ECE 5531 Winter 2020
Exam # 2 Due Date: March 19, 2020
Open textbook, open notes, use of Matlab is allowed. Show all work; no credit
will be given for correct answers without supporting calculations. The use of
Matlab is allowed and encouraged for this exam. Work neatly; if the work is
illegible, it will earn no points. This exam is due back to me in pdf format by
9:00 am EDT on March 20.
Questions 1 and 2 are based on the 6 bus network shown below.
It is a DC network model.The base MVA for the network is 100 MVA.
The network branch data are:

Branch From Bus To Bus Reactance (p.u.) MW Limit (MW)
1 1 2 0.25 100
2 1 3 0.25 100
3 1 6 0.5 100
3 2 4 0.5 100
5 3 5 0.5 100
6 4 6 0.5 100
7 5 6 0.25 100

The load data are:

Bus Load MW
1 0
2 200
3 50
4 200
5 150
6 0

The generator cost functions and MW limits are:
c1 (pg1) = 200 + 50pg1 $/hour 100 pg1 300 MW
c2 (pg2) = 100 + 30pg2 $/hour 100 pg2 300 MW
c3 (pg2) = 100 + 60pg3 $/hour 100 pg2 400 MW
1) Calculate the minimum generation cost OPF solution and identify any lines
that at their line limits.
a) Calculate the MW outputs of G1, G2 and G3. Which generator(s) are at an
operating limit? Which lines are at a line limit?
b) State the n+1 rule for minimum generation cost OPF solutions. Is this OPF
solution consistent with the n+1 rule.
c) Calculate the DC generation shift sensitivity matrix.
DC = 0 Ingv BFb0 A0gv 1 I0nb
which relates incremental generator shifts to incremental bus load shifts.
d) Using the DC generation shift sensitivity matrix, DC, and the generator
costs, calculate the LMP for each bus in the network.
2) This problem deals with state estimation, the network model for this
problem is the same as the DC network model used in problem 1. The measurement data are

Measurement Measurement value Measurement
p12 0.5415 0:005
p13 0.1296 0:005
p16 -0.6611 0:005
p24 -0.3784 0:005
p56 -0.8624 0:005
p65 0.8764 0:005
p3 -0.4800 0:01
p4 -2..700 0:01

a) Calculate the measurement Jacobian matrix H and the gain matrix G =
(b) Calculate xb, the WLS estimate of x = 2 3 4 5 6 T , given the
measurement values and measurement ’s listed in the table above.
(c) Calculate the weighted sum of squares of measurement residuals
J (xb) =
mX i
zi i zbi 2
where m is the number of measurements and zb = Hxb.
(d) Calculate normalized residuals for this measurement set. What, if anything,
do the normalized residuals tell you about the presence or absence of critical
measurements and/or critical pairs of measurements?
(e) Eliminate the measurement with the largest normalized residual and recalculate the state estimate and the normalized residuals. Do you suspect that
there are some bad data among the remaining 7 measurements? How did you
come to your conclusion?

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