Optimizing control for strike point sweeping using lower divertor coil in EAST

In 2021, EAST was equipped with a full-ring divertor coil to facilitate research on the fish tail divertor concept. Initially, it was observed that the coil current had a negligible ability to sweep the strike point. Conversely, when the amplitude and frequency of the alternating current were marginally increased, there was a significant interruption to plasma control. This perturbation was attributed to the poloidal control field’s limited response rate to the coil’s fluctuations. To address this issue, novel control methodologies were devised to ensure stable and effective sweeping of the strike point using the divertor coil. The devised methods are twofold: For high-frequency strike point control, a low-pass filter decoupling technique based on ISOFLUX control strategy enabled achieving a sweeping frequency of 100 Hz. This strategy allowed for consistent plasma management without compromising average stored energy or density regulation. Resulting from this proficient manipulation of the strike point, a reduction in the peak temperature of the divertor plate was observed. For low-frequency sweeping, a static multi-input multi-output decoupling approach was developed, facilitating concurrent sweeping of both the outer and inner strike points.


Introduction
Strike point sweeping is an important way to protect divertor target plate from high heat flux damage [1][2][3][4].The deposition width of the peak of heat flux density is spread evenly.The contact time of the heat flux to the target is reduced (proportional to expansion rate).Strike point sweeping is effective especially for mitigation narrow heat flux damage.Increasing Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. the sweeping frequency is particularly beneficial, as it has minimal impact on divertor tile longevity [4] and is instrumental in mitigating heat flux damage from edge-localized modes (ELMs) [5].
The fish tail divertor (FTD) concept [6,7] represents an innovative approach in dynamic magnetic configuration divertors, aimed at enhancing the high heat flux tolerance of divertor targets through strike point sweeping approach, whilst simultaneously sustaining optimal core plasma performance.This strategy utilizes a coil strategically placed in proximity to the strike point but at a distance from the X point, functioning to produce a localized, alternating poloidal magnetic field in the vicinity of the strike point.The application of this targeted magnetic field enables active manipulation of the strike point, facilitating its oscillatory motion along the divertor plate.This movement serves to periodically displace the location of heat flux peak, broadening the area of heat flux deposition in a dynamic manner.In the realm of FTD technology, both the amplitude and frequency of the coil current can be adjusted over a broad spectrum, allowing for flexible control.The FTD harnesses the full potential of divertor cooling capabilities.
Due to constraints imposed by the water-cooling system [8], the placement of the divertor coil is constrained to a locale relatively distant from the strike point.Consequently, the coil exerts a minimal influence on sweeping the strike point, yet poses a significant risk of interrupting plasma stability.To facilitate effective FTD research utilizing this particular coil configuration, it is imperative to pioneer novel control methodologies.
The paper is organized as follows.In section 2, the divertor coil system is introduced.In section 3, interruption of the coil to plasma control and effects of the coil on sweeping strike point are analyzed.In section 4, optimization control for high frequency strike point sweeping is presented.In section 5, optimization control for low frequency strike point sweeping is presented.Finally in section 6, summary of the work is given.

The divertor coil system
The divertor coil is positioned inside the vacuum vessel beneath the low divertor dome and has a full toroidal ring structure with a single turn.Its design is similar to the vertical stabilizing coils used in EAST and ITER.The coil consists of a water-cooled hollow copper conductor, which is insulated by magnesium oxide (MgO) powder and protected by stainless steel armor.The power supply for the coil is composed of 12 sub-units, which can collectively deliver a maximum output power of 1.2 MW.The power supply is capable of supporting sinusoidal and triangular alternating current (AC) operation, as well as direct current (DC) operation modes.
In AC operating modes, the coil can generate current frequencies (f FTD ) ranging from 1 to 100 Hz, with an amplitude (I FTD ) below 20 kA.When the amplitude exceeds 20 kA, the frequency range is limited to 5-10 Hz.For AC operation, the single run duration is ⩽20 s for I FTD < 20 kA, and ⩽2 s for I FTD ⩾ 20 kA.The system incorporates various diagnostics to ensure safe operation, including monitoring of coil current, voltage, cooling water temperature, coil oscillation amplitude and frequency, as well as the video and audio of the site.These parameters can be monitored in real-time and offline remotely.Additionally, the coil's contribution is considered in the realtime experimental equilibrium reconstruction using pEFIT [9].The system is controlled by the PCS [10].Figure 1 provides an illustration of the coil, and table 1 summarizes its main parameters and those of the power supply.

The problems
During the 2021 campaign, the divertor coil system was commissioned with plasma for the first time.Examination of coil current amplitude and frequency demonstrated that the  coil significantly disrupted plasma control.The stable maximum operating parameters were determined to be 2 kA and 10 Hz, which had negligible effects on strike point sweeping.However, increasing the coil current amplitude or frequency resulted in large-amplitude, low-frequency oscillations in the main plasma that intensified over time.Eventually, these oscillations compromised vertical stabilization, leading to a vertical displacement disruption.
Analyses to determine the cause of the interruption was performed.Firstly, the poloidal magnetic flux perturbations (δΨ DivC ) from the divertor coil to the plasma shape control points were compared with the background poloidal magnetic flux variance within one centimeter (∇Ψ ), as shown in figure 2(b).Among the control points, the low X point (Xpt) exhibited a much larger δΨ DivC than ∇Ψ , since it is closest to the coil and its flux gradient is zero.For the control points that constrain the plasma main body (cnti), the flux perturbations are mild.The DivC current itself has a significant effect on perturbing Xpt, but has a small influence on the shape of the plasma main body.
The low-frequency oscillations are induced by the dynamic control of superconducting poloidal field coils (PFs).In the EAST tokamak, plasma regulation leverages the ISOFLUX algorithm, referenced in [11], with its logic outlined in figure 3. Fast changing of the plasma's vertical position (Zp) is controlled by a pair of anti-serially connected in-vessel coils (ICs), characterized by a swift ∼1 ms response time [12].The overall control of plasma current and shape, however, is managed by the PFs.Control signals, in the form of demanded coil voltages, are instantaneously transmitted to the coil power supply system upon detection of control errors.Yet, the poloidal field effectuated by the PFs manifests with a ∼40 ms delay in impacting the plasma following the command.This postponement is largely due to the time needed for the field to penetrate the vacuum vessel wall and the response time of the PF power supply.
This convention stipulates that the PF coils' current may ramp at a peak rate of 20 kA s −1 for the fleeting span of 100 ms, progressively declining for extended intervals.
Such delays can unintentionally exacerbate control errors during the management of swiftly alternating disturbances.The capped ramp rates pose a risk of rendering control responses insufficient for promptly evolving errors.Collectively, this lag in response suggests that the PFs can amplify control errors when tasked with counteracting rapid, alternating interruptions through feedback control.
Another problem is that the divertor coil alone proves insufficient for effectively sweeping the strike point.To evaluate the impact of the divertor coil current on the lower outer divertor strike point and the lower X point, static plasma equilibria are computed using the tokamak simulation code (TSC) [13].These computations maintain the position of the plasma's magnetic axis using feedback control from PF13 and PF14 coils.Comparisons between plasma boundaries without and with DivC current of ±5 kA (refer to figure 4) demonstrate that the X point position is influenced more than the strike point by the divertor coil current, shifting by a factor of approximately 1.7, significant movement of the strike point is not achievable when the X point position remains static.
The awkward situation is summarized.The DivC primarily influences the lower X point and the PFs are incapable of counteracting rapid variations in the perturbations.Moreover, significant shifting of the strike point is unattainable with a stationary X point solely using DivC actuation.Fortunately, the current within the DivC has a minimal effect on the main body of the plasma.With these in mind, development of control methods for strike point sweeping using the divertor coil have been carried out.The devised control strategy consists of a dualapproach.For high-frequency strike point sweeping (beyond the response bandwidth of the PFs), a low-pass filter is implemented to isolate the DivC's high-frequency perturbations from the PFs' response, thereby preventing unwanted interactions.For low-frequency actuation (within the PFs' response capabilities), a Multi-Input Multi-Output (MIMO) decouple method harmonizing the actions of the PFs and the DivC is applied to control the plasma shape evolution.

High frequency strike point sweeping
Based on the EAST ISOFLUX framework, the time constants of the low pass filter for plasma current and shape control are increased to 10 ms.The control error signals that have been filtered include: the radial and vertical position errors of both the lower and upper X-points (thus relaxing the constraints on X-point positioning at higher frequencies), errors in the poloidal magnetic flux at the six designated control points that define the periphery of the plasma main body, deviations in the overall plasma current, and symmetry error.The fast vertical position control (Zp feedback) remains unaffected by these changes.Under these settings, a ±5 kA, 100 Hz sinusoidal FTD current is applied to sweep the strike point.
As shown in figure 5, the position of the outer strike point (R SP ) is being swept with a consistent amplitude of approximately 3.5 cm, whereas the inner strike point (Z SP ) experiences minimal sweeping.The discrepancy in sweeping amplitudes between R SP and Z SP can primarily be attributed to the varying distances from the respective strike points to the control coil, as well as the differences in the background magnetic flux gradients (∇Ψ) at the two locations.Concurrently, the radial and vertical positions of the lower X point (denoted as R Xpt and Z Xpt , respectively) exhibit oscillations in response to the DivC current, with an observed amplitude of around 2 cm.This behavior is primarily due to the relatively small ∇Ψ in the vicinity of the X point.
As depicted in figure 6, the periodic oscillation of the lower X point induces a corresponding periodic variation in the plasma shape, transitioning through a specific sequence within each cycle.The plasma shape evolves from an initial low single null configuration to a double null, then to an upper single null, back to a double null, and finally returns to the low single null state.This cyclical transformation of the plasma shape creates an opportunity for controlled distribution of particle and heat flux across the four divertor targets.It is important to note that this periodic shape variation is contingent upon the initial condition of the plasma being in a state that closely approximates a double null configuration.
The main body of the plasma is maintained with precision.Figure 7 illustrates the control accuracy for the poloidal magnetic flux, denoted by δΨ i , representing the discrepancy between the actual poloidal magnetic flux at control point i (Ψ cnti ) and that at the last closed flux surface (Ψ LCFS ).The peak-to-peak amplitude of rapid oscillations in the flux error at these points is confined to approximately 3 milliwebers per radian or less.Furthermore, the average values of these flux control errors over time are well-regulated, indicating stable control performance.
The effects of rapid oscillations on critical plasma parameters are assessed and presented in figure 8.The plasma current (I p ) exhibits an oscillation with an amplitude of about ±2 kA (±5% of its mean value), which minimally impacts its mean value.The poloidal beta (β p ) displays oscillations with an amplitude of approximately ±0.03 (±12% of mean value).The internal inductance (l i ) shows oscillations with an amplitude of around ±0.075 (±4.8% of mean value).The safety factor at the 95% flux surface (q 95 ) has an oscillation amplitude of about ±0.15 (±2.4% of mean value).The average plasma elongation (κ) sees a reduction of approximately 0.04 (2.3% of initial value).The area of the poloidal cross section of the plasma (S) experiences an amplitude of oscillation of approximately ±0.015m 2 (±1.5% of mean value).The radial position of the magnetic axis (R mag ) fluctuates around its mean position with an amplitude of roughly ±0.4 cm (±0.2% of mean value).Lastly, the vertical position of the magnetic axis (Z mag ) has an amplitude of about ±0.2 cm.
The impacts of rapid oscillations on plasma stored energy, density regulation, and recycling processes are examined and depicted in figure 9.The stored energy (W MHD ) fluctuates with an amplitude of approximately ±4 kJ without any reduction in its average value.The D α trace indicates that recycling exhibits minor synchronous fluctuations with sweeping motions.Furthermore, these rapid oscillations appear to have a negligible effect on the line-averaged density (N e ).
The implications of strike point sweeping on the temperature of the divertor target plate, as recorded by an infrared camera, are presented in figures 10(a) and (b). Figure 10(a) compares the temporal profiles of the maximum temperature of the lower divertor's outer target plate with FTD current of ±1 kA at 100 Hz in shot 115 134 (represented by the black line) and ±5 kA at 100 Hz in shot 115 136 (indicated by the red line).Compared to shot 115 134, which has a minor influence and can be viewed as a reference, an increase in the sweeping amplitude of the strike point periodically reduces the maximum temperature of the divertor plate (Max.T plate ), which also lowers its time-averaged value.Figure 10(b) displays the spatial distribution and temporal evolution of the target plate's temperature, which demonstrates variations that correspond with the strike point sweeping cycles.The time evolution of the ion saturation current density (j s ) values along the lower outer divertor target plate is shown in figure 10(c).The j s oscillates synchronously with the coil current waveform shown by the black curve below the contour.The strike point location reconstructed from the EFIT is superimposed on the j s graph to illustrate the correlational relationship between them.

Design method
The objective of low-frequency strike point sweeping is to enhance the extent of sweep for both inner and outer strike points while maintaining control over the main plasma body.This is achieved through the coordinated operation of the divertor coil in conjunction with the PFs, collectively referred to as the COILs, to govern the evolution of the plasma shape.This includes managing both the central plasma configuration and the strike point positions.To accomplish this, a series of target plasma equilibria are statically prescribed using the TSC code, wherein the DivC coil and the PFs act in unison to mitigate disturbances to the main plasma shape and to broaden the sweep range of the strike points.The underlying approach to this design methodology is detailed as follows.
Within the framework of TSC, a set of nine control points is utilized to delineate the target plasma shapes: six points (cnt1, cnt3, cnt4, cnt6, cnt8, cnt9) govern the main plasma boundary-consistent with those used in experimental settings-while two points regulate the lower inner and outer strike points (lisp and losp), and one point determines the position of the lower X-point (Xpt).The goal is to iteratively adjust the current through both the DivC and the PFs to achieve these predefined target shapes.
During each iteration, an optimal current configuration for the COILs is computed using the singular value decomposition (SVD) least-squares method.This approach addresses the under-determined problem expressed as min {∥Ax − b∥ 2 }, where A represents a 10 × 13 response matrix that maps the relationship between the currents in the COILs and the positions of the control points, x is a 13 × 1 vector signifying the currents in the COILs, b is the 10 × 1 error vector of the control points, and ∥•∥ 2 symbolizes the Euclidean norm.
The error vector (b) is formulated as The low X point shift induced periodical plasma shape variation.
Here, δΨ cnt1 = (Ψ LCFS − Ψ cnt1 − Ψ 1_coil )_o is the control error in poloidal magnetic flux at control point 1, where Ψ LCFS is the poloidal flux at the last closed flux surface, Ψ cnt1 is the flux at control point 1, and Ψ 1_coil denotes the contribution to the poloidal flux at control point 1 from the COILs.The symbol '_o' signifies the measurement from the previous iteration step.Likewise, δΨ cnt3 through δΨ cnt9 , as well as δΨ lisp and δΨ losp , follow the same definition manner as δΨ cnt1 .The terms dΨ dr Xpt and dΨ dz Xpt represent the radial and vertical components of the gradient of poloidal flux at the targeted low X-point position, respectively.
The optimal coil current combinations are computed using the formula (x = VΣ −1 U T b), where (U), (V), and (Σ) are the SVD matrices resulting from the decomposition of the response matrix (A) into the product (A = UΣV T ).These coil currents are iteratively calculated and integrated into the plasma equilibrium computation process.This iterative process is repeated until the desired plasma shape is achieved, and the criteria for plasma equilibrium convergence are fulfilled, as outlined in [14].

Experimental settings and results
To perform the experiment, initial steps involve the reconstruction of plasma equilibria at 4 s, 5 s, 6 s, 7 s, and 8 s during a reference experimental shot with fixed strike point positions.Following this, adjustments are made to the strike point positions in the 5 s and 7 s equilibria while maintaining the other control points unchanged, resulting in two modified equilibria with newly calculated coil current combinations.
Subsequently, the PF coil currents are optimized such that they minimize both the absolute deviations of each coil's current from its reference experimental value and the cumulative deviation across all coils.This approach ensures ease of experimental implementation and minimizes any potential negative impacts on plasma current control.
After optimization, the determined COILs currents are utilized as feed forward targets during the experimental shot at 4 s, 5 s, 6 s, 7 s, and 8 s, with the strike point sweeping period starting at 4 s.The current targets between each consecutive pair of these time points are set to vary linearly.For time periods outside the 4-8 s interval, the reference shot's experimental PF currents are used as feed forward targets to reduce the demand on feedback control currents.
Throughout the duration of the shot, standard feedback control for plasma current, shape, and vertical position (Zp) remain operational.If necessary, lower hybrid wave (LHW) heating is employed to compensate for any increased voltsecond consumption that may arise due to less favorable discharge conditions.
Figure 11 illustrates that both the inner strike point (designated as Z SP ) and the outer strike point (R SP ) undergo a backand-forth movement or 'sweeping'.The amplitude of sweeping reaches up to 5.5 cm for Z SP and 4 cm for R SP , accompanied by an approximately ±10 kA FTD current depicted by the pink lines.However, it is observed that the strike point displacements do not align consistently with the FTD currents, particularly in the latter half of the time interval.This discrepancy can be attributed to several factors: 1.The discharge conditions during the experimental shots were less favorable compared to the reference shot, resulting in the need for increased volt-seconds to sustain the plasma current plateau.2. The efficacy of LHW heating in maintaining the plasma current diminishes significantly as more power is injected, leading to a reduced contribution from the LHW to the overall volt-second budget.3. Consequently, a larger portion of volt-seconds has to be compensated by the PF coils, which in turn causes the actual PF currents to deviate from their intended targets.
To address this issue, a new temporal evolutionary shape control strategy is under development, which integrates plasma current regulation by the COILs.This method will also account for the plasma conditions, potentially enhancing the congruence with targeted outcomes.The details of this new strategy and its results will be presented in future.

Summarys
This study presents an optimized control method for conducting strike point sweeping on the EAST tokamak by utilizing the divertor coil.Employing a low-pass filter decoupling technique, we have achieved 100 Hz frequency strike point sweeping within the ISOFLUX control framework.During the sweep, the low X point demonstrates an oscillation around its average position.Key plasma parameters are effectively preserved.These rapid oscillations are shown to be nondetrimental to the plasma's stored energy, recycling or lineaveraged density.The practice of sweeping the strike point cyclically reduces the peak temperature of the divertor target and subsequently lowers the time-averaged maximum temperature on the plate.Furthermore, a static MIMO decoupling method for low-frequency strike point sweeping has been initially developed, enabling the simultaneous sweeping of both inner and outer strike points with the divertor coil.
This advancement significantly broadens the utility of the divertor coil for sweeping strike point locations, a critical aspect of FTD research.Moreover, optimized control reduces the erstwhile necessity for maintaining a long separation distance between a single divertor coil and the main plasma to minimize interference with plasma control.This reduction in positional constraints enhances the practicality of employing a solitary divertor coil for strike point sweeping applications.

Figure 1 .
Figure 1.The divertor coil on EAST.

Figure 2 .
Figure 2. The EAST geometry in 2021 (a) and the poloidal magnetic flux perturbations (δΨ DivC ) of the DivC current to the plasma shape control points (b).cnti refers to a control point used to constrain the shape of plasma main body.PFs are the poloidal field coils.Xpt refers to the target position of low X point.∇Ψ is background poloidal flux variance in 1 centimeter.Ip = 400 kA is the plasma current.I FTD = 5 kA is the divertor coil current.

Figure 4 .
Figure 4. Influences of DivC current on lower outer strike point (Sp) and X point (Xp).

Figure 5 .
Figure 5. 100 Hz frequency strike point sweeping by FTD.I FTD is the DivC coil current.R SP and Z SP are the strike point positions on the lower divertor outer horizontal target plate and inner vertical target plate respectively.R Xpt and Z Xpt are radial and vertical positions of the low X point.

Figure 7 .
Figure 7.Control errors of poloidal magnetic flux at control points which constrain the plasma main body shape.

Figure 8 .
Figure 8.Time evolution of some key parameters.

Figure 9 .
Figure 9.The plasma stored energy, Dα emission and line averaged electron density (Ne).

Figure 10 .
Figure 10.FTD effects on the temperature and particle flux of the divertor target plate.(a) The maximum temperature.FTD currents are ±1 kA at 100 Hz in shot 115134 (represented by the black line) and ±5 kA at 100 Hz in shot 115136 (indicated by the red line).(b) Spatial-temporal evolution of plate temperature in shot 115136.The green line shows the strike point position variance.(c) The ion saturation current density (js) measured by Langmuir probes (LPs) for discharge 123780, by applying I FTD = ±6 kA at the frequency of 100 Hz for Ip = 400 kA.The strike point variation is shown by the pink line.

Figure 11 .
Figure 11.Low frequency strike point sweeping using DivC and PFs.Black lines denote values in reference shot.Pink lines denote values in the experiment with about ±10 kA FTD current.Cyan lines denote values in the experiment with about ±5 kA FTD current.The PF1 subplot shows the experimental value in reference shot (black) and designed targets in the experiments (pink and cyan).

Table 1 .
Main parameters of the divertor coil system.