Graphics Processing Units (GPUs) have revolutionized the computing landscape over the past decade. However, the growing energy demands of data centres and computing facilities equipped with GPUs come with significant capital and environmental costs. The energy consumption of GPU applications greatly depend on how well they are optimized. Auto-tuning is an effective and commonly applied technique of finding the optimal combination of algorithm, application, and hardware parameters to optimize performance of a GPU application. In this paper, we introduce new energy monitoring and optimization capabilities in Kernel Tuner, a generic auto-tuning tool for GPU applications. These capabilities enable us to investigate the difference between tuning for execution time and various approaches to improve energy efficiency, and investigate the differences in tuning difficulty. Additionally, our model for GPU power consumption greatly reduces the large tuning search space by providing clock frequencies for which a GPU is likely most energy efficient.