Energy-OptimalCavityFillingOlof Troeng, Bo Bernhardsson

Department of Automatic Control, Lund University, Sweden{oloft,bob}@control.lth.se

Anders J JohanssonDept. of Electrical and Information Tech., Lund University, Sweden

anders_j.johansson@eit.lth.se

OverviewPulsed, high-power accelerators consume significant amounts of power to fillthe cavities, power which does not contribute to particle acceleration. A recentpaper [1] considered how to minimize the reflected power during filling, whichessentially corresponds to minimizing the energy consumption of an ideal ampli-fier. We demonstrate how to find optimal filling profiles for arbitrary efficiencycharacteristics, and also in presence of detuning (assumed to be repetitive andknown). The results on this poster has been previous published in [2].

Example: European Spallation SourceESS Parameters

Beam Current 62.5 APulse length 2.86 msPulse rate 14 HzAverage Beam Power 5 MWFinal Energy 2 GeV# SRF Cavities 146Construction cost ≈2Be

Pulse structure:

Cavityfield [-]

71 msTime

Cavitydrive [-]

RF amplifier outputBeam current

Filling Flat-top(beam acceleration)

2.86 ms≈ 0.3 ms

I Total electricity cost for filling the 84 high-β cavities powered by inductiveoutput tubes (IOTs): 100 ke/year

Optimal Control Problem (Amplitude)Cavity dynamics during filling (no beam), normalized wrt time and amplitude:

V̇ (t) = −V (t) + Ig(t)

Optimal control problem:

minimizeIg, tf

∫ tf

0Pamp(|Ig|) dt

subject to V̇ (t) = −V (t) + Ig(t)|Ig(t)| ≤ Imax

g

V (0) = 0V (tf ) = 1

1Filling

Cavityfield, V [-]

tf

Imaxg

Time [s]

Ig [-]

Energy-optimal control signal:

I?g (t) = argmaxIg

−V (t) + IgPamp(Ig)

Typical efficiency characteristics

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

Relative Output Power [-]

Efficiency,|Ig|2

Pamp(|Ig|)

TetrodeSSAInductive Output TubehejConst. Efficiency(ideal amplifier)Const. Power Draw(e.g., klystron)

Results

Tetrode DohertyArch. SSA

IOT ConstantEfficiency

⇒ Savings for the high-βsection of ESS: 12 ke/year

Filling

Ene

rgy[a.u.]

Minimum Time (standard)Minimum Reflection [1]Minimum Energy (this contribution, [2])

Optimal filling profiles:

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.5

1V [-]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

1

2

Normalized Time [-]

Ig [-]

Min. TimeMin. Reflection [1]Energy Opt. (Tetrode)Energy Opt. (SSA)Energy Opt. (IOT)

Accounting for detuningAccounting for detuning gives complex-valued cavity dynamics:

minimizeIg,tf

∫ tf

0Pamp(|Ig|) dt

subject to d

dtV = (−1 + i∆ω(t))V + Ig

|Ig| ≤ Imaxg

V(0) = 0V(tf ) = 1.

Can decouple problem by introducing polar coordinates, V(t) = V (t)eiφ(t) andIg(t) = Ig(t)eiθ(t)

Optimal angle for control signal:

θ∗(t) = −∫ tf

t

∆ω(t) dt

RemarksI Due to the longer fill time, the cryogenic load is increased. For ESS: ≈

1–2 ke/yearI The proposed filling approach requires good knowledge of the system

parameters, and that the detuning ∆ω(t) is known and repetitive (lowgain feedback would reduce the impact of microphonics)

Bibliography[1] Bhattacharyya, A.K., Ziemann, V., Ruber, R., and Goryashko, V. (2015).

“Minimization of power consumption during charging of superconductingaccelerating cavities”. Nuclear Instruments and Methods in Physics Re-search A, 801, 78–85.

[2] OT, BoB (2017). “Energy-Optimal Excitation of Radio-Frequency Cavities”.Proceedings of the IFAC World Congress 2017.