Equal Area Criterion

The equal area criterion is a graphical energy test for first-swing power stability after a large disturbance. In a classical synchronous-machine power system model, it compares the rotor’s accelerating energy during the disturbance with the decelerating energy available after clearing. This is a direct transient stability method for one machine against an infinite bus or for an equivalent two-machine system. A steady state stability study instead concerns response to small changes near an operating point.
For a specified fault and switching sequence, the Equal Area Criterion indicates whether the first rotor-angle swing has a turning point before the post-fault unstable equilibrium. It does not replace numerical simulation for a general multimachine network.

Equal Area Criterion for Stability

For the classical lossless single-machine model, the electrical power-angle relation is
Before the disturbance, mechanical input and electrical output are equal at the initial rotor angle. During the assumed fault, the transfer relation becomes
Protection clears the fault by opening the required circuit breaker. Clearing time depends on fault detection, relay operation and breaker performance, so it must be calculated or measured for the actual protection scheme.

The classical first-swing model treats mechanical input power as constant over the short disturbance because turbine and governor response is much slower than the electrical transient. The power system is transiently stable when the machine remains in synchronism and its rotor angle reaches a finite first-swing maximum after fault clearing.

Figures 1 and 2 show a generator initially delivering mechanical power P_m at rotor angle δ₀. If the assumed fault reduces electrical transfer to zero while P_m remains constant, accelerating power is
This positive power difference increases rotor kinetic energy, speed relative to synchronism and rotor angle δ.
At clearing angle δ_c, the network changes to its post-fault power-angle curve. When post-fault electrical power exceeds mechanical input, accelerating power P_a is negative and the rotor decelerates. Its angle continues to increase while relative speed remains positive. Stability requires the speed difference to reach zero before the angle passes the post-fault unstable equilibrium.

The swing equation is

P_m = mechanical input power
P_e = electrical output power
δ = rotor power angle
H = inertia constant
ω_s = synchronous speed
The derivative relation is

Substitution in the swing equation gives

Multiplying by the angle-rate term and integrating between δ₀ and an arbitrary angle δ_c gives

At the initial steady state, relative rotor speed is zero at δ₀, as represented by
The fault injects kinetic energy. After clearing, the rotor decelerates until it reaches its first-swing maximum angle δ_max, where relative speed again becomes zero:
The accelerating area obtained from the energy integral is

The available decelerating area is

For a clearing angle δ_c, instability follows if the required accelerating area A₁ exceeds the available decelerating area A₂. The rotor then passes the limiting angle δ_m, where mechanical power again exceeds electrical power, and continues to accelerate out of synchronism.
If the available A₂ is greater than A₁, the rotor reaches zero relative speed before that limiting angle. With physical damping, later swings reduce and the machine can settle.
The boundary occurs when A₂ equals A₁. The associated clearing angle δ_cr is the critical clearing angle.
Setting A₂ equal to A₁ gives

This accelerating-and-decelerating energy balance is the equal area criterion. For the assumed network and disturbance, it determines the critical clearing angle. Critical clearing time also depends on machine inertia and the fault-on trajectory.