Power Flow Control Solutions for a Modern Grid Using SMART Power Flow Controllers. Kalyan K. Sen

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      where the phase‐shift angle,

       (2‐51b)

      or

(2‐51c)

      The magnitude (V_s′s) and the relative phase angle (β) of the series‐compensating voltage (V_s′s) are

      (2‐100)

      and

      (2‐101)

      For a Shunt–Shunt Compensator, the relationship between the active and reactive power flows (P_r and Q_r) at the receiving end is shown by a quarter circle, the radius (a′) of which is varied within the upper and lower limits by the modified sending‐end voltage (V_s′) as shown in Figure 1-27. The relationship is given by the following equation:

(2‐79)

      where

      (2‐61)

Equation (2‐79) represents the locus of a circle, centered at with a radius of a′.

      For a Shunt–Series Compensator, the relationship between active and reactive power flows (P_r and Q_r) at the receiving end is shown by the small circle in Figure 1-27 and is given by the following equation:

(2‐136)

      where P_rn and Q_rn are the natural active and reactive power flows at the receiving end of the line and

(2‐132)

Equation (2‐136) represents the locus of a circle, centered at [P_rn, Q_rn], with a radius of a_r that is given in Equation (2‐132).

An ideal PFC controls the values of the power flow control parameters (line voltage magnitude, its phase angle, and line reactance) to regulate the magnitude and the phase angle of the line voltage simultaneously by adding a series‐compensating voltage to the original voltage with the use of a Shunt–Series Compensator as shown in Figure 1-28. The compensating voltage is variable in magnitude and phase angle with respect to the line voltage. The series‐connected VSC is rated for a fraction of the line voltage, but carries the full line current. The shunt‐connected VSC is rated for the full line voltage, but carries only a fraction of the line current. Therefore, each VSC carries only a fraction of the full transmitted power. In an actual installation, both shunt‐ and series‐connected VSCs are designed to be of the same voltage and current ratings, which reduces the inventory of spare parts. Therefore, their different interface voltages with the line are accomplished with selection of proper turns‐ratios of the respective coupling transformers. For example, in the world’ first UPFC at the AEP Inez substation, the two VSCs were designed identically to be rated at ±160 MVA and 37 kV phase‐to‐phase voltage. However, the shunt VSC was connected to a 138 kV‐line through a coupling transformer and the series VSC was designed to inject 13.33 kV (~16% of the phase voltage) in series with the line through a coupling transformer.