Stellarators are devices that use magnetic fields to optimize the conditions needed for plasmas to undergo fusion. Unlike tokamaks, stellarators do not rely on a plasma current but can produce a helical magnetic field using only external coils. In a stellarator, the coils surrounding the plasma are called modular coils and those that follow the plasma are called poloidal field coils. Modular coils can be difficult to build if they are too complex. An effort is underway to develop coil conditions that meet both physics and engineering constraints. The FOCUS code was developed to flexibly optimize stellarator coil configurations [C. Zhu, et al., Plasma Phys. Contr. Fusion 60, 065008 (2018)]. In this work, we will use the FOCUS code to develop and analyze coil configurations for a plasma boundary that is optimized for particle confinement. We will examine modular coil configurations with and without a set of poloidal field coils

Bader, Aaron

Hegna, Chris

Ware, Andrew

Wilson, Haley

English

University of Montana

University of Montana ScholarWorks at University of Montana Undergraduate Theses, Professional Papers, and Capstone Artifacts 2021 Stellarator Optimization with Poloidal Field Coils Haley Wilson The University Of Montana, hw106283@umconnect.umt.edu Andrew Ware The University Of Montana, andrew.ware@mso.umt.edu Aaron Bader University of Wisconsin-Madison Chris Hegna University of Wisconsin-Madison Follow this and additional works at: https://scholarworks.umt.edu/utpp  Part of the Engineering Physics Commons, and the Plasma and Beam Physics Commons Let us know how access to this document benefits you. Recommended Citation Wilson, Haley; Ware, Andrew; Bader, Aaron; and Hegna, Chris, "Stellarator Optimization with Poloidal Field Coils" (2021). Undergraduate Theses, Professional Papers, and Capstone Artifacts. 349. https://scholarworks.umt.edu/utpp/349 This Thesis is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in Undergraduate Theses, Professional Papers, and Capstone Artifacts by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact scholarworks@mso.umt.edu. Stellarator Optimization with Poloidal Field CoilsHaley Wilson1, A. S. Ware1, A. Bader2, C. C. Hegna21University of Montana2University of WisconsinApril 28, 2021AbstractStellarators are devices that use magnetic fields to optimize the conditions needed for plasmasto undergo fusion. Unlike tokamaks, stellarators do not rely on a plasma current but canproduce a helical magnetic field using only external coils. In a stellarator, the coils surroundingthe plasma are called modular coils and those that follow the plasma are called poloidal fieldcoils. Modular coils can be difficult to build if they are too complex. An effort is underway todevelop coil conditions that meet both physics and engineering constraints. The FOCUS codewas developed to flexibly optimize stellarator coil configurations [C. Zhu, et al., Plasma Phys.Contr. Fusion 60, 065008 (2018)]. In this work, we will use the FOCUS code to develop andanalyze coil configurations for a plasma boundary that is optimized for particle confinement.We will examine modular coil configurations with and without a set of poloidal field coils.1 Introduction & TheoryA stellarator is a type of nuclear fusion reactor. Nuclear fusion is a reaction in which two nucleicombine into one or more different nuclei and subatomic particles. In the process, energy is released.The aim of a fusion reactor is to create as much energy as possible from fusion reactions so that itmay eventually be turned into electricity.1Fusion can only occur between very high energy particles. Hydrogen gas must be heated up totemperatures around 100 million Kelvin to overcome the repulsive force between particles. At theseextreme temperatures, the atoms in a gas break into ions and electrons, and the gas becomes whatwe call a plasma. Plasmas are essentially very hot, energetic, and charged gases. The issue, then,is how to force particles collide in a plasma, because their charge and energy makes them want tobe as far away from each other as possible. One way we can achieve relatively high densities whilemaintaining high energies is using magnetic confinement.Since plasmas are charged, we try to confine them using magnetic fields. The magnetic fieldsare created by running current through coils that surround the plasma. The two most commonmagnetic confinement devices are tokamaks and stellarators. Both are generally shaped like a torus.The boundaries of both are shown in Figure 1, with the direction of the magnetic fields shown by theblack arrows. The magnetic field for both boundaries is composed of toroidal and poloidal magneticFigure 1: The plasma boundaries (blue) and magnetic field lines (black) typical of a tokamak andstellarator.[2]fields. These directions are shown in Figure 2 by φ and θ, respectively. The toroidal direction followsthe plasma around the loop and the poloidal direction circles the plasma. The twisting magneticfields of Figure 1 are due to the superposition of toroidal and poloidal magnetic fields.The key difference between tokamaks and stellarators is that a tokamak depends upon plasmacurrent to create a toroidal magnetic field. That is, the magnetic field must not only come from coilswrapping the plasma, but from within the plasma itself. Stellarators create their helical magneticfield completely using external coils surrounding the plasma. This results in stellarator coils being2Figure 2: Coordinates to describe a toroidal fusion reactor.considerably more complex. In a tokamak, each coil wrapping the plasma is identical to the othersand arranged axisymmetrically around what would be the z-axis as shown in Figure 2. In stellara-tors, the coils wrapping the plasma, called modular coils, bend and twist like shown in Figure 3. InFigure 3: An example of a plasma boundary and the coils that produce a magnetic field surroundingit.this paper, we use a novel code called FOCUS to optimize the shape and placement of modular coilson two stellarator configurations. We focused primarily on optimizing two parameters: coil lengthand the magnetic field normal to the plasma surface Bn. By controlling coil length, we minimizethe amount of twists and turns the modular coils can make and therefore reduce their complexity.This is considered an engineering constraint. By minimizing Bn, we better match a target plasmaboundary. The target plasma boundaries were developed by the WISTELL collaboration at Uni-3versity of Wisconsin and are optimized for particle confinement. Matching the optimized plasmaboundary is considered a physics constraint.2 MethodsAll coil optimizations were done using FOCUS, a novel coil optimization code developed at Princetonby C. Zhu.[1] Previous coil optimization codes relied on what is called a winding surface. This surfacehad to be chosen at an arbitrary distance from the plasma surface and then coils had to be builtupon it. FOCUS treats coils as space curves, so their placement is much less restricted. FOCUS hasvarious cost functions that represent distinct optimization parameters. Cost functions map eventsonto a number that represents the “cost” of that event. The user has the ability to weight costfunctions of various optimization parameters depending on what parameters they want to prioritize.We started with a plasma boundary from the University of Wisconsin called Bila. Bila is the5-field period boundary case shown in Figure 3. Our main goal was to see the effect on Bn ofadding poloidal field coils. Polodial field (PF) coils are coils separate from the modular coils thatonly influence the poloidal magnetic field. We used two graphs made in Python using data fromthe FOCUS code to analyze our data. One is of the change in cost functions over all iterations inthe optimization code. The user specifies the number of iterations the code will go through beforeit stops. The second graph is a contour plot of the surface Bn. An example of both of these plotsis shown in Figure 4.4Figure 4: An example of the cost function and Bn graphs we will use in our analysis.The only lines in the cost function plot that we will be analyzing are that of χ2, fBn , and fL,or the black, red, and blue lines, respectively. The first, χ2, is an overall measure of optimizationfor a non-linear least-squares fit, while fL is the cost function that minimizes coil length and fBn isthe cost function that minimizes Bn. We ideally would like all three to go down. The countour plotshown in Figure 4 is a good example of what we don’t want. In Figure 4, we can see distinct coilripple, but ideally this plot will be much more homogeneous with as low contrast as possible. Thecloser to zero our Bn is, so the whiter our contour plot, the better we’ve matched our target.53 DataWe started with the Bila set of coils and a plasma boundary optimized by the University of Wisconsin.We reduced the number of coils to 40 to allow more coil-to-coil separation, which is an importantparameter in engineering stellarators. We optimized the 40 coils in FOCUS with and without a setof PF coils. The coil shapes and target plasma boundary are shown in Figure 5, viewing down thez axis. Two PF coils are placed at z = 0.6 and z = −0.6 on the inside and outside of the boundary,making a total of 4 PF coils.Figure 5: A view down the z-axis of the Bila boundary with modular coils, both with and withoutPF coils as well.The cost function and Bn analysis graphs are given in Figure 6 for both cases. Note that χ2decreases more in the non-PF case, but that fBn and fL do not decrease much in either case. Wealso notice that fL is relatively higher for the PF case, which we assume is on account of the PFcoils being of a greater length than the modular coils, so our length function has reflected the “cost”of that. The Bn contour plot has changed considerably. Note that the scale on the PF coil casegoes down to only -0.3 as opposed to -0.4 in the first case, so the blue “valleys” in the plot are notas deep. This is a slight improvement in the Bn as it has homogenized slightly more with the PFcoils and does not stray as far from 0, so we have matched our target plasma boundary better.We would prefer our Bn to be at least a tenth smaller than seen in the last case, so we decided tolessen the length constraint by lowering the weight of fL in FOCUS by a factor of 10. That resultedin the coils and boundaries showm in Figure 7. The modular coils have become considerably more6Figure 6: The cost function and Bn analysis graphs for the optimization of Bila with and withoutPF coils.complex and the PF coils have expanded away from the z-axis.The cost function and Bn analysis graphs are shown in Figure 8. The most notable change incost functions is that fBn has decreased significantly more than in any previous run. This is in nodoubt due to the more complex coil shape, and analysis of the Bn contours tell the same story. First,the scale is about a fifth smaller, and the coil ripple is not so pronounced. There are less extremepeaks and valleys as well.7Figure 7: A comparison of the coil shape of the previous optimization with PF coils and a newoptimization with a decreased weight on fL.Figure 8: Cost function and Bn analysis associated with the cases in Figure 7.84 ConclusionOptimizing modular coils in tandem with PF coils changed modular coil shape and led to improvedsurface Bn. The coil optimization code FOCUS allowed flexibility to put priority on minimizing Bninstead of coil length, which led to even further improvements in Bn. While the increased coil lengthled to the coils being more complex to potentially build, the overall decrease in Bn is a worthwhiletrade-off that we’d like to explore further. There are many more cost functions in FOCUS that we’dlike to familiarize ourselves with. Future work will be to test equilibria in the boundaries producedby our optimized coil configurations and to experiment with new PF coil shapes and amounts.9References[1] C. Zhu, et al., Plasma Phys. Contr. Fusion 60, 065008 (2018)[2] Krane, K. S. (1983). Modern physics. New York: Wiley, 87-91.10

Stellarator Optimization with Poloidal Field Coils

https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1370&amp;context=utpp

Stellarator Optimization with Poloidal Field Coils

Abstract

Similar works

Full text

Available Versions

University of Montana