Wind turbine fatigue reduction based on economic-tracking NMPC with direct ANN fatigue estimation

Renewable Energy 147 (2020) 1632e1641

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Renewable Energy journal homepage: www.elsevier.com/locate/renene

Wind turbine fatigue reduction based on economic-tracking NMPC with direct ANN fatigue estimation
bastien Gros a, Axel Schild b Julio Luna a, *, Ole Falkenberg b, Se a b

€gen 11, SE-41296, Go €rsalsva €teborg, Sweden Chalmers University of Technology, Ho IAV, GmbH, Rockwellstrasse, Gifhorn, Germany

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 July 2019 Received in revised form 14 September 2019 Accepted 18 September 2019 Available online 25 September 2019

The aim of this work is to deploy an advanced Nonlinear Model Predictive Control (NMPC) approach for reducing the tower fatigue of a wind turbine (WT) tower while guaranteeing efficient energy extraction from the wind. To achieve this, different Artificial Neural Network (ANN) architectures are trained and tested in order to estimate the tower fatigue as a surrogate of the traditional Rainflow Counting (RFC) method. The ANNs receive data stemming from the tower top oscillation velocity and the previous fatigue state to directly estimate the fatigue progression. The results are compared to select the most convenient architecture for control implementation. Once an ANN is selected, an economic-tracking NMPC (etNMPC) solution to reduce the fatigue of the WT tower is deployed in real-time. The closedloop results are then compared to a baseline controller from a renowned WT simulation tool and a classic etNMPC implementation with indirect fatigue minimisation to demonstrate the improvement achieved with the proposed strategy. Finally, conclusions regarding computational cost and real-time deployment capabilities are discussed, as well as future lines of research. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Fatigue estimation Fatigue reduction Neural networks Wind turbines Real-time control

1. Introduction Renewable electricity generation from wind turbines (WT) is one of the fastest expanding energy sources, with over 486 GW installed around the world by the end of 2016, as reported by the Global Wind Energy Council (GWEC) [1]. Only in Europe, wind energy represents 51% of the newly installed electricity generating capacity [1]. WTs are exposed to variable winds, rotor loads and other structural and mechanical effects. These loads and their interaction affect WTs during their normal operation. When operated properly, WTs can last for decades, however, there are still challenges to overcome. For this reason, in recent years, advanced control strategies and modelling for control applications have considerably grown in importance with the aim of further extending the lifetime of WTs. In the literature, Nonlinear Model Predictive Control (NMPC) and Economic NMPC (ENMPC) are rising as the most appropriate candidates for closed-loop control of very large WTs. NMPC is capable of handling the complex and highly nonlinear dynamics of WTs, and the constraints present in the system. ENMPC tends to

* Corresponding author. E-mail address: [email protected] (J. Luna). https://doi.org/10.1016/j.renene.2019.09.092 0960-1481/© 2019 Elsevier Ltd. All rights reserved.

have superior performance than classic NMPC, especially in the presence of transients [2] because it handles the operating cost in the objective function directly [3]. There is significant research focusing on NMPC and ENMPC for WT control (WTC). The main control objectives that can be found in the literature are: power maximisation [4], cost reduction [5] and fatigue load reduction [6]. When implementing fatigue damage estimation and reduction for WTs, authors have traditionally adopted data-based approaches to handle the fatigue in closed-loop control schemes by lowering the tower oscillation velocity [7] or using individual pitch control to reduce the WT tower loads [8]. While fatigue minimisation is achieved, these approaches might not be optimal since fatigue damage is not dealt with directly, but through proxy signals. In this sense, using models to estimate the WT fatigue damage will aid to develop real-time optimal control solutions that act directly on the damage signal progression, providing better handling of fatigue cost reduction in the control problem. Recent developments in fatigue estimation for WTs include the use of Kalman filters [9] to estimate strain signals from sensors located on the tower. However, these solutions rely on open-loop data to represent fatigue. When deploying closed-loop solutions, feedback from the fatigue signal is needed in the update stage of these predictors. Since these are not available using current sensor technology, a different approach is

J. Luna et al. / Renewable Energy 147 (2020) 1632e1641

needed to tackle closed-loop implementation. A common practice to model damage accumulation produced by cyclic loading is the combination of the Rainflow Counting (RFC) algorithm [10] and a damage-equivalent load approach. While widely used in the industry, these approaches generate long data cycles that are non-smooth and non-differentiable, preventing their utilisation in real-time control applications. To address this issue in the scope of WTC, we propose to use Artificial Neural Networks (ANN) to estimate the fatigue accumulation on the WT and then use the output as a penalty in the cost function of an economic-tracking NMPC (etNMPC) control scheme. The function approximation capabilities of ANNs make them relevant as a surrogate to the RFC [11] in order to deliver an estimation of the damage progression within the NMPC prediction horizon. The first main contribution of this work is the development of a fatigue estimator based on ANNs that delivers a smooth and differentiable damage progression approximations. The ANNs are trained on data stemming from the National Renewable Energy Laboratory (NREL) simulation tool FAST [12]. After training, the ANNs performance is compared employing a set of metrics. The second main contribution is the deployment of the ANN fatigue estimation output as a cost term of a real-time etNMPC to actively reduce fatigue damage in the WT tower and increase its lifetime. The entire control solution is tested using the NREL 5 MW turbine [12] as reference. The results are compared with a baseline controller and an etNMPC controller with an indirect fatigue reduction approach. The paper is organised as follows. In Section 2, the fatigue estimation preliminaries are introduced along with a load and fatigue analysis to justify the signal selection for fatigue estimation. Section 3 presents the proposed ANN for fatigue estimation with a performance comparison of different configurations to select the most appropriate one for control applications. Section 4 introduces the elements to consider when deploying real-time etNMPC schemes for WTs. Closed-loop results are shown and discussed in Section 5. Finally, Section 6 summarises the conclusions of this paper and research lines for future work are proposed. 2. Fatigue preliminaries In this section, the fundamental concepts for fatigue modelling using counting methods are introduced. These counting methods produce cyclic data that will be transformed into temporal data for real-time control implementation. Moreover, a load and fatigue analysis is performed on the WT tower to obtain a suitable signal for fatigue estimation. 2.1. WT tower top simulation data Fig. 1 represents the wind speed profile W and the tower top position (xT ) and velocity (x_T ) along the tower fore-aft axis for an average wind speed of 11 m/s and a total time of 3500 s. The data has been extracted from the WT simulation tool FAST [12] for the NREL 5 MW reference turbine. In the scope of this paper, different average wind profiles from FAST are going to be used to test the robustness of the proposed fatigue estimation and control solutions. 2.2. Fatigue damage computation Fatigue is the weakening of a material subject to stress, specially the cyclic ones. Even with nominal maximum stress values, much lower than the ultimate stress limit of the material, cycling loading and fatigue can lead to the propagation of cracks and eventually, to the fracture of the structure [13].

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Fig. 1. Wind speed profile with an average value of 11 m/s (a), time domain response for x (b) and x_ (c).

While the amount of fatigue that a material has suffered is complex to calculate, there are widely used methods for fatigue estimation in the industry. In the case of WTs, the International Standard IEC 61400-1 [14], suggests the use of the RFC counting method, combined with the Palmgren-Miner linear damage accumulation rule. A general overview of the standard is shown in Fig. 2 (a). Given a time series signal u, the material ultimate design load su and the mean stress jsj, the RFC algorithm is computed as follows:

½Dsi ; ni  ¼ RFCðuðtÞ; su ; jsjÞ;

(1)

where Dsi is the stress cycle ranges amplitude, ni is the number of cycles with amplitude Dsi and RFC is the counting method function, which in this paper is the MLife algorithm from the NREL [12]. In the scope of this paper, the time series signal u in Equation (1) is going to be computed from the axial stress that appear at the tower base due to the oscillations caused by the wind. The complete justification for the selection of the axial stress as the main cause of fatigue accumulation on the WT tower is detailed in Section 2.3. After applying the RFC algorithm, the maximum number of cycles at the i-th stress (Ni ) that the material can tolerate before failure is obtained


Ni ¼

su  jsj 0:5gDsi

m ;

(2)

where g is a safety factor and m the SeN curve slope [15]. With Equation (2), the estimated damage fraction D is obtained using the Palmgren-Miner rule

D¼

X ni i

Ni

2½0; 1:

(3)

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J. Luna et al. / Renewable Energy 147 (2020) 1632e1641

Fig. 2. Industry-standard fatigue computation (a) and extrapolation technique to obtain time-dependant data (b).

The function obtained from the RFC algorithm in Equation (1) is non-smooth, nonlinear, and highly difficult to differentiate. Therefore, the formulation of the RFC output in optimal control problems is extremely complicated. Moreover, the described fatigue damage estimation in Equation (3) depends on cycles rather than time. Each cycle computation requires substantial amounts of computing time, making it impractical for real-time control schemes, such as the one proposed in this paper. To address the aforementioned problems, a two-part solution is proposed in this paper. First, a simple time extrapolation is used to compute a time-dependent damage value, as shown in Fig. 2(b) and summarised in Algorithm 1. Once the data has been expressed in time-dependant values, ANNs are deployed to obtain differentiable and smooth signals that can be implemented in real-time in order to directly reduce the WT fatigue as an associated cost of the optimal control problem.

2.3. Tower base stress calculation Depending on the wind direction, a WT tower can move on a side-to-side or fore-aft direction. Wind speed causes the fore-aft motion, while side-to-side motions are due to electrical and mechanical disturbances [16]. These movements generate mechanical stresses in the WT structure that can lead to blade and tower fatigue. Assuming that the tower base is a thin-walled cylinder structure (see Fig. 3) without accounting the effect of paint, bolts, welds and flanges, a stress concentration factor of 1.0 is considered during the

study [17]. The physical parameters of the NREL-5 MW WT are included in Table 1. When subjected to wind, a WT acts as a cantilever vertical beam. The reactions that appear on the tower base generate axial and shear stresses that affect the lifetime of the system. The total axial stress is computed as a combination of the axial force and the bending moment components. Its maximum component occurs on the outermost surface of the tower base and it is expressed as follows:

smax ¼

Fz My þ ro ; Ab Iy

(4)

being Fz the force applied vertically on the tower base, My and Iy the bending moment and moment of inertia around the y-axis respectively, and ro the tower base outer radius. Ab is the cross sectional area of the tower base on the y-z plane

  Ab ¼ p r 2o  r 2i ;

(5)

where ri is the tower base inner radius. Regarding the total shear stress, it is a combination of the transverse force and the torsional components

t¼

VQ Mz þ ro ; Ix t Jz

(6)

being V the shear force applied on the x-axis, Q is the first moment of area at a given t width from the neutral line of the geometry, Mz the torsional moment, Ix the x-axis moment of inertia, and Jz the

J. Luna et al. / Renewable Energy 147 (2020) 1632e1641

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Fig. 3. Free body diagram of the WT tower base. Fig. 4. Axial and shear stress comparison (a) and detail (b and c) for different wind speeds.

Table 1 Physical properties of the NREL-5 MW WT [17]. Parameter

Value

Power rating Hub height Base diameter Base thickness Young’s modulus Shear modulus Steel density

5 MW 90 m 6m 0.0027 m 210 GPa 80.8 GPa 8500 kg m3

polar moment of inertia around the z-axis. In hollow cylinders, the maximum shear stress appears when t equals the effective width of the cross section

t ¼ 2ðro  ri Þ:

(7)

Knowing Equation (7), the first moment of area is computed as follows:

Q¼

 2 3 r  r 3i : 3 o

(8)

Therefore, Equation (6) for the maximum shear case is expressed as

  V r 2o  r 2i Mz tmax ¼ þ ro : 3Ix Jz

(9)

An analysis using short-term data with wind speeds w2 ½3; …; 25 m/s has been conducted to extract the root mean square (RMS) values for the axial and shear stresses. The results are included in Fig. 4. From the data, it can be concluded that the main stress that appears on the tower base is the one related to the axial forces in Equation (4). Specifically, the bending moment My

Fig. 5. Tower base fatigue due axial and shear stresses.

generates the main contribution to the total axial stress smax , as it can be seen in Fig. 4(b). However, higher axial stress does not always mean larger fatigue values caused in the material when compared to shear stress fatigue. To corroborate which is the signal that contributes the most to the tower base fatigue, the damage fraction for both smax and tmax is computed and compared in Fig. 5. As it can be seen, ultimately, smax has a higher fatigue impact on the material and therefore, will be used to train and deploy the ANNs in Section 3.

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3. Fatigue estimation for WTC using ANNs In this section, the ANN structure for fatigue estimation is selected taking into account the properties of the RFC signal and the control application considerations for deployment in Section 5. Moreover, the training and validation methodology are presented along with results comparing different ANNs.

3.1. Artificial neural networks for fatigue estimation In the field of energy systems, Artificial Neural Networks (ANN) are widely used for numerous applications such as classification, time-series prediction, and function approximation [18]. More recently, they have been attracting the attention of researchers to tackle some of the unsolved problems related to wind power generation [19]. From a mathematical point of view, ANNs can be considered black-box models that have the ability to map given inputs to a desired output using advance learning methods. ANNs can be described as a function of the form f : X/Y, where X is a given set of independent variable inputs or features and Y is the set of expected outputs, mapped through the use of a nonlinear function f. In the scope of fatigue estimation, ANNs play an important role as function approximators to the complex nonlinear phenomena that takes place in the system under study. When considering control deployment for fatigue reduction, a traditional modelling approach could be challenging due to the increase of complexity in the closed-loop solution. In this paper, the ANNs are used as a surrogate to the RFC approach introduced in Section 2 in order to have reliable fatigue predictions for WTC applications. The selection of the neural network structure is based on the knowledge of the system to be described among other considerations. Taking into account that the output of Algorithm 1 is a time series of the WT damage, it has been concluded that nonlinear autoregressive networks with exogenous inputs (NARX) are the best option to predict the tower fatigue. NARX neural networks have feedback connections to obtain nonlinear time series predictions by using past values of the predicted or true time series [20]. A classic NARX implementation can be expressed as follows:

yt ¼ f ðyt1 ; yt2 ; …; ut ; ut1 ; …Þ þ εt ;

(10)

where y and u are the output and input variables at different t time samples respectively. Moreover, ε is the error term and f the nonlinear function used to map the inputs to the predicted output. The architecture of ANNs can be divided into different layers: input, hidden, and output. The optimal tuning of the ANN structure is out of the scope of this paper. However, a standard trial-and-error procedure based on several simulation cases has been carried out to select the most suitable ANN structure for fatigue estimation. In Section 3.4, different ANNs are tested to select the one that will be deployed in the closed-loop scheme, based on performance and computational cost considerations. An overview of the ANN architecture is shown in Fig. 6. As shown in Fig. 6, the proposed ANN has two inputs: the WT x-

_ axis speed (xbu) and the feedback loop from the ANN only output, b_ which is the expected instantaneous damage ( Dby) at different time samples. These inputs have been selected considering the statistical relevance of the signals with respect to the dependant variable output of the ANN. Both ANN inputs are connected to a delay block to take into account current and past information from the WT fatigue signal, which greatly affects the next iteration output. The values of the weights connecting the input, as well as the hidden and output layers are set using the training set defined in Section 3.2 and the training considerations presented in Section 3.3. The activation functions for the hidden layers are hyperbolic tangent functions such as

tanhðzÞ ¼

expz  expz ; expz þ expz

(11)

and the output layer incorporates a linear function. Once the structure is defined, the model has to be trained and tested before its use on the control application. In the following sections, the different steps to train, test, and select the ANN to be deployed are developed. 3.2. Data processing Before training and evaluating the performance of the ANNs, the temporal and time-frequency data has to be correctly preprocessed to obtain training (G 2Rtrr ) and validation (S 2Rsr ) sets, where tr represents the number of training samples, s the number of validation samples and r the number of wind cases used for the training and testing of the ANN. To obtain the training and testing sets, first the instantaneous damage data D_ is obtained off-line from the damage estimation Algorithm 1 using Equation (4) as the relevant fatigue load. Then, D_ is filtered using a simple moving average approach to reduce the noise originated by the RFC in order to remove possible outliers, which could generate false trends when training the ANNs. Once the damage data is filtered, the signals are scaled to standardise the variables to a suitable range. In particular, the tower _ base velocity is restricted to x2½ 1; 1 and the instantaneous _ damage data to D2½0; 1. In the scope of this section, the scaling is performed taking into consideration the maximum and minimum values of all the x_ and D_ datasets respectively. When deploying the proposed methodology in a closed-loop scheme, the physical limits of the signals will be considered when performing the data scaling. Finally, the data is split between training G and validation sets S . 3.3. Training considerations The training process has been carried out on a PC Intel Core i56500 @ 3.20 GHz with 16 GB of RAM with training and validation sets of 80% and 20% of different wind averages stemming from FAST data, similar to the ones shown in Fig. 1. As aforementioned, different ANN structures are going to be analysed to study the effect of the number of neurons in the first layer to estimate fatigue. In particular, the set of neurons to be studied is nn ¼ ½2; 4; 10; 25; 50 with 2 input delays and 2 feedback delays (see Equation (10)). For each value of nn , a different ANN is trained and validated using additional data, completely independent from the training and testing datasets. 3.4. Fatigue estimation validation and discussion

Fig. 6. Proposed ANN architecture.

To validate the trained ANNs, the estimated accumulated damage is going to be computed using the following expression:

J. Luna et al. / Renewable Energy 147 (2020) 1632e1641

    Fig. 7. Accumulated damage DAcc validation set, baseline fatigue x_j2 and estimation b D Acc for the ANNs in Table 2.

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103 MPa. Moreover, the results include the [2 norm of the tower bending velocity (jjx_T jj), which is the most common approximation to estimate fatigue in control applications [21]. Fig. 7 shows the validation results for the trained ANNs, the RFC signal, and the jjx_T jj fatigue approximation. All of the trained ANNs provide satisfactory results with minor disparities with respect to the test data. Meanwhile, using the [2 norm of x_ does not guarantee good results when there are extreme loads that cause big jumps in the fatigue, e.g. between t ¼ 3000 s and t ¼ 3050 s. To quantify the improvement of the proposed methodology versus the classic baseline approach, Fig. 8 shows three metrics to compare the results with the test set from the RFC: [1 , [2 and [∞ norms. The overview of the trained ANNs, along with the simulation results, is presented in Table 2. Fig. 8 clearly shows how the proposed strategy outperforms the classic baseline approach in all of the analysed metrics. Major differences between the test data and the estimations are observed in Fig. 7 at the last stages of the simulation. However, these differences are a consequence of the accumulated error over time. The results in Figs. 7 and 8 indicate that ANN3 is the one with the best performance. This is beneficial for deploying real-time NMPC controllers with fatigue reduction: a good prediction with a low number of neurons (nn ¼ 10) means that the fatigue penalty will hold a smaller state space size and thus, the computation time will not increase dramatically. For all these reasons, ANN3 is selected for control deployment in Section 5 as it provides the best estimation results with a relatively low complexity.

4. Economic-tracking NMPC for WTC Before closed-loop deployment and comparison of the different controllers in Section 5, the basic concepts for real-time economic NMPC (ENMPC) and economic-tracking NMPC (etNMPC) for WTC are given in this section. The indirect implementation of WT tower fatigue reduction is discussed, and the problem formulation for direct fatigue minimisation is derived in view of the results from Section 3.

    Fig. 8. Metrics for the baseline validation set, baseline fatigue x_T j2 and estimation b D Acc ðnn Þ.

b b_ D Acc ¼ Si D i ;

(12)

b_ is the estimated damage at time instant i. The results are where D i going to be discussed using a validation dataset (different from the training dataset) for each one of the trained ANNs. The different ANNs are going to be compared to test data, obtained using the RFC algorithm with the following parameters: m ¼ 4, su ¼ 5:48 

4.1. Real-time economic-tracking NMPC with proxy fatigue damage minimisation Classic ENMPC for WTC aims to maximise the generated WT electric power (P E ) directly in the cost function. However, it has been proven that maximising the aerodynamic power (P A ) is a better surrogate since it can aid in reducing the turnpike effect caused by the stored kinetic energy in the WT rotor [2]. Moreover, in Ref. [2] the ENMPC was rewritten in an equivalent etNMPC formulation that is suitable for real-time deployment, which will be briefly discussed in this section. In classic real-time NMPC formulations, the stage cost function Lðxk ; uk Þ [22] is one of the penalties to be minimised. In etNMPC, the penalty is replaced by a quadratic function of the type

Table 2 Comparison of the different ANN structures for fatigue estimation (the one selected for control deployment is highlighted). Case

Neurons

Input delays

Feedback delays

RMSE

[1

[2

[∞

ANN1 ANN2 ANN3 ANN4 ANN5

2 4 10 25 50

2 2 2 2 2

2 2 2 2 2

1.88 2.16 1.60 1.69 1.46

11.7e8 20.2e8 4.3e8 5.0e8 5.5e8

4.8e10 9.1e10 1.9e10 2.5e10 2.8e10

2.3e12 6.5e12 1.3e12 2.2e12 2.4e12

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J. Luna et al. / Renewable Energy 147 (2020) 1632e1641

 2 b U  Uref ð wÞ b Lðxk ; uk Þ ¼ QU ð wÞ  2 b q  qref ð wÞ b þQq ð wÞ þ Pc ðxk ; uk Þ;

(13)

which utilises a tracking behaviour of the optimal wind-dependent pitch angle qref ðwÞ and rotor-speed Uref ðwÞ. Pc ðxk ; uk Þ are the b competing penalties of the problem. The tracking weights Qq ð wÞ b are chosen in such a way, that they locally approximate and QU ð wÞ the ENMPC scheme described in Ref. [2]. The complete etNMPC scheme with proxy signals as competing economic costs reads as follows:

minfðxN Þ þ x; u

N1 X

 2 b U  Uref ð wÞ b QU ð wÞ

k¼0

 2 b q  qref ð wÞ b þQq ð wÞ þ Qtower x_2T 2 2 þQq_ q_ þ QM_ M_ þ Qflap x_2B ;

s:t

(14)

xkþ1 ¼ Fðxk ; uk ; Wk Þ Fig. 9. Results for the BC, etNMPC, and etNMPC þ NN.

hðxk ; uk Þ  0 b ðti Þ; x0 ¼ x where f is the terminal cost to be minimised, x_T and x_B are the tower and blade speed in the x-axis direction, q_ is the pitch rate velocity and M_ the generator rotatum. Qtower and Qflap are the competing weights for the tower and blade fatigue. Moreover, Qq_ and QM_ are the weights for the pitch rate and generator torque velocity variation. 4.2. Real-time economic-tracking NMPC with direct fatigue damage minimisation While the approach in Section 4.1 is very well suited for realtime NMPC, and allows to write Nonlinear Programs such as the ones described in Ref. [2], the use of a simple quadratic function as a proxy for tower estimation in Equation (14) is debatable as it impacts the fatigue indirectly through x_2T . In this paper, the quadratic term Qtower x_2T is replaced by a more sophisticated approach using a tower fatigue estimation based on ANNs, as described previously. The complete etNMPC scheme with ANN tower base fatigue estimation (etNMPC þ NN) is expressed as follows:

min x;u

fðxN Þ þ

N1 X

 2 b U  Uref ð wÞ b QU ð wÞ

k¼0

 2 2 b_ b q  qref ð wÞ b þQq ð wÞ þ Qtower D 2 2 þQq_ q_ þ QM_ M_ þ Qflap x_2B ;

s:t

(15)

xkþ1 ¼ Fðxk ; uk ; Wk Þ hðxk ; uk Þ  0 b ðti Þ; x0 ¼ x

b_ is the estimated value of the WT base fatigue damage that where D will be obtained from the ANN output from Section 3. 5. Closed-loop results The ANN with the best performance in Table 2 is deployed in this section using the real-time etNMPC scheme with direct fatigue minimisation described in Section 4 to test the fatigue reduction

capabilities of the proposed methodology. It will be compared with a real-time etNMPC with indirect fatigue estimation and with a Baseline Controller (BC) from the NREL simulation tool FAST. 5.1. Parameter study for damage equivalent loads Fatigue is often given in terms of so-called DEL, instead of remaining life-time or damage fraction. A DEL is a constant amplitude, fixed frequency and fixed mean value signal which generates the same amount of damage as the original variablespectrum signal. The remaining life-time of the component can then be easily computed for different material properties. More information about the derivation of DEL datasets for WTs can be found in Ref. [23]. In our analysis we use a frequency of f ¼ 1 Hz. A parameter study has been performed to find the optimal values for the weights ½Qtower ; Qflap ; Qq_ ; QM_  in Problem (15). For this study, wind profiles that comply to DLC1.2 according to IEC 61400-1 [14] are used (extreme conditions such as turbulent wind are out of the scope of this paper). A total of 135 simulations over 13 different wind speeds (each one 300 s long) have been used. Then, a standard damage-analysis using the MLife-program [24] was performed and the resulting short-term DELs were extrapolated to a lifetime of 20 years using a Weibull probability distribution [25] for the wind speed with the parameters k ¼ 2:5 and c ¼ 13. A second study covers an in-depth analysis of one parameter set for the etNMPC and etNMPC þ NN controllers. This analysis is based on 120 simulations with wind speeds ranging between 3 and 25 m/ s. Each case is run over 580 s. From these simulations we again compute life-time DELs. The results are shown in Fig. 9. Note that the figure has been scaled to show a value equal to 1 for the BC in order to extract percentage values more easily for result comparison. 5.2. Closed-loop results discussion As shown in Fig. 9, the etNMPC þ NN (with direct tower fatigue reduction) clearly reduces the DEL for the tower when compared to the etNMPC and BC approaches. Particularly, it improves more than an 8% when compared with the etNMPC controller and over 40% with respect to the well-known NREL controller. The counterpart is that the pitch activity using the etNMPC þ NN is increased.

J. Luna et al. / Renewable Energy 147 (2020) 1632e1641

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Table 3 Pareto front weights for the experiments (the best tower fatigue case controller is highlighted). Controller etNMPC

etNMPC þ NN

Case A#1 A#2 A#3 A#4 A#5 A#6 A#7 A#8 B#1 B#2 B#3 B#4 B#5 B#6

Qtower 5

1.00e 1.00e3 2.50e3 1.25e2 1.25e2 1.00e2 1.00e2 5.05e3 7.50e5 7.50e3 1.25e2 1.25e2 5.05e3 5.05e3

Qflap

Qq_ 2

5.05e 1.00e3 5.05e2 5.05e2 5.05e2 1.00e3 1.00e3 1.00e3 5.05e2 5.05e2 5.05e2 5.05e2 1.00e3 1.00e3

Moreover, a slight increase (less than 1%) in the DEL for the blades can be observed in Fig. 9. This behaviour was expected since the better modelling of the fatigue penalty in Problem (15) translates into a higher blade pitch activity to minimise fatigue. Nevertheless, the damage from the blades is still reduced more than 25% when compared to the BC solution. With respect to the energy harvesting, a minor 1% improvement is achieved when comparing the etNMPC þ NN and etNMPC strategies. It has to be noted that while the energy harvested increases, the improvement could be considered negligible in some cases. Fig. 11(a) shows all the simulation cases that were carried out for the short-term wind profiles using the BC, etNMPC, and etNMPC þ NN controllers. The long-term cases can be obtained by extrapolating to life-time data from the short-term case. Moreover, it shows the Pareto fronts for the best performing cases, which are included in Table 3. The detail of the etNMPC and etNMPC þ NN is shown in Fig. 11(b), both of which greatly improve the BC performance. As expected, most of the weight combinations for the etNMPC þ NN approach greatly reduce the tower fatigue DEL as a result of the improved fatigue estimation characteristics included in the ANN strategy. However, it can be seen that in some cases (e.g., controller B#1), the etNMPC can outperform the tower fatigue reduction. This is due to the small weight set for the Qtower in these cases (see Table 3). A similar weight is used for A#1, which clearly performs worse than B#1. The best case for the tower fatigue DEL is

QM_ 6

1

1.00e 1.00e4 5.00e6 1.00e6 5.00e6 5.05e4 1.00e3 1.00e3 5.05e4 7.50e4 2.50e4 7.50e4 1.00e3 5.05e4

5.05e 1.00e2 5.05e1 5.05e1 5.05e1 1.00e2 1.00 1.00 5.05e1 5.05e1 5.05e1 5.05e1 5.05e1 5.05e1

DELBlade [kN m]

DELTower [kN]

5880 5820 5800 5730 5770 5940 6030 6090 5750 5780 5890 5920 6110 6120

25130 24572 24066 22293 22191 22748 23458 24015 22698 19405 19050 19253 20671 20773

the experiment B#3. When Qtower is set to higher values, the tower fatigue DEL is greatly reduced for the etNMPC þ NN strategy. This also means that the pitch activity is increased and therefore, the WT blades are subject to increased effort which translates into higher values of blade DEL. The pitch activity is defined as the integral of the blades pitch angle velocity b_ over a given period of time T:

ðT P¼

0

jb_ jdt:

(16)

Fig. 10 shows the cloud plot representation of the experiments, comparing the blades DEL with the pitch activity for the short-term simulation case. Most of the etNMPC þ NN results are located on the right side of the figure, denoting a higher pitch activity in order to reduce the fatigue levels on the WT tower. However, higher pitch activity does not always correspond to higher blade DEL (and therefore, higher blade fatigue). In fact, the higher values of blade DEL correspond to low pitch activity cases of the etNMPC. While pitch activity is a metric for how much the blade moved about its chord axis, blade fatigue is evaluated for flap motion. Therefore, it is possible to have a low blade fatigue with a high pitch and vice versa. 5.3. Closed-loop tower base stress calculation In Section 5.2, it has been shown that the etNMPC þ NN controller reduces the fatigue impact on the WT when compared to the etNMPC and BC controllers. Following the analysis presented in Section 2.3, Fig. 12 presents the closed-loop results for the WT load study. The etNMPC þ NN controller notably reduces the axial and shear stresses in the WT base, with the exception of the axial stress at wind speed equal to 3 m=s. This outlier can be neglected as it is not a wind speed that can damage the WT in the long term. Regarding the etNMPC controller, while it works well at high wind speeds, it behaves similarly to the BC for winds under 11 m=s. Thus, it can be concluded that the etNMPC þ NN approach reduces both the fatigue and load signals in the WT tower base, while increasing the energy harvested. The counterpart is an increase in the blade DEL value. 5.4. Computing time

Fig. 10. Pitch activity for the etNMPC and etNMPC þ NN controllers.

As discussed in Section 4, the computation time is one of the most important aspects to take into account when implementing

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Fig. 12. Closed-loop axial (a) and shear (b) stress comparison for different wind speeds.

Table 4 Computation time comparison. Controller #

Mean [ms]

Std. dev. [ms]

Diff. [%]

A#4 B#2

8.003 10.334

1.504 1.305

0.00 29.16

the proposed fatigue estimation strategy in real-time NMPC schemes. Optimisation-based closed-loop control solutions for WTs use complex nonlinear models. Introducing an additional term in the form of fatigue estimation can contribute to increase the complexity of the problem and therefore, reduce the real-time implementation feasibility. The increase of the complexity of the problem was one of the reasons behind the selection of ANN3 in Section 3.4. The lower number of neurons guarantees a feasible computational implementation in a real-time controller. Table 4 shows the computing time for controllers A#4 and B#2 from Table 3. While there is roughly a 30% increase in the computation time, it is still in the milliseconds range. Considering that the sampling time for WTC is within the seconds range, it can conclude that the proposed approach is feasible for real-time implementation.

developed and deployed in real-time closed-loop simulations. The evaluation of the proposed strategy for fatigue estimation shows good performance for differently trained ANNs. A fatigue estimation ANN has been selected based on performance and computation time requirements for its inclusion in the closed-loop etNMPC scheme. The outcome of the simulations shows promising results when deploying the direct fatigue minimisation strategy. It has been shown that the etNMPC controller is capable of reducing the blades and tower fatigue with respect to the classic BC that is included in FAST. Moreover, the addition of the fatigue estimation in the etNMPC þ NN controller has further reduced the tower fatigue and stress loads by means of increasing the pitch activity of the blades, but not necessarily the associated fatigue. While the proposed fatigue estimation approach shows prom    ising results when compared with the classic x_T j2 norm strategy, further work can help to improve the results. For instance, the inclusion of additional features as an input to the ANNs. Moreover, frequency decomposition of the tower velocity signal can help to reduce the computational cost of the etNMPC þ NN scheme. Regarding the controller, future work includes the inclusion of the proposed ANNs for all critical components (tower and blades) in a full etNMPC þ NN scheme.

6. Conclusions

Acknowledgements

Fig. 11. Short-term tower and blade fore-aft DEL results for the BC, etNMPC, and etNMPC þ NN controllers (a) and detail of the etNMPC and etNMPC þ NN controllers (b).

A methodology to estimate fatigue in WTs using ANNs has been

This research has been funded by the German Ministry of

J. Luna et al. / Renewable Energy 147 (2020) 1632e1641

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