This article presents an optimization model whose purpose is to identify the optimal expansion of the gas network to cities currently not met, but that belong to the area of operation of a particular local distribution company. The model maximizes the potential increase to the company's profit on a horizon specified by the user, seeking to link neighboring cities. The problem is defined as a Multi-period Prize Collecting Steiner Tree problem with budget constraints, since there are vertex profits, edge costs and limits to the company's budget per period. The formulation is of an integer linear program on a directed graph model. The Branch-and-Cut technique is used with cuts based on the separation of sets of violated inequalities by a maximum flow algorithm. Instances up to 250 cities are satisfactorily evaluated with the model.