In GitLab by @fmavig on Mar 2, 2017, 15:16

Dear All,
I would like to introduce some elements on the various points:

• for the Constraint 41: There is a preliminary attempt to establish the maximum voltage for the CS and PF coils, and it is reported in the document
  https://idm.euro-fusion.org/?uid=2NEVTT
• Also there is a study carried from CREATE where first ramp-up studies were performed (from xpoint configuration, considered at 5MA to flattop, ignoring for the time being the limiter phase), and it is available at https://idm.euro-fusion.org/?uid=2N9MMH , slide 5, where it is reported the upper limit for plasma current ramprate is dIp/dt = + 0.2 MA/s (instead of 0.4Ejima). This was obtained in collaboration with transport simulations.

Best regards.

Francesco

Current ramp

The plasma current ramp-up time is set as 0.1 MA/s (assuming lpulse=1 and pulsetimings=0):

This is consistent with Impact of the plasma operation on the technical requirements in EU-DEMO M. Siccinio

The calculation of resistive flux consumption during start-up and current ramp is based on the assumption that the ramp-up takes of the order of the resistive current penetration time during the ramp

where $\eta_{CR}$ is the resistivity of the plasma during the current ramp. The flux consumption is therefore independent of the resistivity and the minor radius and is:

where $C_E$ is the empirical Ejima constant, defined by the user (default value 0.4).

With this assumption the flux consumption is independent of the temperature and $Z_{eff}$ during the current ramp.

The 0.1 MA/s limit is probably much the same as the assumption above since the plasma becomes unstable if the ramp is much faster than the resistive penetration time, because a current hole develops. The other two constraints (41 and 66), however, are additional and may be limiting.

Let's assume the ramp time is an input or iteration variable $t_{CR}$ chosen to obey based these constraints (tohs). Assume that the temperature and $Z_{eff}$ during the ramp are user inputs (as fractions of the flat-top values). Multiplying the flux consumption by

where $\tau_{CR}$ is the resistive penetration time during current ramp, and relating the loop resistance to the resisitivity by

gives the proposed resistive flux consumption during the current ramp:

where $\kappa$ is the elongation and $\rho_{CR}$ is the plasma loop resistance at the (assumed) conditions during the current ramp.

To estimate $\rho_{CR}$ we need the typical temperature and effective charge during the current ramp, which can be input as a fraction of the flat-top values:
t_current_ramp = t_current_ramp_ratio * te and
zeff_current_ramp = z_eff_current_ramp_ratio * (zeff - 1) +1
Suggested defaults:
t_current_ramp_ratio = 0.25,
z_eff_current_ramp_ratio = 0.25

Fusion power ramp

The ramp-up for fusion power takes longer than the current ramp, and the current must be maintained throughout this period, using additional flux swing.

We might assume that temperature and $Z_{eff}-1$ are both proportional to time during the power ramp. In this case the loop voltage would decrease as ${1} / {\sqrt{t}}$. (Although the loop resistance would be infinite at time zero, the integral is well-behaved.)

However, this is not what we see in Mattia's plots below, where the loop voltage is very roughly constant during the power ramp (200 to 700 s) and very roughly equal to the flat-top value. (The voltage is outside the range of the plot during the current ramp.)

It is important to distinguish between loop voltage and resistive loop voltage. When the current is constant they should be the same, although this does not seem to be the case for the first minute or so after the current ramp:

Loop voltage for the ramp-up case shown in the previous figure. The total loop voltage and its resistive fraction are shown. Black vertical lines identify the end of the current ramp (200 s) and the end of the density ramp (600 s).

We can therefore just add t_fusion_ramp to burn_time when calculating the resistive flux swing requirement. This already seems to be correct:

Change the default to t_fusion_ramp = 700 following the DEMO example above.

Comments? @ajpearcey @stuartmuldrew @jonmaddock @jmorris-uk @chris-ashe @j-a-foster

t_fusion_ramp = 700: large tokamak works OK. Total volt-second requirement increases by ~10% but bizarrely the burn time has increased slightly and the central solenoid is slightly thinner! This may all go wrong when someone has a really good look at the flux consumption calculations.

Fully implemented.
Resistive current ramp flux swing has dropped from 50 to 14 Vs because the duration of the current ramp is now 166 s instead of, by implication, the resistive diffusion time of 3000 s. However, the inductive flux consumption dominates.

Resistive flux swing during burn has gone up because the relevant duration has gone up from 10 to 700 s.

 Total volt-second requirement (Wb)                                       (vsstt)                   5.406E+02  OP 
 Inductive volt-seconds (Wb)                                              (vsind)                   2.372E+02  OP 
 Ejima coefficient                                                        (gamma)                       0.300     
 Start-up resistive (Wb)                                                  (vsres)                   1.360E+01  OP 
 Flat-top resistive (Wb)                                                  (vsbrn)                   2.899E+02  OP 
...
 Loop voltage during burn (V)                                             (vburn)                   3.669E-02  OP 
 Plasma resistance (ohm)                                                  (rplas)                   3.973E-09  OP 
 Resistive diffusion time (s)                                             (res_time)                3.063E+03  OP 
 Plasma inductance (H)                                                    (rlp)                     1.432E-05  OP 
 Coefficient for sawtooth effects on burn V-s requirement                 (csawth)                      1.000     
 Temperature during the current ramp as a fraction of the flat-top value  (t_cr_ratio)                  0.250  IP 
 (Zeff-1) during the current ramp as a fraction of the flat-top value     (z_eff_cr_ratio)              0.250  IP 
 Plasma loop resistance during the current ramp (ohm)                     (rho_cr)                  1.786E-08  OP 

tburn has gone down to its minimum value, but bizarrely the CS thickness (ohcth) and the bore have both decreased very slightly - both of which tend to reduce the flux provided by the CS.

Minor point: I have tweaked the descriptions here:

 Total V-s capability of Central Solenoid/PF coils (Wb)                   (abs(vstot))              5.574E+02     
 Required volt-seconds during current ramp-up (Wb)                        (vssoft)                  2.508E+02     
 Available volt-seconds during fusion power ramp and flat-top (Wb)        (vsmax)                   2.899E+02