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Technical Description of Thyristor Controlled Series Capacitors (TCSC)

  • Stig NilssonEmail author
  • Marcio Oliveira
Living reference work entry
Part of the CIGRE Green Books book series (CIGREGB)

Abstract

Thyristor-controlled series compensation (TCSC) systems and thyristor switched series compensation (TSSC) systems are power electronic systems developed in the late 1980s and early 1990s in response to the anticipated need for better utilization of existing high voltage overhead transmission lines because of the difficulties in getting approval for building new lines. The actual experience has been that TCSC systems are primarily being applied in areas with high growth rates where there is a need for long, high voltage ac transmission lines. However, even in areas with existing high power lines imbedded in the ac power system, the load carrying capacity of the lines can be improved by using fixed or switched series capacitor compensation systems. The inherent risks associated with increased loading of existing lines is that if the power system were to be subjected to severe disturbances, there might be a wide spread blackout. TCSC systems represent a tool to manage disturbances and to avoid blackouts by quickly rerouting the power flows from the high stressed lines to lines with the ability to carry higher loads and thereby avoiding blackouts. TCSC systems have been applied to enable construction of long ac lines, which would be unstable if the TCSC systems were not installed. That is, TCSC systems have been proven to be a powerful tool to enhance the stability of the ac systems and even to provide damping of subsynchronous oscillations where the use of fixed series capacitor (FSC) installation could have caused subsynchronous resonance endangering the reliability of large steam turbine generators.

The design requirements for the TCSC FACTS controller are discussed in this chapter. The fundamental operating principles of TCSC systems, the key TCSC design aspects, standards, and other documents, which would be useful to have by those who procure, maintain, or operate a TCSC system are also discussed in this chapter.

1 Introduction

When active power flows through a transmission line, a voltage drop between the sending and receiving ends of the line primarily because of the inductance in the line. If the resistance in the line and the capacitive shunt reactance between the conductors and ground are ignored, the active power sent is equal to the power received as described by Eq. 1. However, as described in the chapter “Introduction to Flexible AC Transmission Systems (FACTS) Controllers: A Chronology”, the active power flow also causes magnetic energy to be absorbed in the inductive reactance of an overhead transmission line. Assuming that the sending and receiving end voltages are the same, then the reactive power required to be supplied from the sending and receiving ends of the line is described by Eq. 2. That is, the more active power that flows through the overhead line, the greater is the reactive power demand.

$$ \left|{P}_S\right|=\left|{P}_r\right|=\frac{V_s{V}_r}{X}\sin \left({\delta}_s-{\delta}_r\right)=\frac{V_s{V}_r}{X}\sin \left(\delta \right) $$
(1)
$$ \left|{Q}_S\right|=\left|{Q}_r\right|=\frac{V^2}{X}\left(1-\cos \delta \right) $$
(2)
where
  • Vs is the sending end voltage with an amplitude equal to Vs and an angle equal to δs.

  • Vr is the receiving end voltage with an amplitude equal to Vr and an angle equal to δr.

  • X is the line’s reactance.

  • δ is the electric angle between the sending and receiving ends of the line (δ = δs − δr).

  • Ps is the active power sent from the sending end.

  • Qs is the reactive power demand at the sending end.

  • Pr is active power received at the receiving end.

  • Qr is the reactive power demand at the receiving end.

If the sending end is located in a strong system, with small voltage variations for different power flow levels but the receiving end is a located in a weak system, then the receiving end voltage can be described by Eq. 3, where Iline is the current flowing in the line.
$$ {V}_r={V}_s-X{I}_{\mathrm{line}} $$
(3)
Figure 1 shows that if the power system at the receiving end of an overhead line is not able to provide reactive power, then the receiving end voltage will be reduced at high power transfer levels until it collapses. (This is referred to as the nose curve because the graph has the appearance of a nose). To avoid this, the voltage along the line has to be increased, which is accomplished by inserting equipment that provides capacitive reactive power as the load increases although the opposite can be needed under low load conditions. That is, when the line loading is very low, because of the Ferranti effect (Steinmetz 1971), reactive power might have to be absorbed along the line instead of added. Reactive power control can be accomplished by means of shunt compensation using capacitor banks/FACTS controllers or by means of series compensation by inserting capacitor in series with the line. For long overhead lines, series capacitors inserted into the overhead line is normally the preferred alternative. The compensation can be switched in or out depending on the line loading.
Fig. 1

Voltage collapse example

FACTS controllers used for reactive power control enables continuous, often step-less control of the reactive power flows. This performance advantage can be used to optimize the reactive power compensation in the power system, to enable dynamic support to damp oscillatory modes and can be used to improve the transient stability of the power systems. One of the FACTS controller options used often in conjunction with fixed series capacitor (FSC) banks is the thyristor-controlled series compensation (TCSC) system (CIGRÉ TB 123 1997).

TCSC systems are used to modulate the impedance of the series capacitors. These systems utilize large, high power thyristors as described in the chapter “Power Electronic Topologies for FACTS Controllers”. The thyristor is the preferred semiconductor device for a controlled series compensation based on power electronics (TCSC and TSSC type systems) because of the short circuit performance of thyristor devices is superior to other semiconductors. Thyristor switched series capacitor (TSSC) type systems can also be applied since they enable rapid insertion or bypass of series capacitor banks. A prototype TSSC system was the AEP – ABB Kanawha River system installed into operation in 1991 (Keri et al. 1992). The Slatt multimodular TCSC system can also be used as a combination of switched and controlled series capacitor system (Larsen et al. 1992).

Standards have been developed to assist power system planners and engineers about what is required when specifying a TCSC system and the IEEE has produced a recommended practice for specifying thyristor-controlled series capacitors (IEEE Standard 1534 2009). This standard provides information about design issues, information needed when procuring a TCSC system, as well as recommendations for factory and commissioning tests. IEC has also developed standards for series capacitor installations, which can be used in applicable portions for the specification of TCSC systems (IEC standard 60143-4 2010).

2 TCSC Principles of Operation

All of the installed TCSC systems were built using a reactor in series with a controllable thyristor valve and a metal oxide varistor (MOV) in parallel with a series capacitor as shown in Fig. 1 (CIGRÉ TB 554 2013). The MOV is used for overvoltage protection of the capacitors as well as the thyristors. The main reason for using thyristor devices is that when there are short circuits on the compensated line, the capacitors and the MOV bank are bypassed by switching the thyristors to full conduction mode. In that case, the thyristors will have to carry the full fault currents until the line breakers are opened or the TCSC system is bypassed by means of a spark gap (or any fast protective device) or mechanical bypass switches. Furthermore, during a power system disturbance, the TCSC systems are often required to operate by switching the TCSC system to the maximum compensation mode thereby providing synchronizing torque to stabilize the connected generators. The high current rating that can be achieved using large diameter, high voltage thyristors therefore make thyristors the device of choice for TCSC systems.

Figure 2 illustrates that when the thyristors are not conducting, the system operates as a conventional series capacitor module. When the thyristors are conducting continuously as shown in Fig. 3, the module can be characterized as a small inductance in parallel with a capacitor (CIGRÉ TB 123). That is, in this operating mode, the impedance of the TCSC is primarily inductive.
Fig. 2

Thyristor-controlled series capacitor module with thyristors turned off

Fig. 3

Thyristor-controlled series capacitor module with thyristors conducting continuously

Figure 4 shows the state of the TCSC system when in the vernier control mode with the thyristors conducting for a fraction of a cycle. In that mode, in addition to the line current, currents are also circulating between the capacitor and reactor as shown in Fig. 5.
Fig. 4

Thyristor-controlled series capacitor module with thyristors in the vernier conduction mode

Fig. 5

Capacitor voltage and thyristor currents in the vernier control mode

In the capacitive modulation mode, shown in Fig. 5, the thyristor valve is turned on for a short period of time just prior to the voltage zero crossing at the 180 electrical degree point of the capacitor voltage (shortly before the maximum current through the capacitor). The capacitor will then discharge through the thyristors and the reactor. The effect of this is that the capacitor will appear to be smaller, i.e., it will have a higher impedance. This increases the apparent degree of series compensation for the line thereby boosting the current flow through the line. When operating in this mode, the apparent impedance of the TCSC (X), the average thyristor current (ITAV) and root mean square (RMS) current (ITRMS) in steady state can be calculated as follows (IEEE 1534):
$$ {\displaystyle \begin{array}{ll}& X\left(\alpha \right)\\ {}& =\frac{-j}{\omega C}\left[1-\frac{k^2}{k^2-1}\ast \frac{\sigma +\sin \left(\sigma \right)}{\pi }+\frac{4{k}^2}{{\left({k}^2-1\right)}^2}\ast {\mathit{\cos}}^2\left(\frac{\sigma }{2}\right)\ast \left(\frac{k\ast \tan \left(\frac{k\sigma}{2}\right)-\tan \left(\frac{\sigma }{2}\right)}{\pi}\right)\right]\end{array}} $$
(4)
$$ {I}_{\mathrm{TAV}}=\frac{k^2}{k^2-1}\ast \frac{{\hat{I}}_L}{\pi}\ast \left[\frac{1}{k}\ast \cos \frac{\sigma }{2}\ast \tan \left(\frac{k\sigma}{2}\right)-\sin \left(\frac{\sigma }{2}\right)\right] $$
(5)
$$ {I}_{\mathrm{TRMS}}=\frac{k^2}{k^2-1}\ast {\hat{I}}_L\ast A $$
(6)
where A equals:
$$ A=\sqrt{\frac{\sigma }{4\pi}\left\{1+\frac{\sin \sigma }{\sigma }+\frac{1+\cos \sigma }{1+\cos \left( k\sigma \right)}\left[1+\frac{\sin \left( k\sigma \right)}{k\sigma}\right]-4\ast \frac{\cos \left(\frac{\sigma }{2}\right)}{\cos \left(\frac{k\sigma}{2}\right)}\ast B\right\}} $$
where B equals:
$$ B=\left[\frac{\sin \left(\frac{\left(k+1\right)\sigma }{2}\right)}{\left(k+1\right)\sigma }+\frac{\sin \left(\frac{\left(k-1\right)\sigma }{2}\right)}{\left(k-1\right)\sigma}\right] $$
where
  • σ is 2 (π – α), the thyristor conduction angle.

  • α is the control (firing) angle from capacitor voltage zero.

  • k is λ/ω.

  • ω is 2πf.

  • f is the power frequency.

  • \( \lambda =\frac{1}{\sqrt{LC}}. \)

  • L is the inductance.

  • C is the capacitance.

  • \( {\hat{I}}_L \) is the peak value of the power frequency component of the line current.

As shown in Fig. 6, the degree of compensation using this so-called vernier control mode can be increased up to a maximum advance angle beyond which the firing would be too close to the resonance point for the module. If the firing of the thyristors valve is delayed for some time when the thyristors are in the continuous conduction mode, the effect is similar. In this mode, as shown in Fig. 6, the thyristors are triggered at 90° with reference to the capacitor voltage zero. If in this mode, the triggering of the thyristors is delayed, the thyristor circuit will operate as a thyristor-controlled reactor. That is, the inductive impedance can be modulated in a way that can be used to buck (oppose) the current flow through the line; a function typically reserved for phase angle regulators. Thus, the vernier control mode for the thyristors valve can be used to increase as well as decrease the current flow through the compensated line (CIGRÉ TB 123 1997).
Fig. 6

Control range for a single TCSC module assuming k = 2.5 in Eq. 4. The vertical axis uses the capacitor bank reactance amplitude as base, i.e., the impedance is equal to −1.0 when the thyristors are blocked

3 Operating Range of TCSC Systems

The operating range for a single TCSC module is shown in Fig. 7.
Fig. 7

TCSC capability for a single module controller

Typically, a series compensation system can be overloaded using the long-term and short-term overload capability of the capacitors. (IEEE Standard 824 2005; IEC 60143-1 to 2015), as shown in the figure. The short-term emergency rating is used during and after a short circuit event in the system where the TCSC is installed. This requires very powerful thyristor valves because the capacitors are assumed to be bypassed by the thyristors during the time it takes for the breakers to clear the fault. When the fault is cleared, the TCSC system must provide maximum reactive compensation to provide the needed synchronizing torque across the line where the TCSC system is installed. During this phase, the TCSC system could be operated and overloaded using its long-term overload rated current shown in Fig. 7.

This requires the following:
  • The TCSC system should preferably not be bypassed during an ac system short circuit except on a line in which the TCSC system is installed but if bypassing is needed it must recover immediately after the short circuit is cleared.

  • The TCSC system must not fail or be permanently bypassed as a result of the ac system short circuit event. That is, failures requiring bypass of the TCSC system must be an independent event not associated with any system short circuit or other overload events for which the TCSC system is required to operate.

For these reasons, as shown in Fig. 7, the TCSC system specifications would normally include a 30-min long-term overload rating and a 10 s emergency overload rating. The 30 min overload rating is typically specified for a 35–50% overcurrent and the 10 s rating is typically for 70–100% overcurrent, usually denoted “swing current” as shown in Fig. 8 (IEEE 1534 2009).
Fig. 8

TCSC performance map

The long-term overload rating is needed to redispatch the power flows after some major ac system disturbance, and the short-term overload rating is needed to manage the transient power swings during and immediately after an ac system fault.

TCSC systems are typically combined with fixed or switched conventional series compensation systems (Gama et al. 1998). In some cases, a fully controllable TCSC, i.e., without any fixed series capacitor, is specified depending on planning studies. Impedance control of the high voltage lines using TCSC technologies can be used to fine tune the loading of parallel lines. Connecting several series connected TCSC modules together as shown in Fig. 9 is one way of achieving a large control range.
Fig. 9

Single line diagram of a six-module TCSC system

The control range for a system consisting of four TCSC controllers is illustrated in Fig. 10.
Fig. 10

TCSC transient capability curves for a four module multimodule TCSC controller

By using vernier control in combination with switching in and out of the series connected modules, a large and almost continuous control range can be obtained as shown in Fig. 11. The system illustrated in Fig. 11, can be considered as a combination of TSSC and TCSC systems. That is, it can offer a stepwise change in the overall transmission line impedance as well as impedance modulation control. This capability should make it possible to schedule the power flow on the TCSC compensated line; a capability which might be useful in a deregulated transmission system1. This capability is, however, associated with a higher cost since each TCSC module will have to include its own reactor and the bus work on the platform becomes more extensive. However, the benefit would be an increased power flow control range.
Fig. 11

TCSC and TSSC impedance characteristics

The vernier control method can be applied on each phase independently of the other phases. Therefore, it could be used to balance the impedance between phases in an untransposed system (Nolasco et al. 2014). It could also be used to increase the power flow across two healthy phases in a system with a single phase to ground fault using single pole trip-reclose schemes. In this way, it could provide synchronizing torque even during a single-phase system short circuit event to improve the transient stability between sending and receiving ends connected by means of a single line.

All TCSC systems can be controlled to add damping to the ac system. This has been the major reason for installing TCSC systems around the world (TB 554 2013). However, the modularity shown in Fig. 9 might not be needed where system damping is the major objective for the use of a TCSC system, which might be the case for a TCSC installation in a long radial line. In that case, a single TCSC system as shown in Fig. 7 would be the most economical solution.

Other TCSC requirements are:
  • No dc component should be injected into the line by having asymmetrical firing between the two antiparallel connected thyristors. Therefore, in the system shown in Fig. 9, one module will normally have to be left with the thyristors in the nonconducting mode (capacitor module continuously inserted) to avoid passing small amounts of DC currents through the line. If, however, the line is also equipped with fixed series capacitors any dc component generated by the TCSC will be blocked by the fixed series capacitor.

  • Short circuit current through the TCSC system will be limited by the line impedance. However, if high short circuit currents are encountered that will overstress the capacitors or the overvoltage protection provided by MOV banks, then the thyristors of the TCSC system can be turned on and bypass the capacitors and the MOV bank. When the thermal limit of the thyristors is reached, a protective gap can be triggered or a mechanical bypass switch in parallel with the capacitors must be closed.

  • If the bypass breaker shown in Fig. 9 is closed at the peak of the through-fault current with a fully offset fault current and with the thyristors in the full conduction mode, the current through the thyristors will continue to circulate by free-wheeling in the thyristor branch and the bypass switch (McDonald et al., 1994). Since the resistance in the circuit with the bypass breaker closed is very low and the inductance is relatively high, the time constant of this circuit (L/R) is long and therefore, the decay of the current will be slow. This scenario could well impose the highest thermal stress on the thyristors. The critical design issue for this high stress event is that the maximum junction temperature in the thyristors must be kept below the maximum allowable junction temperature or the thyristors might fail.

4 Power-Transmission Characteristic Controlled by TCSC Systems

As the TCSC device, as shown in Fig. 12, is a serially connected device and acts like a controlled reactance XTCSC, it affects the transmission line reactance directly. The extreme modes of operation for a TCSC module are with the thyristor path either blocked, in which case it is a conventional capacitor (net reactance of XC), or continuously gated where it appears as a small inductance (net reactance of Xbypass). Between these two extremes, partial-conduction or “vernier” control can be used to increase the reactance in either the capacitive or inductive direction (Larsen et al. 1994).
Fig. 12

Power system model including TCSC (left), the corresponding voltage-phasor diagram (right)

It is rather straightforward to write the equation for active power transmission, as the location of the TCSC along the transmission line does not have any effect on transmitted fundamental frequency power2. Therefore:
$$ {P}_1={P}_2=P=\frac{U_1{U}_2}{X_{12}+{X}_{\mathrm{CSC}}}\sin \delta =\frac{U_1{U}_2}{X_{12}\left(1-{K}_{\mathrm{CSC}}\right)}\sin \delta $$
(7)
where KCSC shown in Fig. 13 represents the so-called series-compensation rate or compensation degree (KCSC = −XTCSC/X12).
Fig. 13

Power transmission characteristic modification imposed by TCSC

In the inductive regime of operation, device’s reactance XCSC in Eq. 7 exhibits positive reactance value, whereas in capacitive regime it is negative. The power transmission characteristics for several values of KCSC are depicted in Fig. 13. It is clear that only the characteristic amplitude of the power curve is modified by the series compensation.

5 Cost Benefit of TCSC Systems

As is shown in Fig. 13, the thyristor-controlled series compensation (TCSC) can provide improved stability for interconnected power systems, allowing higher power transfer levels and directing flows on desired transmission paths (EPRI EL-6943 1991; Larsen et al. 1992; Nyati et al. 1993; Christl et al. 1992).

To ensure a reasonably accurate assessment of benefits and costs, it is important to have simulation models, which closely approximate the behavior of a TCSC. Such a model must represent the physical constraints on operation of the TCSC, as they relate to voltage and current ratings of the equipment (CIGRÉ TB 145; Mittelstadt et al. 1992). Study results can then be used with confidence to specify the parameters of the TCSC, which most closely relate cost to the performance benefit seen in the system studies. For further information about the costs and benefits of TCSC systems, see the chapter “Economic Appraisal and Cost - Benefit Analysis”.

6 TCSC Models

Many different mathematical models are used for study of electric power systems. Models are used by planners to study the effect of the power flows and voltage profiles in the power systems, for study of the stability of the system, and for engineering design of equipment under consideration. Numerous computer models have been developed also for TCSC controllers. CIGRÉ has published some theoretical application studies summarized below that illustrate various performance aspects of TCSC systems (CIGRÉ TB 145 1999)3. TCSC application studies and computer models are also described by IEEE (IEEE 1534 2009).

6.1 TCSC Static Models

The reactance limits of the TCSC must be considered for static modeling as well as for dynamics (CIGRÉ TB 145 1999). These limits are relatively complex and time dependent. The characteristic limits are shown in Fig. 14, which shows the limits enforced in very short (one to tens of seconds) time frame. As implied by Fig. 14, these limits become progressively more restrictive in longer time frames.
Fig. 14

Block diagram for the TCSC model for typical stability studies; line current is inferred on the horizontal axis in the figure

The TCSC controller’s operating limitations at very low line currents are not shown in Fig. 14 or any of the other figures shown in this section. That is, it is not possible for the TCSC to control the line impedance if the line current is below a certain threshold level. The main reasons for the low current operating limit are:
  • If the power for the thyristor gate drives is derived from the ac line current through the line, the power needed for the gate drives might also be insufficient to generate gate drive currents sufficient to turn on the thyristors. This might be one reason to transfer gate drive power from ground up to the energized platform or to use light triggered thyristors with built in self-protection.

  • The measuring systems used for the control system need to produce measurements with a sufficiently high signal to noise ratio and with a sufficient resolution to for example, enable synchronization of the thyristor valve firing, which relies on measurement of the capacitor voltages.

  • At low current levels, the voltage across the series capacitors is also very low and may not be high enough to cause current spreading across the entire thyristor surface area, which can cause current crowding on the thyristors’ surface area if the device is subjected to a high current rate of rise (di/dt) and lead to a device short circuit failure (Kinney et al. 1995).

The low current operating limit could be at around 10% of the rated line current (IEEE 1534 2009). The minimum TCSC current limit needs to be considered in all of the model studies since it can affect the applicability and operation of the TCSC especially if SSR damping is one of the requirements.

6.2 TCSC Dynamic Models

Development of dynamic models is intrinsically related to the specific TCSC application. Power flow control, SSR mitigation, and power oscillation damping control have different model needs and representations.

6.2.1 Block Diagram

Figure 14 shows a block diagram for a TCSC model for a typical stability study. The sign convention for this model is positive reactance in ohms for capacitive compensation and negative reactance in ohms for inductive compensation. The model has provisions for an open-loop auxiliary signal (Xauxiliary), which could be, for example, the input from an external power flow controller. The model also has provisions for a small-signal modulation input (Xmodulation). The reference (Xreference) is the initial operating point of the TCSC.

These inputs sum to Xdesired, which is put through signal conditioning into a lag block. This lag is associated with the firing controls and the natural response of the TCSC and is represented by a single time constant (TTCSC). The time constant is application specific and may vary considerably.

The output of the lag block is called XTCSC, which should have non-windup limits associated with the integration function. These limits are variable limits based on the TCSC reactance capability curve and equations as shown in Fig. 14. This value is added to the value of the fixed compensation (Xfixed), if used in a specific application, to obtain a total compensation value called Xtotal. For the network interface, care must be exercised to assure compatible signs and per-unit basis with the system equations used in the calculations.

6.2.2 Dynamic Reactance Limits

Referring to the limits shown in Fig. 15, the TCSC model should permit operation anywhere within the enclosed region except for the area close to the zero line current where triggering of the thyristors is not possible. The boundaries are due to a number of constraints, as subsequently described. (All reactances are in per unit on XC except as noted, all voltages in per unit on ILrated*XC, and all currents are in, or converted to, amperes.)
Fig. 15

Transient reactance limits; the horizontal axis is the line current

In the capacitive region, the constraints are due to:
  1. 1.

    Limit on firing angle, expressed as a constant reactance limit (Xmax0).

     
  2. 2.

    Limit on voltage across the TCSC which is a function of the current and the capacitive reactance. The maximum voltage limit is used during system transients when the maximum boos level is needed.

     
  3. 3.

    Limit on line current during short-term transient events at which point the TCSC will go into a protective bypass mode.

     

Once the TCSC is bypassed on this overcurrent constraint, it is subject to a time delay on reinsertion after line current falls back below current limit. In a multimodule TCSC, it is possible that only some of the modules will bypass, since once one module bypasses the line current will drop, which in turn may allow the remaining modules to stay in capacitive mode. For simplicity in typical stability studies, it is suggested that this nuance be neglected.

There is also a minimum current operating limit for both the capacitive and inductive operating range (not shown in Fig. 15).

On the inductive side similar constraints apply:
  1. 1.

    Limit on firing angle, expressed as a constant reactance limit (Xmin0).

     
  2. 2.

    Limit on harmonic currents circulating between the thyristor branch and the capacitor, approximated as a constant voltage across the TCSC. (See 9.2.2.1 for calculation of the harmonic currents.)

     
  3. 3.

    Limit on thyristor current. As an approximation, the fundamental frequency component of thyristor current is limited to that at which the TCSC can operate in thyristor bypass for the duration of the transient.

     

See CIGRE TB 145 for a detailed description of these limits (CIGRÉ TB 145 1999).

6.2.3 TCSC Model Performance

TCSC models used for system stability studies have been developed and tested in large-scale power system stability analysis programs (Price et al. 1992; Sanchez-Gasca et al. 1993; Paserba et al. 1994). The simulation results described below show examples of the TCSC model performance (CIGRE TB 145 1999). The CIGRE TCSC stability model was tested on a 25-machine, 100 bus test system. This system included several interconnected areas, and therefore, several interarea modes of oscillation. In this study, the TCSC was located in a circuit between two of the areas which experience multiple swing modes for certain system disturbances. The TCSC for this system had a RMS line-to-line voltage rating of 500 kV, RMS line current rating of 2900 amperes, and a reactance rating (XC) of 8 Ω.

Further demonstration of this stability model is included in Mittelstadt (Mittelstadt et al. 1992).

6.2.4 TCSC Model Alternatives

The TCSC model defined above can be called a “voltage limited” model, because for most of the interesting performance scenarios, the limits on the TCSC reactance are determined from the maximum voltage capability of the TCSC equipment. Without having such a constraint included in the simulation model, the next best approximation is with fixed reactance limits. The following sections compare performance of a “fixed reactance limit” model with a “voltage-limited” model and demonstrate that the system performance is sufficiently different to warrant proper modeling.

Three simulation cases are presented here for the CIGRE test system (CIGRE TB 145 1999):
  • Case A – TCSC with 8 Ω XC nominal, voltage-limited model

  • Case B – TCSC with +14/−4 Ω fixed impedance limit model

  • Case C – TCSC with +8/−2 Ω, fixed impedance limit model

For all three cases, the disturbance was a severe system fault between two areas, followed by line clearing. The remaining line between the areas, which includes the TCSC, picks up the additional current and the TCSC modulates its reactance to damp the power swings.

The simulation results are presented in Figs. 16 and 17. In Fig. 16, the results of Cases A and B are plotted. The solid curve is the benchmark and shows performance with an 8 Ω TCSC represented with a voltage-limited model. The dashed curve shows performance with a 14 Ω TCSC represented with fixed reactance limits. On the first swing, the voltage-limited model is more reactance constrained due to the large increase in line current. On subsequent swings, neither model is limited but performance differs due to the different behavior of the first swing.
Fig. 16

Comparison of system dynamic performance with 8 Ω nominal voltage limited TCSC model (solid curves) and +14/−4 Ω fixed reactance TCSC model (dashed curves)

Fig. 17

Comparison of system performance with voltage limited TCSC model (solid curves) and +8/−2 Ω fixed reactance TCSC model (dashed curves)

In Fig. 17, the solid curve shows the same benchmark case and the dashed curve shows performance with an 8 Ω TCSC represented with fixed reactance limits. On the first swing where line current is very high, the two models encounter roughly the same reactance limit, although the voltage-limited model allows the reactance to be over 8 Ω for a short time. In subsequent swings where line current is lower, however, the difference between the two models is more pronounced. The model with fixed reactance limits hits the 8 Ω maximum limit several times while the voltage-limited model shows that the TCSC reactance can exceed 8 Ω and provide greater modulation capability.

In planning studies, the objective is to determine the TCSC rating required to satisfy specific system performance criteria. The examples illustrated in Figs. 16 and 17 show that the dynamic response of the system subject to voltage-limited modeling can be substantially different than those obtained with fixed reactance limits. Thus, by using a voltage-limited model, planning studies can more accurately determine the correct TCSC rating required to meet system performance requirements.

6.2.5 Operating Studies

In operating studies, the objective is to accurately determine system performance with an existing TCSC. The voltage-limited TCSC model accurately represents the performance of the actual equipment while the model with fixed reactance limits does not. Consider again the simulations in Figs. 16 and 17 as examples, with the solid curves showing performance of the actual equipment. If constant reactance limit models (dashed curves) were used to represent the actual equipment, overall system performance is significantly different. Figure 16 shows that if the TCSC is represented by a 14 Ω fixed limit model, the big difference in the first swing performance of the TCSC causes all subsequent swings to be significantly different. Figure 17 shows that if the TCSC is represented by a 8 Ω fixed limit model, the TCSC’s modulation capability is incorrectly restricted.

These cases illustrate the dilemma faced when using a simple fixed-reactance-limit model for the TCSC. It may not be possible to achieve simulation results which are a reasonable representation of the expected TCSC behavior, and even selecting the proper TCSC rating will be subject to some uncertainty.

Regardless of the model used, the engineer should monitor the magnitude of terminal voltages to be certain that other system equipment is not subjected to unacceptable voltages during power swings and other operating considerations.

6.2.6 Modeling Exclusions

The model described here is not suitable for analysis of harmonics, torsional interactions, high frequency transients, or unbalance problems. Each of these problems requires more detailed modeling of the TCSC and the host system.

Electromagnetic transients programs are needed for study of high voltage transients imposed on the TCSC controller modules if a short circuit to ground occurs on the line at either side of the installed TCSC controller. Such transient events require a detailed high frequency model of the TCSC capacitor banks, reactors, thyristor valves, and bus structures. This need to take into account the facts that the capacitance of an ac power capacitor becomes inductive at high frequencies and that the winding capacitances of reactors dominate the high frequency impedance of the reactors. Also, the stray inductances of the bus work surrounding the thyristor structures of the TCSC controllers need to be included in a high frequency time-domain simulation model.

Models for study of the TCSC systems during line short circuit events also need to include the non-linearity of metal oxide varistor (MOV) blocks. A simple model, which can be used if specific data about the nonlinear characteristics are not known, is as follows:

$$ \frac{I}{I_{\mathbf{0}}}={\left(\frac{V}{V_{\mathbf{0}}}\right)}^{\propto } $$
(8)
where
  • I is the current at voltage V.

  • Io and Vo is typically chosen as the 1 mA knee point and the maximum continuous voltage rating of the material.

  • α is an exponent, which varies with the composition and manufacturing of the MOV material and the applied current.

The simplified model is useful for planning studies but not for TCSC design because the exponent α varies from the knee point to the maximum useful surge current through the MOV. It can be established through tests of MOV blocks (Sakshaug et al. 1988). A value for α equal to 33 has been used for simulation purposes (Anderson and Framer 1996a). Nowadays, electromagnetic transient simulation software allows for the direct representation of the MOV blocks by their voltage-current characteristics provided by the manufacturers.

The trade-off when designing the MOV bank for overvoltage protection of series capacitors is between the knee point of the MOV material and the fundamental frequency overvoltage impressed on the capacitors at the maximum fault current in the transmission line in which the TCSC system is installed. CIGRÉ has developed basic information about the performance of MOV-based arresters for various applications including energy absorption capability of MOV arresters (TB 544 2013). The maximum allowable power frequency voltage across the capacitors is according to standards at least twice the rated capacitor voltage (IEEE Standard 1726 and IEC Standard, 143-1). The critical energy dissipation in the MOV material occurs during the 10 s swing current of the TCSC. After having been exposed to the energy injection during a power system external fault, and the corresponding temperature increase, the MOV shall be thermally stable against the swing voltage caused by the power system oscillation. Therefore, the MOV bank must not be bypassed during this interval. The swing voltage will appear as an overload voltage stress on the MOV for the specified duration (usually 10 s). In TCSC systems, the thyristor valves are typically used for thermal protection of the MOV banks.

The compensated line might be reclosed into the fault adding energy dissipation in the MOV bank, which is made up of a number of parallel connected columns, unless the bank is bypassed. This makes it difficult to obtain uniform loss distribution among the MOV columns since with a nonlinearity index α of around 30, a very small difference between the nonlinearity of the several parallel columns will lead to large differences in energy dissipated in the different columns. Therefore, each of the parallel columns must be built to have close characteristics, which are verified during the current distribution test (IEC 60099-4). That is, the MOV blocks in each of the parallel columns must be closely matched. Because aging of the MOV blocks will change the voltage versus current characteristics of the blocks, it is not possible to replace a failed MOV column with another new or spare column and to get even energy absorption. This requires that redundant MOV columns must be installed when the MOV bank is first built and installed (IEC 60143-2 2012).

6.3 TCSC Modeling Considerations for Long-Term Planning Studies

For long-term dynamic stability studies, the time limited overload capability must also be considered. Figures 10 and 11 illustrate the capability curves for a multimodule TCSC. Figure 18 shows typical capability curves for TCSC modules including the time-overload limits for both capacitive and inductive vernier operation.
Fig. 18

Reactance versus line current characteristics for multimodule TCSC including time overload capability

6.4 Validation

Detailed digital and analogue simulations, including those used in the design and commissioning of TCSC hardware must eventually be validated through system tests. The study results shown above have been used for positive sequence, fundamental frequency analysis of TCSC in electric power systems (Nyati et al. 1993; Mittelstadt et al. 1992; Urbanek et al. 1993). Validation of the study assumptions normally takes place during system acceptance testing typically including staged fault testing (Kinney et al. 1995), whenever staged fault tests are accepted by the transmission system operator.

7 TCSC Design

7.1 TCSC Platform Equipment

Platforms insulated from ground are used for series compensation systems on which the capacitors with their associated protection equipment are placed. One platform is used for each ac system phase. The platform for the controller and its equipment placed on the platform has to withstand wind, snow, ice, and seismic stresses (IEC Standard, 143-1; IEEE Standard 1726 2013). The protection systems used for a conventional series compensation system are typically comprised of bypass switches and MOV columns for overvoltage protection and a spark gap (or any Fast Protective Device) for protection of the MOV bank from overload. Information about the status of the series capacitors, switches, etc. is typically transmitted to ground level via fiber optic data links. Bypass switches can be controlled from the ground level if the operating mechanism is placed at the ground level. Alternatively, the operating mechanisms can be placed on the platform level if power to operate the switches is brought up to the platform level. Most of these types of equipment are also used for TCSC systems (IEEE Standard 1534 2009).

For TCSC controllers, thyristor valves with antiparallel connected thyristors as shown in Fig. 2 and their triggering system plus the reactors are added to the equipment on the platform. However, there exist possibilities for cost and size reductions of the capacitor protection systems since the thyristors can act to bypass the capacitors and the MOV columns during line short circuit events (CIGRÉ TB 123). The thyristor valves are placed outdoors on the capacitor platform and therefore, need to be housed in a weatherproof enclosure. This enclosure must also provide protection for electromagnetic interference (EMI) from outside of the enclosure as well as prevent the thyristor housing being an EMI source to external equipment (CIGRÉ TB 123 1997 and IEEE Standard 1534 2009).

Some protection and control systems are also placed on the platform, depending on the manufacturer’s design philosophy. Typically, these systems will communicate through fiber optic links with the control and protection systems located at ground levels. These systems need auxiliary power to operate. The thyristors also require power to turn on and for monitoring of the devices. If the thyristors are electrically gated, this power must be provided to the thyristors at the platform level or from the triggering circuit itself.

Cooling fluids needs to be pumped up to and from the platform level from the ground level. The fluid is typically deionized water with glycol added to avoid freezing of the fluid. The electric field stress on the cooling fluid is a dielectric stress due to the ac applied voltage. The insulating pipes through which the fluid is pumped need to have sufficient creepage distance to avoid surface discharges. Also, these pipes will be exposed to solar radiation and pollutions, which have to be taken into account when selecting material for the cooling pipes and when the surface stresses on the pipes are considered. Furthermore, ethylene glycol might be considered as an environmentally hazardous fluid, which might require leak containment around the pipes.

Fiber optic links for control and protection systems can be similar to those already in use and proven for FSC banks.

7.2 TCSC Thyristor Valves

The thyristor valves are made up from several series connected antiparallel connected thyristors in order to achieve the voltage rating required for the valves (CIGRÉ TB 123 1997 and IEEE Standard 1534 2009). The valve design is similar to those used for SVC systems; see chapter “Technical Description of the Static Var Compensators (SVC)” for information about typical valve designs. However, the high surge current requirements for TCSC valves differ from those of SVC valves because the thyristors in TCSC systems must be able to ride through line short circuit events without being bypassed by a mechanical switch or a spark gap in order to be quickly returned to the vernier control mode to provide transient stability support of the ac system and to provide system damping to prevent unstable oscillations to arise. Thus, there is a trade-off to consider in the design of the valves between the electrical and thermal ratings of the thyristor devices. The design of the thyristor valves has to be verified through tests. IEC has issued a standard for the electrical testing of thyristor valves specifically for TCSC applications (IEC 62823 2015).

7.2.1 Thyristor Devices

The voltage rating of thyristors can be increased by making the thyristor devices thicker but this causes higher bulk resistivity of the devices leading to higher losses and potentially higher junction temperatures in the devices, which is detrimental to short circuit current survivability. Larger diameter devices can be used, which reduces the current density in the thyristor devices leading to lower device losses. Therefore, TCSC valves typically used custom thyristor devices in order to meet the short circuit current duties. For example, the Slatt TCSC system uses 100 mm diameter thyristors rated at 3.3 kV (Urbanek et al. 1992). The forward voltage drop of these thyristors was typically less than 1.4 V when conducting for 8 ms and at a device temperature of 105 °C. This design was required to meet the 20.3 kA short circuit current duty and 60 kA crest asymmetrical fault current duty. Larger diameter devices and lower short circuit current duties would enable the use of higher voltage devices.

Thyristors have limitations on the rate of current rise (di/dt) upon turning on and also for the rate of voltage rise (dV/dt) when the thyristors are blocked (Mohan et al. 1995). In a TCSC system, the inductance in series with the thyristor valves normally limits the di/dt when the thyristors are turned on (Mohan et al. 1995). However, the di/dt which results from the transfer of current from one thyristor into the antiparallel device during the recovery phase after a high current transient event can be very large (McDonald et al. 1994). The di/dt stress in thyristors occurs along the turn-on line (the edge of the gate towards the bulk of the wafer) on the thyristor wafer. That is, the gate should have a long turn-on line to be able to sustain a high di/dt.

One way to achieve a long gate line is to use an amplifying gate as shown in Fig. 19. The center of this wafer is the electrical gate contact. An electric current injected into the center of this wafer will induce larger current flows in the surrounding regions that turns on a second gate area, etc. Finally, the current is flowing through the conductors out to the six three-legged islands clearly visible in Fig. 19. The current flow from the edges of the long gate legs will then cause current flowing through the main thyristor. Because the gate length is substantial, this device can be subjected to a high di/dt without failing. If light were to be injected into the center of the thyristor wafer instead of an electric current, the electron flow resulting from injection of photons into the gate area will result in the turn on of the device in essentially the same way as the electron injection caused by an electric signal injected into the center gate.
Fig. 19

Thyristor wafer design (courtesy of the Silicon Power Corporation)

Thyristors can also be overstressed if the di/dt on turn on of the devices is too low because then the current will not spread over the entire thyristor wafer. This current spreading requires a defined voltage across the thyristor wafer when the turn on pulse is applied. A weak turn on of the thyristor device will also be the consequence of a weak turn on pulse to the gate of the thyristor device. This can be an issue when the TCSC controller is operating with low line currents and if the gate drivers for the thyristors are fed from current transformers (CTs) sensing the line current because then the voltage fed to the gate drivers is low. If power is fed to the gate drivers from a constant voltage source (requiring power from ground), then the risk for weak gate turn on pulses can be eliminated. The thyristors also have to conduct a sufficiently high current after the gate pulse is delivered to latch in the on-state.

Thyristors also have dV/dt limits because a high capacitive current flow through the semiconductor wafers can cause an uncontrolled turn on of the thyristor devices. The current channels arising through the wafers for such a turn on will cause the device to fail. Therefore, emitter shorts are included in the wafer to limit the sensitivity for capacitive turn on and snubber circuits (resistor – capacitor network) are connected across each thyristor device to limit the dV/dt to which the thyristors can be exposed.

The gate drivers typically also include a so-called voltage break-over (VBO) operation function, which will cause the thyristor device to be turned on even if a gate pulse is absent. This is used to turn on the device if a device is exposed to an excessive voltage, which can arise if the gate driver for one device in a string of devices fails and does not deliver a gate turn-on pulse. It can also retrigger the thyristors if the device current drops below zero temporarily. However, it will also turn on all the series connected thyristor devices if the valve is exposed to an excessive overvoltage.

Thyristors can fail if the device is in the process of turning off and a forward voltage is applied across the valve. This forward voltage might be unevenly imposed on one device in a string of devices. Protective firing (turning on) of the devices is then required to avoid device failure during the recovery period, i.e., during the time from the zero crossing of the thyristor current at turn off until the thyristor can block full forward voltage again. This might be accomplished by the VBO function.

The thyristor valve typically also incorporates various monitoring functions with information constantly transmitted via fiber optic links to the station ground level. This includes the operational status of the thyristor devices so that failures of individual thyristors are known as soon as they occur.

7.2.2 Gate Driver Power Issues

There are options to bring power via isolation transformers or capacitive dividers from ground up to the platform to power the platform control and protection systems. However, if the line to ground voltage is used to power the gate drivers, then this power source will be lost, if a line short circuit occurs that reduces the line to ground voltage, unless the gate drivers have built-in energy storage. Gate drivers can also be powered by using currents transformers (CTs) in series with the line current. The drawback with gate power derived from the CTs is that the power that can be pulled from the platform circuits varies with the load current flowing through the line. That is, when the line current is very low, it would not be possible to trigger the thyristors.

7.3 Valve Cooling

The thyristor devices need to be cooled to remove the switching and conduction power losses dissipated in the devices and their snubber circuits. Liquid cooling is the preferred cooling method. The objective of the cooling system is to keep the junction temperature in the thyristor devices as low as required for the application. That is, the temperature rise over the ambient temperature has to be controlled to prevent the junction temperature in the thyristor devices to rise to an unacceptable level during line short circuit events for which most of the thermal stresses will remain in the thyristor wafer and device package since the device heating will not be dissipated by the cooling media during such short-term events (<1 s). Also, if the thyristors are subjected to “freewheeling” current flows in case the capacitors are bypassed at the peak of the line fault currents, the thyristor devices could be destroyed if the heat dissipated in the devices is not removed. This may require designing the cooling system to have sufficient liquid flow rates in order to keep the temperature rise of the device heat sinks low.

The power electronic systems used for TCSC systems are placed on the capacitor platform but the cooling system is placed at ground potential. The cooling fluid is pumped from the ground level up to the platform and back down using hollow insulator columns. The cooling fluid has to sustain the voltage stresses imposed on the capacitor platform, i.e., it has to be a good dielectric. Deionized water is typically used as the cooling media. If the TCSC will be exposed to freezing temperatures then the water is typically mixed with glycol.

Where high ambient temperatures are encountered, forced air cooling or air conditioning of the thyristor compartment might also be needed. In addition, if excessive humidity levels can be experienced, dehumidifiers might be needed to prevent condensation of moisture in the valve enclosure and control cabinets. In these situations, the demand for auxiliary power at the platform level might be significant.

8 Insulation Coordination

The insulation between the platform and ground for a TCSC system is the same as for conventional series compensation systems except there are more connections between ground and the platform. The insulation requirements, however, are the same for TCSC and conventional series compensations systems (IEC 60071 and IEEE 1726, latest editions). The additional requirements relate to the insulation requirements for the thyristor valves, reactors, and controls.

The basic limitation that establishes the maximum voltage across the series capacitors, reactors, and the thyristors is the voltage limiting characteristics of the MOV columns, which have to be protected from being thermally overloaded during the specified emergency overvoltage operation periods.

Capacitors are typically required to operate under short-term overload conditions at between 1.3 and 1.5 pu of the rated capacitor voltage so the 30-min rating at 1.5 pu of the TCSC system is in line with normal capacitor operating duties. The 10 s rating of the thyristor branch of the TCSC system at 100% current brings the capacitor voltage up to 2 pu of the rated capacitor voltage (IEEE 824 2004). The instantaneous overvoltage stress for the thyristor with its series reactor would be a lightning surge across the TCSC (IEC 60071-1 2015). For such fast transients, the equivalent impedance of the capacitors and the capacitor bank will then be inductive. That is, there will be a transient overvoltage impressed upon the thyristor valves because the capacitor bank is not acting as a short circuit for transient current flows. A similar transient overvoltage would arise if the TCSC system suffers a short circuit to ground since that will expose the thyristor/reactor combination to a stress up to the peak phase to ground voltage. The transient short circuit duty stresses have to be addressed in the system specification.

9 TCSC Losses

Estimation of the losses in a TCSC system requires calculation of the losses in individual TCSC subsystems (IEEE Standard 1534 2002). The losses in TCSC systems are similar to those dissipated in SVC systems; see also the chapter “Technical Description of Static Var Compensators (SVC)” for an information about losses in thyristors switched reactors. The evaluation of such losses for comparison of different investment options is fraught with uncertainties.

Compared to a shunt compensator, where a transformer is needed for connecting the compensator branches to the high voltage system, the TCSC has much lower losses.

Losses are typically differentiated into two categories. One is heat dissipated continuously, which for TCSC systems is when the TCSC equipment is energized but not carrying any load. That is the thyristors are blocked and not conducting any load current. In this mode, there will only be a small current through the thyristors’ snubber circuits. These are the so called no-load-losses. The other category is losses dissipated when the thyristor valves are carrying load current. These are the load losses. The losses in a TCSC system are different if the TCSC valves are in the continuous conduction or in the vernier control mode. When the TCSC system is in vernier control mode, the losses will vary depending on the operating point. This complicates further the process of estimating the likely power losses over the expected operating time for the TCSC system. That is, the application of a TCSC system for damping purposes that might be operating almost all the time at a defined boost level versus a possible application of a TCSC system for power flow control might be significantly different.

The electrical power required to operate the TCSC auxiliary systems should be included in the loss evaluation. The valve cooling system is the major contributor for the auxiliary losses, which are dependent on the ambient temperature specified for evaluation and the TCSC operating point. The auxiliary power losses might be considered as no-load losses unless the cooling system is controlled as a function of the TCSC system loading.

Because the losses are typically capitalized and added to the evaluated direct cost of the installed TCSC at the time when the TCSC system is procured, the expected operating times in the three different states have to be estimated before the economic penalty of the loss evaluation can be estimated. The conversion of the estimated losses can be prescribed in regulations but if not, a net present value (NPV) calculation can be used (Weston and Brigham 1981).

9.1 No-Load Losses

The no-load losses in a TCSC system are calculated assuming that the TCSC is energized and connected to the ac line in parallel with the series capacitor in the system but with the thyristors in the TCSC valves blocked, that is in the not conducting operating mode. In this stand-by operating mode, the losses originate from the auxiliary services and the capacitor units.

The snubber circuits connected across the thyristors will be conducting a very low current. However, these losses are typically small enough to be ignored. The losses in the series inductance of the TCSC system in this stand-by operating mode will also be insignificant. The losses in the series capacitors themselves are also typically small but an IEEE standard can be used to evaluate these losses if it is deemed desirable (IEEE 824 2005). Because the voltage applied across the MOV bank is well below the knee-point of the MOV blocks, the losses in the MOV racks can also be disregarded.

Thyristor valves and valve housings may also include fans for cooling and air conditioning. These auxiliary systems require power for their operation. The power demand of these systems should be categorized as no-load and load losses as relevant, depending on the number of fans that are required to run at no-load and at different load points. The power demanded by the cooling system pump is usually constant for all operating points because the thyristor junction temperatures need to be sufficiently low at all steady-state operating points to enable the thyristors to ride through line fault currents when the junction temperatures will be the highest.

9.2 Load Losses

The load losses can be divided into valve and reactor losses when the thyristors are fired in the continuous current mode (Larsen et al. 1994). That is, the series capacitor is short circuited by the thyristor calves. This is the bypass operating mode. The other operating mode is when the thyristor valves are operating in the vernier control mode. The no-load losses are also dissipated when the TCSC system is operating with load, but are not included in the load losses. However, if the cooling system duty varies with the operating point for the TCSC system, the additional cooling system power demand might have to be estimated based on the TCSC load current.

9.2.1 Bypass Operating Mode

Reactor Losses

In the bypass operating mode, losses are dissipated in the resistance of the inductance in series with the thyristor valves. Also, conduction losses in the thyristors themselves are dissipated. This operating mode will be very rarely used so the losses dissipated in the operating mode might be disregarded unless the TCSC is configured as a multimodule controller in which some modules might be bypassed and other modules operated in the vernier control mode.

The current flowing through the inductance is a continuous fundamental frequency current. However, it is amplified slightly because in this operating mode there will be a current circulating between the series capacitor and the parallel inductance as follows (IEEE 1534 2002):
$$ {I}_{\mathrm{TRMS}}=\frac{k^2}{k^2-1}\frac{I_{\mathrm{Line}}}{\sqrt{2}}\, \mathrm{and} $$
(9)
$$ {I}_{\mathrm{TAV}}=\frac{k^2}{k^2-1}\frac{{\hat{I}}_{\mathrm{Line}}}{\pi } $$
(10)
where
  • \( k=\frac{\uplambda}{\upomega}\ \mathrm{and}\ \uplambda =\frac{1}{\sqrt{LC}} \).

  • L is the TCSC inductance.

  • C is the TCSC capacitance.

The losses in the inductance are an ohmic loss. That is, the loss dissipation PRLoad is proportional to the square of the current. That is, the reactor losses in one phase are:
$$ {P}_{R,\mathrm{load}}={r}_R\ast {I}_{\mathrm{TRMS}}^2 $$
(11)

where rR is the resistance in the TCSC reactor.

Thyristor Losses
In this operating mode, there are no thyristor switching losses. However, the conduction loss dissipation in the thyristors is complex because the voltage drop across thyristors is almost constant across a large current range. That is, the thyristor losses can be approximated by a constant voltage times the conduction current (McDonald et al. 1994). However, a more accurate loss estimate is usually adopted by using a two-parameter loss evaluation function for each thyristor (IEEE 1534):
$$ {P}_{T,\mathrm{cond}}=\left({u}_{T0}\ast {I}_{\mathrm{TAV}}+{r}_T\ast {I}_{\mathrm{TRMS}}^2\right) $$
(12)
where
  • PT,cond is the power loss in one thyristor.

  • uT0 is the thyristor threshold voltage, given in the thyristor datasheet.

  • ITAV is the average current flowing through the thyristors, given in Eq. (9).

  • rT is the thyristor slope resistance, given in the thyristor datasheet.

  • ITRMS is the RMS current flowing through the thyristors, given in Eq. (10).

That is, the total losses for the three phases are:
$$ {P}_{T,\mathrm{cond},\mathrm{total}}=3\ast 2\ast N\ast {P}_{T,\mathrm{cond}}\ \left({u}_{\mathrm{T}0}\ast {I}_{\mathrm{T}\mathrm{AV}}+{r}_T\ast {I}_{\mathrm{T}\mathrm{RMS}}^2\right) $$
(13)

where N is the total number of thyristor levels in the valves, including the redundancy.

9.2.2 Vernier Control Operating Mode

In this operating mode, the thyristors conduct for a portion of the fundamental frequency voltage. The vernier control mode can be used to modulate the inductance value in the inductive mode or the series capacitance in the capacitance mode as shown in Fig. 6. In this operating mode, the following takes place:
  • Each thyristor branch of the antiparallel connected thyristor pairs is conducting only for a fraction of the power cycle. In other words, as shown in Fig. 5, the thyristors are blocked for a portion of the ac cycle.

  • During the interval when the thyristors are blocked, power losses are dissipated in the voltage divider circuits of the thyristor valves and in the series capacitors but the snubber circuit and inductance losses are insignificant.

The calculation of the power losses in the TCSC reactor is shown in section “Reactor Losses” and those in the valve during the conduction period are shown in section “Valve Losses”.

Reactor Losses
The losses in the inductance (reactor) of the TCSC should consider both the fundamental frequency component and the harmonic currents through the reactor. The power frequency and harmonic currents in the thyristor reactors can be calculated while the impedance of the reactor at the power frequency and the X/R ratio at the power and harmonic frequencies should be measured during the factory routine tests. Fundamental frequency and harmonic current values must be considered in the reactor loss calculation. The reactor losses in the three phases of a TCSC are thus calculated as (IEEE 1534 2009):
$$ {P}_{\mathrm{TC}-\mathrm{reactor}}={3}^{\ast}\sum \limits_{h=1}^{h=49}\frac{I_h^2\, \ast \, h\, \ast {X}_{L1}\ }{Q_h} $$
(14)
where:
  • PTC-reactor are the total three phase losses for a reactor under rated conditions.

  • Ih is the calculated harmonic current of the hth order.

  • XL1 is the reactor inductive reactance at the fundamental frequency.

  • h is the harmonic order.

  • Qh is the quality factor at the hth harmonic, i.e., the ratio of reactance to effective resistance.

The harmonic currents can be calculated as follows (Ängquist 2002):
$$ \frac{I_{\mathrm{vh}}}{I_L}=\frac{2}{\pi}\ast \frac{{\mathrm{k}}^2}{{\mathrm{k}}^2-1}\ast A $$
(15)
where A equals:
$$ A=\frac{\sin \left[\left(1-h\right)\bullet \frac{\sigma }{2}\right]}{1-h}+\frac{\sin \left[\left(1+h\right)\bullet \frac{\sigma }{2}\right]}{1+h}-\frac{\cos \left(\frac{\sigma }{2}\right)}{\cos \left(\frac{\mathrm{k}\ \upsigma}{2}\right)}\bullet \left\{\frac{\sin\ \left[\left(\mathrm{k}-h\right)\bullet \frac{\sigma }{2}\right]}{\mathrm{k}-\mathrm{h}}+\frac{\sin\ \left[\left(\mathrm{k}+h\right)\bullet \frac{\sigma }{2}\right]}{\mathrm{k}+\mathrm{h}}\right\} $$
(16)
where
  • IL is the peak value of the power frequency component of the line current.

  • Ivh is the peak value of the harmonic frequency component of the valve current.

  • σ is equal to the conduction angle.

  • ω is 2πf

  • \( \lambda =\frac{1}{\sqrt{LC}}. \)

  • L is the inductance.

  • C is the capacitance.

  • k is λ/ω.

For the fundamental frequency component (h = 1), the first term in A will be equal to half of the conduction angle. That is, for the fundamental component, A equals:
$$ A\left(h=1\right)=\frac{\sigma }{2}+\frac{\sin \left(\sigma \right)}{2}-\frac{2\ {\cos}^2\left(\frac{\sigma }{2}\right)}{k^2-1}\left[k\ \tan \left(\frac{k\ \sigma }{2}\right)-\tan \left(\frac{\sigma }{2}\right)\right] $$
(17)
Valve Losses
Snubber Circuit Losses
Although not being mentioned in IEEE 1534, losses in the valve snubber circuits should also be considered since they are not negligible in vernier control mode. When the thyristor is turned off, the snubber capacitor is charged to a voltage equal to the amplitude of the ac voltage at turn off. The stored energy in the snubber capacitors is dissipated in the resistance of the snubber circuit and the thyristor wafer at next turn on of the device. Since the snubber circuit is usually common to a pair of antiparallel thyristors, this process occurs twice per cycle, the overall losses dissipated in the snubber circuit of the three-phase valve is equal to:
$$ {P}_{\mathrm{sn}}=3\ast {f}_n\ast \frac{C_{\mathrm{sn}}{U}_{\alpha}^2}{n}\ast 2=3\ast {f}_n\ast \frac{C_{\mathrm{sn}}}{n}\ast {\left[\sqrt{2}\ast {U}_1\ast \sin \left(\alpha \right)\right]}^2\ast 2 $$
(18)
where:
  • PSN is the snubber circuit losses.

  • CSN is the snubber circuit capacitance per level.

  • Uα is the instantaneous voltage across the snubber capacitors at the firing angle α; note that this voltage is a function of the firing angle as illustrated in Fig. 6.

  • U1 is the RMS fundamental frequency valve connection voltage.

  • n is the number of series connected thyristors per phase of the valve.

  • fn is the system fundamental frequency.

  • α is the thyristor firing angle.

The equation above is the same as the one derived in IEEE 1031, applicable to a TCR valve although the conduction current and blocking transient voltage across the thyristor valve are different in a TCSC application.

Voltage Divider Losses

Thyristors devices in the off-state (nonconducting) have a finite resistance. That is, if a voltage is applied across the device, when the device is turned off, a small amount of current will flow through the device. The resistance of the thyristors in the off-state is temperature dependent and varies from device to device. Thus, if a string of devices is connected in series, a leakage current will flow though the string but the voltage across each individual device will not be identical. Therefore, a resistor is connected across each of the devices in the string to equalize the voltage division between the devices. This voltage divider will dissipate some power and should therefore be included in the overall loss estimate. The losses are present during the intervals when the devices are in the off-state. The estimated power dissipation is then:

The power dissipated in the voltage divider is then:
$$ {P}_{\mathrm{vd}}=\frac{3\ast {U}_{1\alpha}^2}{n\ast {R}_{\mathrm{vd}}} $$
(19)
where
  • Pvd is the voltage divider losses.

  • U is the RMS of the thyristor blocking voltage; note that this voltage is a function of the firing angle as illustrated in Fig. 6.

  • n is the number of series connected thyristor levels.

  • Rvd is the voltage divider resistance per thyristor level.

Thyristor Conduction Losses

Equations (12) and (13) are still applicable to the losses of one thyristor but the average current (ITAV) and RMS current (ITRMS) are given by Eqs. (5) and (6), respectively.

Thyristor Switching Losses

Thyristors do not reach full conduction immediately upon the application of a turn-on pulse to the gate. There is a finite time for the current to begin to flow around the gate area of the thyristor wafer. During the turn-on time, the voltage decays over a few microseconds (μs) as the current increases. The integral of the current times the voltage across the wafer represents energy dissipated in the wafer. This is the turn-on loss.

Similarly, when the current is commutated from the device into circuits surrounding the device, the conduction current through the wafer does not instantaneously go to zero but reverses for a short period of time as the voltage transiently increases because the plasma that developed during the conduction interval needs to be removed before the wafer enters a nonconducting state. The turn-off process is usually represented by the so-called reverse recovery charge, Qrr. For the same reasons, as there are losses dissipated in the wafer during turn on, there are losses dissipated in the device during the turn-off interval during this time interval.

The time for devices to turn on and for the reverse recovery charge to be removed depends on the applied voltage, the current being switched, the diameter of the device, its gate structure, and a number of other device parameters. Therefore, the device and the specific application duties have to be known before an estimate of the turn on and turn off losses can be made. For large devices, these losses can be several joules per pulse.4 However, once the devices have been selected, the losses can be estimated as follows (IEEE Standard 1031 2011):
$$ {P}_{\mathrm{Tsoff}}=3\ast 2\ast {Q}_{\mathrm{rr}}\ast \sqrt{2}\ast {U}_1\ast \sin \left(\alpha \right)\ast {f}_n $$
(20)
where:
  • PTsoff are the turn-off losses for the TCR thyristor valve.

  • Qrr is the thyristor recovery charge.

  • n is the number of series connected thyristors per phase of the valve.

  • fn is the system fundamental frequency.

  • α is the thyristor firing angle.

If it is assumed that the energy loss is 0.2 J for each turn off, then the power losses will be (IEEE 1031 2011):
$$ {P}_{\mathrm{Tswon}}=3\ast 2\ast n\ast 0.2\ast {f}_n $$
(21)

Although IEEE 1534 recommends the calculation of switching losses using a similar approach to the one described above for a TCR valve, it should be considered that the voltages applied to the thyristor valve have a different waveform and amplitude in a TCSC application. For instance, the recovery charge is dependent on the current derivative at turn off, which has a higher slope for a TCSC application. This means that the real switching losses of a TCSC valve can be higher than those calculated using the equations directly taken from IEEE 1031.

Capacitor Losses

When evaluating the capacitor losses in vernier control, it is important to consider that the capacitor current is not equal to the line current, since it is also dependent on the thyristor firing angle.

10 Harmonic Injection

TCSC operation with partial conduction will cause some energy to be injected into the power system at harmonics of the fundamental frequency. The TCSC will generate all odd harmonics of fundamental frequency, with the triplens being zero-sequence when the line currents are balanced (Larsen et al. 1994). However, most of the harmonic current flows when the TCSC operates in the capacitive vernier mode stay within the TCSC system because the series capacitor acts as a high pass filter (Kinney et al. 1995). The harmonic losses dissipated in the TCSC compensated line are therefore negligible in most cases.

11 Torsional Interactions Between Turbo-Generators and TCSC Systems

11.1 Series Capacitor Bank Interactions with Turbo-Generators

Capacitor series compensation systems for long overhead transmission lines evolved early in the twentieth century (Shelton 1928; Alimansky 1930). It was recognized early that there could be undesirable torsional interactions (TI) between capacitor compensated lines and the connected high speed, steam-turbine synchronous generators (Concordia and Carter 1941; Bodine et al. 1943).5 Butler and Concordia concluded that in the case of a single synchronous generator connected to an infinite bus, as was shown by Nickle and Pierce (Butler and Concordia 1937; Nickle and Pierce 1930; Wagner 1930), the ratio of line resistance to line reactance determines whether or not there is negative damping, and that if all rotor circuits except the field winding are neglected, there is a critical operating angle determined by this ratio above per which the machine is unstable. One consequence of this is that if a series capacitor is used in the line, the ratio of line resistance to line reactance is increased, since the reactance has been reduced. Consequently, the tendency toward negative damping is increased. However, as stated by Butler and Concordia, this oversimplifies the situation because hunting among a group of generators and self-excitation phenomena can also lead to dangerous generator shaft stresses although the inherent damping of the system is usually sufficient to prevent severe hunting. In cases of multiple machines and if the existence of shunt loads is taken into account, the resistance to the reactance components of the transfer impedance between the synchronous machines have to be considered. In certain cases, the damping may be either too small to be effective or may actually be negative, i.e., the rotor oscillations may be amplified rather than damped out. While self-excitation can be damped by adding line resistance, this might enhance TI effects. That is, TI, self-excitation phenomena and hunting may exist at the same time, addressing only one of the issues and may lead to solutions that exacerbate one or more of the other phenomena (Butler and Concordia 1937).

Analysis of the subsynchronous characteristics of the power system is complex. In the 1930s, the tools available for identifying SSR risks were limited. Kilgore published some fundamental equations which can be used to show the buildup in resonance that can lead to dangerous torque levels if the applied torque has a component at the resonant frequency (Kilgore et al. 1977). These equations also show that the electrical time constant, mechanical frequency, modal inertia, and mode shape are all important factors in determining peak shaft torques.

After the subsynchronous resonance incidents in 1970 and 1971 involving generators in the Navajo and Mohave power plants in the USA, the understanding of TI phenomena was greatly improved (Anderson and Framer 1996b). Specifically, during the Mohave investigation, the potential for a so-called induction generator effect (IGE) and torque amplification (TA) were discovered.

As shown by Kilgore, the IGE effect is if the equivalent negative resistance during a subsynchronous current flow in the armature circuity of a generator as seen from the power system is lower than the positive resistance at one of the natural modes of the electric circuit (Kilgore et al. 1977). Under these conditions, a self-sustaining subsynchronous oscillation will arise in the system that can cause generator damage.

Torque amplification (TA) arises after clearing a fault on a series compensated power system if trapped charges in the series capacitors discharge through the generators and if the oscillatory frequency of the discharge transient happens to be coincident with one of the torsional modes of the turbo-generator. There is a large amount of published papers on SSR issues leading IEEE to publish a guide for how to study the published documents (IEEE Committee Report 1992). The real issue for those who consider TCSC systems is how the TCSC technology fares regarding SSR.

11.2 Subsynchronous Damping Performance of TCSC Compensated Lines

It was anticipated early during the development of the TCSC systems that modulating the impedance of the series capacitor banks using thyristor switches might not completely eliminate the risk for subsynchronous resonance (SSR) between series compensated lines and generators, but would at least reduce the risk for such events. It was also anticipated that in order to guarantee the elimination of SSR risk in the line where series compensation is planned, all or most of the series capacitor banks in the line might have to be equipped with a TCSC branch. Therefore, the potential for use of TCSC systems for SSR damping was a part of the initial EPRI project scoping study.

Fundamentally, torsional vibrations in a generator results in amplitude modulation of the output voltage of the generator because the output voltage is proportional to the rotor velocity. Furthermore, it results in a frequency modulation of the output voltage where the fundamental frequency (50 or 60 Hz) output from the generator acts as a carrier. That is, if the frequency of the torsional vibration is f1 and the fundamental frequency of the power system generator is f0, two sidebands are formed: f0 − f1 and f0 + f1. That is, if the thyristor switches in the TCSC could be modulated to counteract the subsynchronous f0 − f1 mode, this might be sufficient to prevent SSR. As had already been demonstrated by the development and testing of the so-called NGH SSR damping system, adding a thyristor switch in series with a resistor connected in parallel with the series capacitors could provide damping of subsynchronous oscillations (SSO) (Hingorani et al. 1987). Extending this concept to TCSC systems seemed to be a realistic expectation. Therefore, the potential for use of TCSC systems for SSR damping was a part of the initial EPRI project scoping study.

As a part of the EPRI project, General Electric (GE), the contractor selected by EPRI for the TCSC development, first reviewed the available SSR mitigation methods (EPRI EL-6943 1991). This was followed by further studies specifically to assess how to use TCSC systems for SSR damping (Bowler et al. 1992; Bowler 1992). The potential for using TCSC systems for damping of SSR in more complex systems was also assessed (Hill et al. 1997). The concept for this study was to investigate if TCSC systems, added to a select subset of fixed series capacitor banks in a complex power system, might be used to damp local torsional modes. A few conclusions were drawn from these studies as follows:
  • The TCSC controller could be built to appear as an inductive reactance in the subsynchronous frequency range.

  • The TCSC might be a suitable tool to combat SSR effects. The Slatt TCSC project produced an SSR control scheme that offered greater security with respect to SSR. However, the effectiveness depended on the selection of suitable design criteria, which must include all turbine generators with risk for TI simultaneously.

  • If other fixed series compensation systems are installed in the ac system, then adding the TCSC power electronic package across these fixed series compensation systems might be needed to prevent SSO arising from the uncontrolled, fixed series compensation systems (Bowler 1992).

  • In a complex power system, there could be other installed series capacitor systems that might cause unstable TI. These risks should be eliminated independently.

  • A TCSC can quickly discharge the series capacitors after a power system short circuit event thereby eliminating the TA effect caused by the TCSC equipped series capacitors.

  • Although SSR mitigation using TCSC or other FACTS controllers might be possible, there will be a need to provide SSR protective relaying systems on the potentially affected generators because there could well exist unusual or unforeseen system situations where SSR could arise.

SSR studies were also performed as a part of the developments of the Kayenta project built by Siemens and WAPA. During these studies, it was shown that the TCSC system performed as an inductor in the subsynchronous region (Hedin et al. 1992, 1997). The study results indicated specifically for this TCSC installation that the damping was sufficient to prevent SSR in adjacent generators. The study results were compared with actual transient response measurement from the Kayenta installation.

The control scheme selected for the TCSC installation has an impact on the ability of the TCSC systems to provide damping of the TI modes. One control mode is the constant reactance control scheme. Another possibility is to inject a special modulation signal to counteract the measured TI. The former method was studied and used for the Slatt system. This demonstrated that the TCSC by itself did not excite the torsional modes.

Injection of a measurement to actively counteract torsional modes requires measurements of subsynchronous power flows. This might require a few to many critical measurements, each of which has the potential to fail and thereby potentially making SSR more likely. However, it might be possible to make such measurements fail-safe, but this has yet to be demonstrated.

Another TCSC control scheme has been proposed by Ängquist et al. 1996, 1997; Ängquist 2002. This scheme also make the TCSC appear as resistor/inductor impedance at subsynchronous frequencies. It has been shown through simulations that by using appropriate controls for the TCSC systems they can contribute damping of subsynchronous oscillations (SSO). Such controls have been implemented in some TCSC installations (Holmberg et al. 1998).

The IEEE standard for specification of a TCSC controller states that an advantage of TCSC technology is that at subsynchronous frequencies, the TCSC will provide a degree of SSR mitigation if the TCSC valves are fired on a continuous basis (IEEE 1534). In this mode, the TCSC is operating in the vernier control mode. When the line current is below the level where the thyristor valves cannot be reliably triggered, and it may be necessary to bypass the TCSC, and the TCSC then equivalent to a fixed capacitor.

However, the standard also cautions the potential users of TCSC systems that detailed studies of the power system are required to determine an appropriate design for the SSR mitigation. Such studies should be made using detailed models of the power system, the nearby turbine generators, and the TCSC. When there are fixed series capacitors installed in the network, a detailed SSR study is critical. However, the IEEE standard does not provide any guidance for how to design the TCSC system for SSR damping, if possible, in the intended application.

It is clear from what is known about SSR that it must be seriously considered as a possibility in capacitor series compensated power systems. It is also clear that proven analytical models to use for TI analysis are available (Anderson and Farmer 1996b).

The approach proposed for study of SSR risks is to first do a frequency screening study (Agrawal and Farmer 1979). This technique provides the resistance and reactance from the neutral of the generator under study. If the scans show a potential for SSR, then a time domain program such as the electromagnetic transients program (EMTP) will have to be used to study any torque amplification and fatigue issues. Torsional interactions (TI) and induction generator effects can be studies in a linear frequency domain program used for eigenvalue analysis (Anderson and Farmer 1996b).

The control strategies used in TCSC systems as discussed above can be used to avoid exciting the torsional modes. It can be used to avoid inducing torsional amplification by short circuiting the capacitors during and after an ac system fault. Both time domain and eigenvalue analysis methods have been used and proven in real-time simulations studies to assess how TCSC systems will perform in regards to SSR and TI (Hill et al. 1997). However, a thorough and detailed analysis of the power system would be needed to determine if a TCSC system can be used to provide damping of SSR in a system with distributed series capacitor installation. For insurance, SSR protective relaying systems can be installed in the critical generator plants to mitigate the risk (Anderson and Farmer 1996b).

12 Stability Improvement and Power Oscillation Damping with TCSC Systems

12.1 Transient Stability

Series compensation can improve the transient stability of a power system if located on appropriate transmission lines. Series compensation makes the compensated transmission line appear electrically shorter. This results in increased synchronizing torques between the generators and thus increases the transient stability margin of the power system. However, careful analysis should also be made of the effect on the parallel uncompensated lines and the overall system in case of outage of the compensated lines.

The TCSC can increase the transient stability margin of the power system beyond the level achieved by a comparably rated fixed series capacitor. With a TCSC, the short-time overvoltage rating of the series capacitor elements may be utilized to provide a higher compensation level for the immediate post-fault period (Gama et al. 1998). This further reduces the tie-reactance and improves synchronizing torques. The TCSC is also able to yield additional stability benefits by providing a period of maximum inductive compensation during the subsequent return swing. These control objectives can be achieved using a bang-bang style transient stability control loop which is active for a short period following the fault.

12.2 System Damping Improvement

Power systems often experience undamped low frequency interarea oscillations when the power transfer between two regions in the system exceeds a threshold, e.g., due to a fault in the system. The threshold depends, among other things, on the strength of the interconnecting transmission systems. TCSC topology offers the possibility of improving the damping of power oscillations and in this way allow increased levels of stable power transfer between two regions.

The interarea low frequency oscillation phenomenon is a well-known interaction between distinct groups of machines, which are interconnected by a weak or heavily loaded ac tie lines. The interarea mode is typically in the range of 0.1–1 Hz depending on the power system’s characteristics (CIGRÉ TB 111 1996). The characteristics of interarea modes of oscillation are complex and, in some respects, significantly differ from the characteristics of local modes, also called local plant modes. Local plant mode oscillations are associated with the swinging of units at a generating station with respect to the rest of the system. However, care should be exercised since some “local” problems may also be associated with oscillations between the rotors of a few generators in the same area. These oscillations are termed intermachine or interplant mode oscillations. The local plant mode and interplant mode typically have frequencies in the range of 0.7–2.0 Hz.

The TCSC is a robust and efficient means of providing damping for interarea oscillation modes. By means of thyristor firing control, the TCSC reactance is modulated and a controlled variation that is 90° out of phase with the power swings is performed. This controlled variation damps the power oscillations.

A sudden change of the average power occurs at the onset of the power oscillation and a high-level power control system slowly restores the average power towards a new equilibrium level by intervention of the power dispatch control. For power system damping control applications, the TCSC is supposed to react only to the oscillation. In order to control the TCSC for damping power oscillations, it is crucial that the oscillation part of the measured line power signal can be extracted as fast as possible and separated from the change in average power (Gama et al. 2000). A further requirement is that the correct phase-shift shall be preserved even when the reactance command is being limited to respect the maximum permitted main circuit stress in the TCSC, avoiding also operation close to the resonance frequency according to Fig. 6.

Any implementation of FACTS controllers includes some maximum permitted stress levels when the power swing amplitude and/or the gain is sufficiently high. Its command signal then must be limited accordingly but it is important that the phase of the reactance control signal is not compromised by such limiting actions.

Although the frequencies of low damped power oscillations are usually well-known in a power system, some variations are caused by changes of the network configurations, loading conditions etc. In order to obtain maximum available damping performance of the installed TCSC, the desired phase-shift between the power oscillation and the reactance modulation signal shall be sustained despite small variations of the interarea mode frequency.

In summary, the TCSC control for damping power system oscillations should accomplish two major tasks:
  • To detect the initiation of the power oscillation

  • To generate a reference signal for the TCSC reactance with the correct phase relationship regarding the measured power oscillation

Issues related to the utilization of controlled series compensation for damping of power system oscillations involve the size of the controlled segment and the choice of controlling signals. While both issues are system dependent, a small segment of the series capacitor (e.g., 10–20%) could be sufficient for power oscillation damping (Grünbaum et al. 2006). Local feedback measurements involving line current or power are effective means of detecting oscillations and controlling the TCSC to enhance the damping of power oscillations.

13 Cross-References

Footnotes

  1. 1.

    The stated objective of EPRI’s FACTS initiative was to provide utilities in the USA with methods for systems analysis, design, and operation that would enable better utilization of the existing transmission facilities and improve the operational flexibility (EPRI EL-6943 1991).

  2. 2.

    In the same way as for a fixed series capacitor bank, the voltages on either side of the TCSC system will, however, need to be considered to avoid creating line voltages that will overstress the insulation of the line.

  3. 3.

    A comprehensive treatise specifically for the TCSC but also for some aspects of FACTS technology applications in general can be found in Ängquist 2002.

  4. 4.

    See, for example, data sheet for a device 5STP 42U6500. https://library.e.abb.com/public/c92a9062c3392b1f83257c63004dbb1d/5STP%2042U6500_5SYA1043-07%20Mar%2014.pdf, accessed November 11, 2018.

  5. 5.

    Hydrogenerators and turbines are less susceptible to subsynchronous generators because of the high inertia of these systems and the inherent damping provided by the waterwheel Kundur (1994).

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Copyright information

© Springer Nature Switzerland AG 2019 2019

Authors and Affiliations

  1. 1.Electrical Engineering Practice, ExponentSedonaUSA
  2. 2.ABBVästeråsSweden

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