## Initial Value Theorem

The initial value theorem relates frequency domain expressions to the time domain behavior as time approaches.

The Equal Area Criterion is the method used to study the Transient Stability of a system of two machines or a single machine connected to an infinite bus. Transient Stability analysis helps determine whether synchronism is maintained, meaning whether the load angle δ settles down to a steady-state stable value after a fault or disturbance is cleared.

The Equal Area Criterion states that the power transfer between two sources is inversely proportional to the reactance. During fault conditions like Single Line to Ground Fault or Double Line to Ground Fault, the effective transmission reactance increases, resulting in a reduction in power transfer between the two sources. This reduction in power transmission affects the balance between mechanical torque and electromagnetic torque, leading to the acceleration or deceleration of the generator rotor.

**Proof of Initial Value Theorem**

When a fault occurs, the power output decreases, which causes the generator rotor to speed up and the load angle δ to increase. However, because of the rotor’s inertia, the angle doesn’t immediately go back to its original position. It keeps increasing until it reaches a point where the energy lost during slowing down is equal to the energy gained during speeding up. This point is known as the Equal Area Criterion.

To understand the out-of-step condition and how the Equal Area Criterion is applied, let’s look at two cases:

### Case 1: If δF < δL or Area 1 < Area 2

In this case, the generator rotor will swing back and forth, but the presence of damping causes the size of the swings to gradually decrease until δ settles at a balanced angle called δ0. This situation shows transient stability. The margin for transient stability is determined by subtracting δF from δL, where δL is the angle at which the power output exceeds the mechanical power input.

The Critical Clearing Angle, denoted as δF, is important in determining the transient stability margin. Its value depends on the time it takes to clear the fault, which is influenced by the time it takes for the breaker to open. When the fault clearing time is shorter, δC and δF become smaller, resulting in a larger transient stability margin.

### Case 2: If δF < δL or Area 1 > Area 2

If Area 2 is smaller than Area 1, when the angle reaches δL, any further increase in δ will cause the electric power output to be smaller than the mechanical power input. As a result, the rotor speeds up again, causing δ to increase beyond recovery. This situation represents a transiently unstable condition known as a loss of synchronism or an out-of-step condition in the power system.

In conclusion, the Equal Area Criterion is an important tool for analyzing transient stability in power systems. Understanding the limits and margins of transient stability, as well as the impact of breaker opening time, is important for maintaining a stable and synchronized power system.