Homework #3
(ECE552 students need to finish all questions; ECE422 students who finish the highlighted questions that
are required for ECE552 only will receive up to 10% more points.)
Consider the following equivalent circuits for a 384 MVA, 24kV, 0.85 power factor, 60Hz, 3 phase, 2 pole
synchronous generator.
It has the following parameters:
laa
= 4.743+0.0265 cos(2) mH
lab
= -2.371-0.0265 cos(2+/3) mH
laF (lafd) =25.0 cos mH
LF (Lffd) =211 mH
Ra
=0.0021 
RF (Rfd) =0.04 
Ll
=0.768 mH
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
KD=0
1. Using the machine rated values as the base values for the stator quantities and ignoring magnetic
saturation, determine the following parameters in H.
Lad, Laq, Ld, Lq, MF(i.e. Lafd in Kundur’s book), LF, Lfd
2. Determine the per unit values of the following in the Lad base reciprocal per unit system (assuming
Lad=MF=MR in per unit).
Ll, Lad, Laq, Ld, Lq, MF, LF, Lfd, Ra and Rfd
3. Calculate unsaturated transient and subtransient reactance parameters in per unit values and open-circuit
time constants
X’d, X”d, X’q, X”q, T’d0, T”d0, T’q0, T”q0
4. (Not required for ECE422 students) Assuming the following open-circuit saturation curve for both d- and
q-axis saturation characteristics, draw the saturation curve (refer to Kundur’s Figure 3.30).
Asat= 0.03125
Bsat=6.931
T1= 0.8 pu
T2= 1.0 pu
Lratio= 1.5
5. With the armature terminal voltage at rated value, for each of the following steady-state generator outputs
Output 1:
Pt= 307 MW Qt= 115 MVAr
Output 2:
Pt= 345 MW Qt= -154 MVAr
a. Calculate the following generator parameters
i)
Compute factor Ksd (ECE422 students may use Ksd =0.87 and 0.92, respectively for
Outputs 1 and 2 if you don’t want to answer Question 4 and calculate Ksd), internal rotor
angle i, and per unit values of
at, Eq, ed, eq, id, iq, i1d, i1q, i2q, ifd, efd, fd, 1d, 1q, 2q,
ii)
Draw the following steady-state phasor diagram.
iii)
Calculate accurate transient and subtransient reactance parameters in per unit values and
open-circuit time constants (considering magnetic saturation)
X’d, X”d, X’q, X”q, T’d0, T”d0, T’q0, T”q0
iv)
Calculate steady-state air-gap torque Te in per unit and Nm. How much is Tm in per unit
and Nm?
b. In a table, for the above two generator outputs, list Pt, Qt, at, Ksd, i, ifd, Xd, Xq, X’d, X”d, X’q, X”q,
T’d0, T”d0, T’q0 and T”q0 all in per unit values, and comment on the effects of real and reactive power
outputs on those variables or parameters
c. 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. Xq=Xd, X’q=X’d and X”q and X”d,
i)
Estimate the magnitudes of E” for the Voltage behind X” model and E’ for the classic model.
Comment on the comparison of Eq, E’ and E” in magnitude.
ii)
Calculate inertia H in s and provide the state-space form and the block diagram representation
of swing equations with only r,  and Te unknown (refer to Kundur’s section 3.9.5)
iii)
Considering the classic model of the generator and assuming that the external network seen
from the generator can be regarded as an equivalent load bus connected by reactance Xt =0.1
pu, as shown by the figure below, calculate the per unit voltage magnitude Vt and real power
P at 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]
where P0 and Q0 and Vt0 take the values calculated above and let f0=60Hz. If at two time
points, t1 and t2, real-time measurements of f and Vt are actually
at t1
f=59.85Hz
Vt=0.985 pu
at t2
f=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.