3_Lines_Travelling_waves_VERA_2.pdf

Travelling waves on transmission lines and line modeling in EMTP/ATP Oct 15, 2012 - Bp

Electromagnetic wave propagation along a horizontal, lossless overhead line ΔV I(x+Δx,t) I(x,t)

V(x,t)

V(x+Δx,t) ΔI

x

x+Δx

∂I − ΔV = L Δx ∂t ∂U − ΔI = C Δx ∂t 2 / 64

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Wave eq. (telegraph equations) ∂I − ΔV = L Δx ∂t ∂V − ΔI = C Δx ∂t

∂V ∂I = −L ∂x ∂t

∂ 2V ∂2I = −L 2 ∂x ∂x ∂t

∂I ∂V = −C ∂x ∂t

∂2I ∂ 2V = −C ∂x ∂t ∂t 2

∂ 2V = LC 2 ∂x ∂2I = LC 2 ∂x

∂ 2V ∂t 2 ∂2I ∂t 2 11/6/2012

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Solution of wave equations I ( x, t ) = f1 ( x − vt ) + f 2 ( x + vt ) V ( x, t ) = Z 0 f1 ( x − vt ) − Z 0 f 2 ( x + vt ) where v=

1 LC f1(x,t)

,

Z0 =

L C +v

-v

f2(x,t) x

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Example 10km 10km 10km 10km 10km 10km 10km 10km 10km 10km SRC

V1

V3

V

V

V

V5 V

V7

V9

END

V

V

V

1 p.u.

Line parameters, L=1.6mH/km, C=10nF/km makes Zo=400 ohm, v=250m/us 2.5 [V] 2.0 1.5 1.0 0.5 0.0 -0.5

0.0

0.4

0.8

(file AC2_Tr_waves.pl4; x-var t) v:SRC

v:V1

1.2 v:V3

v:V5

1.6

v:V7

v:V9

[ms]

2.0

v:END

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Example (cont.) SRC V

V1 V

V3

V5

V7

V9

END

V

V

V

V

V

1 p.u. Single phase distr. parameter line, Zo=400 ohm, v=250m/us 2.5 [V] 2.0 1.5 1.0 0.5 0.0 -0.5

0.0

0.4

(file AC2_Tr_waves.pl4; x-var t) v:SRC

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0.8 v:V1

1.2 v:V3

v:V5

v:V7

1.6 v:V9

[ms]

2.0

v:END

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Reflection and refraction of waves V2

V3

V1 ZA

ZB

V2=(ZB-ZA)/(ZA+ZB) V1 = ρ V1 V3=(2ZB)/(ZA+ZB) V1 = β V1 ZB=infinite (open circuit) ZB=zero (short circuit)

β=2, ρ=1 β=0, ρ= -1

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Simulation time step selection z z

Small enough to meet sampling requirements for signal reconstruction (dT < 1 / 2 x fmax) Much smaller than smallest time constant or natural oscillation SRC period in circuit V

H

70 [V] 60

100 [V] 80

50 60

40 30

40

20 20

10 0

0 2 4 6 (file AC2_Surge_DT_selection.pl4; x-var t) v:SRC

Δt=10ns 8 / 64

8

[us]

10

0

0 2 4 6 (file AC2_Surge_DT_selection.pl4; x-var t) v:SRC

8

[us]

10

Δt=3μs 11/6/2012

Simulation time step selection (II) Vcap 10 ohm

1 mH

V

10 nF

200 [V]

fmax=50 kHz

160 [V] 140

160

120 100

120

80 80

60 40

40

20 0 0.00 0.02 0.04 0.06 (file AC2_Surge_DT_selection.pl4; x-var t) v:VCAP

0.08

[ms] 0.10

0 0.00 0.02 0.04 0.06 (file AC2_Surge_DT_selection.pl4; x-var t) v:VCAP

0.08

[ms] 0.10

Δt=10 μs fsample=100kHz

Δt=0.1 μs fsample=10MHz

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Recommended dT (time step) Phenom ena Being Studied Lightning surges Transm ission Line Switching Surges Capacitor switching Short circuits M achine dynam ics

Typical Study D urations 100-200 μs

Typical Tim e Step .1 -1 μs

.2 - 1 m s

1-20 μs

1 - 100 m s

10-100 μs

0.1 - 1 s

10-200 μs

0.5 - 5 s

100-1000 μs

(About 5 - 10 samples within a period of the highest frequency of interest)

Overhead line models (single phase)

R’l/4

Zo,v length/2

R’l/2

Zo,v

R’l/4

length/2

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Line Models in ATP-EMTP z

Nominal-Π model – – – – – – –

Frequency independent Lumped-parameter model Skin effect and earth return corrections Multi-phase lines can be represented No time step limit (often misleading!) Not suitable for transient studies! PI sections can be connected in cascade for transient studies with certain limitations: ¾ ¾ ¾

z z

Produces reflections at the cascading points Computationally expensive Sections must be kept very short { 5-10 km for frequencies up to about 2 kHz}

Frequency independent distributed parameter line model (CPDL) Frequency dependent distributed parameter line model (JMarti)

Line Models for Transient Studies z

Constant parameter distributed line model (CPDL) – – – – – – –

z

Model assumes that per unit R’, L’, & C’ are constant L’ & C’ are distributed Losses R’*l are assumed to be lumped in three places Shunt losses are ignored Use traveling wave solutions and valid over a wide frequency range Require transformations between phase and modal domain Keep track of modal waves traveling at different speeds

Frequency dependent transmission line model (JMarti) – – – – –

Represents distributed nature and freq. dependency of all line parameters accurately Transformation matrix can be real or complex, but constant Most accurate model for transient studies Approximations in the transformation matrix for untransposed lines Can be used with compromise if asymmetry gets stronger

Frq. dependency of line parameters Zero sequence resistance vs. frequency

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Frq. dependency of line parameters Zero sequence inductance vs. frequency

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Transient solution in a larger network For the lossless line with characteristic impedance Z and travel time τ, the expression v + Zi does not change if observer travels forward with the wave (method of characteristics):

or

v 5 ( t − τ ) + Zi 51 ( t − τ ) = v1 ( t ) − Zi15 ( t ) 1 i15 ( t ) = v1 ( t ) + I 15 ( t − τ ) Z

Equivalent circuit:

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Input data needed for line modeling z z z z z z z z

(x,y) coordinates of each conductor and shield wire; bundle spacing, orientations; sag of phase conductors and shield wires; phase and circuit designation of each conductor; phase rotation at transposition structures; physical dimensions of each conductor; DC resistance of each conductor and shield wire Ground resistivity of the ground return path.

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Overhead line modeling (LCC)

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Line/Cable modeling z

Line/Cable Constants, Cable Parameters – Bergeron, PI, JMarti, Semlyen, Noda(?)

z

View – Cross section, grounding

z

Verify – Frequency response, power frequency params.

log(| Z |)

3.9

Line Check – Power freq. test of line/cable sections

z

2.7

1.5

log(freq)

0.4 0.0

2.0

4.0

6.0

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Cable modeling (CC & CP) z

Specify – – – –

z

Automatic ATPexecution: – – – – –

z

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Geometrical data Electrical param. 1-21 phases Max 42 overhead cond.,16 cables

Bergeron PI-equiv. Semlyen JMarti Noda

View and Verify modules 11/6/2012

Specification of data + View module

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Line Check z z

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The user selects a group in the circuit ATPDraw identifies the inputs and outputs (user modifiable)

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Line Check (cont.) z

ATPDraw reads the LIS-file and calculates the series impedance and shunt admittance

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Conclusions z z z

Use PI-exact model for steady state studies Use FD-line models for lines of main interest in your study Use CPDL-line models for lines of secondary interest

Time controlled switches Conventional deterministic switch – – – –

Model circuit breakers Model short circuits Other similar switching devices Modeled as ideal element: I = 0 when open, R = 0 when closed

Statistics switch –

Random closing/opening switch

Systematic switch

Voltage controlled switch – – – – – – –

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Simulate protective gaps Surge arrester gaps Series capacitor gaps Flashovers across insulators Normally open at the start of the simulation Closes when the voltage across the switch exceeds a user-defined flashover value Opening occurs at the first current zero, provided the user defined delay time has elapsed

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TACS controlled switches – – –

Type 11 - Simulates diodes and thyristors Type 12 - can be used to simulate triacs and spark gaps Type 13 - General TACS controlled switch

Load Models Load models should satisfy two requirements – –

The power frequency MVA load should be accurate in order to set up the proper initial conditions The low and high frequency characteristics should also match the physical load to properly represent its effects on harmonics and transients

Series RL and Parallel RL models – – – –

Produce correct initial conditions Inaccurate at higher frequencies Series RL provides very little damping at high frequencies Parallel RL provides a constant damping at higher frequencies

Nonlinear branches type 99 : Pseudo-nonlinear resistance type 98 : Pseudo-nonlinear inductance type 97 : Staircase time-varying resistance type 96 : Pseudo-nonlinear hysteretic inductor type 94 : User-defined component via MODELS type 93 : True, nonlinear inductance type 92 : – Exponential ZnO surge arrester – Multi-phase, piece-wise linear resistance with flashover type 91 : Multi-phase time-varying resistance TACS/MODELS controlled resistance User supplied Fortran nonlinear element

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A non-linear circuit V

R

I L

V L

L

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A piece-wise linear ψ-i characteristic Ψ( t ) Ψ(t - Δt)

Ψ slope

kn

iL iL(t- Δ t) iL(t)

kn[iL (t ) − iL (t − Δ t )] = ψ (t ) − ψ (t − Δ t ) 11/6/2012

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dψ vL = dt

dψ (t ) dt ⇒ ∫ vL (t ) dt = ∫ dt t − Δt t − Δt t

t

1 [vL(t ) + vL(t − Δt )]Δt = ψ (t ) −ψ (t − Δt ) 2

kn[iL (t ) − iL (t − Δ t )] = ψ (t ) − ψ (t − Δ t ) i L (t ) = 32 / 32

Δt 2 kn

v L (t ) + i L (t − Δ t ) +

Δt 2 kn

v L (t − Δ t ) 11/6/2012