Q2Power System 2
Question
Q.2. Apply equal area criterion method for transient stability analysis:
- (i) When mechanical power input is increased suddenly.
- (ii) When three-phase fault occurs at the sending end of the line.
Answer
The equal area criterion, applied to a sudden mechanical power increase, requires the accelerating area (between the new higher mechanical power line and the power-angle curve, from the old to the new equilibrium angle) to be matched by an equal decelerating area beyond the new equilibrium, up to the maximum swing angle; for a three-phase fault at the sending end (reducing transmitted electrical power to zero during the fault), the accelerating area builds up during the fault-on period at constant mechanical power against zero electrical output, and stability requires sufficient post-fault decelerating area to be available before the rotor angle reaches the critical clearing angle.
(i) Equal Area Criterion for Sudden Increase in Mechanical Power Input
When the mechanical power input to a generator connected to an infinite bus is suddenly increased from an initial value Pm0 to a new value Pm1 (with the power-angle curve Pe=Pmax*sin(delta) itself unchanged), the initial equilibrium angle delta0 (where Pm0=Pmax sin(delta0)) is no longer an equilibrium point, since Pm1 now exceeds Pe at delta0, causing the rotor to accelerate and delta to increase. The rotor swings forward until it reaches the new equilibrium angle delta1 (where Pm1=Pmax sin(delta1)), but due to the kinetic energy gained during this acceleration, it does not stop there — it continues to swing further, decelerating (since beyond delta1, Pe now exceeds Pm1) until all the kinetic energy gained during the initial acceleration has been given back, at some maximum angle delta_max.
The equal area criterion states that the accelerating area A1 (the area between the constant Pm1 line and the power-angle curve, from delta0 to delta1, representing the net kinetic energy gained during acceleration) must be exactly equal to the decelerating area A2 (the area between the power-angle curve and the Pm1 line, from delta1 to delta_max, representing the kinetic energy given up during deceleration) at the point where the rotor's angular velocity returns to zero (i.e., at delta_max, the extreme swing angle):
For the system to remain stable, sufficient decelerating area must be available beyond delta1 (up to the point where the power curve intersects Pm1 again, symmetric to delta1 about 90 degrees) to fully absorb the accelerating area A1 — this is precisely the calculation carried out numerically in the corresponding droop/stability numerical problem elsewhere in this paper, where a sudden 30 MW mechanical power increase was shown to satisfy A1 <= A2,max with a substantial margin, confirming stability.
(ii) Equal Area Criterion for Three-Phase Fault at the Sending End
For a three-phase (symmetrical) fault occurring directly at the sending end of a transmission line connecting the generator to the infinite bus, the electrical power transferred during the fault is reduced to zero (since a three-phase fault at the sending-end bus effectively short-circuits the generator's terminals, meaning no real power at all can be transmitted to the infinite bus through the faulted connection during the fault-on period, assuming the fault is a solid, zero-impedance short circuit right at the generator terminals).
During the fault-on period (from the initial equilibrium angle delta0 until the fault is cleared at the clearing angle delta_cr), the generator continues to receive its full mechanical input power Pm (essentially unchanged over this short interval, since turbine governor response is much slower than the electrical transient), but delivers zero electrical output power (Pe=0), so the entire mechanical input power accelerates the rotor throughout the fault duration, and the accelerating area is:
which is simply a rectangular area (since Pe=0 throughout, the integrand is the constant value Pm), directly proportional to the fault-clearing angle (and hence, indirectly, to the fault clearing time) — this is precisely why fast fault clearing (via high-speed protective relaying and circuit breakers) is so critical to transient stability, since it directly limits this accelerating area by limiting delta_cr. Once the fault is cleared (typically by isolating the faulted line/section via circuit breaker operation), the power-angle relationship is restored to its post-fault characteristic (which may differ somewhat from the pre-fault curve if the fault clearing involves permanently removing a line, changing the effective transfer reactance), and the rotor now decelerates (since Pe once again exceeds Pm, for delta beyond the new equilibrium angle) as it continues swinging forward, with the decelerating area A2 available from delta_cr up to the maximum swing angle delta_max (where the power curve again intersects Pm) needing to be at least as large as A1 for the system to remain stable:
The critical clearing angle, delta_cr, is defined as the maximum fault-clearing angle for which A1 still exactly equals the maximum available A2 (i.e., the boundary/limiting case of stability) — if the fault is cleared at any angle less than delta_cr, the system remains stable with a positive stability margin (A2,max > A1), while if fault clearing is delayed beyond delta_cr, the required accelerating area A1 would exceed the maximum available decelerating area, and the system would lose synchronism. This sending-end-fault analysis, and the resulting concept of critical clearing angle/time, is one of the most important practical applications of the equal area criterion, directly informing the maximum permissible protective relay and circuit breaker operating time required to maintain transient stability for a given fault location and power system operating condition.