The nation’s drive toward a clean, secure, and low-carbon energy future puts enormous demands on its electrical energy systems and related infrastructure, including the electrical grid and its interdependencies with water infrastructure, industrial facilities, and commercial/residential buildings. New mathematics are needed to handle the complexities of new technologies, actively controlled electricity distribution, the competing goals of cost efficiency and increased reliability, and policies regarding decentralized generation and renewable energy. These complexities are expressed in various ways including very large system size, system features spanning an enormous range of space and time scales, variability and uncertainty in ambient and external conditions, and astronomical numbers of outcomes stemming from possible abnormal behaviors.

**To analyze, design, plan, maintain, and operate the nation’s electrical energy systems and related infrastructure in an optimal way is a DOE Grand Challenge. **Addressing this challenge requires us to reach deep into the foundations of applied mathematics and to develop deep mathematical understanding and effective algorithms to remove current bottlenecks in analysis, simulation, and optimization, by taking a holistic view of the complex systems application space.