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14 Oct 2011 1
Coupled Magnetodynamic and Electric Circuit Models for Superconducting Fault Current Limiter
Presentation at the COMSOL Conference 2011 Boston, 13 – 15 October 2011
L. Graber1, J. Kvitkovic1, T. Chiocchio1, M. Steurer1, S. Pamidi1, A. Usoskin2
1Center for Advanced Power Systems, Florida State University 2Bruker Energy & Supercon Technologies
Comsol Conference 2011
Presented at the 2011 COMSOL Conference in Boston
Contents
• Basics of superconducting fault current limiters (SFCL) • Shielding properties of superconductors • FEA magnetic model
– Implementation of shielding properties
• Electric circuit model (“SPICE”) • Model validation
– Experiment with benchtop model – Measurements
• Conclusion
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Inductive Superconducting FCL
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Stack of SC rings
Primary coilMagn. flux lines
~ ZLoad
ZSource LSFCL
USource
Fault t
LSFCL
faultnormal
• Problem: Increasing levels of fault currents in power grids
• SFCL limits fault current without negative impact at normal operation
– Low voltage drop during normal operation
– Low reactive power • Inductive SFCL provide
operational advantages: – No heat influx into cryostat
through current leads – No Joule heating in cryostat
Shielding Properties of HTS
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0
20
40
60
80
100
0 2 4 6 8 10
Shie
ldin
g Fa
cto
r [%
]
External Magnetic Field [mTRMS]
8.26 mT
%100ext
intext
B
BBS
• Measured with the exact same ring of HTS as later used for the validation experiment
• Hall probes inside and outside the ring pick up the magnetic field
• Operation frequency differs slightly from 60 Hz to reduce noise
Finite Element Model
• Shielding properties of HTS modeled by
conductivity of HTS – Exact value of conductivity is not critical – Factor 100 lower due to increased thickness in model
• Primary coil: Multi-turn coil domain with 60 turns of 1 mm Cu wire
• Liquid nitrogen (LN2) in open bath at 77 K (µr = 1; σ = 0 S/m)
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LN2
HTS
Prim. coil
0.1
43 35
66.2 72.5
2
AB
JBvHA
e
t
operation.lt quench/fauin S/m 105.2
operation, normalin S/m 105.20
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iSFCL for FEA Validation: Equivalent Circuit
• Electric circuit with lumped elements (“SPICE”), coupled to the FEA model • Transformer ratio: 240 V : 32 V = 7.5
~
25 kW inverter with output filter and resistive grounding
~ ~
Step down transformer
11
1 7
9 10
8
2 3 4 5 6
12
0
iSFCL & protec. resistor
LS
CS RG
Lh
LL LL RW RW
RL LSFCL
1.35 mH 1.35 mH
9.545 H 100 Ω
10 kΩ 60 µF · 7.52
0.6 mH
0...240 VRMS (phase-to-phase)
7.52
7.52
7.5
1.34 Ω 1.34 Ω 7.52 7.52 7.52
7.52
7.52
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~ ~ ~ =
• Input parameters – Applied voltage (pulse of 7× nominal simulates a fault situation) – HTS conductivity
• Output parameters – Inductance of the iSFCL as a function of time
iSFCL for FEA Validation: Pulse Pattern
7
Time [ms]
Pulse/Fault
Voltage 12 VRMS 84 VRMS 12 VRMS
Recovery Normal Normal
Conductivity
Inductance 0.3 mH 0.3 mH 0.7 mH
2.5·1014 S/m 2.5 S/m 2.5·1014 S/m
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iSFCL for FEA Validation: Magnetic Flux Density
8
a) After fault but before recovery of superconduction (quenched)
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b) Normal operation (shielding)
Bmax = 10.2 mT Bmax = 41.5 mT
L = 0.7 mH L = 0.3 mH
• Primary purpose of this small-scale iSFCL: Validation of FEA model (i.e. conductivity-based magnetic shielding)
iSFCL for FEA Validation: Construction
9
LN2
HTS
Prim. coil
0.1
43 35
66.2 72.5
2
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HTS ring (single layer)
Primary coil on G10 former
Comsol Conference 2011
iSFCL for FEA Validation: Voltage, Current, and Inductance
10
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-50
0
50
Quench of Benchtop FCL (Experimental vs Simulation)
FC
L P
rim
ary
Curr
ent,(A
)
time,(s)
FEA Simulation
Experimental
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-20
-10
0
10
20
FC
L P
rim
ary
Voltage,(
V)
time,(s)
FEA Simulation
Experimental
0.4 0.45 0.5 0.55 0.6 0.65 0.7
-20
-10
0
10
20
Recovery of Benchtop FCL (Experimental vs Simulation)
FC
L P
rim
ary
Curr
ent,(A
)
time,(s)
FEA Simulation
Experimental
0.4 0.45 0.5 0.55 0.6 0.65 0.7
-10
0
10
FC
L P
rim
ary
Voltage,(
V)
time,(s)
FEA Simulation
Experimental
0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.71
2
3
4
5
6
7
8
x 10-4
time,(s)
Calc
ula
ted Inducta
nce,(
H)
FEA Simulation
Experimental
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• FEA results compared to measurements around instant of fault (above, left), instant of recovery (above, right), and ratio of inductance (right)
• Convincing agreement of model and measurement
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Conclusion
• HTS conductivity as input parameter to the FEA model is a valid technique to simulate basic magnetic properties
– Model can be used for parametric studies
• Coupling with circuit model allows interaction with grid components – Enables power hardware-in-the-loop tests
• Computational very efficient – A couple of minutes to calculate on a PC
– Would allow to implement geometry of higher complexity
• Regarding the air core iSFCL... – Ratio of inductance is limited to approx. 1.5 ~ 3 (depends on geometry of primary
coil, cryostat wall thickness, and height to diameter ratio)
– Insertion of an iron core (e.g. I-core) boosts the ratio to 5 ~10
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Additional Slides
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Coreless Model: Parameters
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• Geometry – rAir = 2 m; hAir = 5 m – rCoil = 0.5 m; hCoil = 1.5 m; wCoil =
0.02 m – rSC = 0.4 m; hSC = 2 m; wSC = 1 mm – Optional: rCore = 0.42 m; hCore = hSC
– N = 65 – ACoil = 240 mm2
• Material – Air, HTS, and primary coil:
εr = 1; μr = 1; ρ = 10−15 ; 1 Ωm – Iron core:
εr = 1; μr = 4000; ρ = 1 Ωm
• Load current in the coil – f = 50 Hz; Icoil = 0.5 ; 20 kA;
Itot = N·Icoil
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Coreless Model: Equations and BCs
0
0
Jn
An Magnetic and electric insulation
0
0
coiltot AIj DEJ External current density
All vectors are in cylindrical coordinates
Axis of symmety
0
0
0
2
coiltotr AIj BvHA
Multi-turn coil domain
AB
AE
DEJ
JH
J
jV
AIj coiltot
0
0
0
Governing equations
HB
ED
r
r
0
0
Material equations
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Coreless Model: No-Fault Condition
LCoil = 0.96 mH
ρSC = 10−15 Ωm ICoil = 500 A
Model input
Model output
mH 00.1
22
0
2
Coil
SCCoil
h
rrNL
Cross-validation by adapted formula for long cylindrical coil:
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Coreless Model: Fault Condition
ρSC = 1 Ωm ICoil = 20 kA
Model input
LCoil = 2.19 mH
mH 78.2
2
0
2
Coil
Coil
h
rNL
Cross-validation by simple formula for long cylindrical coil:
Model output
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Coreless Model: Inductance Ratio wrt. Gap Distance
• Primary winding: rCoil = 0.5 m (const) • SC stack radius: rSC = 0.30, 0.32, ...0.48 m
Corresponds to 20 cm gap
Corresponds to 2 cm gap
0
0.5
1
1.5
2
2.5
0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48
Radius [m]
Ind
uc
tan
ce
[m
H]
L_nonFault [mH]L_Fault [mH]
0
1
2
3
4
5
6
7
8
9
0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48
Radius [m]
Ra
tio
(L
_n
on
Fa
ult
/L_
Fa
ult
)
ratio
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Model with Iron Core
Quenched state (20 kA; 1 Ωm) Normal state (500 A; 10−15 Ωm) Gap rCoil − rSC = 40 mm rCoil − rSC = 40 mm LCoil = 11.4 mH LCoil = 0.452 mH Inductance ratio: 25.2
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