Integrated model control simulations of the electron density profile and the implications of using multiple discrete pellet injectors for control

Pellet injection is regarded as the only realistic actuator for core density control in future reactors such as ITER and DEMO. However, a control strategy that can reliably regulate the plasma close to operational limits using multiple pellet injectors is not yet available. In this paper, we present the first integrated model control simulations where a dedicated model-predictive controller is included in JINTRAC. We show that, when continuous actuators are considered, a simple transport model with a steady-state disturbance rejection paradigm is capable of capturing the particle transport dynamics for multiple transport models and scenarios. This in turn allows the model-predictive controller to deal with the uncertainty and minimize the control error given the limited actuation space. Furthermore, we show that for ITER and DEMO relevant pellet sizes, the discrete, nonlinear dynamics of pellet injection will limit the control performance and jeopardize the constraints if not accounted for by the controller. Hence, we conclude that for high-performance control on future reactors, controllers will have to be developed that explicitly deal with the discrete pellet dynamics.


Introduction
The next generation of fusion reactors, like ITER and DEMO, will aim to operate at high plasma density in order to maximize the produced fusion power and to minimize the fuel cycle throughput [1][2][3].To achieve this, efficient core fueling actuators and reliable feedback control strategies that can maintain the plasma close to operational limits in the presence of varying plasma and wall conditions will be required [4].
The injection of pellets, mm-sized solid bodies of frozen hydrogen fuel, from the torus high-field side is regarded as the only realistic option for the control of the core density [5][6][7].That is because gas injection, the main actuator used for density control purposes on contemporary devices, is expected to be inefficient and too slow for core fueling [8].
One of the challenges that remains to be solved is the development of a control strategy that can reliably regulate the plasma close to operational limits using pellet injection and deal with the complexity of the particle transport dynamics.
On contemporary devices, feedback control with pellets is performed by neglecting the discrete pellet dynamics from the controller side and using the fueling rate as control input [9,10].The fueling rate is then approximated by a sequence of discrete pellets.The advantage of this method is that linear controllers can be used to determine the required control inputs.Using this method, successful density control has been shown in contemporary devices [11].However, for future reactors, the size of the pellets will be significantly larger and the injection frequency lower [12], and this approach might no longer be valid.Furthermore, future reactors will have multiple injectors available, capable of injecting different pellet types (either size or composition) [13].Given that this greatly increases the actuation space, the controller should be able to deal with multiple actuators.
In this work, we aim to tackle two challenges regarding the control of the density with pellet injection.Namely, dealing with the complex particle transport dynamics and the feasibility of using a continuous representation for the pellet actuator from the controller side when multiple pellet injectors with reactor relevant pellet sizes are used.
For this purpose, a model-predictive controller designed to regulate the density profile using pellet injection [14] has been integrated in the JINTRAC code suite [15].It has been used to control the density profile in integrated model simulations where the high-fidelity 1.5D core transport solver JETTO-SANCO [16] is used to model the plasma behavior.The results of the first dedicated integrated model control simulations are presented here and used to investigate control methods for particle transport in a tokamak.
Furthermore, we present the results of closed-loop feedback control simulations where multiple discrete pellet injectors with reactor relevant pellet sizes are used.In these simulations, the controller considers the actuator as continuous and a pellet model is introduced that approximates the fueling rate by a sequence of pellets.The impact on control performance and safety constraint satisfaction is investigated.
The remainder of this paper is structured as follows.In section 2, a summary of the model-predictive controller used throughout this work is given.In section 3, a short description of the integrated modeling simulation setting and modeling assumptions is provided.This is followed by the a brief discussion of the used configuration of RAPDENS.The results of the integrated model control simulations and an analysis of the control performance are presented in section 4. Subsequently, the analysis with multiple discrete pellet injectors is provided in section 5.The work is discussed and concluded in section 6.

Robust model-predictive controller
In this section, we give a brief overview of the modelpredictive controller and elaborate slightly on the parts relevant for the results presented later.For details about the controller, we refer the reader to the original publication [14].

Model-predictive control (MPC) paradigm and control model
In MPC, a model (called prediction model) of the system is used to formulate the evolution of the state over a prediction horizon (denoted N ∈ N +4 ) as function of the control inputs.This formulation is subsequently used to construct an optimization problem that is solved at each time step in order to determine the optimal control input.We use discrete time MPC, where the controller is called at discrete time instances with a control period T c .The subscript k denotes the discrete time instance where the controller is called.It can be related to real time via t k = kT c .
where θi,k and ξi,k are the predicted state and control input at future time In this work, a linear disturbance augmented prediction model is used of the form with predicted states θi,k = xi,k di,k ⊤ , where x ∈ R nx comprises the parameterization of the density profile and d ∈ R ny are disturbance states (discussed later).The predicted control inputs are chosen as ξi,k = ũi,k , where u ∈ R nu are the fueling rates of the pellet injectors.The predicted outputs are ψi,k = ỹi,k , where y ∈ R ny comprises synthetic density profile measurements.The linear state-space matrices A ∈ R nx×nx and B ∈ R nx×nu model the particle transport dynamics around a desired operating point.As explained in more detail in [14, section 5.3], they are obtained by linearizing the equations of the nonlinear control-oriented plasma simulator RAPDENS [17].The linear state-space is afterwards augmented with a constant disturbance state d and disturbances matrices B d ∈ R nx×ny and C d ∈ R ny×ny .As shown in [18], the use of a disturbance model with an observer in the MPC framework allows to compensate for steady-state mismatches between the prediction model and the plant, and to counteract the effect of slow-varying external disturbances.

Optimization problem
The optimization for the controller used in this work is given by (for compactness the subscript i, k is written short as i): where where ∥ζ∥ 2 T def = ζ ⊤ Tζ for a given matrix T and vector ζ of appropriate dimensions.Furthermore, N a,b = a, a + 1, . . ., b with a, b ∈ N and a ⩽ b.The cost function (3a), the constraints (3c)-(3f ), and the observer (3g) and (3h) are briefly discussed below.The prediction model (3b) was discussed in the previous section.

Cost function
The quadratic cost function (3a) is used to penalize the differences with respect to the desired inputs and state xi and ūi derived from the control reference.Furthermore, ϵ is a soft constraint parameter that penalizes violation of the nonlinear constraint (3f ) (see [14] for more detail).The weight matrices W x , W u , W ϵ determine the performance of the controller and matrix P is chosen to guarantee stability.

Constraints
Three different constraints are present in the controller.An input constraint (3d) and (3e) to enforce the pellet limits, a linear constraint (3c) to account for the Greenwald density limit [19] where ngw the Greenwald density and Z a matrix that calculates the line-averaged density from the states of RAPDENS, and a nonlinear constraint (3f ) that aims to optimize impurity transport via the ratio of temperature and density profiles' logarithmic gradients.Note that each of the constraints can be omitted if desired.

State and disturbance observer
The ability of the controller to deal with plant-controller model mismatch is provided by the augmented disturbance state d(t) and model B d , C d .Since the neither the states x(t) nor d(t) can be directly measured, they are estimated using synthetic density profile measurements y(t) ∈ R ny in a Kalman filter framework [20].Definition 2.2.For a general state θ(t) ∈ R n , with n ∈ N + , we introduce θi|j as the estimate of θ(t) at time t = t i using the diagnostic information of time t = t j with i ⩾ j.
At each call of the controller, the prediction model is initialized with the latest estimates of the state and disturbances, xk|k and dk|k respectively (3g) and (3h) which are given by where L k is the Kalman gain, and C ∈ R ny×nx is an output mapping that translates the states of the model to a density profile.

Interpretation of the disturbance state estimate.
It is important to note that the disturbance states are constant in the prediction model (3h), hence they only change during the update step of the observer (6).Therefore, their rate of change is indicative of the quality of the predictions made by the model.If the prediction model captures the one-step ahead dynamics of the to be controlled system, the residual z k|k−1 will be small and dk|k ≈ dk−1|k−1 .If the disturbance model does not capture the dynamics well, z k|k−1 will be large and the rate of change of the disturbance estimate will also be.

Description of the simulations
In this section, we describe the models and settings used in the integrated model simulations and in the control-oriented model to derive the controller.

Integrated modeling
The integrated model simulations presented in this work were performed using the JINTRAC code suite [15].The modeling of the ITER DT 15MA/−5.3T baseline is based on [21,22].Transport in the core is modeled using the core transport code JETTO+SANCO.At the separatrix, constant values for the ion densities, electron and ion temperatures are prescribed as boundary conditions.Neoclassical transport for the main ion species and impurities is modeled using NCLASS [23].Anomalous transport is determined with the Bohm/gyro-Bohm (BgB) H-mode model together with a collisionality dependent inward pinch term or using the QuaLiKiz neuralnetwork (QLKNN), i.e. a surrogate model of the reduced order gyrokinetic turbulent transport model QuaLiKiz [24,25].The effect of ELMs is included in a time-averaged way by means of the Continuous ELMs model as described in [26].
The plasma is mainly fueled by pellet injection, but particle sources due to neutral beam ionization and gas puffing are also included in the simulation.The neutral beam particle source is calculated by PENCIL [27].For edge gas puffing, a fueling rate of 5 × 10 21 [e/s] is applied and source profiles due to ionizing neutrals are calculated using the FRANTIC code [28].Pellet core fueling is modeled using the continuous pellet model whereby the time-averaged fueling rate is deposited on a fixed Gaussian radial profile.We consider two pellet sources with this modeling format, noting that the possibility to include a second continuous pellet source is a novelty introduced as part of this work.The Gaussian profiles are centered respectively around p 1,cen = 0.88 and p 2,cen = 0.8.The widths of the Gaussian profiles are respectively chosen as p 1,wid = 0.07 and p 2,wid = 0.1.The first pellet fueling profile will be referred to as shallow and the latter as deep.These deposition profiles are based on recent results for pellet deposition profiles in ITER [29].Please note that the selected deep deposition profile would require an injection velocity larger than 500 ms −1 .Such deep fueling however is not achievable via the inner pellet guide tube, which has a maximal injection velocity of 300 ms −1 [30,31].It has nonetheless been used to clearly distinguish the two continuous actuators, as well as prepare for innovations either for ITER, or other devices where such deep deposition could be achievable.
A major novelty in this work is the fact that the fueling rate of each injector is provided to the code by the modelpredictive controller.Feedback modes for both pellets and gas were readily available in JETTO.However, these are based on simple control schemes not relying on dedicated controllers.In this work, for the first time, a dedicated controller was used to provide the actuator commands to the transport code.
The controller is called at a fixed frequency.In between calls, the control input is kept constant.This approach is chosen as it mimics the set-up on a real reactor where the control system runs at a given frequency.For the simulation presented here, a control period of 30 ms is chosen.Note that the control systems usually run at frequencies in the kHz range.However, since the maximum pellet injection frequency is ≈16 Hz [12], it does not make sense to compute the pellet inputs at such high frequencies.Hence, the control period is chosen to reduce the computational burden of the simulations.

Synthetic density profile measurements
For the estimation of the disturbance and controller's states (see section 2.5), synthetic density profile measurements are used.They are obtained by simple interpolation of the JETTO profile to a low resolution spatial grid with 11 equidistant measurement location on ρ ∈ [0, 1].The effect of measurement noise is not taken into account in the simulations.

RAPDENS' configuration
The linear prediction model and the simulation with discrete pellets (see section 5) are obtained using the lightweight plasma simulator RAPDENS.In the latter, the diffusion coefficient D and the drift velocity V are constant in time and their spatial profiles are based on [32] and have been tuned to approximate the transport in the ITER baseline H-mode 15 MA, −5.3 T discharge of [33].To simulate the effect of an opaque SOL, the ionization model in RAPDENS is turned off, meaning that core fueling is achieved solely by pellet injection.A continuous pellet model is used for the two pellet injectors.Their spacial deposition profiles are modeled as parabolic functions with the same characteristics as discussed before (see section 3.1).We note here the discrepancy in the pellet fueling modeling between the continuous pellet models in RAPDENS and JETTO.In the first, deposition of the pellet is modeled as parabolic functions; in the latter as Gaussians.This mismatch will have to be compensated by the controller.

Control of the density profile in integrated model simulations
In this section, we present the results of the first integrated control simulations within the JINTRAC framework.We first provide an overview of the performance over the performed simulations before discussing in more detail two specific cases.

Overview of the performance
The simulations were performed with two different control references; the first being the 'low density' profile, i.e. 1 × 10 20 m −3 in the core with a slightly peaked profile (see black dotted line in figure 2(a); the second being a reference density profile taken from ITER baseline 15 MA −5.3 T H-mode simulation [33].For each reference, the simulations are performed using the BgB and the QLKNN, resulting in four distinct simulations scenarios.
A parameter of merit used to quantify the performance of the controller is the relative root mean square (RRMS) difference between the reference profile and the controlled profile, defined as where N ref,ρi and N ctrl,ρi are the reference and controlled density profile respectively, sampled at the discrete points ρ i ∈ [0, 0.9].The time resolved RRMS error is shown for the four simulations in figure 1.The controlled profiles at the end of each simulation are shown together with the desired reference in figure 2. For the cases using BgB transport model, the controller is capable of regulating the profile with less than 5% error.For the cases using QLKNN, the RRMS error is ≈6% for the low density reference and ≈11% for the ITER baseline case.
For the four simulated cases, the linear disturbance augmented model (2a) with the estimated states by the Kalman filter ( 4) is capable of capturing the particle transport dynamics, this is argued in detail in section 4.2.
The controller has a limited actuation space, i.e. the control inputs it can change are the fueling rates of the pellet injectors of which the actuation takes places between ρ = 0.7 and ρ = 1.Given that it has two inputs, it can locally shape the gradients and control the profile but it has to rely on transport to meet the density outside of the actuation region.In our simulations, the reference profiles have been designed assuming BgB transport with a Gaussian pellet source centered around ρ = 0.85 [33].As a result, the reference and controlled profiles have similar shapes in the simulations using BgB transport.Given that the model in the controller with the estimated states by the Kalman filter is capable of capturing the particle transport dynamics, the controller can provide the required inputs to track the desired profile, as can be seen in figure 2 where For the simulations with QLKNN as transport model, it can be seen in figure 2 that the profiles have a different shape, most notably are less peaked than with BgB.As a result, given the limited actuation space available to the controller, the desired control reference shape is infeasible in these simulations.Nonetheless, since the model in the controller captures the particle transport dynamics, the controller will drive the system to a feasible profile which is as close as possible to the reference and thus minimizing the error.This results in the controlled profiles having a higher pedestal density and a lower core density than the reference (see figure 2).Note that the region that will be tracked can be shaped using the weight matrix W x .For example, if tracking in the core region is more relevant, the entries of W x that penalize the corresponding states can be increased.
In the simulations where the reference profile is infeasible, the resulting control error is not caused by the controller.To illustrate this, control simulations have been performed with a reference designed using the QLKNN model, of which the results are shown in figures A3 and A4.When the QLKNN model is used, the reference is feasible and the controller is capable of regulating the profile with less than a percent RRMS error.Conversely, when the BgB model is used with the new reference, the shape of the profile is different and the controller regulates the system while trying to minimize the error, resulting in an RRMS error is 7.5%.
In these simulations, a control-oriented plasma simulator with constant transport coefficients is used to derive an MPC controller.Together with a steady-state disturbance rejection paradigm, the controller is capable of regulating the density profile within 5% of the desired reference for multiple operating points when the BgB transport model is used.When using QLKNN, the reference shape is not feasible and the controller drives the density to a closest feasible profile.Based on these results, we observe that from a control perspective, under the made assumptions, uncertainty in particle transport during the flat-top can be considered as a 'slow-varying disturbance' for the two transport models and dealt with as such by a dedicated technique.

Detailed analysis of the results
Here we discuss the results of the simulations and the interaction between the controller and the transport code in more details.We start with a discussion of the simulation with the low reference profile, i.e. 1 × 10 20 m −3 in the core with a slightly peaked profile (see black dotted line in figure 2(a)) and using the BgB transport model.
The evolution of the control error is depicted in figures 3(a) and (c).The disturbance state estimates d(t k ) are shown in figure 3(e).Furthermore, the prescribed fueling rates for the pellet injectors are portrayed in figures 4(a), (b) and the achieved profile at multiple time instances in the simulation is shown in figure 4(c).The JETTO density profile is initialized at the desired reference.The states and disturbances of the controller's model are initialized at zero, i.e. no prior knowledge about the current state of the plasma is provided to the controller.
Between t = 400 and t = 404 s, the estimates of the disturbance states d(t k ) are varying rapidly, see figure 3(e).These states are estimated using the discrepancy between the density profile calculated by JETTO and the density profile predicted by the linear model in the controller.Fast changing estimates of the disturbance states indicate large disparity between the actual and predicted profiles.Meaning that the model in the controller does not fully capture the transport dynamics at the beginning of the simulation.Since the controller is using the prediction model to compute the control action, this mismatch explains the increase in error, first seen at the edge (between ρ = 0.7 and ρ = 1) where the actuation takes place, followed by an increase in the core, see 3(a) and (c).From t = 402 s onward however, it can be seen that some of the disturbance states are no longer varying quickly.This goes in pair with a decrease of the control error in the actuation region.
From t = 404 s onward, the estimated disturbance states barely changes and a steady decrease in the control error over the entire profile can be noticed.From this, it can be inferred that the disturbance augmented prediction model captures the particle transport dynamics.Consequently, the controller is capable of providing the required control inputs to drive the density profile to the desired reference.The controlled profile and reference profile are shown together in figure 4(c).At the end of the simulation, the control error does not exceed 5% at any points in the profile and the RRMS is below 4%.
For the ITER baseline case with BgB (appendix), the disturbance states do not reach a steady value for most of the simulation, only reaching a steady-state value around t = 413 s.This hints to changing transport dynamics during most of the simulation.Nonetheless, the MPC controller is capable of dealing with the change in dynamics, keeping the RRMS error below 4% from t = 404 s onward; with a maximum in local control error of ≈5% around ρ = 0.2 and at the edge (ρ > 0.8).
For the ITER baseline simulation with QLKNN, the evolution of the control error is depicted in figures 3(b)-(d) and the disturbance state estimates d(t k ) are shown in figure 3(f ).The low reference density case can be seen in figure A1.
The profiles calculated when using QLKNN as transport model have a higher pedestal density and are significantly less peaked than the ones calculated using BgB (see figure 2).This results in the desired reference profiles being infeasible for these simulations.Nonetheless, the model structure in the controller together with the estimation of the disturbance states with the Kalman filter are capable of capturing the particle transport dynamics for these cases too.This can be seen as the disturbance state estimates reach a steady-state value in figures 4(f ) and A1(c).As explained in section 2.5, this implies that the disturbance model successfully captures the dynamics of the system.Given that the desired reference is not feasible, the controller will aim to regulate the profile to a closest feasible one.As a result, the RRMS error in the ITER baseline case is ≈11% with local control error of 12% in the core and 15% at the edge.
Note that if the MPC controller would be part of a larger control system, with for example an overarching supervisory controller, and a control reference was to become infeasible during a discharge, it would be capable of providing this information to the overhead controller such that appropriate actions could be undertaken.

The impact of multiple discrete pellets on control
In previous sections, we have considered the pellet actuator as a continuous source of particles.However, they are by nature discrete events.Discrete actuators are a lot more complex to handle from the side of control as standard approaches are based on continuity.For feedback purposes on contemporary tokamaks, pellets are therefore often considered as a continuous source and the requested fueling rate is approximated with a sequence of pellets [11,34].It has been shown that good control performances can be achieved using this method [9].
However, the pellets used on contemporary machines are a lot smaller and injected at higher frequencies than the pellets  d(t k ).The cyan dotted lines in (a) denote the time instances where the profile is plotted in figure 4(c).The relation between the estimated disturbances states d(t k ) and the control performance can clearly be observed.For the BgB simulation, between t = 400 and t = 404 s, the estimates of the disturbance states d(t k ) are varying rapidly, indicating a discrepancy between the dynamics in the controller and the JETTO.This leads to the increase in control error seen during that time.From t = 404 s onward, the estimated disturbance states barely changes, indicating a good match between the plant-controller dynamics.A reduction in the control error is observable in this phase.that are foreseen for bigger machines.For example, the pellets used at AUG vary between 1.4 and 4.3 × 10 20 electrons per pellet whereas for ITER, pellets with 6 × 10 21 electrons are envisaged [12].Furthermore, control studies for DEMO [35,36] and nowadays reactors rely on a single pellet source.For next generation reactors, multiple sources with possibly different pellet types will have to be used, as pellets will be the main fueling actuator [13].
As a first step, we therefore investigate here the effect of using multiple discrete pellet injectors on the control performance.The use of discrete pellets with the MPC controller is not yet possible within the JINTRAC suite.Hence the results presented in this section are obtained using RAPDENS as plasma simulator.
The MPC controller presented in section 2 is used as regulator with the ITER baseline profile (see figure 2  reference.A pellet model is introduced between the controller and plasma simulator.It converts a fueling rate in series of discrete pellets.The pellet model integrates the fueling rate and compares the integrated flow with pre-defined pellet sizes.It also includes a model for the maximal injection frequency of a given injector.A pellet is fired if the required amount of particles surpasses the nominal number of particles in a pellet and a pellet is available.A pellet is available when the time since last the last injection is larger than of the maximal injection period.Note that this modeling implies the use of a gas gun type pellet injector as once a pellet is available it can be fired freely at any given time in the simulation rather than at fixed time intervals as would be the case for a centrifuge launcher.
The control simulations were performed for two cases: • Two injectors injecting pellets with a nominal electron content of 6 × 10 21 [#], with a maximal injection frequency of 4 Hz.This corresponds to a 92 mm 3 spherical pellet [29]  and it is the pellet size that is envisioned for core fueling in ITER [12].The pellet deposition profiles are constructed as parabolic functions between ρ = [0.7 − 1] and ρ = [0.8− 1] respectively.These approach fueling profiles for the 92 mm 3 pellet with 500 ms −1 and 200 ms −1 injection velocities [29].• Two injectors injecting respectively pellets with 6 × 10 21 [#] and 2.3 × 10 21 [#] with respective maximal injection frequencies of 4 and 16 Hz.The smaller pellet is envisioned to be primarily used for ELM mitigation on ITER [12].However, since primary studies of the matter injection in DEMO indicate that for controllability reasons, the size of the fueling pellets should be in the order of ≈2 × 10 21 [37], we consider this pellet size for fueling here also.For the 6 × 10 21 pellet, the deposition profile is chosen between ρ = [0.7 − 1] similar as discussed above.For the smaller pellet it is chosen between ρ = [0.8− 1].
The results of the simulations are shown respectively in figures 5 and 6.The discrete injection times of the pellets are shown in (a).The resulting fueling rates are depicted in (b).They are given by where t n denote the arrival time of the n'th pellet and n p is the number of electrons in a pellet.For comparison, the fueling rate from the continuous case (without pellet model) is also shown (in dashed lines).The RRMS error are shown in (c).For the simulation with 6 × 10 21 electron pellets, the average RRMS error for the first simulation is 18.2% and the peak error is 76%.For the simulation with the two pellet types, the average RRMS error is 14% and the peak error is 58.7%.It can be seen that the fueling rate is similar in the cases with and without the pellet model.The spikes in the control error when a pellet is injected are intrinsic to the actuator and the magnitude depends on the pellet size.However, as the controller does not account for the discrete nature of the pellets, it cannot guarantee that the constraint is respected (see figures 6(d) and 5(d).Furthermore, the coupling between the two injectors has a detrimental effect on the performance.This is the case at the instances where both injectors fire their pellets at the same time, leading to large influxes of particles, spikes in the control error and violation of the constraints.Particularly for the large pellets (with 6 × 10 21 electrons), this phenomenon is problematic as such an influx of particles would most likely lead to a disruption on a real-reactor.
Note that if the plasma is operated far from constraints, it might be sufficient to introduce cross-talk between the actuators to avoid the injectors firing at the same time.This approach would avoid the large fluctuations caused by two pellets being injected simultaneously.If constraints are to be accounted for, this method will not be sufficient.
From this analysis, we observe that to control the plasma in feedback with multiple pellet sources, injecting large pellets while constraints have to be accounted for, the actuators cannot be considered as a continuous actuators from the control side and subsequently approximated by a series of discrete pellets.This highlights the need for a control strategy that accounts for the non-linear nature of the pellet actuator.

Conclusions
In this work, we present the results of the first integrated model density control simulations.In these simulations, a dedicated model-predictive controller is included in the JINTRAC code suite and provides the fueling inputs to the pellet module of JETTO-SANCO.The MPC controller uses a linear prediction model that is derived from the RAPDENS plasma simulator.The specific configuration of the model relies on constant transport coefficients.To compensate for the significant simplifications, we use a robust scheme that is capable of dealing with steady-state and slow varying disturbances.We show that, under the assumption of continuous pellets and ELMs, this robust scheme is capable of dealing with mismatch in particle transport modeling between controller and plant in the flat-top of an ITER discharge.This means that the disturbance augmented linear prediction model with estimated states by a Kalman filter is capable of capturing the more complex particle transport dynamics as modeled by BgB and QuaLiKiz neural-network for multiple operating points.
The MPC controller based on this prediction model is capable of regulating the density profile within 5% of the desired reference for multiple operating points and different anomalous transport models (BgB and QuaLiKiz neural-network), given that the reference is feasible.If a certain reference is not feasible, the controller will drive the profile to a close profile while trying to minimize the error.Our results highlight that the limited actuation space provided by pellet injection mean that the shape of the controlled profile depends strongly on the transport.The designed controller does not allow for an arbitrary density profile to be reached.This means that reference design must account for the transport.We note here that the controller could provide the information of the infeasible reference to a overarching supervisory controller for example.
We observe in these simulations that from a control perspective, particle transport in the flat-top of a tokamak reactor can be considered as a slow varying in the regimes where the assumptions are valid.This means that mismatches in transport modeling can be dealt with using dedicated control tools.This implies that simplified control-oriented models can be used to derive efficient controllers as demonstrated here.It needs to be verified if this holds for cases where the made assumptions are not valid.
Furthermore, we present an analysis where we investigate the impact of using multiple discrete pellet injectors in feedback while considering them as continuous actuators from the controller side.We show that the coupling between the pellet injectors has a detrimental effect on the control performance.Furthermore, it is observed that, for the large pellets required on future reactors, the large fluctuations intrinsic to the pellets cause violation of the constraints if the discrete nature is not accounted for.We conclude that for future reactor, where multiple large pellet sources will be used and the plasma will have to be operated close to limits, the approach of considering pellet injection as a continuous actuator from the controller's perspective will be insufficient as satisfying the constraints cannot be guaranteed.
Therefore, to achieve high-performance and reliable control of the density in ITER and DEMO, it is required to use nonlinear control strategies capable of explicitly handling the discrete pellet dynamics.Prime candidate control methods are hybrid control, hybrid MPC, or nonlinear MPC.

Figure 1 .
Figure 1.Time resolved RRMS error (7) for the integrated model control simulations.

Figure 2 .
Figure 2. Controlled and reference profiles for the low density reference in (a) and high density reference in (b).

Figure 3 .
Figure 3. Left column: results from the integrated model simulation with a low reference density profile and the BgB transport model.Right column: results from the integrated model simulation with an ITER baseline reference density profile and the QLKNN transport model.(a) and (b) Control error in the integrated model simulation; (c)-(d) contour plot of the control error in percentage; (e)-(f ) estimated disturbance states d(t k ).The cyan dotted lines in (a) denote the time instances where the profile is plotted in figure4(c).The relation between the estimated disturbances states d(t k ) and the control performance can clearly be observed.For the BgB simulation, between t = 400 and t = 404 s, the estimates of the disturbance states d(t k ) are varying rapidly, indicating a discrepancy between the dynamics in the controller and the JETTO.This leads to the increase in control error seen during that time.From t = 404 s onward, the estimated disturbance states barely changes, indicating a good match between the plant-controller dynamics.A reduction in the control error is observable in this phase.

Figure 4 .
Figure 4. (a) Fueling rate of the first pellet injector (centered around ρ = 0.85); (b) fueling rate of the second pellet injector (centered around ρ = 0.7); and (c) controlled density profiles compared with the reference density profile (dotted black line).The times at which the profiles are portrayed are indicated by the cyan lines in figure 3(a).

Figure 5 .
Figure 5. Results of control simulations with pellets containing 6 × 10 21 electrons.(a) Discrete pellet control inputs; (b) Fueling rate resulting from discrete inputs calculated as Γp(t) ⟨1⟩ from [7].The fueling rate in the continuous case (without pellet model) is depicted with the dotted line; (c) Relative root mean square error with respect to the reference profile; (d) The line-averaged density compared with the Greenwald density limit (red);.

Figure 6 .
Figure 6.Results of control simulations with pellets containing 6 × 10 21 and 2.3 × 10 21 electrons.(a) Discrete pellet control inputs; (b) Fueling rate resulting from discrete inputs calculated as Γp(t) ⟨1⟩ from [7].The fueling rate in the continuous case (without pellet model) is depicted with the dotted line; (c) Relative root mean square error with respect to the reference profile; (d) The line-averaged density compared with the Greenwald density limit (red);.

Figure A2 .
Figure A2.Results from integrated model simulations with the lITER baseline density reference and BgB transport model.(a) Control error in the integrated model simulation; (b) contour plot of the control error in percentage; (c) estimated disturbance states d(t k ).

Figure A3 .
Figure A3.Results from integrated model simulations with a density reference designed using QLKNN.The QLKNN transport model is used in JINTRAC.(a) Control error in the integrated model simulation; (b) contour plot of the control error in percentage; (c) estimated disturbance states d(t k ).

Figure A4 .
Figure A4.Results from integrated model simulations with a density reference designed using QLKNN.The Bohm/gryo-Bohm transport model is used in JINTRAC.(a) Control error in the integrated model simulation; (b) contour plot of the control error in percentage; (c) estimated disturbance states d(t k ).