The HPLWR (High Performance Light Water Reactor) is the European version of the SCWR (Supercritical Water Reactor) and is one of the Generation IV concepts that have enhanced safety, improved efficiency and less nuclear waste compared to current nuclear reactors. A possible way to enhance the safety is by using natural convection as the driving mechanism for the coolant flow. Natural convection is especially interesting because of the large differences in density occurring under supercritical conditions. This is safer because of its independence of mechanical systems (i.e. pumps, which are used in forced convection loops). The goal of this project was to investigate the linear stability of a one-dimensional, simplified version of the HPLWR (without the power-density feedback, but with constant power) around the steady-state solution for a range of operational conditions. A code was written based on one-dimensional equations for mass, energy and momentum transport. The code successfully predicted the steady-state behaviour of a system. The results were benchmarked with data from literature. The code works for natural convection loops as well as forced convection systems. The stability plots do not agree with literature and are therefore considered to be incorrect.