This paper studies the day-ahead prediction of electricity consumption for power supply-demand balance in electric power networks. To handle the uncertainties in weather forecast and the nonlinearity relation between the electricity consumption and the weather conditions, this paper proposes a Radial Basis Function like Artificial Neural Network (RBF-like ANN) model with temperature, humidity, and sampling times as inputs. Then the Least Absolute Deviation, i.e., the L1 norm condition, is employed as the optimization cost which is minimized in the model training process. To solve the L1 optimization problem, two approaches, namely least square (L2) based and alternating direction method of multipliers (ADMM), are utilized and compared. The simulations on real data collected in California shows that the latter approach performs better, and the number of neurons does not affect much to the prediction performance of the latter approach while it does influence on that of the former approach. Further, the proposed RBF-like ANN model equipped with ADMM solving approach provides reasonably good prediction of the electricity consumption in spite of the imprecise weather forecast.