GATE-PULSE GENERATION

SVC Control Components and Models

4.4 GATE-PULSE GENERATION

GATE-PULSE GENERATION 123 where Yn c the present output at t seconds

Y_n−1 c the past output att −T seconds X_n−1 c the past input att −T seconds

For an SVC controller,Xand Ycorrespond to the error signalV_e and desired susceptanceB_ref, respectively. Thus, Eq. (4.45) can be rewritten as

B_ref(n)cK T V_e(n−1)+B_ref(n−1) (4.46) A general difference equation corresponding to a third-orderG(s) having three poles is expressed as

Bref(n)cK[A1Ve(n−1)+A2Ve(n−2)+A3Ve(n−3)]

+B₁B_ref(n−1)+B₂B_ref(n−2)+B₃B_ref(n−3) (4.47)

124 SVC CONTROL COMPONENTS AND MODELS

two rules for transient-free capacitor switching: that the capacitor bank should be switched on when the voltage across the TSC valves is either zero or at a minimum. The firing pulses for TSC are produced in a similar manner as the TCR, except that they are applied incessantly to the TSC thyristors to maintain continuous conduction. A hysteresis is incorporated in the switching scheme of TSCs to prevent rapid switching (chattering) from the small variations of system voltage around the threshold.

The TCR firing angleacomputation corresponding to its calculated suscep- tance is done using a susceptance-to-firing (B-to-a) angle converter or through a lookup table residing in a microprocessor.

4.4.1 The Linearizing Function

Let us assume that the desired susceptance output,B_ref, of the voltage regulator is to be entirely implemented through the TCR, that is, there are no fixed or switchable capacitors. Then the implementation of the voltage-regulator output Bref as an actual installed susceptance BSVC takes place through an interme- diate stage of the firing-angle calculation, as shown in Fig. 4.20. Because the relationship between the firing angle aand the susceptance BSVC—expressed as F1(a)—is nonlinear, it necessitates the inclusion of a linearizing function F2(a) to ensure that

F₂(a)F₁(a)c1 (4.48)

where

F2(a)c[F1(a)]⁻¹ (4.49)

For the case of a single TCR alone,F₁(a) is expressed as F₁(a)cB_SVCc 2p−2a+ sin 2a

p (4.50)

Thus

F₂(a)c p

2p−2a+ sin 2a (4.51)

In physical terms, the functionF₂(a) represents the calculation of the firing angle, corresponding toBrefif all of theBref is to be implemented on the TCR.

B_ref

F₁(a) a

F₂(a)

B_SVC

Figure 4.20 The linearization function.

GATE-PULSE GENERATION 125 Subsequently, when the TCR is fired with an anglea, the net susceptanceBSVC

connected to the SVC bus will become equal toBref.

In a realistic case, however, the SVC will not be a simple TCR but a com- bination of TCR and either fixed capacitors or switched capacitors on the secondary of the coupling transformer. Furthermore, the number of switched capacitors also varies with time; thus the nonlinearity of function F1(a) gets compounded. The linearizing function F₂(a) must therefore accommodate all the nonlinearities of the functionF₁(a) in addition to any other nonlinearity in the voltage regulator, especially the one caused by susceptance feedback instead of the actual SVC current for implementing the current droop. Sometimes, the linearizing function is inherently embedded in the scheme used for firing-pulse generation, as in the cosine-wave–zero-crossing method [2].

In analog implementations of the GPG, the linearizing function is modeled by a piecewise linear circuit that closely resembles the linearizing function. In digital implementations, it is often realized through a lookup table, as mentioned previously.

The firing pulses are produced in the GPG unit at a low voltage level with respect to the ground potential, but they need to be transmitted to thyristors operating at very high potential levels. The present technology, which ensures isolation between the two widely different potential levels, is to transmit the firing-pulse information through fiber-optic cables.

4.4.2 Delays in the Firing System

4.4.2.1 Thyristor Deadtime The concept of thyristor deadtime in SVC controls [3], [4], [13], [41] can be explained on the basis of a single-phase, 2-pulse TCR in which the firing angle can change from 90⁸ to 180⁸. In such a TCR, the firing angle can only be varied once in each half-cycle; when the conduction starts in either half-cycle, any change in the firing angle of the same thyristor will not have any effect, impling that the desired firing-angle signal, aorder, may be sampled just twice in each cycle just before the positive- and negative-voltage peaks. The sampling frequency thus needs to be no faster than twice the fundamental frequency. If theaorderchanges to a new value just before the positive-voltage peak, it can be implemented instantaneously with a zero delay in the forward thyristor of the anti-parallel–connected pair. However, if the update in aorder occurs, in the worst case—just after the positive peak of the voltage across TCR—there will be a time delay of a half-cycle until this update is picked up at the next sampling just before the negative-voltage peak.

This delay, caused by the thyristors inability to respond to changes in aorder

at any arbitrary time instant, is termed thyristor deadtime. For a 2-pulse TCR, the deadtime is a random quantity varying from 0 to T

/

^{2, where T}is the time period of the voltage wave. It is therefore assigned an average value ofT

/

^4.

Extending the preceding argument to a 6-pulse TCR, it can be seen that the frequency of sampling the signala_orderneeds to be only six times the fundamen- tal frequency. The sampling instants occur shortly before the six voltage peaks

126 SVC CONTROL COMPONENTS AND MODELS

corresponding to the 3-phase voltages in one time period. Thus similar to the 2-pulse TCR the average thyristor deadtime is set at half ofT

/

⁶or one-twelfth cycle time, which is 4.17 ms for 60-Hz systems. In general, for appulse TCR, the thyristor deadtime is given byT

/

(2p), whereTis the time period of the fun- damental voltage. Sampling at a frequency 6f0 has the additional advantage of filtering out the 6th harmonic ripple in the controller output introduced by the rectifier in measurement circuit.

4.4.2.2 Thyristor Firing-Delay Time There is another kind of delay caused by the magnitude of the firing angle at which the TCR is operating. It is attributed to the sequential nature of firing of a 3-phase TCR and corresponds to the delay between the sampling instant and the instant when the thyristors commence conduction. If a change occurs in a_order shortly before a voltage peak (i.e., at one of the sampling instants), the TCR current will achieve its steady state after the new firing angle is sampled by all the three phases. The TCR firing angle varies from 90⁸to 180⁸, implying a delay that varies between zero and one-fourth of a cycle time, as measured from the peak of voltage in one phase. The firing angle in the forward thyristors of other phases will be delayed by 60⁸and 120⁸, respectively. The transfer function of the firing delay is therefore expressed as follows [41]:

G_Y(s)c ¹₃(e^{−s T1}+e^{−s T2}+e^{−s T3}) (4.52) When ais close to 90⁸, the time constants are

0<T₁ <T

/

⁶

T

/

^{6<T2 <2T}

/

⁶

2T

/

^{6<T3 <3T}

/

^{6 (4.53)}

and whenais close to 180⁸, these time constants are T

/

^{4<T1 <5T}

/

¹²

5T

/

^{12<T2 <7T}

/

¹²

7T

/

^{12<T3 <9T}

/

^{12 (4.54)}

where

Tc 1 f0

(4.55) The following time constants can be used as a rough approximation:

THE SYNCHRONIZING SYSTEM 127 T₁cT

/

³

T₂cT

/

²

T₃c2T

/

^{3 (4.56)}

In most stability studies, the thyristor deadtime,T_d, and the thyristor firing- delay time,T_Y, are together represented by the following transfer function [4], [11], [13], [14]:

GY(s)c e^{−s Td}

1+s T_Y (4.57)

where

T_d c T

2p (4.58)

and

T_Y c T

4 (4.59)

corresponding to a maximum delay of 180⁸. AsTd is small,GY(s) can also be approximated by

G_Y(s)c 1

(1 +s Td) (1 +s TY) (4.60)