Quench Protection System Optimization for the High Luminosity LHC Nb $_3$Sn Quadrupoles

The upgrade of the large hadron collider to achieve higher luminosity requires the installation of twenty-four 150 mm aperture, 12 T, $Nb_3Sn$ quadrupole magnets close to the two interaction regions at ATLAS and CMS. The protection of these high-field magnets after a quench is particularly challenging due to the high stored energy density, which calls for a fast, effective, and reliable protection system. Three design options for the quench protection system of the inner triplet circuit are analyzed, including quench heaters attached to the coil's outer and inner layer, Coupling-Loss Induced Quench (CLIQ), and combinations of those. The discharge of the magnet circuit and the electromagnetic and thermal transients occurring in the coils are simulated by means of the TALES and LEDET programs. The sensitivity to strand parameters and the effects of several failure cases on the coil's hot-spot temperature and peak voltages to ground are assessed. A protection system based only on quench heaters attached to the outer layer can barely maintain the hot-spot temperature below the target limit and cannot guarantee the coil protection under failure scenarios. On the contrary, systems including either inner quench heaters or CLIQ are adequate to protect the coil under all realistic operation and failure scenarios. In particular, the option including outer quench heaters and CLIQ achieves lowest hot-spot temperatures, and highest redundancy and robustness.