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The shaping of solutions to come

Mathematical optimization provides robust algorithms to find the best answer to decision-making problems that can be described using linear, discrete or convex equations on the problem’s decision variables. For example, it allows utility companies to find the lowest-cost mix of power plants to meet a given demand for electricity. For these algorithms, convexity is critical.

But what happens when you factor in the real-world physics of the electrical grid, and the need to balance the wear-and-tear of ramping generators up and down with the availability of relatively inexpensive renewable energy sources? Modeling these types of phenomena in decision-making problems requires non-convex equations. Moreover, using algorithms for linear, discrete or convex problems in these cases, leads to solutions that might not be best for the problem at hand.

Simplifying the equations of nonlinear phenomena so that decision-making problems can be solved with robust computational techniques already in place is the forte of Luis Zuluaga, assistant professor of industrial and systems engineering. NSF, the U.S. Air Force and the Pennsylvania Infrastructure Technology Alliance (PITA) have supported Zuluaga’s efforts.

Zuluaga’s NSF work centers on the mathematics of finding the best way to model complex physics in optimization problems. “You can construct mathematical problems that are simple to solve and give an approximation of reality,” he says. “I am working to make better approximations.”

Researchers already have algorithms, says Zuluaga, to solve problems that can be framed as the interaction of linear relationships, such as that between the capacity of generators and the demand placed on them.

Modeling the transmission losses and variable routing across the electrical grid, however, “is a highly nonlinear problem,” Zuluaga says, represented by functions that undulate and have many local minima and maxima that are difficult to solve for.

A common approach is to develop more complex algorithms to solve such problems. Zuluaga, however, reframes the situation in terms of polynomial expressions that can be quickly solved using tried-and-true algorithms for linear and convex problems. Instead of developing more solution algorithms, this technique, called “convex optimization,” simplifies the problem so it fits existing methods.

Zuluaga is working to expand this method to difficult problems such as deciding when the costs of ramping generators up and down outweighs the savings from using cheap renewable sources. With its variable conditions—the weather, the distance from a power plant to a wind or solar farm—this kind of problem contains multiple peaks and valleys in its mathematical representation that can’t be solved by traditional convex techniques.

Zuluaga’s technique carves the problem into one that can be solved using convex methods. His research explores the tradeoffs inherent in this process.