The increasing usage of Artificial Intelligence (AI) models, especially Deep Neural Networks (DNNs), is increasing the power consumption during training and inference, posing environmental concerns and driving the need for more energy-efficient algorithms and hardware solutions. This work addresses the growing energy consumption problem in Machine Learning (ML), particularly during the inference phase. Even a slight reduction in power usage can lead to significant energy savings, benefiting users, companies, and the environment. Our approach focuses on maximizing the accuracy of Artificial Neural Network (ANN) models using a neuroevolutionary framework whilst minimizing their power consumption. To do so, power consumption is considered in the fitness function. We introduce a new mutation strategy that stochastically reintroduces modules of layers, with power-efficient modules having a higher chance of being chosen. We introduce a novel technique that allows training two separate models in a single training step whilst promoting one of them to be more power efficient than the other while maintaining similar accuracy. The results demonstrate a reduction in power consumption of ANN models by up to 29.2% without a significant decrease in predictive performance.