Simulations of longitudinal beam stabilisation in the CERN SPS with BLonD

The Super Proton Synchrotron (SPS) at CERN, the Large Hadron Collider (LHC) injector, will be pushed to its limits for the production of the High Luminosity LHC proton beam while beam quality and stability in the longitudinal plane are influenced by many effects. Particle simulation codes are an essential tool to study the beam instabilities. BLonD, developed at CERN, is a 2D particle-tracking simulation code, modelling the longitudinal phase space motion of single and multi-bunch beams in multi-harmonic RF systems. Computation of collective effects due to the machine impedance and space charge is done on a multi-turn basis. Various beam and cavity control loops of the RF system are imple-mented (phase, frequency and synchro-loops, and one-turndelay feedback) as well as RF phase noise injection used forcontrolled emittance blow-up. The longitudinal beam stability simulations during long SPS acceleration cycle ($\sim$20 s)include a variety of effects (beam loading, particle losses, controlled blow-up, double RF system operation, low-level RF control, injected bunch distribution, etc.). Simulations for the large number of bunches in the nominal LHC batch (288) use the longitudinal SPS impedance model containing broad and narrow-band resonances between 50 MHz and 4 GHz. This paper presents a study of beam stabilisation in the double harmonic RF system of the SPS system with results substantiated, where possible, by beam measurements.