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ECE 476 POWER SYSTEM ANALYSIS Lecture 6 Development of Transmission Line Models Professor Tom Overbye Department of Electrical and Computer Engineering Reading For lectures 5 through 7 please be reading Chapter 4 – we will not be covering sections 4.7, 4.11, and 4.12 in detail HW 2 is due now HW 3 is 4.8, 4.9, 4.23, 4.25 (assume Cardinal conductors; temperature is just used for the current rating) is due Thursday 1 Bundle Inductance Example Consider the previous example of the three phases symmetrically spaced 5 meters apart using wire with a radius of r = 1.24 cm. Except now assume each phase has 4 conductors in a square bundle, spaced 0.25 meters apart. What is the new inductance per meter? r 1.24 102 m 0.25 M 0.25 M r ' 9.67 103 m 3 R b 9.67 10 0.25 0.25 2 0.25 0.25 M 0.12 m (ten times bigger!) 0 5 La ln 7.46 107 H/m 2 0.12 2 1 4 Transmission Tower Configurations The problem with the line analysis we’ve done so far is we have assumed a symmetrical tower configuration. Such a tower figuration is seldom practical. Therefore in general Dab Dac Dbc Typical Transmission Tower Configuration Unless something was done this would result in unbalanced phases 3 Transmission Tower Examples 230 kV wood pole H-frame 230 kV lattice steel tower double circuit Source: Tom Ernst, Minnesota Power 4 Transposition To keep system balanced, over the length of a transmission line the conductors are rotated so each phase occupies each position on tower for an equal distance. This is known as transposition. Aerial or side view of conductor positions over the length of the transmission line. 5 Line Transposition Example 6 Line Transposition Example 7 Inductance of Transposed Line Define the geometric mean distance (GMD) Dm d12 d13d 23 1 3 Then for a balanced 3 system ( I a - I b - I c ) Dm 0 1 0 1 I a ln I a ln I a ln a r' Dm 2 r' 2 Hence Dm 0 Dm 7 H/m 2 10 ln ln La r' r' 2 8 Inductance with Bundling If the line is bundled with a geometric mean radius, R b , then 0 Dm a I a ln 2 Rb 0 Dm Dm 7 La ln 2 10 ln H/m 2 Rb Rb 9 Inductance Example Calculate the per phase inductance and reactance of a balanced 3, 60 Hz, line with horizontal phase spacing of 10m using three conductor bundling with a spacing between conductors in the bundle of 0.3m. Assume the line is uniformly transposed and the conductors have a 1cm radius. Answer: Dm = 12.6 m, Rb= 0.0889 m Inductance = 9.9 x 10-7 H/m, Reactance = 0.6 /Mile 10 Review of Electric Fields To develop a model for line capacitance we first need to review some electric field concepts. Gauss's law: A D da = qe (integrate over closed surface) where D = electric flux density, coulombs/m 2 da = differential area da, with normal to surface A = total closed surface area, m 2 q e = total charge in coulombs enclosed 11 Gauss’s Law Example Similar to Ampere’s Circuital law, Gauss’s Law is most useful for cases with symmetry. Example: Calculate D about an infinitely long wire that has a charge density of q coulombs/meter. A D da D D 2 Rh q e qh q 2 R Since D comes radially out integrate over the cylinder bounding the wire ar where ar radially directed unit vector 12 Electric Fields The electric field, E, is related to the electric flux density, D, by D = E where E = electric field (volts/m) = permittivity in farads/m (F/m) = o r o = permittivity of free space (8.85410-12 F/m) r = relative permittivity or the dielectric constant (1 for dry air, 2 to 6 for most dielectrics) 13 Voltage Difference The voltage difference between any two points P and P is defined as an integral V P P E dl In previous example the voltage difference between points P and P , located radial distance R and R from the wire is (assuming = o ) V R R R dR ln 2 o R 2 o R q q 14 Voltage Difference, cont’d With V R R R dR ln 2 o R 2 o R q q if q is positive then those points closer in have a higher voltage. Voltage is defined as the energy (in Joules) required to move a 1 coulomb charge against an electric field (Joules/Coulomb). Voltage is infinite if we pick infinity as the reference point 15 Multi-Conductor Case Now assume we have n parallel conductors, each with a charge density of qi coulombs/m. The voltage difference between our two points, P and P , is now determined by superposition V n R i qi ln 2 i 1 R i 1 where R i is the radial distance from point P to conductor i, and R i the distance from P to i. 16 Multi-Conductor Case, cont’d n If we assume that qi 0 then rewriting i=1 V 1 1 n qi ln qi ln R i 2 i 1 R i 2 i 1 1 n n We then subtract qi ln R1 0 i 1 V R i 1 1 n qi ln qi ln 2 i 1 R i 2 i 1 R 1 1 n R i As we more P to infinity, ln 0 R 1 17 Absolute Voltage Defined Since the second term goes to zero as P goes to infinity, we can now define the voltage of a point w.r.t. a reference voltage at infinity: V 1 n 1 qi ln 2 i 1 R i This equation holds for any point as long as it is not inside one of the wires! 18 Three Conductor Case Assume we have three infinitely long conductors, A, B, & C, each with radius r C B and distance D from the other two conductors. Assume charge densities such that qa + qb + qc = 0 1 1 1 1 Va q ln q ln q ln a b c 2 r D D qa D Va ln 2 r A 19 Line Capacitance For a single line capacitance is defined as qi CiVi But for a multiple conductor case we need to use matrix relationships since the charge on conductor i may be a function of Vj q1 C11 qn Cn1 q CV C1n V1 Cnn Vn 20 Line Capacitance, cont’d In ECE 476 we will not be considering theses cases with mutual capacitance. To eliminate mutual capacitance we'll again assume we have a uniformly transposed line. For the previous three conductor example: Va V Since q a = C Va qa 2 C Va ln D r 21 Bundled Conductor Capacitance Similar to what we did for determining line inductance when there are n bundled conductors, we use the original capacitance equation just substituting an equivalent radius R cb (rd12 1 d1n ) n Note for the capacitance equation we use r rather than r' which was used for R b in the inductance equation 22 Line Capacitance, cont’d For the case of uniformly transposed lines we use the same GMR, D m , as before. ln 2 Dm d ab d ac dbc C c Rb where Dm c Rb (rd12 1 d1n ) n 1 3 (note r NOT r') ε in air o 8.854 10-12 F/m 23 Line Capacitance Example Calculate the per phase capacitance and susceptance of a balanced 3, 60 Hz, transmission line with horizontal phase spacing of 10m using three conductor bundling with a spacing between conductors in the bundle of 0.3m. Assume the line is uniformly transposed and the conductors have a a 1cm radius. 24 Line Capacitance Example, cont’d Rbc Dm C Xc 1 (0.01 0.3 0.3) 3 1 (10 10 20) 3 0.0963 m 12.6 m 2 8.854 1012 1.141 1011 F/m 12.6 ln 0.0963 1 1 11 C 2 60 1.141 10 F/m 2.33 108 -m (not / m) 25 Line Conductors Typical transmission lines use multi-strand conductors ACSR (aluminum conductor steel reinforced) conductors are most common. A typical Al. to St. ratio is about 4 to 1. 26 Line Conductors, cont’d Total conductor area is given in circular mils. One circular mil is the area of a circle with a diameter of 0.001 = 0.00052 square inches Example: what is the the area of a solid, 1” diameter circular wire? Answer: 1000 kcmil (kilo circular mils) Because conductors are stranded, the equivalent radius must be provided by the manufacturer. In tables this value is known as the GMR and is usually expressed in feet. 27 Line Resistance Line resistance per unit length is given by R = where is the resistivity A Resistivity of Copper = 1.68 10-8 Ω-m Resistivity of Aluminum = 2.65 10-8 Ω-m Example: What is the resistance in Ω / mile of a 1" diameter solid aluminum wire (at dc)? 2.65 10-8 Ω-m m R 1609 0.084 2 mile mile 0.0127m 28 Line Resistance, cont’d Because ac current tends to flow towards the surface of a conductor, the resistance of a line at 60 Hz is slightly higher than at dc. Resistivity and hence line resistance increase as conductor temperature increases (changes is about 8% between 25C and 50C) Because ACSR conductors are stranded, actual resistance, inductance and capacitance needs to be determined from tables. 29 ACSR Table Data (Similar to Table A.4) GMR is equivalent to r’ Inductance and Capacitance assume a Dm of 1 ft. 30 ACSR Data, cont’d Dm X L 2 f L 4 f 10 ln 1609 /mile GMR 1 3 2.02 10 f ln ln Dm GMR 1 3 2.02 10 f ln 2.02 103 f ln Dm GMR 7 Term from table assuming a one foot spacing Term independent of conductor with Dm in feet. 31 ACSR Data, Cont. To use the phase to neutral capacitance from table 2 0 1 XC -m where C Dm 2 f C ln r Dm 1 6 1.779 10 ln -mile (table is in M-mile) f r 1 1 1 1.779 ln 1.779 ln Dm M-mile f r f Term independent Term from table assuming of conductor with a one foot spacing Dm in feet. 32 Dove Example GMR 0.0313 feet Outside Diameter = 0.07725 feet (radius = 0.03863) Assuming a one foot spacing at 60 Hz 1 X a 2 60 2 10 1609 ln Ω/mile 0.0313 X a 0.420 Ω/mile, which matches the table 7 For the capacitance 1 1 6 X C 1.779 10 ln 9.65 104 Ω-mile f r 33 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 34 Additional Transmission topics Ground wires: Transmission lines are usually protected from lightning strikes with a ground wire. This topmost wire (or wires) helps to attenuate the transient voltages/currents that arise during a lighting strike. The ground wire is typically grounded at each pole. Corona discharge: Due to high electric fields around lines, the air molecules become ionized. This causes a crackling sound and may cause the line to glow! 35 Additional Transmission topics Shunt conductance: Usually ignored. A small current may flow through contaminants on insulators. 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). 36 DC Transmission Line +/- 400 kV HVDC lattice tower Source: Tom Ernst, Minnesota Power 37