# Novel mathematical model to assess wind stability of solar trackers

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Researchers at the Technical University of Madrid in Spain have introduced a novel way to calculate the effective damping coefficient of single-axis solar trackers. In aerodynamics, damping is the force that opposes the motion of a vibrating or oscillating system, and its effective coefficient is a parameter regarding its stability.

With the proposed approach, developers can estimate the critical speed a tracker can withstand.

“We do not rule out extending our research to two-axis tracking systems,” corresponding author Juan A. Cárdenas-Rondón told pv magazine. “The main challenges to be addressed include dealing with two degrees of freedom in torsion as well as three-dimensional aerodynamic phenomena.”

The researchers are planning to increase the number of tested cases to develop a mathematical model.

In the study “Stability analysis of two-dimensional flat solar trackers using aerodynamic derivatives at different heights above ground,” published in the Journal of Wind Engineering & Industrial Aerodynamics, the research group attempted to create as general as possible a methodology to determine the stability curve of a PV tracker.

The mathematical model they developed is dependent on several parameters, such as the chord length of the solar tracker's supporting structure, the height of the solar tracker axis, and the angle of the air's attack on the rotating PV.

Moreover, the model requires using two derivatives – one related to aerodynamic damping and the other to stiffness. While damping relates to the force that opposes the motion, stiffness refers to the system's resistance of an object to deformation. Those derivatives were obtained with a series of lab tests with different angles of attack and standard axis height (H) to chord (B) ratios.

“The values that have been tested are H/B = 0.3, 0.4, 0.5, 0.6, 1, and 2,” the academics stated. “For each of the height-to-width values, the nominal angle of attack has been varied. The angles tested are 0◦, ±5◦, ±10◦, ±20◦, ±30◦, ±40◦. The tests were limited to angles within the range of ±40◦ because, for lower H/B ratios, the model could not be tested at higher absolute nominal angles as it would collide with the ground.”

After finding both derivatives for all of these different conditions, the scientists were able to calculate the effective damping coefficient of solar trackers as a function of the incident wind speed. They then conducted a series of tests on an additional experimental setup to validate their mathematical model.

“The results obtained with the model exhibit a good agreement with the results obtained in the validation tests performed in the auxiliary experimental setup for the case of large angles of attack,” they stated. “Conversely, for small angles of attack, it can be noted that the value of the maximum critical reduced speed has been accurately determined, but both results do not align regarding the angle of attack at which it is achieved.”

The team also found that, as for the oscillation frequency, the model forecasts the experimental results for low reduced speeds. “But as the system approaches unstable conditions, the predictions become overestimated,” it concluded.

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