ECE4762011_Lect8

# ECE4762011_Lect8 - Lecture 8 Transmission Lines...

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Unformatted text preview: Lecture 8 Transmission Lines, Transformers, Per Unit Professor Tom Overbye Department of Electrical and Computer Engineering ECE 476 POWER SYSTEM ANALYSIS 2 Announcements Start reading Chapter 3. HW 2 is due now. HW 3 is 4.32, 4.41, 5.1, 5.14. Due September 22 in class. “Energy Tour” opportunity on Oct 1 from 9am to 9pm. Visit a coal power plant, a coal mine, a wind farm and a bio-diesel processing plant. Sponsored by Students for Environmental Concerns. Cost isn’t finalized, but should be between \$10 and \$20. Contact Rebecca Marcotte at [email protected] for more information or to sign up. 3 V, I Relationships, cont’d 2 2 Define the propagation constant as where the attenuation constant the phase constant Use the Laplace Transform to solve. System has a characteristic equation ( ) ( )( ) yz j s s s γ γ α β α β γ γ γ = = + = =- =- + = 4 Equation for Voltage 1 2 1 2 1 2 1 1 2 2 1 2 1 2 1 2 The general equation for V is ( ) Which can be rewritten as ( ) ( )( ) ( )( ) 2 2 Let K and K . Then ( ) ( ) ( ) 2 2 cosh( ) sinh( ) x x x x x x x x x x V x k e k e e e e e V x k k k k k k k k e e e e V x K K K x K x γ γ γ γ γ γ γ γ γ γ γ γ----- = + +- = + +- = + =- +- = + = + 5 Real Hyperbolic Functions For real x the cosh and sinh functions have the following form: cosh( ) sinh( ) sinh( ) cosh( ) d x d x x x dx dx γ γ γ γ γ γ = = 6 Complex Hyperbolic Functions For x = α + j β the cosh and sinh functions have the following form cosh cosh cos sinh sin sinh sinh cos cosh sin x j x j α β α β α β α β = + = + 7 Determining Line Voltage R R The voltage along the line is determined based upon the current/voltage relationships at the terminals. Assuming we know V and I at one end (say the "receiving end" with V and I where x 0) we can @ 1 2 determine the constants K and K , and hence the voltage at any point on the line. 8 Determining Line Voltage, cont’d 1 2 1 2 1 1 2 2 c ( ) cosh( ) sinh( ) (0) cosh(0) sinh(0) Since cosh(0) 1 & sinh(0) ( ) sinh( ) cosh( ) ( ) cosh( ) sinh( ) where Z characteristic R R R R R R R c V x K x K x V V K K K V dV x zI K x K x dx zI I z z K I y yz V x V x I Z x z y γ γ γ γ γ γ γ γ γ = + = = + = = ⇒ = = = + ⇒ = = = = + = = impedance 9 Determining Line Current By similar reasoning we can determine I(x) ( ) cosh( ) sinh( ) where x is the distance along the line from the receiving end. Define transmission efficiency as R R c out in V I x I x x Z P P γ γ η = + = 10 Transmission Line Example R 6 6 Assume we have a 765 kV transmission line with a receiving end voltage of 765 kV(line to line), a receiving end power S 2000 1000 MVA and z = 0.0201 + j0.535 = 0.535 87.8 mile y = 7.75 10 = 7.75 10 90 j j-- = + Ω ∠ ° × × ∠ .0 Then zy 2.036 88.9 / mile 262.7 -1.1 c mile z y γ ° = = ∠ ° Ζ = = ∠ ° Ω J 11 Transmission Line Example, cont’d * 6 3 Do per phase analysis, using single phase power and line to neutral voltages. Then 765 441.7 0 kV 3 (2000 1000) 10...
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ECE4762011_Lect8 - Lecture 8 Transmission Lines...

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