Consider a lossy transmission line with V ₁and V ₂as the sending and receiving end voltages, respectively. Z and X are the series impedance and reactance of the line, respectively. The steady-state stability limit for the transmission line will be

Understanding Transmission Line Stability Limits

The steady-state stability limit of a transmission line is crucial for understanding the maximum amount of electrical power that can be transferred from the sending end to the receiving end without the system losing synchronism. This limit ensures the reliable operation of the power grid.

Stability Limit in a Lossless Line

For a simplified case, consider a lossless transmission line. In such a line, the series impedance is purely reactive, meaning $Z=jX$. The maximum power transfer capability, often referred to as the steady-state stability limit, is primarily determined by the sending end voltage $V_1$, the receiving end voltage $V_2$, and the line's reactance $X$. The formula for the maximum power transfer in a lossless line is generally approximated as:

$$P_{max\_lossless}\approx \left|\frac{V_1{V}_2}{X}\right|$$

Here:

- $V_1$ is the sending end voltage magnitude.
- $V_2$ is the receiving end voltage magnitude.
- $X$ is the net reactance of the transmission line.

This value represents the theoretical ceiling for power transfer when only the line's reactance is considered as the limiting factor.

Impact of Losses on Stability Limit

Real-world transmission lines are not lossless. They possess resistance $R$ in addition to reactance $X$, making the series impedance $Z=R+jX$. The resistance causes power dissipation in the form of heat (I²R losses), where I is the current flowing through the line.

The presence of resistance affects the overall voltage drop across the line and reduces the net power delivered to the receiving end for a given current and voltage level. More importantly, for stability, the ability to transfer power is intrinsically linked to the voltage regulation and the phase angle difference between $V_1$ and $V_2$, which is heavily influenced by the reactive component $X$. Resistance introduces additional voltage drop and power loss, which means that the system will become unstable (lose synchronism) at a lower power transfer level compared to a lossless line.

Therefore, the steady-state stability limit for a lossy transmission line, which accounts for the power dissipated due to resistance, will be lower than the theoretical limit calculated for a lossless line using only reactance.

Conclusion Regarding the Stability Limit

Based on the analysis that resistance causes power losses and affects voltage regulation, reducing the maximum power transfer capability before instability occurs, the steady-state stability limit for a lossy transmission line will be less than the value calculated using only the reactance for a lossless line.

This means the stability limit will be:

$${\text{Stability Limit}}_{lossy}<\left|\frac{V_1{V}_2}{X}\right|$$

Comparing this with the given options, the correct statement is that the steady-state stability limit for the transmission line will be less than $\left|\frac{V_1{V}_2}{X}\right|$.