Draft:Equal area criterion

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The equal area criterion is a mathematical relation within engineering of a power system that definies the limits of stability of changes in the power of a synchronous machine . The criterion is useful for defining the critical limits of stability of AC power system.

It is based on mathematical function on how the electric power, P_el depends on the load angle, δ, by P_el=P_maxsin(δ). With the load angle, δ as the angle between the stator reference frame and the rotor reference frame. As described by the Swing equation. The areas in the equal area criterion refer to acceration caused by the difference between a mechanical power P_mech and the electrical system given by P_el=P_maxsin(δ).

Sudden change of load

Equal area criterion of a synchronous machine of an electrical power system

Consider a synchroncous machine described by the intial mechanical power P_mech0 and the electrical system P_el=P_maxsin(δ). A sudden change of the mechanical power from P_mech0 to P_mech1 will move the new stable opertion point from point (0) at δ₀ to point (1) at δ_cr. This does not happend at an instant, and the load angle will accelerate and move from point (0) to point (1). But the angle δ does not stop at point (1), the load angle will continue further beyond point (1) and deaccelerate to point (2) at δ_max. The system will then swing between point (0) and point (2), around point (1). Damping will decrease the occilation and move it closer to the final point (1).

Any movement further than the maxmimum angle δ_max, would lead to instability. Stability is definied by the equal area criterion. That the area (A1) during the acceleration of the load angle that store kinetic energy

$A_1=\underset{{\delta }_0}{\overset{{\delta }_{cr}}{\int }}(P_{mech1}-P_{el})d\delta .$

should be equal to the area (A2) of the already stored kinetic area.

$A_2=\underset{{\delta }_{cr}}{\overset{{\delta }_{max}}{\int }}(P_{el}-P_{mech1})d\delta$

Meaning that both areas are equal as

$A_1=A_2$

Fault and clearance of fault

Equal area criterion of a synchronous machine of an electrical power system, during fault and post fault

A common application of the criterion is for stability of fault clearance. Consider an electrical system described by the curve P_pre=P_max1sin(δ), and the mechanical power P_mech. The operation point of the system (before the fault) is by point (0) where P_pre=P_mech.

The first area (A1) is defined by the acceleration during the fault (defined by the fault curve, P_fault=P_max2sin(δ)), from point (0) to point (1), by

$A_1=\underset{{\delta }_0}{\overset{{\delta }_{cr}}{\int }}(P_{mech}-P_{fault})d\delta .$

this area should be equal to the area (A2) after the fault is cleared (defined by the post-fault curve P_post=P_max3sin(δ)), from point (1) to point (2), by

$A_2=\underset{{\delta }_{cr}}{\overset{{\delta }_{max}}{\int }}(P_{post}-P_{mech})d\delta$

Meaning that both areas are equal as

$A_1=A_2$

usually solved by finding the critical angle δ_cr when the fault must be cleared (or a the corresponding time t_cr of this angle).

See also

• Swing equation
• Electric power system