Nonlinear dynamic modeling for control of fusion devices
Alessandro Beghi, Mario Cavinato and Angelo Cenedese*
Abstract— In this paper the issue of modeling for control
design with application to fusion devices is discussed. Given
the difficulty to provide analytical solutions to the equations
describing the system dynamics, the use of numerical tools for
evaluating different control architectures is required. Although
standard tools for Computer Aided Control System Design
(CACSD) can be usefully employed for both analyzing the fu-
sion device dynamics and designing control laws, dedicated tools
are required to adequately describe the complex interactions
among the different system components. In this paper the use of
the nonlinear equilibrium code MAXFEA to support the control
design task is described. MAXFEA is a finite element code able
to produce quite good approximations of the plasma boundary
location and shape, together with internal distributions of
current and magnetic fields, and other plasma features. The
code provides the simulation of the plasma dynamics, while
all the other elements (diagnostics, controller, actuators) in
the control loop system can be modeled independently and
integrated in the code as external modules, thus making it a
candidate tool for Software-in-the-Loop solutions.
I. INTRODUCTION
One of the most promising and viable approaches to
produce energy from nuclear fusion reactions, on Earth and
in a controlled way, is to resort to the magnetically confined
plasmas, which yield a relevant rate of nuclear reactions
while kept inside a metallic toroidal chamber at extremely
low pressures and high temperatures.
The equation governing the equilibrium of a magnetically
confined plasma is the Grad-Shafranov equation, a two
dimensional nonlinear elliptic partial differential equation [1]
that basically states the equilibrium of a plasma column with
an external magnetic field, in hypothesis of axisymmetry. The
toroidal plasma current density profile J
p
(ψ) is given as a
function of the poloidal flux ψ by the following
J
p
(ψ)= L(ψ)= -µ
0
r
dp(ψ)
dψ
-
1
r
f (ψ)
df
dψ
, (1)
with L being the operator
L =
∂
∂z
1
r
∂
∂z
+
∂
∂r
1
r
∂
∂r
. (2)
The profile is thus parameterized according to the two
functions p(ψ) and f (ψ), respectively the pressure profile
and the poloidal current density flux function.
*Corresponding author.
A. Beghi is with Department of Information Engineering, University of
Padova, Padova, Italy beghi@dei.unipd.it
M. Cavinato is with Consorzio RFX - EURATOM Fusion Association,
Padova, Italy mario.cavinato@igi.cnr.it
A. Cenedese is with the Department of Engineering
and Management, University of Padova, Vicenza, Italy
angelo.cenedese@unipd.it
Fig. 1: Control Loop scheme. The open loop is the cascade
of power supply, plasma-vessel system, and diagnostics.
Solving the Grad-Shafranov equation means to compute
the magnetic flux distribution for a given external coil current
configuration, whence the plasma boundary (its flux value,
location, and shape) is determined. To compute a boundary
for a plasma in equilibrium is known as the free boundary
problem. Actually, the nonlinear nature of the problem arises
from the parametrization chosen for p(ψ) and f (ψ) and the
free-boundary condition itself.
Unfortunately, in general this does not offer an analytical
solution so that a common way to solve the problem is
to employ numerical procedures, as finite element or finite
difference methods, implemented in equilibrium codes.
At the same time, the magnetic control of a fusion device
is required for the control of the macroscopic characteristics
of the plasma mass (shape, position, modes, current) both to
drive the plasma during the various phases of the discharge
and to counteract disturbances and internal instabilities, and
is exerted by acting dynamically on the magnetic field
produced by external coils surrounding the vacuum chamber.
The use of equilibrium codes is of fundamental importance
not only for the inherent level of detail, but also for the
perception of the dynamics of the complete system obtained
through the nonlinear simulation. In fact, the complete closed
loop system can be simulated in the equilibrium code:
The vessel-plasma system, the diagnostics (output measure-
ments), the actuators (input signals). The power supply
block and the diagnostics can be easily understood and
their accurate modeling poses stringent constraints on system
controllability [2]. The vessel-plasma system basically refers
to the complex interaction among the effects of the active coil
currents, the eddy currents flowing in the machine structure
(passive metallic structures, plasma facing components, ...),
and the plasma, to produce the output quantities measured
by the diagnostic sensors.
In particular, the possibility of performing long time steady
state simulations, or even simulations of different phases of
the plasma discharge with the relevant transitions, represents
Proceedings of the
47th IEEE Conference on Decision and Control
Cancun, Mexico, Dec. 9-11, 2008
WeB18.3
978-1-4244-3124-3/08/$25.00 ©2008 IEEE 3133