Jul 19, 2024

# Mathematical Optimization: A Powerful Tool for the Energy Industry

With mathematical optimization, you input your goals, constraints and decision variables into software that tells you exactly how to achieve your goals, given your constraints and the decision variables.

By: Ed Klotz and Nell-Marie Colman

Although artificial intelligence and machine learning have captured the world’s imaginations, a powerful but lesser-known technology is at work all around us, touching our lives every day.

It’s the unseen force behind daily conveniences — calculating the quickest route on your mobile phone, ensuring timely package deliveries, and streamlining your airline and hotel bookings.

It’s at work in the energy industry, too — addressing challenges associated with battery storage, the variability of wind and solar power generation, and the design and operation of complex energy industry processes.

This technology is called “mathematical optimization,” and although it’s not a household name, it has been transforming the way enterprises make decisions for decades.

## What Is Mathematical Optimization?

Some may define it as “a powerful decision-making technology” or “a set of algorithms for solving complex problems.” But at its heart, it involves three key elements: a goal you want to achieve, the constraints you’re facing and the questions you’re asking.

For example, in the electric power industry, providers must produce electricity as efficiently as possible. That’s a goal.

But they face constraints involving customer demand; the physics of electrical power flow; available raw sources of energy (e.g., coal, water, sunlight and wind); and operational resources.

They also face questions about which power sources to use, when to use them, and at what usage level. These are the decision variables.

How can you solve a complex problem like this? With mathematical optimization.

With mathematical optimization, you input your goals, constraints and decision variables into software that identifies the optimal answer to your questions. In other words, it tells you exactly how to achieve your goals, given your constraints and the decision variables.

## Is Mathematical Optimization the Same as Machine Learning?

Machine learning and mathematical optimization are both data analytics technologies. But they’re fundamentally different. Machine learning can predict the future based on the past. Mathematical optimization can identify your best path forward, regardless of what’s happened before or what happens next.

Data scientists are beginning to combine these two technologies. They are using mathematical optimization to identify the best path forward, given machine learning’s predicted future.

For the electric-power example, machine learning can be used to predict fluctuating customer demand and energy produced by more variable sources, such as wind and solar. Given this predicted data, mathematical optimization can then identify the most cost-effective way to meet customer demand while satisfying operational constraints.

Mathematical optimization supports different types of models, but they all share a notion of a goal or objective to be minimized or maximized. For example, the overall cost of power generation could be minimized. The total power generated could be maximized. The models also include constraints imposed by limited resources, often associated with a physical system.

## Taking a Peek Under the Hood

If you’re interested in the science behind mathematical optimization, here’s a quick lesson.

The simplest type of mathematical-optimization model is the linear program. It includes a linear expression of the decision variables to be minimized, as well as constraints involving other linear expressions of decision variables associated with the limitations of the physical system.

Consider a very simple electric-power model. The linear objective could be minimizing the total cost — the sum of the costs per kWh of each type of power source multiplied by the amounts of power that were generated by each source.

A linear constraint could involve meeting overall energy-demand requirements. Specifically, the sum of the amounts of each type of available power source must match or exceed the total demand.

Another example of a linear constraint is ensuring that the power flowing into a node on the grid equals the power flowing out.

With the introduction of discrete decisions, nonlinear objectives and nonlinear constraints, the models can quickly become elaborate and computationally challenging. For example, complexity increases when you include a yes/no decision about placing a solar panel in a particular location on a rooftop^{1} or deciding how many turbines to build on a wind farm.^{2}

In these examples, a fractional value of the decision variable has no practical meaning. After all, you can’t build a fourth of a turbine or place a tenth of a solar panel.

The decision variables in a linear program can sometimes take on fractional values. That’s essential when dealing with continuous variables like the usage level of power sources in an electric-power model — or the amount of fluid flowing through the system of a concentrated solar power plant.

## Mathematical Optimization’s History

In the 1960s, the oil industry was an early adopter of mathematical-optimization software. At the time, software and computers were far less evolved than today — so practitioners often used linear programming, the simplest mathematical-optimization algorithm.

The heavy use of linear programming by the oil industry in the 1960s and 1970s led to significant improvements in linear-programming software for all industries.

The electric-power industry knew about mathematical-optimization software in its early stages, but the software couldn’t meet its needs until later.

For example, unit commitment problems — such as scheduling generators to meet demand at minimum cost^{3 }— involve discrete variables such as yes/no decisions to access a power source. This type of problem is a mixed-integer program, not a simple linear one.

By the late 1980s, state-of-the-art mixed-integer programming was still unable to solve practical unit commitment problems effectively. But the 1990s saw huge improvements, with the software surpassing the performance threshold needed to solve these models.

By 2010, most Independent System Operators and other organizations that manage grids and electricity markets used mixed-integer programming software to solve real-time, practical unit commitment problems for day-ahead and real-time markets.

In the last decade, mathematical-optimization software has continued to improve and can now solve extremely complex problems. Today, the energy industry can model and solve more precise representations of physical systems instead of approximations. The AC Optimal Power Flow Problem, which minimizes the cost of power production while adhering to the physics of the generators and transmission lines, is a noteworthy example.^{4, 5}

Furthermore, the tried-and-true industrial applications of linear programming remain relevant as well. For example, since the transition to electric vehicles will not be immediate, a need remains for efficient blending and pooling of crude oil to produce gasoline.^{6, 7} Blending is a linear problem and pooling is a nonlinear problem.

## Mathematical Optimization Use Cases in the Energy Industry

While continued advances in the core technologies that determine the effectiveness of renewable energy sources are most important, mathematical optimization has a major role to play. Let’s look at some well-established use cases:

## Concentrated Solar Power Plant Operations

Figure 1 illustrates how a common concentrated solar power plant operates.

An array of mirrors concentrates reflected sunlight to heat fluid inside a receiver. The heated fluid creates steam that spins a turbine, which generates electricity for the power grid. When the thermal energy in the fluid is exhausted, the cooled fluid is recycled back to the receiver.

The flow through the system involves products of continuous variables. The system can be optimized to maximize the electricity generated by solving a mathematical-optimization model involving quadratic expressions involving products of continuous decision variables.

These capabilities are relatively new, thanks to significant progress over the past five years in the algorithms and software for solving such quadratically constrained models.

## Recycling of Lithium-ion Batteries^{9}

Lithium-ion batteries are important for electric vehicles. However, they also play an important role in renewable energy storage, which can reduce the variability associated with renewable energy sources like wind and solar.

We will soon face a large number of spent lithium-ion batteries from electric vehicles. Since they still contain valuable critical materials, it will be important to recycle them efficiently — a process that involves multiple steps across the supply chain.

Mathematical optimization can help with decisions about when to build the sorting, acid production and bioleaching facilities that are key components of the supply chain while meeting the constraints associated with facility operation.

This technology can also increase the efficiency of the environmentally friendly bioleaching process, optimize the supply chain, and calculate price ranges of various materials to assess the economic viability of the recycling process.

## Additional Variability Associated with Renewable Energy Sources

We’re all familiar with the variability of sunlight and wind. But variability can stem from other issues, too — like disruptive weather events and changes in customer demand.

Net metering, for example, can result in customers selling energy to utilities from their own solar or wind systems. Improvements in battery-storage technology can help reduce variability, but that is a gradual process. Additional changes in demand may arise as purchases of plug-in hybrid or electric vehicles increase.

So how can a power plant approach short-term daily operations decisions and long-term infrastructure planning, given all of this uncertainty? With mathematical optimization.

For example, machine learning can help predict demand — and mathematical optimization can then identify the best way forward, given that predicted future. Or you can use stochastic mathematical optimization,^{10} which considers all of the variability and then outputs the best solution.

## Long-Term Power-Grid Capacity Planning

The National Renewable Energy Laboratory’s (NREL) Regional Energy Deployment System^{11} (ReEDS) models electricity generation and transmission in the U.S. power grid, providing detailed accounting of the role played by renewable energy sources.

The model supports a time horizon ranging from the present day to 2050 and beyond. Simulation and mathematical optimization are at the core of the model.

The model can help assess the long-term impact of different policies, including renewable energy tax credits or renewable portfolio standards, as well as changes in technology or fuel costs. Technology costs can be varied with different scenarios (e.g., high renewable cost and low natural gas cost or vice versa).

Once the scenarios and other input data have been specified, ReEDS solves a linear program to determine how the power generation and transmission systems evolve over time.

Decision variables include the amounts of new generation, transmission and storage capacity across several representative days in the different regions of the United States.

Constraints include the needs to meet customer demand and provide adequate reserves, operational constraints involving power generation and transmission, emissions constraints that influence power sources used, regulatory constraints and renewable portfolio standards.

The computed values of the decision variables from ReEDS are used to calculate electricity prices, marginal emission rates and several other outputs at the hourly level in a separate component of the ReEDS system, called the Cambium module.

Results from the end of a scenario run provide projections on the effects of the initial policy and economic inputs. The NREL’s Scenario Viewer^{12} facilitates the visualization and analysis of these results.

The open-access ReEDS model currently has over one thousand users — mostly academics, government staff and consultants. ReEDS developers are working on the extensions needed for utility companies to use the tool.

## Location Problems

The “facility location problem”^{13} is a classic mathematical optimization problem. It involves identifying the best location for a facility, given your objectives and constraints.

Although many facility-location problems are generic, operational details can introduce unique constraints. For example, when identifying an optimal location for a plant that produces green ammonia,^{14} you must be sure to have access to renewable energy.

Or suppose you need to optimally locate electric vehicle charging stations.^{15} Your solution will depend on electric vehicle traffic flow and charger speed. Charger speed, in turn, will be affected by budgetary constraints.

## Resources for the Energy Industry

Many other applications of mathematical optimization exist in the renewable energy space. The annual meeting of the Institute for Operations Research and the Management Sciences^{16} is a large conference with tracks on different areas of operations research across numerous industries, including the energy industry and renewables, so it’s a great place to start.

**Sources **

- Hillier, F. S., Lieberman, G. J., & Veatch, M. H. (2024).
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*This article was originally published in Solar Today magazine and is republished with permission.*