Homework, Solutions and Reviews

Updated throughout the semester

Reviews/Equation sheets/Exam materials

Background equations - Equation sheet summarizing basic power system relationships and some more specifics for load flow equations.
Notes on PV curves from Dr. Dimitrovski.
Voltage stability criteria for Western US. and disturbance performance criteria.
Link to general NERC Reliability standards. Some specific standards for frequency control, Bal003 and general balancing criteria. Also, a recent report on frequency response in the Eastern Interconnect.
Example midterms and equation sheet. Note this is a sampling of other years so there may be some problems that we haven't covered in this class yet and you can ignore those. Also, these two exams didn't have anything on state estimation so I'll try to find an old exam with a state estimation question.
Solutions to first midterm.
Machine electromagnetics, circuit equations and dynamics.
Example midterm and solutions. Exam equation sheet.
Solutions to second midterm.
Exam equation sheet with addition from Dr. Dimitrovski.

Assignment 1 - PV curves and continuation power flow

Due Jan. 27 5 PM - New due date

Perform CPF for the 2-bus system from the class.
Find the expressions for Pmax and Qmax that include shunt branches. Set G/2=0 and B/2=10^-4s. Extra credit.
Solutions. Here is the solution with CPF. (Updated)

Assignment 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 Y_bus 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.

Solutions.

Assignment 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 set?

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.

Solutions.

Assignment 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.

Generation Data

Unit	Capacity	FOR
1	150 MW	0.10
2	50 MW	0.05
3	25 MW	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.

Season	Daily Peak Load	Daily Off-Peak Load
Winter	100 MW	80 MW
Spring	120 MW	70 MW
Summer	175 MW	50 MW
Fall	140 MW	80 MW

Problems 12.1, 12.2, 12.3 in H. Saadat. Here in pdf format.

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 a100 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?

Solutions.

Assignment 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 X_d'=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 X_t=0.1 p.u. and the transmission lines each have an impedance of X_l=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.
Solutions.

Assignment 6

Due March 30 5 PM

Given the inductance parameters of a three phase synchronous machine of L_aa0 = 2.7656 mH, L_aa2 = 0.3771 mH, L_afd = 31.6950 mH, L_ab0= 1.3828 mH, L_ffd = 433.6569 mH. The field current is i_fd=4000A and i_a=20,000sin(θ_d - 30^o) with other phases balanced. Find the flux linkages when θ_d = 60^o in both abc and dqo coordinates. Assume steady-state operation (i.e., all damper winding currents have settled to zero).
Given i_a=1,414sin(θ_d- θ_a); i_b=1,414sin(θ_d-120^o-θ_a); i_c=1,414sin(θ_d- 240^o-θ_a). Find the currents in dqo coordinates. Repeat if i_a=1,414sin(θ_d- θ_a); i_b=i_c=0. You can do this symbollically or numerically using values of θ_d = 60^o and θ_a = 0^o.
Find X^'_d and X_d given the inductance parameters of L_aa0 = 2.7656 mH, L_aa2 = 0.3771 mH, L_afd = 31.6950 mH, L_ab0= 1.3828 mH, L_ffd = 433.6569 mH, L_a1d= 3.1523 mH, L_11d = 4.2898 mH, L_f1d= 37.0281 mH.
A generator is supplying a load of P_load=0.8 through a 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: E_t=1.075; d-axis transient reactance: X^'_d=0.2; armature winding resistance: R_a=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 components.
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 reactance: X_d=1.81; q-axis reactance: X_q=1.76; d-axis transient reactance: X^'_d=0.2; armature winding resistance: R_a=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.)
Solutions.

Assignment 7

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 X_t=0.1 p.u. and the transmission lines each have an impedance of X_l=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 the line.

Exciter model:

Exciter gain: K_A = 200
Exciter time constant: T_A = 0.02
Exciter limits: -5<E_fd<5 (simply force Efd equal to upper limit if above derivative to zero if positive and vice versa for below limit)
Terminal reference voltage: E_t=1.05

Power System Stabilizer (PSS) model::

Stabilizer gain: K_STAB= 9.5
Washout filter time constant: T_W= 1.4
Phase compensator: T₁ = 0.154; T₂ = 0.033

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 stability.
To find critical clearing time, repeat the second and third steps for increasing fault times until the system goes unstable.

Solutions.

Assignments 8 and 9 (in pdf)

Assignment 10 (in pdf)

Assignment 8 Solutions.
Assignment 9 Solutions.
Assignment 10 Solutions.