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

are classified as being short lines; lines of lengths between 50 and 150 miles (80.5 and 241.4 km) are classified as medium‐length lines and lines above 150 miles (241.4 km) are considered long lines. Consider a line in the interconnected transmission system, connecting sources and loads as shown in Figure 1-1 as a relatively short line where the capacitive shunt reactance from the line to ground and among the lines can be ignored as shown in Figure 1-4. The resistances and inductive reactances from all the line sections are lumped together as shown in the figure. The natural power flow in an AC transmission line depends on (1) magnitudes of the sending and receiving‐end voltages, (2) phase angle between these voltages, and (3) line impedance.

      The additional symbols shown in the figure are

       VXn is the natural voltage across the line reactance with a magnitude (VXn) and a phase angle (θVXn),

       VRn is the natural voltage across the line resistance with a magnitude (VRn) and a phase angle (θVRn),

       In is the natural line current with a magnitude (In) and a phase angle (θIn),

       Psn is the natural active power flow at the sending end,

       Qsn is the natural reactive power flow at the sending end,Figure 1-4 Power flow along a transmission line between sending and receiving ends.

        Prn is the natural active power flow at the receiving end,

       Qrn is the natural reactive power flow at the receiving end,

       R is the line resistance (R > 0 and represents a positive resistance), and

       X is the line reactance (X > 0 and represents an inductive reactance).

      The natural active and reactive power flows (P_sn and Q_sn) at the sending end are derived in Appendix B as

      (B‐12)

      and

      (B‐14)

      where

      (B‐13)

      and the power angle is given in Chapter 2 as

      (2‐27)

      The natural active and reactive power flows (P_rn and Q_rn) at the receiving end are

      (B‐21)

      and

      (B‐22)

Ignoring the line resistance as shown in Figure 1-5a, the natural active and reactive power flows (P_sn and Q_sn) at the sending end and the natural active and reactive power flows (P_rn and Q_rn) at the receiving end for a relatively short lossless line are

      (2‐40)

      (2‐43)

      (2‐46)

      and

      (2‐48)

      where

      (2‐41)

In addition to using these formulae to characterize a two‐generator/single‐line power system network, they may be used when designing an electrical generator where the V_s and V_r are the generator’s internal voltage and terminal voltage, respectively, and X is the internal reactance of the generator as shown in Figure 1-5b. When designing an inverter, V_s represents the inverter’s output voltage, which is typically created using a Pulse‐Width Modulation (PWM) technique and passed through a filter that consists of an inductor with a reactance (X) and a capacitor (C_f) to create a filtered voltage, V_r, as shown in Figure 1-5c.

Figure 1-5 (a) Electric grid: power flow along a lossless transmission line between sending and receiving ends; (b) equivalent representation of an electrical machine; (c) equivalent representation of an inverter.

The direct way to modify the effective line reactance (jX_eff) between its two ends is to connect a compensating reactance (–jX_se) in series with the line as shown in Figure 1-6. The active and reactive power flows (P_r and Q_r) at the receiving end of the line are given by the following equations:

      (2‐207)

      and

      (2‐208)

      where

      (2‐209)

      (2‐210a)

      Note that X_se > 0 represents a capacitive compensating reactance and X_se < 0 represents an inductive compensating reactance, respectively. However, X_eff > 0 represents an effective inductive reactance and X_eff < 0 represents an effective capacitive reactance, respectively.

Figure 1-6 Power flow in a lossless line with a series‐compensating reactance (X_se).

      Depending on whether the compensating reactance (–jX_se) is capacitive or inductive, the voltage (V_q