numerics Module

ipnvars FIX : total number of variables available for iteration

ipeqns FIX : number of constraint equations available

ipnfoms FIX : number of available figures of merit

ioptimz /1/ : code operation switch:

• = -2 for no optimisation, no VMCOM or HYBRD;
• = -1 for no optimisation, HYBRD only;
• = 0 for HYBRD and VMCON (not recommended);
• = 1 for optimisation, VMCON only

minmax /7/ : switch for figure-of-merit (see lablmm for descriptions) negative => maximise, positive => minimise

lablmm(ipnfoms) : labels describing figures of merit:

• ( 1) major radius
• ( 2) not used
• ( 3) neutron wall load
• ( 4) P_tf + P_pf
• ( 5) fusion gain Q
• ( 6) cost of electricity
• ( 7) capital cost (direct cost if ireactor=0, constructed cost otherwise)
• ( 8) aspect ratio
• ( 9) divertor heat load
• (10) toroidal field
• (11) total injected power
• (12) hydrogen plant capital cost OBSOLETE
• (13) hydrogen production rate OBSOLETE
• (14) pulse length
• (15) plant availability factor (N.B. requires iavail=1 to be set)
• (16) linear combination of major radius (minimised) and pulse length (maximised) note: FoM should be minimised only!
• (17) net electrical output
• (18) Null Figure of Merit
• (19) linear combination of big Q and pulse length (maximised) note: FoM should be minimised only!

ncalls : number of function calls during solution

neqns /0/ : number of equality constraints to be satisfied

nfev1 : number of calls to FCNHYB (HYBRD function caller) made

nfev2 : number of calls to FCNVMC1 (VMCON function caller) made

nineqns /0/ : number of inequality constraints VMCON must satisfy (leave at zero for now)

nvar /16/ : number of iteration variables to use

nviter : number of VMCON iterations performed

icc(ipeqns) /0/ : array defining which constraint equations to activate (see lablcc for descriptions)

active_constraints(ipeqns) : Logical array showing which constraints are active

lablcc(ipeqns) : labels describing constraint equations (corresponding itvs)

• ( 1) Beta (consistency equation) (itv 5)
• ( 2) Global power balance (consistency equation) (itv 10,1,2,3,4,6,11)
• ( 3) Ion power balance DEPRECATED (itv 10,1,2,3,4,6,11)
• ( 4) Electron power balance DEPRECATED (itv 10,1,2,3,4,6,11)
• ( 5) Density upper limit (itv 9,1,2,3,4,5,6)
• ( 6) (Epsilon x beta poloidal) upper limit (itv 8,1,2,3,4,6)
• ( 7) Beam ion density (NBI) (consistency equation) (itv 7)
• ( 8) Neutron wall load upper limit (itv 14,1,2,3,4,6)
• ( 9) Fusion power upper limit (itv 26,1,2,3,4,6)
• (10) Toroidal field 1/R (consistency equation) (itv 12,1,2,3,13 )
• (11) Radial build (consistency equation) (itv 3,1,13,16,29,42,61)
• (12) Volt second lower limit (STEADY STATE) (itv 15,1,2,3)
• (13) Burn time lower limit (PULSE) (itv 21,1,16,17,29,42,44,61) (itv 19,1,2,3,6)
• (14) Neutral beam decay lengths to plasma centre (NBI) (consistency equation)
• (15) LH power threshold limit (itv 103)
• (16) Net electric power lower limit (itv 25,1,2,3)
• (17) Radiation fraction upper limit (itv 28)
• (18) Divertor heat load upper limit (itv 27)
• (19) MVA upper limit (itv 30)
• (20) Neutral beam tangency radius upper limit (NBI) (itv 33,31,3,13)
• (21) Plasma minor radius lower limit (itv 32)
• (22) Divertor collisionality upper limit (itv 34,43)
• (23) Conducting shell to plasma minor radius ratio upper limit (itv 104,1,74)
• (24) Beta upper limit (itv 36,1,2,3,4,6,18)
• (25) Peak toroidal field upper limit (itv 35,3,13,29)
• (26) Central solenoid EOF current density upper limit (ipfres=0) (itv 38,37,41,12)
• (27) Central solenoid BOP current density upper limit (ipfres=0) (itv 39,37,41,12)
• (28) Fusion gain Q lower limit (itv 45,47,40)
• (29) Inboard radial build consistency (itv 3,1,13,16,29,42,61)
• (30) Injection power upper limit (itv 46,47,11)
• (31) TF coil case stress upper limit (SCTF) (itv 48,56,57,58,59,60,24)
• (32) TF coil conduit stress upper limit (SCTF) (itv 49,56,57,58,59,60,24)
• (33) I_op / I_critical (TF coil) (SCTF) (itv 50,56,57,58,59,60,24)
• (34) Dump voltage upper limit (SCTF) (itv 51,52,56,57,58,59,60,24)
• (35) J_winding pack/J_protection upper limit (SCTF) (itv 53,56,57,58,59,60,24)
• (36) TF coil temperature margin lower limit (SCTF) (itv 54,55,56,57,58,59,60,24)
• (37) Current drive gamma upper limit (itv 40,47)
• (38) First wall coolant temperature rise upper limit (itv 62)
• (39) First wall peak temperature upper limit (itv 63)
• (40) Start-up injection power lower limit (PULSE) (itv 64)
• (41) Plasma current ramp-up time lower limit (PULSE) (itv 66,65)
• (42) Cycle time lower limit (PULSE) (itv 17,67,65)
• (43) Average centrepost temperature (TART) (consistency equation) (itv 13,20,69,70)
• (44) Peak centrepost temperature upper limit (TART) (itv 68,69,70)
• (45) Edge safety factor lower limit (TART) (itv 71,1,2,3)
• (46) Equation for Ip/Irod upper limit (TART) (itv 72,2,60)
• (47) NOT USED
• (48) Poloidal beta upper limit (itv 79,2,3,18)
• (49) NOT USED
• (50) IFE repetition rate upper limit (IFE)
• (51) Startup volt-seconds consistency (PULSE) (itv 16,29,3,1)
• (52) Tritium breeding ratio lower limit (itv 89,90,91)
• (53) Neutron fluence on TF coil upper limit (itv 92,93,94)
• (54) Peak TF coil nuclear heating upper limit (itv 95,93,94)
• (55) Vacuum vessel helium concentration upper limit iblanket =2 (itv 96,93,94)
• (56) Pseparatrix/Rmajor upper limit (itv 97,1,3)
• (57) NOT USED
• (58) NOT USED
• (59) Neutral beam shine-through fraction upper limit (NBI) (itv 105,6,19,4 )
• (60) Central solenoid temperature margin lower limit (SCTF) (itv 106)
• (61) Minimum availability value (itv 107)
• (62) taup/taueff the ratio of particle to energy confinement times (itv 110)
• (63) The number of ITER-like vacuum pumps niterpump < tfno (itv 111)
• (64) Zeff less than or equal to zeffmax (itv 112)
• (65) Dump time set by VV loads (itv 56, 113)
• (66) Limit on rate of change of energy in poloidal field (Use iteration variable 65(tohs), 115)
• (67) Simple Radiation Wall load limit (itv 116, 4,6)
• (68) Psep * Bt / qAR upper limit (itv 117)
• (69) ensure separatrix power = the value from Kallenbach divertor (itv 118)
• (70) ensure that teomp = separatrix temperature in the pedestal profile, (itv 119 (tesep))
• (71) ensure that neomp = separatrix density (nesep) x neratio
• (72) central solenoid shear stress limit (Tresca yield criterion) (itv 123 foh_stress)
• (73) Psep >= Plh + Paux (itv 137 (fplhsep))
• (74) TFC quench < tmax_croco (itv 141 (fcqt))
• (75) TFC current/copper area < Maximum (itv 143 f_coppera_m2)
• (76) Eich critical separatrix density
• (77) TF coil current per turn upper limit
• (78) Reinke criterion impurity fraction lower limit (itv 147 freinke)
• (79) Peak CS field upper limit (itv 149 fbmaxcs)
• (80) Divertor power lower limit pdivt (itv 153 fpdivlim)
• (81) Ne(0) > ne(ped) constraint (itv 154 fne0)
• (82) toroidalgap > tftort constraint (itv 171 ftoroidalgap)
• (83) Radial build consistency for stellarators (itv 172 f_avspace)
• (84) Lower limit for beta (itv 173 fbetatry_lower)
• (85) Constraint for CP lifetime
• (86) Constraint for TF coil turn dimension
• (87) Constraint for cryogenic power
• (88) Constraint for TF coil strain absolute value
• (89) Constraint for CS coil quench protection
• (90) Lower Limit on number of stress load cycles for CS (itr 167 fncycle)
• (91) Checking if the design point is ECRH ignitable (itv 168 fecrh_ignition)

ixc(ipnvars) /0/ : array defining which iteration variables to activate (see lablxc for descriptions)

lablxc(ipnvars) : labels describing iteration variables

• ( 1) aspect
• ( 2) bt
• ( 3) rmajor
• ( 4) te
• ( 5) beta
• ( 6) dene
• ( 7) rnbeam
• ( 8) fbeta (f-value for equation 6)
• ( 9) fdene (f-value for equation 5)
• (10) hfact
• (11) pheat
• (12) oacdcp
• (13) tfcth (NOT RECOMMENDED)
• (14) fwalld (f-value for equation 8)
• (15) fvs (f-value for equation 12)
• (16) ohcth
• (17) tdwell
• (18) q
• (19) beam_energy
• (20) tcpav
• (21) ftburn (f-value for equation 13)
• (22) NOT USED
• (23) fcoolcp
• (24) NOT USED
• (25) fpnetel (f-value for equation 16)
• (26) ffuspow (f-value for equation 9)
• (27) fhldiv (f-value for equation 18)
• (28) fradpwr (f-value for equation 17), total radiation fraction
• (29) bore
• (30) fmva (f-value for equation 19)
• (31) gapomin
• (32) frminor (f-value for equation 21)
• (33) fportsz (f-value for equation 20)
• (34) fdivcol (f-value for equation 22)
• (35) fpeakb (f-value for equation 25)
• (36) fbetatry (f-value for equation 24)
• (37) coheof
• (38) fjohc (f-value for equation 26)
• (39) fjohc0 (f-value for equation 27)
• (40) fgamcd (f-value for equation 37)
• (41) fcohbop
• (42) gapoh
• (43) NOT USED
• (44) fvsbrnni
• (45) fqval (f-value for equation 28)
• (46) fpinj (f-value for equation 30)
• (47) feffcd
• (48) fstrcase (f-value for equation 31)
• (49) fstrcond (f-value for equation 32)
• (50) fiooic (f-value for equation 33)
• (51) fvdump (f-value for equation 34)
• (52) NOT USED
• (53) fjprot (f-value for equation 35)
• (54) ftmargtf (f-value for equation 36)
• (55) NOT USED
• (56) tdmptf
• (57) thkcas
• (58) thwcndut
• (59) fcutfsu
• (60) cpttf
• (61) gapds
• (62) fdtmp (f-value for equation 38)
• (63) ftpeak (f-value for equation 39)
• (64) fauxmn (f-value for equation 40)
• (65) tohs
• (66) ftohs (f-value for equation 41)
• (67) ftcycl (f-value for equation 42)
• (68) fptemp (f-value for equation 44)
• (69) rcool
• (70) vcool
• (71) fq (f-value for equation 45)
• (72) fipir (f-value for equation 46)
• (73) scrapli
• (74) scraplo
• (75) tfootfi
• (76) NOT USED
• (77) NOT USED
• (78) NOT USED
• (79) fbetap (f-value for equation 48)
• (80) NOT USED
• (81) edrive
• (82) drveff
• (83) tgain
• (84) chrad
• (85) pdrive
• (86) frrmax (f-value for equation 50)
• (87) NOT USED
• (88) NOT USED
• (89) ftbr (f-value for equation 52)
• (90) blbuith
• (91) blbuoth
• (92) fflutf (f-value for equation 53)
• (93) shldith
• (94) shldoth
• (95) fptfnuc (f-value for equation 54)
• (96) fvvhe (f-value for equation 55)
• (97) fpsepr (f-value for equation 56)
• (98) li6enrich
• (99) NOT USED
• (100) NOT USED
• (101) NOT USED
• (102) fimpvar # OBSOLETE
• (103) flhthresh (f-value for equation 15)
• (104) fcwr (f-value for equation 23)
• (105) fnbshinef (f-value for equation 59)
• (106) ftmargoh (f-value for equation 60)
• (107) favail (f-value for equation 61)
• (108) breeder_f: Volume of Li4SiO4 / (Volume of Be12Ti + Li4SiO4)
• (109) ralpne: thermal alpha density / electron density
• (110) ftaulimit: Lower limit on taup/taueff the ratio of alpha
• (111) fniterpump: f-value for constraint that number
• (112) fzeffmax: f-value for max Zeff (f-value for equation 64)
• (113) ftaucq: f-value for minimum quench time (f-value for equation 65)
• (114) fw_channel_length: Length of a single first wall channel
• (115) fpoloidalpower: f-value for max rate of change of
• (116) fradwall: f-value for radiation wall load limit (eq. 67)
• (117) fpsepbqar: f-value for Psep*Bt/qar upper limit (eq. 68)
• (118) fpsep: f-value to ensure separatrix power is less than
• (119) tesep: separatrix temperature calculated by the Kallenbach divertor model
• (120) ttarget: Plasma temperature adjacent to divertor sheath [eV]
• (121) neratio: ratio of mean SOL density at OMP to separatrix density at OMP
• (122) oh_steel_frac : streel fraction of Central Solenoid
• (123) foh_stress : f-value for CS coil Tresca yield criterion (f-value for eq. 72)
• (124) qtargettotal : Power density on target including surface recombination [W/m2]
• (125) fimp(3) : Beryllium density fraction relative to electron density
• (126) fimp(4) : Carbon density fraction relative to electron density
• (127) fimp(5) : Nitrogen fraction relative to electron density
• (128) fimp(6) : Oxygen density fraction relative to electron density
• (129) fimp(7) : Neon density fraction relative to electron density
• (130) fimp(8) : Silicon density fraction relative to electron density
• (131) fimp(9) : Argon density fraction relative to electron density
• (132) fimp(10) : Iron density fraction relative to electron density
• (133) fimp(11) : Nickel density fraction relative to electron density
• (134) fimp(12) : Krypton density fraction relative to electron density
• (135) fimp(13) : Xenon density fraction relative to electron density
• (136) fimp(14) : Tungsten density fraction relative to electron density
• (137) fplhsep (f-value for equation 73)
• (138) rebco_thickness : thickness of REBCO layer in tape (m)
• (139) copper_thick : thickness of copper layer in tape (m)
• (140) dr_tf_wp : radial thickness of TFC winding pack (m)
• (141) fcqt : TF coil quench temperature < tmax_croco (f-value for equation 74)
• (142) nesep : electron density at separatrix [m-3]
• (143) f_copperA_m2 : TF coil current / copper area < Maximum value
• (144) fnesep : Eich critical electron density at separatrix
• (145) fgwped : fraction of Greenwald density to set as pedestal-top density
• (146) fcpttf : F-value for TF coil current per turn limit (constraint equation 77)
• (147) freinke : F-value for Reinke detachment criterion (constraint equation 78)
• (148) fzactual : fraction of impurity at SOL with Reinke detachment criterion
• (149) fbmaxcs : F-value for max peak CS field (con. 79, itvar 149)
• (150) REMOVED
• (151) REMOVED
• (152) fgwsep : Ratio of separatrix density to Greenwald density
• (153) fpdivlim : F-value for minimum pdivt (con. 80)
• (154) fne0 : F-value for ne(0) > ne(ped) (con. 81)
• (155) pfusife : IFE input fusion power (MW) (ifedrv=3 only)
• (156) rrin : Input IFE repetition rate (Hz) (ifedrv=3 only)
• (157) fvssu : F-value for available to required start up flux (con. 51)
• (158) croco_thick : Thickness of CroCo copper tube (m)
• (159) ftoroidalgap : F-value for toroidalgap > tftort constraint (con. 82)
• (160) f_avspace (f-value for equation 83)
• (161) fbetatry_lower (f-value for equation 84)
• (162) r_cp_top : Top outer radius of the centropost (ST only) (m)
• (163) f_t_turn_tf : f-value for TF coils WP trurn squared dimension constraint
• (164) f_crypmw : f-value for cryogenic plant power
• (165) fstr_wp : f-value for TF coil strain absolute value
• (166) f_copperaoh_m2 : CS coil current /copper area < Maximum value
• (167) fncycle : f-value for minimum CS coil stress load cycles
• (168) fecrh_ignition: f-value for equation 91
• (169) te0_ecrh_achievable: Max. achievable electron temperature at ignition point
• (170) beta_div : field line angle wrt divertor target plate (degrees)
• (171) casths_fraction : TF side case thickness as fraction of toridal case thickness
• (172) casths : TF side case thickness [m]
• (173) EMPTY : Description
• (174) EMPTY : Description
• (175) EMPTY : Description

sqsumsq : sqrt of the sum of the square of the constraint residuals

Description of the objective function

Normalised objective function (figure of merit)

epsfcn /1.0e-3/ : finite difference step length for HYBRD/VMCON derivatives

epsvmc /1.0e-6/ : error tolerance for VMCON

factor /0.1/ : used in HYBRD for first step size

ftol /1.0e-4/ : error tolerance for HYBRD

boundl(ipnvars) /../ : lower bounds used on ixc variables during VMCON optimisation runs