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

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


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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 (Psn and Qsn) at the sending end are derived in Appendix B as

      (B‐12)equation

      and

      (B‐14)equation

      where

      (B‐13)equation

      and the power angle is given in Chapter 2 as

      (2‐27)equation

      The natural active and reactive power flows (Prn and Qrn) at the receiving end are

      (B‐21)equation

      and

      (B‐22)equation

      (2‐40)equation

      (2‐43)equation

      (2‐46)equation

      and

      (2‐48)equation

      where

      (2‐41)equation

Schematic illustration of (a) Electric grid: power flow along a lossless transmission line between sending and receiving ends; (b) equivalent an electrical machine; (c) equivalent an inverter.

      (2‐207)equation

      and

      (2‐208)equation

      where

      (2‐209)equation

      (2‐210a)equation

      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.

Schematic illustration of 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


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