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

This method is extended to the optimization of rotors with respect to annual energy production (AEP), in which the thrust characteristics

From the inception of the wind energy industry, it has been a clear trend that rotor sizes have been
increasing. However, as discussed in

Another concept that is relevant in the context of this paper is thrust clipping (also known as peak
shaving or force capping). For turbines, it is often the case that the maximum thrust is reached just
before reaching the rated power, resulting in a so-called thrust peak. When using thrust clipping,
this peak is lowered at the cost of power. It is used with many contemporary turbines for load
alleviation but is often added as a feature after the design process.

In this paper, we investigate the relationship between the load, power and structural response of wind turbine rotors. Simple analytical models, based on 1-D aerodynamic momentum theory and Euler–Bernoulli beam theory, are introduced to establish the first-order relationship between these responses. This provides a useful framework for the initial rotor design, especially when high-level design parameters such as the rotor radius need to be fixed or there is a need to understand how load/structural responses will change with rotor size. The effect on the power curve and the related load/structural response with the variation in wind speeds is also investigated, which is useful for the initial design of the highly coupled aeroservoelastic rotor design problem.

The relatively simple models used in this paper do not capture the full complexity needed for detailed wind turbine rotor design and should be considered a tool for early-stage rotor design and overall exploration only. For example, the underlying theories (of 1-D aerodynamic momentum and Euler–Bernoulli beams) assume steady-state conditions, while designs are often constrained by load cases that are linked with extreme, unsteady or non-normal operational events, e.g., extreme turbulence, gusts, emergency shutdowns, subsystem faults or parked conditions. This is a limitation of the model developed here, but if there is a relation between the steady-state loads and the extreme loads, which is very likely, then the results are still valid.

As mentioned before, the overall target for current turbine design is to lower the CoE, but a cost model is not used, which is also a limitation of this study. However, cost models use several assumptions made in the design process such as the price of components in the design or composite lay-up of the blades, so a predicted cost will always be made with some uncertainty. Instead, load constraints are considered, much like in the above-mentioned LIR example. As was found by

This study is carried out in order to obtain an overview of how the rotor design is more fundamentally influenced by different types of aerodynamic loading. Thus, an issue like the self-weight is important for modern turbines but is not directly included in this study; the static-mass moment especially has an impact on contemporary turbines. It could be included, but it was excluded to keep the study as simple as possible. Further discussion about the limitations and possible improvements of the study is given later in Sect.

This section will introduce the variables and the basic relationships used in this paper. It is split into two subsections, in which
Sect.

The theory underlying this Aerodynamics section is found in

For wind turbine aerodynamics non-dimensional coefficients are often introduced and some of the common ones are for the rotor thrust (

These definitions can be applied for any wind turbine rotor, but in this paper, we will use a simplified relationship between

Relationship between normalized rotor load

One way to understand the power yield of a rotor is to consider
Eq. (

When considering turbine design over the range of operational conditions, annual energy
production (AEP) is introduced as an integral metric representing the energy produced per year
given some wind speed frequency distribution. It can be computed as the power production (

In this paper, we will use a dimensionless measure for AEP which is equivalent to the so-called capacity factor, defined as follows:

The work here aims to demonstrate an improved rotor performance compared to a baseline design. This baseline design is chosen to be a turbine operating at the Betz limit below the rated wind speed and keeping a constant power above the rated power.

In this paper, all results are presented as the change in performance relative to that of the baseline
rotor. For this reason, all of the relevant variables (denoted with a zero in the subscript) will be
normalized by the corresponding baseline rotor values.

In this section, examples of static aerodynamic design-driving loads (DDLs) will be presented. These examples are not meant to be exhaustive but include several of the key considerations that constrain the practical design of wind turbine rotors. From the scaled loads, design-driving load constraints (DDLCs) are introduced, which limit loads so that these do not exceed the levels of the baseline rotor. Based on the DDL examples, it is shown that DDLCs can be elegantly put in a generalized form.

Thrust typically does not limit the design of the rotor itself but more likely is a constraint imposed from the design of the tower and/or foundation. The thrust scaling and the associated DDLC is given by

The root flap moment is the bending moment at the rotational center in the axial flow direction. To compute

Tip deflection is a common DDLC for contemporary utility-scale turbines, where tip clearance between tower and blade may become critical because of the relatively long and slender blades. To get an idea of how tip-deflection scales with changes in loading and rotor radius Euler–Bernoulli beam theory;

With the equation for

The final example of a DDL is also based on tip deflection but includes a condition to maintain a constant mass of the load-carrying structure of the blade. To this end, the stylized spar-cap layout depicted in Fig.

Assumed spar-cap structure with dimensions:

Considering the four DDLC examples presented above, there appears to be a pattern in the scaling relations that may be written as follows:

If the constraint limit is met, the following relationship can be written

Based on the performance and constraint relationships outlined in the previous section, this section will present the formulation for rotor design as optimization problems. Two different classes of problems are introduced, namely

The optimization problem can be stated as

It should be noted that this optimization problem is similar to the problem that is given by

In contrast to the above mentioned optimization of power capture, optimization with respect to
AEP requires the determination of

This section discusses the solutions to the rotor design optimization problems introduced in the previous section.

The constrained optimization problem maximizing power capture, as stated in Sect.

This unique solution is a maximum, which is apparent from the always-positive value
of

When maximizing power capture for a given thrust (

Even though meeting the constraint limits means that the chosen DDL will be the same as the baseline, it is interesting to know what happens to loads that scale differently than the DDL. As an example, if the DDLC is

To investigate it we will introduce a

An equation for the relative change

Relative change in different rotor load parameters (

Sketch of a turbine with the load/structural response outlined. The zoomed-in figure shows the radius increase (

The concept in this section was mentioned in the Introduction since it has had some attention over the recent years. The low-induction rotors (LIR) are rotors designed with a lower axial induction

Power and thrust curves for a low-induction rotor (solid lines), designed using the present method with the DDLC exponent

To investigate such an LIR design, it was chosen to fix the

The LIR is illustrated by the examples in Figs.

Power and thrust curves for rotor with the DDLC exponent

In both cases, the resulting power curves are slightly above the equivalent baseline ones, and the thrust peaks are reduced compared to the baseline. The relative change in AEP results in a smaller change than the change in power at the design point. For the case with DDLC(

Sketch of a turbine with the load/structural response outlined. The zoomed-in figure shows the radius increase (

As mentioned in Sect.

operation with maximum power coefficient (

operation at constraint limit (constant thrust

operation at the rated power.

Power and thrust curve for an AEP-optimized rotor (solid lines) where the DDLC exponent is

Power and thrust curve for an AEP-optimized rotor (solid lines) where the DDLC exponent is

The only free parameter that needs to be determined to find the optimal AEP is

Examples of the resultant power and thrust curves can be seen in Figs.

DDLC exponent (

DDLC

A more realistic design for modern turbines is found in Fig.

In Fig.

Sketch of a turbine with the load/structural response outlined. The zoomed-in figure shows the radius increase (

In Fig.

In Table

Overview of the optimization results from optimizing power capture (Opt. PC), low-induction rotor (Opt. LIR) and annual energy production (Opt. AEP).

As seen from the tables, the largest increase in

In all three optimization cases, the optimization of the design with thrust constraint (DDLC(

The study shows that for a rotor constraint by a static aerodynamic DDL there is a benefit to lowering the loading and increasing the rotor size in terms of power/AEP. But, as it was found by

Another issue that is not taken into account in this study is the influence of the turbines self-weight.
As was found by

The fidelity of the models is also a limitation. Even though 1-D aerodynamic momentum theory is a common approximation to do for first-order studies in rotor design, it is well known that the constantly loaded rotor is not possible to realize, and when losses are included the constantly loaded rotor is not the optimal solution anymore. At the same time, if it was possible to decrease the load at the tip more than at the root, it would lead to less tip deflection than a constantly loaded rotor with a similar

For modern turbine design, it is often the case that the structural design is determined by the aeroelastic extreme loads, such as extreme turbulence or gusts. With the simplicity of the models in this study, this is not taken into consideration. But if the extreme load happens in normal operation there will likely be a direct relationship between the steady and extreme loads, meaning that a decrease in steady loads will also lead to a decrease in the extreme load. This is an assumption that should be tested in future work. If the design-driving load is happening in nonoperational conditions, e.g., extreme wind in parked conditions, grid loss or subcomponent failure, then the analysis tool cannot be directly applied.

A first-order model framework for the analysis of wind turbine rotors was developed based on aerodynamic 1-D momentum theory and Euler–Bernoulli beam theory. This framework introduces the concept of design-driving load (DDL) for which a generalized form has been developed in which loads only differ by a scaling exponent

The optimization of power capture determines the best possible design for a given wind speed. By considering the annual energy production (AEP), an optimal design across the range of operational wind speeds can be found for a given wind speed frequency distribution. Optimal AEP was considered with two different approaches, namely the low-induction rotor (LIR) and full AEP optimization. For LIR, the

For the full AEP optimization,

operation with maximum power coefficient (

operation at constraint limit (constant thrust

operation at the rated power.

In spite of relatively crude model assumptions made, this paper provides profound insight into the trends of rotor design for maximum power/energy, e.g., the use of thrust clipping. As wind turbine rotors continue to develop towards larger diameters with slender (more flexible) blades, the type of design-driving load constraint also evolves. With the present model framework, the conceptual implications of this development become clearer; an increase in AEP of up to 5.7 % is possible compared to a traditional

No data sets were used in this article.

KL came up with the concept and main idea, as well as made the analysis. All authors interpreted the results and made suggestions for improvements. Also, some modeling has changed based on discussions between the authors. KL prepared the paper with revisions from all coauthors.

The authors declare that they have no conflict of interest.

We would like to thank Innovation Fund Denmark for funding the industrial PhD project that this article is a part of. We would like to thank all employees at Suzlon Blade Science Center for being a great source of motivation with their interest in the results. We would like to thank all people at DTU Risø who came to us with valuable inputs.

This research has been supported by the Innovation Fund Denmark (grant no. 7038-00053B).

This paper was edited by Mingming Zhang and reviewed by two anonymous referees.