DI INGEGNERIA CORSO DI LAUREA MAGISTRALE IN INGEGNERIA DELL AUTOMAZIONE

Transcription

1 UNIVERSITÀ DEGLI STUDI DI ROMA TOR VERGATA FACOLTÀ DI INGEGNERIA CORSO DI LAUREA MAGISTRALE IN INGEGNERIA DELL AUTOMAZIONE A.A. 21/211 Tesi di Laurea Modeling and nonlinear control for MAST tokamak RELATORE Dott. Daniele Carnevale CANDIDATO Antonio De Paola CORRELATORE Dott. Luigi Pangione

2 Contents Abstract 1 1 Nuclear fusion and MAST Nuclear fusion Tokamak Spherical tokamaks and MAST CREATE model General description Implementation Change of coordinates Vertical stability Order reduction of the model Plasma current parameter Feedforward currents simulations Model of the coils Feedforward voltages simulations PCS: Plasma control system General description CONTENTS I

3 CONTENTS 3.2 Implementation Simulations The input allocator General introduction Design of the allocator Allocation on the currents Allocation on the voltages Design of the allocator on the closed-loop system Comparison between the two allocators Conclusions Possible future developments List of figures 83 Bibliography 88 CONTENTS II

4 Abstract This thesis has been developed in the context of the scientific research on controlled thermonuclear fusion. The final aim of the scientists devoted to this field is to achieve the necessary knowledge to create a thermonuclear fusion reactor. This device would allow commercial production of net usable power by a nuclear fusion process. This source of energy, with respect to nuclear fission energy production, is cleaner and safer. More specifically, this work has been realized through the collaboration between Università degli studi di Roma Tor Vergata and Culham Centre for Fusion Energy that runs MAST experiment, the tokamak considered for this thesis. The professionals who have made this collaboration possible are the professors Luca Zaccarian and Daniele Carnevale from Dipartimento di Ing. Informatica, sistemi e produzione, doctor Luigi Pangione and Graham McArdle from Culham Center for Fusion Energy. The subject of this work has been the design of a nonlinear system, to be added to the existing shape controller of MAST, which would avoid current saturation on the electric circuits of the poloidal coils. In order to do so, a model of the plant has been realized using as a starting point the precious work of the CREATE team, that has developed the linearized model of MAST, and of Graham McArdle, which has created a model of the plasma shape controller (PCS). This preliminary modeling phase has made possible to design the nonlinear control and to test its performances in a simulative environment. The first chapter of this work represents Abstract 1

5 Abstract a general introduction to the physical principles of thermonuclear fusion, it also describes the purposes of magnetic confinement and how this confinement is performed by tokamaks. There is also a general overview of spherical tokamaks and a presentation of the MAST experiment. Chapter 2 contains the description of the CREATE-L model, the linearized model which has been used in the creation of the simulation environment. The mismatchings and the problems that have been experienced during its implementation, together with the proposed solutions, are discussed. In Chapter 3 there is a detailed description of the plasma shape controller (PCS) which is used at the moment on MAST: the control law is analyzed and the saturation phenomena that are desired to be avoided are discussed. There is also the description of the system used to model the PCS, which has been implemented in the simulation environment and tested. Chapter 4 addresses the problem of the currents saturation on the coils and describes the proposed solution: a nonlinear subcompensator which allocates the inputs in order to minimize a cost function and achieve a trade-off between output performances and input allocation. Different versions of the allocator are described and tested through the simulation environment and their performances are compared and discussed. In the last chapter the results are summarized and possible future developments and applications for the present work are considered. Abstract 2

6 Chapter 1 Nuclear fusion and MAST 1.1 Nuclear fusion Nuclear fusion is, in a sense, the opposite of nuclear fission. Fission, which is a mature technology, produces energy through the splitting of heavy atoms like uranium in controlled energy chain reactions. Unfortunately, the by-products of fission are highly radioactive and long lasting. In contrast, fusion is the process by which the nuclei of two light atoms such as hydrogen are fused together to form a heavier (helium) nucleus, with energy produced as by-product. This process is illustrated in Figure 1.1 where two isotopes of hydrogen (deuterium and tritium) combine to form a helium nucleus plus an energetic neutron. Figure 1.1: The process of nuclear fusion. 3

7 Cap. 1 Nuclear fusion and MAST 1.1 Nuclear fusion In this reaction a certain amount of mass changes form to appear as the kinetic energy of the products, in agreement with the equation E = mc 2. Fusion produces no air pollution or greenhouse gases since the reaction product is helium, a noble gas that is totally inert. The primary sources of radioactive by-products are neutronactivated materials (materials made radioactive by neutron bombardment) which can be safely and easily disposed of within a human lifetime, in contrast to most fission by-products which require special storage and handling for thousands of years. The primary challenge of fusion is to confine the plasma, a state of matter similar to gas in which most of the particles are ionized, while it is heated and its pressure increases to initiate and sustain fusion reaction. There are three known ways to do so: Gravitational confinement: the method used by the stars. The gravitational forces compress matter, mostly hydrogen, up to very large densities and temperatures at the star-centers, igniting the fusion reaction. The same gravitational field balances the enormous thermal expansion forces, maintaining the thermonuclear reactions in a star, like the sun, at a controlled and steady rate. Unfortunately huge gravitational forces, not available on Earth, are required. Inertial confinement: a fuel target, typically a pellet containing a mixture of deuterium and tritium, is compressed and heated through high-energy beams of laser light to initiate the nuclear fusion reaction. This method has not reached the efficiency and the results that were expected in the 197s but new approaches and techniques are currently experimented in some research centers such as the NIF (National Ignition Facility) in California and the Laser Me`gajoule in France. Magnetic confinement: hydrogen atoms are ionized, so that magnetic fields can 4

8 Cap. 1 Nuclear fusion and MAST 1.2 Tokamak exert a force on them, according to the Lorentz law, and confine them in the form of a plasma. The magnetic confinement is the most promising technique and it is worth spending a few words to describe it in more detail. In normal conditions the gas is unconfined and free to move, if the gas is ionized and subject to a magnetic field the forces imposed by the field cause the ions to travel along the magnetic fields lines with a radius known as the Larmor radius. Ions and electrons have opposite charges, these particles move in opposite directions along the field lines under the influence of an electric field. Since positively charged ions are more massive than electrons, the positive ions rotate in a much larger radius circle. The number of rotations per second at which the ions and electrons rotate around the field lines are the ion cyclotron frequency and electron cyclotron frequency, respectively. Figure 1.2: The trajectory of ionized gas subject to a magnetic field. 1.2 Tokamak The most promising device for magnetic confinement of plasma is the tokamak (Russian acronym for Toroidal chamber with axial magnetic field ), a device shaped as a torus (or doughnut) that has been originally designed in Russia during the 195s. The general structure of the device is shown in Figure

9 Cap. 1 Nuclear fusion and MAST 1.2 Tokamak Figure 1.3: General structure of the tokamak. The main problem with the magnetic confinement described in the previous section is that the particles remain confined by the magnetic field until the field lines end or dissipate, contrary to the desire of keeping them confined. To solve this problem, the tokamak bends the field lines into a torus so that these lines continue forever. The magnetic fields that create and confine the plasma in the tokamak are generated by electric coils which can be located outside the chamber, such in JET and most of the tokamak, or inside, as in MAST experiment. Since the plasma is ionized and confined inside the toroidal chamber, it can be considered as a coil circuit, the secondary side of a coupled circuit whose primary side is the central solenoid. Figure 1.4 displays the currents and fields that are present inside the tokamak. All existing tokamak are pulsed devices, that is, the plasma is maintained within the tokamak for a short time: from a few seconds to several minutes. There is no agreement yet among fusion scientist on whether a fusion reactor must operate with truly steady-state (essentially infinite length) pulses or just operate with a succession of sufficiently long pulses. The main reason for this limitation is that, in order to 6

10 Cap. 1 Nuclear fusion and MAST 1.2 Tokamak Figure 1.4: Currents and magnetic fields of the tokamak. sustain constant values of plasma current, the derivative of the current on the central solenoid must be constantly ramping up (or down), rapidly reaching a structural limit on the coil which cannot be exceeded. To avoid this limitation, different methods to sustain the plasma current have been studied and introduced, such as LH/ECRH antennas or neutral beams injectors, currently used at MAST. All tokamak produce plasma pulses (also referred to as shots) with approximatively the same sequence of events. Time during the discharge is measured relative to t=: the time when the physical experiment starts after all the preliminary operations. The toroidal field coil current is brought up early to create a constant magnetic field to confine the plasma when this is initially created. Just prior to t= deuterium is puffed into the interior of the torus and the ohmic heating coil (primary coil in Figure 1.4) is brought to its maximum positive current, in preparation for pulse initiation. At t= the primary coil is driven down to produce a large electric field within the torus. This electric field accelerates free electrons, which collide with and rip apart the neutral gas atoms, thereby 7

11 Cap. 1 Nuclear fusion and MAST 1.3 Spherical tokamaks and MAST producing the ionized gas or plasma. Since plasma consists of charged particles that are free to move, it can be considered as a conductor. Consequently, immediately after plasma initiation, the primary coil current continues its downward ramp and operates as the primary side of a transformer whose secondary is the conductive plasma. At the end of the downward ramp of the primary coil the plasma current is gradually driven to zero and the shot moves towards its conclusion. The separate time intervals in which the plasma current is increasing, constant and decreasing are referred to, respectively, as ramp-up, flat-top and ramp-down phase of the shot. At the moment the tokamak technology has reached a point such as the quantity of energy produced by these devices is almost as much as the one used in heating and confining the plasma. The next step is the construction and operation of the proposed International Thermonuclear Experimental Reactor (ITER) which, supported by an international consortium of governments, will provide major advancements in fusion physics and constitute a testbed for developing technology to support high fusion levels. 1.3 Spherical tokamaks and MAST MAST (Mega Amp Spherical Tokamak) is the fusion energy experiment, based at Culham Centre for Fusion Energy, which has been used for the present thesis. Its main difference from a classical tokamak is the shape: since the origin of tokamak in the 195s, research is mainly concentrated on machines that hold the plasma in a doughnut-shaped vacuum vessel around a central column. MAST belongs to a different category of tokamak, named spherical tokamak, which presents a more compact, cored apple shape and a lower aspect ratio. Spherical tokamak hold plasmas in tighter magnetic fields and could result in more economical and efficient fusion power for many reasons: 8

12 Cap. 1 Nuclear fusion and MAST 1.3 Spherical tokamaks and MAST plasmas are confined at higher pressures for a given magnetic field. The greater the pressure, the higher the power output and the more cost-effective the fusion device. The magnetic field needed to keep the plasma stable can be a factor up to ten times less than in conventional tokamak, also allowing more efficient plasmas. Spherical tokamaks are cheaper, since they do not need to be as large as conventional machines and superconducting magnets, which are very expensive, are not required. Spherical tokamaks, at the moment, are at a very early stage of development and they will not be used for the first nuclear fusion power plants but they can be very useful for component test facilities and they are providing insight into the way changes in the characteristic of the magnetic field affect plasma behaviour. These informations have been very useful for the development of ITER, the advanced experimental tokamak which is being built in France. MAST, along with NSTX at Princeton, is one of the world s two leading spherical tokamak.table 1.1 and Figure 1.5 give an idea of its dimension, structure and technical specifications. Plasma Vacuum vessel Current 1, 3, amps Height 44.4m Core up to Diameter 4m temperature 23,, C Pulse length up to 1 second Material Stainless steel 34LN Plasma 8m 3 Toroidal field 24 turns,.6 tesla radius Density 1 2 particles/m 3 Total mass 7 tonnes of load assembly Diameter approximatively 3m Neutral beam 5,, watts heating 75, volts Table 1.1: Technical specifications of MAST experiment. 9

13 Cap. 1 Nuclear fusion and MAST 1.3 Spherical tokamaks and MAST Figure 1.5: Section of MAST. A cross-section of the MAST vessel and the position of the six PF (poloidal field) coils is shown in Figure 1.6. Since the present thesis has focused on the control system on the PF coils which confine and shape the plasma, it is worth describing them in more detail: Solenoid (P1): Provides the magnetizing field used to control plasma current, it is analogous to the primary winding of a transformer, where the plasma itself acts as a single-turn secondary winding. It is composed of four layers (152 turns per layers), 2.7 meters long. Its power supply (P1PS) is four quadrant and it normally drives current in the range [ 45kA, +45kA] although its maximum current range is [ 55kA, +55kA] Divertor coil (P2): It is composed of two independent windings in each coil pack, it can be used to achieve the desired plasma configuration and compensate the 1

14 Cap. 1 Nuclear fusion and MAST 1.3 Spherical tokamaks and MAST Figure 1.6: Cross-section of the MAST vessel and position of the six PF coils. stray field from the solenoid. Its power supply (EFPS) has a single direction, although this direction can be reversed during pulse. The maximum current value that can be driven is 27 ka. Start-up coil (P3): It is a capacitor bank used for the pre-ionization of the plasma. It has no power supply or feedback, just a switch that starts the discharging of the capacitor hence it cannot really be considered an actuator from the plasma shape controller point of view. Vertical field/shaping coils (P4 and P5): Both coils contribute to the main vertical field for radial position control. The shape and elongation depend both on the plasma internal profile and on how the total vertical field current is divided between P4 and P5. Each of them is driven by a bank which provide the rapid initial vertical field rise and by power supplies (respectively SFPS 11

15 Cap. 1 Nuclear fusion and MAST 1.3 Spherical tokamaks and MAST and MFPS), which provide controlled flat-top current. Both power supplies can drive current in a single direction. The maximum value of the current is 17kA for P4 and 18 ka for P5. Vertical position coil (P6): There are actually two coils in one can, each of them with two turns. These coils provide the radial field for vertical position control. Since the vertical dynamics are much faster than the time scale of the existing MAST PCS, they are independently driven by a separate analogue controller. The time behaviour of the currents on the PF coils for a standard shot, together with the associated value of the plasma current, is shown in Figure 1.7 Figure 1.7: Typical PF current evolution. 12

16 Chapter 2 CREATE model 2.1 General description The first step for the realization of a simulation environment has been the choice of the model of MAST. The model that has been adopted is the CREATE-L model, developed by the CREATE team. This model, which has already been successfully tested on various tokamaks (TCV, FTU and JET), is a linearized model about an equilibrium point. It is obtained from the following set of equations: dψ dt + RI = U [Ψ, Y ] T = η(i, W ) Circuit equations Grad-Shafranov constraint (2.1.1) I Poloidal field (PF) circuit currents and plasma current I p Ψ Fluxes linked with the above circuits U Applied voltages R Resistance matrix W Poloidal beta (β p ) and internal inductance (l i ) Y Most remaining quantities of interest (plasma shape descriptors and current moments) Table 2.1: List of phisical quantities in eq The Grad-Shafranov constraint is the equilibrium equation in ideal magnetohydrodynamics (MHD) for a two dimensional plasma. This set of equations is linearized 13

17 Cap. 2 CREATE model 2.1 General description using incremental ratios or Jacobian matrix and the result is the eq (L is an inductance matrix modified by the presence of the plasma which, differently from many similar models, is not included in the state space). L di dt + Ri = u L E y = Ci + F w dw dt (2.1.2) with L = Ψ I L E = Ψ W C = Y I F = Y W From the equation it is quite straightforward to obtain a state-space form of the model with dx dw = Ax + Bu + E dt dt Y = Cx + F w (2.1.3) x = i A = (L ) 1 R B = (L ) 1 E = (L ) 1 L E In the starting configuration of the model, the signal of interests are the following: Inputs: Disturbances: Outputs: State Variables: - Active PF circuit voltages - Poloidal beta - Internal inductance - Active PF circuit currents - Passive PF circuite currents - Plasma shape descriptors - Magnetic signals - Plasma current moments - Active and passive PF circuit currents 14

18 Cap. 2 CREATE model 2.2 Implementation 2.2 Implementation The CREATE team has developed a graphic interface which makes very easy to obtain the desired model. Initially the number of the shot is chosen and all the data needed by the tool to generate the model are downloaded from the database. In order for the linearization performed by the CREATE tool to be reliable, the chosen shot must have a long flat-top phase and no important nonlinearities which may be caused, for example, by plasma disruptions. It has to be considered that the MAST top-flat phase lasts at most.3 seconds and the signals are sampled at 2Khz so only about 7 measurements are available for the modeling of the experiment. The next step is the choice of the settings of the model: it is possible to create models for plasmaless shots, take in account the presence of eddy currents, choose a double null or limiter configuration. In the limiter configuration the border of the confined region of the plasma (LCFS) is limited by inserting a barrier a few centimetres into the plasma, in the double null configuration there is a different shaping of the plasma which leads to the formation of two poloidal field nulls, above and below the plasma column. In the first simulations used to test the CREATE-model, a plasma model with eddy currents and plasma in double null configuration of the shot n (a standard shot with a long top-flat phase) has been used. The tool returns the matrices L,R andl E of the eq but it is necessary to do some preliminary modifications, described in the next sections, in order to correctly run the simulations. 15

19 Cap. 2 CREATE model 2.2 Implementation Change of coordinates The first step towards an implementation of this model has been the change of its inputs. The eq is an equivalent representation of the state-space model: [ L11 L 12 L 21 L 22 ] [ ] [ ] [ ] [ x1 R1 x1 + = x 2 R 2 x 2 S 2 ] [ LE1 U L E2 ] Ẇ (2.2.1) The components of the state vector x can be divided in x 1 (passive currents generated by inductive phenomena) and x 2 (active currents on the coils). A problem experienced by the CREATE team when the linear model of MAST has been realized is that the voltages signals (the original inputs of the model) are too noisy hence they cannot be used in the simulations. The problem has been solved in the following way: the dynamics of the coil circuits have been removed from the model, considering the currents on the coils as new inputs. If the new state vector p 1 is defined as follows: p 1 = L 11 x 1 + L 12 x 2 + L E1 w (2.2.2) it holds the following: p 1 = L 11 x 1 + L 12 x 2 + L E1 ẇ x 1 = L 1 11 p 1 L 1 11 L 12 x 2 L 1 11 L E1 w (2.2.3) and it is straightforward to obtain a new set of state-space equations where p 1 is the new state variable: ṗ 1 = R 1 L 1 11 p 1 + [ R 1 L 1 11 L 12 R 1 L 1 11 L E1 ] [ x 2 w y = C 1 L 1 11 p 1 + [ C 1 L 1 11 L 12 + C 2 C 1 L 1 11 L E1 + F ] [ x 2 w ] ] (2.2.4) which can be easily rewritten in state-space form, considering the vector ξ = [x 2 w], containing the currents on the poloidal coils and the disturbances, as the new input: 16

20 Cap. 2 CREATE model 2.2 Implementation ṗ 1 = Ãp 1 + Bξ y = Cp 1 + Dξ (2.2.5) This new model has a lower order than the original one (p 1 has the same dimension of x 1 ) because it totally ignores the electric dynamics on the poloidal coils and assumes that the value of the currents can be arbitrarily imposed. A model of the electric circuits of the coils which receives the applied voltages and returns the correspondent values of currents has been created and will be described in a later section Vertical stability The next step in the implementation of the CREATE model has been its closed-loop stabilization. In the model, as well as in the plant, there is an unstable mode relative to the vertical instability of the plasma. During the shot the plasma is elongated, pulled along its vertical direction by the magnetic field generated by the coils: in elongated plasma it is easier to achieve higher values of current and better performances. The more the plasma moves in one direction, the bigger is the attraction towards that very same direction and the smaller in the opposite one: the plasma, if adequate control is not applied, crashes on the wall of the vessel and it disrupts. To avoid this, the coil P6 is used to generate a magnetic field which balances the vertical displacement of the plasma: the signal ZIP, which represent the z position of the plasma current centroid, is used as a controlled variable for a PD controller that returns the value of the voltage to be applied on P6. As pointed out before, the vertical controller, shown in Figure 2.1, is not included in the PCS and it is implemented separately through analogue circuits since very fast time response is needed to achieve stability. In order to run any kind of simulation, it is necessary to preemptively stabilize the model. In the simulations run by the CREATE team the A matrix of the model has 17

21 Cap. 2 CREATE model 2.2 Implementation Figure 2.1: Diagram of the vertical control on MAST. been diagonalized, the stable modes have been normally simulated forward while the unstable mode has been simulated backward in order to grant stability. In this way the unstable model exponentially converges to zero. This solution is inadequate for the purpose of this work since it is not feasible for feedback simulations. The method that has been adopted instead is conceptually similar to the one actually implemented on MAST: a feedback on the ZIP signal and the implementation of a PD controller. The first step has been the identification of the electric circuits of the P6 coil; it has to be kept in mind that at the moment our model receives currents as inputs but the controller returns voltages. An initial attempt has considered the coil n.6 as an R-L circuit described by the equation V P 6 = R P 6 I P 6 + L P 6 di P 6 dt (2.2.6) which is equivalent to the first order system described by I P 6 = K P 6 τ P 6 s + 1 V P 6 (2.2.7) Given the low number of parameters used to describe the system, a manual tuning on their values has been used to achieve a satisfactory fitting between the measured values of currents and the output of when the correspondent voltages are inputted. After a few attempts, the results shown in Figure 2.2 have been obtained. 18

22 Cap. 2 CREATE model 2.2 Implementation 5 est. data currents [A] Figure 2.2: Results of the identification on P6 coil. The coil identification has been considered acceptable and the next step has been the design of the PD controller and the tuning of its two parameters. The reference has been the filtered (low-pass) reference of the actual controller, in order to avoid abrupt changes in the output voltage signal due to initial high value of the tracking error. The ZIP signal, the voltage on P6 and the resultant current on P6 are shown in fig. 2.3, 2.4 and 2.5. There is a mismatch between the measured signals and the results of the simulation and it is mostly caused by measurement noise and nonlinear phenomena which are not considered by the CREATE-model. It has to be underlined, though, that there is a decoupling between the unstable mode of the vertical position of the plasma and the other dynamics: they are actually controlled by two independent controllers and the only purpose of the implemented vertical controller is to stabilize the system and make simulations possible. 19

23 Cap. 2 CREATE model 2.2 Implementation 1 ZIpl MAST shot #24542 exp. data 1 ZIP [m * A] Figure 2.3: ZIP signal for the simulation of shot n P6 voltage [V] P6 current [V] Figure 2.4: Measured current and voltage on P6 for the shot n Order reduction of the model Once the system has been stabilized, it has been possible to run feed-forward simulations in order to test the performances and the reliability of the new model. Currents 2

24 Cap. 2 CREATE model 2.2 Implementation 1 5 P6 voltage [V] P6 current [V] Figure 2.5: Current and voltage on P6 for the simulation of shot n on the coils for a certain shot are retrieved from the database and are used as inputs of the CREATE-model, whose outputs are compared with the measured data. The output of these simulations fit the experimental data but they show a high-frequency oscillation which is not present in the input. This has been thought to be caused by a very low eigenvalue in the à matrix of the eq which would also explain the long time needed for the simulation, since Matlab has to shorten the integration time in order to simulate the fast dynamic relative to this eigenvalue. To solve this problem a change of coordinates has been performed on the model and a new matrix Ā, diagonal, has been obtained. The eigenvalue which was thought to cause the oscillation has been removed from the state dynamics and simulations have been repeated. In Figure 2.6 the plasma current for the two cases, during the whole flat top phase, is shown: the values of the signal appear to be identical. If a shorter time interval is considered, as in Figure 2.7, the oscillation on the output signal of the model without reduction and its total absence in the new model are evident. The model reduction by truncation has allowed to achieve a ten times shorter simulation time and the 21

25 Cap. 2 CREATE model 2.2 Implementation elimination of the high frequency component on the outputs, without introducing any error in the simulation results. 9.9 x ord. red.= ord. red.=1 9.7 plasma current [A] Figure 2.6: Plasma current during the flat top phase for the original model (blue) and for the reduced one (red). x ord. red.= ord. red.= plasma current [A] Figure 2.7: Plasma current for the original model (blue) and for the reduced one (red) during a shorter time interval. 22

26 Cap. 2 CREATE model 2.2 Implementation Plasma current parameter Once all the procedures described above had been implemented, there still were differences between the measured and the simulated signals, especially in the plasma current. In the attempt to understand the source of the problem, the row of C relative to that output in the eq has been analyzed and it has been noticed that the plasma current is, with a good approximation, dependant from just one state which, furthermore, is independent in its evolution from the others. Basically in the CREATE model, considering I pl as the value measured at the starting time of the simulation, the expression of the plasma current can be approximated as follows: ẋ pl = a pl x pl + B pl u I pl = c pl x pl + I pl (2.2.8) If we consider the value of a pl initially set by the CREATE model for the shots taken into account, it is usually in the range [ ]. This means that, if u is set to, x pl converges exponentially to and the plasma current remains constant. It is known this is not the case since the plasma current presents a resistive effect that makes it decay if the coils are not powered. This explains why, in the simulations, if currents measured from a shot are used as input of the model, the plasma current increases instead of being constant, as can be seen in Figure 2.8 In order to solve this inaccuracy of the model, the value of the parameter a pl has been changed and tested with new simulations, trying to minimize the least square error between the measured plasma current and the simulated one. It has been noticed that, for all the analyzed shots, the error has only one minimum with respect to the value of a pl as it can be seen in figure 2.9. It is worth saying that the new parameter ā pl which minimizes the error is a 23

27 Cap. 2 CREATE model 2.2 Implementation 9.8 x plasma current [A] Figure 2.8: Measured plasma current (blue) and simulated plasma current(red) when no correction on the plasma parameter is applied. 7 x quadratic error a pl Figure 2.9: Value of the quadratic error between measured and simulated plasma current with respect to the parameter a pl. positive number: this means that in the modified model the plasma current decreases exponentially (c pl is negative) if currents are not applied on the coils. Simulations run with the modified parameter ā pl have led to an improvement of the fitting of the 24

28 Cap. 2 CREATE model 2.2 Implementation simulated data, especially for the plasma current, as can be seen in figure x plasma current [A] Figure 2.1: Measured plasma current (blue) and simulated plasma current(red) when the parameter a pl is modified in order to minimize the quadratic error. 25

29 Cap. 2 CREATE model 2.3 Feedforward currents simulations 2.3 Feedforward currents simulations After the model has been modified in the way described in the previous sections, it is possible to test it running the first simulations. Currents on the coils are retrieved from the database and fed in the model, the results are then compared with the correspondent measured signals. The currents of the shot n during the flattop phase which are used as inputs are shown in Figure 2.11, while in Figures 2.12, 2.13 and 2.14 there is the comparison between measured and simulated signals. It can be observed that the outputs of the CREATE-L model, especially the plasma current, have a good fitting with the correspondent measured signals. A mismatching can be noticed in the fields measurements but the relative error is below 5% and it has been considered acceptable. 3 x P1 current [A] P2 current [A] P4 current [A] P5 current [A] Figure 2.11: Measured currents on the coils P1,P2,P4 and P5 for the shot n during the flat-top phase. 26

30 Cap. 2 CREATE model 2.3 Feedforward currents simulations 9.2 x plasma current [A] Figure 2.12: Measured plasma current (blue) and simulated plasma current(red) for the shot n ccbv11 field [T] ccbv16 field [T] ccbv2 field [T].41.4 ccbv24 field [T] Figure 2.13: Measured fields (blue) and simulated ones (red) for the shot n

31 Cap. 2 CREATE model 2.4 Model of the coils flcc3 flux [Wb] flcc7 flux [Wb] flp4u4 flux [Wb] flp4l4 flux [Wb] Figure 2.14: Measured fluxes (blue) and simulated ones (red) for the shot n Model of the coils The model whose simulative results have been shown in the previous section assumes that any value of the coil currents can be obtained instantaneously. This is not the case: in reality a voltage signal is applied to the electric circuits of the coils and then the resultant value of current is measured. Since the dynamics of the coils have been excluded from the CREATE model because of the inaccuracy of the voltage signals, it is now necessary to independently model them. It should be pointed out, though, that the model will involve only the coils P1,P2,P4 and P5 which are the ones used for the feedback control, P6 has been pre-emptively modelled (eq ) and P3 is used in feed-forward. The first adopted approach has been to consider the coils as R-L circuits described by the following equations: V cm1 = R cm1 I cm1 + L cm1 di cm1 dt (2.4.1) 28

32 Cap. 2 CREATE model 2.4 Model of the coils which can be easily expressed in the state-space form: di cm1 dt = L 1 cm1r cm1 I cm1 + L 1 cm1v cm1 = A cm1 I cm1 + B cm1 V cm1 (2.4.2) The matrices R c and L c that have been used to test the model have been retrieved, for each shot, from the database of the controller which uses them to convert its currents requests in voltages. Voltages measurements from a certain shot are used as input of the model and the resulting currents are compared with the measured ones. The results are shown in Figure x 14 4 est. data 15 1 est. data P1 current [A] 2 2 P2 current [A] est. data 1 est. data P4 current [A] P5 current [A] Figure 2.15: The measured current for the shot n (blue) are compared with the estimation of the R-L model (green). It is clear from the graph that, although a certain fitting of the currents is achieved, there are still some inaccuracies. The main ones are thought to be the following: The discharge of the bank of capacitors on P3 at t = causes an induction 29

33 Cap. 2 CREATE model 2.4 Model of the coils effect which cannot be ignored: this is likely the cause of the increase of error in the current estimation error at that time. the presence of the plasma (and its induction effect) is not considered. In order to achieve a better estimation of the currents, a model error has been introduced: y em = ζ(u em ) (2.4.3) The vector u em includes the voltages on the six coils (as to take in account the inductive phenomena between P3, P6 and the other four coils) and the plasma current (in order to consider the presence of the plasma) while y em is a vector composed by the current error estimation on P1, P2, P4 and P5 and the current on P3 and P6. This black-box system has then been identified through the Matlab Identification Toolbox, using the data of four shots retrieved from the database (n , 24533, and 24538). Iterative prediction-error minimization and subspace method have been tried as well as different orders of the system. On the basis of the simulative results, the model chosen to identify the estimation current error has been a tenthorder state-space model obtained with the subspace method and described by the following equations: ẋ em = A em x em + B em u em ȳ em = C em x em (2.4.4) The model obtained through the identification has been tested with a validation shot (n.24542): in Figure 2.16 the error of the first coil model is compared with the estimation performed by the error model and in Figure 2.17 the new error on the current estimation is compared with the original one. Other simulations have been run for different shots (specifically n , 24567, an 24572) showing the same performances for the error identification model. 3

34 Cap. 2 CREATE model 2.4 Model of the coils meas. error est. error 2 1 meas. error est. error P1 error [A] P2 error [A] meas. error est. error 5 meas. error est. error P4 error [A] 5 1 P4 error [A] Figure 2.16: Estimation errors of the currents on the coil P1,P2,P4 and P5 (blue) and new estimate of the current errors(red). P1 error [A] P2 error [A] P4 error [A] P5 error [A] Figure 2.17: Estimation errors on the currents using the first model of the coils (blue) and the second (green). 31

35 Cap. 2 CREATE model 2.4 Model of the coils The system described by the eq is used to improve the current estimation subtracting from it the estimated error: I cm2 = I cm1 y em (2.4.5) This version of the coils model, represented in Figure 2.18, leads to a general improvement of the results, as can be seen in Figure V cm 1 R-L model Icm Icm 2 I pl V 3 V 6 Error model _ yem Figure 2.18: Scheme for the coils model with current error estimation. 6 x 14 4 est. data 15 1 est. data P1 current [A] 2 2 P2 current [A] est. data 1 est. data P4 current [A] P5 current [A] Figure 2.19: The measured current for the shot n (blue) are compared with the results of the coils model which includes the error estimation (green). 32

36 Cap. 2 CREATE model 2.5 Feedforward voltages simulations 2.5 Feedforward voltages simulations Once the model of the coils has been created, it has been possible to run feed-forward simulations of the cascade coils-create model in order to validate its behaviour in the final model. Since these simulations are only run during the flat-top phase, whose time interval will be henceforth expressed as [t in, t fin ], it has been necessary to correctly set the initial conditions on the coils model. If only the first part of the model had been used, the initial value of its states would have been the value of the measured coil currents at t in. It is slightly more complicated to set the initial conditions if the error model is used: its ten states have been obtained through a black-box identification and do not correspond to any physical parameter. To set them correctly, a feed-forward simulation of the coils is preemptively run: the value of x em at t = t in is used as initial condition of the cascade simulation and the initial condition for the R-L model are such that the estimation error of the currents at t = t in is equal to : I cm1 (t in ) = I(t in ) + C em x em (t in ) (2.5.1) It is now possible to properly run the simulation of the cascade, whose results are shown in Figures 2.2, 2.21, 2.22, 2.23 and The simulated values of the currents, compared in Figure 2.21 with the measured ones, can be considered satisfactory: the highest error is on the coil P5 and it is lesser than 5%. The plasma current in Figure 2.22 shows a good fit with the actual one if the measurement noise is not considered while the differences on fields and fluxes are considered acceptable. 33

37 Cap. 2 CREATE model 2.5 Feedforward voltages simulations voltage on P1 [V] voltage on P2 [V] voltage on P4 [V] voltage on P5 [V] Figure 2.2: Measured voltages on the coils P1,P2,P4 and P5 for the shot n during the flat-top phase. 3 x P1 current [A] P2 current [A] P4 current [A] P5 current [A] Figure 2.21: Measured currents on the coils P1,P2,P4 and P5 (blue) and simulated currents for the shot n during the flat-top phase. 34

38 Cap. 2 CREATE model 2.5 Feedforward voltages simulations 9.2 x plasma current [A] Figure 2.22: Measured plasma current (blue) and simulated plasma current(red) for the shot n ccbv11 field [T] ccbv16 field [T] ccbv2 field [T] ccbv24 field [T] Figure 2.23: Measured fields (blue) and simulated ones (red) for the shot n

39 Cap. 2 CREATE model 2.5 Feedforward voltages simulations flcc3 flux [Wb] flcc7 flux [Wb] flp4u4 flux [Wb] flp4l4 flux [Wb] Figure 2.24: Measured fluxes (blue) and simulated ones (red) for the shot n

40 Chapter 3 PCS: Plasma control system 3.1 General description The plasma control system (PCS) is the device which is used to control and configure the plant. It can be roughly schematized in two main sections which operate in different times and perform different operations. The first section is a configuration tool that allows to set the parameters and the waveforms to be used during the shots and synchronizes with the Machine Control System, the machine that actually runs the shot. The second part operates in real-time: it gets data from the plant via analogue and digital inputs and, on the basis of the data it receives and the parameters and waveforms which have been set before the starting of the shot, it makes control decisions and drives the plant via analogue and digital outputs. For the purpose of this thesis, we will focus our attention on this last part: in particular on the controlled variables and the way the output of the PCS (voltages driven on the coils) is calculated. The general scheme of the plant is represented in fig The PCS receives as input the value of the currents (I), a feedforward reference (I ff ), the error on the current feedback reference (I ref SM I) and the error on the plasma current and on the flux Ψ. Each input is used to calculate a different vector of 37

41 Cap. 3 PCS: Plasma control system 3.1 General description Ip y ref ref + - Iref + - Iff PCS V Coils model I MAST CREATE model Ip y SM Figure 3.1: General scheme of the plant. voltage requirements on the coils which are then summed to calculate the final output of the controller. It is now described in more detail how this is done: Resistive term: the drops on the voltages due to resistive effect have to be taken in account. A matrix R P CS which describes the resistance of the coils is stored as a PCS parameter and used to calculate the voltages required to compensate the resistive losses: V res = R P CS I (3.1.1) Current Feedforward: at the beginning and the ending of the shot, respectively when plasma is created and ramped down, a reliable model of the system is not currently available and it is difficult to design a feed back controller. That is why in these phases of the shot, and in minor part during the flat-top phase, the value of currents are preemptively calculated and driven in feed forward. The requested derivative on the currents are set offline in the PCS ( I ff ). The corresponding values of voltages are calculated as follows: V ff = I ff L F F coeff (3.1.2) In the expression above L is the matrix which describes the mutual inductance between the coils (in a similar way to the R described in the previous point) 38

42 Cap. 3 PCS: Plasma control system 3.1 General description while F F coeff is used to take in account a reduction of the mutual inductances caused by the presence of the plasma. Current Feedback: it is necessary, in order to drive the plant, a feed back component in the controller which compensates disturbances and model uncertainties. References are set offline in the PCS on P1, P2-.5P1 and on the sum and difference of P4 and P5. The currents on P4 and P5 are expressed in sum/difference terms because of the strong mutual coupling between the two coils. The feedback control on each coil can be enabled or disabled at any time during the shot setting offline the relative gain function. The voltages that aim at correct the error on the current references have the following expression: V fb = I err τ 1 L (3.1.3) I err is the tracking error vector, L is the mutual inductance matrix of the coils and τ is a diagonal matrix which contains the time constant for each coil (design parameter). Plasma current: the plasma current is controlled in feedback with a PI controller. The tracking error and its integral (respectively Ip err and Ip Ierr ) are calculated and the voltages are obtained as follows: V pl = Ip Ierr τ IpInt + Ip ( err τ Ip L pl M pl sol ) L + c fac τ Bv dψ err (3.1.4) The first factor calculates the required value of Ipl which is then converted in the correspondent required value of I P 1 by the inductance ratio and finally the voltages are obtained through the inductance matrix L. The second term of 39

43 Cap. 3 PCS: Plasma control system 3.1 General description the sum is used to take in account the shape of the plasma which influences the amount of current that needs to be driven. Radial position feedback: Since the difficulties to reconstruct the radial position of the plasma in real-time, the control of the radius uses flux signals. Two isoflux lines are considered: one on CC, supposedly close to the isoflux line on the plasma boundary (therefore representing a good estimate of the flux in that point), and the other at a chosen control point R C. The value of the flux reference dψ ref is calculated multiplying the radius reference dr ref by the factor Ψ, which is estimated through a linear combination of magnetic R measurements. The controlled output is calculated through a similar linear combination of measurements and the expression of the tracking error is the following: ( nm ) dψ err = dψ ref dψ = a i m i dr ref The voltages are calculated similarly to the previous cases: V R = dψ err K Ψ i=1 n m j=1 b i m i 1 L + Sh Ip err (3.1.5) τ P si τ Ip The flux error is converted through the gain K Ψ into a current error which is then multiplied by a time costant and by the inductance matrix. The second term of the sum represents correction estimated by the Shafranov equation. It is worth saying that, during real shots, the radial position of the plasma can be measured through a camera positioned inside the chamber, converted in a magnetic measurement multiplying it by the factor Ψ and used as an alternative R estimation of dψ. Unfortunately it is not possible to implement this case in the simulations because the CREATE-model does not return the camera signal. 4

44 Cap. 3 PCS: Plasma control system 3.2 Implementation Once the four voltages term, each one relative to a different input of the PCS, are calculated, their values are summed to obtain the voltage output of the PCS: V = V res + V ff + V fb + V pl + V R 3.2 Implementation The PCS has been implemented in the simulation environment using a Simulink model developed by G. McArdle which accurately reproduces the control laws described above. It also takes in account and models the power supplies which are schematized by a clipping function followed by a one-pole low-pass filter. The model has been tested with feedforward simulations, throughout the whole length of the shot and not only in the top-flat phase: input data of the PCS from previous shots have been retrieved from the database and used as input of the model. The results of the simulation have then been compared with the measured output of the PCS as can be seen in Figure 3.2 This simulation has been crucial for two reasons: 1. It proves that the PCS model correctly reproduces the actual control signals, there is only some marginal disagreement in the preliminary phase. As we can see from the graphs the measured data (blue) fit with the simulated data (red) apart from the considerable measurement noise. 2. It provides all the parameters that are needed to initialize correctly the simulations with the CREATE model. Since these simulations are only run in the top-flat phase, the initial values of all the plant outputs are initialized with the measured values at the starting time of the simulation. If the initial conditions 41