Sohail Ansari

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Swing Equation

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Let us consider a three-phase synchronous alternator that is driven by a prime mover. The equation of motion of the machine rotor is given by

 1

where

   J is the total moment of inertia of the rotor mass in kgm2
  Tm
is the mechanical torque supplied by the prime mover in N-m
  Te is the electrical torque output of the alternator in N-m
  θ is the angular position of the rotor in rad

Neglecting the losses, the difference between the mechanical and electrical torque gives the net accelerating torque Ta . In the steady state, the electrical torque is equal to the mechanical torque, and hence the accelerating power will be zero. During this period the rotor will move at synchronous speed ωs in rad/s.The angular position θ is measured with a stationary reference frame. To represent it with respect to the synchronously rotating frame, we define

 2

where δ is the angular position in rad with respect to the synchronously rotating reference frame.

Defining the angular speed of the rotor as we can write as

 3

 Differentiating the equation 2 w.r.t. time & using the equation 3, we can write :

We can therefore conclude that the rotor angular speed is equal to the synchronous speed only when dδ / dt is equal to zero. We can therefore term dδ / dt as the error in speed. Taking derivative of above equation , we can then rewrite as

Multiplying both side by ωm we get

 4

where Pm Pe and Pa respectively are the mechanical, electrical and accelerating power in MW.

We now define a normalized inertia constant as

Substituting the value of H in above equation 4, we get

In steady state, the machine angular speed is equal to the synchronous speed and hence we can replace ωr in the above equation by ωs. Note that  Pm Pe and Pa are given in MW. Therefore dividing them by the generator MVA rating Srated we can get these quantities in per unit. Hence dividing both sides  by Srated in above equation we get

  per unit

The above equation is known as Swing Equation in P.U.

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