How do you apply the equal area criterion to determine the critical clearing angle and time of a fault?

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The equal area criterion is a graphical method to analyze the stability of a power system after a fault. It helps you to find the critical clearing angle and time, which are the maximum values of the rotor angle and the fault duration that still allow the system to return to a stable equilibrium. In this article, you will learn how to apply the equal area criterion to a simple two-machine system with a single transmission line.

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1 Fault model

To use the equal area criterion, you need to model the fault as a change in the electrical power output of the generators. The electrical power output is proportional to the sine of the rotor angle difference between the machines. Before the fault, the system is in steady state, meaning that the electrical power output is equal to the mechanical power input of each machine. During the fault, the electrical power output is reduced by a factor of k, which depends on the type and location of the fault. After the fault is cleared, the electrical power output is restored to its original value.

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Dr. DIPANSHU NAWARE

Assistant Professor, EEE, NIT Tiruchirappalli
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One point I would like to add in the fault model is sudden change/loss in mechanical input. This situation will also create a scenario for stability analysis.

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Ahmad F. Abdelhadi, EIT, MIET

Power Systems Engineer | BESS Plant Controls | IBRs Grid Compliance Studies | Ph.D. candidate | Towards a 100% RE Future | Electrical Design | M.Sc. | PMP?
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The equal area criterion is applied by plotting the power-angle curve and determining the area under the curve for both pre-fault and post-fault conditions. The critical clearing angle is identified when these areas are equal. The corresponding time of fault clearance is estimated based on system dynamics. This criterion helps ensure stability by indicating the maximum fault clearing time before instability occurs.

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2 Swing equation

The swing equation describes the dynamic behavior of the rotor angle of each machine. It relates the change in the rotor angle to the difference between the mechanical and electrical power. The swing equation is a second-order differential equation that can be solved by numerical methods or by using the equal area criterion. The equal area criterion simplifies the swing equation by assuming that the mechanical power is constant and that the electrical power is piecewise linear.

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3 Equal area construction

The equal area construction is a graphical representation of the swing equation. It plots the electrical power output as a function of the rotor angle difference. The equal area construction has three regions: before the fault, during the fault, and after the fault. The equal area criterion states that the system is stable if the area under the electrical power curve before and after the fault is equal to the area above the curve during the fault. This means that the net change in kinetic energy of the rotors is zero.

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Shair Hassan Barech

Transmission Planning, Forecasting Engineer at Quetta Electric Supply Company
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Power System Stability analysis is carried out in PSSE software. for a specific case. The training is carried by US-AID Pakistan , under Power Sector Improve Activity PSIA at Lahore.

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4 Critical clearing angle

The critical clearing angle is the maximum value of the rotor angle difference that still allows the system to be stable. It is the point where the electrical power curve before and after the fault intersects. The critical clearing angle can be found by drawing a horizontal line from the mechanical power level to the electrical power curve and finding the points of intersection. The smaller of the two angles is the critical clearing angle.

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5 Critical clearing time

The critical clearing time is the maximum duration of the fault that still allows the system to be stable. It is the time required for the rotor angle difference to reach the critical clearing angle. The critical clearing time can be found by solving the swing equation for the time variable, using the initial conditions and the critical clearing angle. Alternatively, it can be estimated by using a geometric approximation, such as the tangent method or the extended equal area criterion.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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Marina Zvytsevich

Senior Power Protection Advisor at HITEC
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It's important to also consider the impact of system inertia and damping on stability. Higher system inertia can help absorb disturbances and improve stability margins, while damping helps to mitigate oscillations. Additionally, conducting sensitivity analysis on fault location and severity can provide deeper insights into system robustness. Finally, integrating real-time monitoring and advanced control strategies can enhance fault detection and improve response times, further ensuring system stability.

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How do you apply the equal area criterion to determine the critical clearing angle and time of a fault?

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1 Fault model

2 Swing equation

3 Equal area construction

4 Critical clearing angle

5 Critical clearing time

6 Here’s what else to consider

Power Systems

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