The power flow calculation on an electric network consists of determining the network s state, power flows and electrical losses on the lines, and total losses on the feeder. In this type of problem, the system s modeling is static, and the network is represented by a set of algebraic equations and inequations. Different solution methods were proposed in the literature to perform power flow calculations. However, for distribution networks, these methods must be able to model, with sufficient details, some unique features of these systems, such as their near radial structure, the unbalanced nature of the loads, and distributed generators insertion. Besides that, modeling the consumption pattern in distribution systems is more complex, and the line parameters are more difficult to be obtained when compared to the transmission system. Hence, applying traditional methods for power flow calculations in distribution networks may lead to divergent solutions. Within this context, this work proposes a new approach for power flow calculations in distribution systems based on Machine Learning. The proposed models use Artificial Neural Networks (ANNs) to predict the active internal losses of a distribution network and the power flows at the borders with the transmission system. Numerical simulations demonstrate the effective performance of the proposed approach, as well as its computational advantages over benchmark software programs since, once trained, ANNs can approximate power flow calculations extremely fast, as only matrix operations are needed. Moreover, the work presents an application of the ANN methodology proposed: predictions of the flows at the borders with the transmission network were used to generate optimal demand contracts for a real distribution system in Brazil.