In the power-angle curve for equal area criterion shown below, the area of acceleration is defined as:

Explanation:

Equal Area Criterion in Power-Angle Curve

Definition: The equal area criterion is a method used to analyze the stability of a synchronous machine under transient conditions. It is based on the principle that the area of acceleration (A₁) and the area of deceleration (A₂) in the power-angle curve should be equal for the system to remain stable after a disturbance. The power-angle curve represents the relationship between the electrical power output of a synchronous machine and the rotor angle (δ).

Working Principle: When a disturbance occurs, the rotor angle changes, causing a temporary imbalance between the mechanical input power (P_m) and the electrical output power (P_e). The rotor accelerates or decelerates depending on whether P_m is greater than or less than P_e. The equal area criterion ensures that the energy gained during acceleration is equal to the energy lost during deceleration, allowing the rotor to settle at a new equilibrium point.

Correct Option Analysis:

The correct option is:

Option 3: \(\rm A_{1}=\int_{\delta_{o}}^{\delta_{c}}\left(P_{m}-P_{e}\right) d \delta=0\)

This option accurately defines the area of acceleration (A₁) in the power-angle curve. The integral represents the energy gained by the rotor during acceleration, which occurs when the mechanical input power (P_m) exceeds the electrical output power (P_e). The limits of integration are δ_o (the initial rotor angle at the start of the disturbance) and δ_c (the critical rotor angle where the system transitions to deceleration). The condition for stability is that the total energy gain (A₁) during acceleration equals the total energy loss (A₂) during deceleration.

Mathematical Representation:

The energy gained during acceleration is given by:

\(A_{1} = \int_{\delta_{o}}^{\delta_{c}} (P_{m} - P_{e}) \, d\delta\)

The energy lost during deceleration is given by:

\(A_{2} = \int_{\delta_{c}}^{\delta_{max}} (P_{e} - P_{m}) \, d\delta\)

For stability:

\(A_{1} = A_{2}\)

This ensures that the rotor angle returns to a stable operating point after the disturbance.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \(\rm A_{1}=\int_{\delta_{c}}^{\delta_{o}}\left(P_{m}-P_{e}\right) d \delta=0\)

This option incorrectly reverses the limits of integration. The area of acceleration (A₁) is always defined from the initial rotor angle (δ_o) to the critical rotor angle (δ_c), not vice versa. Reversing the limits would result in a negative value for A₁, which is physically incorrect in the context of energy gained during acceleration.

Option 2: \(\rm A_{1}=\int_{\delta_{o}}^{\delta_{c}}\left(P_{e}-P_{m}\right) d \delta=0\)

This option swaps the terms inside the integral, using (P_e - P_m) instead of (P_m - P_e). This representation corresponds to the area of deceleration (A₂), not acceleration (A₁). The area of acceleration must be defined by the difference (P_m - P_e), which represents the net power causing the rotor to accelerate.

Option 4: \(\rm A_{1}=\int_{\delta_{c}}^{\delta_{o}}\left(P_{e}-P_{m}\right) d \delta=0\)

This option combines the errors from options 1 and 2. It reverses the limits of integration and swaps the terms inside the integral. As explained earlier, the limits of integration for A₁ must be from δ_o to δ_c, and the integrand must be (P_m - P_e). This option is therefore doubly incorrect.

Conclusion:

The equal area criterion is a vital tool for analyzing the transient stability of synchronous machines. By ensuring that the areas of acceleration (A₁) and deceleration (A₂) are equal, we can determine whether the system will return to a stable operating point after a disturbance. The correct representation of A₁ is:

\(A_{1} = \int_{\delta_{o}}^{\delta_{c}} (P_{m} - P_{e}) \, d\delta\)

This integral accurately describes the energy gained during acceleration, and its equality with the energy lost during deceleration ensures stability. Understanding the limits of integration and the terms inside the integral is crucial for correctly applying the equal area criterion in power system analysis.