alt="equation"/>
where
(2‐209)
(2‐210a)
Note that Xse > 0 represents a capacitive compensating reactance and Xse < 0 represents an inductive compensating reactance, respectively. However, Xeff > 0 represents an effective inductive reactance and Xeff < 0 represents an effective capacitive reactance, respectively.
Figure 1-6 Power flow in a lossless line with a series‐compensating reactance (Xse).
Depending on whether the compensating reactance (–jXse) is capacitive or inductive, the voltage (Vq = jVq) across the compensating reactance lags or leads the prevailing line current (I) by 90°. This leads to the concept of an emulated reactance, which is defined as
(1‐3a)
or
(1‐3b)
and is shown in Figure 1-7. In this concept, a compensating voltage
The concept of an emulated reactance can be further extended to represent an emulated impedance when the compensating voltage is not restricted to be in quadrature, but at any phase angle with respect to the prevailing line current (I). That means the compensating voltage
Figure 1-7 Power flow in a lossless line with a series‐compensating voltage (Vs′s).
Figure 1-8 (a) Power transmission system with a series‐compensating voltage (Vs′s); (b) four‐quadrant emulated impedance.
The series‐compensating voltage (Vs′s) is related to (Vdq), such that
(1‐4)
and
(1‐5)
where Vd = Vd and Vq = jVq are the respective active or direct and reactive or quadrature components of the compensating voltage with load convention, meaning the line current (I) enters at the higher potential terminal of the voltages (Vd and Vq) as shown in Figure 1-8a.
The natural or uncompensated power flow through a transmission line in a power system network is, in general, not optimal. Any of the power flow control parameters (line voltage magnitude, its phase angle, and line reactance) can be regulated with the use of the following equipment:
Voltage‐Regulating Transformer (VRT), shunt or parallel‐connected switched reactor/capacitor, also known as Shunt Reactor (SR)/Shunt Capacitor (SC), Static Var Compensator (SVC), or STATic synchronous COMpensator (STATCOM) for voltage regulation as shown in Figure 1-9
PAR or Phase‐Shifting Transformer (PST) for phase angle regulation as shown in Figure 1-10
Thyristor‐Controlled Series Capacitor (TCSC) or Static Synchronous Series Compensator (SSSC) for series reactance regulation as shown in Figure 1-11.
The dynamic performance of a VRT is limited by the speed of operation of the mechanical Load Tap Changers (LTCs), which respond in seconds; this level of response time is acceptable in most utility applications. However, if a faster response is desired, the mechanical LTCs can be upgraded with power electronics‐based LTCs, called Thyristor‐Controlled (TC) LTCs as discussed in Chapter 2. More on LTCs can be found in the book, titled “On‐Load Tap‐Changers For Power Transformers: Operation, Principles, Applications and Selection,” by A. Krämer, Maschinenfabrik Reinhausen, 2000. The power electronics‐based solutions can be divided into two categories, based on the type of semiconductor switches: naturally commutated switch, such as a thyristor and forced‐commutated switch, such as Insulated Gate Bipolar Transistor (IGBT). Each of these solutions is based on engineering trade‐offs.
Figure 1-9 Transmission line Voltage Regulators: (a) Two‐winding Transformer, (b) Autotransformer, (c) Switched Reactor, (d) Switched Capacitor, (e) TSC + TCR = SVC, and (f) STATCOM.
