The evolution of a high burnup structure (HBS) in a light water reactor (LWR) $UO_2$ fuel was simulated using the Potts model. A simulation system for the Potts model was defined as a two-dimensional triangular lattice, for which the stored energy was calculated from both the irradiation damage of the $UO_2$ matrix and the formation of a grain boundary in the newly recrystallized small HBS grains. In the simulation, the evolution probability of the HBS is calculated by the system energy difference between before and after the Monte Carlo simulation step. The simulated local threshold burnup for the HBS formation was 62 MWd/kgU, consistent with the observed threshold burnup range of 60-80 MWd/kgU. The simulation revealed that the HBS was heterogeneously nucleated on the intergranular bubbles in the proximity of the threshold burnup and then additionally on the intragranular bubbles for a burnup above 86 MWd/kgU. In addition, the simulation carried out under a condition of no bubbles indicated that the bubbles played an important role in lowering the threshold burnup for the HBS formation, thereby enabling the HBS to be observed in the burnup range of conventional high burnup fuels.