{"title": "Real-Time Control of a Tokamak Plasma Using Neural Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 1007, "page_last": 1014, "abstract": "", "full_text": "Real-Time Control of a  Tokamak  Plasma \n\nUsing Neural Networks \n\nChris M  Bishop \n\nNeural  Computing Research  Group \nDepartment of Computer Science \n\nAston  University \n\nBirmingham, B4  7ET, U.K. \n\nc.m .bishop@aston .ac.uk \n\nPaul S  Haynes,  Mike  E  U  Smith, Tom  N  Todd, \n\nDavid L  Trotman and Colin G  Windsor \n\nAEA Technology,  Culham Laboratory, \n\nOxfordshire OX14  3DB \n\n(Euratom/UKAEA Fusion Association) \n\nAbstract \n\nThis  paper  presents  results  from  the  first  use  of neural  networks \nfor  the real-time feedback  control  of high  temperature plasmas in \na  tokamak fusion  experiment.  The tokamak is  currently  the  prin(cid:173)\ncipal  experimental  device  for  research  into  the  magnetic  confine(cid:173)\nment  approach  to  controlled  fusion. \nIn  the  tokamak,  hydrogen \nplasmas,  at  temperatures  of  up  to  100  Million  K,  are  confined \nby  strong  magnetic  fields.  Accurate  control  of the  position  and \nshape  of the plasma boundary  requires  real-time feedback  control \nof the magnetic field  structure on  a  time-scale of a  few  tens of mi(cid:173)\ncroseconds.  Software simulations have demonstrated that a  neural \nnetwork  approach  can  give  significantly  better  performance  than \nthe linear technique currently  used  on  most tokamak experiments. \nThe practical  application of the neural  network  approach  requires \nhigh-speed  hardware,  for  which  a  fully  parallel  implementation of \nthe  multilayer perceptron,  using  a  hybrid  of digital and  analogue \ntechnology,  has been  developed. \n\n\f1008 \n\nC.  Bishop,  P.  Haynes,  M.  Smith,  T.  Todd,  D.  Trotman,  C.  Windsor \n\n1 \n\nINTRODUCTION \n\nFusion  of the nuclei  of hydrogen  provides  the energy source  which  powers  the sun. \nIt  also  offers  the  possibility  of a  practically  limitless  terrestrial  source  of energy. \nHowever,  the harnessing of this power  has proved to be a  highly challenging  prob(cid:173)\nlem.  One of the  most promising approaches  is  based on  magnetic confinement of a \nhigh temperature (107  - 108  Kelvin) plasma in a  device  called a  tokamak (from the \nRussian  for  'toroidal magnetic  chamber')  as  illustrated  schematically in  Figure  1. \nAt  these  temperatures  the highly  ionized  plasma is  an  excellent  electrical  conduc(cid:173)\ntor,  and  can  be  confined  and  shaped  by  strong  magnetic  fields.  Early  tokamaks \nhad plasmas with  circular  cross-sections,  for  which  feedback  control  of the  plasma \nposition and shape is  relatively straightforward.  However,  recent tokamaks, such as \nthe COMPASS experiment at Culham Laboratory, as  well  as most next-generation \ntokamaks, are  designed  to  produce  plasmas whose  cross-sections  are strongly  non(cid:173)\ncircular.  Figure  2  illustrates  some  of the  plasma shapes  which  COMPASS  is  de(cid:173)\nsigned to explore.  These novel cross-sections  provide substantially improved energy \nconfinement  properties  and  thereby  significantly  enhance  the  performance  of the \ntokamak. \n\nz \n\nR \n\nFigure  1:  Schematic cross-section  of a  tokamak experiment  show(cid:173)\ning the toroidal vacuum vessel  (outer  D-shaped curve)  and plasma \n(shown shaded).  Also shown are the radial (R)  and vertical (Z) co(cid:173)\nordinates.  To a good approximation, the tokamak can be regarded \nas axisymmetric about the Z-axis, and so the plasma boundary can \nbe described  by its cross-sectional shape at one  particular toroidal \nlocation. \n\nUnlike circular cross-section plasmas, highly non-circular shapes are more difficult to \nproduce and to control accurately,  since currents  through several control coils must \nbe adjusted simultaneously.  Furthermore,  during a  typical plasma pulse,  the shape \nmust  evolve,  usually  from  some initial  near-circular  shape.  Due  to  uncertainties \nin  the  current  and  pressure  distributions  within  the  plasma,  the  desired  accuracy \nfor  plasma control  can only be  achieved  by  making real-time measurements of the \nposition and shape of the boundary, and using error feedback to adjust the currents \nin the control coils. \n\nThe physics  of the  plasma equilibrium is  determined  by force  balance  between  the \n\n\fReal-Time Control of Tokamak Plasma  Using  Neural Networks \n\n1009 \n\ncircle \n\nellipse \n\nO-shape \n\nbean \n\nFigure 2:  Cross-sections  of the COMPASS vacuum vessel  showing \nsome examples of potential plasma shapes.  The solid  curve  is  the \nboundary  of the  vacuum  vessel,  and  the  plasma is  shown  by  the \nshaded regions. \n\nthermal  pressure  of the  plasma and  the  pressure  of the  magnetic field,  and  is  rel(cid:173)\natively  well  understood.  Particular  plasma configurations  are  described  in  terms \nof solutions of a  non-linear partial  differential  equation  called  the  Grad-Shafranov \n(GS)  equation.  Due  to  the  non-linear  nature  of this  equation,  a  general  analytic \nsolution  is  not  possible.  However,  the  GS  equation  can  be  solved  by  iterative  nu(cid:173)\nmerical methods,  with  boundary  conditions  determined  by  currents  flowing  in  the \nexternal  control  coils  which  surround  the  vacuum vessel.  On  the  tokamak itself it \nis  changes in these  currents  which  are used  to alter the position and cross-sectional \nshape  of the  plasma.  Numerical solution  of the  GS  equation  represents  the  stan(cid:173)\ndard  technique  for  post-shot  analysis  of the  plasma,  and  is  also  the  method  used \nto  generate  the  training  dataset  for  the  neural  network,  as  described  in  the  next \nsection.  However , this  approach  is  computationally very  intensive  and  is  therefore \nunsuitable for feedback  control purposes. \n\nFor  real-time  control  it  is  necessary  to  have  a  fast  (typically:::;  50J.lsec.)  determi(cid:173)\nnation  of the  plasma boundary  shape.  This  information can  be  extracted  from  a \nvariety  of diagnostic  systems,  the  most  important  being  local  magnetic  measure(cid:173)\nments  taken  at  a  number  of points  around  the  perimeter  of the  vacuum  vessel. \nMost  tokamaks have several tens or hundreds of small pick up  coils located at care(cid:173)\nfully  optimized points  around  the  torus for  this  purpose.  We shall represent  these \nmagnetic signals collectively as  a  vector  m . \n\nFor  a  large  class  of equilibria,  the  plasma boundary  can  be reasonably  well  repre(cid:173)\nsented  in  terms of a  simple parameterization, governed  by  an  angle-like  variable B, \ngiven by \n\nR(B) \nZ(B) \n\nRo + a cos(B + 8 sinB) \nZo  + a/\\,sinB \n\n(1) \n\nwhere  we  have defined  the following  parameters \n\n\f1010 \n\nC.  Bishop,  P.  Haynes,  M.  Smith,  T.  Todd,  D.  Trotman,  C.  Windsor \n\nRo \nZo \na \nK \n6 \n\nradial distance of the plasma center from  the major axis of the torus, \nvertical distance of the plasma center from  the  torus  midplane, \nminor radius measured in  the plane Z = Zo, \nelongation, \ntriangularity. \n\nWe  denote these  parameters collectively by Yk.  The basic  problem which has  to be \naddressed,  therefore,  is  to  find  a  representation  for  the  (non-linear)  mapping from \nthe magnetic signals m  to  the  values  of the geometrical parameters  Yk,  which  can \nbe implemented in suitable hardware for  real-time control. \n\nThe  conventional  approach  presently  in  use  on  many  tokamaks involves  approxi(cid:173)\nmating the  mapping between  the  measured  magnetic signals  and  the geometrical \nparameters  by  a  single linear  transformation.  However,  the  intrinsic  non-linearity \nof the mappings suggests  that a  representation  in terms of feedforward  neural  net(cid:173)\nworks should give significantly improved results  (Lister  and Schnurrenberger,  1991; \nBishop  et  a/.,  1992;  Lagin  et  at.,  1993).  Figure  3  shows  a  block  diagram  of the \ncontrol  loop for  the neural  network approach  to tokamak equilibrium control. \n\nNeural \nNetwork \n\nFigure  3:  Block  diagram  of the  control  loop  used  for  real-time \nfeedback  control of plasma position and shape. \n\n2  SOFTWARE SIMULATION  RESULTS \n\nThe  dataset  for  training  and  testing  the  network  was  generated  by  numerical so(cid:173)\nlution of the  GS  equation using  a  free-boundary  equilibrium code.  The data base \ncurrently  consists  of over  2,000  equilibria spanning  the  wide  range  of plasma po(cid:173)\nsitions  and  shapes  available  in  COMPASS.  Each  equilibrium  configuration  takes \nseveral  minutes to generate  on  a fast  workstation.  The boundary of each  configu(cid:173)\nration is  then fitted  using the form in equation 1, so  that the equilibria are labelled \nwith  the  appropriate  values  of the shape  parameters.  Of the  120  magnetic signals \navailable on  COMPASS  which  could  be  used  to  provide  inputs  to  the network,  a \n\n\fReal-Time  Control o/Tokamak PLasma  Using  Neural Networks \n\n1011 \n\nsubset  of 16  has  been  chosen  using  sequential forward  selection  based  on  a  linear \nrepresentation  for  the mapping (discussed  below) . \nIt is important to note that the transformation from magnetic signals to flux surface \nparameters involves an exact linear invariance.  This follows from the fact that, if all \nof the currents are scaled by a constant factor, then the magnetic fields will be scaled \nby  this factor,  and  the geometry of the  plasma boundary  will  be unchanged.  It is \nimportant to take advantage of this prior knowledge and to build it into the network \nstructure,  rather  than  force  the  network  to  learn  it  by  example.  We  therefore \nnormalize the  vector  m  of input signals  to the  network  by  dividing  by  a  quantity \nproportional  to the  total  plasma current.  Note  that  this  normalization has  to  be \nincorporated into the hardware implementation of the network, as  will be discussed \nin  Section  3. \n\n4 \n\n01  2 \nc .5. \nCIS \n:E \n1iI \n~ -2 \n::J \n\n0- \u00b0 \n\n-4 \n\n2 \n\n01 c .5. \n:E  \u00b0 \n~ \n1iI \nCD c \n::J \n\n-2 \n\n1.2 \n\ngo.8 \n.5. \n0-\nCIS \n:E \n1iI0.4 \nCD c \n::J \n\n\u00b0 \n\nDatabase \n\nDatabase \n\nDatabase \n\n4 \n\n2 \n\n-2 \n\n-4 \n\n~ \n\n~  \u00b0 \n\nCD \nZ \n~ \n:::I \nCD z \n\n1.2 \n\n~O.8 \n~ \nCD z \n~O.4 \n:::I \nCD \nZ \n\n\u00b0 \n\n.2 \n\n.2 \n\nDatabase \n\nDatabase \n\nDatabase \n\nFigure  4:  Plots  of the  values  from  the  test  set  versus  the  values \npredicted  by  the linear mapping for  the 3 equilibrium parameters, \ntogether  with  the corresponding  plots for  a  neural network  with  4 \nhidden units. \n\nThe results presented in this paper are based on a multilayer perceptron architecture \nhaving  a  single  layer  of hidden  units  with  'tanh'  activation functions ,  and  linear \noutput units.  Networks are trained by minimization of a sum-of-squares error using \na  standard  conjugate gradients  optimization algorithm, and  the  number of hidden \n\n\fJ012 \n\nC.  Bishop,  P.  Haynes,  M.  Smith,  T.  Todd,  D.  Trotman,  C.  Windsor \n\nunits is  optimized by  measuring performance  with  respect  to  an  independent  test \nset.  Results  from  the  neural  network  mapping are  compared  with  those  from  the \noptimal  linear  mapping,  that  is  the  single  linear  transformation  which  minimizes \nthe same sum-of-squares error  as  is  used  in  the neural  network  training algorithm, \nas  this represents  the method currently used  on a number of present  day  tokamaks. \n\nInitial results  were  obtained  on  networks  having  3 output units,  corresponding  to \nthe  values  of vertical  position  ZQ,  major radius  RQ, and elongation  K;  these  being \nparameters which  are  of interest  for  real-time feedback  control.  The smallest nor(cid:173)\nmalized test set  error of 11.7 is  obtained from the network having  16  hidden  units. \nBy comparison, the optimal linear mapping gave a normalized test set error of 18.3. \nThis represents  a reduction in error of about 30%  in going from the linear mapping \nto the neural  network.  Such  an improvement, in the context of this application, is \nvery  significant. \n\nFor  the experiments on  real-time feedback  control  described  in  Section  4  the cur(cid:173)\nrently available hardware only permitted networks having 4 hidden units, and so we \nconsider  the results from  this  network  in  more detail.  Figure 4  shows  plots of the \nnetwork  predictions  for  various  parameters  versus  the  corresponding  values  from \nthe  test  set  portion  of the  database.  Analogous  plots  for  the optimal linear  map \npredictions  versus  the  database  values  are  also  shown.  Comparison of the  corre(cid:173)\nsponding  figures  shows  the  improved  predictive  capability  of the  neural  network, \neven  for  this sub-optimal network topology. \n\n3  HARDWARE IMPLEMENTATION \n\nThe hardware implementation of the  neural  network  must have  a  bandwidth of 2: \n20  kHz  in  order  to cope  with  the fast  timescales  of the  plasma evolution.  It must \nalso  have  an  output  precision  of at least  (the the analogue equivalent of)  8  bits in \norder to ensure that the final accuracy which is attainable will not be limited by the \nhardware system.  We have chosen to develop a fully parallel custom implementation \nof the multilayer perceptron,  based  on  analogue signal  paths with  digitally stored \nsynaptic  weights  (Bishop  et  al.,  1993).  A  VME-based  modular  construction  has \nbeen  chosen  as  this  allows flexibility  in  changing the network  architecture,  ease  of \nloading network  weights,  and simplicity of data acquisition.  Three separate  types \nof card  have been  developed  as follows: \n\n\u2022  Combined 16-input buffer  and signal normalizer. \n\nThis  provides  an  analogue hardware  implementation of the input  normal(cid:173)\nization described  earlier. \n\u2022  16  x  4 matrix multiplier \n\nThe  synaptic  weights  are  produced  using  12  bit  frequency-compensated \nmultiplying DACs  (digital to analogue converters)  which  can  be configured \nto allow 4-quadrant multiplication of analogue signals by  a  digitally stored \nnumber. \n\n\u2022  4-channel sigmoid module \n\nThere  are  many  ways  to  produce  a  sigmoidal  non-linearity,  and  we  have \nopted for  a  solution  using  two  transistors  configured  as  along-tailed-pair, \n\n\fReal-Time  Control of Tokamak Plasma  Using  Neural Networks \n\n1013 \n\nto generate  a  'tanh ' sigmoidal transfer  characteristic.  The principal  draw(cid:173)\nback  of such  an  approach  is  the  strong  temperature sensitivity  due  to the \nappearance of temperature in the denominator of the exponential transistor \ntransfer  characteristic.  An elegant solution to this  problem has been found \nby exploiting a  chip  containing 5 transistors in close thermal contact.  Two \nof the  transistors  form  the  long-tailed  pair,  one  of the  transistors  is  used \nas  a  heat  source,  and  the  remaining two  transistors  are  used  to  measure \ntemperature.  External  circuitry  provides  active thermal feedback  control, \nand stability to changes in ambient temperature over the range O\u00b0C to 50\u00b0C \nis  found  to be  well  within the acceptable range. \n\nThe  complete  network  is  constructed  by  mounting  the  appropriate  combination \nof  cards  in  a  VME  rack  and  configuring  the  network  topology  using  front  panel \ninterconnections.  The  system  includes  extensive  diagnostics,  allowing  voltages  at \nall key  points within the network to be monitored as a function of time via a  series \nof multiplexed output channels. \n\n4  RESULTS  FROM  REAL-TIME FEEDBACK  CONTROL \n\nFigure  5  shows  the  first  results  obtained  from  real-time  control  of the  plasma in \nthe COMPASS  tokamak using  neural  networks.  The evolution of the plasma elon(cid:173)\ngation,  under  the  control  of the  neural  network,  is  plotted  as  a  function  of time \nduring  a  plasma pulse.  Here  the  desired  elongation  has  been  preprogrammed  to \nfollow  a  series  of steps  as  a  function  of time.  The  remaining  2  network  outputs \n(radial position  Ro  and vertical  position Zo)  were  digitized for  post-shot  diagnosis , \nbut were  not used  for  real-time control.  The solid curve shows  the value of elonga(cid:173)\ntion  given  by  the  corresponding  network  output,  and the  dashed  curve  shows  the \npost-shot  reconstruction  of the elongation obtained from  a  simple  'filament'  code, \nwhich gives relatively rapid post-shot  plasma shape reconstruction  but with limited \naccuracy.  The circles  denote the elongation values given by the much more accurate \nreconstructions  obtained from  the full  equilibrium code.  The graph  clearly  shows \nthe network  generating  the required  elongation signal  in  close  agreement  with  the \nreconstructed  values.  The typical residual error is  of order 0.07 on elongation values \nup  to around  1.5.  Part of this error  is  attributable to residual offset  in the integra(cid:173)\ntors  used  to  extract  magnetic field  information from  the  pick-up  coils,  and  this  is \ncurrently  being corrected  through modifications to the integrator design.  An  addi(cid:173)\ntional  contribution  to  the  error  arises  from  the  restricted  number  of hidden  units \navailable with the initial hardware configuration.  While these results represent  the \nfirst  obtained using closed  loop control, it is  clear from earlier software modelling of \nlarger network architectures  (such  as 32- 16-4) that residual errors of order a few  % \nshould be attainable.  The implementation of such larger networks  is  being persued, \nfollowing the successes  with the smaller system. \n\nAcknowledgements \n\nWe  would  like  to  thank  Peter  Cox,  Jo  Lister  and  Colin  Roach  for  many  useful \ndiscussions  and technical  contributions.  This  work  was  partially supported  by  the \nUK  Department of Trade and Industry. \n\n\f1014 \n\nC.  Bishop,  P.  Haynes,  M.  Smith,  T.  Todd,  D.  Trotman,  C.  Windsor \n\n1.8 \n\nshot 9576 \n\nC) \n\nc: o \n~  14 \nc: o as \n\n\u2022 \n\n1.0 \n\n0.0 \n\n0.1 \n\n0.2 \n\ntime (sec.) \n\nFigure  5:  Plot  of the  plasma elongation  K.  as  a  function  of time \nduring shot no.  9576 on the COMPASS tokamak, during which the \nelongation was being controlled in real-time by the neural network. \n\nReferences \n\nBishop C M,  Cox P,  Haynes P S,  Roach C M,  Smith M E U, Todd T  N and Trotman \nD  L,  1992.  A neural  network  approach  to tokamak equilibrium control.  In  Neural \nNetwork Applications,  Ed.  J  G Taylor,  Springer  Verlag,  114-128. \n\nBishop C M,  Haynes  P  S,  Roach C M,  Smith ME U,  Todd T  N,  and Trotman D L. \n1993.  Hardware implementation of a  neural network for  plasma position control in \nCOMPASS-D. In  Proceedings  of the  17th.  Symposium  on  Fusion  Technology,  Rome, \nItaly.  2  997-1001. \n\nLagin  L,  Bell  R,  Davis  S,  Eck  T,  Jardin  S,  Kessel  C,  Mcenerney  J,  Okabayashi \nM,  Popyack  J  and  Sauthoff N.  1993.  Application  of neural  networks  for  real-time \ncalculations  of plasma equilibrium  parameters  for  PBX-M,  In  Proceedings  of the \n17th.  Symposium  on  Fusion  Technology,  Rome,  Italy.  21057-106l. \n\nLister  J  Band  Schnurrenberger  H.  1991.  Fast  non-linear  extraction  of  plasma \nparameters using  a neural network mapping.  Nuclear Fusion.  31,  1291-1300. \n\n\f", "award": [], "sourceid": 1005, "authors": [{"given_name": "Chris", "family_name": "Bishop", "institution": null}, {"given_name": "Paul", "family_name": "Haynes", "institution": null}, {"given_name": "Mike", "family_name": "Smith", "institution": null}, {"given_name": "Tom", "family_name": "Todd", "institution": null}, {"given_name": "David", "family_name": "Trotman", "institution": null}, {"given_name": "Colin", "family_name": "Windsor", "institution": null}]}