Section III: Equal Area Criterion

The real power transmitted over a lossless line is given by (9.4). Now 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 circuit breakers take about 5/6 cycles to open and the subsequent post-fault transient last for another few cycles. The input power, on the other hand, is supplied by a prime mover that is usually driven by a steam turbine. The time constant of the turbine mass system is of the order of few seconds, while the electrical system time constant is in milliseconds. Therefore, for all practical purpose, the mechanical power is remains constant during this period when the electrical transients occur. 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 Fig. 9.3. Suppose the system of Fig. 9.1 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 power P_a becomes 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 re-closes 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.

Note that

Fig. 9.3 Power-angle curve for equal area criterion.

Hence multiplying both sides of (9.14) by  and rearranging we get

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

Now suppose the generator is at rest at δ₀. We then have dδ / 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 at δ_c , at which point we again have dδ / dt = 0. Thus the area of accelerating is given from (9.15) as

In a similar way, we can define the area of deceleration. In Fig. 9.3, the area of acceleration is given by A₁ while the area of deceleration is given by A₂ . This is given by

Contd...