There is something wonderfully simple about a wind turbine gently turning in the breeze.

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As the wind flows by the blades of the turbine, a rotating force is created that spins the giant assembly. The rotation is then converted into electricity, just like conventional power generation.

A wind turbine consists of a set of three blades defined by twisting and bending teardrop shapes. The turbine blades are cylindrical on one edge and sharper at the other.

This blade graphic shows how the wind turbine shapes change as they move from the hub of the turbine blade (left) to the tip on the right. Credit: NIST

The blade surface changes over its long, slender axis. It begins with blunt, thick shapes at the hub and transforms into slender, thin teardrop shapes at the tip. (See the graphic above for a closer look at the changing sequence of shapes moving from the hub on the left side to the tip on the right side.) A turbine spinning in the wind may seem simple, but designing and measuring the blades of a wind turbine using a sequence of changing cross-sectional shapes requires a lot of sophisticated geometry.

I’m part of a team studying abstractions of geometric methods to design better wind turbines with mathematics. In the future, we hope to use our methods to measure how well wind turbines work in the field and how well they are made.

Design and measurement are intertwined, so addressing the challenges of both will push us toward a cleaner energy future.

Using the laws of physics, we can build computer models to predict how air flowing around the blades of a spinning turbine generates power. When the surface of the blade changes, the forces that rotate the turbine also change.

Modeling how these forces change when the surface changes is a subject of aerodynamics. Aerodynamic design of a turbine blade’s shape is crucial to maximizing the power of each turbine, just as an airplane’s wings must be designed to keep it aloft during flight.

Two of the most important forces acting on aerodynamic surfaces are *lift* and *drag*. The force defying gravity using an aircraft wing is the same that spins a turbine—we call it lift. Perpendicular to lift, the force that works to slow the aircraft or turbine’s rotation is called drag.

With physics-based models, we can study how to best design the cross-sectional shapes of turbine blades, called *airfoils*. You can see some examples of wind turbine airfoils shown in the graphic. The solid, black outlines of cross-sections are airfoils. They are like slices taken from a giant blade-shaped loaf of bread. Better airfoil shapes create more lift with less drag.

While an aircraft wing has control surfaces that the pilot can change, a turbine blade’s airfoil shapes remain fixed. For wind turbines, the outer portion generates the most lift, while the inner portion supports the spinning blade. So improved airfoil shapes with greater lift near the tip and less drag near the hub equal more power.

The final blade surface, defined by a sequence of airfoil shapes, then needs to be manufactured, put into operation, and measured during operation. How we represent the shape for design has big implications for how we should measure it in operation. And tackling challenges through measurement science is what NIST does.

At NIST, the big question is: How do we measure and mathematically define shape? Can our work be used to design and measure the best possible wind turbine?

### Geometry makes the turbine go round

Wind turbine design requires physics and math. Computer models use the laws of physics to predict how air flowing around the blades of a spinning turbine generates power. Credit: Nick Brundle Photography/Shutterstock.

Working alongside the National Renewable Energy Laboratory (NREL), we’re developing new methods to describe and generate these complicated airfoil shapes to create next-generation designs and augment measurements in the field.

The first step is tackling the turbine blade design problem. Our goal is to get the most efficient power generation while satisfying size constraints to meet structural requirements. I’ll explain this using a few highlights from the scientific method.

The hypothesis

Separating rotation and size of each airfoil from all other changes in the shapes can help improve designs and measurements, with fewer total numbers that modify the shape’s representation in the computer, which we refer to as *parameters*. The parameters are what we modify to change the shapes. The more you have, the more complex the math problem becomes. Traditional approaches need anywhere from 100–200 parameters to define a blade.

The concept is simple. If we want precise control over specific changes, like rotation and size of each airfoil, we then “divide these effects out” mathematically and study whatever is left over as learned from data. Fewer parameters make it significantly easier for us to understand changes in the design.

Our applied geometric approach provides a new method for defining and exploring this concept with airfoil shapes learned from NREL shape data.

The experiment

The design approach taken at NREL involves solving one of the hardest problems in math—an inverse problem. An inverse problem sets the desired outputs by saying we need “XYZ” aerodynamics. We then work backward through the modeled physics to determine a collection of airfoils that satisfy the requested aerodynamics. Inverse problems are something NIST mathematicians consider all the time for measurement, and they can be challenging problems to solve—if we can solve them at all.

First, we generate tens of thousands of new shapes from our geometry model trained using NREL shape data. Then, we use physics-based models to predict aerodynamics of each shape. Finally, our colleagues at NREL train the AI to tell the geometry model how to achieve new designs with requested aerodynamics based on the generated data. This effectively “closes the loop” in the analysis.

The results

NREL has successfully used the geometric methods in a tool called G2Aero and generated new wind turbine designs to reduce the cost of energy. (More on this later.) Additionally, we trust the results of the AI because we have carefully constructed the description of the shapes, so the AI only makes recommendations about new shapes represented in our way. Most important, our geometry method prevents the AI from creating wild wiggling shapes that aren’t well suited to creating desirable aerodynamics or physics modeling.

Not only that, but the traditional approach involving hundreds of parameters was reduced to using just* four *parameters to describe an entire blade! This is due entirely to our data-driven geometry model—not the AI. With such a dramatic reduction in the total number of parameters, we can more easily train the AI. We can also put better checks and balances (constraints) on the AI recommendations and compare with alternative approaches for mapping requirements to parameters.

### Next steps

This research has a lot of exciting future applications. For example, wind turbines are often monitored remotely and serviced only when a problem is discovered—especially if those turbines are offshore. Using our geometry and physics-based modeling, we hope to augment remotely measuring turbines in operation to make sure they are working efficiently.

If we see something has changed with the power generation, we could work backward through the physics to see what has changed in the environment. For example, the blade could be damaged, or there could be a buildup of dirt/salt/sediment affecting the turbine blade’s aerodynamics.

This geometry research goes well beyond designing and measuring the big spinning blades. We believe we can improve basically every aspect of wind-generated power for lower cost—from designing floating offshore platforms for future wind turbines to measuring microscopic shapes hidden in the materials used to manufacture blades.

Problems requiring precise definition and measurement of shapes are everywhere. There’s a diverse set of applications for geometry-based research beyond designing and measuring wind turbines.

### Using math to make things better

I grew to love applied math during my undergraduate degree at Embry-Riddle Aeronautical University. After getting my degree, I started working at Rolls-Royce designing jet engines—no, not the cars.

One of my jobs was teaching designers and analysts how to incorporate statistical considerations into their complex physics-based modeling. These experiences further fueled my love of using applied math to achieve different perspectives on problems and make things more robust.

What I find most fascinating about studying applied math is that universal principles stay the same while the technology and applications of those principles are constantly changing. In many ways, for me, mathematics is the art of assigning language to logic. Math offers explanations and interpretations of so many different things in the natural world. If you seek an explanation or interpretation of any physical or artificial process, the answer is almost certainly hidden in the mathematics.

Applying universal principles of math in different contexts is one of the most enjoyable aspects of my work. For example, the same geometry methods we used for wind turbines can also be used to explain the complex microstructures found in steel and lithium-ion batteries. Other geometric perspectives can describe more optimal processes for telecommunications.

This type of research allows me to use these abstract, unfamiliar concepts and apply them to real-world problems and questions, such as:

• How do we build the best wind turbine?

• Why did that AI model make that decision?

• What is statistically different in the measurement of one material from another?

This geometry-focused research gives us rich explanations and interpretations of many different problems to unfold the complicated nature of shape in data.

### Calculating our way toward a clean energy future

In addition to studying challenging math problems, I’m excited to be a part of how we can generate clean energy in the future.

About 40% of the nation’s population (128 million people, excluding Alaska) live in coastal counties. One of the biggest challenges facing future energy generation is how we produce the energy, but also how we deliver it.

There are a growing number of wind turbines in the center of the country where the wind blows the most on land, but infrastructure challenges make it hard to get that wind power to the coastal areas. Not only that, but offshore turbines can be much larger, with nearly three times the potential power generation vs. land-based turbines. The wind is also expected to blow faster and more often offshore near coastal areas.

These considerations create an obvious demand for offshore wind power. But they also create more challenging design and measurement problems. We want to create the most efficient wind turbine systems, and we want to make them robust and resilient for future generations of offshore applications—a significantly more volatile environment with lots of severe weather.

Helping design next-generation turbines is only the first step. We must also measure the hard-to-reach offshore turbines while in operation to make sure they are holding up to the test of time and can still provide power to those who need it.

If we could redesign existing wind turbines with our geometry model, NREL estimates new designs could achieve about a 1% reduction in cost of energy. That may not seem like much, but if we could go back and redesign all the existing turbines, we could save approximately $288 million in 2021 alone.

However, wind energy production is expected to quadruple by 2050. In a very rough estimate, scaling these seemingly small 1% savings over the next 30 years of wind energy deployment could accumulate billions in total savings, compared to using existing methods.

Based on current energy goals, we expect our new methods can save Americans $60 million per year in energy costs if we assume offshore wind turbines are only generating electricity half the time. That’s a conservative estimate; the overall savings could be closer to $120–$180 million per year in potential savings achieved with our next-generation wind turbine designs.

Solving the problems of design and measurement for wind turbines blades has dramatic implications for improving energy costs. And NIST mathematicians, alongside NREL scientists, are equipped to tackle these issues to achieve the promise of clean, cost-effective energy.

It feels great to be part of building a future with clean, reliable energy resources. We face many challenges with a changing global economy and climate. And although wind energy is only one piece of a very complicated puzzle, I’m proud of the contributions we’ve made to the wealth of scientific discoveries working toward a safer, more robust, and cleaner future for the nation.

*Author’s note: Special thank you to Andrew Glaws and Ganesh Vijayakumar at NREL for providing estimates summarizing results of applying our geometry methods, as well as the rest of the ARPA-E DIFFERENTIATE team, including: Ryan King, Olga Doronina, James Baeder, and Bumseok Lee.*

*Published Jan. 10, 2024, by NIST.*

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