Lattice Optimization Using Multi-Objective Genetic Algorithm
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1 Lattice Optimization Using Multi-Objective Genetic Algorithm Vadim Sajaev, Michael Borland Mini-workshop on ICA in Beam Measurements and Genetic Algorithm in Nonlinear Beam Dynamics March 14, 2012
2 Introduction Present APS is a highly symmetric storage ring (40-fold symmetry) APS Upgrade lattice has been worked on for some years now (recently DOE approved CD-1 for APS upgrade) Main features of the lattice upgrade are: Provision of Long Straight Sections (LSS) Generation of Short-Pulse X-rays (SPX) using deflecting cavities Small number of Reduced Horizontal Beam size (RHB) insertions (if possible) LSS is provided by removing a pair of quadrupoles adjacent to the Insertion Device (ID) straight section Increases useful ID length from 5 to 8 m Provides room for SPX cavity cryostats Breaks beta function symmetry SPX insertion requires specific linear and nonlinear optics Deflecting cavities are separated by 2 sectors need exactly 2 phase advance Sextupoles are locally adjusted to minimize coupling on deflected trajectories 2
3 Constraints on New Lattices Effective emittance presently 3.15 nm Constrain to 10% increase No new magnets or power supplies Only an issue with S2 sextupoles Run at 250 A (150 A design) Pole noses added to increase strength Maximum beta functions: 40m (H) and 30m (V) Preserve physical aperture Limit effective transverse impedance Limit sensitivity to nonlinearities Horizontal beta functions at IDs: Presently 20m, constrain to 20±5m Only one request for smaller beam size 3
4 Constraints on New Lattices Vertical beta functions at IDs: Keep within 20% of ideal L/2 value Value at transition/aperture limit then within 2% of ideal Sufficient chromaticity ξ to stabilize beam Assume the use of bunch-by-bunch feedback system ξ=7 should allow 200 ma/24 bunches ξ=8~9 needed for 16 ma hybrid mode Dynamic aperture should be similar to present to ensure good injection efficiency Presently, have 90% injection efficiency Top-up puts minimum limit on the lifetime: Works out to 3.8 hours at 200mA Single bunch droop is 10% 4
5 Optimization Technique It was clear from the beginning that linear and nonlinear optics need to be optimized together and that number of optimization variables could be big Use a direct method based on tracking and a multi-objective genetic algorithm Well suited to tuning dynamics with large chromaticity Penalty function includes contributions from Injection area computed from Dynamic Aperture (DA) Touschek lifetime computed from Local Momentum Aperture (LMA) and tune vs momentum Method in brief Generate N randomized configurations of optimization variables Match lattice, perform tracking, determine best configurations Breed and randomize the best configurations to make new configurations Repeat from step 2 until convergence 5
6 Simulation Tools Due to volume of simulations, need highly scriptable, parallel/concurrent simulation elegant1 and Pelegant2 used for accelerator simulation, e.g., Lattice matching Dynamic and momentum aperture tracking Coupling correction SPX sextupole optimization elegantringanalysis3 used for ensemble evaluation Runs on APS cluster for fast turn around geneticoptimizer3 used for overall optimization Runs elegant or Pelegant touscheklifetime4 used to compute beam lifetime 1: 2: 3: 4: M. Borland, APS LS-287. Y. Wang et al., Proc. AAC M. Borland, H. Shang. A. Xiao et al. 6
7 Computing Hardware APS weed cluster1 400 Nehalem cores Infiniband network Lustre filesystem Argonne Laboratory Computing Resource Center's fusion cluster 2560 Nehalem cores Infiniband network GPFS filesystem Argonne Leadership Computing Facility's intrepid cluster 200k PPC cores PVFS filesystem Use of completely open source software lowers barriers to using many different systems 1: Created and maintained by R. Soliday 7
8 Dynamic Aperture Modeling Dynamic aperture used as a proxy for injection efficiency In some optimizations we used direct injection simulation Line-scan method used to find the stability boundary Search from origin out to reduce chance of being mislead by a stable island After initial search, interval is sub-divided to improve resolution DA for present APS lattice is larger on the inboard side, where we inject 8
9 Dynamic Aperture Details After finding DA boundary, apply clipping algorithm Eliminate lobes or islands that don't represent useful DA Further clip the boundary to remove non-useful DA Vertical DA beyond 1 mm Horizontal DA beyond +7mm Compute the area A inside clipped boundary The penalty value is just A Example of clipping algorithm applied to a particularly non-regular dynamic aperture 9
10 Lifetime Modeling Touschek scattering is the dominant lifetime limiting mechanism for APS Gas scattering lifetime at 100 ma is ~60h Compute lifetime from beam sizes and LMA using Piwinski formalism1 Bruck's method significantly overestimates APS lifetime2 Use measured bunch length vs current LMA is determined by tracking3 Scan positive and negative momentum deviation at the exit sextupoles in 4 or more sectors Assumed representative of other sectors Scan from 0 to avoid islands LMA for present APS lattice is limited by the rf acceptance to ±2.35% 1: A. Piwinski, DESY : A. Xiao et al., PAC07, : M. Belgrounne, PAC03,
11 Common Features of Aperture Simulations Longitudinal motion included Essential for LMA Some effect on DA (path-length changes from betatron motion) Radiation damping included May increase DA/LMA by overcoming slow growth rates May decrease DA/LMA by sweeping beam over resonances Track for 400 turns 13~25% of initial amplitude decays away Helps choose the spacing for aperture search Results for more turns essentially identical Physical apertures included Elliptical for main chamber One-sided super-elliptical for ID chambers Include lengthened ID chamber for LSS 11
12 Physical Apertures small gap chamber septum 12
13 Common Features of Tracking Simulations Errors Ideally, should include realistic errors and correction schemes Correction schemes Reduce orbit errors to few 100 microns Reduce beta function errors to ~1% rms Reduce emittance ratio to ~1% For expediency, simulate errors at level that gives post-correction performance Quad and sextupole rolls of 0.05 mrad rms Quad and sextupole strength errors of 0.02% rms Methods Symplectic hard-edge magnets with exact Hamiltonian Zero-length rf cavities with exact phase dependence Lumped synchrotron radiation 13
14 LSS Placement Traditionally, only symmetric arrangements considered viable Pro: Easier to obtain large injection aperture and lifetime Con: Increases cost by requiring experimental programs to move Using MOGA, we are able to tune non-symmetric lattices Variables include Tunes 25~50 sextupoles APS and ANL computing resources (fusion, intrepid) invaluable Have developed three basic lattices: 4 non-symmetric LSS (4NLSS) 4NLSS + SPX in sector 7 4NLSS + SPX + RHB in sector 20 14
15 LSS Lattice Functions SSS LSS SSS Horizontal lattice functions are little changed Vertical beta function increases from 3m to 5m 3.3 nm effective emittance 15
16 Incorporation of Linear Optics Knobs Originally, the only linear optics knobs at the disposal of MOGA were the tunes Linear lattices interpolated from a (νx,νy) grid of solutions that all satisfy common constraints Reasonable beta functions at center of straights Reasonable maximum beta and dispersion in a cell Problem surfaced with difficult lattices (e.g., RHB insertion) Two lattices with the same tune that respected the same reasonable constraints had significantly different optimized performance Hypothesized that we needed more detailed control of linear lattice Added linear optics knobs for MOGA algorithm E.g., for each type of sector, we might include Target values for beta and eta at straight sections Target values for beta and eta at selected quads and sextupoles Target values for maximum beta and eta The first step in any function evaluation is to rematch the linear optics to attempt to satisfy trial values for these parameters This makes use of the parallel hybrid simplex optimizer in elegant 16
17 RHB Example Incorporating RHB into the APS-U lattice proved very difficult The original method was even unable to provide a configuration for APS operations that was as good as present operations Struggled to reduce the beam size below 180 microns Comparing the linear optics of the (poor) optimized configuration to the (good) operational configuration led to the realization that more detailed linear optics knobs where needed We decided to develop this capability by looking at RHB-only lattices Linear optics knobs: Max beta, eta in regular and RHB sectors (6 knobs) Betas and eta at selected quads or sextupoles (20 knobs) Result is dramatically better 17
18 RHB Result Yielded Several Surprises Larger maximum lattice functions in RHB area Greater variation in dispersion between straights and intra-bend section Beam size at target sector below 80 microns 18
19 SPX Optimization The Short Pulse X-ray (SPX) system will employ two sets of crab cavities Cavities separated by two sectors Special linear optics (e.g., vertical phase advance of 360 deg) Special sextupole tuning to suppress emittance blow-up Originally, we fixed the SPX linear optics and sextupoles, but simulations predict poor lifetime with cavities on Recently, gave MOGA control of SPX linear optics (max. and straight-section beta and eta values, 6 knobs) SPX sextupoles (14 knobs) Added minimum vertical emittance growth as one of the objectives This is measured using an additional tracking step, simulating one pass through the SPX sectors Incorporated this into a mockup 4NLSS lattice MOGA succeeded in restoring DA and lifetime while keeping emittance growth ~1pm 19
20 SPX Optimization Result (ξ=7) Lifetime increased from <5 hours to nearly 10 hours, which is normal for this chromaticity. 20
21 Evaluation of Optimized Configurations Optimization is performed with a single error ensembles, and might be misleading Optimized configurations are checked for robustness by ensemble evaluation Procedure: Check linear optics, chromaticities without errors Generate coupling correction matrix Generate 50 error ensembles For each ensemble Correct coupling Add vertical dispersion to get desired vertical emittance Determine lattice functions Determine equilibrium emittances Determine DA Determine LMA Compute lifetime for various bunch patterns 21
22 Comparison with APS Today Today ξ=7 Today ξ=7 4NLSS ξ=9 4NLSS ξ=9 22
23 Frequency Map 4NLSS, ξ=9 23
24 Phase Space Distortion Increases Acceptance Horizontal phase space distortion in 4NLSS Minimum aperture is -15mm at S4, but DA exceeds this Results from distortion of phase space going through the aperture limit at S4 24
25 Dynamic Apertures for Several Lattices +SPX 4NLSS +SPX +RHB 4NLSS has ξ=9, +SPX has ξ=8, +SPX+RHB has ξ=7 Sextupole knobs: 8NLSS: 24 +SPX: 38 +SPX+RHB: 52 25
26 LMA for Several Lattices 4NLSS +SPX +SPX +RHB 4NLSS has ξ=9, +SPX has ξ=8, +SPX+RHB has ξ=7 Sextupole knobs: 8NLSS: 24 +SPX: 38 +SPX+RHB: 52 26
27 Lifetime Predictions (50 Ensembles) 4NLSS+SPX+RHB has acceptable lifetime with =7 Work continues on higher-chromaticity solutions 27
28 Experimental tests of lattices 4NLSS lattice test is completed Injection efficiency is 80% and lifetime is 6 hours Expected lifetime is 8 hours (5th percentile is 5.6 hours) 4NLSS+RHB 5 hour lifetime is achieved Injection efficiency needs more work (50% presently) 4NLSS+SPX 6 hour lifetime is achieved Injection efficiency needs more work (60% presently) Overall, experimental tests show slightly lower lifetime and dynamic aperture Don t have much time for detailed optimization and correction of the test lattices Still, lattices are acceptable or nearly acceptable for operation 28
29 Another optimization example extreme SPX (X-SPX) X-ray pulse length in SPX is defined by: Ec s V id rf y rad id 2 L u rf is increased from 4m to 10m, optimization of nonlinear dynamics if ongoing: rf 29
30 X-SPX optimization Started with optimized 4NLSS lattice Used only sextupoles between deflecting cavities 14 variables Evaluation of optimized configuration is under way Next step would be to increase rf to 15 m 30
31 One more example sextupole failure APS has separate power supplies for all magnets (680 PS) Recently, we had an incident when a sextupole power supply tripped during user top-up operation The beam was not lost but lifetime and injection efficiency decreased Power supply was later reset It can happen again, and if the power supply would no reset, we would need to operate with lower lifetime until next intervention 31
32 Optimization results We used the same optimization procedure to adjust sextupoles around the failed one to recover lifetime and injection efficiency DA for A:S4 LMA for A:S2 DA measurement Black initial lattice Red after failure Blue after correction 32
33 Different example low-alpha lattice Standard way to develop a low momentum compaction lattice: Start with original lattice Use simplex minimization of some penalty function (for example, simply momentum compaction) Used this approach, came up with the usual low-alpha lattice where dispersion crosses zero inside the dipole Concluded that low alpha and low emittance cannot be achieved together 33
34 Low-alpha lattice genetic optimization Ask optimizer to minimize momentum compaction and emittance Use 5 quadrupoles to preserve reflection symmetry inside the sector (but mostly to limit the number of variables) Unlike nonlinear dynamics optimization, function evaluation here is very fast just beta function calculations Used personal 8-core desktop to generate 100,000 evaluations in a few hours Sorting becomes limiting factor Lattice with =5nm and =
35 Low-alpha lattice genetic optimization Genetic optimization lets you Scan multi-dimensional space faster than grid scan See dependencies and trade-offs that are hidden when simplex is used 35
36 Conclusion APS-U requires several lattice changes Long straights SPX insertion RHB in one location Major challenges Non-symmetric locations for LSS Symmetry breaking SPX and RHB insertion Integrating all of these Using multi-objective genetic optimization, good progress has been made Experimental tests of the lattices are under way 4NLSS is acceptable for operation Other lattices need more work but results are encouraging Started using MOGA in linear lattice development for low-alpha lattice 36
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