The maximum power flow for lossy transmission lines is derived using ABCD parameters in phasor form. These parameters create a matrix relationship between the sending-end and receiving-end voltages and currents, allowing the determination of the receiving-end current. This relationship facilitates calculating the complex power delivered to the receiving end, from which real and reactive power components are derived.

For a lossless line, simplifications streamline the calculation of real power.

The theoretical maximum real power, or steady-state stability limit, occurs when the phase angle between sending-end and receiving-end voltages equals the total series impedance angle.

However, if the series impedance exceeds the reactance, the maximum real power delivered decreases.

In practical scenarios, power lines do not deliver their theoretical maximum power due to limitations based on rated terminal voltages and a 90-degree angular displacement. Instead, the practical line loadability curve, plotted below the theoretical steady-state stability limit, considers a voltage-drop limit of 0.95 or less and a maximum angular displacement of 30 to 35 degrees. These practical constraints ensure the reliable operation of the power system by maintaining acceptable voltage levels and limiting angular displacement to avoid instability.

For short transmission lines, typically under 25 kilometers, loadability is not constrained by voltage drop or stability concerns but by the thermal ratings of the conductor or terminal equipment. These thermal limits ensure the equipment does not overheat under high current conditions, which could lead to damage or failure.

In summary, theoretical and practical considerations, including impedance characteristics, voltage limits, and thermal ratings, influence the maximum power flow and line loadability of transmission lines. Managing these factors is essential for the safe and efficient operation of electrical power systems.