Standard, 1992) are used in simulation studies. The active power deviation, ~p., is sampled with 50 ms sampling interval (20Hz) for parameter identification and control computation.

,(/+1)

0

-I)

A(z u(/)

COlltroUer G(z -I)

F(z -I)

6.1 PSS on One Unit A 0.10pu step increase in input torque reference of G 3 is applied at 1 s at the operating point #1 as given in Appendix A.1. At lOs, the system returns to its initial condition. The proposed APSS was first installed on G3 only. None of the other generators had any PSS. The power deviation of G 3 was sampled at the rate of 20 Hz. As shown in Fig.5, the APSS damps out the local mode oscillations effectively. However, as expected, it has little influence on the inter-area mode oscillations. This is because the rated capacity of G3 is much less than G l and G 2 , and the interarea mode oscillations are introduced mainly by these large generators. G 3 does not have enough power to control the inter-area mode oscillations. For comparison, a conventional PSS with the following transfer function (IEEE Standard, 1992) was installed on G 3 :

Closed-Loop Syskm

Fig. 4. Closed-loop system block diagram

(13) In the PS algorithm, Q, a scalar, is the only parameter to be determined and its value reflects the stability of the closed-loop system. Supposing>. is the absolute value of the largest characteristic root of A( Z-l), then Q.>. is the absolute largest characteristic root of T(z-l). To assure the stability of the closed-loop system, Q ought to satisfy the following inequality (stability constraint): 1

->: <

Q

<

1

>:.

(14)

This implies that the magnitudes of all the poles of the closed-loop system are adjusted by a shifting factor Q and hence all map inside the unit circle. It can be seen that once T(z-l) is specified, F(z-l) and G(z-l) can be determined by eqn. (12), and thus the control signal u(t) can be calculated from eqn. (11).

s :- 1 + sTl 1 + sT3 ~P(s) U(s) = K. _s_71_ 1 + sTs 1 + sT2 1 + sT4

:'" '.. ~ . . ~ i~~ '.. :

i

0

.

-4.05

= E[Y(t + 1) -

'"

""

"

',' .

,,".

o

.

2

..~'

.....

"

,.

16

,-'

~.

,,::

To consider the time domain performance of the controlled system, a performance index J is formed to measure the difference between the predicted system output, y(t + 1) and its reference, Yr(t + 1): J

(17)

..

•

•

10

12

11

20

r."..•

llE: ' : ~-: :,

+ I)F. (15) y(t + 1) is determined

Yr(t

o

2

..

S

•

10

12

14

'I

la

1 20

~~E, ,-~=~~

E is the expectation operator. by system parameter polynomials A (z -1), B (z -1 ) and past y(t) and u(t) signal sequence. Considering that u(t) is a function of the pole-shift factor Q, the performance index J becomes

o

2

..

6

•

'0

r.,., •

12

1.

16

18

20

Fig. 5. System response with APSS installed on generator G 3

J

= W[A(z-l), B(z-l), C(z-l), u(t), y(t),

Q,

Yr(Hl)].

Following parameters were chosen for CPSS on G 3 :

(16) The pole-shift factor Q is the only unknown variable in eqn. (16) and thus can be determined by minimizing J.

6.

K. = 1.0, Tl

=T

3

= 0.3, T 2 = T 4

= 0.10,

Ts

= 0.4 (18)

6.2 PSS on Three Units

SIMULATION RESULTS

To damp both the local and the inter-area modes of oscillation, two additional APSS are additionally installed on G l and G 2 . Responses given in Fig.6 show that both modes of oscillations are damped out effectively. If CPSSs are to be installed additionally on

A non-linear fifth-order model is used to simulate the dynamic behavior of the multi-machine power system (Yu, 1983). The differential equations used to simulate the synchronous generators and the parameters used in simulation studies are given in (Shamsollahi

162

. ,. , , !JE< :'_. ~~:•-,:

r:tl?~,\\~~'~.:~,I

G l and G 2 to damp the inter-area modes of oscillation, their parameters have to be re-tuned. Following parameters are obtained for the CPSS on G l and G 2 :

K.

= 0.3,

= T3 = 0.07,

Tl

T2

= T 4 = 0.03,

Ts

o

= 0.3 (19)

2

•

6

10

12

I.

16

,.

20

1

fli' :: 50 ,:: 1!~[f:\':'": '1':- ,~~ o

2

..

6

8

10

12

'4

16

18

o

2

..

6

•

o

2

•

6

•

10 Tftw••

12

'4

16

"

20

10

12

,.

16

,.

20

20

Tme,'

~~t!: o

Fig. 6.

2

..

Fig. 8. System response to a three phase to ground fault with PSSs installed on G 1 • G2 and G3

:: ~:' ,I: ~~ •

•

10

rme.•

12

14

1.

System response with PSSs installed on G l

1.

,


-0.'0 2 . 6 8

•

10

12

1.

1&

"

20

2 - . . . .----'-.-~.--"::10---"-:-'2--'...,..--''":-.----:,:--. --:'20 r."..•

!j~~........ '~:, 1-: ~'1 Fig. 7.

2

I

10 Tftw.•

12

'4

16

"

o

2

o

2

Fig. 9.

1

~·"o

:

:

j

'0

12

16

18

20

•

8

8

'0

12

"

IS

18

1 20

!~l&:": ~', :I~ ~~

Tme.•

o

•

lJi' ,:,:' ::

ftf2i: : .~ ,::] !Ji' ,: y: : : ' e

•

TfTW, •

G 2 and

One of the important features of the APSS is its selfcoordinating property. The proposed APSS can c~ ordinate itself with existing PSSs in the system automatically due to its on-line learning property. To demonstrate this fact, the APSS is installed on Gl and G 3 and CPSS on G 2 , G 4 and G s . Figure 7 shows the response for 0.10 pu step increase in torque reference of G 3 at 1 s and return to initial condition at 10 s. It can be seen that all PSSs work cooperatively to achieve a good performance.

..

:.. . -

........"

6.3 Self-Coordination Ability of APSS

. 2

/".:-:~;

}~-~2:::6=:',;,\ .....•

~

G3

-O.OS0

nn..•

•

6

8

10

nn..•

12

I.

16

18

20

System response to a three phase to ground fault with PSSs installed on GI. G2 and G3 for the new operating point

6·4 Three-Phase to Ground Fault Test

With the power system operating at the same operating conditions (operating condition #1, Appendix A.l), a three phase to ground fault was applied at the middle of one transmission line between buses 3 and 6 at 1 s and cleared 50 ms later by removing the faulted line. At 11 s, the faulted transmission line was restored successfully. Response of the system under this disturbance with no PSS, with the proposed APSSs only and with CPSSs only installed on G l , G 2 and G 3 is shown in Fig. 8.

6.5 New Operating Conditions Test To test the behavior of the proposed APSS under other operating conditions, the operating point of the system is set to operating point #2 as given in Appendix A.2. Figure 9 shows the response under the three-phase to ground fault described in Section 6.4 for no PSS (OPEN), CPSS and APSS on G l , G 2 and

20

System response with APSS installed on generators

G 1 and G3 and CPSS on G2, G. and G s

163

Apparatus and Systems, PAS-103, pp. 19831986. Hagan , M.T., H.B. Demuth and M.H. Beale (1996) Neural Network Design, PWS Publishing, Boston. He, J. and O.P. Malik (1997). An adaptive power system stabilizer based on recurrent neural networks. IEEE Trans. on Energy Conversion, 12(4), pp. 413-418. Larsen, E.V. and D.A. Swann (1981). Applying Power System Stabilizers, Part I-Ill. IEEE Trans. on Power Apparatus and Systems, PAS100, pp. 3017-3046. Mills, P.M., A.Y. Zomaya and M.O. Tade (1996). Neuro-Adaptive Process Control - A Practical Approach, John Wiley & Sons, NY, U.S.A. Pierre, D.A. (1987). A Prespective on Adaptive Control of Power Systems. IEEE Trans. on Power Systems, PWRS-2(5), pp. 387-396. Shamsollahi, P. and O.P. Malik (1997). An AdaJr tive Power System Stabilizer Using On-Line Trained Neural Networks. IEEE Trans. on Energy Conversion, 12(4), pp. 382-387. Shamsollahi, P. and O.P. Malik (1999). Application of Neural Adaptive Power System Stabilizer in a Multi-Machine Power System. IEEE Trans. on Energy Conversion, 14(3), pp. 731-736. IEEE Standard 421.5 (1992). IEEE Recommended Practice for Excitation Systems for Power System Stability Studies. Yu, Y.N. (1983). Electric Power System Dynamics, Academic Press. Zhang, Y., G.P. Chen, O.P. Malik and G.S. :Uope (1993). An artificial neural network based adaJr tive power system stabilizer. IEEE Trans. on Energy Conversion, 8(1), pp. 71-77.

G3 . It is shown that APSSs can damp out the oscillations very effectively even when the operating conditions change. 7.

CONCLUSION

An APSS using an ADALINE-Identifier and PS control technique is described in this paper. The proposed APSS has the following advantages because of ADALINE modeling and PS feedback: • The basic model used in this paper to represent the power system is a third-order ARMA model. The ADALINE is modeled so that its weights have a one-to-one relationship with the ARMA parameters. Thus the weights have physical interpretations unlike the "black-box" characteristic of the neural network. • The ADALINE weights store 'a priori' knowledge because of off-line training. Further, the weights are updated in an on-line mode by the same learning algorithm (Widrow-Hoff procedure) used for off-line training. Thus the network updates adaptively to follow the different operating conditions and disturbances. The learning algorithm is much simpler and computationally efficient than learning by back-propagation algorithm in feed-forward neural networks. • The PS control uses the on-line updated ARMA parameters to calculate the closed-loop poles of the system. The unstable poles are moved inside the unit circle in the z-plane and the control is calculated so as so optimize the output performance. Simulation results show that the proposed APSS is effective for both the local mode and the inter-area mode of oscillations under different operating conditions. Also, the self-coordination ability of the proposed APSS with other CPSS is demonstrated.

ApPENDIX A.l Operating Point #1

REFERENCES

P (pu)

Chen, G.P., O.P. Malik, G.S. Hope, Y.H. Qin and G.Y. Xu (1993). An adaptive power system stabilizer based on the self-optimizing pole shifting control strategy. IEEE Trans. on Energy Conversion, EC-8, pp. 639-646. Chen, G.P. and O.P. Malik (1995). Tracking Constrained Adaptive Power System Stabilizer. lEE Proc. - Generation, Transmission and Distribution, 142(2), pp. 132-138. Etxebarria, E. (1994). Adaptive Control of Discrete Systems using Neural Networks. lEE Proc. Control Theory Appl., 141, pp. 209-215. Fi~ch, J.W., K.J. Zachariah and M. Farsi (1996). Self-Tuning Control Applied to Turbogenerator AVRs. lEE Proc. - Generation, Transmission and Distribution, 2(3), pp. 492-498. Ghosh, A., G. Ledwich, O.P. Malik, and G.S. Hope (1984). Power system stabilizer based on adaJr tive control techniques. IEEE Trans. on Power

Q (pu) V (pu)

5 (rad.)

G1

G2

5.1076 6.8019 1.0750 0.0000

8.5835 4.3836 1.0500 0.3167

G3

G4

Gs

0.8055 0.4353 1.0250 0.2975

8.5670 4.6686 1.0750 0.1174

0.8501 0.2264 1.0250 0.3051

Loads in admittances in pu

LI = 7.5 - j5.0

L 2 = 8.5 - j5.0

L 3 = 7.0 - j4.5

A.2 Operating Point #2

P (pu) Q (pu) V (pu)

5 (rad.)

G1

G2

G3

G4

3.1558 2.9260 1.0500 0.0000

3.8835 1.4638 1.0300 0.1051

Gs

0.4055 0.4331 1.0250 0.0943

4.0670 2.1905 1.0500 0.0361

0.4501 0.2574 1.0250 0.0907

Loads in admittances in pu

L 1 = 3.755 - j2.5

164

~

= 4.25 - j2.5

L3 = 3.5 - j2.25