New theory could improve design and operation of wind farms

Propeller and wind turbine blades are designed according to aerodynamic principles first described mathematically more than a century ago. But engineers have long understood that these formulas don’t work in all situations. To compensate, they have added ad hoc “correction factors” based on empirical observations.

For the first time, MIT engineers have developed a comprehensive physics-based model that accurately represents airflow around rotors, even under extreme conditions, such as when the blades are operating at high speeds and forces, or are oriented in certain directions. This model could improve the design of the rotors themselves, but also the way wind farms are built and operated.

The new findings are described in the journal Nature Communicationsin an open-access paper by MIT postdoc Jaime Liew, MIT doctoral student Kirby Heck, and Michael Howland, the Esther and Harold E. Edgerton Assistant Professor of Civil and Environmental Engineering.

“We’ve developed a new theory for rotor aerodynamics,” Howland says. The theory can be used to determine the forces, flow velocities and power of a rotor, whether it’s extracting energy from the airflow, as in a wind turbine, or applying energy to the flow, as in a boat or airplane propeller. “The theory works both ways,” he says.

Because the new model is based on fundamental mathematics, some of its implications could be applied immediately. For example, wind farm operators must constantly adjust various parameters, including the orientation of each turbine, its rotation speed and the angle of its blades, to maximize energy production while maintaining safety margins. The new model can provide a simple and rapid way to optimize these factors in real time.

“That’s what we’re so excited about: it has immediate and direct potential to impact the entire wind energy value chain,” Howland says.

Momentum modeling

Known as impulse theory, the previous model of how rotors interact with their fluid environment (air, water, or other) was originally developed in the late 19th century. Using this theory, engineers can start with a given rotor design and configuration and determine the maximum amount of energy that can be derived from that rotor, or conversely, if it is a propeller, the amount of energy required to generate a given amount of propulsive force.

The equations of momentum theory “are the first thing you read in a wind energy textbook, and they’re the first thing I talk about in my classes when I teach wind energy,” Howland says. From this theory, physicist Albert Betz calculated in 1920 the maximum amount of energy that could theoretically be extracted from wind. Known as the Betz limit, this amount represents 59.3 percent of the kinetic energy of the incoming wind.

But a few years later, other researchers discovered that the momentum theory broke down “pretty dramatically” for higher forces corresponding to faster blade speeds or different blade angles, Howland says. It fails to predict not just the amount, but even the direction, of changes in thrust force at higher speeds or different blade angles: While the theory said the force should start to decrease beyond a certain speed or blade angle, experiments show the opposite: The force keeps increasing. “So it’s not just a quantitative error, it’s a qualitative error,” Howland says.

The theory also falls apart when there is misalignment between the rotor and the airflow, which Howland says is “ubiquitous” in wind farms, where turbines are constantly adjusting to changes in wind direction. In fact, in an earlier paper published in 2022, Howland and his team found that deliberately slightly misaligning some turbines relative to the airflow entering a wind farm significantly improves the wind farm’s overall power output by reducing wake disturbances from downstream turbines.

In the past, when designing the rotor blade profile, the layout of wind turbines in a wind farm or the day-to-day operation of wind turbines, engineers have relied on ad hoc adjustments added to the original mathematical formulas, based on some wind tunnel tests and experience gained in wind farm operation, but without any theoretical basis.

To arrive at the new model, the team analyzed the interaction between airflow and turbines using detailed computer modeling of aerodynamics. They found, for example, that the original model assumed that a drop in air pressure immediately behind the rotor would quickly return to normal ambient pressure just a little downstream. But it turns out, Howland says, that as thrust continues to increase, “that assumption is increasingly inaccurate.”

And the inaccuracy occurs very close to the point of the Betz limit that theoretically predicts the maximum performance of a turbine, and therefore corresponds exactly to the desired operating regime for the turbines. “So we have the Betz prediction of where we should be operating the turbines, and within 10% of that operating set point that we think maximizes power, the theory completely breaks down and doesn’t work,” Howland says.

With their modeling, the researchers also found a way to compensate for the original formula’s reliance on one-dimensional modeling that assumed the rotor was always precisely aligned with the airflow. To do this, they used fundamental equations that were developed to predict the lift of three-dimensional wings for aerospace applications.

The researchers developed their new model, which they call the unified impulse model, from a theoretical analysis and then validated it using computational fluid dynamics modeling. In follow-up work that has not yet been published, they are conducting further validation using wind tunnel and field tests.

Basic understanding

One interesting result of the new formula is that it changes the calculation of the Betz limit, showing that it is possible to extract slightly more power than the original formula predicted. While this is not a significant change (on the order of a few percent), “it is interesting to see that we now have a new theory and the Betz limit, which has been the rule of thumb for a hundred years, is actually changed because of the new theory,” Howland says. “And that is immediately useful.”

The new model shows how to maximize the power of turbines that are not aligned with the airflow, something the Betz limit cannot account for.

The aspects related to the control of individual turbines and turbine groups can be implemented without requiring modifications to existing hardware in wind farms. In fact, this has already happened, based on previous work by Howland and his collaborators two years ago that addressed wake interactions between turbines in a wind farm, and which was based on existing empirical formulas.

“This breakthrough is a natural extension of our previous work on optimizing large-scale wind farms,” he says, because in conducting this analysis, they saw the shortcomings of existing methods for analyzing the forces at play and predicting the energy produced by wind turbines. “Existing empirically-based modeling simply didn’t achieve what we were looking for,” he says.

In a wind farm, individual turbines absorb some of the energy available to neighboring turbines, due to wake effects. Accurate wake modeling is important both for designing the layout of turbines in a wind farm, and also for the operation of the wind farm, determining at each moment how to adjust the angles and speeds of each turbine in the array.

Until now, Howland explains, even wind farm operators, manufacturers and designers of wind turbine blades had no way of predicting how a wind turbine’s power output would be affected by a given change such as its angle to the wind without using empirical corrections.

“That’s because there was no theory for it. So that’s what we worked on here. Our theory can tell you directly, without any empirical correction, for the first time, how you actually need to operate a wind turbine to maximize its power,” he explains.

Since the fluid flow regimes are similar, the model also applies to propellers, whether for aircraft or ships, as well as to hydro turbines such as tidal or river turbines. Although this aspect is not addressed in this research, “it is naturally present in theoretical modeling,” he explains.

The new theory exists as a set of mathematical formulas that a user could integrate into their own software, or as an open-source software package that can be downloaded for free from GitHub.

“This is an engineering model developed for rapid prototyping, control and optimization tools,” Howland says. “The goal of our modeling is to position the wind energy research field to move more aggressively in developing the wind capacity and reliability needed to address climate change.”

This article is republished with kind permission from MIT News (web.mit.edu/newsoffice/), a popular site covering the latest research, innovation, and teaching at MIT.