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Transient analysis of transformersTransient analysis of transformers and cables for offshore wind
tiBjørn Gustavsen
connectionBjørn Gustavsen
SINTEF Energy ResearchTrondheim, Norway
Wind power R&D seminar – deep sea offshore wind: Trondheim, Norway, 20-21 January, 2011.
SINTEF Energy Research
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OutlineOutline
Modeling of transformers and cablesModeling of transformers and cables
High-frequency transformer-cable resonanceHigh frequency transformer cable resonance
Wind powerWind power
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PART I :Modeling of transformers and cables
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High-frequency transformer modelingHigh frequency transformer modeling(black box)
1. Characterize the transformer by its frequency domain behavior at its external terminals
2. Identify a model which emulates the behavior of the transformer, as seen from the terminals.transformer, as seen from the terminals.
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Terminal characterization by admittance
t l t i l
ymatrix
external terminals
n=i Y v
1 11 12 1 1( ) ( ) ( ) ( ) ( )nI Y Y Y Vω ω ω ω ω⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥
1( )I ω 1( )V ω
2 21 22 11 2( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
I Y Y Y V
I Y Y Y V
ω ω ω ω ω
ω ω ω ω ω
⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
2 ( )I ω
( )I ω
2 ( )V ω
( )nV ω: Transformer
1 2( ) ( ) ( ) ( ) ( )n n n nn nI Y Y Y Vω ω ω ω ω⎣ ⎦ ⎣ ⎦ ⎣ ⎦( )nI ω ( )n
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Measurement of admittance matrixeasu e e t o ad tta ce at
Network analyzerConnection boxCoaxial cablesCoaxial cablesCurrent sensor
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Built-in current sensor (Pearson)B. Gustavsen, 2010
Modeling via rational functions g
1 11 12 1 1( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
nI Y Y Y VI Y Y Y V
ω ω ω ω ω⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥
1( )I ω 1( )V ω( )V Measurement
2 21 22 11 2
1 2
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
I Y Y Y V
I Y Y Y V
ω ω ω ω ω
ω ω ω ω ω
⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
2 ( )I ω
( )nI ω
2 ( )V ω
( )nV ω: Transformer
Measurement
1 2( ) ( ) ( ) ( ) ( )n n n nn nI Y Y Y Vω ω ω ω ω⎣ ⎦ ⎣ ⎦ ⎣ ⎦( )n
Yn
Model extraction(fitting)
1( )
Nm
nm mj a
ωω=
≅ +−∑ RY D
Rational model
The rational model is compatible with EMTP type circuit simulators
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The rational model is compatible with EMTP-type circuit simulators
Procedure for rational fittingProcedure for rational fitting
1 Calculate a rational model using Vector Fitting1. Calculate a rational model using Vector Fitting
1( )
Nm
j aω
ω≅ +
−∑ RY D
2. Enforce passivity by residue perturbation
1m mj aω=
1
Nm
m ms a=
Δ= + Δ ≅
−∑ RΔY D 0
N Δ∑ R1
(Re{ })m
m m
eigs a=
Δ+ >
−∑ RY 0
( )eig + Δ >D D 0
Matrix Fitting Toolboxh // i f / d k /VECTFIT/i d
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http://www.energy.sintef.no/produkt/VECTFIT/index.asp
High frequency cable modelingHigh-frequency cable modeling
1 Characterize the cable by its per-unit-length series1. Characterize the cable by its per unit length series impedance matrix Z and shunt admittance matrix Y
( ) ( ) ( )jω ω ω ω= +Z R L
2 F Z d Y l l t t f f
( ) ( ) ( )jω ω ω ω= +Z R L
( ) ( ) ( )jω ω ω ω= +Y G C
2. From Z and Y, calculate parameters for a frequency-dependent traveling wave model.
This modeling capability is available in EMTP-type programs
Main challenge: calculate Z− Analytical expressions
Fi it El t
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− Finite Element
PART II :Cable-transformer high-frequency resonance
B. Gustavsen, “Study of transformer resonant overvoltages caused by cable-transformer high-frequency interaction” IEEE Trans Power Delivery vol 25 no 2 pp 770-779 April
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high frequency interaction , IEEE Trans. Power Delivery, vol. 25, no. 2, pp. 770 779, April 2010.
Demonstrate that transients on the high voltage side of aDemonstrate that transients on the high-voltage side of a transformer can cause excessive overvoltages on the low-voltage side.
Identify critical cable-transformer configurations that lead t hi h ltto high overvoltages
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Voltage ratio from high to lowVoltage ratio, from high to low
41230 V11 kV
4411230 V11 kV
456
123
456
456
123
123
300 kVA
At high frequencies, the voltage ratio is governed by stray capacitances and inductances, not ampere-winding balance.
50 Hz voltage ratio ≈0.022MHz voltage ratio ≈ 2
A 2 MH i id l lt ld d 100 lt
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A 2 MHz sinusoidal voltage would produce a 100 p.u. overvoltage
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Laboratory measurementLaboratory measurementHigh-voltage
Low-voltage30 Ω
High-voltage
Low-voltage30 Ω
Step voltage
voltage voltage30 Ω456
123
Step voltage
voltage voltage30 Ω456
456
123
123
Cable (27 m)Cable (27 m)
Before connecting cable to transformer After connecting cable to transformer
24 p u overvoltage !!~24 p.u. overvoltage !!
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Measurement-based model of transformer230 V11 kV 230 V11 kV
300 kVA456
123
230 V11 kV456
456
123
123
230 V11 kV
– Frequency sweep measurements of Y(ω)
63 6633
– Model extraction by Matrix Fitting Toolbox
N R
1( ) m
mm j aω
ω=≈ +
−∑ RY D
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Measurement-based model of 27-m cable
A AAAA AAA
v1 v2
CableSR
Cable50 Ω1̂v 2v̂
SRv1 v2
CableSR v1 v2
CableSRSR
Cable50 Ω1̂v 2v̂
SR
Cable50 Ω1̂v 2v̂
SRSR
ba
yy
a= −
( 1)b
bybR
=a( 1)R
a−
Model extraction by Matrix Fitting ToolboxN
ya , yb
1( )
Nm
mm j aω
ω=≈ +
−∑ RY D
ya , yb
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Simulation vs. measurementHigh-voltage
Low-voltage30 Ω
1
High-voltage
Low-voltage30 Ω
11
Cable (27 m)
Step voltage 456
123
Cable (27 m)
Step voltage 456
456
123
123
Cable (27 m)Cable (27 m)
Before connecting cable to transformer After connecting cable to transformer
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Max. overvoltage vs. cable lengthMax. overvoltage vs. cable length– State-of-the art frequency-dependent traveling-wave
type model obtained from geometry.
– Compute max. overvoltage on low-voltage side for alternative cable lengths
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Ground fault.M lt bl l thMax. overvoltage vs. cable length
High-voltage
Low-voltage30 Ω
High-voltage
Low-voltage30 Ωideal
Step voltage
g g30 Ω456
123
Step voltage
g g30 Ω456
456
123
123
ideal
Cable (27 m)Cable (27 m)20 m cable (open LV)Overvoltage in p.u. of applied voltage
~43 p.u. overvoltage !!
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Switching overvoltages (1)g g ( )
300 m 5000 m 20 mcablecable300 m 5000 m 20 mcablecable
cable
close@ t=0
overhead line1 p.u.
clos
cable
close@ t=0
overhead line1 p.u.
clos
– Several parallel cables connected to bus
– Combined characteristic impedance much lower than that of connection cable
– Bus appears as ”stiff” voltage seen from connection cable
Closing CB results in oscillating voltage on cable
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Closing CB results in oscillating voltage on cable
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Switching overvoltages (1)g g ( )300 m cable
5000 m overhead line1 p u
20 mcablecable300 m
cable5000 m overhead line1 p u
20 mcablecableA
cable
close@ t=0
overhead line1 p.u.
clos
cable
close@ t=0
overhead line1 p.u.
clos
~20 p.u. overvoltage !!
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Switching overvoltages (2)g g ( )– Two cables of equal length coupled to the same busbar
One cable is liveT1
close@ 0clos
5000 m 20 m cableoverhead line
1 p.u.
T1
close@ 0clos
5000 m 20 m cableoverhead line
T1
close@ 0clos
5000 m 20 m cableoverhead line close
@ 0clos
5000 m 20 m cableoverhead line
1 p.u.
– One cable is live
@ t=0
20 m cable
T2@ t=0
20 m cable
T2@ t=0
20 m cable
@ t=0
20 m cable
T2
T1
~25 p u overvoltage !!~25 p.u. overvoltage !!
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Switching overvoltages (3)g g ( )5000 m
20 m cableoverhead line1 p.u. 5000 m
20 m cableoverhead line1 p.u.
20 m cable
close@ t=0clos 1 µF
20 m cable
close@ t=0clos 1 µF@@
~43 p.u. overvoltage !!
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Note:Other transformers may have resonances at much lower frequenciesfrequencies.
Overvoltages will occur with longer cables.
Voltage ratio for a 410 MVA generator transformer (434 kV / 21 kV)
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PART III :PART III :Relevance to wind power
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Switching overvoltagesg g
Radials of nearly equal lengthEnergizing a branchEnergizing a branch
oscillating overvoltage on WT transformer HV sideIn the case of short radials (< 1km) the oscillatingIn the case of short radials (< 1km), the oscillating overvoltage has frequency above 50 kHzHigh overvoltages may result on WT transformer LV
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High overvoltages may result on WT transformer LV side by resonance
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Ground fault initiation
Ground fault initiation can cause an oscillating overvoltage in the cableovervoltage in the cable. Frequency depends on fault locationHigh overvoltages may result on WT transformer LVHigh overvoltages may result on WT transformer LV side by resonance
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Notes
The actual overvoltage on the WT LV side is stronglyThe actual overvoltage on the WT LV side is strongly dependent on the network on the LV side
A complete model must be developed
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ConclusionsConclusionsHigh-frequency interaction between the wind turbineHigh frequency interaction between the wind turbine transformers and the cables can lead to excessive overvoltages the transformer LV side. The phenomenon can be triggered by ground fault initiation and by switching.
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Electromagnetic Transients inElectromagnetic Transients inFuture Power Systems.Future Power Systems.
Phenomena, Component Stresses, Modeling
A JIP project (KMB) between SINTEF and industry partners(2011 2015)(2011-2015)
New partners are welcome !Contact: [email protected]
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ObjectiveObjective
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