A novel wind turbine rotor optimization methodology is presented. Using an assumption of radial independence it is possible to obtain an optimal relationship between the global power (

Applying the optimization methodology to maximize power (

With a simple cost function and with the same setup of the problem, a power-per-cost (PpC) optimization resulted in a power-per-cost increase of

Wind turbine design optimization has been an integral part of wind turbine design since the start of the wind turbine industry.
The target for such optimization has varied greatly from pure aerodynamic optimization with the target to maximize the power extraction (see

Lately, some research has been performed within preliminary rotor design which seems to have started with the concept of low-induction rotors

Common to these studies is the assumption of constant axial induction along the rotor span. There have also been some studies to investigate the impact of allowing the axial induction to change along the rotor span.

In this paper, an optimization methodology is presented which aims to maximize the power (

This paper is split into two sections: the “Optimization methodology” section, where the optimization problem is presented and the process of solving the optimization problem with the assumption of radial independence is then given; and then the “Results and discussion” section, where the results from solving the optimization problem are presented and discussed.

In this section, we will present an optimization methodology for wind turbine rotor optimization. It is named Wind turbine Optimization with Radial Independence (WOwRI). Before presenting WOwRI a discussion of the assumptions as well as the terminology is given, ending with a short discussion of the aerodynamic solver used. Then WOwRI is presented for power optimization with a fixed radius increase as well as wind speed. WOwRI is then extended for AEP optimization with a fixed radius increase, and at last WOwRI is extended for optimization with a simple cost function to determine optimal rotor size.

The core assumption for WOwRI is the assumption of radial independence. An important concept in
this relation is the difference between global and local variables. Global rotor variables have a scalar value for the whole rotor (e.g., power, thrust), whereas local rotor variables have a scalar at a given rotor radius (

An assumption that is related to the radial independence is a direct relationship between the local thrust loading and the local power at the same radial location. It means that if the local thrust loading is given the local power can be computed. This is further discussed in Sect.

Throughout this paper, the flow is assumed to be steady state. As a consequence, when the optimization is made with load constraints (e.g., thrust and flap moment), it is the steady-state load that is constrained. But for the current utility scale wind turbine design, it is common that the design is driven by the dynamic extreme loads. It means that the underlying assumption for this optimization methodology is that a constraint steady-state load is in some way connected with the dynamic extreme load. This assumption is, however, not tested in this paper.

WOwRI is based on power (

The aerodynamic solver (Radially Independent Actuator Disc model – RIAD) used though out this paper is further described in Part 1, and therefore only a brief overview is given here. It makes an explicit relationship between the local-thrust coefficient (

Diagram showing a diagram for the Radially Independent Actuator Disc (RIAD) model.

In this section, the optimization methodology that allows for the fast and very efficient solution to the optimization is derived. It finds the optimal power for a fixed rotor increase. In principle, the rotor radius could also be an optimization parameter, but as is shown later, the optimal global power turns out to be unbounded, and having the solution for the fixed rotor radius increase allows for optimization with a simple cost function, which is further explained later. The main outcome of this section is a function that through solving an optimization problem gives the optimal power for a given set of constraints with a fixed radius increase and fixed wind speed (

The optimization problem is maximizing power (

Mathematically the problem can be stated as

Using the same normalization as in Part 1

The optimization problem stated in the previous sections has a solution that needs to satisfy the Karush–Kuhn–Tucker (KKT)

The key point for rewriting the optimization as a Lagrange objective function is to be able to solve the optimization of the

In this section we will apply the assumption of radial independence to show that the optimal solution for the trade-off between global power (

Flowchart for the loading optimization with a given set of inputs. Aerodynamic input:

Introducing the local Lagrange objective function (

With the

Flowchart for power optimization. Note that the loading optimization is nested within the optimization loop. The optimizer needs to adjust the

The purpose of this section is to extend the optimization methodology to include optimization for maximum annual energy production (AEP) with load constraints across all wind speeds as well as fixed rated power and a fixed radius increase.

AEP is computed as the average power over a year multiplied by the time of a year. The average power can be computed from the wind distribution (

It can be further simplified as

Flowchart for the AEP optimization. The optimization is simply a power optimization for each wind speed in the power curve.

The optimizations presented so far have been for a fixed radius increase, but in this section the optimization for rotor radius will be presented. The power optimization and AEP optimization could in principle easily be extended for radius optimization as well by simply adding the rotor radius as a design variable, but as discussed in Sect.

The current work focuses on preliminary wind turbine rotor design, and a detailed cost function like the one in

The cost function will roughly estimate the mass increase associated with the increase in rotor radius, with the underlying assumption that mass and cost scale roughly in the same way. It is important to note here that it is not the whole turbine and associated components that need to be scaled with the change in rotor radius; as the optimization is a load-constrained optimization, the loads do not change and the associated components, therefore, do not need to be scaled.

The cost model is simply based on a cost fraction, which is the fraction of the cost that is affected by changes in radius, as well as the cost exponent, which describes how the cost (or mass) for this cost fraction scales with changes in radius. If the components affected by the radius increase are assumed to be the blades, tower and foundation, the cost fraction is found to be 39 % (using the number from

The outcome from Sect.

In this section, the result of applying the WOwRI optimization methodology is presented. At first, the result of pure power optimization at a single wind speed is presented and discussed, and then the result of including a cost function for the so-called power-per-cost optimization, which leads to a turbine blade planform design, is presented and discussed. The AEP optimization is then presented, and then at the end the AEP-per-cost optimization is presented, which leads to the optimal power curve. At the very end, how close it is possible to get to the optimal power curve with common wind turbine technology is tested.

The following shows how the WOwRI methodology can easily be applied for large investigations of the design space, which would otherwise be very computationally expensive with methods where simulation tools are coupled. The results presented here only consider the two constraints (thrust and flap moment) as presented earlier, but they can be extended to more constraints (like max chord, tip deflection, tower bottom bending moment) but are omitted here as the focus is on presenting the model.

This section shows the result of applying the optimization methodology described in Sect.

The input for the aerodynamic solver (Part 1,

In Fig.

Optimal relative power (

For the case of

Common to both cases is that the curve is seen to increase again beyond the saddle point/local optimum, and the curves are seen to still increase at

The unbounded behavior of

This section will show the result of applying WOwRI for power-per-cost (PpC) optimization at a single wind speed (assumed to be

For the rotor design, the aerodynamic losses will be included (i.e., wake rotation loss, viscous loss, tip loss). To include viscous loss, the glide ratio (

Aerodynamic input based on the polars from the 10 MW DTU reference turbine.

With the glide ratio from Fig.

A plot of the relative power per cost

Relative power per cost (PpC) vs. radius (

A comparison of the loading distribution is shown in Fig.

The rotor planform (blade chord and twist) can be found from the loading distribution (

For the twist (Fig.

In this section the result of solving for the optimal annual energy production (AEP) is shown, as explained in Sect.

When solving the optimization problem in Eq. (

Optimal AEP (

The solution for solving the AEP optimization problem can be seen in Fig.

As was the case for the power optimization the global optimum for AEP optimization is found to be a similar asymptotic limit with the optimum as

We turn to the power and load curves for the four highlighted points in Fig.

max

maximizing power with one or more active constraints, and

rated power.

In this section, the result of solving for the optimal AEP per cost (AEPpC) is presented. At first, the optimal power curve is presented, and at the end common wind turbine technology is used to see how close it can get to the optimal power curve.

The optimization will use the same aerodynamic input as in Sect.

Relative AEP-per-cost (AEPpC) vs. relative radius increase (

Normalized power (

AEPpC for increasing values of

Figure

The presented optimal power curve can not be made into a blade design as was done in Sect.

Power and load curves (top curves) as well as blade pitch (

For current utility scale wind turbines, there are two common parameters for altering the loading with changing wind speed, namely the blade pitch (

To compute the aerodynamic performance for a turbine where the control parameters are the blade pitch and rotational speed, the classical blade element momentum (BEM) theory is well suited. As shown in Part 1

In order to directly compare the rotor design with the optimal power curve, the radius increase is assumed to be the same as for the AEPpC optimized power curve (

The result of the optimization can be seen in Fig.

The optimal BEM rotor design is seen to be achieved through a

A novel wind turbine optimization methodology was presented. The crucial assumption that allows for this nested optimization approach is the assumption of radial independence, which is similar to the assumption made in the blade element momentum theory. It allows solving the optimal relationship between the global power (

Applying the optimization methodology for power (

With a simple cost function a power-per-cost (PpC) optimization resulted in a power-per-cost increase of

Variables that are scalars for the whole rotor. Boldface variables indicate the variable is a function or vector that changes with wind speed.

Variables that are scalars at a given radius location (

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 and figures 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 part of the industrial PhD project which this article is a part of.

We would like to thank all employees at the former Suzlon Blade Sciences Center (Vejle, Denmark) for giving valuable feedback in the initial phase of the development.

We would like to thank Antariksh Dicholkar from DTU Risø for many good discussions and input regarding the work.

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

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