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Computational demand has brought major changes to Advanced Cyber-Infrastructure (ACI) architectures. It is now possible to run scientific simulations faster and obtain more accurate results. However, power and energy have become critical concerns. Also, the current roadmap toward the new generation of ACI includes power budget as one of the main constraints. Current research efforts have studied power and performance tradeoffs and how to balance these (e.g., using Dynamic Voltage and Frequency Scaling (DVFS) and power capping for meeting power constraints, which can impact performance). However, applications may not tolerate degradation in performance, and other tradeoffs need to be explored to meet power budgets (e.g., involving the application in making energy-performance-quality tradeoff decisions). This paper proposes using the properties of AMR-based algorithms (e.g., dynamically adjusting the resolution of a simulation in combination with power capping techniques) to schedule or re-distribute the power budget. It specifically explores the opportunities to realize such an approach using checkpointing as a proof-of-concept use case and provides a characterization of a representative set of applications that use Adaptive Mesh Refinement (AMR) methods, including a Low- Mach-Number Combustion (LMC) application. It also explores the potential of utilizing power capping to understand power- quality tradeoffs via simulation.