plasma profile and shape optimization for the advanced tokamak power plant, aries-at

Fusion Engineering and Design 80 (2006) 63–77

Plasma profile and shape optimization for the advancedtokamak power plant, ARIES-AT

C.E. Kessela,∗, T.K. Maub, S.C. Jardina, F. Najmabadib

a Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton, NJ 08543, USAb Center for Energy Research, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 93037, USA

Accepted 1 June 2005Available online 27 September 2005

Abstract

An advanced tokamak plasma configuration is developed based on equilibrium, ideal MHD stability, bootstrap current analysis,vertical stability and control, and poloidal field coil analysis. The plasma boundaries used in the analysis are forced to coincidewith the 99% flux surface from the free-boundary equilibrium. Using an accurate bootstrap current model and external currentdrive profiles from ray tracing calculations in combination with optimized pressure profiles,βN values above 7.0 have beenobtained. The minimum current drive requirement is found to lie at a lowerβN of 6.0. The external kink mode is stabilized by atungsten shell located at 0.33 times the minor radius and a feedback system. Plasma shape optimization has led to an elongationof 2.2 and triangularity of 0.9 at the separatrix. Vertical stability could be achieved by a combination of tungsten shells locatedat 0.33 times the minor radius and feedback control coils located behind the shield. The poloidal field coils were optimized inl uction int hke, D.A.E nt, FusionE©

K

1

p

f

s firstdiventhewerced

xper-

0

ocation and current, providing a maximum coil current of 8.6 MA. These developments have led to a simultaneous redhe power plant major radius and toroidal field from those found in a previous study [S.C. Jardin, C.E. Kessel, C.G. Bathst, T.K. Mau, F. Najmabadi, T.W. Petrie, the ARIES Team, Physics basis for a reversed shear tokamak power plang. Design 38 (1997) 27].2005 Elsevier B.V. All rights reserved.

eywords: Advanced tokamak; Ideal MHD stability; Power plant studies

. Introduction

The simultaneous achievement of highβ (at highlasma current), high bootstrap fraction, and the trans-

∗ Corresponding author. Tel.: +1 609 243 2294;ax: +1 609 243 3030.

E-mail address: [email protected] (C.E. Kessel).

port suppression consistent with these features washown in Ref.[1]. An overview of experimental antheoretical results, excluding the most recent, was gin Ref. [2]. The reversed shear configuration fortokamak has the potential to be an economical poplant [2], and its features, referred to as the advantokamak, are being pursued in several tokamak eiments [3–11]. Previous work reported in Ref.[2]

920-3796/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.fusengdes.2005.06.350

64 C.E. Kessel et al. / Fusion Engineering and Design 80 (2006) 63–77

obtained attractiveβ values and reasonable currentdrive requirements. However, it is of interest to furtherimprove such configurations to understand the poten-tial benefits and identify the highest leverage researchareas. The present studies will show that accurate boot-strap models are necessary to determine MHD stabilityand current drive requirements. In addition, requiringthe plasma boundary to coincide with the 99% fluxsurface of the free-boundary plasma enforces consis-tency and has led to higherβ limits. The plasma pres-sure profile is optimized to provide high ballooningβlimits and bootstrap current alignment simultaneously.Plasma shaping has been utilized, within engineeringconstraints, to increase theβ, both through increases inβN (β =βNIP/aBT, a = minor radius,BT = toroidal field,and IP = plasma current) and plasma current. Verticalstability and control analysis have found a reasonablesolution for the passive stabilizer and feedback controlpower. Poloidal field (PF) coil optimization has beenperformed and meets all engineering constraints.

2. Equilibrium and stability studies

The fixed boundary flux-coordinate equilibriumcode JSOLVER[12] is used with 257 flux surfaces and257 theta points from 0 toπ. The n =∞ ballooningstability is analyzed with BALMSC[13], and PEST2[14] is used for low-n external kink stability. For allcases reported here, a conducting wall is assumed tos ca-ti

owni alp -Ap ilib-r qui-l inw acea sible,l b-r thea canhd onea

Table 1ARIES-RS[2] and ARIES-AT global plasma parameters

ARIES-RS ARIES-AT

IP (MA) 11.3 12.8BT (T) 7.98 5.86R (m) 5.52 5.20a (m) 1.38 1.30κa 1.70 2.15δa 0.50 0.78κ (X-point) 1.90 2.20δ (X-point) 0.70 0.90βp 2.29 2.28β (%) 4.98 9.07βa (%) 6.18 11.0βN (%) 4.84 5.40βmaxN (%) 5.35 6.00

q0 (axis) 2.80 3.50qmin (minimum) 2.50 2.40qe (edge) 3.52 3.70IBS (MA) 10.0 11.4I∇p/IP 0.91 0.91ICD (MA) 1.15 1.25q 2.37 1.85li (3) 0.42 0.29n0/〈n〉 1.36 1.34T0/〈T〉 1.98 1.72p0/〈p〉 2.20 1.93(b/a)kink 0.25 0.33

a Value corresponds to fixed boundary equilibrium.

where the elongation and triangularity are plotted for aseries of flux surfaces from the 95% to 99.5%. Thecases shown have a plasma internal self-inductance(li ) of 0.46. Both elongations and triangularities wouldincrease with lowerli , and decrease with higherli .

In order to calculate free-boundary equilibria for theadvanced plasmas being considered, it was necessaryto make modifications to the methods used. The free-boundary equilibrium equation to be solved is given bythe Grad–Shafranov equation:

�∗ψ = −µ0Rjϕ (1)

jϕ = −R dp

dψ− 1

2µ0R

dg2

dψ(2)

whereψ is the poloidal flux function (equal to theactual poloidal flux divided by 2π), jϕ the toroidal cur-rent density,p the pressure, andg is the toroidal fieldfunction (equal toRBT). For reversed shear plasmasat high pressure the two terms in the definition ofjϕhave opposite signs cancelling to some degree. Physi-

tabilize the external kink modes, and this wall loion is determined. In addition,n = 0 vertical stabilitys assessed with Corsica[15].

The final optimized reversed shear plasma is shn Fig. 1, with various equilibrium profiles, and globlasma parameters are given inTable 1, under ARIEST, where they are compared to ARIES-RS[2]. Thelasma boundaries used in the fixed-boundary equium calculations are taken from free-boundary eibria at the 99% poloidal magnetic flux surface,hichq95 is kept >3.0. We attempt to use a flux surfs close to the diverted plasma separatrix as pos

imited by the ability of the fixed-boundary equiliium and ideal MHD stability codes to handlengular boundary near the X-point. This approachave a strong impact on the calculated stableβ valuesue to progressively stronger plasma shaping aspproaches the separatrix. This is illustrated inFig. 2

C.E. Kessel et al. / Fusion Engineering and Design 80 (2006) 63–77 65

Fig. 1. Equilibrium profiles for ARIES-AT, showing the plasma pressure, safety factor, parallel current, temperature, density, and poloidalflux.

cally, the toroidal current density is low in the plasmacore due to the hollow current profile, but is also shift-ing to the outboard side as the pressure increases. Thiscancellation between the two terms is critical to obtain-ing proper force balance, and is not easily achieved forarbitrary choices of the functionsp(ψ) and g(ψ). Inorder to produce these equilibria and have them repre-sent the desirable fixed boundary configurations withhighβ and high bootstrap current fractions, we recast

the toroidal current density in terms of the pressure andparallel current density:

jϕ = −R dp

dψ

(1 − B2

ϕ

〈B2〉

)+ Bϕ

〈j · B〉〈B2〉 (3)

The first term is the combination of the Pfirsch–Schluter and diamagnetic currents, and the second termis any driven parallel current that exists (i.e. ohmic,


Fig. 2. Variation of the plasma elongation and triangularity as oneapproaches the separatrix, where the elongation is 2.2 and triangu-larity is 0.9.

bootstrap, RFCD). Now both terms are positive and donot require cancellation at high pressure, and specify-ing the parallel current density is how fixed boundaryequilibria are calculated in JSOLVER. This specifi-cation of thejϕ requires flux surface averages to becalculated at each iteration, which slows the calcula-tion down, however, with this approach the pressure,current, and safety factor profiles can be accuratelyrepresented between fixed and free-boundary equilib-ria. In addition, one can obtain difficult equilibria notaccessible with the original formulation.

There are two types of fixed boundary equilibriumcalculations used, target equilibria and self-consistentequilibria. Target equilibria have the total parallel cur-rent density and pressure profile prescribed. These areused to scan MHD stability more efficiently for plasmashape and profile optimization. The bootstrap currentprofile is calculated[16] at the end of the equilib-rium calculation, and is monitored for alignment withthe prescribed current profile. Self-consistent equilibriahave only the pressure profile and the parallel cur-rent density profiles from current drive sources (i.e.FWCD and LHCD) prescribed. The bootstrap currentdensity is calculated at each iteration of the equilibrium,which is added to the current drive sources to providethe total parallel current density. These equilibria areused to develop final configurations, by prescribing the

required current drive profiles and interating with theactual current drive deposition profiles from ray tracingcalculations reported in Ref.[17].

3. Current profiles and bootstrap current

The parallel current density profile used in targetequilibrium calculations is given by the following form:

〈j · B〉〈B · ∇ϕ〉 = j0(1 − ψ) + j1

d2ψa1(1 − ψ)b1

(ψ − ψ0) + d2(4)

whereψ is the normalized poloidal flux that is 0 atthe magnetic axis and 1.0 at the plasma edge. The firstterm is a centrally peaked contribution, while the sec-ond peaks off-axis and is zero at the center and theedge. The coefficientsψ0, d, aj, andbj are the peaklocation, width, and fall-off exponents. The parame-ters are chosen to generate a hollow current profile. Theparticular form is chosen by considering the ability toreproduce the profile shape with bootstrap current andthe need for external current drive. The location of theqmin (minimum safety factor) is chosen as close to theplasma edge as possible to obtain high ballooningβ-limits, while avoiding the degradation of the magneticshear near the plasma edge, and to stay within externalcurrent drive limitations (i.e. LH current penetration).

For the self-consistent equilibrium calculations onlythe external current drive profiles are input, and theya econdtw theo n.T ly inF

apc la-t thet adew ot-s ame,wl aturen e theb f thec ro le,

re described by terms that are the same as the serm in Eq.(4). An example of these appears inFig. 1,here the parallel current density is shown, andn-axis FWCD and off-axis LHCD profiles are showhe bootstrap current is calculated self-consistentig. 1.

It was found by comparing full collisional bootstralculations[16] to the single ion collisionless calcuion [18], that the external current drive required inwo cases is quite different. A comparison was mhere two equilibria with approximately 100% botrap current, all other plasma parameters the sith βN of 5.35, and is shown inFig. 3. The collision-

ess case ignores the effect of decreasing temperear the plasma edge, which would strongly reducootstrap current there, and has an overall shift ourrent profile outward. This causesqmin to be furtheut in minor radius, which for a fixed pressure profi


Fig. 3. Comparison of collisionless (a) and collisional bootstrap (b) formulations for nearly 100% bootstrap fraction, showing that the collisionlessform provides for a minimum safety factor that is closer to the plasma edge, resulting in no current drive requirement.

will give a higher ballooningβ-limit. The collisionlessmodel is misleading because it indicates that no off-axiscurrent drive is required to achieve ballooning stabilityat thisβN. The collisional model, on the other hand,would require off-axis current drive of about 10% ofthe total plasma current to obtain ballooning stabilityat thisβN. The difference in the location ofqmin for thetwo bootstrap models is critical to assessing the idealβ-limit and the magnitude of off-axis current drive. Theuse of a collisionless model should be avoided in con-figuration development because of its overly optimisticpredictions for bootstrap current near the plasma edge.

The bootstrap current calculation requires knowl-edge of the pressure profile and either the density ortemperature profile. In the present work particle and

energy transport calculations were not done. Rather,the density and temperature profiles were constrainedto have their dominant gradients in the vicinity of theqmin consistent with the presence of an internal trans-port barrier[3–11]. This approach was chosen due tothe widely varying characteristics observed on variousreversed shear tokamak experiments, with the excep-tion of this dominant gradient. Beyond this, the profilesare chosen to optimize the bootstrap current alignment,and typical profiles for these plasmas are shown inFig. 1. Ref.[17] presents a comparison of the assumedprofiles and those from GLF23 theoretical predic-tions indicating reasonable agreement. The densityand temperature profiles assumed in this study showgradients that are spread out more than is observed


experimentally, and may require some form of ITBcontrol to achieve, but this is beyond the scope of thepresent work.

4. Pressure profile optimization

The pressure profile is described by the following:

p(ψ) = p0[c1(1 − ψb1)a1 + c2(1 − ψb2)

a2] (5)

where the coefficients are chosen to reduce the pres-sure gradient in the region outside theqmin to improveballooning stability, and to provide bootstrap currentalignment to reduce the magnitude of external currentdrive. The pressure gradient is zero at the plasma edgefor these profiles, making these typical of an L-modeedge. Previously[2], the pressure was parameterizedby a simpler form:

p(ψ) = p0(1 − ψb1)2

(6)

which was restrictive in optimizing both the ballooningstability and bootstrap current simultaneously.

A sequence of pressure profiles was determined,using Eq.(5), that had progressively higherβ-limits,due to a progressively smaller pressure gradient in theballooning unstable region. This is shown inFig. 4,where the pressure gradient is plotted as a function ofpoloidal flux. The region of ballooning instability isnoted, and the remaining part of the pressure profile ism res-ss rnalc on,t thep ofilet lemn e cur-rs ntb tion,w ur-r arei veryp theb edu nallyd ,

Fig. 4. A series of pressure gradient profiles as a function of poloidalflux showing how the gradient is reduced in the ballooning unstableregion to obtain successively higherβN values, while keeping boot-strap alignment.

and this is shown inFig. 5. From ray tracing calcula-tions[17] theβN of 6.0 was chosen as the best tradeoffbetween maximizingβ and minimizing current drivepower, for the reference ARIES-AT configuration.

Fig. 5. Total externally driven current required as a function ofβN,showing that a minimum exists, and that the highestβ is not associ-ated with the lowest current drive.

ade to provide bootstrap alignment. The smaller pure gradients near the plasma edge, to increaseβ, led tomaller bootstrap current there causing higher exteurrent to be required to provide stability. In additihe highestβs led to excessive bootstrap current inlasma core, which requires a broader density pr

o eliminate it, exacerbating the current drive probear the plasma edge. These two factors cause thent drive requirement to begin increasing asβ risesufficiently high. At lowerβ values there is insufficieootstrap current to provide a large bootstrap fracithin our transport constraints, leading to large c

ent drive requirements. If no transport constraintsmposed, one can find very broad temperature andeaked density profile solutions that can provideootstrap current at lowβ, but these are considernphysical. Due to these effects the required exterriven current as a function ofβN shows a minimum


The impact of a finite edge pressure gradient wasnot examined systematically, but was determined fora particular case. The pressure profile in Eq.(5) wasexpanded by another term of the form (1− ψb)

a, with

a = 1.0 andb = 6.0 and a relative magnitude of 5% ofthe peak pressure, to provide a pedestal near the plasmaedge. The plasma temperature and density profiles weremodified slightly near the edge to accommodate this.More bootstrap current was produced, reducing thetotal current drive from 1.15 to 0.85 MA. The high-n ballooning was improved slightly, as described inRef. [17], although the kink stability was unchangedsince the stabilizing wall dominates. These results arenot conclusive since the plasma density at the sepa-ratrix would likely increase, from the value assumedin these studies, with such an H-mode edge condition,which would worsen the current drive requirements.Since the current drive is close to the plasma edge, theprecise plasma properties there are very important, andfurther experiments are needed to establish the accessi-bility to edge conditions that optimize the current drive,MHD stability, and divertor solutions.

5. Plasma shape optimization

The plasma triangularity and elongation have astrong impact on the ideal MHD stability. In additionto providing higherβN values they also allow higherps d by

R

Hr tri-a e as too re. Ina lan-k ratec Thiss thes all)t othe t thei out as the

heat load. This allows us to take advantage of highertriangularities. To assess the benefits, a scan of the trian-gularity was done, for two elongations. For this scan theplasma boundary is given analytically, the edge safetyfactor is held fixed at 3.5, the aspect ratio is 4.0, andone of the pressure profiles described above was used.Shown inFigs. 6 and 7is theβN andβ as a functionof triangularity, forn =∞ ballooning modes. Includedis the result given earlier[2], with a more restrictivepressure profile (Eq.(6)). It is clear thatβN improveswith higher triangularity, but a rollover occurs beyond atriangularity of 0.65. However, theβ continues to risedue to the continued increase in the plasma current.The combination of higher elongation creates an evenstronger increase inβ.

The plasma elongation is limited by then = 0 verti-cal instability, through the ability to provide conductingstructure close enough to the plasma and a verticalposition feedback system with reasonable power. Forpower plant designs the location of a conducting mate-rial is limited to lie outside the blanket, and typicallyresides in the region between the blanket and shield.The plasma elongation can substantially increaseβ

due to a 1 +κ2 scaling. Recently, optimization of theblanket[19] has allowed the conducting structure to bemoved closer to the plasma, and actually be in the blan-ket, although, it must exist in a very high temperatureenvironment (>1000◦C). The benefits of increasing theplasma elongation above the values previously found[2] for a conducting structure behind the blanket weree smab itiesb edges pro-fiam lon-g hertt e theβ ionsa r thet ser nc is sos -t vert

lasma current to be driven in the plasma forq95 con-trained to be >3.0. The plasma shape is describe

= R0 + a cos(θ + δ sinθ), Z = κa sinθ (7)

ereκ is the elongation,δ the triangularity,a the minoradius, andR0 is the major radius. The plasmangularity has been limited by the need to providufficiently long slot divertor on the inboard sidebtain detached plasma and radiate the power thepower plant, the divertor slot must cut through b

et and shielding, which allows neutrons to penetlose to the superconducting toroidal field magnet.ets a limit on the triangularity (or the angle thateparatrix flux line can make with the inboard wo provide neutron shielding. However, recently bxperimental and theoretical results indicate tha

nboard plasma detachment can be obtained withlot, although a short slot is still used to disperse

xamined. As for the triangularity scan, analytic plaoundaries were used, with a range of triangularetween 0.4 and 0.85, aspect ratio of 4.0, theafety factor fixed at 3.5, and one of the pressureles described by Eq.(5). Shown inFigs. 8 and 9areβNndβ as a function of elongation, forn =∞ ballooningodes. TheβN shows a decrease with increasing eation at lower triangularity, and the opposite at hig

riangularity. At higher triangularity the increase inβNurns over and begins to decrease. The point wherN increase turns over moves to higher elongats the triangularity is increased. As was seen fo

riangularity scan, theβ values continue to increaegardless of the structure in theβN versus elongatiourves, because the increase in plasma currenttrong. The slope of the increase inβ versus elongaion continues to improve with higher triangularity ohe whole range.


Fig. 6. βN for high-n ballooning modes as a function of the plasma triangularity.

The low-n external kink stability was also analyzed,and is shown inFig. 10. Here the marginal wall is givenas a function of elongation for toroidal mode numbersfrom 1 to 6. The wall locations are only resolved to

Fig. 7. β for high-n ballooning modes as a function of the plasmatriangularity.

0.025 times the minor radius. Higher toroidal modenumbers require closer walls for stabilization, and allmode number wall locations move much closer to theplasma as the elongation exceeds a value of 2.3. The

Fig. 8. βN for high-n ballooning modes as a function of the plasmaelongation.


Fig. 9. β for high-n ballooning modes as a function of the plasmaelongation.

triangularity for this case was fixed at 0.7. In orderto observe the impact of triangularity on the kinkmode wall stabilization, three other values were ana-lyzed; 0.4, 0.55, and 0.85. Shown inFig. 11 are the

Fig. 10. Stabilizing wall location as a function of plasma elongationwith the triangularity fixed at 0.7, and for toroidal mode number from1 to 6. These are evaluated at theβN values determined by high-nballooning stability.

Fig. 11. Stabilizing wall location as a function of plasma elongationfor then = 1 kink mode, with various plasma triangularities. These areevaluated at theβN values determined by high-n ballooning stability.

marginal wall locations for toroidal mode numbern = 1,as a function of elongation. It is clear that higher tri-angularity results in marginal wall locations that arefarther from the plasma at the lower elongations, how-ever, the stability at the highest elongations is rapidlydegrading. Shown inFig. 12are the marginal wall loca-tions for toroidal mode numbersn = 1–5, at a fixedelongation, and varying triangularity, showing that theimprovement with triangularity persists at highern. Aswas pointed out in Ref.[1], when a stablizing wallis present there is typically a toroidal mode numbergreater than 1 that is the most limiting toβN. Thecurves inFigs. 10–12were done with analytic plasmaboundaries to identify trends in shaping on the low-nstability. Analysis for a separatrix triangularity of 0.9,and elongation of 2.2, the reference ARIES-AT values,was done up to toroidal mode numbers of 9 to resolvethe limiting mode. Shown inFig. 13 is the marginalwall location as a function of toroidal mode number, attwo differentβN values, withβN = 6 corresponding tothe final plasma configuration. The plot shows that then = 4–6 are limiting the ARIES-AT configuration, andthe wall is located at that position. As the pressure isincreased theβ-limiting toroidal mode number movesto higher values and the wall must move closer to theplasma to stablize alln.


Fig. 12. Stabilizing wall location as a function of toroidal mode num-ber and various plasma triangularities, with fixed plasma elongationof 2.2. These are evaluated at theβN values determined by high-nballooning stability.

Fig. 13. Stabilizing wall location as a function of toroidal mode num-ber for two values ofβN, showing the shift of the limiting modenumber as the pressure increases. TheβN = 6 case is the referenceoperating mode for ARIES-AT, showing thatn = 4–6 provide thelimiting wall location at theβN limit from the high-n ballooningstability.

Based on the plasma shape analysis, values for theelongation and triangularity at the separatrix were cho-sen to be 2.2 and 0.9, respectively. This integrated theballooning and kink stability behavior, requiring a shellfor the kink mode stabilization at 0.33 times the minorradius, and avoiding the rapid degradation in kink sta-bility at higher elongations. As will be shown in thenext section, the shell for vertical stability is placed atthe same location, and therefore provides full cover-age on the plasma outboard side as required for kinkstabilization. The plasma elongation choice was alsodependent on vertical stability analysis that follows.The full stabilization of the external kink mode requireseither plasma rotation[20] or feedback control[21].Plasma rotation is difficult to provide externally forreactor size plasmas so the feedback approach is taken,however plasma rotation is not well understood andexperimental results may show that the plasma rotatesin the absence of external sources of momentum. The-ory indicates[21] that if plasma rotation is present witha feedback control system, it aids the controller allow-ing smaller gains. For the feedback approach the shellis necessary to slow the mode growth rate to time scaleswhich the feedback coils can respond with reasonablepower. This is discussed in Ref.[17].

6. Axisymmetric stability

The plasma elongation is ultimately limited by thev n-d andp ablep tioni ver-t ac wayf Thes

f

w dτ on-s hellu ide,a y ont ttom

ertical instability, through the ability to locate coucting structure sufficiently close to the plasmarovide a feedback control system with reasonower. Due to the high leverage of plasma elonga

n increasingβ, scans were done to determine theical instability growth rate, and stability factor asonducting shell is moved progressively further arom the plasma, for various plasma elongations.tability factor is defined as

s = 1 + τg

τL/R(8)

hereτg is the vertical instability growth time, anL/R is the longest up–down asymmetric time ctant of the surrounding structure. The conducting ssed in the analysis is vertical on the inboard snd approximately contouring the plasma boundar

he outboard side, with gaps at the top and bo


Fig. 14. Example of the generic structure used in the vertical stabilitycalculations, and the final structure used in the design.

for the divertor.Fig. 14 shows this generic structuremodel, which is toroidally continuous. For the curvesshown inFig. 15, the conductor was tungsten, 0.035m thick, with a resistivity of 8× 10−8m. In addi-tion, the plasma hasβp of 0.25 andli of 0.8, typical of

Fig. 15. Vertical instability growth rate versus the distance of atungsten shell, normalized to the minor radius, for various plasmaelongations.

Fig. 16. Vertical stability factor as function of the distance of atungsten shell, normalized to the minor radius, for various plasmaelongations.

plasmas during rampup which are the most unstable.Fig. 15shows that as the plasma elongation is increasedthe instability growth rate increases, and that the shellmust be located closer to the plasma to provide anyinfluence on the plasma. As the shell is moved fur-ther away, initially the instability growth rate changesslowly, but later begins increasing rapidly. Where thiscurve asymptotes is called the critical ideal wall loca-tion, and the stability factor is approaching 1.0. If theshell is located outside this location, it will not influ-ence (slow down) the vertical instability even if it weresuperconducting. If the shell is located at this locationor closer to the plasma, and it were superconducting itwould stabilize the plasma. However, actual structuralmaterials are resistive, so a more useful shell locationis the critical resistive wall location. This is definedas the wall location that provides a stability factor of1.2, and coincides with a shell location at the knee inthe growth rate versus shell distance curve.Fig. 16shows the stability factor for the same plasma elon-gations and wall distances withfs = 1.2 denoted on theplot.

The typical location for a conducting shell in powerplant designs is between the blanket and the shield,which is roughly at a normalized distance of 0.45


times the minor radius, the precise value dependingon the design. However, recent optimizations of theblanket for neutronics[19] has allowed the conductorto be located inside the blanket, closer to the plasma.This location is found to be about 0.3–0.35 times theminor radius. For a shell at this location, elongationsat the separatrix up to 2.2–2.3 can be considered, asopposed to those in Ref.[2] which could not exceed1.9. The shells used in the scoping studies above arenot practical since they surround the plasma and willadversely affect the neutronics. The shells are reducedin size by removing the section closest to the midplane,which has the weakest affect on vertical stability. Thisis done until the vertical stabilization provided by theshells starts to significantly degrade. The dark lines inFig. 14through the inboard and outboard shells showhow much of the shells is removed for a final shelldesign, which in this case is 60◦ on the outboard mea-sured from the midplane. The inboard shell is reducedso that it has the same vertical extent as the outboardplate.

Once the plasma elongation is chosen,κ(X-point) = 2.2 in this case, and the final shell design isprovided, more detailed vertical stability analysis isdone to examine the impact of pressure and currentprofile on the growth rate. In addition, the vertical con-trol calculations are done with this configuration todetermine the maximum current and voltage expectedduring control, using the Tokamak Simulation Code[22]. It should be noted that since the stabilizing shellsa hight per-ar y ofavβ ‘A ucto dis-t smav er.A backp umv chi surea nga-t wthr tside

Fig. 17. Vertical instability growth rate as a function of the plasmainternal self-inductance,li (3), andβp, for the final design verticalstabilizing structure.

the range can be produced but only with lower elonga-tion (which reduces the growth rate) or lower plasmacurrent (which reduces the feedback power for a givengrowth rate). It should be noted that approximately 85%

Fig. 18. Feedback power for vertical position control as a functionof the vertical instability growth rate, for the final vertical stabilizingstructure.

re located in the blanket they must operate atemperatures and cannot be actively cooled. The oting temperature turned out to be about 1100◦C. Theesulting shell was 0.04 m thick, and had a resistivitbout 35× 10−8m. Shown inFigs. 17 and 18is theertical instability growth rate as a function ofli andp, with the final reference plasma denoted by the×’.lso shown is the feedback control power (the prodf peak current and peak voltage) for a random

urbance with 0.01 m RMS displacement of the plaertically, giving 30 MVA for the peak feedback powgain, the reference plasma is denoted. The feedower that can be tolerated determines the maximertical instability growth rate that is allowed, whis 45 s−1. Consequently, the range of plasma presnd current profiles that can be produced at full elo

ion is limited, and not to exceed the maximum groate. Plasmas with pressure and current profiles ou


of the power for vertical position control is reactive,and therefore can be recovered with a suitable energystorage system, so that this power is not included in therecirculating power for the power plant.

7. Poloidal field coil optimization

The poloidal field (PF) coil currents are determinedto force the plasma boundary to pass through specifiedpoints in space, the desired outboard and inboard majorradii, and produce a zero poloidal field at the desired X-point. This is accomplished with a least-squares solu-tion. Otherwise the plasma boundary is allowed to takeon the shape that minimizes the coil stored energy. Inaddition, the PF coil locations are optimized to mini-mize an energy measure equal to the sum of the coilmajor radius times the coil current squared, which isfound to scale with the coil cost. This is done by sur-rounding the plasma by a large number of PF coils,avoiding regions where coils cannot exist (i.e. outboardmidplane for maintenance). The coils are eliminatedone by one and the resulting increase in the coil energymeasure is determined. The coil that increases this mea-

F oils,s eases.

Fig. 20. Layout of the optimized poloidal field coils for ARIES-AT.

sure the least is eliminated, and the process repeatedwith the remaining coils until a certain number of coilsis obtained or the coil energy measure begins increas-ing rapidly.Fig. 19shows the coil energy measure asfunction of the number of coils, and it begins to increasestrongly below about 14 coils. It should be noted thatthere are 7 coils used to provide a thin solenoid on theinboard side, for inductive startup and plasma shapingthat are not included in the elimination. The final PFcoil arrangement is shown inFig. 20. The maintenancerequires that entire sectors be removed through the out-

Table 2ARIES-AT poloidal field coil parameters

Coil # R (m) Z (m) I (MA)

1 2.25 0.25 −0.6202 2.25 0.75 −1.0533 2.25 1.25 −1.5134 2.25 1.75 −0.6655 2.25 2.25 −0.6656 2.25 2.75 1.1847 2.25 3.25 3.3608 3.25 5.75 6.3489 3.75 6.00 6.518

10 5.25 6.30 4.81011 5.75 6.25 3.64312 7.50 5.65 −3.27613 8.00 5.40 −5.87714 8.50 5.10 −8.624

ig. 19. Poloidal field coil energy as function of the number of chowing that the energy increases as the number of coils decr


board midplane, which forced the PF coils to be above5.0 m. The maximum current of 8.6 MA occurs in thelargest radius coil, andTable 2shows the coil locationsand currents.

8. Conclusions

The advanced tokamak plasma configuration has thepotential to provide a highβ and high bootstrap cur-rent fraction, resulting in a more compact economicalpower plant. The present work has utilized high resolu-tion equilibrium and ideal MHD stability calculationsin combination with a full velocity space bootstraptreatment to analyze the impact of pressure profile andplasma shape optimization. In addition, plasma bound-aries used in fixed boundary equilibria are providedby the 99% flux surface of the corresponding free-boundary equilibria.

Pressure profile optimization allowed theβNfor ballooning instabilities to be maximized, whilesimultaneously providing bootstrap current alignment.Optimization of the plasma shape allowed theβNto be increased, but more importantly allowed theplasma current to be increased, leading to aβ nearlytwice that in previous studies[2]. It was found thatthe minimum current drive requirement did not occurat the maximumβ due to a combination of bootstrapunderdrive near the plasma edge and overdrive nearthe plasma center. The external kink mode is assumedt ell,l backc eld.T ings nd af Ther nga-t hem f-dc ent theβ ingβ avem int tede alc

References

[1] C.E. Kessel, J. Manickam, G. Rewoldt, W. Tang, Improvedplasma performance in tokamaks with negative magnetic shear,Phys. Rev. Lett. 72 (1994) 1212.

[2] S.C. Jardin, C.E. Kessel, C.G. Bathke, D.A. Ehst, T.K. Mau,F. Najmabadi, T.W. Petrie, ARIES Team, Physics basis for areversed shear tokamak power plant, Fusion Eng. Design 38(1997) 27.

[3] X. Litaudon, R. Arslanbekov, G.T. Hoang, E. Joffrin, F.Kazarian-Vibert, D. Moreau, et al., Stationary magnetic shearreversal experiments in tore supra, Plasma Phys. Contr. Fusion38 (1996) 1603.

[4] C. Gormezano, High performance tokamak operation regimes,Plasma Phys. Contr. Fusion 41 (1999) 367.

[5] A. Becoulet, L.-G. Eriksson, Yu.F. Baranov, D.N. Borba, C.D.Challis, G.D. Conway, et al., Performance and control of opti-mized shear discharges in JET, Nucl. Fusion 40 (2000) 1113.

[6] S. Ishida, JT-60 Team, JT-60U high performance regimes, Nucl.Fusion 39 (1999) 1211.

[7] T. Oikawa, JT-60 Team, High performance and prospects forsteady state operation of JT-60U, Nucl. Fusion 40 (2000)1125.

[8] V.S. Chan, C.M. Greenfield, L.L. Lao, T.C. Luce, C.C. Petty,G.M. Staebler, et al., DIII-D advanced tokamak researchoverview, Nucl. Fusion 40 (2000) 1137.

[9] C.C. Petty, T.C. Luce, P.A. Politzer, M.R. Wade, S.L. Allen,M.E. Austin, et al., Advanced tokamak physics in DIII-D,Plasma Phys. Contr. Fusion 42 (2000) 75.

[10] R.C. Wolf, O. Gruber, M. Maraschek, R. Dux, C. Fuchs, S.Gunter, et al., Stationary advanced scenarios with internal trans-port barrier on ASDEX upgrade, Plasma Phys. Contr. Fusion41 (1999) 93.

[11] O. Gruber, R.C. Wolf, H.-S. Bosch, R. Dux, S. Gunter, P.J.McCarthy, et al., Steady state H-mode and Te≈ Ti opera-

ucl.

[ tricem,

[ bility

[ il-ms,

[ erti-ure,

[ n 34

[ adi,mak

[ cur-

[ nce10.

o be stabilized by a combination of a conducting shocated at 0.33 times the minor radius, and a feedontrol system with coils located behind the shihe vertical instability is slowed down by a conducthell also located at 0.33 times the minor radius, aeedback control coils located behind the shield.esulting plasma configuration has a separatrix eloion and triangularity of 2.2 and 0.9, respectively. Taximumβ is 10%, withq95 > 3.0, resulting in a selriven (bootstrap + diamagnetic + Pfirsch–Schluter)urrent of 91%. In order to provide margin betwehe operating pressure and the ideal MHD limit,

is reduced to 90% of its maximum value, givN = 5.4. The physics improvements noted here hade a significant contribution to the reduction

he power plant cost of electricity (COE) reporlsewhere, by increasingβ and reducing the externurrent drive power.

tion with internal transport barriers in ASDEX upgrade, NFusion 40 (2000) 1145.

12] J. Delucia, S.C. Jardin, A.M.M. Todd, An iterative memethod for solving the inverse tokamak equilibrium problJ. Comp. Phys. 37 (1980) 183.

13] J.M. Greene, M.S. Chance, The second region of staagainst ballooning modes, Nucl. Fusion 21 (1981) 453.

14] R.C. Grimm, R.L. Dewar, J. Manickam, Ideal MHD stabity calculations in axisymmetric toroidal coordinate systeJ. Comp. Phys. 49 (1983) 94.

15] S.W. Haney, L.D. Pearlstein, R.H. Bulmer, J.P. Friedberg, Vcal stability analysis of tokamaks using a variational procedPlasma Phys. Rep. 23 (1997) 789.

16] C.E. Kessel, Bootstrap current in a tokamak, Nucl. Fusio(1994) 1221.

17] S.C. Jardin, C.E. Kessel, T.K. Mau, R.L. Miller, F. NajmabV.S. Chan, et al., Physics Basis for the Advanced TokaPower Plant, ARIES-AT, Fus. Eng. Des. 80 (2006) 25–62.

18] S.P. Hirshman, Finite aspect ratio effects on the bootstraprent in tokamaks, Phys. Fluids 31 (1988) 3150.

19] L.A. El-Guebaly and The ARIES Team, Nuclear performaassessment of ARIES-AT, Fus. Eng. Des. 80 (2006) 99–1


[20] A. Bondeson, D.J. Ward, Stabilization of external modes intokamaks by resistive walls and plasma rotation, Phys. Rev.Lett. 72 (1994) 2709.

[21] Y.Q. Liu, A. Bondeson, C.M. Fransson, B. Lennartson, C. Bre-itholtz, Feedback stabilization of non-axisymmetric resistive

wall modes in tokamaks. Part 1. Electromagnetic model, Phys.Plasma 7 (2000) 3681.

[22] S.C. Jardin, N. Pomphrey, J. Delucia, Dynamic modeling oftransport and position control of tokamaks, J. Comp. Phys. 66(1986) 481.

plasma profile and shape optimization for the advanced tokamak power plant, aries-at

Documents