We develop an automated controller tuning procedure for wind turbines that uses the results of nonlinear, aeroelastic simulations to arrive at an optimal solution. Using a zeroth-order optimization algorithm, simulations using controllers with randomly generated parameters are used to estimate the gradient and converge to an optimal set of those parameters. We use kriging to visualize the design space and estimate the uncertainty, providing a level of confidence in the result.

The procedure is applied to three problems in wind turbine control. First, the below-rated torque control is optimized for power capture. Next, the parameters of a proportional–integral blade pitch controller are optimized to minimize structural loads with a constraint on the maximum generator speed; the procedure is tested on rotors from 40 to 400 m in diameter and compared with the results of a grid search optimization. Finally, we present an algorithm that uses a series of parameter optimizations to tune the lookup table for the minimum pitch setting of the above-rated pitch controller, considering peak loads and power capture. Using experience gained from the applications, we present a generalized design procedure and guidelines for implementing similar automated controller tuning tasks.

In this article, we present a data-driven, simulation-based optimization procedure for tuning wind turbine controllers using measures that are directly related to component design. Controller tuning influences the power capture and structural loading on wind turbines, which are directly related to the cost of the wind energy generated. At the same time, different turbine models require different control parameters. As rotor designs are iterated upon and also customized, e.g., with larger towers, tip extensions, or for site-specific turbulence, an updated (and ideally optimized) controller is required for component design and cost specification. Given the aeroelastic turbine model, the algorithm presented in this article automatically finds the optimized parameters of the predefined control architecture, reducing the effort required of the control designer.

The wind turbine control tuning can be automated, but design choices for the various parameters often require expert knowledge of the controller and turbine operation. An automated procedure to determine these choices could reduce the design cycle time of a manufacturer's research and development process or aid researchers in other disciplines of wind engineering that require a well-tuned controller without worrying about its finer details. Several control parameters are directly related to the performance of the turbine and must be tuned for each design iteration or model update. The simplest method to determine these design choices using simulation results is to exhaustively search the design space and then make an educated design choice of the parameter. However, exhaustive search may become computationally intractable for fine discretizations of the search space; on the other hand, coarse discretizations may lead to suboptimal design choices.

A systematic, simulation-based parameter search of the pitch control gains for generator speed control was first published in

As the wind industry has matured and computational cost has decreased, wind turbine design increasingly relies on simulation of power capture and structural loads for design analysis.
As a result, system engineering tools for wind turbine design have been developed and refined, leading to updated efforts in automated controller development, with the aim of deploying tuning methods for many different turbines.
One approach is to use a model-based control scheme in order to limit the control tuning effort

A scalar cost function was presented in

Simulation-based optimization has been used to solve these problems, where solving for the value of the cost function is expensive compared to the optimization procedure.
One approach to solving these types of problems is using response surface methodology (RSM)

We use a Gaussian smoothing approach to generate samples, estimate the gradient, and identify a (possibly local) minimum point. Then we use the samples to visualize the design space and provide a level of confidence in the result. Previous work in controller optimization usually only provides the cost function and goals of the optimization, whereas this work explicitly details the method for determining the sample simulations and how their results are used to iterate on control designs.

Instead of using cost functions directly related to overall wind turbine performance, our work solves specific wind turbine control problems that are related to the cost of energy. First, the optimization procedure is demonstrated on below-rated torque control to increase power capture. Next, the pitch control parameters for above-rated pitch control are optimized to reduce fatigue or extreme loads on the tower or blades with a maximum generator speed constraint. Finally, the minimum pitch setting of the pitch controller is optimized in a series of parameter optimizations aimed at reducing peak blade loads.

This article is organized as follows.
The optimization algorithm and visualization method are presented in Sect.

Superscript notation will be used to index the stage

The zeroth-order optimization algorithm uses

The algorithm begins with an initial guess

A search sample

The number of stages and samples per stage must also be chosen by the designer.
A large number of samples per stage gives the best estimate for the gradient but requires more simulations.
During the development of this work, it was found that a smaller number of samples per stage and more stages resulted in better convergence using the same total number of simulations (e.g., in Sect.

At each stage

From the estimated gradient, the possible descent direction is computed:

A decreasing step size rule ensures convergence and a line search is used so that the cost function does not increase in successive iterations.
After choosing a base step size

Once an adequate step size is found, the parameter

If the maximum number of step size simulations (

To provide confidence in the result of the zeroth-order optimization, we visualize the cost, and the measures associated with it, over the parameter space. If the minimum of the zeroth-order parameter optimization matches that of the visualization, the user can be confident in the result. The visualization method also provides a quantitative measure of the uncertainty of the estimated cost over the parameter space.

To estimate the cost and its variance over the parameter space, we use ordinary kriging. Kriging was originally developed for mining applications, where sparsely sampled information over a geographical space was used to estimate the quantity over the whole area. More recent applications of kriging include engineering design and computer experiments.

Kriging, or Gaussian process regression, is a method of interpolation that incorporates uncertainty in the area between samples.
Using all the observed data from the zeroth-order parameter search at stage

The mean squared error, or variance, of the cost at

The correlation function parameters

To measure the number of stages the optimization procedure requires to find the minimum of the cost function, we define the settling function

In this section, we present three examples of using zeroth-order parameter optimization to tune the parameters of wind turbine controllers. As an initial demonstration, we optimize a one-dimensional parameter to maximize power capture through torque control in below-rated operation. Next, we present the motivating example for this work, a two-dimensional parameter optimization for a standard pitch controller, with the goal of regulating generator speed so that loads are minimized, subject to a constraint on the maximum generator speed. Finally, we demonstrate how a series of one-dimensional parameter optimizations can be used to determine the minimum pitch setting of the pitch controller for controlling peak blade loads.

In below-rated (region II) operation, the generator torque is typically controlled using

The goal of this optimization procedure is to find the gain

At each stage,

The first iteration of the one-dimensional parameter tuning for the optimal torque control gain. Starting with the initial parameter

The optimal value of

Design choices for 1-D parameter search to optimize the torque gain in below-rated control.

One-dimensional parameter tuning for the optimal torque control gain of the 5 MW reference turbine in Class A turbulence using the negative mean generator power (

The parameter optimization was performed on the NREL-5MW reference turbine with the standard lookup-table-based torque controller in

In this section, we optimize the parameters of an above-rated blade pitch controller for load reduction and generator speed regulation.
Each time a new rotor is designed, the pitch controller should be tuned so that the structural loads can be computed to design the various hardware components of the wind turbine.
As will be seen, the pitch controller affects the loads that drive turbine design.
The procedure for tuning the gain-scheduled proportional–integral (PI) controller is detailed in Appendix

In general, changing the bandwidth of the pitch controller via

First, we reformulate the constrained optimization

To adapt the cost function to different rotors and load measures,

Parameters used for the 2-D speed regulator control tuning procedure for all rotors tested. The effects of the number of stages

In most cases, the initial parameter set

First iteration of the zeroth-order parameter optimization algorithm for the two-dimensional pitch control tuning. Random samples

The cost function is more sensitive to changes in

Summary of test cases and results from a single zeroth-order parameter tuning for speed regulation control using the parameters in Table

The algorithm was tested on a range of rotor models with different wind classes and load measures.
First, the pitch control parameters of the NREL-5MW reference model are optimized, starting from the parameters specified by the NREL-5MW reference manual

Results of using the zeroth-order optimization for tuning the pitch control regulator mode (natural frequency

The optimization procedure was also performed for each rotor design in the Segmented Ultralight Morphing Rotor (SUMR) project

Cost

For most of the rotors in this section, the baseline rotor speed proportional–integral (PI) control parameters are optimized to have a regulator mode with a lower natural frequency and higher damping ratio than the initial guess (Table

PI gains derived from a regulator mode with a low natural frequency result in less pitch actuation and thus less change in the load. Higher natural frequencies result in faster and more frequent pitch control variations, which translate to the structural load signals and increase fatigue loading. A controller with a high natural frequency can also be problematic when the wind speed decreases. Because the underlying controller is trying to regulate the generator speed, the pitch will decrease during a wind lull to maintain the generator speed at its rated value, which can also lead to large peak loads, especially when an increase in wind speed follows.

High damping ratios are also found to be optimal when using the described cost function.
A generator speed response and pitch control response with a high damping ratio lacks any overshoot and secondary transients when the system is subjected to a disturbance (wind).
Secondary transients and overshoot in the pitch command result in load transients.
The original NREL-5MW controller (where

When comparing the PI gains of the original versus optimized controller, we see that the proportional gains are of similar magnitudes, but the integral gain is much less in the optimized set of gains.
The original NREL-5MW gains are

To quantify the performance of the zeroth-order optimization (ZOO) for this pitch control application, we compare it, in terms of the number of simulations and optimal cost, with a grid search optimization for tuning the pitch controller of the NREL-5MW reference turbine in Class 1A turbulence.
The same area spanned by the hard bounds of the zeroth-order method (Table

Panels

The ZOO procedure outlined in Sect.

Compared to

We should note that the comparison presented in this section applies only to this pitch control tuning application. To compare the efficacy of the zeroth-order optimization with a grid search more generally would require comparing functions of different complexities and dimensions, which is outside the scope of this article and we leave for future work.

In this final example, a series of one-dimensional parameter optimizations will be used to tune the minimum pitch setting of the above-rated pitch controller described in Appendix

During the load analysis of a control design,
a number (

A zeroth-order optimization for the minimum pitch setting at 14 ms

A wind speed estimate, which can be found using, e.g., one of the methods in

Minimum pitch setting as a function of wind speed. Each active breakpoint is tuned in a series of one-dimensional parameter optimizations. There is an additional high wind speed breakpoint at 50 ms

The algorithm (presented in Algorithm 1) is initialized by choosing an initial lookup table for the minimum blade pitch,

In Step 2, the initial guess that is used by the optimization procedure (

Parameters used to optimize the minimum pitch lookup table.

Algorithm 1 is used to optimize the minimum pitch table in Fig.

Changing the minimum pitch setting of the controller can have an effect on the below-rated power production (optimized in Sect.

In future work, a multiobjective optimization might be more suitable, where all the tuning procedures are simultaneously performed; the zeroth-order method is suitable in this case. A potential challenge would be determining what simulations should be used to efficiently optimize all of the control parameters. Currently, each control tuning procedure requires running different simulations. Additionally, the goal of this work was to automate design choices, rather than having to choose from a set of possible choices that would result from a multiobjective optimization.

However, with additional resources, our goal could shift from efficient optimization of smaller problems to larger optimizations of the overall turbine system. Within a system engineering framework, more information might determine which simulations and loads are sensitive to control parameter changes. In this article, we focused on minimizing the peak blade loads of the SUMR rotors because those were the design driving load of those blades. Other loads could certainly be used, but all loads are not important to the overall design: some components are overdesigned, and others drive design; this information depends on the specific design but could be determined using detailed system engineering tools.

Our goal was to reduce the design cycle times for processes that already occur during control design.
Rather than solving all problems at once, we propose solving them in sequence, in the order they are presented: first optimizing the torque and pitch controllers and then tuning the minimum pitch setting for peak loads.
Then with the new minimum pitch table, a designer could optionally reoptimize the torque and pitch gains; we have done this and witness little change.
Solving smaller problems tends to be more efficient in terms of the number of simulations and more transparent in terms of how control parameters affect different performance measures during specific simulations.
We discuss setting up similar optimization problems for future work in Sect.

In this section, we present guidelines for performing similar optimization procedures. Experience gained in problem formulation, the usefulness of performing a preliminary offline analysis, and the determination of the parameters of the solver is shared.

Using the zeroth-order optimization procedure described in this article for determining control parameters through simulation requires effort in setting up the problem and developing software.
In order to justify the up-front effort, the task would ideally be one that is repeated for many different rotor models, like the examples in Sect.

It is important to determine how the turbine should be simulated in order to generate the measures that are used for the optimization; they should highlight some problematic or indicative case that the control solution is trying to solve.
For example, when optimizing the torque control gain for below-rated operation in Sect.

It is often helpful to perform a preliminary offline analysis to fine-tune the cost function and optimization parameters.
In an offline analysis, a grid search of the optimization parameter is used to estimate the output space of the simulations (e.g., maximum generator speed and blade loads), using a linear or quadratic estimate of the cost function.
To clarify, the results in Sect.

As the examples of Sect.

The smoothing parameter

When optimizing over multiple parameters, the

For example, in the pitch controller tuning (Sect.

The initial step size

The parameters used in the Armijo step size rule were the same for all examples. Conservative values were used, which essentially only ensures a nonincreasing cost function without a requirement on the rate of descent of the cost function.

Enough stages should be evaluated so that the cost function converges to some value; this is typically learned through offline analysis or by trial and error in online tests.
For example, when analyzing the pitch control tuning results of Fig.

Hard constraints on the parameter should reflect the set of feasible parameters for the control task being optimized. However, the bounds should not be so small as to restrict the space and possibly miss nonobvious control solutions. The initial guess provided to the algorithm should also allow for the space to be adequately searched.

After performing initial, offline analysis and running the zeroth-order optimization algorithm using online simulation data, the whole procedure should be evaluated with the following questions:

Does the algorithm converge to a feasible solution?

Does the optimized parameter appear to be near the minimum of the visualized cost over the parameter space?

In this article, we developed a data-driven approach for optimizing controller parameters using simulation results. By using a zeroth-order optimization algorithm, random samples are generated near an initial guess, which are used to compute the local gradient. A standard gradient descent method ensues, where a step size rule is used to ensure convergence and attempt to decrease in the cost function before the next guess is chosen and the process is repeated. We also use ordinary kriging to visualize the design space and its uncertainty to provide a level of confidence in the optimized result.

The zeroth-order algorithm was applied to three different applications in wind turbine control. To demonstrate the process on a one-dimensional parameter optimization, the torque control gain was tuned to optimize power capture in below-rated operation. The baseline pitch controller parameters were tuned in a two-dimensional optimization problem with the goal of minimizing structural loads and include a constraint on the maximum generator speed. Using an adaptable cost function and step size, the algorithm was able to tune the baseline rotor speed control for rotors ranging from 40 to 400 m in diameter. We compare the results, in terms of accuracy, convergence, and number of function evaluations (simulations) for different optimization parameters and against the standard grid search method. In a series of one-dimensional parameter optimizations, we also determined the settings of a lookup table for the minimum pitch limit of the pitch controller, reflecting the overall blade design process and system-level goals.

Since each optimization procedure depends on the specific control problem, we have provided a set of guidelines based on the experience gained during this study for developing future, similar optimization procedures. The methods presented in this article automate a usually manual process, reduce designer effort, and require fewer simulations compared with grid searching methods. These methods can be used for repeated control tuning processes that are required for continually updating designs that must be evaluated in simulation using a well-functioning controller.

The pitch controller described in Sect.

To derive the PI gains for a generic rotor model, a rigid model of the drivetrain is used:

Proportional–integral control with antiwindup scheme used for above-rated control. The difference between the pitch control set point

By defining a new state,

By defining the desired properties of the generator speed dynamics,

Both the proportional Eq. (

Sensitivity of power to pitch used for gain scheduling the pitch controller for a selection of rotors in this article. The sensitivity values obtained from FAST linearizations are fit linearly and used to determine the gain-scheduling parameters for the pitch controller.

Because of the near-linear relationship with blade pitch

To derive the parameters from simulations and tune the regulator mode, we use the following procedure:

Simulate the operating points in FAST using a steady wind input across above-rated wind speeds. Choose a

Linearize the turbine in FAST, disabling all of the degrees of freedom, at the wind speeds and pitch angles found from the previous step. Use the element of the input–output matrix that corresponds to the pitch input and power output matrix to determine the sensitivity of power to pitch at the various pitch angle operating points. Plot the values and fit the parameters

Tune the regulator mode

The turbine models summarized in Table

Summary of turbine models used in this study.

The data from this study can be made available upon request.

DSZ developed the optimization software, developed the example problems, and prepared the visualizations and manuscript. ED reviewed the zeroth-order optimization theory. LYP guided the study, helped formulate the article concept, and reviewed multiple drafts of the article.

The authors declare no competing interests.

The views and opinions of the authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Support from a Palmer Endowed Chair Professorship is also gratefully acknowledged.

This research has been supported by the Advanced Research Projects Agency-Energy (grant no. DE-AR0000667).

This paper was edited by Katherine Dykes and reviewed by Alan Wai Hou Lio and Luca Sartori.