Optimal relationship between power and design-driving loads for wind turbine rotors using 1-D models

• Main Authors: K. Loenbaek, C. Bak, J. I. Madsen, B. Dam
• Format: Article
• Language: English
• Published: Copernicus Publications 2020-01-01
• Series: Wind Energy Science
• Online Access: https://www.wind-energ-sci.net/5/155/2020/wes-5-155-2020.pdf

<p>We investigate the optimal relationship between the aerodynamic power, thrust loading and size of a wind turbine rotor when its design is constrained by a static aerodynamic load. Based on 1-D axial momentum theory, the captured power <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M1" display="inline" overflow="scroll" dspmath="mathml"><mover accent="true"><mi>P</mi><mo stretchy="false" mathvariant="normal">̃</mo></mover></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="9pt" height="12pt" class="svg-formula" dspmath="mathimg" md5hash="a2ad6297981715cca1b30423afa073ec"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="wes-5-155-2020-ie00001.svg" width="9pt" height="12pt" src="wes-5-155-2020-ie00001.png"/></svg:svg></span></span> for a uniformly loaded rotor can be expressed in terms of the rotor radius <span class="inline-formula"><i>R</i></span> and the rotor thrust coefficient <span class="inline-formula"><i>C</i>_T</span>. Common types of static design-driving load constraints (DDLCs), e.g., limits on the permissible root-bending moment or tip deflection, may be generalized into a form that also depends on <span class="inline-formula"><i>C</i>_T</span> and <span class="inline-formula"><i>R</i></span>. The developed model is based on simple relations and makes explorations of overall parameters possible in the early stage of the rotor design process. Using these relationships to maximize <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M6" display="inline" overflow="scroll" dspmath="mathml"><mover accent="true"><mi>P</mi><mo mathvariant="normal" stretchy="false">̃</mo></mover></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="9pt" height="12pt" class="svg-formula" dspmath="mathimg" md5hash="1541391186c380c03e6e292cc4f0556a"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="wes-5-155-2020-ie00002.svg" width="9pt" height="12pt" src="wes-5-155-2020-ie00002.png"/></svg:svg></span></span> subject to a DDLC shows that operating the rotor at the Betz limit (maximum <span class="inline-formula"><i>C</i>_P</span>) does not lead to the highest power capture. Rather, it is possible to improve performance with a larger rotor radius and lower <span class="inline-formula"><i>C</i>_T</span> without violating the DDLC. As an example, a rotor design driven by a tip-deflection constraint may achieve 1.9&thinsp;% extra power capture <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M9" display="inline" overflow="scroll" dspmath="mathml"><mover accent="true"><mi>P</mi><mo stretchy="false" mathvariant="normal">̃</mo></mover></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="9pt" height="12pt" class="svg-formula" dspmath="mathimg" md5hash="36f0243125da618735c253ab94171173"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="wes-5-155-2020-ie00003.svg" width="9pt" height="12pt" src="wes-5-155-2020-ie00003.png"/></svg:svg></span></span> compared to the baseline (Betz limit) rotor.</p> <p>This method is extended to the optimization of rotors with respect to annual energy production (AEP), in which the thrust characteristics <span class="inline-formula"><i>C</i>_T(<i>V</i>)</span> need to be determined together with <span class="inline-formula"><i>R</i></span>. This results in a much higher relative potential for improvement since the constraint limit can be met over a larger range of wind speeds. For example, a relative gain in AEP of <span class="inline-formula">+5.7</span>&thinsp;% is possible for a rotor design constrained by tip deflections, compared to a rotor designed for optimal <span class="inline-formula"><i>C</i>_P</span>. The optimal solution for AEP leads to a thrust curve with three distinct operational regimes and so-called thrust clipping.</p>