Transmission Line Models

transmission line models \| \|\|
line equations \| \|\|		π equivalent circuit \| \|\|
with losses	lossless model	exact π equivalent \| \|\|		approx. π eqivalent \| \|\|
		with losses	lossless model	with losses	lossless model

Line Equations

The line equations give values U(z) and I(z) for any point z along the line.

here: U(z) , U_A , U_E : line-to-ground-voltages

with losses

U( z )	=	U_E · cosh( γ z )	+	I_E · Z₀ · sinh( γ z )

I( z )	=	I_E · cosh( γ z )	+	( U_E / Z₀ ) · sinh( γ z )

Z₀ = sqrt(	R' + j ω L'	) ; ω = 2 π f

	G' + j ω C'

Z₀	:	characteristic impedance
R'	:	resistance	[ Ohm / km ]	L'	:	inductance	[ mH / km ]
G'	:	conductance	[ mhO / km ]	C'	:	capacitance	[ nF / km ]

γ = sqrt( ( R' + j ω L' ) · ( G' + j ω C' ) ) = α + j β

γ = α + j β : propagation constant
α : attenuation constant [ 1 / km ] β : phase constant [ ° / km ]

cosh γ = ( e^γ + e^-γ ) / 2 ; sinh γ = ( e^γ − e^-γ ) / 2

e^γ = e^{α + j β} = e^α · e^{j β} = e^α · ( cos β + j sin β )

lossless line

R' = 0 ; G' = 0 =>

Z₀ = sqrt(	R' + j ω L'	) = sqrt( L' / C' ) = Z₀ (pure resistance)

	G' + j ω C'

γ = sqrt( ( R' + j ω L' ) · ( G' + j ω C' ) ) = α + j β = 0 + j ω · sqrt( L' · C' )

γ = j β ; cosh( j β ) = cos β ; sinh( j β ) = ( e^jβ - e^-jβ ) / 2 = j sin β ;

U( z )	=	U_E · cos( β z )	+	j I_E · Z₀ · sin( β z )

I( z )	=	I_E · cos( β z )	+	j ( U_E / Z₀ ) · sin( β z )

Z₀ = sqrt( L' / C' ) ; β = ω · sqrt( L' · C' )

Power ( 3-phase )

S( z )	=	P( z ) + j Q( z ) = 3 · U( z ) · I^*( z )

	=	3 · ( Re{ U(z) } + j Im{ U(z) } ) · ( Re{ I(z) } - j Im{ I(z) } )

	=	3 · ( Re{ U(z) } · Re{ I(z) } + Im{ U(z) } · Im{ I(z) } )

	+ j	3 · ( Im{ U(z) } · Re{ I(z) } − Re{ U(z) } · Im{ I(z) } )

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400 kV overhead line ( 200 km )

zero load

P_E = 0 MW ; Q_E = 0 Mvar

| U_E | = 400 kV > | U_A | = 390.8 kV

Q_A = −141 Mvar < 0 ( capacitive )

P_A = 281 kW ; δ = 0.17°

lossless line

P_A = 0 kW ; δ = 0°

dU / dP ; dU / dQ

ΔP_E	ΔQ_E	\| U_A \|	ΔU_A
0 MW	0 Mvar	390.8 kV	0 kV
+100 MW	0 Mvar	392.6 kV	1.8 kV
0 MW	+100 Mvar	403.4 kV	+12.6 kV
0 MW	−100 Mvar	378.2 kV	−12.6 kV

=> dU / dP << dU / dQ

natural load / surge impedance load

Q(z) = 0 Mvar ; I_N = 2.58 kA ; I / I_N ≈ 40 %

low load:	Q_A ( − Q_E ) < 0	capacitive
high load:	Q_A ( − Q_E ) > 0	inducitive

maximum load:

Q_loss > 0 inductive

220 kV (underground) cable ( 20 km )

zero load

Q_A = −84 Mvar ; I_N = 0.315 kA ; I / I_N ≈ 71 %

maximum load:

Q_loss < 0 capacitive

π equivalent circuit

The equivalent circuit gives values only for the (receiving and sending end) terminals of the line.

exact π equivalent

U_A = U( z = l ) ; g = γ · l ; l = line length

U_A = U_E · ( 1. + Z_L / Z_Q )	+ I_E · Z_L

U_A = U_E · cosh g	+ I_E · Z₀ · sinh g

cosh g = 1. + Z_L / Z_Q ; Z₀ · sinh g = Z_L

Z_Q =	Z_L	= Z₀ ·	sinh g

	cosh g - 1.		cosh g - 1.

Z_L = Z₀ · sinh g ; Z_Q = Z₀ ·	sinh g

	cosh g - 1.

lossless

b = β · l ; Z_L = j X_L = j Z₀ · sin b ; Z_Q = j X_C = j Z₀ ·	sin b

	cos b - 1.

approx. π equivalent

Simplified calculation of the impedances without trigonometrical functions for l < 200 km

sinh g ≈ g for small values of g (Taylor series)

Z_L = Z₀ · sinh g ≈ Z₀ · g = ( R' + j ω L' ) · l

cosh g ≈ 1. + g² / 2 for small values of g (Taylor series)

Z_Q = Z₀ ·	sinh g	≈ Z₀ ·	g	= 2 · Z₀ / g =	2

	cosh g - 1.		g² / 2		( G' + j ω C' ) · l

Z_L ≈ ( R' + j ω L' ) · l ; Z_Q ≈	2

	( G' + j ω C' ) · l

lossless

R' = 0 ; G' = 0 =>

Z_L = j X_L ; X_L ≈ ω L' · l ; Z_Q = j X_C ; X_C ≈ - 2 / ( ω C' · l ) < 0