At what line power angle does the MW transfer equal ½ of the path's PMAX?

The line power angle (also known as the mechanical angle or the power angle) is crucial in understanding the transfer of real power (MW) between two points in a power network. In power systems, the maximum power transfer capability, represented as PMAX, is determined by the impedance of the line and the voltage levels.

The relationship between the line power angle and the amount of power transferred can be defined mathematically. The formula that describes the power transfer with respect to the line power angle (δ) is expressed as:

[ P = \frac{V_s V_r}{X} \sin(δ) ]

Where ( P ) is the real power transferred, ( V_s ) is the sending end voltage, ( V_r ) is the receiving end voltage, ( X ) is the line reactance, and ( δ ) is the power angle.

When the power transfer is equal to half of the maximum power transfer (½ PMAX), the sine of the power angle will correlate with that amount. Specifically, it is known that the maximum power transfer occurs at a power angle of 90 degrees (where ( \sin(90^\circ) = 1 )). To achieve half of that maximum power transfer,