Optimization of dynamic aperture by TESLA for double mini-by lattice in TPS Mau-Sen Chiu 2012/03/14 Beam Dynamics Group, NSRRC

Outline Introduction Multi-objective genetic algorithm - Double mini-by lattice in TPS. Multi-objective genetic algorithm Optimization of dynamic aperture (DA) by TESLA - Check 268 sets of solutions after completion: DA, tune shift with momentum and tune shift with amplitude. - Pick up the best one to calculate frequency map and Touschek lifetime. Conclusion

Introduction TPS Storage ring TPS Storage Ring Parameters Locations of the double mini-by lattice Circumference 518.4 (m) Nominal energy 3.0 GeV Betatron tune 26.18/13.28 Natural chromaticity -75/-26 RF frequency 499.654 Harmonic number 864 Natural emittance 1.6 nm-rad Energy spread 8.86E-04 Energy loss per turn 853 keV TPS Storage ring 7m X 18 12m X 6 Purpose: Install two small gap IDs in tandem to obtain 4 times brightness. 3

Double mini-by lattice Use 240 quadrupoles to match a lattice (nx/ny =26.18/12.82). 13.26-0.44 Add three sets of quadrupole triplet at the center of three long straights, respectively. Apply p trick to match the double mini-by lattice (nx/ny = 26.18/14.26). Q2 Q5 Q2 4.9 m 4.9 m Q3 Q1 Q4 Q4 Q1 Q3 ny = 12.82 ny = 14.26 (m) L (m) K1 Q1 0.3 -1.217349 Q2 0.6 1.384064 Q3 -1.526270 Q4 0.5 -1.431159 Q5 1.681944 Purpose: Install two small gap IDs in tandem to obtain 4 times brightness. 4

Multiobjective Optimization A general multiobjective optimization problem consists of a number of objectives and is associated with a number of inequality and equality constraints. Mathematically, the problem can be written as follows Minimize/Maximize fk(x) k = 1, 2, …, K Subject to gj(x) j = 1, 2, …, J hm(x) m= 1, 2, …, K with i = 1, 2, …, N The variable vector x represents a set of variables xi, i = 1, 2, …, N Ex: Optimization of dynamic aperture (DA) for double mini-by lattice in TPS Objectives: f1(S1, S2, …, S6, SA, SB): DA area in (x-d) plane. f2(S1, S2, …, S6, SA, SB): Tune shift with amplitude terms. Constraints: Chromaticity are corrected at fixed values with SF and SD through out DA optimization. Sextupole integral strength: for each family.

Genetic Algorithm A genetic algorithm (GA) is routinely used to generate useful solutions to optimization and search problems using techniques inspired by natural evolution, such as inheritance, crossover, mutation, and selection. Crossover during meiosis Mutation of gene What’s a gene? Type of mutation: point mutation, substitution, insertion, deletion, A gene is a segment of DNA needed to contribute to a function. This diagram labels a region of only 50 or so bases as a gene. In reality, most genes are hundreds of times larger.

Multi-Objective Genetic Algorithm (MOGA) 1: Initialize population (first generation, random) 2: for ( int i = 2; i <= gen; i++) { - crossover: Within crossover probability, apply crossover to two parents to generate two children. These two parents are randomly chosen from the survivals of the last generation. Otherwise, copy parents to children. - mutation : Within mutation probability, apply mutation to parents to generate children. Otherwise, do nothing. - evaluate (children): calculate objective functions - merge ( parents, children): - non-dominated sort (rank): - select half of (parents, children) for next generation. }

TESLA Author: Dr. LingYun Yang, NSLS-II, Brookhaven National Laboratory Algorithm: NSGA-II (Non-dominated Sorting Genetic Algorithm II) Parameters: -Number of individual: 2000. -Number of generation: 50 -Number of sextupole family: 8 -Lower and Upper bound of each sextupole family: 3 33 # L H SA -33 -3 # L H SB 0 30 # L H S1 -30 0 # L H S2 18 48 # L H S3 -48 -18 # L H S4 18 48 # L H S5 -48 -18 # L H S6 -Crossover probability: 0.8 -Mutation probability: 0.9 -Distribution index for crossover: 3 -Distribution index for mutation: 0.3 Reference: Tracking code development for beam dynamics optimization, L. Yang, BNL, PAC11 A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II, K. Deb et al. IEEE, 2002

Flow of MOGA in TESLA GA loop evaluate_pop (child) GA loop 1. Master prepare the input for each individual in the child population then sent it to slaves 2. Each slave : Execute DA tracking (128 turns) in (x-d) plane, calculate the DA area and tune shift with amplitude terms. Chromaticity are corrected to a fixed value before DA tracking. After completion, send the results back to the master. 3. Once master receive the results sent by some slave. This slave will receive another individual. 4. This process will continue till the whole population are distributed to slaves completely.

PC Cluster in NSRRC Hardware for running TESLA node28 ~ node37: CPU: Intel Xeon X5550, quad core 2.66 GHz * 2 RAM: 16G ECC DDR2 RAM node38 ~ node40: RAM: 18G ECC DDR3 RAM Total: 104 cores / 26 CPUs 2000 individuals and 50 generations takes about 10 days. Hardware for running TRACY 2.6 node19 ~ node23: CPU: Intel Xeon X5550, quad core 2.66 GHz * 2 RAM: 18G ECC DDR3 Total: 40 cores / 10 CPUs

Objective functions in TESLA f1: The inverse of the sum of the on- and off-momentum DA area S in (x-d) plane. (To transfer the maximum problem to minimum problem.) f2: Square sum of tune shift with amplitude. The Sextupole Scheme for the Swiss Light Source (SLS), 1997, Johan Bengtsson

Objective functions after generation 2 nature log Total number of points: 2000 Each point represents a set of solution of sextupole strength.

Objective functions after generation 50 nature log 1. 286 points inside 3 regions are checked: DA (d =0, 3%, -3%), tune shift with momentum, tune shift with amplitude (horizontal and vertical direction). Total number of figures: 286 * 6 = 1716. 2. Pick up the best solution by inspection for further analysis by frequency map and Touschek lifetime.

DA, Tune shift with momentum and amplitude calculated at x = 0, y = 0 calculated at the long straight center Vertical Horizontal calculated at x = 0, d = 0 1% Emittance coupling Multipole errors Chamber limit ID kick map ━ calculated at y = 0, d = 0

Frequency map analysis (x – d) 1% Emittance coupling Multipole errors Chamber limit ID kick map ┿ ━ 1% Emittance coupling Multipole errors Chamber limit ID kick map ┿ ━

DA & FMA (dp/p = 0) 3nx + ny = 93 3nx + ny = 93 3ny = 43 1% Emittance coupling Multipole errors Chamber limit ID kick map ┿ ━ 1% Emittance coupling Multipole errors Chamber limit ID kick map ┿ 16

DA & FMA (dp/p = 3%) 4nx = 105 1% Emittance coupling Multipole errors Chamber limit ID kick map ┿ ━ 1% Emittance coupling Multipole errors Chamber limit ID kick map ┿ 17

DA & FMA (dp/p = -3%) 3nx – 2ny = 50 3ny = 43 1% Emittance coupling Multipole errors Chamber limit ID kick map ┿ ━ 1% Emittance coupling Multipole errors Chamber limit ID kick map ┿ 18

Momentum Acceptance & Touschek lifetime Ex: 1.58E-9 Ey: 1.59E-11 Bruck’s formula: T: 18.72/ Tp: 12.86/ Tn: 34.39 (hrs) Bunch current: 400 mA / 800 bunches Bunch length: 2.86 mm 1% Emittance coupling Multipole errors Chamber limit ID kick map ┿ ━

Momentum Acceptance & Touschek lifetime Ex: 1.58E-9 Ey: 1.60E-11 Bruck’s formula: T: 18.64/ Tp: 12.84/ Tn: 34.03 (hrs) Bunch current: 400 mA / 800 bunches Bunch length: 2.86 mm 1% Emittance coupling Multipole errors Chamber limit ID kick map ┿

Conclusions 1. After DA optimization by TESLA, you still use tracking code to do DA tracking, then pick up the best one. Finally, plot frequency map to decide whether the solution could be accepted or not. 2. We can see that the frequency map show resonance line when ID kick maps are included. It should slightly move working point to avoid resonance line. 3. Maybe it should add the tune optimization in the near future to search ,

Appendix

Crossover: Simulated Binary Cross-Over (SBX) SBX is used to create child solutions xc1 , xc2 from parents xp1 , xp2. if u <= 1/a1 , if u <= 1/a2 , otherwise, otherwise, u : a random number between [0, 1]. , assume Comput. Methods Appl. Mech. Energ. 186 (2000) 311-338, K. Deb h (distribution index for crossover): It control the shape of probability distribution function of crossover. Within the crossover probability, apply crossover, Otherwise, copy the parents to children. Application multiobjective genetic algorithm in accelerator physics, ICAP09, L. Yang, et al.

Polynomial Mutation Polynomial mutation is used to create a child solution xc in the vicinity of a parent solution xp. u = a random number between [0, 1] if u <= 0.5, otherwise, Comput. Methods Appl. Mech. Energ. 186 (2000) 311-338, K. Deb hm (distribution index for mutation): It control the shape of probability distribution function of mutation. The smaller the hm, the far away from the parents the child. Within the mutation probability, apply mutation, Otherwise, do nothing. Application multiobjective genetic algorithm in accelerator physics, ICAP09, L. Yang, et al.

SR multipole tolerances Multipole errors SR multipole tolerances DM QM SM Bn/BM @ 25mm (*E-4) B1/B0 ±5 A2/B1 ±3 B0/B2 B2/B0 -5±2 B0/B1 B1/B2 ±10 B3/B0 ±2 B2/B1 B3/B2 B4/B0 5±2 B3/B1 B4/B2 B5/B0 ±1 B4/B1 B5/B2 ±0.5 B6/B0 -2±0.2 B5/B1 0±1 B6/B2 B8/B0 -0.6±0.6 B9/B1 B7/B2 ±0.1 Each rest term B13/B1 B8/B2 B17/B1 B14/B2 B21/B1 B20/B2 B26/B2 Note: n=0 is dipole term, n=1 is quadrupole term and so on. Bn is normal term, An is skew term. 25

Min(Chamber size, ID gap) (Aperture) IU22 IU22 IU22 IU22 EPU48 EPU46 Y(mm) S (m) Injection point (80 cm, down stream of the long straight center) Vertical: Horizontal: IU22 EPU48 EPU46 Beam Pipe 26