Control of elongated plasmas in superconductive tokamaks in the absence of in-vessel coils

The roadmap for the commissioning and first operations of superconductive tokamaks envisages the possibility of running discharges with fairly elongated plasmas before the complete installation of the in-vessel components, including vertical stabilization coils, or any other specific sets of coils to be used for the magnetic control of fast transients. In the absence of dedicated actuators, the magnetic control system shall perform the essential fast control actions by using the out-vessel superconductive coils, if needed. These are typically less efficient in reacting to fast transients, due to the shielding effect of the vessel and imply a coupling with other control tasks relying on the same actuators, such as plasma current, position, and shape control. Hence, effective actuator-sharing strategies must be put in place. This paper presents an architecture and a possible control strategy that is able to cope with vertically unstable elongated plasmas subject to fast varying disturbances, in the absence of dedicated in-vessel coils. The architecture exploits a model-based actuator-sharing approach to effectively accomplish the main magnetic control objectives while minimizing the cross-couplings among the various tasks. The effectiveness of the approach is demonstrated by means of nonlinear simulations of realistic JT-60SA scenarios. In particular, an isoflux plasma shape controller is integrated with plasma current control and vertical stabilization. The proposed control approach proves to control vertical displacement events and plasma deformations due to fast variations of poloidal beta with satisfactory performance.


Introduction
Several control problems need to be solved simultaneously to robustly control high-performance plasmas during tokamak operation [1,2].Among the various challenges to be faced, the so-called magnetic control one [3,4] plays an essential role since the very first operations of every magnetically confined fusion device [5][6][7][8][9].Indeed, some basic components of the magnetic control systems, such as the coils currents controller, are needed even before plasma operation, during the commissioning phase, to put the coils in safe operation [5,10].
The main magnetic control tasks during plasma operations include tracking of the plasma current, position, and shape.Furthermore, since the shapes typically pursued are elongated, and therefore vertically unstable [11], an active Vertical Stabilization (VS) control system is needed to stabilize the plasma column [3, chapter 7].VS systems typically rely on dedicated actuators, which are toroidal coils either placed inside the vessel (in-vessel coils [12][13][14]) and/or outside (out-vessel coils [15]), and that are fed with dedicated power supplies.When dedicated actuators are available, different strategies are possible and have proved to be effective in practice (among the various, see [14,[16][17][18][19][20][21]).Moreover, the availability of dedicated actuators for the VS system recently enabled the investigation of data-driven and/or model-free control approaches [22][23][24].
Nevertheless, the roadmap of superconductive tokamaks, such as ITER [25] envisages the possibility of running discharges with elongated plasmas before the complete installation of the in-vessel components, and hence without dedicated actuators for the VS [6,26].However, when a currentdriven control scheme is adopted (see [6,8,17,27,28]), the presence of the Poloidal Field (PF) current controller itself is equivalent to an increase of the coil resistances, which in turn implies a reduction of their passive stabilization effect, and therefore to an increase of the growth rate.This worsening of the vertical instability makes the VS task even more challenging in the absence of a dedicated fast actuator, calling for an effective control system that relies only on the superconductive coils.
As for the ITER case, the first operations of JT-60SA do not envisage the presence of dedicated in-vessel coils for the VS.Specifically, JT-60SA is a superconductive tokamak constructed within the Broader Approach agreement [29] that will operate as a ITER 'satellite' facility.Indeed, it will develop ITER relevant operational scenarios and address the corresponding physics issues.Furthermore, it is expected that the operation of JT-60SA will complement the ones of ITER in R&D areas relevant for the design and realization of DEMO [26].
The present work is motivated by the need for an architecture able to cope with all magnetic control tasks in the absence of in-vessel coils, as arisen in [6], where realistic scenarios relevant for the first operation of JT-60SA have been considered.In these scenarios, there is the need not only to accomplish the usual magnetic control tasks, but also to both ensure vertical stability and counteract fast plasma deformations due to poloidal beta variations, while avoiding voltage saturation of the power supplies that feed the coils.In [6], the authors propose an Adaptive Voltage Allocation (AVA) scheme that relaxes the plasma current control to avoid such saturation and to guarantee closed-loop vertical stability.The architecture proposed in this paper represents a possible alternative to such AVA scheme, which is based on the offline identification of practically decoupled control directions to be assigned to each control task, i.e. plasma current, shape and VS control.Although saturation management is out of the scope of this paper, it is worth to notice that the proposed approach has similarities with real-time allocation techniques such as those proposed in [30,31], hence they can be integrated to increase the robustness against actuator saturation.
Furthermore, the proposed architecture is anyhow machineagnostic and extends the one currently proposed for the ITER tokamak [8], by explicitly taking into account a VS system that shares the same superconductive circuits used as actuators for the other magnetic control tasks, i.e. shape/position and plasma current control.Actuator sharing among the various control tasks is achieved by means of a model-based approach that relies on a linear approximation of the plasma response.Indeed, as has been proved in several tokamaks, the use of linearized models of the plasma/circuit dynamics is an effective means for model-based design of magnetic control actions; the interested reader can refer to [32] for an application of H ∞ plasma magnetic control at TCV, to [33] for the plasma shape control at JET, and [14] for the VS at EAST.As usual, when relying on a model-based design approach, the robustness of the control architecture is first assessed via linear simulations, by using a collection of models that are different from the one used for the design.After such a preliminary assessment, the proposed control algorithms are then validated via nonlinear simulations based on a free boundary Grad-Shafranov equations solver.
The remainder of this paper is structured as follows: section 2 describes the proposed control architecture, by focusing on the main features of the control algorithms that enable effective actuator sharing between the three plasma magnetic control tasks, i.e. plasma current, shape and VS control.The effect of a current-driven control architecture on the growth rate is briefly discussed in section 3. Finally, the effectiveness of the proposed control strategy is shown by considering nonlinear simulations of realistic scenarios for JT-60SA as a case study, similar to those considered in [6].

Magnetic control architecture
The proposed magnetic control architecture is presented in this section, together with the control algorithms implemented by the various components.The proposed setup allows an effective sharing of the PF coil actuators among the main plasma magnetic control objectives, which are plasma current and boundary control, as well as the counteraction of fast plasma movements and the VS.More in detail, the proposed actuator sharing policy is based on a geometric approach that relies on a linear description of the plasma behavior, which in turn allows to define a set of virtual circuits to be used by each magnetic control task.
A simplified block diagram that reports the main functional components of the proposed architecture is reported in figure 1.The architecture relies on the so-called current-driven approach [8], which consists of a nested control system, whose inner loop is the PF Current (PFC) Decoupling Controller.This block ensures the tracking of the required currents in the superconductive PF circuits whose references are obtained as the sum of the scenario (i.e. the nominal) currents and the corrections requested by the outer control loops to track the desired plasma shape and current.Indeed, the proposed architecture includes the following additional components: • the Plasma Current Controller, which tracks the plasma current reference by sending the correspondent requests to the PFC Decoupling Controller; • the Plasma Boundary Controller, which controls the shape of the Last Closed Flux Surface (LCFS) within the vacuum chamber by minimizing the differences of the flux at a set of control points chosen along the desired plasma boundary and the flux at the X-point.This block also computes the PF current corrections to be tracked by the PFC Decoupling Controller; • The Plasma Fast Boundary Controller, which assures a faster response against the plasma deformations that are typically induced by the poloidal beta variations due to the switch on and off of the additional heating, such as the Neutral Beam Injectors or the Electron Cyclotron (EC).• the VS Controller, which is needed to vertically stabilize the plasma column and counteract relevant disturbances.
Given the characteristic time of the vertical instability, this controller must act on a time scale that is faster than the dynamic imposed by the PFC.Therefore, the VS is designed to directly compute the additional voltage request for the PF coils.
As anticipated in section 1, the peculiarity of the proposed architecture is the sharing of the superconductive PF circuits as actuators for both the VS and the other magnetic control tasks.Indeed, in figure 1 the VS, although with voltage requests, acts on the same actuators as the PFC Decoupling Controller, therefore there is a need to adopt a model-based design approach that ensures effective decoupling between the various tasks, to avoid interactions that may lead to poor closed-loop performance and, in the worst case, instability.
In the following sections, a detailed description of the control algorithms implemented by each of the blocks shown in figure 1 is given.Since we propose a model-based approach to design the control algorithms, we conclude this section by briefly introducing the linear model of the plasma response, which we will refer to in what follows.Indeed, it is well known that around a given plasma equilibrium is specified by means of • nominal plasma current I p eq ∈ R; • nominal currents in the n PF PF circuits4 I PFeq ∈ R nPF ; • expected values of poloidal beta β p eq and plasma internal inductance l ieq , whose variation can be regarded as disturbances, i.e. exogenous inputs, as far as the plasma magnetic control is concerned; the behavior of the plasma and of the currents flowing in the surrounding conductive structures can be described by the state space model [3,34] where: T is the vector that holds the variations of the currents in the PF circuits δI PF , in the passive structures δI e , and of the plasma current δI p ; • u is the vector of the voltages applied to the superconductive PF circuits; • w = δβ p δli T is the disturbances vector, i.e. the variations of β p and l i ; • y is the output vector that contains the variations with respect to the equilibrium values of all the quantities of interest; e.g. the plasma shape descriptors like flux differences or plasma-to-wall distances (gaps), the magnetic field at the Xpoint, radial and vertical position of the X-point, PF circuit currents, plasma current, plasma centroid vertical position 5 .

PFC decoupling controller
The proposed architecture relies on a current-driven control scheme, therefore a controller for the currents flowing in the PF circuits is needed.The design of the PFC Decoupling Controller exploits a well-assessed approach for the design of a Multi-Input-Multi-Output (MIMO) controller based on the plasmaless model that describes the behavior of the mutually coupled PF superconductive circuits [4].Let us now consider where V PFC , I PF ∈ R nPF are the voltages applied by the PFC Decoupling controller and the currents flowing in PF superconductive circuits, respectively, while M ∈ R nPF×nPF is the inductance matrix computed in absence of plasma, i.e. the socalled plasmaless mutual inductance matrix.Let I PF ref be the reference waveforms to be tracked by the PFC Decoupling Controller, and let us consider the following diagonal matrix If the PFC Decoupling controller output is computed as follows where the matrix gain K PF is set equal to then the closed-loop behavior of the n PF PF currents becomes equal to i.e. decoupled control on each PF circuit is achieved, and the closed-loop first order time constant can be set by means of the design parameter Λ.It is worth noticing that, although the design of the control matrix K PF is based on plasmaless mutual inductance when mutual coupling with passive toroidally continuous conducting structures is neglected, the robustness of such an approach has also been assessed in the presence of various plasma operating tokamaks, such as JET and EAST [35,36].However, the adoption of a current-driven approach constraints the currents flowing in the active coils, whose contribution to the passive stabilization of elongated plasmas is not negligible in the absence of in-vessel coils and or passive stabilizers.As will be discussed in more detail in section 3, the presence of a PFC Decoupling Controller significantly contributes to the increase of the actual plasma growth rate.

VS controller
Similarly to what is currently implemented at the JET tokamak [12], and to what is envisaged for ITER once the in-vessel coils will be installed [8,14], the proposed VS controller aims at control to zero the plasma vertical velocity, by letting the plasma boundary control to keep the desired shape and position.However, since the control architecture shown in figure 1 is meant to operate without in-vessel coils, the proposed VS controller acts on the plasma column by directly computing additional voltages V VS ∈ R nPF to be summed to the output voltages V PFC computed by the PFC Decoupling Controller, and then applied to the superconductive circuits by the power supplies.
Such voltages V VS are chosen to ideally drive the PF currents along a specific direction given by a the linear combination I z ∈ R nPF .The direction I z is chosen to somehow maximize the radial field able to exert a vertical force on the plasma column, and can be regarded as a virtual circuit to be used by the VS system.Given the typical number of degrees of freedom available on existing tokamaks, which are equal to the number of independent PF circuits, there are different possible choices for the vector I z .
A possible approach to compute I z consists in choosing a point P in the center of the vacuum chamber, a pair of updown symmetric coils and the corresponding currents so that the vertical field in P be equal to 0. However, in this paper, we briefly describe an alternative approach based on the solution of a Quadratic Programming (QP) optimization problem, which gives more flexibility on the choice of the control coils, and allows to set constraints both on the PF currents and on the field in more than one point inside the vacuum chamber.To this aim, let where C Bz , C Br ∈ R m×n are the linear relationship between the n ⩽ n PF currents in the selected PF circuits ĨPF and both the vertical B z and radial components B r of the poloidal magnetic field on a grid of m > n points, typically chosen in the plasma centroid region.The two matrices C Bz , C Br can be derived from the linear model ( 1) of the reference equilibrium, which includes the presence of the passive structures.Given the linear relationship (3), the vector I z can be obtained as the solution of the following constrained QP problem subject to where δI PF and δI PF can be used to weight the control effort on some circuits and/or to force some of the PF coils currents to be positive or negative (e.g.outer currents above/below the poloidal mid-plane forced to be positive/negative, respectively).
Thanks to the objective function ( 4), the solution of ( 4) and ( 5) minimizes the vertical field on the grid while trying to obtain a corresponding radial field which is greater than the chosen threshold B min , as required by constraint (5a).It is worth to remark that the threshold B min does not limit the magnetic field that can be obtained with the control, since the current in the obtained virtual circuit is scaled by the controller as required to achieve the objective.
As an example, we apply the proposed approach to the JT-60SA 1 MA upper single-null equilibrium described in [26,37] and considered for first plasma operation, whose poloidal cross-section is shown in figure 2(a).
As far as the PF coils system is concerned, 10 superconductive coils are installed, 4 in the Central Solenoid (CS), and 6 Equilibrium Field (EF) coils.Each coil is independently fed by a dedicated power supply, and therefore 10 PF circuits are available for the magnetic control system.In particular, only the n = 6 EF coils are considered in the static map (3), while the 8 × 9 grid shown in figure 2(b) is used to constrain the magnetic field components, hence m = 72.Moreover, B min in (5a) is set equal to 0.1 T, while up-down symmetry is enforced across the mid-plane, letting EF1-EF3 positive and EF4-EF6 negative by means of constraint (5b).By solving the QP problem ( 4) and ( 5) with the Matlab 2022b quadprog solver on a MacBookPro, the resulting normalized vector δ ĨPF , i.e. with unitary norm, turns out to be equal to δ ĨPF = 0.0069 0 0.76 −0.65 0 0 T .Therefore, once it is recognized that the contribution given by EF1 in the computed solution is negligible, the corresponding virtual circuit for VS control can be set equal to 6I z = 0 0 0 0 0 0 0.76 −0.65 0 0 which correspond to: if we consider the turns of each JT-60SA coil.Another possible choice for the VS virtual circuit is obtained by solving the QP on the same optimization grid chosen for I z , but removing the constraints that enforce updown symmetry.In this case the solution is obtained, which allow to further minimize the vertical components of the magnetic field on the optimization grid with respect to I z , as summarized in table 1, while obtaining practically the same vertical displacement, as shown in figure 3.
It is worth to remark that the proposed approach to compute VS virtual circuits is just one of the possible options.Indeed, also the complementary alternative that maximizes the (a) Plasma equilibrium shape for the linearized model referred to as Eq#2 in table 2 and quantities controlled by a possible implementation of the proposed Plasma Boundary Controller for JT-60SA.The chosen controlled variables are 9 flux differences plus the radial and vertical components of the magnetic field at the X-point.In the proposed architecture shown in figure 1, these control quantities are sent also to the Plasma Fast Boundary Controller.(b) Plasma displacement and the magnetic field generated by the VS virtual circuit when set equal to Iz.The black dots show the optimization grid used to solve problem (4) and (5).
Table 1.Minimum and maximum magnetic field components on the optimization grid used to solve the QP problem (4) and (5).The current in each virtual actuator has been set equal to the value that corresponds to the plasma vertical movement shown in figures 2(b) and 3, respectively.

VS virtual circuit
Br min (mT) Br max (mT) Bz min (mT) Bz max (mT) radial field on the grid while constraining the vertical one to be below a given threshold is viable.For a given choice of the vector I z , the simplified scheme for the VS controller is shown in figure 4, and the corresponding control action is given by where • I PF are the currents flowing in the superconductive circuits; • I scenario ∈ R nPF are the reference scenario currents; • Żc ∈ R is the vertical velocity of the plasma centroid; • K v , K I ∈ R are the velocity and current gains, respectively; • I p ref ∈ R is the reference plasma current which is used to scale the velocity gain K v ; • the vector V z ∈ R nPF is computed exploiting the plasmaless mutual inductance matrix M as follows Indeed, since the VS controller acts directly on the PF voltages, the V z vector allows to generate voltages that produce currents in the directions specified by the vector I z .
Since in the proposed architecture VS is achieved by using slow circuits/power supply as actuators, which are typically designed to perform other tasks, such as I p and equilibrium control, neglecting the passive structures when deriving V z has no major impacts, as it is shown in the case study presented in section 4. Furthermore, it is also worth noticing that other to bring the plasma velocity to zero, the control law ( 7) aims also at regulating to zero the projection along the direction specified by I z of the difference between the actual currents I PF and the scenario ones I scenario , being I T z • (I PF − I scenario ) the projection of I PF − I scenario along the I z direction.In other words, the VS controller aims at controlling to zero the control effort while stabilizing the plasma.This objective is achieved thanks to the presence of the Plasma Boundary Controller described in section 2.4, which brings the plasma to an equilibrium position on a slower timescale compared to the one of the VS controller.Moreover, it is also important to remark that, if there is a need to improve the stability margins, the control law (7) can be augmented with a Single-Input-Single-Output (SISO) dynamic compensator, typically a lead compensator; more details can be found in [14,38].

Plasma fast boundary controller
The Plasma Fast Boundary Controller is included in the proposed architecture to react to fast plasma shape variations due to external disturbances.This dedicated control loop is designed to induce a specific deformation to balance the one caused by the foreseen disturbance.Such a deformation depends on both the disturbance to be rejected and the reference plasma equilibrium.In our architecture, we exploit this control loop to promptly counteract the Horizontal Displacement Events (HDE) induced by the switching of the additional heating systems, which in turn correspond to a poloidal beta variation.
The proposed control algorithm for the Plasma Fast Boundary Controller receives as input the same quantities controlled by the Plasma Boundary Controller.As it will be discussed in section 2.4, when controlling single-null divertor plasmas, such control input vector, referred to as y sh ∈ R n sh , holds • both B rX and B zX that are the variations of the radial and vertical components of the poloidal magnetic field at the X-point, respectively; • the flux differences ∆ψ i , i = 1 , . . ., p between the poloidal flux at the p control points, that are chosen along the desired plasma boundary, and the poloidal flux at the X-point.
As an example, figure 2(a) shows a possible choice of the components of y sh for the upper single-null equilibrium considered.
It is worth noticing that, given the above choice for y sh , the reference values for both the Plasma Fast Boundary and the Plasma Boundary controllers should be set equal to the null vector, since both the poloidal magnetic field at the X-point and the flux differences should be kept as small as possible to track the desired plasma shape and position.
For the considered JT-60SA equilibrium, the plasma model is exploited to compute the linear combination of PF currents I r ∈ R nPF corresponding to a virtual circuit that induces a plasma deformation capable to counteract the one caused by the β p variation that corresponds to the switching of the additional heating systems.
Moreover, if C sh ∈ R n sh ×nPF is the linear relationship between the currents in the PF circuits and the controlled plasma shape descriptors y sh , then Y r = C sh • I r is an estimation of the plasma deformation due to I r in the n shdimensional space defined by the controlled variables y sh .In other words, Y r represents the direction in the space of the chosen plasma shape descriptors that is controlled by the linear combination I r , and is chosen to be as equal as possible to the mainly outboard deformation induced by the β p variation.
The following choice for I r has been computed for the 1 MA upper single-null equilibrium of JT-60SA shown in figure 2(a) when we consider the ampere-turn.Moreover, the corresponding plasma outboard deformation is reported in figure 5.
The control direction defined by the couple (I r , Y r ) allows to attain the control along the desired direction in the space of the controlled plasma boundary descriptors by means of a SISO controller.Indeed, by properly projecting the control error along the direction defined by Y r , and by properly distributing the control effort along the direction defined by I r , the Plasma Fast Boundary Control is given by the following relationship in the Laplace domain, that relies on the SISO dynamic controller K fast (s) where Y sh (s) ∈ R n sh is the Laplace transform of the plasma shape descriptors vector y sh , and ∆I fast PF (s) is the Laplace transform of the corrections to be added to the scenario PF currents, i.e. is the Laplace transform of the Plasma Fast Boundary Controller output.
Although different solutions can be adopted once the fast boundary control problem is reduced to a SISO one, a possible choice for the controller K fast (s) is a Proportional Integral (PI) one, i.e.K fast (s) = K P + K I s , which turned to be satisfactory for the case study considered in section 4.

Plasma boundary controller
Different strategies are possible to control the shape of the plasma LCFS, the most common being the isoflux control [6,13,27] and the plasma-to-wall gap control [17,39].Although the proposed strategy relies on the so-called eXtreme Shape Controller (XSC) algorithm [40] applied to the case of isoflux control, it can be easily extended to the case of gap control, or the control of any other plasma shape descriptor, as long as a reliable linear model linking the currents in the PF circuits to the controlled variables is available.Motivated by the considered JT-60SA case study, in this section, we focus on the case of isoflux control for a single-null plasma configuration.Therefore, as anticipated in section 2.3, the proposed approach aims to control at zero the poloidal magnetic field at the X-point, as well as the flux differences between the flux at the control points and the flux at the X-point.
Since it relies on the XSC approach (more details can be found in [40]), the design of the proposed plasma boundary control algorithm is based on the following output equation of the linear model (1), that links the variations of the plasma  (8).This is mainly a plasma outboard deformation, which is the one needed to counteract HDEs.shape descriptors y sh to the variations of the currents in the PF circuits: The main advantage of resorting to the XSC approach is that we can set n sh > n PF , i.e. we can control a number of plasma shape descriptors that is greater than the number of independent PF control circuits.However, this implies that, at steady state, the error on the controlled variables y sh is minimized in the least-mean-square sense.Moreover, such a minimization is performed on a weighted version of the shape descriptors by considering the pseudo-inverse of the following matrix where Q ∈ R n sh ×n sh and N ∈ R nPF×nPF are two diagonal matrices that are used to differently penalize the various shape descriptors (Q) and to distribute the control effort among the various PF circuits (N).It turns out that the following steadystate objective function is minimized The pseudo-inverse C † XSC is obtained via the singular value decomposition C XSC = U • S • V T , where the matrix S ∈ R nPF×nPF is a square diagonal matrix that holds the singular values in decreasing order and U ∈ R n sh ×nPF and V ∈ R nPF×nPF are unitary matrices.As a matter of fact, minimizing the performance (10) index by retaining all singular values leads to a high control effort at steady state in terms of currents in the PF coil, which can lead to actuator saturations.For this reason, a trade-off condition is achieved, by minimizing the cost function while penalizing both the error on the controlled shape descriptors and the control effort.This is achieved by controlling to zero the error only for the n < n PF linear combination related to the largest singular values [40].
However, to achieve effective actuator sharing between the plasma boundary and the other magnetic control tasks, the standard XSC approach needs to be extended by further exploiting the linear model.Indeed, the control directions set for both the Fast Boundary Control and the VS must be taken into account.In particular, to achieve decoupling with respect to the Plasma Fast Boundary Controller, the control error −y sh is projected on the subspace orthogonal to Y r (see section 2.3).Furthermore, to decouple the behavior of the Plasma Boundary Controller also from the VS, the pseudo-inverse matrix C † XSC is projected in the orthogonal complement of span (I z ).
As a result, the Plasma Boundary Controller is a MIMO controller that computes the corrections to be applied to the scenario currents as follows: where ×n , while Y ⊥ r is a matrix whose columns form an orthonormal basis of ker(Y r ) and I ⊥ z is a matrix whose columns form an orthonormal basis of ker(I z ).The diagonal transfer matrix K boundary (s) contains a set of n PF regulators needed to improve the dynamic response of the controller, specified in the Laplace domain.As for the Plasma Fast Boundary Controller, a set of PI is typically sufficient to achieve the desired performance.

Plasma Current Controller
The Plasma Current Controller achieves robust control of the plasma current against uncertainties and disturbances by computing an additional correction to the scenario currents, which is then tracked by the PFC Decoupling Controller, together with the other requests from the outer loops.Similarly to the VS and the Plasma Fast Boundary Controller, the output of the Plasma Current Controller moves along a particular linear combination of currents in the PF circuits, namely I Ip , that provides a transformer field, which is a field that maximizes the effect on the plasma current by minimizing the effect on the plasma shape.
Similarly to what has been done for Plasma Boundary Control (see section 2.4), to achieve effective actuator sharing, the output of the Plasma Current Controller is projected in the orthogonal subspace of span [(I r , I z )], in order to decouple it from both the VS and the Plasma Fast Boundary Controller.Therefore, by denoting with I ⊥ a matrix whose columns form an orthonormal basis of ker I z I r , it turns out that the output of the Plasma Current Controller is given by where, similarly to all the other outer loops, K Ip (s) is a SISO dynamic regulator which is typically set equal to a PI controller.

Assessment of the effect of a current-driven control architecture on plasma vertical instability
As anticipated in section 2.1, the adoption of a current-driven control scheme contributes to a worsening of the plasma vertical instability, which makes even more challenging the VS task in the absence of a dedicated and relatively fast actuator.As a case study, in this section we discuss the vertical stability of the considered JT-60SA equilibrium.
The analysis is carried out with reference to the 1 MA upper single-null and elongated equilibrium used during the first operation of JT-60SA, in the absence of in-vessel coils [6,26], shown in figure 2(a).The following are examples of vessel time constants for the considered model: 0.0297 s, 0.0392 s, 0.0551 s, 0.0995 s.A set of different linear models (1) corresponding to small variations of plasma parameters around such an equilibrium have been considered.The main plasma parameters of the considered equilibria are reported in table 2. Although, the shapes corresponding to the equilibria considered are similar, as shown in figure 6, the resulting growth rates vary and cover a wide range.
It can be noticed that, under the assumption used to derive linear models with CREATE-NL [41], in some cases, the effect of the superconductive coil is sufficient to passively stabilize the equilibrium (as an example, see Eq#5 in table 2).However, such a stabilizing effect practically vanishes as a consequence of the parasitic circuit resistances.
Moreover, as anticipated in section 1, the presence of the PFC Decoupling Controller makes the VS task more challenging, by significantly reducing the passive stabilizing effect of the active coils.Indeed, by driving the desired currents in the coils, the PFC Decoupling Controller counteracts the currents induced within the coils themselves by plasma movements.Therefore, in practice, this effect causes an increase in plasma growth rate, calling for the deployment of an effective VS system.
To estimate such an effect on the considered JT-60SA equilibria, two different analyses have been carried out by exploiting the linear model (1).First, the superconductive coils have been removed, by removing the corresponding rows and the columns both from the mutual inductance matrix M and from Table 2. Plasma parameters for the linear models corresponding to the considered 1 MA upper single-null equilibrium of JT-60SA.Poloidal beta βp, internal inductance l i , elongation κ, aspect ratio A, and an estimation of the growth rate γ, are reported.

JT-60SA equilibria
Eq#1 0.78 0.14 the resistance matrix R that appears in the following circuital version of the of the linear model ( 1) in which the disturbance vector has been neglected, for the sake of simplicity.It is worth to remark that M accounts not only for the active coils, but also the presence of plasma and eddy currents.Once the effect of the superconductive coils has Table 3. Nominal growth rate (γ), growth rate obtained by removing the rows and columns of the matrices M and R that correspond to the PF coils (γ NC ), and the growth rate obtained by increasing the PF coil resistances (γ R ) are reported.
Growth rates comparison Eq#1 0. been removed, obtaining the two matrices M mod and R mod , the dynamic matrix in (1), given by A = − M −1 mod • R mod , is computed.A similar result would be obtained by considering a fictitious resistance in the superconducting coils.This additional resistance on the PF coils is an alternative way to emulate the effect of the PFC Decoupling Controller, since, as described in section 2.1, it is designed to impose a given time constant on the various PF circuits-see also (2).The results of both analyses are reported in table 3.
From the results shown in table 3, we can conclude that the presence of the PFC Decoupling Controller can cause an increase of γ up to about one order of magnitude, calling for the presence of an effective VS system, even when this should share relatively slow actuators with the other magnetic tasks.

The JT-60SA case study
In section 2 the model-based control algorithms to be deployed in each of the components of the proposed architecture shown in figure 1 have been introduced.As discussed in that section, the design of the various algorithms exploits the linear model (1); in particular the linear model corresponding to Eq#2 in table 2 has been used for the design of the controllers for the JT-60SA case study.The model has been obtained with the CREATE equilibrium codes tool-suite, which has been exploited to design and validate several magnetic control systems in the last two decades [14,28,[41][42][43].In what follows, we present the nonlinear assessment of the proposed magnetic control system, aimed at showing its effectiveness.In particular, the following two cases are considered: • the rejection of Vertical Displacement Events (VDEs); • the scenario similar to the one considered in [6], where a sudden increase of poloidal beta β p due to the switch-on of the heating systems is included during the plasma current ramp-up from 1 MA to 2.5 MA.Such a β p variation induces a HDE, which is rejected by the Plasma Fast Boundary Controller, as will be shown in section 4.2.
All the results have been obtained by running nonlinear simulations with the CREATE-NL equilibrium code [41] in  1.2 kV 100 V ms −1 3.08 ms 1.07-3.08ms EF2-EF3-EF4-EF5 0.9 kV 100 V ms −1 1.58 ms 0.53-1.58ms the Matlab environment.A comparison between simulations obtained with the nonlinear equilibrium code and the linear model (1) used for controller design is also presented for the VDE case.
In all the cases considered, the model considered for the CS and PF power supplies is the series of a saturation, a rate limiter, and a time delay, as reported in figure 7.With the only exception of the saturation, the values of the parameters for the power supply model change with the operative conditions, as reported in [44].In table 4 the chosen values are reported; the time delay has been chosen equal to the largest reported in [44], while the rate limit has been taken from [6].Finally, an additional time delay of 350 µs has been introduced between the control output voltage and the PS model, to account for the delay due to both the real-time computation of the equilibrium and the controller sampling time, as specified in [6].

VDE rejection
A 1 cm VDE, i.e. displacement of the vertical plasma centroid position along the unstable mode is considered as a test case to show the reliability of the linear model used to design the controller by comparing its behavior with the nonlinear equilibrium code used for the assessment when small perturbation are considered (see figure 8).In particular, the results presented in this section have been obtained using the VS control direction (6).
Nonlinear assessment of VDE of 6 cm, which corresponds to a displacement normalized to the minor radius of ∆Z c /a ∼ 4%, has been performed to test a challenging scenario for the VS. Figure 9 reports the shape snapshots obtained during It should be noticed that in all considered cases, other than initial voltage saturation on the EF3 and EF4 coils, which are the only ones used by the VS system when the direction I z1 is exploited, the voltage requests are within the prescribed ranges.

Plasma current ramp-up scenario
In this section, we consider the simulation of the ramp-up scenario similar to the one considered in [6].In particular, a plasma current ramp-up from 1 MA to 2.5 MA in 6 s is considered, with a constant ramp-up rate equal to 0.25 MA s −1 .Similarly to what has been done in [6], during the ramp-up, a switchon of the auxiliary heating at 2.0 s, and a switch-off at 2.5 s has been considered.When the heating system is turned on, the poloidal beta β p is assumed to increase linearly from 0.14 to 0.5.Moreover, a minor disruption is considered at 3.5 s causing a 20% β p drop in 10 ms, followed by a linear recover to the nominal value in 0.15 s.
The simulation is performed considering a resistive flux consumption corresponding to CEjima = 0.45 [45].
The results presented in this section show that the proposed control architecture is able to perform the I p ramp-up-see figure 11-while maintaining the plasma shape as shown in figures 12 and 13.The voltages remain within the prescribed limits as shown in figure 14(a).The total request to the PFC Decoupling Controller I PF ref is reported in figure 14(b).
Moreover, a simulation without the Plasma Fast Boundary Controller has been performed to assess its necessity in the proposed control architecture.Note that this also means that the input to the Plasma Boundary Controller is not projected into the subspace orthogonal to Y r .As shown in figure 15(a), without the Plasma Fast Boundary Controller, during the β p ramp the plasma touches the first wall due to saturation of the the PF voltages reported in figure 15(b).
Finally, given the nominal value βp = 0.5, we performed nonlinear simulations including a poloidal beta variation ∆β p spanning from 0.2 • βp = 0.1 to 0.6 • βp = 0.3, to assess the maximum perturbation that the closed-loop control can handle.The time traces of both radial and vertical positions of the plasma centroid are reported in figure 16, showing that the proposed system is capable to reject a beta drop up to 0.5 • βp = 0.25, while when ∆β p = 0.6 • βp = 0.3 the plasma is vertically lost.It is worth to remark that the capacity of the proposed architecture to reject a β p drop is affected by the sharing of relatively slow actuators for both radial and vertical control.Indeed, when the disturbance amplitude becomes too big, the plasma is practically vertically lost.This is not surprising, since the proposed architecture is aimed at carrying out the first operation of a superconductive tokamak.Dedicated in-vessel coils are usually envisaged to deal with more challenging plasma scenario, which, if equipped with independent actuators, allow to deploy a faster centroid position control by exploiting a geometric decoupling similar to the one described in section 2.

Conclusions
An architecture for magnetic control able to explicitly vertically stabilize the plasma column and to counteract fast plasma movements in the absence of a dedicated in-vessel actuator is presented in this paper.The proposed architecture is suitable for the first operation of superconductive tokamaks, when not all in-vessel components are yet installed.In order to effectively share the available superconductive actuators among various magnetic control tasks, a model-based geometric control design approach has been adopted.However, the adoption of a current-driven control scheme increases the growth rate of the plasma vertical instability, requiring the presence of an efficient VS system also in the absence of a dedicated fast actuator.Therefore, in the proposed control architecture, the controller for the VS task is the first one designed: indeed, since the stabilization of the plasma is paramount, this controller must act along an effective direction, without interference from the other magnetic control loops.To achieve such an objective, in the proposed architecture, the other magnetic control tasks act in directions orthogonal to the one chosen for the VS.Specifically, the Fast Boundary Controller acts along a direction that maximizes the vertical field, while the Plasma Boundary and Plasma Current Controller, since it is possible to obtain satisfactory performances even acting along sub-optimal directions, are designed to achieve effective actuator sharing with the previously designed controllers.
The effectiveness of the approach is proved by means of nonlinear validation against a possible operative scenario for the first JT-60SA operation, but the architecture and the corresponding algorithm can be applied to any superconductive tokamak, including ITER.Furthermore, the proposed modelbased geometric decoupling is shown to be able to effectively deal with all the magnetic control tasks.

Figure 1 .
Figure 1.Block diagram of the proposed plasma magnetic control architecture that shares the PF superconductive coils system to perform all the magnetic control tasks.

Figure 2 .
Figure 2.(a) Plasma equilibrium shape for the linearized model referred to as Eq#2 in table 2 and quantities controlled by a possible implementation of the proposed Plasma Boundary Controller for JT-60SA.The chosen controlled variables are 9 flux differences plus the radial and vertical components of the magnetic field at the X-point.In the proposed architecture shown in figure1, these control quantities are sent also to the Plasma Fast Boundary Controller.(b) Plasma displacement and the magnetic field generated by the VS virtual circuit when set equal to Iz.The black dots show the optimization grid used to solve problem (4) and(5).

Figure 3 .
Figure 3. Plasma displacement and the magnetic field generated by the VS virtual circuit when set equal to I ′ z .

Figure 4 . 15 TA
Figure 4. Simplified scheme of the proposed VS controller.

Figure 5 .
Figure 5. Plasma deformation corresponding to the current vector Ir given by(8).This is mainly a plasma outboard deformation, which is the one needed to counteract HDEs.

Figure 7 .
Figure 7. CS and EF power supplies model used for the simulation of the JT-60SA test cases considered in section 4.

Figure 8 .
Figure 8.Time traces of the plasma centroid vertical position variation with respect to the equilibrium value δZc and of I Z = I T z • (I PF − I scenario ) in case of 1 cm VDE.The results of both linear and nonlinear simulations are reported.

Figure 9 .
Figure 9. Snapshots of the plasma boundary at t = 0 s and t = 1.5 s for a nonlinear simulation of a 6 cm VDE applied at t = 0 s.

Figure 10 .
Figure 10.(a) Control errors at the control points shown in figure 2(a).(b) Voltages applied to both CS and EF coils during a nonlinear simulation of a 6 cm VDE.

Figure 11 .
Figure 11.Time traces of the plasma current and poloidal beta for a nonlinear simulation of the Ip ramp-up scenario considered in section 4.2.These time traces refer to the case of closed-loop compensation of the fast disturbance via the Plasma Fast Boundary Controller.

Figure 12 .
Figure 12.Plasma shapes obtained during a nonlinear simulations of the Ip ramp-up scenario considered in section 4.2, when the fast disturbance are compensated by the Plasma Fast Boundary Controller.

Figure 13 .
Figure 13.Control errors at the control points shown in figure 2(a) for a nonlinear simulation of the Ip ramp-up scenario considered in section 4.2.These time traces refer to the case of closed-loop compensation of the fast disturbance via the Plasma Fast Boundary Controller.

Figure 14 .
Figure 14.Time traces of the PF voltages (a) and currents (b) for a nonlinear simulation of the Ip ramp-up scenario considered in section 4.2.These time traces refer to the case of closed-loop compensation of the fast disturbance via the Plasma Fast Boundary Controller.

Figure 15 .
Figure 15.Plasma shapes snapshots (a) and PF voltages time traces (b) for a nonlinear simulation of the Ip ramp-up scenario considered in section 4.2.These time traces refer to the case without the closed-loop compensation of the fast disturbance via the Plasma Fast Boundary Controller.

Figure 16 .
Figure 16.Time traces of the plasma centroid vertical and radial position for nonlinear simulations of the Ip ramp-up scenario considered in section 4.2 with a poloidal beta variation ∆βp spanning from 0.2 • βp = 0.1 to 0.6 • βp = 0.3.These time traces refer to the case of closed-loop compensation of the fast disturbance via the Plasma Fast Boundary Controller.

Table 4 .
[44]mum output voltages, rate limits, time delays and possible ranges chosen for the models of CS and EF power supplies as reported in[44].