Adaptive Nonlinear Control with Disturbance Attenuation for Multimachine Power Systems

Adaptive Nonlinear Control with Disturbance Attenuation for Multimachine Power Systems

IFAC Copyright © IFAC System Structure and Control, Prague, Czech Republic, 200 I C:Oc> Publications www.elsevier.comllocatelifac Adaptive Nonlinea...

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IFAC

Copyright © IFAC System Structure and Control, Prague, Czech Republic, 200 I

C:Oc> Publications www.elsevier.comllocatelifac

Adaptive Nonlinear Control with Disturbance Attenuation for M ultimachine Power Systems·

Tielong Shen t and Katsutoshi Tamura Department of Mechanical Engineering, Sophia University, Kioicho 7-1, Chiyoda-ku, Tokyo, 102-8554, Japan Shengwei Mei, Qiang Lu and Wei Hu Department of Electrical Engineering, Tsinghua University, Beijing, 10084, China

Abstract. This paper presents a design approach to non linear feedback excitation control for multimachine power systems with disturbance. It is shown that a decentralized nonlinear feedback control law, which stabilizes the power system with desired L2 gain from the disturbance to a penalty signal, can be designed by a recursive constructing approach. First, a state feedback law is presented for the case of the system with !mown parameters, and then the control law is extended to adaptive version for the case when the parameters of the electrical dynamics in the power system are unknown. Simulation result demonstrate that the proposed controllers guarantee transient stability of the system regardless of the system parameters and a large sudden fault . Copyright @200J IFAC Keywords. Power system control, nonlinear control, decentralized control, disturbance attenuation, adaptive control.

and parameters caused by faults (Hill et al., 1993). In this case, it is difficult to exactly linearize the system with nominal parameters. Several robust excitation control schemes have been proposed very recently by using linear robust control theory such as Hoo design (Ahmed et al., 1996; Taranto et al., 1993) and Loo-stability theory (Vittal et al., 1993). However, these control schemes are essentially based on the linearized model so that it is only deal with small disturbance and modeling uncertainty about an operating point. When a large fault or a large load change occur, the behavior of power system changes in a large extent and a linear controller can not stabilize the system adequately (Wang et al., 1993).

1 INTRODUCTION Recently, advanced nonlinear control techniques have been used to excitation control of power systems (Lu and Sun, 1989; Wang et al., 1997; Chapman et al., 1993; Ortega et al. , 1998). The most successful non linear excitation scheme is based on the exact feedback linearization (Lu and Sun, 1989; Gao et al., 1992). It was shown in the literatures that the dynamics of the power system can be exactly linearized by employing nonlinear precompensation. Then one can use the conventional linear control theory to provide good performance (see Gao et al., 1992; Hill et al., 1993 and the references quoted therein).

This paper presents a decentralized control scheme for multimachine power systems without linearization. Our aim is to design a decentralized nonlinear excitation controller ensuring asymptotical stability and the L2 disturbance attenuation

However, we are often faced with uncertainties in practical power systems. A major source of uncertainty is unmodelled dynamics and parameter uncertainty, which are due to loads level variation, modeling error or variations of network structure

705

performance. The case with known parameters is addressed first, and then the design approach will be extended to adaptive non linear controller for the case where the system parameters are unknown . Simulation results will be given to support the theoretical claims.

We suppose that the physical parameters Pil and Pi2 are not exactly known, and we consider the decentralized adaptive controller of the form

where Bi is the estimate of unknown parameter vector Oi which will be defined later. 2

CONTROL PROBLEM We define the penalty signal

Consider a large-scale power system consisting of n generators interconnected through a transmission network. The dynamical model of the i-th machine can be presented as follows.

Zt. -_

[qil Xil] , Qi2 Xj2

Zi

(i

= 1,2, . . . ,n) by

Qil > 0, Qi2 >

°

(8)

and consider the performance index as follows :

Ji

= Wi(t) - WiO

(1) n

+

L Si(XiO, Bo),

Vdi E L 2[O, T]

(9)

i=1

°

where T > is any given constant, xf{t) = [ XiI Xi2 Xi3 ], dT = [d il di2 ] and Sj(') are a positive definite functions . Our goal is as follows : Given 'Yi > 0, Qil and Qi2 (i = 1,"" n), find a decentralized adaptive controller (7) such that the closed loop system is asymptotically stable at Xi = and (9) holds.

where n

E~i

Pei(t)

L E~jBjj sin(di -

dj),

°

(4)

j=1 n

Idi(t)

=

LE~jBijCos(6i -dj) .

(5)

j=1

3

and the notation for the model is given in Appendix A. d l (t) and d 2 (t) denote the disturbances. SI and S2 are scaling parameters.

Suppose the parameters of the system are exactly known, and consider the decentralized law

Let the desired operating points for each generators be given by (diO,WiO,E: o) (i = 1,2,'·',n) . Define the state variable by XiI == 6i - djO, Xi2 == Wj - WiO and Xj3 = E~i - E:o' Then, the system can be represented by XiI Xi2 Xi3 where Vio

CONTROLLER DESIGN

VIi

= (}:i(Xil, Xj2, Xi3),

(i

= 1, 2"", n)

(10)

and the following coordinate transformation

= Xi2

= -ail Xj2 - ai2[Pei - Pmi] + sild;1 =~ilXi3-Pi2Idi+bi(Vli- Vio)+si2di2

(6)

°

are any given numbers and smooth functions to be designed.

= ~E:o(i = 1, 2"", n), a;2

WiO

Our goal is to find a feedback law (}:i and
1

= Mt', Pil -- -T' , di

706

Then, along any trajecto ries of the system we have

holds, which is the nonada ptive version of (9). From the dissipa tive systems theory, (12) holds if there excists a storage function U(XI' X2, ... ,xn ) such that the differential dissipation inequal ity 1

(; < 2

n

bllldil1 L i=1

2

.

= - ~fkek -1I~di -

(13)

IIZiW}

-

2 2 2I{ IIZill 2 - , IIdill } + Ui3

=

H i3 :

~i Fr~i3112

~i3 { ~i2 + fi3~i3 + ~i F;Fr ~i3 + Ui}

+

where holds along any trajecto ries of the closed loop system (van der Schaft, 1996) . In the following, we show that a desired control law can be obtaine d by constru cting the storage function in a recursive way. Step 1. Define Ui1 (~id = T~;I' where the number ai > 0 will be chosen later. It is easy to show that along any trajecto ry of the system

H il :

1{

=

2 Ilzill

=

-17i k i X

2

-'i IIdill 2

2}

Ui

-/-Li2 a i2(Pei - Pmi)

(18)

yields 3

+

Hi3

!~;2·

Step 2. Then, noting that IIz;l12 = q;I~;1 +q;2(~i2-ki~id2, we get from the step 1 that

-'i

<

-fil~;1 - f;2~~ - ~'illdill2 + ~i2

IIdill }

+ /-Li2 X i2 1

2

/-Ll

=

ai+(fi 2+2--2 qi2 k i •

/-L2

=



(20)

qil -2qi2k i),

,;

1 2)

2

2.2

+ k ,· -

al ,

and fi2 > 0 is any given number , and fil we choose ai to be sufficient large. Choose the functio ns

i.e. the differential dissipayion inequal ity

Define the storage function for the overall system as U(Xl,X 2. ·,xn ) = L~=I U i3 · Then we get

(ai ki -

'i1 sri f·2+- +-q ~i3

(19)

k=1

2

=

S;I

L fk~lk·

- ai2(Pei - Pmi)}

2

fil

2

-

+ U i2

2{lIzill

1

~

.

2

=

{J.LilXil

(17)

Hence, setting

'I

2

+ ai2Pi2 I qi 1di

-ai2E~Jqi - bi a i2 I qi(Vji - \1;0) ·

;I- IIdiW+~lIziIl2+a~il~i2

2

/-Li2 a il )Xi2 + ai2Pil I qi x i3

+ U. 1

Define Ud~iJ, ~d = U i1 (~id

1

= (/-Lil -

>

0 if

The disturb ance attenua tion perform ance by integratin g both sides of the inequality. When d i = 0, it follows by the inequal ity that ~ik --t 0 (k = 1,2,3) at t --t 00, and so Xii --t 0, Xi2 --t 0 and Pei - Pmi --t 0 (i = 1,2,·" ,n).

= i as

From (17) and (18), we get the feedback law Then, we obtain that 2

Hi2

~ -filel-fi2~;2- ~ IIdiIl2+~i2~i3 . (I5)

Step 3. Constru cting the storage function as (22)

707

Substituting (Ji in the control law (24) by Oi, it can be easily seen that

where

Hi

1

-'i IIdill

.-

. Vi

+ 2{lI zill

<

-

2: iik{fk +

2

2

2}

3

.

/J + Birj 101..26)

{i3 W i

k=1

We choose the parameter update law as (27)

which yields the dissipation inequality

V; + ~{llziI12 -,71IdiIl 2 } S -

,i

Proposition 1. For any given > 0 (i = 1,2,···,n), a decentralized control law such that the overall system is stable at (<5iO , WiO, Eio) and the disturbance attenuation performance (9) holds are given by (22) Furthemore, under the controller, XiI ~ 0, Xi2 ~ 0 and Pei - Pmi ~ 0 (i = 1,2,··· ,n).

3

2: iik{lk·

(28)

k=1

For the whole system, we construct the storage function as follows: n

V(Xl>X2,···,X n )

= 2:Ui(X;).

(29)

i=1

We now suppose that Pil and Pi2 (i = 1,2,···, n) are unknown and separate VIi as follows:

Then, from (28), we get

(23)

where V;o is calculated by the nominal values of the related parameters, and unknown constant Piv denotes the uncertain variation.

When d i = 0 (i inequality yields

Fortunately, the feedback control law (22) is linear on the unknown parameter vector Bi .

n

VS -

= 1,2,···, n),

this dissipation

3

2:2: i ik{fk SO

(31)

i=1 k=1

VIi

= VOi+Si(Xil, Xi2, Xi3)+ Wi(Xil, Xi2, Xi3)Bi (24)

Si Ci3] a i2 iqi I ) [ Ti +9i3 Ti (Pei-Pmi)-y(Xi3+ E iO ,

Wi = [~Xi3 ~h 1].

which guarantees the stability of the equilibrium XiI = Xi2 = Xi3 = 0, Oi = Bi (i = 1,2,···,n), and hence, the boundedness of XiI (t), xdt), Xi3(t) and Bi (t). Furthermore, by LaSalle's invariance theorem, all the trajectories converge to the set where (; = 0, i.e.

ttiik{fk =0. i=1 k=l

Consider the positive definite storage function

This means Xil(t) ~ 0, Xi2(t) ~ 0, {i3(t) ~ 0, and hence, Pd(t) - Pmi ~ 0 (i = 1,2,···, n).

(25)

On the other hand, for nay given T ing (30) we get

where r i = diag{Pil,Pi2,Pi3}, Pik > 0 (k = 1,2,3), Bi = Oi - Bi , Oi is estimate of the unknown parameter Bi (i = 1,2,···, n).

708

> 0, integrat-

where Vo = V(Xl(0),X2(0),X3(0) ,8(0» . We have the disturbance attenuation performance (9). Proposition 2. For any given 'Yi > 0 (i 1, 2, · ·· , n) , a decentralized adaptive controllers such that the whole system is asymptotically stable at (tliO,wio,E;o) and the disturbance attenuation performance (9) holds are given by

be pointed out that the power system is not of the strict-feedback form, and the proposed design method dos not requires to solve any HJI inequality. Simulation result shows that the proposed controllers can guarantee the transient stability of system regardless of the system parameters and a large sudden fault .

Reference

4

Ahmed,S. S. , L. Chen and A. Petroianu (1996). Design of suboptimal Hoo excitation controllers, IEEE Transaction on Power Systems, 11, 312-318.

SIMULATION RESULTS

We will illustrate, through simulations results, postfault performance of the systems with the excitation control proposed in Section 3. A 6machine power system shown in Fig.1 was employed as example system. The No.6 machine is a synchronous condenser and No.1 generator itself actually represents an equivalent of a large power system. The fault considered in this simulation is a threephase temporary short-circuit fault occurs at No.l1 bus during the time period 0-0.15(sec.) . Also, the control input limitations are

O.Op:u.

:s Vfi(t) :s 4.0p.u.

(i

= 2,3, · . . , 5).

Gao,L., L. Chen, Y. Fan and H. Ma (1992) . A nonlinear control design for power systems Automatica 28, 975-979. Hill,D. J., I. A. Hiskens and Y. Wang (1993). Robust, adaptive or non linear control for modern power systems. In: Proceedings of the 32th IEEE CDC, San Antonio, pp.2335-2340. Lu, Q. and Y. Z. Sun(1989). Nonlinear stabilization control of multi machine systems. IEEE Trans., on Power Systems 4,236-241.

Case 1. The generators from No.2 to No.5 are equipped with a conventional controllers. The parameters of all controllers are chosen as same as the single machine case. The responses are shown in the left column of Figure 2.

Lu, Q ., Y. Z. Sun, Z. Xu and T. Mochizuki (1996). Decentralized nonlinear optimal excitation control. IEEE Transaction on Power Systems. 11, 1957-1962.

Case 2. The generators from No.2 to No.5 are equipped with the decentralized adaptive nonlinear excitation controller. The disturbance attenuation levels are chosen as 'Yi = 3.0 (i = 2, 3,4,5), and the design parameters are selected as same as the single machine case. The simulation result is shown in the right column of Fig.2.

Lu, Q., Y. Z. Sun and S. Mei (2000). Nonlinear control systems and power system dynamics. Kluwer Academic Publishers, Boston(to be published) . van der Schaft,A. J. (1996). Loo-gain and passivity techniques n nonlinear control, Lecture notes in control and information sciences, Springer, New York.

Comparing the results obtained in the two cases, we can see that the postfault performance of the system can be improved remarkably by employing the proposed nonlinear excitation controller.

5

Chapman, J. W ., M. D. liic , A. C. King, L. Eng and H. Kaufman (1993). Stabilizing a multimachine power system via decentralized feedback linear excitation control. IEEE trans. on Power Systems, 8, 830-838.

Vittal,V., M. H. Khammash and C. D. Pawloski (1993) . Analysis of control for stability robustness of power systems. In: Proceedings of the 32th IEEE CDC, San Antonio, pp.23412348.

CONCLUSIONS

Generally, solution of HJI inequality is required for L 2 -gain synthesis, or the structure of system under consideration should be of the strictfeedback form for the recursive design. It should

Wang,Y., G. Guo and D. J. Hill (1997) . R0bust decentralized nonlinear controller design for multimachine power systems. A utomatica 33, 1725-1733.

709

Wang,Y. , D. J. Hill , H. Middleton and 1. Gao (1993). Transient stability enhancement and voltage regulation of power systems. IEEE trans. on Power Systems 8 , 620-627.

Appendix A. Notations

6(t) wet)

the rotor angle of the generator the relative speed of the generator the transient EMF in the quadrature axis the synchronous machine speed the per unit damping constant the per unit inertia constant the infinite bus voltage the time constant of the excitation circuit the direct axis transient short circuit time constant the input power of generator (constant) the direct axis reactance of the generator the direct axis transient reactance

E~(t) Wo

D M

v. TdO

T~

Pm Xd

x~

110. 3

110. 5

110. 4

Fig. 1. Six-machine example system.

-

,'\ It. "

... :ir

,/" ~t1

~ 50 ' \ 1\:

0 21 \1",..,.---------------____ _ " <>.,

~-5: 7.\f·-::;........·------=\:;S;;-'i;:o

5

10

0

5

time(sec)

10

time(sec)

1.02 r-----~------,

s-

E:

a

ro.

0.98 L-'-......::.'--_ _ _ _ _ _ _.....J 0.98 o 10 5 0

5

time(sec)

10

time(sec)

10

r.

/ p"

,I If.

P.t'f,

Si; : \:-IL-.---------------------\I \~"w-.,------~---5 ____ _ -

~

5

10

,1'11\,

\

.,.~,+---

o

time(sec) The Conventional Controller

5

10

time(sec) The Adaptive Nonlinear Controller

Fig. 2. Responses of the nonlinear control

710