Power Angle Equation of Synchronous Machine

Equation for the Power Angle of a Synchronous Machine Certain simplifications are typically assumed when solving the swing equation (Eq. (12.10)). Which are:

1. The machine’s mechanical power input (Pm) remains constant during the interesting electromechanical transient. In other words, the effect of the turbine governing loop being much slower than the transient speed is ignored. This presumption produces a pessimistic result—the governing loop aids in system stabilisation.
2. Rotor speed variations are negligible; the swing equation has already taken them into account.
3. The effect of the voltage regulating loop is disregarded during the transient, resulting in a constant machine emf. The results of this supposition are also pessimistic because the voltage regulator aids in system stabilisation.

It is necessary to establish the dependence of the electrical power output (Pe) upon the rotor angle before the swing equation can be solved.

Simplified Machine Model

The steady-state per phase induced emf-terminal voltage equation for a non-salient pole machine is

where

and usual symbols are used.

Under transient condition

but

Equation (12.19) during the transient modifies to

The phasor diagram corresponding to Eqs. (12.21) and (12.22) is drawn in Fig. 12.2.

Since under transient condition, X′_d < X_d but X_q remains almost unaffected, it is fairly valid to assume that

Equation (12.22) now becomes

The Power Angle Equation of Synchronous Machine model corresponding to Eq. (12.24) is drawn in Fig. 12.3 which also applies to a cylindrical rotor machine where X′_d = X′_q = X′_s(transient synchronous reactance)

The simplified Power Angle Equation of Synchronous Machine of Fig. 12.3 will be used in all stability studies.

Power Angle Curve of Synchronous Machine

While V, the terminal voltage determined by the generator, is a dependent variable for stability studies, the independent variable |E′|, transient emf of the generator motor, remains constant or is determined by the voltage regulating loop. As a result, as shown in Fig. 12.4, the nodes (buses) of the stability study network relate to the machine’s emf terminal, while the machine’s reactance (X′d) is absorbed in the system network, which is different from a load flow study. In addition, equivalent static admittances will be connected in shunt between the reference bus and the transmission network buses to replace the loads (aside from large synchronous motors). This is because during a stability study, load voltages change. (in a load flow study, these remain constant within a narrow band).

For the 2-bus system of Fig. 12.5

Complex power into bus is given by

At bus 1

But

Since in solution of the swing equation only real power is involved, we have from Eq. (12.26)

A similar equation will hold at bus 2.

Let

and

Then Eq. (12.27) can be written as

For a purely reactive network

Hence

where

where

• X = transfer reactance between nodes (i.e., between E′₁ and E′₂)

The graphical plot of power angle equation (Eq.(12.29)) is shown in Fig. 12.6.

The swing equation (Eq. (12.10)) can now be written as

which, as already mentioned, is a nonlinear, damping-free second-order differential equation.