Control co-design is a promising approach for wind turbine design due to the importance of the controller in power production, stability, load alleviation, and the resulting coupled effects on the sizing of the turbine components. However, the high computational effort required to solve optimization problems with added control design variables is a major obstacle to quantifying the benefit of this approach. In this work, we propose a methodology to identify if a design problem can benefit from control co-design. The estimation method, based on post-optimum sensitivity analysis, quantifies how the optimal objective value varies with a change in control tuning.

The performance of the method is evaluated on a tower design optimization problem, where fatigue load constraints are a major driver, and using a linear quadratic regulator targeting fatigue load alleviation. We use the gradient-based multi-disciplinary optimization framework Cp-max. Fatigue damage is evaluated with time-domain simulations corresponding to the certification standards. The estimation method applied to the optimal tower mass and optimal cost of energy show good agreement with the results of the control co-design optimization while using only a fraction of the computational effort.

Our results additionally show that there may be little benefit to using control co-design in the presence of an active frequency constraint. However, for a soft–soft tower configuration where the resonance can be avoided with active control, using control co-design results in a taller tower with reduced mass.

Control co-design (CCD) is a sub-field of dynamic system design where the controller is designed simultaneously with the structure. Wind turbine design is a promising field of study within CCD because these structures are driven by load constraints, while at the same time control is important for optimal energy production and for reducing loads

Though CCD is not yet widely used in the field of wind energy, several research groups have shown the potential of the method.

Most wind turbine optimization frameworks rely heavily on steady-state analysis

From a mathematical point of view, the difference between a CCD and a standard physical design optimization problem can be seen as the addition of the design variables describing the controller action. A promising problem for CCD applications is one that is likely sensitive to control tuning. Indeed, an integrated design approach is recommended when the physical system and control system are strongly coupled

The proposed estimation method is applied to the design of a wind turbine tower driven by fatigue damage constraints. Several authors have developed control strategies to reduce fatigue damage

Another important constraint in the design of wind turbine towers is the frequency constraint, which prevents resonance with the rotor rotational frequency. Recent developments in control design have allowed researchers to design towers without this constraint, called soft–soft towers, where resonance avoidance is managed by active control. Soft–soft towers generally have a lower mass than standard ones (also called soft–stiff configuration), and their designs can also be driven by fatigue damage

The paper is organized as follows. Section

We consider the control co-design problem (Eq.

Instead, it is possible to solve an optimization problem with frozen control, represented by Eq. (

The change in optimal objective value due to a change in the control parameter

First, the satisfaction of the constraints means that

The stationarity condition is reformulated by post-multiplying it by

Illustration of the estimator on a quadratic problem, with one scalar design variable

A purely linear estimator only takes into account the linear variation in the problem with

the objective function and constraints are linear in

there are no couplings between

the active set does not change with a small variation

In this section, we present the tower design case study used to evaluate the estimator. We first describe the optimization problem on which the estimator is applied. The second part of this section focuses on the adopted linear quadratic regulator (LQR) control law and its parametrization. A description of the analysis and fatigue damage models concludes the section.

We consider a wind turbine tower optimization problem with the objective of reducing the cost of energy (CoE). Two configurations of the tower design are considered: a standard configuration, where the natural frequencies of the structure are required not to interact with the rotor rotational frequency, and a soft–soft configuration, where the natural frequencies can be lower than the passing frequency, and resonance is avoided through active control. The tower design is parameterized with the tower height

The optimization is represented by Eq. (

The following two sets of constraints,

We use a wind-scheduled multi-input multi-output (MIMO) LQR controller with integral action

The tuning of an LQR controller is done through the choice of the entries of the weight matrices associated with the states and inputs, noted

Figure

Impact of the control tuning on the mean fatigue damage and at three locations along the tower.

The numerical experiments presented in this work are conducted using the multi-disciplinary wind turbine design optimization framework Cp-max. The details of the framework can be found in the available literature

The tower is modelled as a steel tubular structure, divided into

The fatigue load analysis is performed according to certification standards

The cost of energy is calculated following the NREL cost model

This section describes how the first-order and high-order estimation formulas derived in Sect.

The estimation formulas presented in Sect.

The objective function for the considered problem is

While the estimator formula cannot be applied directly to the outer optimization problem, it can inform on the sensitivity of CoE with regard to control changes. In Eq. (

This function can be used to gauge the optimal CoE that would have been obtained by solving the minimization problem including control tuning as a design variable, i.e. using CCD. This is done by minimizing the CoE estimate with respect to

In this section, the estimation method presented in Sect.

All optimization problems are solved using the active-set optimization algorithm implemented in the

In this section, the change in optimal tower mass due to a control tuning variation is estimated. Then, this estimate is compared to the solution of the tower mass optimization problem run for different variations in the control parameter at the reference tower height.

Characteristics of the optimal standard and soft–soft tower designs for the reference height

We first look at the importance of the different constraints on the design by solving the inner tower optimization problem with fixed control tuning

Using the value of the Lagrange multipliers, the first-order and high-order estimators are calculated and reported in Fig.

Comparison between the optimum mass change

In this section, we want to understand if the CoE can be reduced by the combined action of control load alleviation and changed tower height through CCD and if the proposed estimation method can predict the CCD results.

Relative change in CoE as a function of the tower height change and control tuning parameter calculated using the first-order and high-order estimators, for the standard and soft–soft configuration. The reference CoE is the optimal value for the non-CCD problem with

Figure

In order to assess the accuracy of the CoE estimator, we solve the tower optimization problem with a non-CCD formulation (corresponding to Eq.

Percentage improvement in the optimal CoE using a CCD approach, calculated with optimization results and the estimation method.

Table

Characteristics of the optimal design for the non-CCD and CCD problems and for the standard and soft–soft configuration. The percentage change between the CCD and the non-CCD cases is reported in parentheses.

In terms of computational costs, calculating the high-order estimator requires evaluating (i) the Lagrange multipliers by solving the optimization problem at the reference control and (ii) the constraints for different values of the control parameter. In this section, we compare this computational effort to the one needed to solve the CCD optimization problem, applied to the CoE.

Table

Computational effort for the CoE estimator and for the CCD optimization: number of iterations for the outer optimization

A CCD approach can incur major computational costs when compared to the simpler non-CCD optimization. At the same time, our results show that CCD is not always guaranteed to provide benefits to the final design compared to a more straightforward non-CCD approach. Without knowing a priori the potential benefit, there is a significant risk, in terms of engineering time, code development, and computational resources, in attempting a CCD optimization. This work suggests that results from the simplified optimization problem can be used in conjunction with the high-order estimator to determine whether a given problem can benefit from a CCD approach. The first-order estimator shows similar results, with a reduced precision. Furthermore, the analysis of the Lagrange multipliers and constraint sensitivity in the proposed method gives a justification for why a CCD approach would fail. This information is generally not readily available when running a CCD optimization directly because optimization algorithms can fail for technical reasons (inadequate parameters, scaling, or problem formulation).

The method is applicable to similar problems where the optimum design is driven by a load constraint, when loads can be alleviated by control action (for example, the design of wind turbine support structures or blades). The computational cost reduction should be similar in problems where the fatigue damage constraints are driving the design. In cases where the driving constraints are easier to evaluate, there should be a greater reduction in computational effort, since the estimator would be less expensive to compute. In addition, while the estimation method was developed to target CCD applications, the mathematical derivations and associated assumptions are developed in the general case, where

The precision of the high-order estimator depends on several assumptions on the objective functions and constraints. When the assumptions are violated, the estimator can under-predict the benefits of CCD, as shown in our results. In addition, the estimator uses local sensitivity information of the non-CCD optimum, and therefore it will be inaccurate when a CCD approach significantly changes the design. Consequently, there may still be a benefit of using a CCD approach, even if the estimator fails to show it.

In this study, we perform CCD using one tuning parameter of the LQR controller. However, the proposed method is general and does not depend on the control architecture. The applicability of the method to parametrizations with a large number of design variables is left for future work on the topic.

Finally, this work shows how CCD can be used for the design of wind turbine towers. In the presence of an active frequency constraint, CCD may not give significant improvements. Instead, the use of active load alleviation enables a taller and lighter-mass tower compared to the non-CCD design. Our results are specific to one particular wind turbine and may not be generally applicable. Notwithstanding these limitations, the results reported here highlight the importance of performing a thorough analysis of the driving constraints through the use of Lagrange multipliers before attempting a complex and computationally expensive optimization.

This study shows how design sensitivity analysis can be used to estimate the change in optimal objective value caused by a change in control. Using the solution of an optimization problem with fixed control, we can characterize the results of the more complex control co-design problem without the associated computational effort. Two estimators are presented, based on first-order and high-order approximations, where the latter captures non-linear effects.

The proposed estimation method is applied to the redesign of a wind turbine tower driven by fatigue loads, using an LQR controller targeting fatigue load alleviation. High computational resources are required to calculate fatigue damage accurately, which makes this problem an ideal application for the estimator. Two design configurations are considered: a standard configuration, where a frequency constraint is enforced to avoid resonance with the rotational frequency of the rotor, and a soft–soft configuration, where resonance is avoided using active control. The proposed first-order and high-order estimators are applied to the optimal tower mass and optimal CoE problems. We have shown that the high-order estimator accurately predicts how the tower mass changes with control tuning compared to optimization results. The first-order estimator is inaccurate for large values of control tuning but captures the difference between the standard and soft–soft configurations. Combined with a simple CoE model, the high-order estimator predicts a 0.45 % reduction in optimal CoE for the soft–soft tower, while running the CCD optimization gives an improvement of 0.53 %. The proposed estimation method is accurate and uses only a fraction of the computational resources of the CCD optimization. Our results additionally show that the standard tower configuration does not benefit from a CCD approach due to the presence of an active frequency constraint. Changing the control is beneficial for the soft–soft tower because the fatigue damage constraint is the primary design driver and can be alleviated by control action. In this case, the use of CCD yields a taller tower with lower mass, which impacts the CoE significantly.

As shown in this work, design sensitivity analysis allows one to identify relevant design problems for CCD from the results of a simplified non-CCD solution. In a context where computational effort is an obstacle to the wide use of CCD, the proposed method can help identify and quantify the benefits of this approach for wind energy applications.

In this appendix, we derive the high-order estimator expressed by Eq. (

We consider the following non-linear optimization problem:

The change in optimal objective value due to a change in the control parameter

We assume that the objective function

The expression for

The first term of the formula can be expanded to all constraints instead of the set

In this section, we illustrate how the assumptions associated with the high-order estimator impact its validity. For this purpose, we study the simple quadratic programme below, with

In order to represent problems with objective functions linear or non-linear in

Figure

In order to represent the coupling between

To study how a change in the active set impacts the validity of the estimator, a constraint is added so that it is not active for

Figure

Contour plot of the objective function with the optimal value marked with an asterisk (

Comparison of the optimal objective value with the first-order estimator and the high-order estimator for objective functions with varying degrees of non-linearity in

Contour plot of the objective function with the optimal value marked with an asterisk (

Comparison of the optimal objective value with the first-order estimator and the high-order estimator, for problems with varying degrees of coupling between

Contour plot of the objective function with the optimal value marked with an asterisk (

All figures and the data used to generate them are available upon request.

JI developed the proposed method and implemented the numerical experiments in Cp-max. CLB supervised the research. JI wrote the paper, with inputs from CLB and MKM. All authors provided important input to this research work through discussions, feedback, and by writing the paper.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors would like to acknowledge H. Doruk Aktan and Helena Canet for their valuable help with Cp-max. In addition, the authors would like to thank Mathias Stolpe for his valuable input, as well as Erik Quaeghebeur and the anonymous referee for reviewing this work and providing feedback on the work.

This research has been supported by the Technical University of Denmark through the Multi-Disciplinary Design Optimization of Wind Turbines with Smart Blade Technology PhD project, under co-supervision of the Wind Energy Institute of the Technical University of Munich.

This paper was edited by Jennifer King and reviewed by Erik Quaeghebeur and one anonymous referee.