The Power-Angle Curve is a graphical representation of the relationship between the electrical power output (\(P_e\)) of a synchronous machine and its power angle (\(\delta\)). It is a fundamental tool for analyzing the steady-state stability of a power system.
The electrical power transferred from a generator to an infinite bus is given by:
$$P_e = \frac{|E| \cdot |V|}{X} \sin(\delta)$$
where:
\(|E|\): Internal excitation voltage of the generator
\(|V|\): Voltage of the infinite bus
\(X\): Total reactance between generator and bus
\(\delta\): Power angle (phase difference between \(E\) and \(V\))
Since electrical power depends on \(\sin(\delta)\), the curve follows a sinusoidal nature starting from zero.
The maximum power occurs when \(\delta = 90^\circ\), and is given by: $$P_{max} = \frac{|E| \cdot |V|}{X}$$
Under normal operating conditions, the generator operates at an angle \(\delta_0\) where the mechanical input power equals the electrical output power.
The curve can be divided into two regions based on its slope. In the range \(0^\circ < \delta < 90^\circ\), the slope is positive, indicating stable operation. If the load increases, the power angle increases accordingly, allowing more power transfer and maintaining equilibrium.
In the range \(90^\circ < \delta < 180^\circ\), the slope becomes negative. In this region, any increase in \(\delta\) leads to a decrease in power output, causing the rotor to accelerate further and eventually lose synchronism, making the system unstable.
The slope of the curve at the operating point indicates the stiffness of the system. A steeper slope means better ability to withstand disturbances. The maximum power represents the steady-state stability limit of the system.
The stability of the system can be improved by increasing the excitation voltage \(|E|\), which increases the height of the curve and enhances the stability margin.