We present an analytical model for assessing the aerodynamic performance of a wind turbine rotor through a different parametrization of the classical blade element momentum (BEM) model. The model is named the Radially Independent Actuator Disc (RIAD) model, and it establishes an analytical relationship between the local thrust loading and the local power, known as the local-thrust coefficient and the local-power coefficient respectively. The model has a direct physical interpretation, showing the contribution for each of the three losses: wake rotation loss, tip loss and viscous loss. The gradient for RIAD is found through the use of the complex step method, and power optimization is used to show how easily the method can be used for rotor optimization. The main benefit of RIAD is the ease with which it can be applied for rotor optimization and especially load constraint power optimization as described in

Wind turbine rotors are with their increasing size subject to continuous optimization, with the overall objective of reducing the cost of energy. Such optimizations are very complex because both the aerodynamic and the structural performance need to be included in the optimization setup. Combining both aerodynamics and structural performance has shown very promising trends indicating that a further cost reduction is possible; see, e.g.,

The development of aerodynamic models for wind turbines is closely linked to that of propellers and helicopters. The first theoretical model for predicting the aerodynamic performance of a rotor was the so-called 1D momentum theory developed by Betz

Later Glauert developed the blade element momentum (BEM) theory

In this paper, we present an aerodynamic rotor performance model which we refer to as the Radially Independent Actuator Disc (RIAD) model. It establishes a direct analytical relationship between the local thrust loading and the local power, which is a useful simplification for rotor optimization. The model is equivalent to BEM but reduces the rotor design space to only two independent variables at each radial station, i.e., the local-thrust coefficient (

The paper starts by presenting the derivation of the RIAD model, which then leads into computing the gradients for RIAD. These gradients are used for power optimization, leading to a simple optimization method. In the end, the relationship between RIAD inputs and blade chord and twist is then established, and RIAD is validated against a BEM solver. This is Part 1 of a two-part paper. Part 1 describes an aerodynamic model for a wind turbine rotor and the use of the model for power optimization. Part 2 is described in

In the following the Radially Independent Actuator Disc (RIAD) model is presented, which starts by establishing the relationship between global and local parameters for a wind turbine rotor as well as introducing normalization. The relationship between the local forces is then established, leading to an implicit equation for the local power. A set of approximate closure equations is then used to establish an explicit equation. The physical interpretation of the different factors and terms is then presented, and at the end some details regarding the tip loss factor and the exclusion of drag from the induced velocity are discussed.

Starting with the fundamental theorem of calculus the following equation can be created for the global values for thrust (

The classical non-dimensional relations for thrust (

Diagram of the relationship between local coefficients (

To establish a relationship between the local coefficients

Diagram of the relationship between the airflow and the forces at each span location.

Combining Eqs. (

The local flow angle

Equation (

There does not exist a general set of model closure equations, but different approximate closures have been proposed. The most widely used set is referred to as the Glauert closure, which is an implicit assumption made for most BEMs. The closures are given as

Diagram showing a graphical representation of the losses and the mathematical origin. The input is for spanwise constant local thrust and glide ratio.

Equation (

The 1D power is the classical 1D momentum theory result by Betz and Joukowsky

Significance of wake rotation loss.

The effect of changing the local loading is seen to have a limited effect. From Fig.

The tip loss is the power loss associated with the rotor having a finite number of blades and not acting as an actuator disc with an infinite number of blades. This effect is captured in the tip loss factor (

The viscous loss is simply the loss associated with the viscous drag from the airfoil profile. The viscous loss is found to be linear in inverse glide ratio (

The tip loss factor is commonly implemented for BEMs, and although some different tip correction has been proposed, the tip loss model by Glauert is a common one to use, and it is also the one used here. It is given as

The Glauert tip loss model breaks the explicit relationship between

Although the Prandtl tip loss model is much simpler and easier to implement, the Glauert model is used throughout this paper as it is the one used for the BEM validation later in Sect.

Whether or not to include drag when computing the thrust loading (including drag in Eq.

As a consequence of excluding drag from Eq. (

In this section, a method for computing the gradients for RIAD is presented. The gradients are then used for power optimization. First, it is applied for loading optimization for maximum power, and it is then further extended for optimization with respect to tip speed ratio and loading. In the end, a discussion of how optimization with RIAD fits within the current state of the art is given.

The local-power equation (Eq.

The complex step method is based on the observation that the Taylor series expansion of an analytical function with a complex step (or perturbation) gives the following (taking Eq. (

The problem of maximizing

The

Applying the optimization with similar input as for Fig.

Optimal local thrust (

The loading optimization in Sect.

Using the same assumption as for the loading optimization the optimization for

In Fig.

Optimal tip speed ratio (

The associated

Optimal power coefficient (

The results of

Section

An equation for chord can be found from Eq. (

An equation for twist can be found in much the same way by using Eq. (

To show that RIAD is an equivalent formulation of the BEM equations, a planform design is created through Eqs. (

Running CCBlade requires an airfoil polar (

The resulting planform design can be seen in Fig.

A rotor performance model called the Radially Independent Actuator Disc (RIAD) model was presented. It is a different parametrization of the blade element momentum (BEM) equations which is found to be better for wind turbine optimization. The model relates the local rotor power output (local-power coefficient –

A method to compute gradients for RIAD was presented, through the use of the complex step method, which allows us to compute the gradient to machine precision with minimum additional work required.

The gradients were used for classical power coefficient (

The relationship between local thrust along the span and the blade chord and twist was presented, and they were used to create the input for validation with a BEM solver

Variables that are scalars at a given radius location (

Variables that are scalars for the whole rotor.

Code is not publicly available and can not be shared.

The validation data are from

KL came up with the concept and main idea, as well as performed the analysis. All authors have interpreted the results and made suggestions for improvements. KL prepared the paper with revisions from all co-authors.

The authors declare that they have no conflict of interest.

We would like to thank Innovation Fund Denmark for funding the industrial PhD project which this article is a part of.

We would like to thank all the former employees at Suzlon Blade Sciences Center for being a great source of motivation with their interest in the results.

We would like to thank Mads Holst Aagaard Madsen from DTU Risø for the inspiration to use the complex step method.

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

This paper was edited by Alessandro Bianchini and reviewed by Peter Jamieson and two anonymous referees.