Switching a Series Connection of a Capacitor and Reactor To overcome the problems discussed in the preceding list, a small damping

Reactive-Power Compensators

3.8 THE THYRISTOR-SWITCHED CAPACITOR (TSC)

3.8.2 Switching a Series Connection of a Capacitor and Reactor To overcome the problems discussed in the preceding list, a small damping

reactor is added in series with the capacitor, as depicted in Fig. 3.27. Let the source voltage be

v(t)cVsinq0t (3.36)

where q0 is the system nominal frequency. The analysis of the current after closing the thyristor switch att c0leads to the following result.

THE THYRISTOR-SWITCHED CAPACITOR (TSC) 73

v_C

v_T i

v C

v_L

Figure 3.27 A TSC with a series reactor.

i(t)cI_ACcos(q₀t+a)−nB_C冢^{VC0 − n2n−2 1 Vsina}冣

. sinqnt −IACcos acosqnt (3.37) where the natural frequency is

qncnq₀ c 1

fLC (3.38)

and the per-unit natural frequency is qnc

h

|

^XC

|

|

^XL

|

^(3.39)

I_AC cV BCBL

B_C+B_L (3.40)

Here, VC0 is initial capacitor voltage att c0. It is well-worth discussing this result in some detail. Note that no damping is considered in the circuit.

3.8.2.1 The Term Involving Fundamental Frequency,␻0 This term rep- resents the steady-state solution. As expected, the current leads the voltage by 90⁸. The current magnitude,IAC, as obtained in the foregoing equation, can be alternatively expressed as

I_ACcVB_C n²

n² −1 (3.41)

A magnification in current by a factor of n²

/

^{(n2 −} 1) is seen as compared to the case without reactor. The same magnification factor is also inherent in the magnitude of the capacitor voltage.

74 PRINCIPLES OF CONVENTIONAL REACTIVE-POWER COMPENSATORS

1 2 3 4 5 6 7 8

00 0.5 1 1.5 2 2.5 3

n² n²− 1

n =q_n q₀

Figure 3.28 The magnification factor for the fundamental-frequency quantities in a TSC.

VCcI XC cV n²

n²−1 (3.42)

It is interesting to study this magnification as a function of the tuning of the TSC branch. The result is given in Fig. 3.28. For LC circuits tuned to resonance frequencies of three times the supply frequency and higher, the magnification factor is close to 1.0; for tuning below 3q₀, the magnification factor increases very rapidly. For practical schemes, therefore,n should be chosen higher than 3(typically, between the 4th and 5th harmonic).

3.8.2.2 The Terms Involving Natural Resonance Frequency,␻_n These terms constitute the oscillatory transients. Let us first discuss if these terms can be made equal to zero. From Eq. (3.27), it is seen that the following two conditions need to be fulfilled simultaneously to avoid the transients:

cos ac0esin ac±1 (3.43)

V_C0 c±V n²

n² −1 cI_ACX_C (3.44)

The first condition, Eq. (3.43), expresses that to avoid transients, the switch must be closed either at the positive or the negative peak of the supply-voltage sine wave. The second condition, Eq. (3.44), shows that the capacitor must be charged to a predetermined value. In practice, there are many problems in the realization of a firing strategy that avoids the transients, including the following:

1. At places where SVCs are installed, usually the voltages are not purely

THE THYRISTOR-SWITCHED CAPACITOR (TSC) 75 sinusoidal and constant, which makes the decision of when to switch on less predictive than the ideal condition. A certain amount of transients is to be expected, even with very precise firing strategies.

2. Keeping the capacitor charged at Vn²

/

^{(n2 −} 1) asks for extra charging equipment. It is easier to keep a capacitor charged atV, but then a certain amount of transients will occur anyway, for the supply voltage V, line inductanceX, and, consequently, ncan change randomly during system operation.

3. Large ac capacitors are not designed to withstand prolonged dc stress of precharging.

If losses are considered in the analysis of the circuit, the oscillatory terms will decay, but the principal conclusions made in the preceding list will still remain valid.

3.8.2.3 Practical Switching Strategies The switching strategies pre- sented here limit the transients to acceptable limits and are based on very simple processes to decide when the thyristors should be fired. The control hardware is simple and the performance is very robust. The following two simple firing schemes are the basis for the switching strategies:

• Firing when the initial capacitor voltage is equal to the supply voltage;

henceVC0cv(t). Firing, therefore, takes place when the valve voltage has a zero crossing:

VsinacVC0 (3.45)

and the firing angle is

acsin⁻¹ VC0

V (3.46)

Equation (3.37) with a from Eqs. (3.45) and (3.46) gives the magnitude of the oscillatory term,I_osc, as follows:

I_osc I_AC c

i

1−冢^VVC0 冣²冢^{1− n12} 冣 ^(3.47)

Figure 3.29(a) shows this relation for different tuning of the TSC branch and for different precharge conditions of the capacitor. It can be noted that the magnitude of the oscillatory term never exceeds the magnitude of the steady-state current. With a higher precharge on the capacitor, the transients become smaller, especially for branches that are tuned to higher-resonance frequencies. The highest transients occur if the capacitor is completely dis-

76 PRINCIPLES OF CONVENTIONAL REACTIVE-POWER COMPENSATORS

0.2 0 0.4 0.6 0.8 1

n (a)

1.0 0.9 0.75 0.5 0

I_OSC I_AC

V_{C 0} V

No Transients n (b) 0

0.5 1 1.5 2

0 0.5

0.75

1.2

0.9 1.1

1.0 2.5

3 I_OSC

I_AC

V_{C 0} V

2 4 6 8 10

2 4 6 8 10

Figure 3.29 The magnitude of the oscillatory component for different tuning of the LC branch and different precharging onC: (a) firing criterion,v(t)cVC0(VT c0) and (b) firing criterion, crest ofvt.

charged. This firing strategy cannot be used if the capacitor is overcharged toV_C0>V.

• Firing at the crest of the supply-voltage sine wave; hence cosac0. Again, Eq. (3.37) gives the magnitude of the oscillatory term as follows:

Iosc

I_AC cn冢^{n2n−2 1 VVC0 −1}冣 ^(3.48)

Figure 3.29(b) demonstrates this result. Note that this scheme is applicable even if the capacitor is overcharged. However, it gives worse transients if the capacitor initial voltage is below the crest of the supply voltage (e.g., V_C0

/

^{V c}0.75). The fewest transients are expected if tuning is in the range nc2to 5. For an initial capacitor voltage of 1.0, the two firing schemes are identical, which can be seen by comparing the characteristics forVC0

/

^Vc

1.0 in Figs. 3.29(a) and (b), respectively.

Based on these firing schemes, the following two firing strategies are used in practice:

Strategy A

• The capacitors discharge if they are not connected to the system. Therefore, any initial voltage across the capacitor is possible.

• If V_C0 < V at the time of demand for the capacitor, it is switched on according to the first explained firing scheme, that is, as soon as the volt- age across the valve reaches zero and the capacitor voltage is equal to the supply voltage.

THE THYRISTOR-SWITCHED CAPACITOR (TSC) 77

• If the capacitor is overcharged (VC0 > V) at the time of demand, it is switched on according to the second firing scheme, that is, when the supply voltage reaches the crest and the voltage across the valve is minimal. This scheme is also called a forced switch-on.

This firing strategy uses both of the two aforementioned firing schemes to min- imize the current transients. It requires no special capacitor-charging strategy and can operate with conventional ac power capacitors. It is widely used in TSC plants for transmission systems. The strategy is presented in Fig. 3.30 in a simplified manner for four cases.

v_C

i_C

(a) (b)

v_C

i_C

v_C i_C

(c)

v_C i_C

(d) TSC

v_C

v_T i

C v

Figure 3.30 The switching strategy for a TSC: (a) firing at the minimum valve voltage, v_C>v; (b) firing at the zero valve voltage,v_Ccv; (c) firing at the zero valve voltage, v_C<v; and (d) firing at the zero valve voltage,v_C c0.

78 PRINCIPLES OF CONVENTIONAL REACTIVE-POWER COMPENSATORS

Strategy B

• The capacitors are charged to the crest of the supply voltage by firing just one of the two thyristors. It is noted that they are not charged to the optimal voltageVn²

/

^{(n2 −1).}

• Firing always takes place at the crest of the supply voltage, where the valve voltage is minimal.

This firing strategy is based on the second firing scheme. It makes use of the relatively small transients if the capacitor is charged to 1 pu voltage, and it can be found in some industrial TSC applications. There are, however, many prob- lems related to its use. The problem of the prolonged dc stress on the capacitor has already been mentioned. To overcome this problem, the capacitor is peri- odically charged to opposite polarity, which eventually becomes tolerable in industrial plants but, because of the high current when the polarity is reversed, is hardly acceptable for transmission compensation. Furthermore, the capacitors are not properly isolated from the system when they are turned off, and interac- tions with the power system can still occur. Because of the voltage transients, it is also possble for the capacitor to get charged to voltages above the ideal value, which is undesirable.