Pareto-Optimization of HTS CICC for High-Current Applications in Self-Field

The ENEA superconductivity laboratory developed a novel design for Cable-in-Conduit Conductors (CICCs) comprised of stacks of 2nd-generation REBCO coated conductors. In its original version, the cable was made up of 150 HTS tapes distributed in five slots, twisted along an aluminum core. In this work, taking advantage of a 2D finite element model, able to estimate the cable’s current distribution in the cross-section, a multiobjective optimization procedure was implemented. The aim of optimization was to simultaneouslymaximize both engineering current density and total current flowing inside the tapes when operating in self-field, by varying the cross-section layout. Since the optimization process involved both integer and real geometrical variables, the choice of an evolutionary search algorithmwas strictly necessary.The use of an evolutionary algorithm in the frame of a multiple objective optimization made it an obliged choice to numerically approach the problem using a nonstandard fast-converging optimization algorithm. By means of this algorithm, the Pareto frontiers for the different configurations were calculated, providing a powerful tool for the designer to achieve the desired preliminary operating conditions in terms of engineering current density and/or total current, depending on the specific application field, that is, power transmission cable and bus bar systems.


Introduction
Today's commercial uses of the HTS technology are mainly limited to high-field magnets in the scientific field [1].However, HTS cables are thought to have the potential to significantly reduce the amount of electricity lost in transmission and distribution in the electrical grid, estimated in about 10% of all generated electricity in the US, with this percentage growing year after year [2].Also, in the perspective for commercial fusion reactor, HTS cables are considered the possible candidate magnetic system, because of the larger temperature margin in the coil section more exposed to nuclear heating that could be an advantage for the cryogenic and cooling system [1].Moreover, the compactness of the HTS section would save more space for the blanket systems [3].
In 2013, ENEA proposed [4] a novel design of a HTS 2G wire-based CICC with twisted stacked tapes on slotted core (Figure 1), allowing a fully industrial process implementation.The cable consists of 150 coated conductors, or 2G-wires, inserted in 5 stacks (30 tapes per stack) and included into ducts created in an aluminum core.The ducts have square cross-section to allocate 4 mm width tapes.The manufacturing process is fully compatible with available existing cabling technologies.
For such a configuration, the current density  eng takes into account the total cross-sectional area of the cable including superconducting and nonsuperconducting components, with the exclusion of the jacket section.Depending on the intended use of the HTS cable, the design should provide optimized value of the engineering current density  eng or of the total current   cable flowing into the cable.For example, for high-field magnets a high  eng is a key feature, whereas for power transmission and/or distribution  eng does not constitute a real constraint.Other applications may require a compromise optimal trade-off between these two objectives.Optimization is the mathematical process used to determine minima (or maxima) x * of l objective functions  = [ 1 , . . .,   ] by varying a number of n input design parameters x = [ 1 , . . .,   ], so that a number m of so-called constraint equations  are simultaneously satisfied.In this sense, optimization becomes a crucial tool to achieve the best performance for the HTS cable as a function of the specific application.In particular, multiobjective optimization (also called Pareto-optimization) is useful in optimizing multiple, even conflicting, objectives giving full evidence to trade-off configurations using the Pareto frontier.The Pareto frontier is the graphical representation of the Pareto optimality conditions by showing the front of all the nondominated solutions.A solution is nondominated where another feasible solution better than the current one does not exist in some objective function without worsening other objective functions.On the contrary, dominated solutions are a set of design points performing worse than some other better points.Then, the Pareto frontier gives the designer an exact measure about how different choices in the design may produce changes in the trade-off among multiple conflicting objectives.So, depending on the specific application for the HTS cable, the designer may use the Pareto frontier to drive the design into a configuration which is best suited for one particular application rather than another.In other words, borrowing information from the Pareto frontier, the HTS cable design is customized depending on the relative "importance" of the engineering current density versus the total current.
In the frame of superconductors, Parametric optimization was recently used to design the highly homogeneous superconducting magnets for magnetic resonance imaging [5], or in beam-optics design of a gantry for carbon ions in cancer therapy accelerators [6], or in the optimal design of saturated iron-core superconducting fault current limiter [7], or to support designers with a guideline for selecting a suitable machine topology for a 10 MW wind turbine application [8].
In this work, a novel approach to the design of the HTS cable cross-section, based on the Pareto optimality criterion, is proposed that may represent a useful tool for the designer to choose among different cable's configurations for electric power transmission, depending on the relative "importance" given to the current density  eng with respect to the total current   cable .The validity of this approach has been firstly assessed in self-field operating conditions by taking into account the effects of single tape   dependence on magnetic field strength and direction at 77 K.
In the following, Section 2.1 is dedicated to the description of the numerical model used to evaluate the current distribution within the cable's cross-section.In Section 2.2, the reasons for choosing the selected optimization algorithm are explained comparing pros and cons of the cutting edge optimization techniques, clarifying the advantage of the chosen approach.Section 2.3 is devoted to the detailed explanation of the optimization process in order to make a reproduction of this study possible.In Section 3.1, the results achieved with the optimization process are reported.In Section 3.2, these results are widely commented and discussed.And finally, Section 4 is a brief mention to possible future work based on this study even in different operating conditions.

Magnetostatic Model Description.
Dealing with the complex problem of calculating the current distribution and the magnetic field inside the superconductor in a variety of geometries, such as those of stacked tapes configurations addressed in this paper, can be done using finite element method (FEM).Since the critical current I c of superconducting tapes depends on both the magnetic field's intensity and direction, it is of fundamental importance to optimize the tape distribution within the CICC in order to reduce the selffield degradation on I c .Here, a 2D magnetostatic model of the commercial package COMSOL Multiphysics [9] was used based on the magnetic vector potential A, thus solving the following equation with the suitable boundary conditions: The self-field of stacked tapes has been investigated in a 2D cross-section of the cable under the assumption of ideal current sharing conditions (i.e., currents are free to redistribute with zero resistance in nonsuperconducting areas).In this approach, the stacks are assumed to be infinitely long and straight, and the current is perpendicular to the plane.Tapes have been modeled with their real dimensions: a thickness of 100 m and width of 4 mm.For the description of the   dependence on the local magnetic field B and its direction  (defined as the angle between the magnetic field and the normal vector to the superconducting tape), the following expression was used [10,11]: Although ( 2) is not based on a physical model, as the original anisotropic Kim model [10], it is able to reproduce the experimental data with greater accuracy.  as a function of angle  at different magnetic field in liquid nitrogen was experimentally measured on patterned SuNAM SCN04150 tapes at 77 K (Figure 2).The experimental multiple data sets of Although being strongly dependent on the considered cross-section configuration, which varies along the optimization process, each single simulation took CPU time of about 300 sec on Pentium Dual Core e6300 @ 2.80 GHz RAM 4 GB.
In the original cable's cross-section configuration [4], the mesh is made up of 145,000 elements (see Figure 3) and the contour plot of the magnetic field intensity, in self-field conditions, is shown in Figure 4. [12][13][14] and Nonlinear Programming [15][16][17].Since the conversion of nonlinear problems into linear models is often not achievable, it is not always possible to apply Mathematical Programming algorithms.Whenever first-or second-order derivatives are available, Nonlinear Programming algorithms can be applied to solve the problem.Unfortunately, derivatives are often not easily available in practical problems.As a ground rule, Mathematical Programming and Nonlinear Programming methods constitute an effective gradient-based optimization set of algorithms for their intrinsic ability to find optimal solutions in few iterations.Nevertheless, gradient-based methods are not thought to deal with discontinuous objectives or design constraints or discrete design variables and they show a reduced exploration capability of the search domain with respect to evolutionary algorithms.The mentioned drawbacks of these methods have induced researchers to introduce another class of algorithms known as evolutionary algorithms (EAs).EAs imitate some natural mechanism of interaction among individuals with the aim of achieving a common goal.The common goal is something single agents could never be able to attain by themselves.So, cooperation plays a key role in detecting the optima.EAs such as Genetic Algorithms [18,19] (GA) and the recent methods like Particle Swarm Optimization [20] (PSO) and Ant Colony Optimization [21] (ACO) are just some algorithms that have successfully exploited natural mechanisms for optimization purposes.

Optimization Algorithm Choice. Many methods have been proposed to solve optimization problems such as Mathematical Programming
In this work, an enhanced version of the original PSO algorithm, called Minimized Computational Effort Particle Swarm Optimization (MCEPSO) [22], was used to the aim of maximizing current density at 77 K in self-field conditions with tapes arrangement in the cross-section as design parameters.MCEPSO has been structured in order to reduce, as much as possible, the computational effort of optimizations.This aspect is particularly useful when dealing with moderate to severe CPU simulation times.And also, MCEPSO introduces modification to the original PSO algorithm allowing dealing with constrained optimization and discrete design variables.As EAs typically involve thousands of function evaluations, it is substantially impossible to deal with computationally intensive simulations.MCEPSO proved to reduce significantly the number of required function evaluations to converge to an optimal design [22], thus allowing the optimization process in several applications where standard EAs may involve unaffordable computational times.MCEPSO was implemented to deal with single-objective optimization processes.However, as already mentioned, when dealing with cable's cross-section layout, two objectives are taken into consideration depending on the desired use of the cable: current density  eng and total flowing current   cable .In these cases MCEPSO is modified by implementing the so-called minimax approach [23], performing a certain number of optimizations while considering varying the relative "importance" (or "weight") of one objective with respect to the other.
With the mentioned simulation time of the order of magnitude of about 300 sec, the adoption of a proven fast convergent algorithm like MCEPSO was an obliged choice to keep the multiobjective optimization times for the 4 configurations (3, 4, 5, and 6 slots) within feasible values.
So, MCEPSO was selected with respect to gradient-based algorithms for its ability to deal with discrete variables, which are an intrinsic aspect of the considered optimization problem.MCEPSO was preferred to other EAs because it demonstrated, on widely approved benchmark test problems [22], being able to find the optimal configuration in a minimized number of FE simulations, which may become a bottleneck when dealing with simulation times of the order of about 6/7 minutes in the frame of a multiobjective search (typically involving hundreds of thousands of FE function evaluations).per slot, the off-center distance r of the innermost tape, and intertape spacing d between two consecutive tapes, as graphically described in Figure 5.The aim of the optimization process is to maximize, at the same time, both the current density  eng (, , ) and the total current   cable (, , ) which are both dependent on the mentioned design variables d, n, r, keeping the number of slots  slots as a parameter.

Optimization Process Description.
The simultaneous effect of varying the entire set of the design variables (the off-center distance r, the intertape spacing d, and the number of tapes per slot n) and the number of slots  slots on the total current   cable and current density  eng is not trivial to be estimated in advance and needs an optimization tool to be exhaustively explored.
The multiobjective optimization was repeated, varying the number of slots  slots in the cable (from 3 to 6 slots) and with the tapes' widths kept fixed at 4 mm and considering SuNAM tapes.SuNAM tapes were characterized as described in [11] in terms of the dependence of the critical current when varying the angle of the applied magnetic field.Among the design variables, the off-center distance r was considered to be continuous whereas the number of tapes in the stack n and intertape spacing between consecutive tapes d were discrete variables.For the sake of precision, the number of tapes n could vary between 22 and 80 while the intertape spacing d could assume the following values: 149, 170, 190, 210, 230, 250, 270, or 300 m.Off-center distance of the inner tape r was constrained to vary continuously between a lower and an upper value defined as a consequence of manufacturing constraints for the cable's core, as detailed in Table 1.Based on previous design experiences and industrial feasibility limits [4], the side limits for the other design variables were decided, as detailed previously and in Table 1.
The multiobjective optimization, as detailed in the previous part of this section, is mathematically summarized by the following equation: (3)

Results and Discussion
3.1.Results.As explained, the optimization was repeated varying the cable's configuration ( slots ) from 3 to 6 slots in order to compare the results obtained in terms of  eng and   cable .Results are summarized in the Pareto frontiers depicted in Figure 6.The obtained Pareto frontiers allow the designer to evaluate the incremental benefits associated with the enhancement of the design optimality.In other terms, the Pareto frontiers play a fundamental role in the design since the designer can evaluate exactly how much more cost is incurred on a certain objective if another objective was favored by a certain amount.It is important to underline that any of the infinite solutions lying on the Pareto frontiers is an optimal solution for the HTS cable design.Moving from one side to the other on a single Pareto frontier, the relative "importance" of one objective with respect to the other varies from 0% to 100%.This means that optimal solutions on the left hand of the single Pareto line are achieved giving 100% of relative "weight" to total current ∑  (and 0% to the current density  eng ).On the contrary, optimal points on the right hand are characterized by 100% of relative importance given to the current density  eng (and 0% to the total current   cable ).Any intermediate point on each Pareto frontier is yet optimal but with a different relative percentage of "importance" between the two objectives.Each point belonging to the Pareto frontiers is also characterized by the respective optimal values of the design variables (n, r, and d) that are not represented in the Pareto plot which is a representation in the objective domain ( eng versus   cable ) rather than in the design variables domain.
The optimal solutions have also been represented, in Figures 7-10, as a function of the relative "weight" given along the multiobjective optimization process (upper panels) and in terms of the design variables versus the same "weight" (lower panels).
Moving from the left to the right in the design variables' plots (from Figures 7-10 Figures 7-10 are useful to interpret these infinite optimal sets in terms of the respective the design variables.

Discussion.
Under the simplifying assumptions introduced in the FEM model, by means of the multiobjective optimization process clear trends were identified to assist the designer's choice in the definition of the best cross-section configuration for HTS CICC cables.
In general terms, from Figure 6, it seems that optimized values for   cable are obtainable by increasing the number of slots  slots up to 6 (upper left side of the plot).On the contrary, highest values for the  eng are achieved using the 3slot configurations (lower right side of the plot).
Also, in the central area of the plot in Figure 6, the same optimal configuration, in terms of the objectives (  cable and  eng ) is achievable with different cross-section configurations, as the Pareto fronts intersect each other.In these cases, the cost of the different layouts may represent the decisive criteria for the layout's choice.
Taking into consideration plots from Figures 7-10, it is firstly clear that a more closely spaced configuration allows an increase in the cable's current density  eng while a wider tape distribution in space is more favorable to larger current values.In fact, the cable's cross-sections layouts which are closer to the origin in plots from Figures 7-10 are characterized by higher values of the current density  eng and lower values of the total current   cable , while also being characterized by smaller values of all the three design variables n, r, and d.In the same plots, moving towards the right hand corresponds to increasing the relative "importance" of   cable with respect to  eng in the multiobjective optimization.As a consequence, moving towards the right hand of the plots causes the optimal solutions to be characterized by increased value of   cable and smaller values of  eng , also corresponding to design variables n, r, and d to increase to their respective upper limits.Secondly, all the plots from Figures 7-10 are characterized by two plateaus: the first one on the left hand of the plots, where design variables have the lowest values, and the second on the right hand, for the highest values of n, r, and d.Along these two plateaus, the values of the design variables are kept fixed by the optimizer so the corresponding objective values on Figure 6 are coincident multiple points laying exactly on the right and left end of the Pareto lines.This means that, despite the fact that the optimizer is increasing the relative "importance" of   cable with respect to  eng , the best configuration is still the same as it was considering when giving no "importance" at all to   cable with respect to  eng .As the relative "importance" is further increased, the very first design variable to be changed by the optimizer is the number  10, where the design variables are increased from each lower plateau to each higher one.This means that an effective change in the objectives' balance is achieved giving a minimum threshold of relative "importance" of to   cable with respect to  eng .And this threshold seems to be changing when considering the 3 slots' configuration (where the threshold is about 0.5), the 4 slots' configuration (threshold is about 0.65), the 5 slots' configuration (threshold is about 0.75), and the 6 slots' configuration (threshold is about 0.65).Also it is interesting to notice that, as the number of slots  slots increases in Figures 7-10, the left plateau for the number of tapes (design variable n) increases from about 25 (Figure 7) to about 58 in Figure 10.On the contrary, the left plateau for the design variable d is fixed to about 150 m, regardless of the number of slots  slots .
A change of slope is easily detectable in Figure 6 for each of the Pareto contours.Focusing the attention on the 6 slots' Pareto line in Figure 6, the change of slop is detected for   cable ≈ 38 KA and  eng ≈ 20 A/mm 2 .The corresponding point in the upper panel of Figure 10 is characterized by an abscissa (relative "importance" of to   cable with respect to  eng ) of about 0.7.For this specific value of the abscissa, the third panel from the top in Figure 10 indicates that design variable n has reached the top plateau and cannot be increased anymore by the optimizer.From now on, to further increase the total current   cable (at the expense of  eng ) in Figure 6, the optimizer will increase the other two design variables: in particular, for an abscissa of 0.7, the optimizer starts to increase design variable d (second panel from the top in Figure 10) and then, for an abscissa of about 0.83, it starts increasing also r.So, in Figure 6, as long as the optimizer has the possibility of increasing the number of tapes n, an increase in total current   cable is achieved with a steep slope with respect to  eng .When the number of tapes n reaches its top side range value, the total current   cable continues to increase but with a much smaller slope than before.This residual increase of   cable , from the Pareto knee to the max value of

Figure 1 :
Figure 1: Artistic view of the assembled ENEA HTS 2G wire-based conductor.

Figure 2 :
Figure 2: Dependence of the critical current on the angle of the applied magnetic field for several values of applied magnetic field.The symbols are experimental data; the lines show the results of the fits to (2).

Figure 3 :
Figure 3: Finite element model mesh for the original configuration [4]: 30 tapes distributed in 5 slots with a spacing between consecutive tapes of 99 m and a radius of the inner tape of 4.95 mm.The mesh consists of about 145,000 finite elements.

Figure 4 :
Figure 4: Contour plot in gray scale of the magnetic field intensity for the 30 tapes, 5 slots' original configuration [4] in self-field operating conditions.

Figure 5 :
Figure 5: Simplified schematic of the cable's cross-section used to describe the design variables n (number of tapes per slot), d (spacing between two consecutive tapes), and r (off-center distance of the inner tape) used in the optimization process.

Figure 6 :
Figure 6: Pareto frontier, in the objectives domain, representing the full set of optimal solutions found for 3, 4, 5, and 6 slots' configurations.

Figure 7 :
Figure 7: Pareto frontier represented as a function of the relative "weight" of one objective with respect to the other for the 3 slots' configuration ( slots = 3).Top panel: values of the two objectives.Bottom panels: values of the respective design variables.

Figure 8 :
Figure 8: Pareto frontier represented as a function of the relative "weight" of one objective with respect to the other for the 4 slots' configuration ( slots = 4).Top panel: values of the two objectives.Bottom panels: values of the respective design variables.

Figure 9 :
Figure 9: Pareto frontier represented as a function of the relative "weight" of one objective with respect to the other for the 5 slots' configuration ( slots = 5).Top panel: values of the two objectives.Bottom panels: values of the respective design variables.

Figure 10 :
Figure 10: Pareto frontier represented as a function of the relative "weight" of one objective with respect to the other for the 6 slots' configuration ( slots = 6).Top panel: values of the two objectives.Bottom panels: values of the respective design variables.

Table 1 :
The optimization variables considered in this work were the number of tapes n Design constraints for the off-center distance.
Side constraints posed for design variable off-center distance r.