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RP.MnROH E L S E V I E R Electric Power Systems Research 33 (1995) 211 218

A discrete-time model of thyristor controlled series compensators

Arindam Ghosh l, Gerard Ledwich Department o1" Electrical and Computer Engineering, The University Of Queensland, Brisbane 4072, Australia

Received 11 January 1995

Abstract

The current trend in power systems is to use power electronic devices to enhance both system stability and power transfer limits. A recent point of interest is the use of thyristor control devices in series compensation of AC transmission lines. In this paper a linearized, discrete-time model of a thyristor controlled series compensated line is presented. The samples for these discrete models are obtained at the peaks of the capacitor voltage of the series compensator. Since these peaks are broad rather than sharp spikes they are quite good indicators of variations in the fundamental component. During transients this will result in variation from uniform sampling. However, the system degenerates to a uniform sampled system in the steady state and thus all the standard tools for the analysis of the standard discrete-time model can be applied. The model developed is validated through digital computer simulation studies.

Keywords: Power electronic devices; Flexible AC transmission systems; Thyristor controlled devices; Series compensation

I. Introduction

Power electronic systems are finding increasing appli- cation in power systems for both transient and steady operation requirements. Examples of application benefits include increased power transfer limits, voltage and transient stability. The high voltage DC transmis- sion system [1] based on thyristors has been used for a long time and the static var compensator (SVC) [2] is now well established. The latest of the power electronic systems to be installed in power systems is the thyristor controlled series compensator. This series compensator is an example of a flexible AC transmission system (FACTS) [3] along with other concepts such as the unified controller [4]. Among the advantages offered by thyristor controllers is the fast speed of response such that the conduction period can be varied every half cycle.

Most papers on power system stability represent the transmission system as operating in an electrical steady state. Even though this is a valid approach for studying the effects of the excitation controls, since the line dynamics are much faster than that of the exciter, the omission of line dynamics of series compensated lines when the transmission network eigenvalues are close to those of the electromechanical systems can lead to highly erroneous models. The models in Refs. [5,6] represent

~On leave from the Department of Electrical Engineering, Indian Institute of Technology, Kanpur 208 016, India.

0378-7796/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved

SSDI 0378-7796(95)00948-H

the series compensator as a variable-mains-frequency reactance. The omission of the line model means that the oscillation envelope of the transmission line in response to changes of firing angle is omitted. This will result in an overestimated response which will be coupled with the deterioration of the phase response introduced by any series resonance, thereby resulting in overoptimistic control design.

Most studies on power electronic controllers include the simulation of the nonlinear dynamics, but the con- trol designs are usually based on the mains frequency model. Most of the control designs used in power systems are linear. Thus we wish to obtain a linearized model of the series compensated system. Linearization of a nonlinear system about a steady operating point is well understood. However, to model line dynamics we need to consider explicitly the mains frequency sinu- soidal drive term due to which every point in the periodic voltage and current waveforms will vary in response to a change in thyristor firing angle. This variation will be for a range of voltage and current states and it will be impossible to obtain a standard linearized model. Thus, if we sample the periodic sig- nals at exactly the same periodic frequency, then the steady-state sample values will be constant and a stan- dard linearization procedure can be applied. Again, if the sample values are somehow to be representative of the variation in the waveform, then these samples need to be at characteristic points of the waveform, such as

212 A. Ghosh, G. Ledwich / Electric Power Systems Resear~41 33 (1995) 211 218

a peak. The method used in this paper is to sample at the peaks of the capacitor voltage of the series compen- sator. Since these peaks are broad rather than sharp spikes they are quite good indicators of variations in the fundamental component. During transients this will result in variation from uniform sampling. However, as the system degenerates to a uniform sampled system in the steady state, the standard discrete-time analysis and design tools can be readily applied. In summary, sam- pling is about the only method to give an accurate linearization of the response of a nonlinear system to a periodic signal, and peak sampling represents a sample which is a better representative of the response than uniform sampling.

In Ref. [8] tools were developed which were applied in Ref. [7] to a model of the operation of the thyristor controlled reactor (TCR) operating as a series compen- sator. The equations in Ref. [7] are derived for state perturbations around a given operating point for a firing angle fixed with respect to the zero crossings of the drive voltage. However, the paper falsely assumes that the zero crossings of the line current are exactly half a mains period apart, which is a serious drawback. A linearized model is typically used for designing a feedback control but in Ref. [7] the sensitivity to firing angle perturbations is not derived. As such, the varia- tion of the strength of the control influence as a func- tion of operating point is not obtained. Also, an arbitrary decision is made concerning the starting point of the periodic sampling. The sample values resulting from such a decision need not be representative of the key aspects of the system response, such as when sam- ple values are at the half-cycle zero crossings of the sampled wave. The analysis assumes a fixed firing with respect to current or voltage zero but infers a sensitivity to requested firing width. The control sensitivity is computed for a particular nature of control feedback rather than recognizing that the system is in standard form and that classical control designs apply.

This paper deals with the development of a lin- earized, discrete-time model of a series compensated transmission line where a TCR is used as the compen- sating device. The discretized model takes on sample values at the zero crossings of the line current once in each half cycle. As mentioned earlier, this has the advantage of synchronizing the circuit behavior with the firing point. The model developed is tested through digital computer simulation studies for both prediction and control design.

2. Transient model of a TCR controlled transmission line

A thyristor controlled reactor (TCR) is a device in which a thyristor switched inductor is connected in parallel with a fixed capacitor (Fig. 1). The effect of the

- ?_

I I ' I I I l

1--~ ~ L R I I ] ICI

TCR I

v

q l N F I N I T E

BUS

Fig. 1. The schematic diagram of a TCR compensated transmission line.

variation of the firing angle of the thyristor can be construed as a fundamental frequency variation in the value of the capacitive reactance. The system consid- ered is a generator connected to an infinite bus through a long transmission system (Fig. 1). The transmission line is represented by lumped resistance and inductance. The capacitance effects are ignored. The generator is assumed to maintain a constant terminal voltage.

Let us denote the current through the transmision line, the voltage across the capacitor and the current through the parallel inductor as IL, Vc and Ip, respec- tively. Then the current and voltage waveforms of the circuit in the steady state are shown in Fig. 2 where V D is the forcing voltage (i.e. the voltage difference between the generator and infinite bus, VD = E - V). Note that the zero of the line current IL coincides with the peak of the capacitor voltage Vc. Also note that the negative peak of the parallel inductor current Ip coincides with the positive peak of the line current and these two currents peak when the voltage across the capacitor is zero.

Let us define a state vector as

x = [IL, Vc, Iv] T

Then the state equation when the switch is open is given as

..g

\ /

~ ' ~ y \ \ 0 ~

/ r

/

\ \ i / ~

0 I0 20 Time (ms)

Fig. 2. The voltage and current waveforms of the compensated transmission line.

A. Ghosh, G. Ledwich /Electric Power Systems Research 33 (1995)211 218 213

Jc= I - R / ' L - 1 / L 0

Io/ C 0 0

0 --K

1/L

x+ LOol v~

= A~x + B1VD (1)

where x is a large positive number. This is equivalent to modeling the open switch as a large series resistance, which is required to ensure numerical accuracy in mod- eling the zero-current state. The state equation when the switch is closed is given by

- - R / L - 1 / L

2 = 1/C 0

0 1/Lp

= A2x + Bz VD

0

- 1/C

- Rp/Lp

1/L]

(2)

Since BI = B 2 we drop the subscripts for the remainder of the paper.

Now, referring to Fig. 2, we define the instant t o, when the line current crosses zero, as our reference instant. We also define three other instants as follows: t~ and t2 respectively define the beginning and the end of conduction of the parallel inductor and t 3 defines the next zero crossing of the line current. Let the state vector at instant to be defined by X(to). Then the state transition diagram for the other instants is shown in Fig. 3, which also depicts the state matrices with which these transitions occur. The quantities c~, fl, and ~, are time differences between the instants, i.e.

0 ~ = t 1 - - to, ,/] = t 2 - t l , ) ' = t 3 - t 2

Let us assume that the drive voltage is given by V D = cos(cot), where co is the line frequency in rad/s. Then the state transition equations for the three differ- ent instants are given as

g *

x(tt) = exp(Al :~) X(to) + texp[Al(c~ -- r)] ~ d

0

x B cos[q5 o + co(z + ~b)] dr (3)

t l

x(t2) = exp(Azfl) x(q) + ~ exp[A2(~ + fl -- r)]

x B cos(q~0 + co(v + ~b)] dr (4)

x(t3) = exp(A~ 7) x(t2) + t exp[A~ (~ + fl + 7 -- r)] L /

~.+/~

x B cos[~bo + co(r + ~)] dr (5)

where q~o is the phase of the drive voltage at instant to. In the steady state the zero crossings of the line cur- rent are exactly one half cycle of the mains frequency apart. During transients, however, the zero-crossing

I A2 I A1 I

t o t l t 2 t 3 x(t 0) ~ x(t 1) -----~.-x(t2) ~ x(t 3)

Fig. 3. The state t rans i t ion d iagram.

instant of the line current can change with respect to the mains. The change in time of the zero crossing from its steady-state value is denoted by ~. To explain the transient behavior of the circuit, let us first assume that the circuit is in the steady state. Then a change in from its steady-state value will cause a change in x(t~) from its steady-state value. This will perturb both fl and x(t2) which, in turn, will cause perturbation in both 7 and x(t3). Now the time difference t3 - to = ~ + fl + 7 is constant ( = re/co) during the steady state but not dur- ing a transient as :~, fl, and 2 are perturbed. Thus (k will be the sum of the change in :~, fl, and 7.

3. Linearized model of the TCR controlled transmission line

Let us denote the perturbed value of any variable v by i~. We also denote the steady-state values of the states at the instants to, t~, t2, and t3 by Xos~, x~s~, Xzs,,, and x3s~, respectively. The corresponding steady- state values of the drive voltage are denoted as VDO, VDj, VD2, and VD3- Let the steady-state values of e, fl, and 7 be ~o, flo, and ?'o, respectively. We shall now linearize the system by expanding Eqs. (3)-(5) in Taylor 's series around their nominal (steady-state) values.

3.1. At instant tl

From Eq. (3) we observe that X ( t l ) is dependent on X(to), ~, and ~, which are the independent variables of the system. Everything else is dependent on these quantities. We thus have to find the sensitivity x(tl) to a change in any of these independent variables. This is accomplished by taking derivative of Eq. (3) with respect to each of these quantities and evaluating the resulting equations at the nominal values. For brevity we denote dx(t0) as dxo and so on, and com- pute these as

d x l - exp(Ai ~0) (6a)

dxo

Before we proceed further, let us state an important result which will be used in many places. This is the derivative of a convolution integral with respect to one of its limits. Let us define such an integral as

,u + ,9 #

l(kt, 9, t) = ~ exp[A(/z + 9 - r)] B cos(q~o + o r ) dr

214 A. Ghosh, G. Ledwich/Electric Power Systems Research 33 (1995) 211 218

then

d d/~ l(/& 0, t) = AI(~, ~9, t) + B cos(q~o + co/z + coO)

- exp(Aoa) B cos(~b o + e)/t)

Using this result we have

dx~ = A1 exp(Al %) Xo~ + A1 [x~ -- exp(Al %) Xos~]

de

+ B V D I

= A l X l s s + BVDI (6b)

;¢0

dXl f do5 = - ~ exp[A1 (Co -- r)] B sin(q~o + ~or) dr (6c)

3.2. At instant t 2

From Eq. (4) we observe that X(t2) is dependent on x(fi), e, fl, and 0. Then we have

d x 2 OX 2 dxl 0X2 dfl 0x2 d~ 0x2 do5 dxo 0fl dxo &¢ dxo ~3(~ dxo

Since 7 and q5 are independent variables, their deriva- tives with respect to any other variable are zero. We then have

d x 2 0X2 dxl Ox 2 dfl dxo - 0x! dxo F ~ dxo

where

0x2 = exp(A2flo) c x 1

Similarly,

d x 2 Ox 2 d x 1 Ox 2

de - Oxl de +

where

0x2 a n d ~- = A2x2s s -1- B VD2

(7a)

0x2 dfl + - - - - (7b)

0fl de

~__~2 = A2[x2~s - exp(A2flo) x, ] + VD2 -- exp(A2flo) VDI B B

and finally,

d x 2 0X 2 d x I OX 2 dfl OX 2 - - - - ~ + ( 7 c )

do3 0x, do5 ~?fl dO 0O3

where °tO + []0

0X2 __ f c~O3 co exp[A2(eo + flo -- z)] B sin(q~o + e)r) dr

~0

To compute the sensitivity of fl with respect to the independent variables, we note that the instant t2 is defined as the instant when the parallel inductor current is zero. Defining a row vector C~ = [0, 0, 1], from Eq. (4) we can write

C~ x ( t : ) = 0

which defines t2 and hence ft. Taking the derivative of the above equation with respect to the independent variables we obtain

_ 0x2 //;, (8a) dfl CI ~ - , d x I (;x I /

where ,2, = Cl~X2/Ofl. Similarly,

dfl C , ( ('x2dx' ~ Z ) / 2 , (8b, d~= - \Ox, ~ + ce / /

C?x:dx: 0x2"~/, dfldo5 - -C1 ~xl -d-~ + ~ - ~ ) / z , (8c)

We now combine Eqs. (6)-(8) to obtain the derivative of x(t2) with respect to the independent variables.

3.3. At instant t3

From Eq. (5) we observe that X(t3) is dependent on x(t2), c¢ fl, 7, and O3. Then we have

d x 3 ~X 3 dx2 0x3 de 0X 3 dfl 0x 3 d7 0x3 do3

= + dxo ev dxo eo5 dxo

The derivatives of the independent variables with re- spect to any other variable are zero. Hence,

d x 3 0 x 3 d x 2 0 x 3 dfl 0x3 d7 - - - - - + - - + - - - ( 9 a ) dxo 0X 2 dxo 0fl dxo 07 dxo

where

0x3 0x3 = exp(Aj 7o), - AIx3ss + BVD3

~x2 0,,

and

0 x 3

off = A1 [X3ss - - exp(A170) X2ss] -}- B VD3

- - exp(A 1 )%) BVD2

Similarly,

dx3 0x3dx2 ~?x3 ~?x 3 dfl 0x3 d7 de - 0x2 dV

where

(9b)

0X 3 0X 3

Finally,

d x 3 Ox 3 d x 2 0x3 dfl 0x3 d~ ~x3 -t ~ t- ( 9 c )

do3 c3x 2 do3 c3fl do3 87 do3 c~o3

where

A. Ghosh, G. Ledwich /Electric Power Systems Research 33 (1995)211-218 215

~o -- [Io + 7o

~x~ f - o) exp[A1 (c% + flo + 70 - r)]

~o+/Jo

× B sin(~b o + e)r) dr

To compute the sensitivity of 7 with respect to the independent variables, we note that the instant t3 is defined as the instant when the line current is zero. Defining a row vector C2 = [1, 0, 0], from Eq. (5) we can write

C2x(13) = 0

which defines t 3 and hence 7. Taking the derivative of the above equation with respect to the independent variables we obtain

C X I / d?' _ C~= -~ /)-2 (10a) dx 2 - CX 2 /

where 22 = C2?~x3/~y. Similarly,

d7 c {Cx3 de- 2 \ ~ x 2 ~ + - ~ + ~ ) / (10b)

dO = 2 \?x2-d-~ 0fl dO ~--~)/"t2 (10c)

We now combine Eqs. (6)-(10) to obtain the derivative of x(t3) with respect to the independent variables.

3.4. Discrete-time system model

Let us define the instant t o as the sampling instant k and the instant t 3 as the sampling instant k + 1. The linearized state equation is then given as

dx3 . . . . dx3 d× 3 £(k + 1) = ~x0 X(K~ + ~ ~(k) + ~ 0(k) (11)

Note that in the representation the samples are not equally spaced during any transient. Now, by defini- tion, both the line current and the parallel inductor current are zero at the sampling instants. Thus it is necessary to retain these quantities in our discrete-time representation. We can then easily modify Eq. (11) to obtain a linearized representation of a variation in the capacitor voltage at the sampling instants. This is given by

Fdx3 l z ( . ( k + l ) = [ dX3] V c ( k ) + 0~(k)

Ldxo]2, 2 L de ]2, 1

[ dx3 ] 0(k) (12) + [dO ]2.1

where [A]~./indicates the element in the ith row and j th column of matrix A.

We now have to include the time shift dynamics of the inductor current crossing in our discrete model. The

incremental time shift between two successive instants is given by

0(k + 1 ) - 0(k) = ~(k) + fi(k) + ~(k)

The lengthening or shortening of any given half cycle depends only on the magnitude o f the phasor o f V¢:.

Thus a positive increase in V c in the positive half cycle has the same effect on 0 as a negative increase in V c at the beginning of a negative half cycle. Defining S = + l for the positive half cycle and S = - 1 for the negative, the state equation descriptor must then be based on S * Vc. We can then write the above equation as

( aft a T ) (d f l d T \ 0 ( k + l ) = 1 + ~ - ~ + ~ - ~ O(k)+\dx,+-d-~xo),. 2

( dfl ~ ) × s , Pc(k)+ s (13)

We now combine Eqs. (12) and (13) to obtain our discrete-time representation of the system. Let us define the new state vector as

~(k) = IS • Vc(k), 0(k)] T

Again note from Fig. 2 that the instantaneous value of the capacitor voltage reverses every half cycle, i.e. it goes through its positive and negative peaks in succes- sive half cycles. Thus, to maintain the correct phasor direction, we define a matrix E = d iag[- 1 1] and then define the discrete-time state equation in a descriptor variable form [9] as

Eg(k + l) = F:~(k) + ( ~ ( k ) (14)

= Ldx0j2.2 Ld5 J2.,

dfl d 7 ] dfl dr ~Xo + ~X~XoJ,, 2 1 + ~-~ + ~

d; = Lde ]2,, (14)

l +

Noting that E - ' = E we can rewrite Eq. (14) in a standard state space form as

~(k + 1) = Fg(k) + G,2(k) (15)

where

F=E~" and G = E I ~

The output equation can then be given in terms of the change in the peak of the capacitor voltage as

)~(k) = S • fZc(k) = [1 0]if(k) = Hff(k) (16)

where

216 A. Ghosh, G. Ledwich /'Electric Power Systems Research 33 (1995) 211 218

4. Numerical results

The model developed in the previous sections is tested through digital computer simulation studies. The simulations are performed using the MATLAB soft- ware package where the model is discretized for 2 ° and 0.05 ° step sizes for higher numerical accuracy. The system parameters chosen are given in Appendix. To be consistent with our model, we have assumed that the firing angle of the TCR is generated from the zero crossing of the line current. For all the tests performed here, we have chosen a firing angle of 70 ° , i.e.

= 3.889 ms.

0.2

L , / / 0.1 11R I II i4¢i-Jl i h M1"l"fi~ iJ i'i ~

o o+ h'/llllllllllll~ I/lll/lllllllllllll Illlllllll/lllllll o F V I/III~IIIIIIWUIUIII/IIIfllII/I/iII/IIIIIIIIIIIIIII/IIIII

°°'1.o., wuunlvvvvlvvlvvllvvlvlvliVlllllll .o.1, ~ I 1 " 1 " " 1 . . . . . I I -0.2 T aclual ~ -- pre?icted

0 0.1 0.2 0.3 0.4 0.5

Time (see)

Fig. 6. C h a n g e in the c a p a c i t o r vo l t age for a one degree decrease in

firing angle.

4.1. Prediction

We shall now use the model derived for predicting the system behavior for a step change in the firing angle. With the system operating in the steady state, the firing angle is suddenly increased by 1 °. The actual and the predicted values of the change in the capacitor voltage peaks are shown in Fig. 4. The predicted values are obtained from the solutions of Eq. (15) from a zero initial condition, while the actual values are obtained from the digital simulation program. It is to be noted that the actual curve alternates as the peak of the capacitor voltage alternates. However, the predicted value predicts only the envelope of the actual wave as

0.04

0.02

0

-0.02

-0.04

-0.06

-0.08

I

II

w

0 0.1

,,x, _ I . . . . . . .

actu predicted II] ! I

0.2 0.3 0.4 0.5

Time (see)

Fig. 7. Shift in zero crossings for a one degree decrease in firing angle.

0.15

0.1

0.05

0

-0.05

-0.1

-0.15 0 0.1 0.2

Fig. 4. C h a n g e in the c a p a c i t o r

f i r ing angle .

I ill, I i l l i l l l l l l / I IA$1, ,d l l i l l l l i l L A/IIIIIIII/IAAAIAfllI/I~AA~I/IA~II/IIA~II~I F, VIIIIIIIVVVIIIIIIII/IVVI/VIVIIIIIIIIIIIIIll I q il I; 4,[I [J [JJJil_tU U t

,+p, f-.+,+., _-++,++j 0.3 0.4 0.5

Time (see)

voltage for a one degree increase in

0.08

0 +

0.04

0,02

.0,02 ~,,] , - - actual predicted

-0.04 , i i

0 0.1 0.2 0.3 0.4 0.5 Time (sec)

Fig. 5. Shif t in zero c ros s ings fo r a o n e degree increase in f ir ing angle .

we have already stripped the sign of this waveform. The actual and predicted changes in the shift in the zero crossings are shown in Fig. 5. To test the prediction behavior for a decrease in the firing angle, the firing angle is decreased by 1 ° when the system is in the steady state. The corresponding plots are shown in Figs. 6 and 7. It can be seen from these figures that the prediction is fairly accurate for a small change in the vicinity of the nominal operating point, especially for about 0.1 s (20 samples or cycles) immediately follow- ing the disturbance.

4.2. State feedback control

A state feedback controller with an integral control action is now designed based on the proposed model to test for its validity for control design. To do that we first define an extended state vector

w(k) = [~V(k), v(k)] v

where

v(k) = v(k - 1) +i f (k) - fR(k) (17)

)~R being the reference input. Note that Eq. (17) defines a recursive model of the integral of the w error between the output and its reference set point. We then combine

218 A. Ghosh, G. Ledwich /Electric Power Systems Research 33 (1995) 211 218

[4] L. Gyugyi, Unified power-flow control concept for flexible AC transmission systems, lEE Proc. C, 139 (1992) 323 ..... 331.

[5] F.P. de Mello, Exploratory concepts on control of variable series compensation in transmission systems to improve damping of intermachine oscillations, 1EEE Trans. Power Syst., 9 (1) (1994) 102 108.

[6] Y. Wang, R.R. Mohler, R. Spee and W. Mittelstadt, Variable- structure FACTS controllers for power system transient stability, IEEE Trans. Power Syst., 7 (1) (1992) 307 313.

[7] S.G. Jalali, R.H. Lasseter and 1. Dobson, Dynamic response of a thyristor controlled switched capacitor, IEEE Trans. Power Delit:- ery, 9 (1994) 1609 1615.

[8] S.G. Jalali, I. Dobson and R.H. Lasseter, Instabilities due to bifurcation of switching times in a thyristor controller reactor, PESC '92 Record, 23rd Annu. IEEE Power Electronics Specialists Con/i, Toledo, Spain, 1992, Cat. No. 92CH3163-3, IEEE, New York, pp. 546-552.

[9] D.G. Luenberger, Boundary recursion in descriptor variable sys- tems, 1EEE Trans. Autom. Control 34 (1989) 287 292.