AVERAGED MODELING AND NONLINEAR CONTROL ... - IEEE Xplore

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Flexible AC Transmission System (FACTS) devices are a very innovative and interesting application of power elec- tronics to the control of power networks.
AVERAGED MODELILING AND NONLINEAR CONTROL OF AN ASVC (Advanced Static VAR Compensator) P.PETITCLAIR, S.BACHA, J.P.ROGNON Laboratoire d'Electrotechnique de Grenoble (INPG/UJF - CNRS.UMR 5529)

L.E.G ENSIEG, BP46, 38402 Saint Martin d'Hkres, cedex, France tel : (;33) 76 82 62 99 Fax : (33) 76 82 63 00 email : Pat rice. Petit [email protected] Abstract - This paper deals with the modelling and control of an Advanced Static Var Compensator (ASVC) or STATCON. A time invariant model is obtained. This model is used to design an efficient nonlinear control law. Simulations are carried out to validate both the time invariant model and the nonlinear control law.

L, and a resistor R, on the AC side ; L,$ represents the leakage inductance of the transformer, R, represents the inverter and transformer conduction losses.

- An inductance

E1 E2 E3

I.

INTRODUCTION

I

C

Flexible AC Transmission System (FACTS) devices are a very innovative and interesting application of power electronics t o the control of power networks. This paper considers the modelling and control of one ofthem : a STATic CONdenser (STATCON) also called Advanced Static VAR Compensator (ASVC). This converter is a current or voltage inverter which can provide or absorb reactive power depending on the network state. The Generalized Averaged Method [3], [5] has been used to get a time invariant (TI) continuous, but non linear model of the system. This model has been validated by comparison with the topological model which accurately describes the successive configurations of the STATCON. Because oil the variations of t h e zeros frequency with the operating point, a PI (Proportional Integrator) controller with constant parameters is not, robust enough. This is verified in iohepaper. Therefore, a nonlinear controller based on a linearization via feedback associated to a proportional controller has been chosen.

Figure 1: STATCON and its environment The line voltages are called E l , Ez, E3. VI,Vz, V3 are the are the DC voltage inverter output voltages, Vdc and and current. The AC system is described by equations (1) and (2), and the DC part by the equations (3) and (4)

VI23

[ -"1

=

-1

11.

DESCRIPTION A N D BASIC E Q U ATIO N 5;

d v d c - 1dc -

dt

A study of various STATCON structures has been done in [l], [2]. In this paper a simplified STATCON model is considered and is described in Figure (1) ; it is composed of :

A voltage inverter A capacitor C and a resistor R on the DC side ; the resistor R, represents the inverter switching losses

-

-

0-7803-3500-7/96/$5.00

01996 IEEE

C

-1 2

-1

-1 -1 2 Vdc

RC

]

U123

(2)

(3)

= ug31123 (4) Where U123 is the switching vector, U123 = [UI uz US]^ with, for j= 1,2,3 u j = 1 for switch j, closed and switch j, open u j = -1 for switch j, open and switch j, closed The angular velocity of the AC voltage and current vectors is equal to w . Let us consider a system of reference (d,q)

753

1dc

111. AVERAGED MODELUSEDFOR CONTROL

rotating ar the same speed, and let us note p the angle -+

between d axis and the line voltage vector E (Figure 2). The model of the AC side in this system of reference is

Ed

d

Figure 2: dq Rotating frame description given by equation ( 5 ) to (7), C32 is the transform matrix such that :

with < u1 > I = This model is nonlinear. A linear small signal model could be obtained by differentiation around a considered operating point. Figure (3) shows the transfer function for three the operating points very close to each other. It can be verified that the resonant pole frequency W O is constant. However, the complex zero significantly moves around W O . Therefore, a linear controller cannot be both robust and performant on the whole operating range.

Thus :

C32 =

[

+ p) sin(wt + p) + p - 9 ) sin(wt + p - %) + p - F) &(ut + p - 9)

cos(wt

cos(wt cos(wt

The DC equation (3) is unchanged. dvde -

dt

with

Id, C

There are two possible kinds of solutions for contolling this device - Variable structure control such as sliding mode control. In this case the previous model can be used t o verify the sliding conditions. The switching frequency is high : this is why we have not adopted this solution. Its avoid important commutation losses and emc problems. - Continuous control associated to full wave commutation of the inverter. The control variable Q is the firing angle with reference to the network voltage Ej zero crossing. In this case, the previous model may be used for simulation purpose, but not to choose and tune the controller. A Generalized Averaging method (see [3, 4, 51) is used t o get a continuous time invariant model of the converter. The averaged equations are :

1

Vde RC

IV.

: Idc

= uE3c32Idq

The nonlinear control law implemented is based on the theory of the exact linearization via feedback [6]. The nonlinear system is described by equations (17) and (18). It is of relative degree r if equations (20) and (21) are verified for all z and all k < r - 1. L j h ( z ) is called h ( z ) derivative along f ; it is defined by (19).

The powers are expressed by equation (12).

If p is chosen equal to zero, the E, voltage is equal to zero and the reactive power becomes proportionnal to EdIq. To control the reative power Q, it is sufficient to control I, (equation 13).

3 Ed = E , E , = 0, Q = -EdIq 2

NONLINEARCONTROLLER

(13)

754

ln

;i: = f(x)

+

gi(z)Ui i=l

(17)

Internal Dynamics Figure 4: Original system splitting up derivative expression of the output I , (see equation 14). Then the relative degree is one. The vector U is equal t o [u1 u1 = &(a) and y = I,, equation (22) leads to the nonlinear feedback equation ( 2 3 ) .

d-lT.

Frequency (Hz)

TLs U1

(2Iq

= sin(a) =

+ UId +

U)

(23)

2Vde

The new input U is the output of a controller acting on current IQ and its reference I q Y e f (equation 24). A proportionnel controller (gain A) is chosen. The model of the system with the nonlinear control law and the P controller is described on figure (5). 21

= A(I,,ef

- I,)

(24)

The stability of the internal unobservable state variables Frequency (Hz)

Figure 3:

AI

Iq ref

___9

for different operating points

I

[ L , I L j h i ( X ) L g 2 L j h i ( X )... LgmLjh&)] = 0

1,g3 . L r z - l hi(.)

# 0 for at

least one 1 15j 5 m

The control law is defined below : U =

1 (-L?h(.) L, L ; - l h ( z )

t- U )

Note than, in our case u2 = cos(o) = J1 - U : ; the system is really a Simple-Input Simple-Output one. By using this control law, the original nonlinear ndimension system is transformed into two :subsystems (Figure 4). The first one is an integrator of degree r, v is the new input and y is the output. The second one is of dimension n-r ( it may be nonlinear). It is unobservable ; therefore its dynamics does not affect the output y. In the STATCON case, the control a appeara in the first

Intemal dynamics ( IdandVdc)

a

I

( Id and Vdc )

Figure 5: Structure of the nonlinear controller

(Id,

Vde)

can be estimated. The expression of their varicorresponding t,o I,,,,

ation around the operating point

755

(and therefore ao) is given by equation ( 2 5 ) .

Iq =

lowing equations :

&&ref

(25)

Id and I 9 T d

transfer functions are easily obtained. 1974

600

400

200

0

-200

\

Responses t o positive and negative steps I p r e f oare studied. Two cases are considered : - Reference case : The STATCON is controlled by a PI controller tuned for a = 0 operating point (Figure 7). - The STATCON is controlled by the proposed nonlinear controller (Figure 8). The responses of the averaged model and of the topological one are also compared. In the reference case, it clearly appears that the dynamic

.............................................

............................................

......,...................

......................

.....................................................

-400 03

0 35

0 4

0 45

04

0 45

I

0 5

TW"

-600

cunrnt Iq

:

:

-1

Figure 6: Root locus of the closed loop system

-2

A , the complex conjugate poles do not depend on A , but they depend on the operating point. The root locus according to I r e f o is shown on figure (6). The system is internally unstable for the highest reachable values of the reference : this limits the operating domain. - All the zeros depend on X ; this effect will be analysed in the following.

- There are three poles. The real pole is equal t o

V.

SIMULATION RESULTS

For the simulations, a per-unit representation has been adopted, according t o equation (27). It leads to the fol-

15

1 0 5 0

-0 5

-1

'

-1 5

--z

0 25

0 3

0 35

I 05

Time

Figure 7: a: I , response with PI controller (averaged model) b: Iq response with PI controller (topological model) c,: First phase current response (topological model) behavior is quite different for positive and negative transitions. In particular, the system is poorly damped during

756

negative transitions (Figure 7-a). In the second case, the first order response desired is obtained for both positive and negative transitions (Figure 8a>. In the two clmes, we can observe that thie averaged model

0.2 0

-0.2

- O 16/ -0

Figure 9: Evolution of the internal dynamic I d

-1'5

-15-

k

Oi3

025

03

'

0.53

035

04 Time

0.45

0!5

0.55

04

045

05

055

Time

Figure 8: a : Iq response with nonlinear controller (averaged model) b : Iq response with nonlinear contr'oller (topological model) c : First phase current response (topological model) matches the real system (Figures 7-b anid 8-b). For the proposed non linear controller it is interesting to observe the evolution of the internal dynamics with A (Figures 9 and 10). The two components of the response clearly appear (first order evolution) linked to the real pole, and damped oscillations linked to the complex conjugate poles. As previously said, the damping and the frequency of the oscillations are function of the operating point, but not of A . The amplitude of the oscillations, linhed to the zeros of the transfer function, depend on A.

VI.

0.80k.'

'o$'

' '

6.$5 ' '04' '

'

ai5 Time '

' '

'0:s'' ' 6.55 ' ' ' O i

' '

O L

Figure 10: Evolution of the internal dynamic

vdc

on linearization via feedback is better suited t o this kind of system than a standard constant parameter PI controller. The internal stability has been investigated. It is guaranteed on the whole STATCON operating range. The whole process of modelling and controller design can be easily generalized to a large scale of power electronics converters.

REFERENCES Hochgraf C., R.Lassetier, D. Divan, T.A. Lip0 : "Comparaison Of Multilevel Inverters For Static Var Compensation" IEEE IAS October 1994 pp 921-928

CONCLUSIOI?TS

Topological and averaged models of a STATCON have been presented in this paper. The averaged model is used to define and tune this controller. It correctly represents the dynamic and static behaviour of the system. As it clearly appears in the result, a non linear controller based

757

Wuest D., HStemmler, G. Scheuer : "A Comparaison Of Different Circuit Configurations For An Advanced Static Var Compensator (ASVC)" IEEE PESC'92 pp.521-529

[3] S.R. Sanders, J.M. Noworolski, X.Z. Liu, G.C. Verghese : “Generalized Averaging Method for Power Conversion circuits” IEEE Transactions on Power Electronics, vol 6 , No 2, pp 251-258, 1991 [4] S. Bacha, A. Hassan, M . Brunello :“ General Nonlinear Control Law for DC-DC Symmetric Switching Converters” IEEE PESC’93 Records, pp 222-228. [5] J . Sun, H. Grotsollen :“Averaged modelling and analysis of resonant converters” IEEE PESC’93 Records, pp 707- 713. [6] A. Isidori : “Nonlinear control systems - an intraduction” Springer-Verlag ; second edition, 1989.

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