Barrier Functionals for Output Functional Estimation of PDEs

We propose a method for computing bounds on output functionals of a class of time-dependent PDEs. To this end, we introduce barrier functionals for PDE systems. By defining appropriate unsafe sets and optimization problems, we formulate an output functional bound estimation approach based on barrier functionals. In the case of polynomial data, sum of squares (SOS) programming is used to construct the barrier functionals and thus to compute bounds on the output functionals via semidefinite programs (SDPs). An example is given to illustrate the results.Comment: 8 pages, 1 figure, preprint submitted to 2015 American Control Conferenc

Bootstrap Current Optimization in Tokamaks Using Sum-of-Squares Polynomials

International audienc

Model-based Optimization and Feedback Control of the Current Density Profile Evolution in NSTX-U

Unlike nuclear fission in present nuclear power plants, where energy is generated by splitting heavy atoms like uranium, nuclear fusion generates energy by fusing light nuclei like hydrogen isotopes under high-temperature and high-pressure conditions, at which the reactants (hydrogen isotopes) separate from their electrons and form an ionized gas called plasma, which is considered as the fourth state of matter. Contrary to fission, fusion provides more energy density, poses almost no risk of a catastrophic nuclear accident, and produces mostly short-term, low-level radioactive waste.The main difficulty in maintaining fusion reactions is the development of a device that can confine the hot plasma for sufficiently long time while preventing it from hitting the walls of the confining device. Among several techniques, magnetic confinement appears as the most promising approach. In particular, the tokamak device is a toroidal device surrounded by large magnetic coils responsible for the magnetic fields that confine the plasma. A spherical tokamak, or a spherical torus (ST), is a variation of the conventional tokamak concept. Compared to a standard tokamak, the ST device extrapolates to a more compact, potentially lower-cost reactor with higher efficiency of confinement. Nuclear fusion research is a highly challenging, multidisciplinary field seeking contributions from both plasma physics and multiple engineering areas. As an application of plasma control engineering, this dissertation mainly explores methods to control the current density profile evolution within the National Spherical Torus eXperiment-Upgrade (NSTX-U), which is a substantial upgrade based on the NSTX device, which is located in Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ. Active control of the toroidal current density profile is among those plasma control milestones that the NSTX-U program must achieve to realize its next-step operational goals, which are characterized by high-performance, long-pulse, MHD-stable plasma operation with neutral beam heating. Therefore, the aim of this work is to develop model-based, feedforward and feedback controllers that can enable time regulation of the current density profile in NSTX-U by actuating the total plasma current, electron density, and the powers of the individual neutral beam injectors.Motivated by the coupled, nonlinear, multivariable, distributed-parameter plasma dynamics, the first step towards control design is the development of a physics-based, control-oriented model for the current profile evolution in NSTX-U in response to non-inductive current drives and heating systems. Numerical simulations of the proposed control-oriented model show qualitative agreement with the high-fidelity physics code TRANSP. The next step is to utilize the proposed control-oriented model to design an open-loop actuator trajectory optimizer. Given a desired operating state, the optimizer produces the actuator trajectories that can steer the plasma to such state. The objective of the feedforward control design is to provide a more systematic approach to advanced scenario planning in NSTX-U since the development of such scenarios is conventionally carried out experimentally by modifying the tokamak’s actuator trajectories and analyzing the resulting plasma evolution.Finally, the proposed control-oriented model is embedded in feedback control schemes based on optimal control and Model Predictive Control (MPC) approaches. Integrators are added to the standard Linear Quadratic Gaussian (LQG) and MPC formulations to provide robustness against various modeling uncertainties and external disturbances. The effectiveness of the proposed feedback controllers in regulating the current density profile in NSTX-U is demonstrated in closed-loop nonlinear simulations. Moreover, the optimal feedback control algorithm has been implemented successfully in closed-loop control simulations within TRANSP through the recently developed Expert routine

Model-based Control of the Current Density Profile in the Experimental Advanced Superconducting Tokamak (EAST)

As worldwide energy consumption increases, the world is facing the possibility of an energy shortage problem. While several approaches have been proposed to slow down this process, which include the improvement of the combustion efficiency of fossil fuels and the introduction of nuclear energy and renewable energy, such as solar, wind, and geothermal energy, a replacement for fossil fuels will eventually be needed. The energy that comes from a nuclear reaction, which includes nuclear fission and nuclear fusion, has a high energy production density (rate of energy produced divided by the area of the land needed to produce it) and produces no air pollution or greenhouse gases, which makes it a strong and attractive candidate. Compared with nuclear fission, the radioactive waste from nuclear fusion can be more easily disposed, the reactants in a nuclear fusion reaction are abundantly available in nature, and nuclear fusion poses no risk of a nuclear accident. For all these reasons, nuclear fusion is a potential solution for the energy shortage problem. However, there are many challenges that need to be conquered to achieve nuclear fusion. The primary challenge is to confine the hot reactants, whose temperatures are about one hundred million degrees Kelvin. At these temperatures, the reactants are in the plasma state and have enough kinetic energy to overcome the repelling electrostatic forces and fuse. One of the most promising approaches to confine the fusion plasma is magnetic confinement, where magnetic fields are used to confine the plasma through the Lorentz force. The tokamak is one of the fusion devices that exploit magnetic confinement. To demonstrate the viability of a nuclear fusion power plant, the International Thermonuclear Experimental Reactor (ITER) tokamak project is aimed at producing 500 megawatts power with 50 megawatts of input power, which will make it the first tokamak with net energy output. To be able to obtain the desired fusion gain, the ITER tokamak will need to operate at a temperature and a pressure so high that the plasma has a good chance of becoming unstable and difficult to confine. To address this issue, extensive research has been conducted on different fusion tokamaks around the world to find high performance operating scenarios characterized by a high fusion gain, good plasma confinement, plasma stability, and a dominant self-generated plasma current with the goal of developing candidate scenarios for ITER. The shape of the toroidal current density profile, or the safety factor profile ($q$-profile), impacts steady-state operation, magnetohydrodynamic (MHD) stability, and plasma performance. The plasma $\beta$, which is the ratio of the kinetic pressure of the plasma to the magnetic pressure (pressure exerted on plasma by the magnetic field), acts as an important economic factor in fusion power generation. Therefore, active control of the toroidal current density profile and plasma $\beta$ is one path towards advanced scenarios. This dissertation focuses on developing control solutions for regulating the current density profile, and to some extent the normalized plasma $\beta$ (denoted as $\beta_N$), on the Experimental Advanced Superconducting Tokamak (EAST) located at the Institute of Plasma Physics, Chinese Academy of Sciences (ASIPP), in Hefei, China. Towards this goal, a control-oriented, physics-based model has been developed for the current density profile evolution in EAST in response to available heating and current-drive (H\&CD) systems. The feasibility of reconstructing the internal plasma states, which may be crucial for feedback control, from measurements at the magnetic axis and at the plasma edge has been studied by using experimental data and exploiting the response model. Target scenarios (characterized by desired $q$-profile and $\beta_N$) have been developed by following a model-based finite-time optimization approach. Feedback controllers ranging from simpler Proportional-Integral-Derivative (PID) controllers to more complex model-based optimal controllers, derived from Linear-Quadratic-Regulator (LQR), $H_\infty$, and Model Predictive Control (MPC) theories, have been synthesized to counteract deviations from the desired target scenario. The overall control solution has been implemented in the Plasma Control System (PCS) and closed-loop $q$-profile regulation has been demonstrated for the first time ever in EAST in disturbance rejection and target tracking experiments

Physics-model-based Optimization and Feedback Control of the Current Profile Dynamics in Fusion Tokamak Reactors

As the demand for energy continues to increase, the need to develop alternative energy sources to complement (and one day replace) conventional fossil fuels is becoming increasingly important. One such energy source is nuclear fusion, which has the potential to provide a clean source of energy and possesses an abundant fuel supply. However, due to the technological difficulty in creating the conditions necessary for controlled fusion to occur, nuclear fusion is not yet commercially viable. The tokamak is a device that utilizes magnetic fields to confine the reactants, which are in the plasma state, and it is one of the most promising devices capable of achieving controlled fusion. The ITER tokamak project is the next phase of tokamak development and will be the first tokamak reactor to explore the burning plasma (one with a significant amount of fusion reactions) operating regime.In order for ITER to meet its demanding goals, extensive research has been conducted to develop advanced tokamak operating scenarios characterized by a high fusion gain, good plasma confinement, magnetohydrodynamic stability, and a significant fraction of noninductively driven plasma current to maximize the plasma performance and potentially enable steady-state operation. As the dynamics of the tokamak plasma magnetic and kinetic states are highly coupled, distributed, nonlinear systems that exhibit many instabilities, it is extremely difficult to robustly achieve advanced operating scenarios. Therefore, active control of the plasma dynamics has significant potential to improve the ability to access advanced operating regimes. One of the key plasma properties investigated in the development of advanced scenarios is the plasma current profile because of its intimate relationship to plasma energy/particle transport and to plasma stability limits that are approached by increasing the plasma pressure. The plasma density and temperature profiles are also important parameters due to their close relationship to the amount of generated fusion power, to the total plasma stored energy, and to the amount of noninductive current drive. In tokamaks, the current and electron temperature profiles are coupled through resistive diffusion, noninductive current drive, and plasma energy/particle transport. As a result, integrated algorithms for current profile and electron temperature profile control will be necessary to maintain plasma stability, optimize plasma performance, and respond to changing power demand in ITER, and eventually a commercial, power producing tokamak reactor.In this work, model-based feedforward and feedback algorithms are developed to control the plasma current profile and thermal state dynamics with the goal of improving the ability to achieve robust tokamak operation. A first-principles-driven (FPD), physics-based approach is employed to develop models of the plasma response to the available actuators, which provides the freedom to handle the trade-off between the physics accuracy and the tractability for control design of the models. A numerical optimization algorithm to synthesize feedforward trajectories for the tokamak actuators that steer the plasma through the tokamak operating space to achieve a predefined target scenario (characterized by a desired current profile and total stored energy), subject to the plasma dynamics (described by the developed physics-based model), actuator constraints, and plasma state constraints, is developed. Additionally, robust feedback control algorithms for current profile, combined current profile + total stored energy, and simultaneous current profile + electron temperature profile control are synthesized for various tokamaks by embedding a FPD model into the control design process.Examples of the performance of the controllers in simulations (DIII-D, ITER, and TCV tokamaks) and DIII-D experiments are presented to illustrate the potential and versatility of the employed control methodology. The DIII-D experimental tests demonstrate the potential physics-model-based profile control has to provide a systematic approach for the development and robust sustainment of advanced scenarios. The ITER simulations demonstrate the ability to drive the current profile to a stationary target while simultaneously modulating the amount of fusion power that is generated. Finally, the TCV simulations demonstrate the ability to drive the current and electron temperature profiles to a self consistent target, as well as to maintain the current profile in a stationary condition while simultaneously modulating the electron temperature profile between equilibrium points