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THE OPTIMIZATION METHOD OF SUPERCONDUCTING MAGNET DESIGN
T. Fujioka, T. Uchiyama, H. Ichikawa
To cite this version:
T. Fujioka, T. Uchiyama, H. Ichikawa. THE OPTIMIZATION METHOD OF SUPERCON- DUCTING MAGNET DESIGN. Journal de Physique Colloques, 1984, 45 (C1), pp.C1-885-C1-888.
�10.1051/jphyscol:19841180�. �jpa-00223656�
JOURNAL D E PHYSIQUE
Colloque CI, supplgment au no 1, T o m e 45, janvier 1984 page Cl-885
THE OPTIMIZATION METHOD OF SUPERCONDUCTING MAGNET DESIGN
T. Fujioka, T. Uchiyama and H. Ichikawa
Fusion ~echnology DeveZopment Dept., Toshiba, 13-12, Mita 3-chome, Mimto-ku, Tokyo, Japan
RBsumd
-
On propose une mdthode d'optimisation des aimants supracon- ducteurs et on pr6sente quelques r6sultats de simulation par calcu- lateur numbrique.Abstract - The optimization method of superconducting magnet design is proposed and results of some computer simulations are presented.
1. Introduction
Recently the optimum design methods have been prevalent not only in the engineering fields but also in other fields like economics. The purpose of optimization is to maximize certain performances and/or to minimize manufacturing cost of an object designed, while satisfying at the same time all design constraints that are usually related in trade-off relationships with each other. Such a optimization problem is usually converted into a nonlinear programming problem, and is solved by computer algorythm. In this paper, we propose the optimization method of the magnet design. Then we analyze an optimum criterion and its formulations, and finally present results of some computer simulations.
The purposes of the proposed method are as follows.
(I) To calculate easily basic parameters which optimize the magnet design using the computer leaving only a small modification in the following detail design.
Conventionally the optimization of the magnet design has been time consuming work because it has been done in a cut and try method. Moreover it has been doubtful whether the obtained design has been exactly optimum or not.
( 2 ) To investigate the future direction required for material development of
the magnet by means of sensitivity analysis concerning each property of the material.
Although as a first step in developing the method, the magnet with circular section and with a one-graded winding is analyzed, we can easily extend our method to the magnet with non-circular section and/or with multi-graded windings with some modifications in the computer software.
2. Optimum Criterion
A schematic diagram of optimum criterion is shown in Fig. 1. It means that the optimum design is to realize such a magnet whose conductor current density J attains as high as possible, while satisfying outer and inner design constraints at the same time. Detail explanations of Fig. 1 are as follows.
2.1 Outer Design Constraint
This constraint requires that the magnet should be designed to have central bore field Bo, inner diameter Di and longitudinal length H all specified from the viewpoint of magnet utilization. In other words, it means the outer diameter of the magnet Do should be determined so that the magnet attains Bo with Di, H and also J where J is given in each optimization procedure. At the same time, we can calculate maximum experienced field Bmax and magnet inductance L.
2.2 Inner Design Constraints
These constraints are to be satisfied in order to secure the stable and safe operation of the magnet, and are related in trade-off relations to each other as shown in Table 1.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19841180
JOURNAL DE PHYSIQUE
<Outer Oesign Canrtraint >
<Desired Condition >
r I
Maximization of conductor current density
cost and most compact magnet
Following parameters specified for the purpose of the magnet should be achieved in the design.
0 Contral field : Bo 0 Inner diameter of winding : Di 0 Longitudinal length of winding : H
<Inner Design Constraints >
Constraint 1
Enough critical current margin should be secured.
I
~~~~~~~
i o u l d be thermally stabiiized.1
+I
~ " ' ~ n d u c t o r stress should be with," atlowable deskan h i t .I
Constrain! 5
Feasibility of manufacturing should be guranteed.
(Conductor aspect ratio, total turn number)
I
-
Fig. 1 Optimum Criterion for Superconducting Magnet Design
Magnet should be safely protected when it quenches.
(Temperature rise, diyharging voltage)
C o n s t r a i n t 1 : C o n s t r a i n t 1 r e q u i r e s t h a t a r a t i o , c r i t i c a l c u r r e n t I c / o p e r a t i n g c u r r e n t I o p , s h o u l d be o v e r a p r e d e t e r m i n e d c o n s t a n t i n o r d e r f o r t h e magnet t o be o p e r a t e d s a f e l y a g a i n s t t h e i n c r e a s e o f c u r r e n t c a u s e d by s u c h d i s t u r b a n c e s a s a n e x t e r n a l f l u x change and s o on.
C o n s t r a i n t 2: C o n s t r a i n t 2 r e q u i r e s t h a t t h e r m a l s t a b i l i t y of t h e c o n d u c t o r s h o u l d be g u a r a n t e e d i n Maddock c r i t e r i o n ; a r a t i o , e q u a l a r e a h e a t f l u x Q e / J o u l e h e a t f l u x Qg, s h o u l d be o v e r a t l e a s t 1.0. Here t h e r e i s a l i t t l e d i f f i c u l t y t h a t a r e a l v a l u e of Q e i s known o n l y by a c t u a l measurements. But we p r o p o s e t o s o l v e t h i s d i f f i c u l t y by t h e f o l l o w i n g p r o c e d u r e , b a s e d on a f a c t t h a t g e n e r a l l y Qe i s dominated by t h e s u r f a c e t r e a t m e n t o f t h e c o n d u c t o r o n l y . For t h e f i r s t o p t i m i z a t i o n , we e s t i m a t e Qe by r e f e r r i n g r e a s o n a b l e c o o l i n g h e a t f l u x d a t a measured p r e v i o u s l y . Then we perform t h e o p t i m i z a t i o n a g a i n u s i n g r e a l Qe o b t a i n e d by a c t u a l measurements and i t e r a t e t h i s p r o c e d u r e i f n e c e s s a r y .
C o n s t r a i n t 3: We p r o p o s e t o a p p r o x i m a t e maximum c o n d u c t o r s t r e s s a s SBmax . D i
.
Jb e c a u s e s u c h a c o m p l i c a t e d method a s FEM i s n o t s u i t a b l e i n t h i s o p t i m i z a t i o n . The above e s t i m a t i o n i s b a s e d on a f a c t s t h a t u s u a l l y maximum s t r e s s o f t h e c o n d u c t o r o c c u r s a t a n i n n e r m o s t t u r n and a l s o a f a c t t h a t i t i s j u s t a rough e s t i m a t i o n b u t w i t h s a f e t y m a r g i n f o r t h e s t r e s s d e s i g n . T h e r e f o r e , c o n s t r a i n t 3 r e q u i r e s t h a t r a t i o , a l l o w a b l e d e s i g n l i m i t of c o n d u c t o r s t r e s s o D / @ Bmax -J-Di) , s h o u l d be o v e r a t l e a s t 1.0.
C o n s t r a i n t 4: The e n e r g y of t h e magnet s h o u l d be d i s c h a r g e d t h r o u g h a dumping r e s i s t o r i n t h e c a s e o f i t s quench s o a s t o p r o t e c t i t from damage. I n t h i s c a s e , maximum t e m p e r a t u r e r i s e of t h e c o n d u c t o r i s g e n e r a l l y p r e f e r a b l e n o t t o exceed a b o u t 60 K from t h e v i e w p o i n t o f t h e t h e r m a l s t r e s s . On t h e o t h e r h a n d , maximum d i s c h a r g i n g v o l t a g e o f t h e magnet Vmax s h o u l d n o t exceed t h e a l l o w a b l e d e s i g n l i m i t of d i e l e c t r i c s t r e n g t h o f t h e i n s u l a t i o n VD. A s i s e a s i l y u n d e r s t o o d , t h e s e two c o n s t r a i n t s c o n f l i c t w i t h e a c h o t h e r . Then c o n s t r a i n t 4 r e q u i r e s t h a t t h e s e two c o n s t r a i n t s a r e s a t i s f i e d a t t h e same t i m e .
C o n s t r a i n t 5 : C o n s t r a i n t 5 i s t o check t h e f e a s i b i l i t y o f m a n u f a c t u r i n g t h e magnet. Although t h e r e a r e some l i m i t a t i o n s c o n c e r n i n g m a n u f a c t u r i n g , we s e l e c t e d two o f them a s t h e e s s e n t i a l o n e s b e i n g r e l a t e d d i r e c t l y w i t h t h i s method. Namely c o n s t r a i n t 5 r e q u i r e s t h a t c o n d u c t o r a s p e c t AR and t o t a l t u r n o f
t h e magnet N s h o u l d be w i t h i n e a c h a l l o w a b l e l i m i t s ARD and ND , r e s p e c t i v e l y . 2.3 D e s i r e d C o n d i t i o n
A f t e r a n a l y z i n g what c o n d i t i o n s a r e t o be d e s i r e d i n t h e magnet d e s i g n , we concluded t h a t t h e f o l l o w i n g two c o n d i t i o n s a r e c o n s i d e r e d t o be e s s e n t i a l .
( 1 ) The manufacturing cost of T a b l e 1 Trade-off R e l a t i o n s h i p b e t w e e n E a c h l n n e r Design C o n s t r a i n t
the magnet is desired to be as low as possible.
( 2 ) The structure of the magnet
is desired to be as compact as possible.
Now in the case of the magnet with the one graded winding, it is clear that (1) and (2) have the same meanings,
Namely both conditions can be realized at the same time by designing J to be as high as
possible. Consequently we can define that the desired condition is to maximize J itself.
3. Formulations
We can formulate the above optimum criterion as follows. Definitions of param- eters are shown in Table 2.
3.1 Outer Design Constraint
The outer diameter Do of the magnet is expressed by a following equation.
1. To increase the design margin of critical current
2. To increase the design margin of thermal stability
3. To increase the design margin of stress
4. To increase the design margins of temperature rise of conductor and discharge voltage when magnet quenches
3.2 Inner Design Constraints
J -*smaller CR -r smaller
J -r smaller. CRflarger,lc + larger
Paa+Pbb-+ larger, p (Ernax, oo ) 'smaller a.b
J -+smaller
p (Bmax, O D ) -larger (hardening of Cu) J -r smaller, CR -r larger,
a.b f larger, plBmax, cr. ) f smaller
Constraint 1 : -!!!.- Jc(Bmax)
l o p Je(CR+I) > + OL1
(1 ) T a b l e 2 D e f i n i t i o n of Parameters
Constraint Qe(Bmax, l c ) - - P(Pa.a+Pb.b).CR.Qe(Bmax, Ic) > 1 Qg a.b.h,.J2.p(Brnax, a, ).(CR+l)
It is to be noticed that Qe is a function of Rmax and Ic while p is a function of Bmax and OD.
Here the improvement of OD means hardening the stabilizer (copper) that results in the increase of p.
Constraint 3 : = 2oD > 1
a,, Bmax.J.Di
.j@
-/.C(T) dTConstraint 4
:x
= 2VD 4.2k p(Brnax, O D , T)V m a x ('cR+~).J 2 > 1 ( 5 )
L'[? 1 .a.b.A,-J
where hot spot model is applied to the calculation of the temperature rise of the conductor.
Constrain 5 : A R D >-> b AD; > 1
a
C
b-A
i t 2Fig. 2 D e f i n i t i o n s of
a.b (a+t, 1-(b+t, )
=- S a . b
Cross sectional area of conductor a n d A2
Width o f cross section of conductor Height o f cross section of conductor Over all copper ratio
Current density of conductor (metal Outer diameter o f winding area) lnner diameter o f winding Central bore field Permeability of conductor Packing factor 1 (see Fig. 2 ) Longitudinal length of winding Critical current
Operation current Critical current density A constant
Qe : Equal area heat flux in Maddock Criterion
Qg : Joule heat flux Bmax :
Pa, Pb :
A2 :
P :
a,, :
Vmax. :
c :
L :
ARD , ARD' '
Experienced maximum field Perimeter ratio of a and b, respectivel) Packing factor 2 (see Fig. 2) Specific resistance of stabilizer (Cu) Allowable design limit of conductor stress
Maximum stress of conductor Allowable design limit of dielectric strength of magnet insulation Maximum discharge voltage Specific weight o f conductor Specific heat o f conductor Self inductance of magnet Allowable design limits of aspect ratio o f conductor
N : Total turn number ND . Allowable design limits of N ND' '
C1-888 JOURNAL DE PHYSIQUE
3.3 Desired Condition
The formulation of the desired condition is as follow.
J + maximum
3.4 Independent Variables for Optimization
From equations (1) % (81, it is clear that at least a, b, CR and J should be selected as independent variables for the optimization procedure. Finally we can conclude that the optimization problem can be converted into the following nonlinear programming.
Search a, b, CR and J which maximize J while satisfying inequalities (1) % ( 7 ) at the same time
.
4. Algorythm
We applied
sum1)
(Sequential Unconstrained Minimization Technique) to the algorhythm which is the typical one for the nonlinear programming problem. As is well known, it is inevitable that an obtained solution by the above algorythm does not guarantee a global optimum one because of its nonlinearity. However, we can cope with this difficulty by parameter survey with respect to the initial values of a, b , CR and J, and can usually find a global optimum solution.Table 3 Comparison between JoPT Optmized and JD Previously Designed 5. Simulation Result
5.1 Simulation 1
As simulation models, taking the magnet A and B manu- factured by Toshiba previ- ously, comparison are made between J O ~ T optimized by this method and those of former
design. The results are shown in Table 3 which BmaXc"t'ca'
indicates the effectiveness of this method.
5.2 Simulation 2
Next, the imaginary magnets C , D and E are chosen as
the simulation models in order to examine the Stabaltfyand dm2t.tc.J lntulatlon
sensitivities of such properties as Qe, p , Jc, VD and
UD in the optimization. The results are shown in Table ~ f a b ~ ~ ~ f y
4. Based on the results of further simulations, we can
Stored energy
conclude that the following inclinations exist in the Flg'3
magnet (1) design The improvement which is also of illustrated OD will be in the Fig. most 3. important Dominating
k
subject factors in in the order magnet design to realize such gigantic magnets as the ones used for the fusion reactor.( 2 ) The thermal stability design is the most severe problem in the design of
middle sized magnets today. However, the electrical insulation design also becomes a severe problem in the case of higher field magnets with larger size than those of today.
( 3 ) The effectiveness of the improvement of Jc is especially remarkable in the
design of high field magnets with a relatively small bore where CR is forced to have a very small value, that is, less than about 2.0.
Table 4 Increase AJ of J o p ~ when 10% improvement of each property of material is assumed
Reference
1 ) Fiacco, A.V. and G.P. McCormick: "Nonlinear Programming: Sequential
Unstrained Minimization Techniques", John Wiley & Sons, Inc., New York, 1968.
