Equal area Criterion, Factors Affecting Transient Stability and Stabiity Enhancement Techniques.

Equal area Criterion:

consider the situation in which the synchronous machine is operating in steady state delivering a power P_e equal to P_m when there is a fault occurs in the system. Opening up of the circuit breakers in the faulted section subsequently clears the fault.The transient stability study, therefore, concentrates on the ability of the power system to recover from the fault and deliver the constant power P_m with a possible new load angle δ.

Consider the power angle curve shown in above. Suppose the system is operating in the steady state delivering a power of P_m at an angle of δ₀ when due to malfunction of the line, circuit breakers open reducing the real power transferred to zero. Since P_m remains constant, the accelerating powerP_abecomes equal to P_m. The difference in the power gives rise to the rate of change of stored kinetic energy in the rotor masses. Thus, the rotor will accelerate under the constant influence of non-zero accelerating power and hence the load angle will increase. Now suppose the circuit breaker recloses at an angle δ_c. The power will then revert back to the normal operating curve. At that point, the electrical power will be more than the mechanical power and the accelerating power will be negative. This will cause the machine decelerate. However, due to the inertia of the rotor masses, the load angle will still keep on increasing. The increase in this angle may eventually stop and the rotor may start decelerating, otherwise, the system will lose synchronism. This criterion helps to check the transient stability of the system after being subjected to large disturbance for the short time.

We have,

Multiplying both sides by

We get,

Multiplying both sides of the above equation by dt and then integrating between two arbitrary angles δ0 and δc we get,

suppose the generator is at rest atδ₀. We then havedδ/dt= 0. Once a fault occurs, the machine starts accelerating. Once the fault is cleared, the machine keeps on accelerating before it reaches its peak atδ_c, at which point we again havedδ/dt= 0. Thus, the area of accelerating is given as

In a similar way, we can define the area of deceleration. The area of acceleration is given byA₁while the area of deceleration is given byA₂. This is given by

Now consider the case when the line is reclosed atδ_csuch that the area of acceleration is larger than the area of deceleration, i.e.,A₁>A₂. The generator load angle will then cross the pointδ_m, beyond which the electrical power will be less than the mechanical power forcing the accelerating power to be positive. The generator will, therefore, start accelerating before is slows down completely and will eventually become unstable. If, on the other hand,A₁<A₂, i.e., the decelerating area is larger than the accelerating area, the machine will decelerate completely before accelerating again. The rotor inertia will force the subsequent acceleration and deceleration areas to be smaller than the first ones and the machine will eventually attain the steady state. If the two areas are equal, i.e.,A₁=A₂, then the accelerating area is equal to the decelerating area and this is defined the boundary of the stability limit.The clearing angleδ_cfor this mode is called theCritical Clearing Angle and is denoted byδ_cr. We then get from Fig. by substitutingδ_c=δ_cr

We can calculate the critical clearing angle from the above equation. Since the critical clearing angle depends on the equality of the areas, this is called the equal area criterion.

Critical Clearing Angle:

Since we are interested in finding out the maximum time that the circuit breakers may take for opening, we should be more concerned about the critical clearing time rather than clearing angle. Furthermore, notice that the clearing angle is independent of the generalized inertia constantH. Hence, we can comment that the critical clearing angle, in this case, is true for any generator that has a d-axis transient reactance of 0.20 per unit. The critical clearing time, however, is dependent onHand will vary as this parameter varies.

To obtain a description for the critical clearing time, let us consider the period during which the fault occurs. We then haveP_e= 0. We can, therefore, write from

Integrating the above equation with the initial acceleration being zero we get

Further integration will lead to

Replacing δ by δcr and t by tcr in the above equation, we get the critical clearing time as

Factors affecting transient stability:

Transient stability is very much affected by the type of the fault. A three phase dead short circuit is the most severe fault; the fault severity decreasing with two-phase fault and single line-to-ground fault in that order.
If the fault is farther from the generator the severity will be less than in the case of a fault occurring at the terminals of the generator.Power transferred during fault also plays a major role. When part of the power generated is transferred to the load, the accelerating power is reduced to that extent.Theoretically, an increase in the value of inertia constant M reduces the angle through which the rotor swings farther during a fault. However, this is not a practical proposition since, increasing M means, increasing the dimensions of the machine, which is uneconomical.The dimensions of the machine are determined by the output desired from the machine and stability cannot be the criterion. Also, increasing M may interfere with speed governing system.

Some of the factors are described below:

Initial operating condition:

If the initial operating angle is less, it may handle the disturbance in the system and hence the chance of stability is more.

Fault clearance time:

If a fault or any disturbance is cleared fast than the chance of stability will be high. So fault clearance time affects transient stability.

Type of fault and its location:

Transient stability depends on its type ie. If the disturbance is severe the chance of stability is less. The total equivalent impedance of the system depends on the fault location so do the current. So transient stability depends on the location.

Post fault condition of the system:

If the fault is cleared before the operation the chance of stability is more. After the operation the fault is severe and hence chance of stability is less.

Inertia of system:

If the inertia is large the fluctuation of the rotor is minimized. So the increase in inertia increases the chance of stability even though the energy increases due to high inertia.

Stability enhancement techniques:

The possible methods that may improve the transient stability are:

(i) Increase of system voltages, and use of automatic voltage regulators.
(ii) Use of quick response excitation systems
(iii) Compensation for transfer reactance X_I2so that P_eincreases and P_m- P_e= P_areduces.
(iv) Use of high-speed circuit breakers which reduce the fault duration time and hence the accelerating power.

When faults occur, the system voltage drops. Support to the system voltages by automatic voltage controllers and fast acting excitation systems will improve the power transfer during the fault and reduce the rotor swing.

Reduction in transfer reactance is possible only when parallel lines are used in place of a single line or by use of bundle conductors. Other theoretical methods such as reducing the spacing between the conductors and increasing the size of the conductors are not practicable and are uneconomical.

Quick opening of circuit breakers and single pole reclosing is helpful. Since the majority of the faults are a line to ground faults selective single pole opening and reclosing will ensure transfer of power during the fault and improve stability.