By **Robert D. Castro**, University of Southern California

Minimizing permanent faults’ negative effects on power system stability can significantly increase the system’s power transfer capability. One way to minimize negative effects is to calculate the affected breakers’ optimal reclosing time. Following is a discussion on using optimal reclosure on permanent faults to increase transient stability. It begins by reviewing the swing equation to introduce the concept of the load angle limit. It then presents the transient energy function through a single machine illustration of these concepts. Finally, it discusses a multi-machine scenario.

## The Swing Equation and Load Angle

Suppose that a synchronous machine is generating in steady state. In this case, the electrical power delivered by the machine, PE, is equal to the mechanical power delivered by the prime mover of the machine, PM. If a fault occurs at time t = 0, the circuit breakers would open the circuit and the fault would typically be cleared in 10 cycles or less. Since the electrical time constant is a few milliseconds, PE would soon become 0.

Because the time constant for the turbine mass system is a few seconds (as opposed to milliseconds for PE ), the inertia of the mechanical system would continue to generate mechanical power, PM. With the load, PE, disconnected, PM would accelerate the machine, increasing the load angle àŽ´. The following swing equation depicts the idea that power differences between PE and PM would cause changes in the load angle and this change in load angle would cause acceleration and deceleration of the machine rotor.

- Where,
- H = inertia constant of the generator
- àâ€°
_{S}= synchronous speed of the machine - àŽ´ = load angle
- P
_{M}= mechanical power input to the generator via the prime mover - P
_{E}= electrical power output of the generator - t = time from the fault

If the breaker is reclosed at a load angle àŽ´_{C}, the electrical load, P_{E}, would be back online (not zero like it was when the breakers were open) and, because this electrical power is now more than the mechanical power, the machine would start to decelerate. Due to the mechanical inertia of the rotor masses, however, the load angle would continue to increase and the rotor would continue to accelerate. àŽ´_{Limit} represents the load angle reached when the rotor stops accelerating.

The swing equation shows that as àŽ´ increases, P_{M} increases, and if allowed to continue, P_{M} would reach a point where its inertia can’t be sufficiently slowed down by P_{E} when the load is brought back online. In that case, the generator would lose synchronism with the system and be removed from the system. In other words, if the load angle is allowed to increase to some point àŽ´Max, the mechanical power of the rotor would becomes greater than the electrical load power. The generator would, therefore, accelerate before slowing down completely and the generator circuit breaker would isolate it from the system. So, as long as àŽ´Limit is less than àŽ´Max, the generator would remain synchronized with the system.

This concept is captured by the equal area criterion, which, as depicted in Figure 1, illustrates that if the decelerating area, A2, is greater than the accelerating area, A1, the machine would decelerate completely before starting to accelerate again, achieving stable operation to remain synchronized.

The key point being that the breaker closure point àŽ´_{C}, must occur before the critical clearing angle, àŽ´_{CR}, or A1 will become greater than A2. This discussion introduces some relevant concepts, however, in reality modern governor controls preclude the worry of meeting the critical clearing angle before the generator needs to be removed from the system.

## The Single Machine Illustration

Consider the simple system in Figure 2, assuming generator No. 2 represents an infinite busbar.

Suppose generator No. 3 experiences a permanent fault causing the breaker at No. 3 to open. Typically, the generator breaker would remain open until crews inspect it. Reclosing the generator breaker, even though it is closing on the permanent fault and would certainly reopen in four to six cycles, could assist the system’s stability by reducing the magnitude of system oscillations introduced by the fault. This is because the oscillations produced by the fault add to the load angle, àŽ´, making the total load angle seen by the generator closer to the critical clearing angle, àŽ´CR. If the oscillations are large enough, they might be sufficient to cause àŽ´ to exceed àŽ´CR, causing instability. Studies have demonstrated that if reclosing is timed correctly, generator swing angle oscillations are decreased enough that the connected line’s transmission capacity could be increased 20 percent.

The idea is to minimize the system’s oscillations introduced by the fault by minimizing the fault-caused transient energy. Reclosing the breaker at the optimal time onto the permanent fault would lower the transient energy function to some minimal level available on the system under the existing operating conditions.

The transient energy function after a fault for a single machine infinite busbar system is given as:

- Where,
- V = transient energy introduced by the machine into the system
- H = inertia constant of the generator
- àâ€° = speed of the machine
- àŽ´
_{S}= stable equilibrium point of the load angle - P
_{M}= mechanical power - P
_{E}= electrical power - t = time from the fault

If the potential energy terms are maximized by evaluating at the critical load angle, the kinetic energy term can be eliminated and the transient energy function can be evaluated as:

Most utilities could not implement this scheme because closing a generation breaker on a permanent three-phase fault could result in serious generator shaft fatigue, making it necessary to replace the shaft. However, this methodology could be applied to larger multi-machine systems.

## Multi-machine Scenario

In the multi-machine implementation, the optimally timed reclosure of a permanently faulted system would still increase the area transmission lines’ capacity by improving the system stability. In this case, instead of reclosing a generator relay, a transmission subsystem relay would be reclosed. This would provide the same effect to stability as outlined in the single machine model without potentially inducing fatal fatigue on generator shafts.

System operating conditions affect the optimal reclosing time, which is calculated by the multi-machine Lyapunov function. Real time implementation of optimal reclosure time involving Lyapunov function calculation is beyond even the most microprocessor-laden relay system’s capability. While implementing this methodology is beyond existing protection technology, other off-line computations are available to roughly approximate optimal reclosing time.

Most utilities’ generators do not have automatic reclosure capabilities nor do their transmission lines. They prefer to have operators reclose the lines manually. This tactic would preclude taking advantage of the optimal reclosure scheme described here, resulting in lower limits in transmission capacity.

Robert Castro teaches graduate courses in power at the University of Southern California and develops wind generation for a local utility. Reach him at robert.castro@alumni.usc.edu.

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