Lecture 11 Transmission Line Parameters

Transcription

1 EE 333 POWER SYSTEMS ENGINEERING Lecture 11 Transmission Line Parameters Reading: ; Homework 3 is due on Feb. 20 th. Dr. Lei Wu Department of Electrical and Computer Engineering

2 Outline Develop simple model for transmission lines Line resistance (R) Line conductance (G) Line inductance (L) Line capacitance (C) R dc, T = µ ρtil L = 0 ln 2π A R L G C GMD GMR ( )( ) ( ) 2 GMR = n D D D D D D D D D n n n1 n 2 nn GMD nm xy = ( D11' D12 ' D1 m )( D21' D22 ' D2m ) ( Dn 1' Dn 2 ' Dnm ) Analyze how the geometry of the transmission lines will affect the model parameters 2

3 Line inductance conductor is stranded, r=1.8cm and conductor spacing in a bundle is 45cm. Find La. 3

4 Line inductance conductor is stranded, r=1.8cm and conductor spacing is 45cm. Find La. GMR of one stranded conductor D12 = D16 = D17 = 2r D13 = D15 = 2 3r D14 = 4r D71 = D72 = D73 = D74 = D75 = D76 = 2r D11 = D77 = r ' = r 6 ( ) ( ) 7*7 GMRs = D11 D12 D13 D14 D15 D16 D17 D71D72 D73D74D75D76 D77 = r = cm 4

5 Line inductance conductor is stranded, r=1.8cm and conductor spacing is 45cm. Find La. GMR of one bundle 4 GMRb = GMR4 = GMRsdd d GMRsd = = GMR of each phase 2 cm 2 2 ( ) GMR1 = GMRbD1,1' = * = 2.678m GMR2 = GMRbD2, 2 ' = * 24 = m 2 2 ( ) GMR3 = GMRbD3,3 ' = * = 2.678m 5

6 Line inductance conductor is stranded, r=1.8cm and conductor spacing is 45cm. Find La. GMR of the three-phase double circuit GMR = 3 GMR GMR GMR = m

7 Line inductance conductor is stranded, r=1.8cm and conductor spacing is 45cm. Find La. GMD of the three-phase double circuit GMD = 3 GMD GMD GMD = GMD = D D D D ' 1' 2 1' 2 ' m = = GMD D D D D m 4 23 = ' 2 '3 2 '3 ' = GMD = D D D D ' 1'3 1'3 ' = + + = m 7

8 Line inductance conductor is stranded, r=1.8cm and conductor spacing is 45cm. Find La. µ 0 7 La = ln = *10 H / m 2π GMD GMR X = 2π fl 8

9 Line capacitance Transmission line conductors exhibit capacitance with respect to each other due to the potential difference between them. The relationship between charge q and potential V is represented by the capacitance C = q V V 12 = q 2πε 0 ln D D where ε =8.85*10-12 Farad/m, permittivity of free space 9

10 Line capacitance Transmission line conductors exhibit capacitance with respect to each other due to the potential difference between them. The relationship between charge q and potential V is represented by the capacitance C = q V Assuming that n q i = i 0 V ij = 1 where D 2πε k = 0 ii n = 1 r i q k ln D D kj ki 10

11 Line capacitance For a single-phase two-wire line q 1 + q2 = 0 V = 1 D q ln + q ln πε 0 D11 D21 1 D r 1 D = q ln q ln q2 ln 2πε r D = 2πε r D 0 0 C 12 = 2πε 0 D 2 ln r 11

12 Line capacitance General formula for calculating capacitance 2πε 0 C = GMD ln c GMR c where GMR is similar to GMR except that r is used instead of r' 12

13 Line capacitance conductor is stranded, r=1.8cm and conductor spacing in a bundle is 45cm. Find Ca and admittance. 13

14 Line capacitance conductor is stranded, r=1.8cm and conductor spacing is 45cm. Find Ca and admittance. GMD of the three-phase double circuit GMD = 3 GMD GMD GMD =

15 Line capacitance conductor is stranded, r=1.8cm and conductor spacing is 45cm. Find Ca and admittance. GMR of one stranded conductor D12 = D16 = D17 = 2r D13 = D15 = 2 3r D14 = 4r D71 = D72 = D73 = D74 = D75 = D76 = 2r D11 = D77 = r 6 ( ) ( ) 7*7 GMRs = D11 D12 D13 D14 D15 D16 D17 D71D72 D73D74D75D76 D77 = r = cm 15

16 Line capacitance conductor is stranded, r=1.8cm and conductor spacing is 45cm. Find Ca and admittance. GMR of one bundle 4 GMRb = GMR4 = GMRsdd d GMRsd = = 2 cm D 14 = 4 r GMR of each phase 2 2 ( ) GMR1 = GMRbD1,1' = * = m GMR2 = GMRbD2,2 ' = * 24 = m 2 2 ( ) GMR3 = GMRbD3,3 ' = * = m 16

17 Line capacitance conductor is stranded, r=1.8cm and conductor spacing is 45cm. Find Ca and admittance. GMR of the three-phase double circuit GMR = 3 GMR GMR GMR = m Capacitance πε GMD ln c GMR 0 11 Ca = = 3.03*10 F / m 11 8 Ya = j2π fc = j2 π * 60 * 3.03*10 = j1.14 *10 S / m 17

18 Line inductance & capacitance conductor is stranded, r=1.8cm and conductor spacing is 45cm. Single circuit Double circuit Inductance (H/m) 7.75* *10-7 Capacitance (F/m) 1.44* *

19 Additional Transmission Topics Multi-circuit lines: Multiple lines often share a common transmission right-of-way. This DOES cause mutual inductance and capacitance, but is often ignored in system analysis. Cables: There are about 3000 miles of underground ac cables in U.S. Cables are primarily used in urban areas. In a cable the conductors are tightly spaced, (< 1ft) with oil impregnated paper commonly used to provide insulation inductance is lower capacitance is higher, limiting cable length 19

20 Additional Transmission topics DC Transmission: Because of the large fixed cost necessary to convert ac to dc and then back to ac, dc transmission is only practical for several specialized applications long distance overhead power transfer (> 400 miles) long cable power transfer such as underwater providing an asynchronous means of joining different power systems (such as the Eastern and Western grids). 20

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