Power Systems Engineering - I

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1 Power Systems Engineering - I Presented by Professor Douglas C. Hopkins, Ph.D. 332 Bonner Hall University at Buffalo Buffalo, NY , fax: Transmission Lines (con d) 1

2 Review Nomenclature Characteristic impendence, Z c = (z/y) Propagation constant, γ = (yz) = α+jβ Attenuation constant is α Phase constant is β For lossless line, R=G=0, i.e. Z c = (L/C) γ = jω (LC) = jβ Also wavelength, λ = 2π/β Closer Look at Transmission Lines The exact equations (p218) give the V(x) and I(x) at any point along the line. V ( x) = VR + ZcIR VR ZcIR e e + e 2 2 e I( x) = VR + ZcIR VR ZcI e e R e 2Z 2Z e where γ = α+jβ c α x jβ x α x jβ x α x jβ x α x jβ x e αx changes the magnitude with distance. e jβx changes the phase angle with distance. c 2

3 Incident Wave S R x jβ x = X=0 V ( ) R + ZcI V x R α e e 2 The first term diminishes in magnitude and retards in phase moving from source end to receiving end. This is a traveling wave, specifically the incident wave. Reflective Wave V + R Z I 2 c R e α x jβ x e S X=0 R The second term increases magnitude and advances phase as moving from source end to receiving end. This is a traveling wave, specifically the reflective wave. 3

4 Line Termination with Z c If a line is terminated into its characteristic impedance, Z c, there are no reflections. The line is called a flat line or infinite line. Usually, power lines are not terminated into Zc, though communication lines most often are. Typical Z c =400Ω for single-circuit overhead lines with a phase angle between 0 and 15. The characteristic impedance, Z c, is called the surge impedance if the line is considered lossless. Surge Impedance Loading (SIL) SIL is the power delivered by a lossless line to a load resistance equal to the Z c = (L/C) [real ohms] At SIL the I R =V R /Z c, then V(x) = cos(βx) V R + j Z c sin(βx) (V R /Z c ) =(cos(βx) + j sin(βx)) V R =e jβx V R [volts] V(x) = V R I(x) =(cos(βx) + j sin(βx)) V R /Z c =e jβx V R /Z c [amps] 4

5 SIL Power The complex power flowing at any point along in line is S(X) = P(x) = jq(x) = V(x) I*(x) = ( e jβx V R ) (e jβx V R /Z c )* = V 2 R / Z c The real power delivered at rated line voltage, SIL, is SIL = V 2 rated / Z c where Z c = (LC) Maximum Power expressed in SIL Power system stability may be defined as that property which enables the synchronous machines to adequately respond to a disturbance from a normal operating condition. Or: The generator cannot deliver more power, otherwise it looses synchronism. The maximum power flow per unit SIL is P max / SIL = V S,pu V R,pu / sin(2π l λ ) where l λ is the fraction of line length relative to wavelength (λ = 2πβ) 5

6 Maximum Power Flow per unit SIL Graph scales with V S and V R. A 95% line drop directly affects the max flow. Maximum Power/SIL Maxflow(W)/SIL(W) Fraction of Wavelength (m/m) Line Loading 6

7 Practical Limit on Line Loading Thermal limits limit is due to heating of conductor and hence depends heavily on ambient conditions Angle limits while the maximum power transfer occurs when line angle difference is 90 degrees, actual limit is substantially less due to multiple lines in the system Voltage stability limits as power transfer increases, reactive losses increase as I 2 X. As reactive power increases the voltage falls, resulting in a potentially cascading voltage collapse. Review Examples 2003 Tom Overbye, University of Illinois at Urbana-Champaign, All Rights Reserved Reactive Compensators Compensation affects steady-state and dynamic control Passive compensators Shunt inductors and capacitors Active compensators Rotational Switched passive elements FACTS Review example 7

8 Good Bye You have all been a great class. Your interest, questions and humor have made this one of the most interesting classes I have taught. You are all above average, if only in the fact that you seek personal improvement. I wish you all well in your future studies and I stand ready to continually help you meet your goals. Dr. Doug Hopkins Farwell 8

Transmission and Distribution of Electrical Power

KINGDOM OF SAUDI ARABIA Ministry Of High Education Umm Al-Qura University College of Engineering & Islamic Architecture Department Of Electrical Engineering Transmission and Distribution of Electrical