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REGULATION AND EFFICIENCY OF LONG TRANSMISSION LINE

Line length of more than 150 km and line voltage of greater than 100kV is considered Long Transmission Line. In this case, line constants (ie line impedance and shunt admittance) are considered uniformly distributed over whole stretch of the line. A rigorous method is used to solve this problem but not shown here. Simplified formulas of sending end voltage (Vs) and sending end current (Is) are as follows:.

Vs=VR cosh root(YZ) + IR root (Z/Y) sinh root (YZ)
Is= VR root (Y/Z) sinh root (YZ) + IR cosh root(YZ) where

cosh root(YZ) = [ 1 + 1/2 (ZY) + 1/24 ( Z2Y2)+............]

sinh root (YZ) = [root(YZ) + 1/6 (YZ) 3/2 +.............]

 

 

Typical example

A 3-phase 132kV overhead transmission line delivers a balanced load 85MW (power factor 0.8 lagging) from City X to City Y having distance of 170km long has these constants:
Resistance, R =0.1 Ohm per km per phase
Inductive reactance, XL=0.2 Ohm per km per phase
Capacitive admittance, Y=0.005 mMho per km per phase

Using rigorous method, find
(a) sending end voltage Vs
(b) sending end Is

Answers

R=0.1 Ohm/km x 170km=17 Ohm
XL=0.2 Ohm/km x 170km = 34 Ohm
Y=0.005 mMho/ km x 170km= 0.85 mMho
Z= R + jX = 17 + j34 = 38 /_63.4° Ohm

YZ = 0.00085/_90° x 38 /_63.4° = 0.0323 /_153.4°

Y2Z2 = [0.00085/_90° ]2 x [38 /_63.4°]2 = 1.04 x 10-3 /_306.8°

root [YZ] = root [0.00085/_90° x 38 /_63.4°] = root [0.0323 /_153.4°]= 0.18 /_76.7°

root [Z/Y] = root [38 /_63.4° / 0.00085/_90° ] =root [44705/_-26.6°] = 211.4 /_-13.3

root [Y/Z] = root [0.00085/_90° / 38 /_63.4° ] =root [2.24 x 10-5 /_26.6°]=1.49 x 10-5 /_13.3°

cosh root(YZ) = [ 1 + 1/2 (ZY) + 1/24 ( Z2Y2) = 1 + 1/2 [0.0323 /_153.4° ] + 1/24 [ 1.04 x 10-3 /_306.8°] = 0.986/_0.42°

sinh root (YZ) = [root(YZ) + 1/6 (YZ) 3/2 =root [0.0323 /_153.4°] + 1/6 [0.0323 /_153.4°] 3/2 = 0.178 /_76.8°

 

Vr= 132kV/1.732 /_0°= 76.2 /_0° kV

Ir = (1/3 x 85MW)/ ( 76.2 x0.8) =464.7 /_-36.9° Amps

(a) Sending end voltage Vs=VR cosh root(YZ) + IR root (Z/Y) sinh root (YZ) = [76,200 /_0° ] [0.986/_0.42°] + [464.7 /_-36.9°] [ 211.4 /_-13.3] [ 0.178 /_76.8°] = 91.15 /_5.3° kV
Vs (line-to-line) = 1.732 x 91.15 =157.9 kV

(b) Sending end current Is= VR root (Y/Z) sinh root (YZ) + Ir cosh root(YZ) = [76,200 /_0° ] [1.49 x 10-5 /_13.3°] [0.178 /_76.8°] + [464.7 /_-36.9° ] [0.986/_0.42°] = 458.06 /_-35.5° Amps

Webmaster engr sobri   21 September 2007