Homework #3: Generator and Load Modeling

Consider the following equivalent circuits for a384 MVA, 24kV, 0.85power factor, 60Hz, 3 phase, 2 pole synchronous generator.

It has the following parameters:

Ll=0.193pu

Lad=1.6049pu

Laq=1.5849pu

Ra=0.0014 pu

Lfd=0.7017 pu

Rfd=0.0012 pu

L1d= 0.0845 pu

R1d= 0.0222 pu

L1q= 0.401 pu

R1q= 0.00523 pu

L2q= 0.0641 pu

R2q= 0.01431 pu

Stored energy at rated speed =1006.5 MWs

Damping coefficient KD=0

  1. Calculate unsaturated transient and subtransient reactance parameters in per unit values and open-circuit time constantsusing the formulas in Slide #37

X’d, X”d, X’q, X”q, T’d0, T”d0, T’q0, T”q0

  1. With the armature terminal voltage at rated value, consider two operating conditions with the following steady-state generator outputs

Output 1: Pt= 307MWQt= 115MVAr

Output 2: Pt=345MWQt= -154 MVAr

For each of the two conditions,

i)Compute internal rotor angle i, and per unit values of

Eq, ed, eq, id, iq, i1d, i1q, i2q, ifd, efd, fd, 1d, 1q, 2q,

ii)Draw the following steady-state phasor diagram based on the calculated quantities.

iii)Calculate steady-state air-gap torque Te in per unit and Nm. How much is Tmin per unit and Nm?

  1. Assume that the generator is currently operated with the first steady-state output (Pt=307MW and Qt=115MVAr). Ignoring the saliency of the generator, i.e. let Xq=Xd, X’q=X’d and X”q=X”d,

i)Use the calculated terminal quantities (e.g. terminal voltage and current) to estimate the magnitudes of E” for the Voltage behind X” model and E’ for the classic model(slides #49-50). Comment on the comparison of Eq, E’ and E” in magnitude.

ii)Calculate inertia Hin s

iii)Considering the classic model of the generator and assuming thatthe external network seen fromthe generator can be regardedasan equivalent load bus connected by reactance Xt=0.1 pu (representing the transmission), as shown by the figure below, calculate the per unit voltage magnitude Vt, real power Pand reactive power Qat the load bus.

If the load of the bus under the current condition can be described by a frequency dependent exponential load model:

P=P0(Vt/Vt0)0.9×[1+1.2×(f-f0)/f0]

Q=Q0(Vt/Vt0)2.0×[1-1.5×(f-f0)/f0]

whereP0and Q0andVt0take the values of Vt, P and Qcalculated aboveand let f0=60Hz. If at two time points,t1 and t2, real-time measurements of f and Vt are actually

att1f=59.85Hz Vt=0.985 pu

att2f=60.01Hz Vt=0.972 pu

Assume Tm does not change under this condition. Calculate dr/dt in rad/s2 at t1 and t2.