It’s hard enough solving math problems of the last century’s grid, with its central generation and one-way power flows. But it’s a lot harder to solve the math problems of the renewable energy-powered grid we’re building this century.

That’s largely because the new grid requires a far more thorough treatment of the variability and uncertainty of supply and demand in a system where more and more energy is affected by the vagaries of sun and wind. This switch from deterministic to stochastic models is a bit like gaming out the odds of every possible combination of outcomes -- a task that requires a lot of computation power and clever ways of applying it.

The Department of Energy’s ARPA-E program has been funding projects aimed at solving these problems for years now, starting in 2012 with its Green Electricity Network Integration (GENI) program. The blue-sky research agency has since made grid integration a significant focus, with investments in distributed-energy-enabled grid controls, open-source grid power flow modeling software, and other key pieces to understanding the new grid.

Take NewGrid, a startup recently spun out of Boston University based on three years of R&D funded by the GENI program. At the ARPA-E Innovation Summit earlier this month outside Washington, D.C., the startup was showing off how it has applied high-speed computation techniques to tap a latent grid-balancing resource for transmission grid operators -- the thousands of circuit breakers scattered across the high- and medium-voltage network.

These circuit breakers are there to prevent catastrophes, Pablo Ruiz, NewGrid’s president and CTO, said. They’re also set differently for different seasons and grid conditions to optimize the flow of power across the grid at large, and to avoid congestion in some areas or disuse in others -- a task known as “topology control.”

While all of these circuit breakers are connected by real-time SCADA systems, they aren’t typically run in response to real-time conditions, Ruiz said. That’s largely because grid operators lack the technology to analyze, interpret, and react fast enough.

The math problems to be solved are simply too huge, with too many variables, he added -- Mid-Atlantic grid operator PJM, for instance, has 220,000 possible combinations of circuit breakers to compute, or trillions of options.

NewGrid’s approach manages this huge mathematical task via reformulated mixed integer programming, or as Ruiz put it, “how to optimize things that are discrete -- zeros or ones, open or closed. Our technology identifies those paths, and suggests interruptions to that flow, while also making sure there are enough alternative routes to allow power to flow reliably,“ he said. “We can find them in a matter of minutes, safely and effectively.”

NewGrid’s booth at ARPA-E featured a video-screen map of 15-minute wholesale power prices across a swath of PJM territory, stretching from Illinois to New Jersey, with red spots emerging in some areas, indicating congestion. As the software searches out, solves for and throws circuit breakers in different combinations, the red splotches of congestion clear up, and wind power curtailments drop, as this before-after display from a July 2015 presentation indicates.

NewGrid hasn’t had a chance to test its software out in real-world grid operations yet, Ruiz said. But PJM has provided its proprietary market data and grid models to simulate the software’s operations during typical congestion scenarios -- and those show the potential to relieve between 30 percent and 50 percent of the costs of congestion, he said.

That’s all without the need to install new hardware, he noted. But NewGrid’s software could make use of hardware from other companies funded by ARPA-E’s GENI program, such as Smart Wires and Varentec, that bring additional flexibility to how electricity is routed through modern grid networks, Ruiz said.

This isn’t the only example of putting reformulated mixed integer programming to work on transmission computational challenges. Another ARPA-E project, featuring the Midcontinent Independent System Operator (MISO), software provider Gurobi Optimization, and GE Grid Solutions (formerly Alstom Grid), is applying it to faster and more accurate solutions to the Security-Constraint Unit Commitment problem – the calculation that allows grid operators to rank generation commitments based on a combination of reliability and lowest cost.

Pacific Northwest National Laboratory (PNNL) was demonstrating its own transmission grid math at the ARPA-E Summit, with a focus on delivering dynamic line rating capabilities to grid operators. The idea is to replace today’s static models with real-time calculations to know how much power transmission lines can handle at any one moment, PNNL’s Ruisheng Diao explained. That could unlock otherwise wasted capacity -- up to 30 percent, according to models run with the Bonneville Power Administration, he said.

Tim Heidel, ARPA-E program director, noted in an interview last week that a new ARPA-E program should make this kind of grid modeling and simulation a lot easier in years to come. It’s called GRID DATA, or “Generating Realistic Information for the Development of Distribution and Transmission Algorithms,” and last month it awarded $11 million to seven projects meant to create “large-scale, realistic, validated and open-access power system models that have every last detail you want,” he said.

Today, publicly available grid models aren’t realistic enough to test new software for real-world applications, Heidel said. But the models that are detailed and realistic enough, typically those developed by grid operators and utilities, come encumbered with proprietary data restrictions and non-disclosure agreements, which can prevent today’s grid analyses from being shared with the rest of the world, he added.

“My hope is that these models are really useful to the new startup that’s trying to achieve industry traction,” he said. As these open-source models are validated and become available through the GRID DATA program, startups like NewGrid working on new ideas “could speed up their development time substantially,” he said.