Original Research
21 June 2023
10.3389/fphy.2023.1170175
TYPE
PUBLISHED
DOI
OPEN ACCESS
EDITED BY
Subrata Das,
National Institute for Interdisciplinary
Science and Technology (CSIR), India
REVIEWED BY
Alexander Scheinker,
Los Alamos National Laboratory (DOE),
United States
Igor Pogorelsky,
Brookhaven National Laboratory (DOE),
United States
Feedforward resonance control
for the European X-ray free
electron laser high duty cycle
upgrade
Andrea Bellandi*, Julien Branlard, Holger Schlarb and
Christian Schmidt
Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany
*CORRESPONDENCE
Andrea Bellandi,
andrea.bellandi@desy.de
20 February 2023
30 May 2023
PUBLISHED 21 June 2023
RECEIVED
ACCEPTED
CITATION
Bellandi A, Branlard J, Schlarb H and
Schmidt C (2023), Feedforward
resonance control for the European X-ray
free electron laser high duty
cycle upgrade.
Front. Phys. 11:1170175.
doi: 10.3389/fphy.2023.1170175
COPYRIGHT
© 2023 Bellandi, Branlard, Schlarb and
Schmidt. This is an open-access article
distributed under the terms of the
Creative Commons Attribution License
(CC BY). The use, distribution or
reproduction in other forums is
permitted, provided the original author(s)
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accepted academic practice. No use,
distribution or reproduction is permitted
which does not comply with these terms.
The High Duty Cycle (HDC) upgrade is a proposed improvement to the existing
European X-ray Free Electron Laser (EuXFEL) to extend the pulsed RF duty factor from
the actual value of around 1% to more than 5% up to Continuous Wave (CW). To
implement this upgrade, the loaded quality factor (QL) of the superconducting cavities
will increase by more than one order of magnitude. This will result in shrinking the
cavity bandwidth to values as low as a few Hertz. Since the Lorentz Force Detuning
(LFD) experienced during the accelerating field buildup is of hundreds of Hertz, the
Low-Level RF (LLRF) system has to accurately track and control the cavity resonance
frequency to obtain the desired accelerating gradient. Moreover, ponderomotive
instabilities have to be suppressed to achieve stability during beam acceleration. Since
LFD is a repetitive disturbance in cavity frequency, the correction to its effects can be
implemented as a feedforward compensation on the piezoelectric tuners of the
cavity. Initial results on the simulation of feedforward resonance control in the HDC
regime are discussed in this proceeding.
KEYWORDS
particle accelerators, LLRF, superconducting RF, free electron laser, control systems
1 Introduction
The European X-ray Free Electron Laser (EuXFEL) is a hard X-ray FEL based on
superconducting TESLA-type cavities operating in pulsed mode [1, 2]. The proposed High
Duty Cycle (HDC) upgrade would make the operation of the machine in Continuous Wave
(CW) mode possible, with a final beam energy of 8 GeV [3]. Due to the requirements of the
experimental community, the EuXFEL HDC upgrade will also enable a Long Pulse (LP) mode of
operation. In such a mode of operation, the experiments will benefit from beam energies higher
than 10 GeV and, at the same time, produce a number of bunches per second comparable to
machines operated in CW [4]. The LP mode is realized by extending the duty factor of the
EuXFEL pulse flattop from 0.65% to 5–50%. For this mode of operation, it is planned to use the
same RF amplifiers as for the CW mode and not increase the cryogenic heat load compared to the
continuous case. The use of either a CW or LP mode on EuXFEL will be scheduled depending on
the requirements of the experimental campaign. From the Low-Level RF (LLRF) perspective, the
most significant change is the requirement to drive the RF superconducting accelerating cavities
with a loaded quality factor (QL) of 6 · 107, one order of magnitude higher than the current value
of 4.6 · 106. This change is required to lower the cavity RF power consumption to the kilowatt level
[5, 6]. Such a change will result in an RF half bandwidth (f1/2) of 10.8 Hz. Due to this change, the
superconducting cavities will be an order of magnitude more sensitive to mechanical
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Bellandi et al.
10.3389/fphy.2023.1170175
FIGURE 1
Simulation of the ratio between the target and achieved Vacc without any active detuning compensation. The effect of LFD-generated
ponderomotive instabilities increases at higher gradients. The cavity parameters are taken from Czarski et al. [7].
In this paper, the mechanical system model used in the simulations
is given in Section 2. A feedforward control approach is presented in
Section 3. Additionally, simulations using the Iterative Learning Control
(ILC) algorithm are presented in Section 4 [13]. Such an algorithm is
already in use in particle accelerators for RF and beam trasient
compensation [14–17]. The advantage of using ILC is that repetitive
un-modeled detuning disturbances can be rejected in an adaptive way.
A challenge of using ILC for detuning compensation in high QL cavities
is the presence of strong nonlinearities and hysteretic behaviour that
might worsen the convergence properties of the algorithm. Therefore, in
this paper, some simple modifications on the ILC algorithm were
studied to limit the effect of ponderomotive instabilities. The final
considerations are given in Section 5.
disturbances. These disturbances originate from external
(microphonics) and internal Lorentz Force Detuning (LFD)
vibrations. A particular concern is the LFD-generated
ponderomotive instabilities when driving the resonators at their
maximum foreseen accelerating voltage (Vacc) of 20 MV.
Considering a static LFD value klfd normalized to the active
electrical length of the TESLA cavity (1.038 m) of up to 1.5HzMV−2,
the ratio between the LFD and the cavity half bandwidth will be.
−
klfd V2acc
55.3.
f1/2
(1)
k
V2
As explained in Schulze [8], such a value for lfdf1/2acc would make
the accelerating system affected by the monotonic instability. Such
an effect would prevent the RF accelerating field build up inside the
cavity as shown in Figure 1 [9].
Therefore it is of extreme importance to first develop a detuningcorrecting method to drive the field of the cavities at a level where
closed-loop operation is possible and, at the same time, not do not
exceed the RF power budget of (6 kW) foreseen for the HDC
upgrade. Once such a condition is realized, the residual field
error will be then regulated by the RF feedback controller which
operates at faster timescales compared to typical mechanical
disturbances. Such scheme is necessary to realize a field
regulation of 0.01% on amplitude and 0.01 deg on phase required
by the EuXFEL experiments [10]. The detuning control method has
to be implemented in a reliable and automated way due to its
criticality in the correct operation of the accelerator. For this, the
LLRF system must control the piezoelectric cavity tuners to
compensate for LFD-generated effects [10, 11]. To simulate the
control algorithms, an accurate mechanical model of the
accelerating cavity is needed. Previous work already proved the
feasibility of implementing an LFD-compensating scheme for
gradients up to 14MVm−1 [12].
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2 System model
The dynamics of the detuning in superconducting cavities can
be described as the result of the excitation of a series of second-order
mechanical resonances. These mechanical resonances can be either
excited by the radiation pressure by the accelerating field or by
external microphonics disturbances
_ (μ) A(μ) Δf (μ) + B(μ) u(t) + E(μ) V2 (t) + F(μ) ](μ) (t),
Δf
acc
0
1
μ)
(
⎢
⎡
⎤⎥⎥⎥
Δf (t) ⎤⎦
⎢
⎢
μ)
(μ)
(
⎣
⎢
⎡
⎢
, A ⎣
Δf
2
ω(μ) ⎥⎦⎥⎥,
⎢
(μ)
μ)
(
_
−
−
ω
Δf (t)
Q(μ)
0
0
B(μ) (μ) (μ) 2 , E(μ) (μ) (μ) 2 ,
δ ω
klfd ω
0 2
F(μ)
,
ω(μ)
dΔf (μ)
_
Δf (μ)
.
dt
02
(2)
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Bellandi et al.
10.3389/fphy.2023.1170175
Where μ is the mechanical mode index, Δf(μ) is the detuning
contribution of the mode, ω(μ), Q(μ) and δ(μ) are the angular resonance
frequency, quality factor and coupling with the piezoelectric actuator. u(t)
is the piezoelectric input variable, controllable by the LLRF system, while
](μ)(t) is the microphonic noise. ](μ)(t) might be not synchronized with the
accelerator timing. In general, an effective resonance control system
requires driving u(t) so the detuning Root Mean Square (RMS) is smaller
than the half bandwidth
⎛
ΔfRMS ⎝
μ
⎞
Δf(μ) ⎠
< f1/2 ,
Figure 2 shows the shape of the feedforward piezoelectric tuner
drive of Eq. 5.
Figure 3 shows a simulation of an accelerating cavity when the
zero-order feedforward correction is used. The LFD effects are
significantly reduced compared to Figure 1 even though the
dynamics of the cavity is not taken into account in the
feedforward signal compitation. The sharp transition between the
different pulse regions produces the residual excitation of the
mechanical resonances.
(3)
RMS
4 Iterative learning control
To minimize the RF power required to drive the cavity at a
certain accelerator voltage [5].
In this work, we ignore the problem of rejecting unsynchronous
detuning originating from ](μ)(t) since the repetitive LFD-generated
disturbance is around two orders of magnitude higher [18]. The
compensation of the unsynchronized part of ](μ)(t) will be addressed
in future studies.
The correction of the residual mechanical oscillations resulting
from the use of Eq. 5 and unmodeled detuning disturbances in
TESLA-like cavities is challenging using white-box approaches [21].
Therefore, an adaptive gray-box ILC method is used. The idea is that
future RF pulses can be corrected adaptively using the detuning
information acquired in past pulses. An ILC-like method is already
implemented at EuXFEL [22]. However, the difference in pulse
length and QL with respect to the HDC upgrade makes it unsuitable
in LP mode due to the absence of strong nonlinear ponderomotive
effects. For the simulation tests in LP mode, a modified discrete
Arimoto-like update law
3 Feedforward control
During the nominal operation of EuXFEL HDC, the accelerating
field Vacc(t) will be required to have a well-defined periodic structure
repeated in time with a repetition rate of 1 Hz. Such a pulse will have
three distinct regions. These are the filling, flattop, and decay regions
[19]. In the HDC upgrade, the filling region will last 20 ms. The
flattop region length will be between 50 ms and 500 ms depending
on the resulting dynamic heat load and cryogenic capacity [20]. The
reference trajectory Vref(t) is then defined
⎪
⎧
⎪
⎪
⎪
⎪
⎪
⎨
Vref (t) ⎪
⎪
⎪
⎪
⎪
⎪
⎩
Vflat
1 − e−2πf1/2 (t−tfill )
1 − e−2πf1/2 (tflat −tfill )
Vflat
−2πf1/2 (t−tdecay )
Vflat e
for
tflat > t ≥ tfill
for
tdecay > t ≥ tflat
for
t ≥ tdecay
∞
Δuk+1 (n)
q−∞
filling
flattop
decay
d argn max |h(n)|,
Where tfill, tflat, tdecay are the starting times for the filling, flattop,
and decay regions, Vflat is the desired accelerating voltage at
flattop. Since Eq. 4 describes the desired field, it can be inserted
in Eq. 2 to find the radiation pressure-induced detuning excitation
and, consequently, the function for u(t) that achieves a perfect
disturbance compensation. To do this ω(μ), Q(μ) and δ(μ) have to
be determined. Alternatively, a transfer function approach can be
used by measuring the step response to V2acc variations. However, the
characterization of cavity mechanical parameters is still matter of
study and the presence of high quality factor mechanical resonances
complicates the identification process. A more simplistic approach is
to use a zero-order approximation of Eq. 2, by setting the time
derivatives to zero. Then it is possible to solve the equations the
actuator drive u0(t)
klfd V2ref (t)
.
δ (μ)
(7)
with h(n) the impulse response of Eq. 2. For the set of
parameters used in the simulations d = 3. The controller gain
γ is empirically found to be ≃ − 0.15 by looking at the value that
gives the fastest algorithm convergence. Q(q) is a second order
lowpass filter with a bandwidth of 1 kHz synthetized and applied
using the routines butter and sosfiltfilt of the package SciPy [23].
Q(q) prevents the growth of high-frequency noise at increasing k.
A benefit of the method of Eq. 6 is the low computational
complexity. A mechanical model-aware ILC might give better
results in terms of disturbance rejection. However it would
require further theoretical study. The reduction in detuning
resulting from the application of Eq. 6 is simulated for a
cavity driven at 20 MV for the following cases.
(5)
•
•
•
•
μ
Eq. 5 can be used in the simulations to see if it is sufficient to
prevent the occurrence of ponderomotive instabilities.
Frontiers in Physics
(6)
Is chosen. In the above formulation, the classical Arimoto
algorithm was changed to take into account the time-delay in the
step response that originate from the high quality factor mechanical
resonances. The adaptive correction component Δuk(n) is summed
with Eq. 5 to give the piezo driving signal for a pulse identified with
k. n is the sample within the pulse. The simulations use a sample rate
of 5 kHz. Such a sample rate is roughly ten times the highest
resonance frequency of the set of parameters used in the
simulations [7]. d is the sample delay and is chosen by
(4)
u0 (t)
QqΔuk n − q + γΔfk n + d − q,
03
Static detuning of 0 Hz (tuned)
Static detuning of −5.4 Hz (−f1/2/2) (under-tuned)
Static detuning of 5.4 Hz (f1/2/2) (over-tuned)
Gaussian microphonics with a standard deviation of 0.5 Hz
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Bellandi et al.
10.3389/fphy.2023.1170175
FIGURE 2
Accelerating voltage pulse structure at 20 MV and piezo feedforward drive generated using Eq. 5. Additionally, the learning thresholds for the clipped
ILC algorithm are shown. The ILC correction magnitude is constrained to be within ± 3% the feedforward value at flattop.
FIGURE 3
Simulation of the accelerating voltage error (A) and detuning (B) when the zero-order piezo feedforward is applied. The cavity system is simulated
with an open RF loop. The simulation shows that in the flattop, an amplitude error of 200 V over an accelerating field of 20 MV. δ(μ) = 0.33 Hz. The residual
detuning oscillation is ±2 Hz.
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10.3389/fphy.2023.1170175
FIGURE 4
Detuning RMS in the filling and flattop region as a function of the pulse number for different types of detuning disturbances. The system is adapted
using the ILC algorithm of Eq. 6.
∞
• Gaussian microphonics with a standard deviation of 0.5 Hz
plus a sinuisoidal component with a frequency of 50 Hz and an
amplitude 5 Hz
Δuk+1 (n)
q−∞
(8)
Where
The detuning disturbances of the above list are chosen to be
similar to the ones observed in working particle accelerators and test
facilities [18].
Figure 4 shows the convergence of the modified Arimoto ILC. For
the tuned and under-tuned case, the detuning converges monotonically
to values of 0.02 − 0.03 Hz RMS. The uncorrelated microphonics
contributions dominate the detuning RMS for the trace affected by
gaussian noise. Therefore, in this case, the ILC cannot decrease the pulse
detuning noticeably. Still, the detuning remains in the order of 1 Hz
RMS, fulfilling the requirement of Eq. 3. The most interesting cases are
the over-tuned simulation and the one affected by sinusoidal
microphonics. The initial detuning RMS is over 300 Hz and stays
constant at a level of 40 Hz for both cases between k = 25 and k = 100.
Then the detuning RMS starts decreasing and reaches a value of 1 Hz at
k = 120. The explanation for this behaviour is that the simulated cavity,
when over-tuned due to static detuning or microphonics, experiences a
static drop [8] that detunes positively the cavity by several hundred
Hertz. Therefore the ILC learns an invalid correction for the trace points
affected by the static drop.
After the ILC under-tunes the cavity enough to prevent the static
drop from happening, it has to ’unlearn’ the invalid part of the
correction for the RMS detuning value to decrease again (Figure 5).
At a repetition rate of 1 Hz, the tuning might reach acceptable RMS
levels only after 2 min. This delay would complicate the machine
setup operations because the operators would need to wait until the
algorithm converges to continue the accelerator commissioning. A
possible improvement is to modify Eq. 6 to constrain the values in
the ILC correction table within a pre-determined range
Frontiers in Physics
Qqclip−Δulearn , Δulearn , Δuk n − q + γΔfk n + d − q,
⎪
⎧
⎨ a for a > x
clip(a, b, x) ⎪ x for b > x ≥ a
⎩
b for x ≥ b
.
(9)
Δulearn determines the maximum learning range for the ILC
algorithm. Therefore, even in the case of a static drop, the controller
does not learn an excessive tuning correction. From the simulations,
Δulearn = 10 is enough for the final correction to be represented in the
tables. Using Eq. 8, a much faster convergence is achieved for the
over-tuned case and the one affected by sinusoidal microphonics.
For pulses with k > 3, a detuning ≈ 1 Hz RMS is achieved for both
cases. No significant differences to the simulations performed with
the original ILC algorithm were found for the other cases.
5 Final remarks
This article presented a study of a possible feedforward detuning
compensation scheme for superconducting cavities. The simulation of
the compensation of the ponderomotive instabilities in TESLA cavities
driven at Vacc = 20 MV and QL = 6 · 107 was possible by using a simple
zero-order approximation. Additionally, the use of an Arimoto-like ILC
algorithm allowed the compensation of the disturbances generated by the
residual un-modeled mechanical dynamics in the presence of
microphonics and ponderomotive instabilities. Clipping the values of
the ILC controller not only allowed reaching an acceptable detuning root
mean square values after just 3 s of operation but limited the amplitude of
05
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10.3389/fphy.2023.1170175
FIGURE 5
Detuning (A) and ILC table (B) at different values of k for the first version of the ILC algorithm. The simulation is performed with a static predetuning of
5.4 Hz. The evaluated k corresponds to the ones highlighted in Figure 4. As can be seen in plot (B), due to the presence of the monotonic instability, the
ILC algorithm learns initially an incorrect state. Such an incorrect state requires around 125 ILC iteration to be cleared.
Author contributions
the piezo driving signal within ± 3% the value already provided by the
static feedforward correction of Eq. 5. Since real piezoelectric tuners can
be damaged when driven at high voltage values, such a technique could
help increase the lifespan of these actuators. The approach of limiting or
stopping the correction learning of adaptive controller when an
anomalous condition is met is already in used at European XFEL for
RF. These results show for the first time the feasibility of performing
resonance control in high-duty cycle pulsed mode superconducting
accelerators even when the cavity bandwidth is of few Hertz. Further
research will concentrate on using a more accurate model-aware
feedforward compensation scheme to minimize the mechanical
excitation between pulse region transitions. Also, the ILC will need a
more sophisticate exception handling mechanism to avoid learning
invalid corrections during other anomalous conditions, like cavity
quenches or amplifier trips, that might happen during operations.
The presented compensation methods will be experimentally tested in
the forthcoming year at DESY module superconducting test facilities.
AB contributed to the idea, design, and simulation of the
feedforward and ILC control scheme for resonance control in
long pulse mode of operation. JB and HS contributed to the
requirements of the resonance controller for the European XFEL
HDC upgrade. CS contributed with consulting on the use of the ILC
technique. All authors contributed to the article and approved the
submitted version.
Funding
This work is supported by the Helmholtz institute and by
European XFEL GmbH.
Conflict of interest
Data availability statement
The authors declare that the research was conducted in the
absence of any commercial or financial relationships that could be
construed as a potential conflict of interest.
The raw data supporting the conclusion of this article will be
made available by the authors, without undue reservation.
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10.3389/fphy.2023.1170175
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All claims expressed in this article are solely those of the authors
and do not necessarily represent those of their affiliated
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