The COoler SYnchrotron in Jülich accelerates and stores unpolarized and polarized proton or deuteron beams in the momentum range between 0.3 GeV/c and 3.65 GeV/c [*,**]. This, in combination with its diverse capabilities of phase space cooling and the flexibility of the lattice with respect to ion-optical settings makes COSY an ideal test facility for accelerator technology development. High demands on beam control and beam based measurements have to be fulfilled for future experiments such as the proposed precursor experiment for a direct measurement of the electric dipole moment of the deuteron (see [***] and references within). The analysis of measured orbit response matrices (ORM), which com- prise the focussing structure of the ring, allows for a better understand- ing of machine imperfections such as gradient errors and misalignments of quadrupole magnets. This contribution presents the development of a MAD-X based LOCO (Linear Optics from Closed Orbits) algorithm [****] in a C++ program aiming to calibrate and correct linear optics as well as improving beam control at COSY

Bai, Mei

Hinder, Fabian

Lorentz, Bernd

Weidemann, Christian

Juelich Shared Electronic Resources

MODEL DRIVEN MACHINE IMPROVEMENT OF COSY BASED ONORM DATAC. Weidemann∗, M. Bai, F. Hinder, and B. LorentzIKP, Forschungszentrum Jülich GmbH, Jülich, GermanyAbstractIn order to improve the existing model of the COolerSYnchrotron (COSY) Jülich a technique called Linear Op-tics from Closed Orbit (LOCO) [1], originally used at lightsources, is favored. For the use at COSY, a hadron stor-age ring, the LOCO algorithm was implemented in a newlydeveloped C++ program, which uses MAD-X1 for optics cal-culation and ROOT2 for data analysis and illustration. Firstresults of the benchmarking of the program with simulateddata are discussed.INTRODUCTIONCOSY’s diverse capabilities of phase space cooling andthe flexibility of its lattice with respect to ion-optical set-tings makes it an ideal test facility for accelerator technologydevelopments [2–4]. High demands on beam control andbeam based measurements have to be fulfilled for futureexperiments such as the proposed precursor experiment fora first measurement of the electric dipole moment of thedeuteron (see [5] and references within).One major task toward reaching this goal is the improvementof the model of COSY, which currently reaches a precisionof ∆β/β ≈ 30 − 50% [6]. A technique called LOCO, thatwas successfully applied for the calibration and correctionof linear optics at light sources [1, 7, 8], will be used. Forthe application on a proton storage ring that operates at non-relativistic energies a new program, utilizing the LOCOalgorithm, was written.LOCO PROGRAMAlgorithmLOCO is based on the analysis of a measured orbit re-sponse matrix (ORM), which contains thousands of datapoints reflecting the focusing structure of the ring [1]. Theclosed orbit ORM is defined by:(~x~y)= M ·(~Θx~Θy), (1)where (~x, ~y) are the measured horizontal and vertical shiftsof the closed orbit at all beam position monitors (BPMs) fora change in the deflection strength of the steering magnet by(~Θx , ~Θy). A typical ORM measurement at COSY utilizesapproximately 40 steering magnets and about 60 BPMs (30horizontal, 30 vertical) along the ring, resulting in about2400 ORM entries.∗ c.weidemann@fz-juelich.de1 The MAD-X Program, http://madx.web.cern.ch2 The ROOT data analysis framework, https://root.cern.chThe model response matrix, for comparison with the mea-surement, is calculated by the MAD-X accelerator opticsprogram. One example ORM, calculated with the COSYmodel is shown in Fig. 1.The goal of the LOCO algorithm is to improve the COSYmodel by comparing the model ORM and the measuredORM. A χ2-minimization is performed by adjusting themodel parameters until the ORMs are equal. The χ2 isdefined as the squared sum of the differences betweenthe model and the measured ORM entries (Mmod, Mmeas),weighted with the inverse of their measurement errorssquared (σMmeas, i j ):χ2 =∑i, j(Mmod,i j − Mmeas,i j)2σ2Mmeas, i j=∑k=i, jE2k . (2)The indices i and j denote the BPM and the steerer magnet,respectively. The model parameters Kl under investigationso far are listed in Tab. 1. By varying these, the error vectordEk/dKl can be determined for every parameter:−Ek = dEkdKl · ∆Kl . (3)Combining dEk/dKl for all parameters results in the Jaco-bian matrix dEk/dK . Applying a singular value decompo-sition (SVD) to this non-square matrix allows for the deter-mination of its pseudoinverse and the direct recalculation ofthe parameter error ∆K :dEkdK= USVT =∑~ulsl~vTl and (4)Steerer0 510 1520 2530 3540BPM0102030405060 (mm/mrad) mod,ijM10−5−0510Figure 1: Calculated orbit response matrix of COSY.THPMB009 Proceedings of IPAC2016, Busan, KoreaISBN 978-3-95450-147-23240Copyright©2016CC-BY-3.0andbytherespectiveauthors05 Beam Dynamics and Electromagnetic FieldsD01 Beam Optics - Lattices, Correction Schemes, TransportTable 1: Parameters under investigation.Parameter name NumberBPM calibration 60BPM roll (ψ), shift (s) 2 · 60Steerer calibration 40Steerer roll (ψ), position (s) 2 · 40Deflection angle (offset) 40Gradient of quadrupole families 14Gradient of individual quadrupoles 56Quadrupole rotations (φ, θ, ψ)and misalignments (x, y, s) 6 · 56Quadrupole coefficient of dipoles (K1) 24Sextupole coefficient of dipoles (K2) 24Sextupole coefficient of quadrupoles (K2) 56∆K = −∑~vl1sl~uTl · Ek . (5)Since the response matrix is not linear to most of the param-eters LOCO must be iterated until it converges to the bestset of parameters.ImplementationThe implementation of the LOCO algorithm is visual-ized in Fig. 2. At the beginning of the program, all relevantdata such as input parameters and settings required for theinitialization of the LOCO procedure are read from the con-figuration file. Depending on the choice of the user a refer-ence ORM (Mmeas) is prepared from real data or generatedusing random parameter settings. In the latter case, the pa-rameter generation is correlated to a user-defined Gaussiandistribution. The ORM is calculated using a specially de-veloped MAD-X script that provides orbit data for differentdeflection strengths of every single steering dipole. TheseFigure 2: Flow chart (top to bottom) of the new LOCOprogram.simulated orbit data are analyzed in the same way as themeasured orbit data [9].The LOCO algorithm now starts with the calculation of anundisturbed ORM (Mmod) using the existing COSY latticewithout parameter errors. The resulting matrix Ek derivedaccording to Eq. 2 forms the starting point of the first itera-tion of the LOCO algorithm.Subsequently, one parameter is varied in three to five stepsand a new ORM is calculated for every step. The change ofevery ORM entry with respect to the reference entry is plot-ted in a separate graph as function of the parameter variation.The slope of a linear fit to these data points yields the entriesof the error vector dEk/dKl . This procedure is repeated forevery parameter. All resulting error vectors are combinedto the Jacobian matrix dEk/dK . The pseudoinverse of thismatrix, calculated using Armadillo libraries1, is than mul-tiplied to Ek (Eq. 5) yielding the new parameter error ∆Kfor the next iteration. After every iteration the collected dataare stored into a ROOT file containing the optical functions,the ORM data and a ROOT Tree with all parameter settings.BENCHMARKINGFor benchmarking the newly developed program hundredsof simulations have been performed for various sizes andtypes of parameter errors as well as for combinations ofparameter types. In addition, the sensitivity of the algorithmto the BPM errors, the step sizes of parameter variations andthe truncated rank of the matrix in the SVD analysis havebeen studied.Exemplary discussed here is the LOCO analysis of simulatedgradient errors of all 56 quadrupole magnets along COSY.The example shown in Fig. 1 is the calculated ORM for aGaussian distributed (σ = 1%) change of the gradients.Executing the LOCO algorithm adjusts the parameters ofthe lattice model in order to minimize the χ2 (Eq. 2). Ahistogram of the differences ∆Mi, j between the calculatedand the reference ORM is displayed in Fig. 3. The rms ofthe deviation is significantly improved with the number of1 Armadillo C++ linear algebra library, http://arma.sourceforge.net (mm/mrad)i,j M∆1.5− 1− 0.5− 0 0.5 1 1.5 2Entries110210310iteration 0 (rms = 0.28 mm/mrad)iteration 1 (rms = 0.033 mm/mrad)iteration 2 (rms = 0.0312 mm/mrad)iteration 3 (rms = 1.9e-03 mm/mrad)iteration 4 (rms = 3.6e-04 mm/mrad)iteration 5 (rms = 9.6e-05 mm/mrad)Figure 3: Histogram of the difference ∆Mi, j between thecalculated and the reference ORM for five iterations. Adecreasing rms indicates an improving agreement.Proceedings of IPAC2016, Busan, Korea THPMB00905 Beam Dynamics and Electromagnetic FieldsD01 Beam Optics - Lattices, Correction Schemes, TransportISBN 978-3-95450-147-23241 Copyright©2016CC-BY-3.0andbytherespectiveauthorsQuadrupole0 10 20 30 40 50 K∆0.08−0.06−0.04−0.02−00.020.040.060.08 iteration 0 (max = 2.7e-02, rms = 9.3e-03)iteration 1 (max = 2.0e-01, rms = 4.4e-02)iteration 2 (max = 1.3e-02, rms = 3.8e-03)iteration 3 (max = 1.4e-03, rms = 4.8e-04)iteration 4 (max = 2.7e-04, rms = 7.9e-05)iteration 5 (max = 4.8e-05, rms = 1.6e-05)Position in COSY (m)0 20 40 60 80 100 120 140 160 180xβ/xβ∆0.4−0.2−00.20.40.6iteration 0 (max = 0.50, rms = 0.23)iteration 1 (max = 0.09, rms = 0.03)iteration 2 (max = 0.04, rms = 0.03)iteration 3 (max = 5.9e-04, rms = 2.6e-04)iteration 4 (max = 1.6e-04, rms = 6.6e-05)iteration 5 (max = 1.4e-04, rms = 6.4e-05)Figure 4: Top panel: ∆K = Kmod −Kmeas for 56 quadrupolegradients is shown for the starting conditions (red curve) andafter up to five iterations.Bottom panel: The ∆β/βmeas in the horizontal plane canbe improved by about four orders of magnitude assumingalmost perfect BPMs.iterations. Figure 4 displays in the top panel the deviation ofthe reconstructed parameter settings with respect to the mea-surement (∆K = Kmod − Kmeas) and in the bottom panel theβ-beat for five iterations. As visible ∆K is improved from9.3 · 10−3 to 1.6 · 10−5 meaning that the randomly generatedvariations are very well detected. In the case of gradienterrors K is a dimensionless scaling factor. The β-beat ofinitially 0.23 was decreased to 6.4 · 10−5. These numberswere achieved assuming an error of 10−9m for the positionmeasurement of the BPMs, this corresponds to almost idealBPMs.The effect of the error of the position measurement on thereconstruction of different model parameters was carefullystudied. In Fig. 5 the resulting rms of ∆βx/βx for the vari-ation of quadrupole gradients is plotted as function of theiteration number assuming BPM errors from 1 nm up to1mm. Summarizing this graph, the optical functions canstill sufficiently be reproduced assuming ∆x = 10−5m. Thereconstruction of the parameter settings is usually a bit worse.This statement is true for most of the parameters named inTab. 1. Sometimes good optics correction is achieved withincorrect parameter settings (e.g. transverse quadrupole mis-alignment). This effect is still under investigation.An example of the sensitivity of the algorithm to the stepsize of the parameter variation is shown in the bottom panelof Fig. 5. Here, the reconstruction of the roll (ψ) of all BPMsis plotted for step sizes ranging from 0.001 to 2 rad. Onlyfor very large steps a significant effect was seen. In general,the sensitivity to the step size is different for all parameters.Other ongoing investigations deal with the sensitivity for thedifferent parameter types and to the truncated rank of theJacobian matrix in the SVD analysis.Iteration0 1 2 3 4 5 6rms)xβ/xβ∆(7−106−105−104−103−102−101−101 x = 1e-3 m∆ x = 1e-5 m∆ x = 1e-7 m∆ x = 1e-9 m∆Iteration0 1 2 3 4 5 6 (rad)rms k∆7−106−105−104−103−102−101−101 K step size = 0.001 rad∆ K step size = 0.01 rad∆ K step size = 0.1 rad∆ K step size = 1 rad∆ K step size = 2 rad∆Figure 5: Top: The determination of ∆βx/βx for adjustingthe gradients of all 56 quadrupole magnets clearly dependson the error of the beam position measurement, which isvaried from 1 nm up to 1mm.Bottom: Reconstruction of the rolls of 60 BPMs for differentstep sizes of parameter variation.SUMMARY AND OUTLOOKA new LOCO program for the use at COSY was success-fully developed and brought into operation. Benchmarkingtests show promising results with respect to the use of theprogram with measured data. Besides finalizing the bench-marking and the implementation of additional features intothe program, such as an automatic step size finder or a secondminimization algorithm, a first analysis of measured data isplanned. An automated ORM measurement system, whichwas implemented in the past year, allows to collect one fullORM data set within 30 minutes [9]. Several of these mea-surements with different settings of the quadrupole gradientswere accomplished in 2015. The goal of the LOCO analysisof these data is the determination of the applied changes.THPMB009 Proceedings of IPAC2016, Busan, KoreaISBN 978-3-95450-147-23242Copyright©2016CC-BY-3.0andbytherespectiveauthors05 Beam Dynamics and Electromagnetic FieldsD01 Beam Optics - Lattices, Correction Schemes, TransportREFERENCES[1] J. Safranek, Nucl. Instrum. Meth. A 388, 27 (1997),doi:10.1016/S0168-9002(97)00309-4.[2] S.A.Martin et al., Nucl. Instrum.Meth. A 236, 249-255 (1985),doi:10.1016/0168-9002(85)90157-3.[3] R. Maier, Nucl. Instrum. Meth. A 390, 1 (1997),doi:10.1016/S0168-9002(97)00324-0.[4] C. Weidemann et al., Phys. Rev. ST Accel. Beams 18, 020101(2015), doi:10.1103/PhysRevSTAB.18.020101.[5] D. Eversmann et al. [JEDI Collaboration], Phys. Rev. Lett. 115,no. 9, 094801 (2015), doi:10.1103/PhysRevLett.115.094801.[6] D. Ji et al., “First Experience of Applying LOCO for OpticsMeasurement at COSY”, presented at IPAC’16, Busan, SouthKorea, May 2016, paper TUPMR026.[7] G. Benedetti, D. Einfeld, Z. Mart and M. Munoz, Conf. Proc.C 110904, 2055 (2011).[8] L. S. Nadolski, Conf. Proc. C 0806233, THPC064 (2008).[9] F. Hinder et al. “Automation of the Orbit ResponseMatrixMea-surement at COSY”, Annual Report 2015, IKP, Forschungszen-trum Juelich GmbH (2015), available from: http://www.fz-juelich.de/ikp/EN/Service/Download/download_node.html.Proceedings of IPAC2016, Busan, Korea THPMB00905 Beam Dynamics and Electromagnetic FieldsD01 Beam Optics - Lattices, Correction Schemes, TransportISBN 978-3-95450-147-23243 Copyright©2016CC-BY-3.0andbytherespectiveauthors

Model Driven Machine Improvement of COSY Based on ORM Data

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Model Driven Machine Improvement of COSY Based on ORM Data

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