Homework, Solutions and Reviews
1 - PV curves and continuation power flow
Due Jan. 27 5
PM - New due date
2 - More on PV curves and convergence properties of load flow
Due Feb. 8 5
PM - New due date
- For the 5-bus system in this diagram, find
the Ybus matrix. Then assume a flat start
(i.e., all the bus angles are all zero, the generator voltage
magnitudes as given and all load bus voltage magnitudes as 1.0 p.u. ),
to calculate initial bus mismatches (both real and reactive power).
- Continue with this 5-bus system to find the load flow solution using:
(a) Fast Decoupled Newton-Raphson and (b) Full Newton-Raphson. Include
the line flows in your solution as well as the voltages and compare the
solutions in terms of number of iterations to converge and computations
(use Matlab function profile report to generate a breakdown of
computation time). Note: You can use my Matlab code from here.
Also for my routine, you need to re-number the buses so that the
generators and load buses are in sequence. You don't have to use my
program but it will probably be easier.
- For this 5 bus system, find a capacitor size and location to bring all
the voltages within 5% of 1 p.u. Then draw a PV curve at bus 2. You
don't need to use a continuation power flow, just incrementally increase
the load at this bus and try to get close to the nose. Repeat for a load
with Unity power factor.
- Now, consider the three bus multiple solution example from class with
instead B=3.0 and P=-7.0, then
- Find a region of contraction, and thus convergence, for the
decoupled load flow algorithm. The region doesn't have to be exact but
it has to be true that everywhere within the region, the contraction
condition holds. Here are some notes and Matlab routines to help you
with these calculations. Namely, derivation of contraction
mapping formulation for this three bus system and Matlab routines
(one is exact and one is approximate) associated with
convergence properties using the Fast Decoupled approach.
- What is the fixed point of this region?
- Find, if possible, a region of contraction for the low voltage
solution using the decoupled load flow algorithm.
3 - State estimation and a simple secure system problem
Due Feb. 15 5 PM
Consider the real power state estimation problem usingthe DC load flow
model. The IEEE 30-bus system one line
diagram (also called the New England Test System) is sketched once
in full and again showing only the lines
with real power measurements. In addition to these measurements,
real power injections at generator buses 1,2,5,8,11 and13 and load buses
18, 20, 26 are measured. For this system of measurements, answer the
following questions: (Note: you do not need to know the line admittances
to answer the questions.)
(a) Is the system observable?
(b) If the system is not observable, what is the minimal number of
measurements needed to make the system observable?
(c) Using the minimum number of measurements, provide all possible
measurement sets which will make the system observable.
(d) What measurement errors are detectable with this minimal measurement
For the 5-bus system from the previous homework including the
capacitor, assume thatevery transmission line and transformer is
physically two pieces of equipment. A contingency(outage) will then only
represent a loss of one half the original line or transformer while the
impedance for that component doubles. Determine for which contingencies
(transformer and line outages) the system is insecure. Note voltages
should be within 5% of 1.0 p.u and equipment ratings are as given in the
original problem description.
4 - Reliability calculations and AGC
Due Feb. 27 5 PM
Find the LOLE and EUE for the system described below. Determine which
would improve LOLE the most: (a) adding a 25 MW unit with an FOR of 0.01;
or (b) adding a 75 MW unit with an FOR of 0.05.
Load Data - Represented simply as seasonal loading conditions. Assume on
peak lasts for 6 hours and off peak lasts for 18 hours each day.
Daily Peak Load
Daily Off-Peak Load
Problems 12.1, 12.2, 12.3 in H. Saadat. Here in pdf
Consider a power system with three generating units rated at 100, 200,
and 600MVA. The governor droops for these three units are 4%, 5% and 6%,
respectively. Each unit is initially operating at 50% of its rated output.
The load is then increased from 450 MW to 550 MW. Find: a)
The unit area frequency response characteristic β on
MVA base; b) the steady-state increase in frequency; and c) the
output increase of each unit in both MW and per unit.
Consider a small two area power system. Both areas have three
generating units of identical size (200, 400, 500 MVA). Area A has its
governors set for a 5% droop while the Area B governors are set for a 10%
droop. The system is operating at 60 Hz when a load of 75 MW is added in
Area B. Answer the following: a) After governor action stabilizes
and before supplemental control, what is the new system frequency?
; b) How many MW's are picked up in AreaA? and Area B? What is the
schedule error in the tie line flow?; c) What does this say about
Area B's practice of using 10% droop?; d) What would be the new
system frequency if both areas had 5% droop?; and e) What would be
the new system frequency if both areas had 10% droop?
Use the original system in the previous problem. Then: a)
Calculate the ACE for Areas A and B given that the Bias constant is 50
MW/0.1 Hz in A and B; b) Calculate the ACE for Areas A and B given
that the Bias constant is 25 MW/0.1 Hz in A and B; c) What would be
appropriate values of Bias in these areas and the resulting ACE values?;
and d) Other than ACE being wrong, what observations can you make
about poorly chosen Bias values?
5 - State estimation and transient stability
Due March 16 5 PM
- For the five bus problem in the previous homework, create and solve a
DC load flow model for the 5 bus system assuming all voltages are 1.0
p.u. Now with this system, assume the following measurements are given
(numbers for reordering of buses). Power injections: P1= 1.0
p.u., P2= 1.15 p.u. and P3 =-1.95;
Power flows P43= 0.8p.u., P53= 1.3 p.u and P45=
0.4 p.u. For these set of measurements, find the best estimate of
the bus angles assuming all measurements are rated at 5% accuracy.
Show that this system of measurements is observable. For which of
the measurements, can one detect errors? For which of the measurements,
can one identify errors?
- A generator with a transient reactance of Xd'=0.3
p.u. is connected through a lossless transformer and lossless parallel
transmission lines to an infinite bus. The equivalent reactance of the
transformer is Xt=0.1
p.u. and the transmission lines each have an impedance of Xl=0.5
p.u. The voltage behind the reactance is E'=1.4
p.u. and that of the infinite bus is 1.0 p.u. Find the rotor angle
during steady state if delivering a load of 1.30 p.u. A fault to ground
occurs halfway down one of the transmission lines. The fault is
cleared when circuit breakers open at the two ends of the line. Find
expressions for the power output of the generator during fault
conditions and post fault conditions. What is the maximum angle
that can be reached without loss of stability for the post fault system?
- A generator with an inertia constant of H=
3.0 seconds is delivering power of 1.2 p.u to an infinite bus through a
transmission line. The magnitude of the terminal voltage of the
generator is 1.1 p.u. and at the infinite bus it is 1.05 p.u. A fault
occurs at the terminal of the generator reducing the real power output
to zero. (Clarification: The implication here is that after the fault
clears the system returns to the pre-fault state, otherwise there will
be no solution). If during normal conditions the maximum power
deliverable to the load is 2.2 p.u., then find: a) the critical
clearing angle; and b) the critical clearing time.
- A generator with H=5.0 sec is supplying 70% of maximum output to an
infinite bus through a reactive network. A fault occurs and increases
the reactance by 200%. The fault then clears but the overall maximum
power output has been reduced by 10%. Find the critical clearing time
using Runge-Kutta integration implemented in the Matlab function ode45.
- For the system in the previous problem, assume the fault clears in 0.1
seconds. Plot the angle response for the first 10 seconds following the
fault. Repeat for a clearing time of 0.15 seconds.
Due March 30 5 PM
- Given the inductance parameters of a three phase synchronous machine
of Laa0 = 2.7656 mH, Laa2 =
0.3771 mH, L afd = 31.6950 mH, Lab0=
1.3828 mH, L ffd = 433.6569 mH. The field
current is ifd=4000A
30o) with other phases balanced. Find the flux linkages
when θd = 60o
in both abc and dqo
coordinates. Assume steady-state operation (i.e., all damper winding
currents have settled to zero).
- Given ia=1,414sin(θd- θa); ib=1,414sin(θd-120o-θa); ic=1,414sin(θd-
Find the currents in dqo
coordinates. Repeat if ia=1,414sin(θd- θa); ib=ic=0.
You can do this symbollically or numerically using values of θd
60o and θa
- Find X'd and X d given
inductance parameters of Laa0 = 2.7656 mH, L
aa2 = 0.3771 mH, Lafd = 31.6950
mH, L ab0= 1.3828 mH, L ffd
= 433.6569 mH, L a1d= 3.1523 mH, L11d
= 4.2898 mH, Lf1d= 37.0281 mH.
- A generator is supplying a load of Pload=0.8 through
lossless transmission line of 0.3 impedance to an infinite bus with 1.0
p.u. voltage. Use a classical model (model 3) based on the following
parameters: Terminal voltage: Et=1.075; d-axis
reactance: X'd=0.2; armature
winding resistance: Ra=0;. With the infinite bus
as the reference point, find the rotor angle, voltage behind the
reactance and the generator terminal voltages and currents in d-q axes
- The generator model is now to be modified with a flux decay model (our
model 2). Use the following parameters in addition to, or modified
from, the previous problem: d-axis
Xd=1.81; q-axis reactance: Xq=1.76; d-axis
reactance: X'd=0.2; armature
winding resistance: Ra=0.003. Sketch a
phasor diagram for this operating point (i.e., first find the
rotor angle, voltage behind the reactance and the generator terminal
voltages and currents in d-q axes components again and use this to
sketch the phasor diagram.)
Due April 6 5PM
This is a continuation of the problem above. A number of other files are
given here for your convenience in answering the
below. These calculate the derivatives and use ODE45 to perform the
numberical simulation. The routines for the three
models are classical, decay and sixth (for the system with no PSS
and an exciter you can just zero out the PSS gain). Another file (HW7.m)
lists off the input parameters to clear up any confusion on which parameters
are being used from the previous problems. A generator with is
connected through a lossless transformer and lossless parallel transmission
lines to an infinite bus. The equivalent reactance of the transformer is Xt=0.1
p.u. and the transmission lines each have an impedance of Xl=0.5
p.u. A fault to ground occurs halfway down one of the transmission
lines. The fault is cleared when circuit breakers open at the two ends of
- Exciter gain: KA = 200
- Exciter time constant: TA = 0.02
- Exciter limits: -5<Efd<5 (simply
force Efd equal to upper limit if above derivative to zero if positive
and vice versa for below limit)
- Terminal reference voltage: Et=1.05
- Power System Stabilizer (PSS) model::
- Stabilizer gain: KSTAB= 9.5
- Washout filter time constant: TW= 1.4
- Phase compensator: T1
= 0.154; T2
- You are to simulate 10 seconds of response for each of the four models
(classical, flux decay, flux decay + exciter, full model) showing rotor
angle, power output and terminal voltage for a) a fault cleared in 0.1
seconds and b) repeat with a load of 0.9 p.u. and 0.5 p.u. Compare and
discuss these solutions. Other parameters you need from above are H=5s
and T'd0=8s. You should also find
the CCT for each system (if it exists)
and three loading conditions, including the phase plot (frequency vs.
rotor angle). I am providing you with all the code you need to solve
this. You will merely have to update the parameters but I want you to do
your best to follow the differences btween each model. The basic
structure of my code is:
- Solve the steady-state initial conditions.
- Simulate the fault on conditions (use the Matlab routine ode45)
based on the equations developed in class.
- Simulate the post fault conditions for several seconds and check for
- To find critical clearing time, repeat the second and third steps
for increasing fault times until the system goes unstable.