Washington State UniversityEE 582 • Fall 2012 • Homework 2

Due Wednesday, Oct. 10, 2012, 11:59 PMReactive Power Control

Please prepare a report in the ieee two-column format (see the course outline for templates for both Wordand LATEX; bonus will be given for LATEX.). Submit as a pdf file through the following website http:

//form.jotform.us/form/22750964894164. Name your pdf file only according to the following format:EE582 HW2 Lastname Firstname.PDF.

1. Midpoint Shunt Compensator. In lectures we studied a shunt compensator placed in the middle ofa purely inductive line with VS = V ] δ

2 and VR = V ]−δ2 . We also saw that the current injected from

the compensator is

IC = j4V

X

(1− cos(

δ

2)

).

Show that this equation is valid.

2. Power Transfer. Consider a line with a purely inductive impedance (Zline = jX). Assume the voltageat the receiving end is kept constant. Consider two cases: (i) an inductive load Zload = |Z|]φ and (ii)a capacitive load Zload = |Z|] − φ. Note that both loads have the same impedance magnitude, butthe power factor of one load is lagging and the other is leading. What is the relationship between realpower transfer in the line in these two cases? Which case leads to a larger power transfer (or are theythe same)?

3. Line Compensation with Short and Long Models. Per unit-length parameters of a 60 Hz, three-phase, balanced, 300 mi (note the unit), 220 kV line are as follows. Base values for per unitization areVAb = 100 MVA and Vb = 220 kV.

r = 0.028 Ω/km

x = 0.325 Ω/km

b = 5.02× 10−6/(Ω km)

g = 5× 10−8/(Ω km)

The line equations are

V (x) = VR cosh(γx) + IRZC sinh(γx)

I(x) = IR cosh(γx) +VRZC

sinh(γx)

where x is the distance from receiving end.

(a) Assume that the voltage at both ends of the line is regulated at 1.0 pu. Also assume that thereference phasor is the receiving end voltage.

1. Assume that the line can be approximated as a short line; neglect losses (A = 1∠0 andB = XL∠90). Plot PS(δ), PR(δ), Pline(δ), QS(δ), QR(δ), and Qline(δ). Also plot the voltageprofile of the line for δ = 0, δ = 25, δ = 50, and δ = 75. Comment on the realistic amountof the real power that can be delivered to the receiving end.

2. Repeat part 1 if the line is represented as a long line.

3. Compare the corresponding results of parts 1 and 2 and identify maximum error for eachvariable due to the short-line approximation. Are the errors acceptable?

(b) Assume that the sending end voltage of the line is adjusted at 1.0 pu, and the receiving end is opencircuit (with no voltage control).

EE 582 • Fall 2012 • Homework 2 Page 1 of 2 V.1.11.AM; Updated Oct. 4, 2012

1. Calculate the receiving end voltage and plot the voltage profile of the line if the line is approx-imated by a short line.

2. Repeat part 1 if the line is represented as a long line. Also calculate the sending end current(i.e., charging current).

3. Plot the difference between the voltage profiles of parts 1 and 2.

4. Is the voltage profile of part 2 (no load) acceptable? If not, why and how the voltage profileshould look like, and how can your proposed voltage profile be achieved?

4. Line Compensation. A 60 Hz, 13.8 kV, three-phase distribution line supplies a load at the end of theline. The load is rated at 1 pu at 0.93 lagging power factor. The line impedance is 0.30 pu. Assume thatthe line X/R ratio can be 10, 5, 1, or 0.5 while the impedance value remains unchanged.

(a) Determine if the line can supply the full load at the rated voltage for each X/R ratio. The maximumpermissible voltage deviation from 1 pu at each end of the line is 5%.

(b) Propose a solution to resolve the problem for any scenario that the line cannot supply the full loadwithin the required voltage constraints. Comment on the applicability of the solution. Can youfind a solution in all cases? Why?

EE 582 • Fall 2012 • Homework 2 Page 2 of 2 V.1.11.AM; Updated Oct. 4, 2012