Problems in energy systems analysis require making decisions in the presence of different sources of uncertainty. On a finer time scale, we have to decide whether to use or store wind and solar energy, where to store it, and how to allocate energy resources in the presence of uncertainty in wind, weather, demand, and prices. On longer time scales, we need to make decisions about optimal investment decisions in the presence of uncertainty in rainfall and changes in energy supplies and prices, as well as changes in battery technologies, improvements in our ability to sequester carbon, the introduction of carbon pricing, and advances in our understanding of climate change.
We need models and algorithms that will solve these design and control problems, often because we need to understand how different components will interact, and how changes in technology and policy may affect the economics of different technologies. The fundamental equation guiding these algorithms is known as Bellman’s equation, but exact algorithms are limited in their capacity to solve massive and complex equations. In energy applications, we may face the need to solve time-dependent problems with hundreds of thousands of time periods and variables with tens of thousands of dimensions. In fact, it is not hard to create relatively small problems that would require centuries on a modern supercomputer to produce an exact solution.
We have developed a new class of algorithms based on approximate dynamic programming that combine tools from mathematical programming, machine learning, and simulation. These ideas were used in SMART, a stochastic, multi-scale model for the analysis of energy resources, technology, and policy. SMART is a multi-decade model for planning energy investments while capturing hourly variations in wind, solar output, demand, and prices. This model featured almost 200,000 time periods, which is well beyond the capabilities of commercial solvers running on supercomputers. By contrast, SMART produces high-quality solutions in a few hours running on a laptop.
In a separate study, the work of two undergraduates, Jessica Zhou ’10 and Ahsan Barkatullah ’12, was used to show that it is not enough to model the variability in the energy generated from wind; it is important to model the uncertainty in how much wind will be available. A new class of stochastic search algorithms, called the knowledge gradient, was used to produce near-optimal solutions for optimization models that control the use of generation plants in the presence of large supplies of energy from wind. This model showed that the error from ignoring uncertainty is substantial, and would distort studies that analyze the value of new storage technologies.