Understanding Power System Analysis: Key Concepts and Models
Explore the fundamentals of power system analysis through lectures by Dr. Kassim Al-Anbarri. Topics include modeling power system components, single-phase transformers, transmission lines, and more. Gain insights into transformer equivalent circuits, transmission line models, and voltage calculations for efficient power delivery.
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Lectures on Power System Analysis 01 Dr. Kassim Al-Anbarri B.Sc., M.Sc., Ph.D (Electrical Power Engineering) Affiliations: Electrical Engineering Department,College of Engineering , Mustansiriyah University Bab Al-Muadham Campus, Al Mustansiriya 10052, Baghdad, Iraq Email: alanbarri@uomustansiriyah.edu.iq alanbarri@yahoo.com kassimalanbarri@gmail.com References: 1. Saadat,H., Power System Analysis, McGraw-Hill book company, NewYork, 1999 2. Stevenson,W.D., and Grainger,J.J., Power System Analysis, McGraw-Hill book company, NewYork, 1995 3. Glover,J.D.Sarma,M.S, and Overbye,T.J., Power System Analysis and Design, Cengage Learning,USA,2012 4. Weedy,B.M.,Cory,B.J.,Jenkins,N.,Ekanayake,J.B.,and Strbac,G., Electric Power Systems, John Wiley & Sons, Ltd., UK,2012 5. Overbye,T.J, Lecture notes, University of Illinois,USA
Modelling of Power System Components Transformer(single phase, three phase) Transmission Lines(Over head , Under ground Cables ) Load ( Con. Power, Current, Impedance) Generators
Single phase transformer High voltage(tension) side HV Low voltage(tension) side LV Zone 2 Zone 1 33kV 132 kV n1 X2 R1 R2 X1 n2 Load V1 V2 Power source Referring to one side yield R1 X 2 X1 R 2 Xm Rm Load Power source Neglecting the magnetization branch yield X1 R1 X 2 R 2 Load Power source
X=X1+ X2 R=R1+ R 2 Load Power source Since R<<X, the simplified equivalent circuit of a single phase transformer X=X1+ X 2 Load Power source Where X is the equivalent transformer reactance referred to one side
Transmission Line model R L VR VS C/2 C/2 Receiving end voltage Sending end voltage R= Resistance of the line = Length of line * resistance per unit length L= Inductance of the line = Length of line * inductance per unit length C= Capacitance of the line = Length of line * capacitance per unit length For short lines <50 km , Capacitance and resistance are neglected
L VR= VR R R Receiving end voltage VS= Vs s s Sending end voltage L= Inductance of the line = Length of line * inductance per unit length ?? ?? ?? ??????????? ????? ????? ? ???? = ???(?? ??)
I + - R 12 E 60 V There are four quantities E,I,Z(R,X),S(P,Q) In per unit system, we have to select the bases for only two quantities, the remaining two should be calculated by the available laws(Ohm,s law) *for the above circuit using actual units ? =? ?=?? ??= ? ??????? The power dissipated in the resistance ? = ?? ? = ?? ?? = ??? ?????
*for solving the previous problem by per unit system Suppose that we select the EMF source and the resistance as a base quantities ??= ??? ????? ; ??= ?? ? The base values for the remaining quantities should be calculated as: ??=?? =??? ??= ? ??????? I ?? ?? = ?? ??= ??? ? = ??? ????? + - For the previous circuit R 0.4 pu E 0.5 pu ? ??=?? ??= ?.? ?? ? ?? ??? ???? = ? ?? ?? ???= ?.? ?? ? ?? ??? ???? = = To calculate the current I ? ?=?.? ?? ? = ?.? ??= ?.?? ?? ? = ?? ? = ?.?? ? ?.? = ?.??? ?? Converting the results obtained from per unit to actual units ? = ? ?? ??? ???? ??= ?.?? ? = ? ??????? ? = ? ?? ??? ???? ??= ?.??? ??? = ??? ?????
Problem 2.10 , pp.83 of Ref[2] A single-phase system shown below has two transformers A-B And B-C connected by a line B feeding a load at the receiving end C. The ratings and parameter values of the components are: Transformer A-B: 500 V/1.5 kV, 9.6 kVA, leakage reactance= 5% Transformer B-C: 1.2 kV/120 V, 7.2 kVA, leakage reactance= 4% Line B: series impedance =(0.5+j 3.0) Load C: 120 V,6 kVA at 0.8 power factor lagging 1) Determine the value of the load impedance in ohms and the actual ohmic impedances of the two transformers referred to both their primary and secondary sides. 2) Choosing 1.2 kV as the voltage base for circuit B and 10 kVA as the system wide kVA base, express all system impedances in per unit 3) What value of sending end voltage corresponds to the given loading conditions?
1.2 kV 120 V n1 500 V 1.5 kV n1 n2 n2 Line Zone B Load Zone A Zone C TA-B TB-C Fig. Problem 2.10
*The load impedance in Ohm can be calculated as follow: ? = ?? ..(?) Where S is the complex power in VA ? =? ? (?) ? =? ? ? Substitute (3) in (1) yields: ? = ?? calculated from the following equation: ?? ? ?The Load impedance in Ohms can be ? = ?? ? (?) ? = The complex power at the load is 6 kVA with 0.8 PF lagging S=6000 (cos-1 0.8) VA =6000 36.8 VA ???? ? = ???? ??.? = ?.? ??.? ?
*The Ohmic impedance of transformer TA-B: 500 V 1.5 kV n1 1 3 n2 Zone B Zone A The data of transformer A-B: 500 V/1.5 kV, 9.6 kVA, leakage reactance= 5% We have to calculate the base impedance ZB for each zone. The power rating and the voltage rating of the unit(transformer) have to be taken as the Base power(SB) and Base voltage(VB), respectively. For the low voltage side of the transformer TA-B (i.e. Zone A) SB= 9.6 kVA, VB=500 V ? =???? ????= ??.??? ? ??=?? ?? The Ohmic impedance of TA-B viewed from low voltage side ZLV ZLV =ZTin per uint * ZB = j0.05*26.042= j1.30208 For the high voltage side of the transformer TA-B (i.e. Zone B) SB= 9.6 kVA,VB=1.5 kV ? =????? ????= ???.??? ? ??=?? ?? The Ohmic impedance of TA-B viewed from high voltage side ZHV ZHV =ZTin per uint * ZB = j0.05*234.375= j11.7187 ? ? ??? ???= ?? ?? ?.????? ??.????= ? ? Verification
*The Ohmic impedance of transformer TB-C: 1.2 kV 120 V n1 10 1 n2 Zone C Zone B The data of transformer B-C: 1.2 kV/120 V, 7.2 kVA, leakage reactance= 4% We have to calculate the base impedance ZB for each zone. The power rating and the voltage rating of the unit(transformer) have to be taken as the Base power(SB) and Base voltage(VB), respectively. For the high voltage side of the transformer TB-C(i.e. Zone B) SB= 7.2 kVA,VB=1.2 kV ? =????? ????= ??? ? ??=?? ?? The Ohmic impedance of TB-C viewed from high voltage side ZHV ZHV =ZTin per uint * ZB = 0.04*200= 8 For the low voltage side of the transformer TB-C(i.e. Zone C) SB= 7.2 kVA,VB=120 V ? =???? ????= ? ? ??=?? ?? The Ohmic impedance of TB-C viewed from low voltage side ZLV ZLV =ZTin per uint * ZB = 0.04*2= 0.08 ? ? ??? ???= ?? ?? ? ?? ? ?.??= Verification
ReferringZLoadto Zone B by multiplying it with square of turns ratio of TB-C ? ???? ??? = ??? ??.??=???.??+? ???.?? ? ??? ????? = ?.? ??.? Referring ZT B-C to Zone B by multiplying it with square of turns ratio ? ???? ??? = ? ???=?? ? ??? ?? ? ? = ?.?? ?? 500 V 1.5 kV n1 ZLine 0.5+j 3.0 XrefTB-C ?? ? n2 Zone A ZrefLoad ??? ??.?? Zone B TA-B
Referring ZrefLoadto Zone A by multiplying it with square of turns ratio of TA-B ? ??? ???? = ??.??? ??.??=??.??+? ??.?? ? ??? ????? = ??? ??.? Referring ZT B-C to Zone A by multiplying it with square of turns ratio of TA-B ? ??? ???? = ?.???? ???=??.???? ? ??? ?? ? ? = ? ?? Referring Z Line to Zone A by multiplying it with square of turns ratio of TA-B ? ??? ???? = ?.??? + ??.???? ? ???= ?.? + ? ?.? ????? Referring ZT A-B to Zone A by multiplying it with square of turns ratio of TA-B ? ??? ???? = ?.??? ???=??.??? ? ??? ?? ? ? = ??.???? ?? ZrefLine .+. XrefTB-C . XrefTA-B . VrefLoad Zone A Vs ZrefLoad . .
By taking the voltage at the load (at zone C)as a reference phasor??= ???? ????? Referring VRto Zone B then referring the result to Zone A yields: ???? ??? ??? ???? ???= ?? ?? = ??? ? ?? ?.??? = ??? ????? ?????=????? =???? ??.? ??? ? = ?? ??.? ??????? ????? Referring ILoadto Zone B then referring the result to Zone A yields: ???? ??? ??? ???? ??? ????? = ?????/ / ??? ????? = ?? ??.?/??/?.???? ??? ????? = ?? ??.? ???????
By taking the voltage at the load (at zone C)as a reference phasor??= ???? ????? Referring VRto Zone B then referring the result to Zone A yields: ???? ??? ??? ???? ???= ?? ?? = ??? ? ?? ?.????? = ??? ? ????? ?????=????? =???? ??.? ??? ? = ?? ??.? ??????? ????? Referring ILoadto Zone B then referring the result to Zone A yields: ???? ??? ??? ???? ??? ????? = ?????/ / ???? ??? ??? ???? ??? = ?? ??.? / ????? / ??? ????? = ?? ??.? ??????? ???+ ????? ??? ??? ???+ ?? ? ? ??? ??= ?? ?? ? ? + ????? ??.???? +?.??? + ??.???? +??.??? ??= ??? ? + ?? ??.? ??= ???.?? ?.?? ?????
By Choosing 1.2 kV as the voltage base for circuit B ( VBB)and 10 kVA as SB, the voltage base for other zones can be calculated as follow: *For Zone C ? ?=?? ?? ?? ???? ?? 1.2 kV 120 V n1 500 V 1.5 kV n1 n2 n2 Line ? =???? ??? ?? ?= ??? ????? Zone B Load Zone A ?? Zone C *For Zone A ? ?=?? ?? ?? TA-B ?? TB-C ? ?? ????= ??? ???? ?= ??? ????? ??
The per unit impedances of power system components should be corrected to the new bases by using the following formula ? ??? ??? ?? ?? ?? ?? ????= ???? ??? ??? *for Load ?? ?? ?? ?= ?? ???? ?????= ?.?? ? ?= ?? ????? ?? ??? ???? =????? =?.? ??.? ?.?? = ?.??? ??.? ?? ? ?? 1.2 kV 120 V n1 500 V 1.5 kV n1 n2 n2 Line Zone B Load Zone A Zone C TA-B TB-C
The per unit impedances of power system components should be corrected to the new bases by using the following formula ? ??? ??? ?? ?? ?? ?? ????= ???? ??? ??? *for Load (Another approach) ????? ?? ??? ???? =????? =???? ??.? ????? = ?.? ??.? ?? ?? ???? ??? ???? =?? ?=??? ???= ?.? ? ( ????????? ??????) ?? ?? ? = ?? ? = ?.? ??.?= ?.??? ??.? ?? 1.2 kV 120 V n1 500 V 1.5 kV n1 n2 n2 Line Zone B Load Zone A Zone C TA-B TB-C
*for transformer TB-C ? ??? ??? ?? ?? ?? ?? ????= ???? ??? ??? ? ????? ???? ??? ??? ?? ? ?= ??.?? = ??.??? ?? or ? ????? ???? ???? ???? ?? ? ?= ??.?? = ??.??? ?? 1.2 kV 120 V n1 500 V 1.5 kV n1 n2 n2 Line Zone B Load Zone A Zone C TA-B TB-C
*for Line Since the impedance of the line is given in actual units, The impedance have to be converted to per unit values by calculating the Base Impedance as follow: ?? ?? ?? ?= ?? ????? ????? ?= ?? = ??? ? ????? ?? ??? ???? =????? ?.? + ? ?.? ??? = = ?.????? + ??.???? ?? ? ?? 1.2 kV 120 V n1 500 V 1.5 kV n1 n2 n2 Line Zone B Load Zone A Zone C TA-B TB-C
Per unit system procedure for scaling 3 phase power system
