This paper presents a method for detecting anomalous power consumption patterns attacks, using a discrete wavelet transform, as well as the variance fractal dimension (VFD) and an artificial neural network (ANN) for a smart grid. The main procedure of the proposed algorithm consists of the following steps: (i) Finding normal and anomalous patterns of power consumption to train the proposed method, (ii) Applying wavelet transform to power consumption patterns to extract features, (iii) Applying the VFD to the extracted features from Step 3 as an input, (iv) Training an ANN with the extracted features from Step 3, and (v) Launching the trained ANN from Step 4 to detect the anomalous power consumption attack based on a threshold. The proposed method can detect an anomalous power consumption attack with 51% accuracy in the worst case scenario.