EECE 471 Final, Fall 2003

S. KARAKI AMERICAN UNIVERSITY OF BEIRUT FALL 2003/ 2004 FACULTY OF ENGINEERING AND ARCHITECTURE FINAL EXAM POWER SYSTEMS FUNDAMENTALS (EECE 471) CLOSED BOOK (3 HOURS) January 29, 2004 FIRST READ CAREFULLY THE QUESTION SHEET AND THEN START: � PROGRAMMABLE CALCULATORS ARE NOT ALLOWED. � THIS QUESTION SHEET MUST BE RETURNED WITH THE ANSWER BOOKLET. � BRIEFLY EXPLAIN CALCULATIONS BY SHOWING FORMULAE OR OTHERWISE AS NEEDED. � ON YOUR ANSWER BOOKLET, USE THE RECTO FOR ANSWER AND THE VERSO FOR SCRATCH NAME: _________________________ ID#: _________________________ 1. Consider the following power system shown in Fig. 1 below. The load will be modeled as a constant impedance load taking 32+j16 MVA at nominal voltage. The given percent impedances are with respect to corresponding equipment rating. For this exercise calculations are to be done with SB= 10 MVA. 13.2kV 50 MVA XG= 120% Load Y-∆ G ∆-Y 13.2:66kV 50 MVA XT1= 12% 66:11kV 40 MVA XT2= 12% 0.08+j0.4 Ω/ km 3 4 2 1 80 km Fig. 1: 4-Node System for Problem 1. a) Calculate the load impedance in ohms. b) Select the generator rated voltage as a base voltage on generator side. What should be the base voltages on the transmission and load sides? What are base impedances on the generator, transmission, and load sides? c) Draw an impedance diagram showing the generator equivalent circuit, the transformers, transmission line and load, and label it properly with per unit impedances. Show the per unit calculation of the generator and transmission line impedances and for the other just show the per unit values on the diagram. d) If the generator terminal voltage is regulated to 1 pu, calculate the complex power delivered by the generator, the voltage at node 4 (V4), and the internal generator voltage Ea. e) If the rated power factor of the generator is 0.85, use its capability curve to determine its reactive power limits when it delivers the active power calculated in Part e) above? Explain whether the generator is operating satisfactorily. EECE 471 FINAL EXAM - FALL 2003/ 2004 2/ 6 2. Consider the power system shown in Fig. 2 below for which a load flow is to be solved using the Gauss-Seidel iterative approach. The powers are given in MVA and the line reactances are given in per-unit on SB= 100 MVA base. Node 1 is the reference node with a voltage V1= 1.05∠0. G 3 2 1 G SD3=40+j20 SG2=40+j20 0.04+j0.2 0.05+j0.25 0.02+j0.1 Fig. 2: 3-Node system for Problem 2 a) Determine the nodal admittance matrix (Y) of the above network. Show the calculation for one diagonal and one off-diagonal elements, and for other elements just provide the vales on the matrix. b) Write down the Gauss-Seidel iterative expressions for this problem for V2 and V3. Write down the expressions in variable form then in numeric for replacing appropriate quantities by their values. c) Starting the iterative process at nominal values (1+j0 pu), calculate the values of V2 and V3 at the end of the first and second iterations. 3. Consider a generation system consisting of three units for which it is required to perform an economic load dispatch study. The units have the following fuel-cost curves and limits. C1(PG1)= 250 + 25 PG1 + 0.010 PG1 2 $/h C2(PG2)= 350 + 20 PG2 + 0.020 PG2 2 $/h C3(PG3)= 240 + 20 PG3 + 0.025 PG3 2 $/h 50 ≤ PG1 ≤ 150 MW 80 ≤ PG2 ≤ 250 MW 50 ≤ PG3 ≤ 150 MW a) Write down the equations needed to determine the optimum point of system operation when the demand is 400 MW. b) Solve the equations set up in Part a) and determine the power delivered by each generator and the incremental cost of the system. c) What is the optimum operating point of the system, its incremental cost and the power delivered by each unit when the demand is raised to 500 MW and then to 520 MW? EECE 471 FINAL EXAM - FALL 2003/ 2004 3/ 6 4. Consider the system shown in Fig. 3 below, which has the following data in per unit on a 10 MVA base: V1= 1 ∠0 ZL= j 0.2 SD1= 0.4 + j 0.2 SD2= 0.8 + j 0.4 The generator rating is 15 MVA at 0.9 PF. jQ2 S1 SD1 SD2 IL ZL G Fig. 3: System of two nodes for Problem 4 a) Use a Gauss-Seidel iterative method to determine V2 when Q2 is 0. Set up the iterative equation of V2 and carry out 4 iterations only. b) What are the line current IL and the complex power S1 supplied by the generator? Is the complex power supplied within the capability limit of the generator? Explain. c) What would be the reactive power Q2 if |V2| is to be regulated to 1? Under these conditions what is the line current IL and the complex power S1 supplied by the generator? d) If you were a system designer in a power company, would you recommend installing a capacitor bank to supply Q2? Can Q2 be made smaller? If so how small can Q2 be made approximately? Explain. 5. Write a 100 to 150 words abstract explaining what was your project about, the scope of work that you have done, the methodology followed or main topics discussed, the main results, conclusions, or recommendations that you achieved. Then explain with just enough details to specifically clarify your personal contribution to the project. EECE 471 FINAL EXAM - FALL 2003/ 2004 4/ 6 Fundamentals of Power Systems Analysis (EECE 471) FORMULAE • Ch.4: Transmission Line Modeling � Complex Power Flow on Medium Line: 12 * 2 1 * 2 1 2 1 * 12 2 jθ e Z V V Z V V Y S − + = For S21 exchange indices 1 and 2 in above equation � Power Flow on a short loss-less line: 12 2 1 21 12 sinθ X V V P P = = − jX R l z Z + × = = 12 2 1 2 1 12 cos = θ X V V X V Q − l y Y × = 12 2 1 2 2 21 cos = Q θ X V V X V − l: length of line � Power Circle Diagram Both circles have a radius: Z V V B 2 1 = , Z Z V ∠ 2 1 C1 = Z Z V ∠ 2 2 2 - = C � Power transmission capability: l P l Z V P SIL c β θ β θ sin sin sin sin = 12 12 2 1 12 = ( ) γ β = Im C1 C2 S12 -S21 θ12 θ12 EECE 471 FINAL EXAM - FALL 2003/ 2004 5/ 6 • ∗Ch.5: Transformers and the Per-Unit System B B B ll B B S V S V Z 2 3 2 = = Φ B pu actual Z Z Z = n B o B o pu n B actual n pu Z Z Z Z Z Z = = • *Ch.6: Generator Capability Limit m s a a G X E V P sinδ = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = a m a s a G V E X V Q cosδ s a a X E V max = ρ f a k i E = • ∗Ch.9: Nodal Admittance Matrix I = Y V � Yii is the sum of all admittances connected to node i; � Yik is the negative of the series admittance between nodes i and k. A transformer between nodes i and k can be modeled using a Π-equivalent as shown below: y Ii 1 : n Ii/n Vi n Vi Vk Ik + Vi _ + Vk _ n y y (n2-n) y (1-n) s a X V 2 − rated φ G Q G P ρ max 2 tan δ s a X V EECE 471 FINAL EXAM - FALL 2003/ 2004 6/ 6 • Ch.10: Power Flow Analysis * * 1 * k ik n k i i i i Y V V V I S = ∑ = = i= 2, 3 …n Di Gi i S S S − = ∆ ik ik ik jB G Y + = k i ik θ θ θ − = ( ik ) ik ik ik k i n k ic B VV G P θ θ sin cos 1 + = ∑ = ( ik ) ik ik ik k i n k ic B VV G Q θ θ cos sin 1 − = ∑ = • Gauss-Seidel Load Flow: ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − ∑ = ≠ = k ik i k n k i i ii i Y V V S Y V 1 * * 1 i= 2, 3 … n • Ch.11: Economic Operation of Power Systems Minimize ( Gi ) i m i T C P C =1 = ∑ Subject to D Gi m i P P = ∑ =1 and max min Gi Gi Gi P P P ≤ ≤ i=1 … m ( ) 2 Gi Gi Gi i P P C P γ β α + + = iV Gi Gi Gi jQ P S + = Di Di Di jQ P S + = Sic = * * 1 k ik n i k Y V V = ∑