Turbines in wind power plants experience significant power losses when wakes from upstream turbines affect the energy production of downstream turbines. A promising plant-level control strategy to reduce these losses is wake steering, where upstream turbines are yawed to direct wakes away from downstream turbines. However, there are significant uncertainties in many aspects of the wake steering problem. For example, infield sensors do not give perfect information, and inflow to the plant is complex and difficult to forecast with available information, even over short time periods. Here, we formulate and solve an optimization under uncertainty (OUU) problem for determining optimal plant-level wake steering strategies in the presence of independent uncertainties in the direction, speed, turbulence intensity, and shear of the incoming wind, as well as in turbine yaw positions. The OUU wake steering strategy is first examined for a two-turbine test case to explore the impacts of different types of inflow uncertainties, and it is then demonstrated for a more realistic 11-turbine wind power plant. Of the sources of uncertainty considered, we find that wake steering strategies are most sensitive to uncertainties in the wind speed and direction. When maximizing expected power production, the OUU strategy also tends to favor smaller yaw angles, which have been shown in previous work to reduce turbine loading. Ultimately, the plant-level wake steering strategy formulated using an OUU approach yields 0.48 % more expected annual energy production for the 11-turbine wind plant than a strategy that neglects uncertainty when considering stochastic inputs. Thus, not only does the present OUU strategy produce more power in realistic conditions, but it also reduces risk by prescribing strategies that call for less extreme yaw angles.

This work was authored in part by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under contract no. DE-AC36-08GO28308. The US Government retains and the publisher, by accepting the article for publication, acknowledges that the US Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for US Government purposes.

A key determinant in the profitability of a wind power plant is its annual energy production (AEP). The traditional strategy for increasing AEP has been to control each turbine in the plant such that single-turbine power generation is maximized, irrespective of the generation by other turbines. Plant-level control, by contrast, is an innovative approach that has the potential to further optimize wind plant performance and increase AEP

Recently, researchers from the National Renewable Energy Laboratory (NREL) have partnered with utility-scale wind power plants to demonstrate the potential benefits of the wind plant control strategy known as wake steering

There are also various sources of uncertainty that can have substantial impacts on the success of wake steering strategies. For example, thermally driven vertical mixing in the atmospheric boundary layer is difficult to measure and characterize

Physical uncertainties associated with engineering wake models used for the design of wind plant control strategies pose additional difficulties. For example, although several studies have reported significant gains in AEP using plant-level control strategies under the assumption of perfect (i.e., certain) information

Uncertainty in the design process can be addressed using optimization under uncertainty (OUU), a technique that has been used in several prior wind plant optimization studies to provide a robust solution under varying levels of uncertainty

In this paper, we extend prior work on OUU and plant-level control to address uncertainty in turbine yaw positions and the direction, speed, shear, and turbulence intensity of the wind inflow during the optimization of turbine yaw offsets for wake steering strategies. Our objective is to understand how uncertainty in the examined parameters influences wake steering optimization and to quantify the effect of uncertainty on the performance of a hypothetical wind power plant. Using a polynomial chaos expansion (PCE) approach, which has not been employed in previous OUU studies of wake steering strategies, we show that direction is the most important uncertain input, effectively smearing out the paths of wakes and reducing the expected velocity deficit. We further show that uncertainty generally reduces optimal yaw offsets, in agreement with the results of

In this study, we first examine a two-turbine test case to explore how different magnitudes of uncertainty impact the efficacy of wake steering schemes, with a particular focus on the trade-off between the power produced by the front and back turbines. Assuming standard uncertainty distributions based on available information, we find that the inflow speed and direction are the most influential parameters for the wake steering design problem. In a more realistic 11-turbine wind plant test case, we further demonstrate the benefits of the OUU formulation. In particular, in addition to yielding more robust designs, the OUU formulation results in less extreme prescribed yaw offsets than a deterministic problem formulation.

The paper is organized as follows. In the next section, we outline details of the engineering wake model, the formulation of the OUU problem, and the specific application examined. Results are outlined for two-turbine and wind plant test cases in Sect.

In this study, we applied the FLOw Redirection and Induction in Steady State (FLORIS) engineering wake model (NREL,

We used the FLORIS implementation of the steady-state Gaussian wake model

In this study, we limited the value of the thrust coefficient to be strictly less than 1. Without this modification, wake calculations for low wind speeds may result in inaccurate predictions (in particular, the calculation of

Power (solid line) and thrust coefficient (dashed line) as functions of wind speed for the NREL 5 MW reference turbine

Using the steady-state FLORIS wake model, the deterministic power production of a wind plant can be predicted given turbine-specific yaw positions,

During plant operation, low-frequency variation, spatial variability, and measurement errors are inevitable complications that may be represented in a stochastic setting. We envision wake steering strategies changing approximately every 10 min. As a result, we introduce the stochastic expected power, denoted

The energy production may be estimated for a whole year (i.e., the expected AEP) as a linear sum of each speed- and direction-specific expected power production, weighted by speed- and direction-specific probabilities and multiplied by 8760 h yr

Using Eq. (

We used four metrics to assess the quality of different solutions for

Summary of AEP-based metrics used to assess the quality of solutions for given values of

In the present demonstration tests, we considered the effects of uncertainty in turbine yaw offsets and wind inflow speed, direction, turbulence intensity, and shear. We envision wake steering strategies changing every 10 or 20 min, so we identify reasonable uncertainty values to represent spatial and temporal variations, as well as measurement errors, in each of these uncertain parameters over that time span. Together, these variations comprise the joint pdf,

Errors in the yaw position,

To estimate the yaw position uncertainty, we compared operational data from an NREL turbine with a nearby meteorological measuring mast

Based on the shape of the distribution in Fig.

Estimated values for

The resultant distribution choices and hyperparameter estimates are provided for each uncertain variable in Table

Probability distributions and hyperparameter values describing the uncertainty associated with various inputs to the wake model.

We approximated the integral in Eq. (

PCE was used to facilitate computing

During computation of the AEP via Eq. (

To demonstrate the benefits of OUU in the development of wake steering strategies, we considered a two-turbine layout as well as a larger 11-turbine layout. We used the two-turbine layout to explore the basic trade-off between the power production of front and back turbines as well as the sensitivity to different levels of uncertainty. The wind plant problem was used to assess the potential benefits of OUU in a more realistic wind plant design problem. The mean values, scale parameters, and upper and lower bounds associated with each input considered in the two-turbine and wind plant cases are shown in Table

Mean and scale values and lower and upper bounds used in the two-turbine and wind plant OUU problems. Sequences are expressed as

Contours of wind speed for a simple two-turbine test case with an inflow speed of 8 m s

In the two-turbine case, the front turbine directly wakes the back turbine when flow is from the north, as shown in Fig.

The wind plant wake steering optimization problem is intended to provide insights into the benefits of OUU in more realistic scenarios. The plant layout is shown in Fig.

It should be noted that a relatively limited range of inflow directions and relatively large direction increments were considered in the present study, as shown in Fig.

The expected power production was maximized during the optimization. The COBYLA optimization driver in DAKOTA

Contours of wind speed for the 11-turbine wind farm with the baseline yaw configuration (

Annual wind speed and direction probability mass function used in the 11-turbine wind plant optimization study.

In the following, we present results for OUU of the simple two-turbine case as well as the 11-turbine wind plant. It will be shown from an analysis of the two-turbine case that wind speed and direction are the most influential parameters, and so we performed the plant OUU using only these two uncertain variables, assuming

Figure

Uncertainty in the wind direction affects the path that wakes behind wind turbines will follow. This can be thought of as spreading out the wake. This effect is explored in Fig.

Maximum VSS across all speeds and directions considered, given the reference standard deviation values in Table

Power production for the front

It is interesting to note that the deterministic solution may be worse than the baseline solution if there is large uncertainty in the inflow wind direction. This is shown in Fig.

Summary statistics from Table

Uncertainty in the incoming wind speed,

Power production for the front

Overall, we found this two-turbine case to be less sensitive to uncertainties in yaw misalignment, turbulence intensity, and wind shear. Uncertainty in the turbine yaw positions generally reduces the rotor-swept area and spreads out the path of the turbine wake. As a result, the power of the back turbine may be increased or decreased by yaw misalignment uncertainty, depending on which dynamic dominates. Yaw position uncertainty does not dramatically affect the solution at the reference uncertainty (

Turbulence intensity affects the wake expansion geometry, which effectively smears out the path of wakes, decreasing the velocity deficit felt by waked turbines. Although we did not find turbulence intensity uncertainty to be significant here, we found a maximum VSS of 1.29 % when

To quantify the benefits of OUU in a more realistic scenario, we also performed OUU to design wake steering strategies for an 11-turbine wind power plant. Given the maximum VSS results summarized in Table

Expected and deterministic AEP of OUU, deterministic, and baseline plant-level wake steering strategies for the 11-turbine wind plant test case.

The stochastic average and deterministic AEP associated with the OUU, deterministic optimization, and baseline (i.e., no wake steering) solutions are provided in Table

Summary statistics from Table

Optimization histories for deterministic

Probability mass distributions of

It is interesting to note that the uncertain expected AEP is greater than the deterministic AEP in Table

Figure

In general, we found that by incorporating uncertainty in the wake steering problem formulation, less extreme yaw offsets were required to optimize AEP. We show the aggregate of yaw positions suggested by the OUU and deterministic optimization approaches in Fig.

In this study, we examined how uncertainty affects wake steering strategies and what benefits may be associated with designing these strategies in the presence of operational uncertainty using OUU.
While previous approaches

Uncertainty in yaw positions is epistemic and may be reduced with more accurate yaw position detection methods. Uncertainty in inflow conditions is more nuanced. While there are issues with accurately measuring these quantities, fundamentally, there may not be a single characteristic direction, speed, turbulence intensity, or shear associated with the wind flowing into a utility-scale wind plant. For example, a wind power plant may be built downstream of a mountain, causing wind to enter from multiple directions. So, the uncertainties in these inflow parameters may be thought of as a combination of epistemic and aleatoric, irreducible, or model-form uncertainties.

The fact that OUU results in more expected power production with less extreme yaw offsets makes a strong case for designers to move toward OUU formulations in plant-level control strategies. In particular, OUU results in wake steering strategies that are more conservative than the deterministic approach – the magnitude of the turbine yaw offsets determined by OUU is diminished, compared to those found using deterministic optimization. Assuming that the inflow uncertainties were precisely quantified, we have shown that wake strategies formulated with the OUU approach can produce up to roughly 4 % more power (for the 5 m s

We are optimistic for the future of plant control strategies and anticipate that uncertainty will become increasingly incorporated in future plant control analysis. In future work, we plan to further quantify typical levels of uncertainty in input parameters, explore higher-fidelity flow models, and include fatigue loading in the OUU objective function. There are several other sources of uncertainty that may be injected into this problem. For example, we assumed perfect knowledge of the turbine power and thrust curves. Typical levels of uncertainty in turbine power and thrust curves probably would have resulted in somewhat different optimum wake steering strategies. Uncertainty in power would change the shape of the mean power curve, altering the trade-off point, and uncertainty in thrust would affect the magnitude of the velocity deficit behind the turbine. Quantifying fatigue loading is an attractive prospect, though it requires a more advanced wake model. Partial waking may be more detrimental than full exposure to a wake, complicating the fundamental trade-offs that we explored. Finally, the effect of correlations between different sources of uncertainty also deserves further future consideration.

The speed and direction probability mass function and an example FLORIS input file are included in the Supplement.

The supplement related to this article is available online at:

KD, JK, and JQ designed the experiments. RK, PH, and JQ developed the problem formulation. JQ and JK performed the simulations. JQ and PH prepared the manuscript with contributions from all co-authors.

The authors declare that they have no conflict of interest.

The views expressed in the article do not necessarily represent the views of the DOE or the US Government.

Funding was provided by the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office. The research was performed using computational resources sponsored by the U.S. Department of Energy's Office of Energy Efficiency and Renewable Energy and located at the National Renewable Energy Laboratory.

This research has been supported by the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office (contract no. DE-AC36-08GO28308).

This paper was edited by Michael Muskulus and reviewed by Erik Quaeghebeur and one anonymous referee.