# Factors that limit maximum power transfer through transmission lines?

I have read that the maximum power transfer is limited by (1) Thermal Limit, (2) Voltage Regulation and (3) Transmission angle (Stability).

Thermal limit is self-explanatory. Is voltage regulation the voltage drop across the line and transmission angle reactive power losses (if uncompensated)? And what do they mean by Stability in terms of transmission angle?

• Arc-over is the limit. – analogsystemsrf May 3 '18 at 5:22

## 1 Answer

The following figure gives you an idea which stability is to be considered based on transmission system voltage level and length. Since Thermal limit is fine, I will give a brief idea about voltage stability and rotor angle stability. Source

Voltage stability

Let us take the following simple single load-infinite bus system The active/reactive power received by the load is given by,

$$\P=-\frac{E V}{X} sin(\theta)\$$

$$\Q=-\frac{V^2}{X}+ \frac{E V}{X} cos(\theta)\$$

After doing some math you can get the receiving voltage relation with the system parameters and load condition as follows,

$$\V=\sqrt{\frac{E^2}{2}-QX\pm\sqrt{\frac{E^4}{4}-X^2P^2-XE^2Q}}\$$

By plotting PV curve for different power factor of load we can get, Where $$\tan \phi=Q/P\$$.

Note the following points,

• The system is stable as long as it operates at the upper part of the PV curve. When the load withdraw more power, operating point move toward the "nose point". If the system is overloaded so that the operating point cross the "nose point" the system losses its stability.
• Reactive power compensation improve voltage stability significantly as shown in the figure. You can notice at unity power factor ($$\PF=1\$$ or $$\tan \phi=0\$$), the system can be loaded much higher than when load has lagging PF before it lose its stability. Similarly, for leading power factor (negative $$\tan \phi\$$) improves the stability further. This answer your question related to impact of voltage regulation on system stability.
• Note that the value of $$\X\$$ is multiplied by the power drawn by load in this figure (x axis). This means that increasing the line length has significant impact on pushing the operating point toward the "nose point". However, in practice

Source

Transmission angle stability

Assuming you have the same previous system but with connecting a synchronous generator instead of load. The active power transfer between the synchronous generator and infinite bus is given by,

$$\P=\frac{EV}{X}sin(\delta)\$$

Where $$\\delta\$$ represents rotor angle (load angle in some references). If we plot $$\(P,\delta)\$$ for different generator terminals we get, P1 in this figure represents mechanical power of generator (power set point). The intersection between P1 and ($$\P,\delta\$$) curve represents the operating point of the generator. Now as long as the operating point is such that $$\\delta\ <90^\circ\$$, the system is stable. When $$\\delta\$$ cross the $$\90^\circ\$$ limit the system becomes unstable. Note the impact of generator terminal voltage on improving the stability.

Source

Note that this is very basic idea about stability issues, it is more complex in reality. So it is recommended to read further if you are interested in this topic.

• :thank you for this very helpful comment. I am wondering if you still know where you took the first graph (about how the line load changes with varying line length) from? – Paul Jun 19 at 15:32
• @Paul Sorry, I forgot to cite the source. Now I added it under the figure. – Hazem Jun 20 at 2:55