Optimal Design of Superconducting Magnets

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Optimal Design of Superconducting Magnets
CRIS
V. Cavaliere  CRIS Ansaldo Via Nuova delle
Brecce, 260 80127 Napoli cavaliere.enzo.cris_at_ansal
dobreda.it
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OUTLINE
Superconducting magnets for high homogeneity
field. Definition of design goals and
constraints Definition of the multiobjective
optimal design problem for Superconducting
magnets. Pareto optimality. Monte Carlo
analysis and Genetic Algorithms. Quench analysis
and protection of superconducting magnets
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MAGNETS OPTIMAL DESIGN

• Superconducting magnets for high homogeneity
  field.
• For NMR spectroscopy application magnetic
  induction up to 20 T with field homogeneity
  better than o.1 ppm in a few cms diameter Volume
  Of Interest (VOI) are required.
• MRI requires polarising field of few tesla in a
  VOI of several cm diameter
• Robust design solutions, less sensitive to
  fabrication defects, should be preferred.

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The desired performance are obtained opportunely
sizing and positioning a set of superconducting
coils
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Split coil configuration of MRI magnets
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The performance requested to the magnet to be
designed could be summarized with the following
figures
Central field B0 ? 1.5 T VOI diameter 0.15
m Field Homogeneity 1 ppm Field Stability
better than 0.1 ppm/h Region allowable for the
coils 0.2 lt r lt 0.4 m -0.4lt z lt 0.4 m
Region allowable for the coils 0.2 lt
r lt 0.4 m 0.2lt z lt 0.4 m
0.5
0.5
1
1
0.45
0.45
2
2
Region
allowed
Region
allowed
0.4
0.4
for
the
coils
for
the
coils
0.35
0.35
0.3
0.3
r axis m
r axis m
0.25
0.25
3
3
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
VOI
VOI
0
0
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
z axis m
z axis m
2D view of the split coils configuration
2D view of the split coils configuration
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Layout of an NMRS Magnet
The higher field requirements of NMRS magnets are
achieved adopting the sketched layout. A number
of coaxial solenoids made of different
superconducting cable serve to produce the main
field. A further set of coils compensate the
inner solenoid end effects in order to achieve
the desired field homogeneity.
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Stray Field reduction requirements
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Tipical NMRS Magnet Specification
Central field B0? 12 T VOI Diameter 1.5
cm Field Homogeneity 0.1 ppm Stability better
than 0.01 ppm/h Region allowable for the
coils 0.06 lt r lt0.6 m -0.4 lt z lt 0.4 m 5 gauss
line at 2.0 m
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Inside the VOI the field may be expressed in
terms of the harmonic expansion
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Peak to peak field homogeneity is expressed by
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Field expansion coefficients An are efficiently
computed by means of analytical formulae. The
field homogeneity requirements may be fulfilled
making vanish the first terms of the field
expansion
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Stray field reduction may be achieved minimising
the filed modulus in set of points at the desired
distance from the magnet centre.
5 gauss line of a 12 T NMR magnet with and
without active shielding
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Further design goals.
Reduce the overall superconductor amount
Achieve compact design
Reduce the stored energy
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Superconducting Materials properties and field
stability
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Design constraints arising from the field
stability requirements
Superconducting cables shows a voltage current
characteristic described by the well known power
law expression. Ec value is for convention equal
to 1 ?V/cm, the index n and Jc are material
depending parameters.
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High critical Temperature Superconductors (HTS)
I and II generation HTS superconducting
tapes Bi2Sr2CaCu2Ox (Bi-2212), Bi2Sr2Ca2Cu3Ox
(Bi-2223) YBa2Cu3Ox. (YBCO),
Cross section of an Ag/Bi-2223 HTS tape
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Crystal structure ed anisotropy in HTS
superconductors
Critical current versus B? and B//
Biscco-2212 Bi2Sr2CaCu2Ox
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Design peculiarity of HTS magnets
The plot shows the critical current behaviour
inside a 4 T Ag/Bi-2223 solenoid. The presence
of a high radial field component at the coil ends
strongly limits the magnet performance.
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Shape optimisation of HTS coils
HTS coils are usually wound in forms of double
pancakes, modifying the shape of the outer
pancakes is possible to reduce the field radial
component increasing the magnet critical current.
This additional goal, introduce difficulties in
computing the objective function.
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Design goals.
Field homogeneity
Stray field reduction
Reduce the overall superconductor amount
Achieve compact design
Reduce the stored energy
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Design constraints.
Geometrical constraints arising from
technological issues Minimal distance between
adjacent coils Constraints imposed by cryogenic
issues Coil thickness
Working current limitation
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MULTIOBJECTIVE OPTIMISATION
F(x) (f1(x), ... , fk(x)) x (x1, ... , xn) ?
S ? Rn
Scalar Objective Function OF
wi ? 0 

• Poor smoothness properties of OF and multiple
  local minima.
• Computationally efficient global search
  algorithms needed.

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• Two main approach to optimized design,
  deterministic and stochastic
• 0-th order and higher orders methods
• Solution is found in a relatively small number
  of steps
• Solution depends on the initial guess ?
  optimal only in a local sense
• stochastic algorithms
• Do not follow a deterministic path
• Global rather than local search for optimal
  solution(s)
• Large number of function calls are required to
  obtain the solution
• The genetic algorithms
• Parameters are coded with a string of binary
  digits chromosomes
• The initial population is composed by a certain
  number of random configurations
• Algorithm evolves by applying randomly three
  different operators.
• Reproduction the individual is simply passed to
  the next generation
• Cross over Two individuals (parents) are
  chosen, and their genetic codes are exchanged in
  a random point of the string. The resulting
  individuals (children) are then inserted in the
  population
• Mutation One or more of the bits in the genetic
  code of the individual are randomly chosen, and
  then are switched. The resulting individual is
  inserted in the population.

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GENETIC ALGORITHMS

• Evolutionary principle best individuals survive
  and reproduce.
• Search space landscape defined by the OF.
• Set a population of individuals in the parameters
  space
• Evaluate fitness of each individual
• For each generation
• Randomly apply the genetic operators Selection,
  Crossover, Mutation
• Evaluate fitness
• Update generation

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PARETO OPTIMALITY
A vector a ?S is said to dominate a vector b ? S
(also written as ) iff
The set X ? S of the non dominated solutions is
said the Pareto Set. The image of X in the
objective space is the Pareto Front.
A way to generate the Pareto Front is to minimise
scalar OF for various choices of wi
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MULTI POPULATIONS GA

• Global search
• Multiple populations concurrently evolving
• Implemented strategy
• Migrations amongpopulations shared information
• Biodiversity the populations aim to the same
  objectives which have different relative
  importance.

Island 1
Island 2
Weights set 1
Weights set 2


Master process



Island 3
Weights set 3

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SENSITIVITY ANALISYS
Errors in coil winding process and assembly
tolerances will result in a pourer field
Homogeneity of the real magnet.
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SENSITIVITY ANALISYS

• Monte Carlo Method
• Random generation of a large number of different
  configuration inside tolerance intervals.
• realistic figure of the device sensitivity with
  respect to the mechanical tolerances
• the statistical parameters give information about
  sensitivity.
• (Pseudo-)Worst case analysis.

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Tolerances Modelling

• Mechanical tolerances and assembly errors
  modelled as Gaussian random variables
• Assessment of the Statistical behaviour of the
  magnets performances

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Initialise the random number generator
Generate N random quantities
Perturb the reference configuration
Compute the field harmonic expansion coefficients
Compute the field homogeneity
Compute the output statistical moments
Check for convergence
no
yes
Stop procedure and plot results
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Statistical moments estimators
Monte Carlo Method
mean
variance
covariance
correlation
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EXAMPLE OF APPLICATION
nominal configuration central field 3 T field
homogeneity 5 ppm over a 10 cm dsv
Field Homogeneity Distribution mean 40 ppm std.
deviation 24 ppm worst case 162 ppm A30
normally distributed around its nominal value
Cross-correlation coeffs.
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ROBUSTNESS IMPROVEMENT

• Sensitivity analysis by the Monte Carlo method
  alongside the Pareto Front
• allows to select the most robust points in the
  Pareto Set
• does not require the early selection of weigths
• requires quite consistent computational effort.

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Non dominated solution ranking
Pareto Front and Montecarlo sensitivity analysis.
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Quench analysis of superconducting magnets
Sketch of a superconducting coil cross section
LTS cables for NMR and MRI magnets
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• When part of the SC winding looses SC properties,
  the current redistributes in the stabilizing
  conductive matrix, determining a thermal
  production.
• Effects of Quench Possible damages due to
• Over Temperature.
• Over Voltages
• Over Currents
• It is therefore necessary to assess the magnet
  behaviour and design a suitable magnet protection
  system.

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Superconducting magnet circuital scheme
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Mathematical Model
The mathematical model is composed by a complex
system of coupled integral and differential
equation. It includes 1) Magnetostatic Equation
(necessary to describe the flux density map B,
whose local value contributes to determine the
critical current density and, consequently, the
losses) 2) Thermal Diffusion Equations
(necessary to describe the temperature profile
and its evolution) 3) Circuit Equations
(necessary to describe the current evolution in
the protection circuit) 4) Non linear and
coupled constitutive relationships of the
materials.
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Fundamental equations
Magnetostatic
Thermal Diffusion
Circuit
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Decoupling the magnetostatic equation
The magnetic field on the superconducting cable
can be evaluated by means of the superposition of
the coil unitary contribution weighted by the
instantaneous currents
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Material Characteristic
Superconducing cable critical current is
expressed by means of empirical formulae.
At the generation temperature current starts
flowing in the normal conductor stabilising
matrix.
In the temperature range of the phenomenon copper
resistivity is strongly variable
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Coupling Terms
SC state 0
NC state
Current Sharing
Coil resistance
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Numerical Simulation
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CONCLUSIONS

• The design of superconducting magnets can be
  achieved adopting suitable multiobjective
  optimization strategies.
• The adoption of Pareto Optimality allows to delay
  the choice of the relative objective importance
• The statistical sensitivity analysis can be used
  to choose robust solution among non dominated
  solutions
• Once a suitable magnet design has been obtained
  it is necessary to protect the windings against
  quench damages, to verify the efficacy of the
  protection circuit a coupled problem has to be
  solved.