Page 298 - Fundamentals of Magnetic Thermonuclear Reactor Design

Plasma Control System Chapter | 8 277

the best point where the flux function Ψ can be decomposed in series using
basis functions. The toroidal loops are located in these points. The flux function
value, Ψ= Ψ x (), is calculated for the X-point whose coordinates are found in Ψ=Ψx
the divertor region. The coordinates of other special points are calculated from
Ψ= Ψ x (). Ψ=Ψx
The movable current filament method is based on a model that assumes that
the plasma current is a discrete set of N current filaments distributed inside the
plasma, so that

N
r rz zI , j )
j ( r z ∑ δ( − j , − j (8.67) jr,z=∑j=1Nδr−rj,z−zjIj,
, ) =
ϕ
j 1 =
th
where r , z and I are the coordinates and the current of the j filament, and
j
j
j
rr zz )
δ( − j , − j is the Dirac delta function. In a general case, system (8.67) δr−rj,z−zj
has 3N free parameters: 2N coordinates and N currents. Its advantage is that
it allows any magnetic magnitude yr z, ( ) (flux, field and current momenta) yr,z
to be computed using a Green’s function Q for current filaments, which is
known:
N
yr z ∑ ( , ,, ) j (8.68) yr,z=∑j=1NQr,z,rj,zjIj.
) =
, (
Q rz rz I .
j
j
j 1 =
The model underlying this method has fixed currents in all filaments, while
the filament coordinates are unknowns. These coordinates are found by the best
agreement between the Eq. (8.68) calculations and magnetic signals measured
with the sensors:
y (r zr z,, , j ) = y (rz, i ), i = 1,..., M, (8.69) yri,zi,rj,zj=y¯ri,zi, i=1,⋯,M ,
i
j
i
i
where M is the number of measurements that must be equal to or exceed 2N.
The (8.69) non-linear system is solved using Newton’s iterative method:
k
k
k
k
r j k 1+ = r + ∆ rz k 1+ = z +∆ z , rjk+1=rjk+∆rjk, zjk+1=zjk+∆zjk,
,
j
j
j
j
j
N ∂ Q k ∂ Q k
∑ ij I ∆+ ij I ∆ z = y − y i, = 1,..., M. (8.70) ∑j=1N∂Qijk∂rjIj∆rjk+∂Qijk∂zjIj∆z
k
k
r
i
j
i
j
j
j
j 1 = r ∂ j z ∂ j jk=y¯i−yi, i=1,...,M.
The problem is considered to be solved if increments ∆r, and ∆z become
smaller than the given small value.
Two points that remain theoretically unclear in the solution of the system
of nonlinear equations by iterative methods are the computational convergence