Swing Equation of a Single Machine Infinite Bus System

Transient Stability Analysis:

Assumptions:

1. P_m= Constant. i.e. Actions of the turbine governor is neglected.
2. P_e= Constant. i.e. Actions of AVR is neglected.
3. The disturbance is Supposed to be so large that linearization of the system is impossible.
4. Effect of damper winding & effect of saliency (non-smoothness) of the rotor pole is neglected.
5. The resistance of the system is normally neglected i.e. assumptions of the lossless system.

Consider the single-machine-infinite-bus system shown in Fig. below. In this the reactanceXincludes the reactance of the transmission line and the synchronous reactance or the transient reactance of the generator. The sending end voltage is then the internal emf of the generator. Let the sending and receiving end voltages be given by:

from the above system, we have,

whereP_max=V₁V₂/Xis the maximum power that can be transmitted over the transmission line. The power angle curve is shown in Fig. below

The sending end real power and reactive power are then given by

or,

Since the line is loss less, the real power dispatched from the sending end is equal to the real power received at the receiving end. We can therefore write;

whereP_max=V₁V₂/Xis the maximum power that can be transmitted over the transmission line. The power angle curve is shown in Fig. below

From this figure, we can see that for a given power P0 . There are two possible values of the angle

δ - δ0 and δmax .

Acceleration or retardation of rotor synchronous machine is described by an equation known as swing equation. The equation governing rotor motion of a synchronous machine is based on the elementary principle i n dynamics which states that accelerating torque is the product of the moment of inertia of the r o t o r times i t s angular acceleration....(1)

Where,

J= the total moment of inertia of t h e rotor masses

θm = the angular displacement of the rotor with respect to a stationary axis

t = time, i n seconds

Tm= the mechanical or shaft torque supplied by the prime mover less retarding torque due to rotational losses, i n N-m

Te = the net electrical or electromagnetic torque, in N-m

Ta = the net accelerating torque, i n N-m

The rotor will accelerate if Tm is greater than Te and decelerates if Te is greater than Tm but the rotation is always in the direction of input mechanical torque.

Let us suppose Tm is greater than Te due to some disturbance in the system then the rotor accelerates. The phase diagram for this case is shown in fig below.

From the phasor diagram,.......(2)

Where θm is angular position of the rotor

ωsm is the synchronous speed of the machine

ωm is actual speed of the motor

δm is the angular displacement of the rotor

The derivatives of Eq. (2) with respect to time :.......3

Again differentiation with respect to time :

...........4

From Eq. (1) and (4)

.........5

Multiplying by ωm on both side.......6

Where, Pa =Ta ωm = acceleration power

Pm =Tm ωm = mechanical input power.

Pe =Te ωm = electrical output power........7

Where, M= Jωm = angular momentum

I n machine data supplied for stability studies another constant related to inertia is often encountered. This is the so-called H constant , which is defined by

where Smach is the three-phase rating of the machine

Solving for M:

And substituting in eq.(7)

We have ωsm =2πf , so the equation becomes:

So this is the swing equation in term of inertia constant.

If the equation is solved for δ , obtain a solution in terms of ‘t’ and constants which can be interpreted to check whether the system is stable or not.