Final Report Summary - SOC-MP-ES (Stochastic Optimal Control of multiscale processes with applications to energy systems) The energy challenge consists of the set of technologies and policies that need to be developed in order to enable the transition to a low-carbon economy. With the introduction of new technologies, combined with the need for optimal utilization of existing infrastructures and the optimal integration of new technologies, it is becoming more and more apparent that finer time-scale processes have important feedbacks on the optimal strategy for longer time-scales that are not captured by the conventional simplifications. The transition from a fossil-based economy to the post carbon economy can only happen once, it is, therefore, paramount to use a rigorous, and transparent modeling framework in order to maximize the benefits brought forward during this transition. New methods grounded in sound engineering principles are needed to address these challenges.The first objective of the project is to define the representational structures that will facilitate rigorous quantitative multiscale models to be developed. Our second objective is to develop algorithmic techniques for simulating, optimizing and controlling stochastic multiscale systems. Lastly, our third objective is to apply the analytical and computational tools to realistic case studies arising from complex energy systems models.We showed that the major benefit of dimensionality reduction techniques is that the reduced order model is numerically well behaved. This was a surprising result since intuition would suggest that the reduced order model can be solved more efficiently because it has fewer degrees of freedom. However we showed that this is not necessarily the case. This result has shed new light on the role of reduced order models and gave us several ideas about how to develop new algorithms based on this insight. Our second main result concerns an exact mathematical estimation of error bounds for stochastic models that capture multiscale dynamics across both time and space. Using modern optimisation techniques we were able to show rigorous bounds for spatial models with singular perturbations across space. It is important to note that while some of these models have been studied for over 30 years, the magnitude of the error due to the dimensionality reduction was (until now) impossible to estimate. During the second part of the project, we made two important contributions. The first concerns the global optimization of problems arising in energy systems. Here we developed a multigrid algorithm based on semidefinite programming relaxations of polynomial optimizations problems. By taking advantage of sparsity and the geometric information between the different levels we were able to solve much larger models than existing methods.The algorithm above can be used to solve global optimization problems. We also developed new algorithms for convex models, that are extremely large scale and therefore only first order information can be used. Using recent advances in proximal algorithms we developed new methods for large-scale models that can compute local minima of nonconvex models.Finally, we applied our algorithms to large-scale calibration and integrated energy models. For the calibration models, we showed that our methodologies can be used to obtain stable models for large models using several years of data. Finally, using the new algorithms we were able to solve integrated energy models that were currently unsolvable with state of the art algorithms.
