Prognostics-based adaptive control strategy for lifetime control of wind turbines

for lifetime control of wind turbines Edwin Kipchirchir1, M. Hung Do2, Jackson G. Njiri3, and Dirk Söffker4 1,4Chair of Dynamics and Control, University of Duisburg-Essen, Lotharstr. 1-21, 47057, Duisburg, Germany 2School of Heat Engineering and Refrigeration,Hanoi University of Science and Technology 3Department of Mechatronic Engineering, Jomo Kenyatta University of Agriculture and Technology, 62000-00200, Nairobi, Kenya Correspondence: Edwin Kipchirchir (edwin.kipchirchir@uni-due.de)


Introduction
Growing demand for wind energy has led to the development of large wind turbines. However, these turbines are less tolerant to system performance degradation and faults (Gao and Liu, 2021). To ensure utility-scale wind turbines operate with respect to their design lifetime, advanced control strategies have been developed in recent years to reduce structural loading of blades 20 and tower. Most of these incorporate additional objectives such as power optimization and rotor speed regulation. The objective of lifetime control of wind turbines using prognostics-based load mitigation strategies has become more important in recent years. Most of the proposed methods focus on controlling the lifetime of one structural component of a wind turbine, typically the rotor blade or the tower, without considering the fatigue damage level in other components. These lifetime controllers are also designed to be valid about specific operating points.

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A control strategy for extending the maintenance interval of wind turbine blades under assumed crack initiation, detected using a data filtering algorithm, is proposed (Beganovic et al., 2015). In (Beganovic et al., 2018;Njiri et al., 2019), a set of switching controllers with varying degrees of load mitigation are engaged sequentially based on the accumulated damage obtained from an online damage evaluation model to extend the lifetime of rotor blades. An adaptive lifetime controller is proposed in (Do and Söffker, 2019) to guaranty the desired lifetime of the tower. Depending on the damage accumulation and 30 the predicted lifetime provided by a online damage evaluation model, the weights of the lifetime controller are varied. However, in (Beganovic et al., 2015(Beganovic et al., , 2018Njiri et al., 2019;Do and Söffker, 2019) fatigue damage is considered in only one turbine component. The lifetime controllers used are not adaptive to varying wind conditions. In recent times, resilient control has been proposed in (Acho et al., 2016;Azizi et al., 2019;El Maati and El Bahir, 2020;Jain and Yamé, 2020) to minimize the effect of unanticipated faults or unexpected dynamics to maintain the operation of a wind turbine within a limited degradation tolerance 35 bound. However, resilient control does not address the problem of controlling life consumption in wind turbine components to avoid early fatigue failures. Although new concepts like operational modal analysis (OMA), which relies on measurement data to analyze vibrating structures are becoming the industry standard for condition monitoring and diagnosis especially for offshore wind turbines (Kim et al., 2019;Bajric et al., 2017;Dong et al., 2018;Pegalajar-Jurado and Bredmose, 2019), these concepts are yet to be integrated for prognosis and lifetime control of wind turbines.

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In this work an adaptive lifetime control strategy is proposed for controlling ageing of rotor blades to guaranty a desired lifetime while considering damage accumulation level in the tower. A robust disturbance accommodating control (RDAC) proposed in (Do and Söffker, 2021) is used for rotor speed regulation and load mitigation in the tower, while a prognosticsbased adaptive independent pitch control (aIPC), which adapts to wind speed variation, is used for lifetime control of rotor blades. By monitoring the accumulated damage using an online structural health evaluation model, the load mitigation level 45 in the blades is controlled by varying the control gains in the respective IPC controllers based on a threshold evaluation of the estimated lifetime. As an improvement to the approaches in the aforementioned contributions, the proposed adaptive lifetime control strategy regulates fatigue loading in the rotor blades to reach a predefined damage limit at the desired lifetime with subsequent reduction in tower damage accumulation. This is realized without trade-off in speed/power regulation performance.
The paper is organized as follows. In section 2, a theoretical background on wind turbine health monitoring is given. In 50 section 3, design of the primary RDAC controller for rotor speed regulation and tower load mitigation, and the prognosticsbased aIPC lifetime controller for controlled ageing of rotor blades is outlined. The proposed prognostics-based adaptive lifetime control strategy, which incorporates the primary and lifetime controllers, and an online damage evaluation model is described in section 4. In section 5, simulation results based on performance evaluation of the proposed prognostics-based adaptive lifetime control strategy on a reference wind turbine are discussed. Lastly, summary and conclusions are given in 55 section 6.
Wind speed variability subjects wind turbine components like blades and tower to cyclic loading. This causes damage to be accumulated in these components overtime causing gradual degradation until failure occurs. Therefore, structural health monitoring of wind turbines is important in preventing occurrence of fatigue failure before reaching related design lifetime.

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Information on the damage evolution in a component can be utilized as a health indicator for failure detection as well as for developing control measures to guaranty desired lifetime. This section outlines the methods used for estimating the damage accumulation in wind turbine components.

Evaluation of damage accumulation
A Wind turbine endure varying and complex load conditions over its lifetime. Fatigue analysis is therefore important in determining the consumed lifetime of its components. Component degradation starts at micro-scale as micro-cracks resulting from irreversible changes in the microstructure, and propagates gradually until it fails. Assumptions of underlying damage evolution laws are often used to estimate the actual damage level as well as to predict the remaining useful life (RUL) of a component.
Component-specific high-cycle fatigue experiments are used to generate S-N curves (Wöhler curve), which describe the relationship between applied stress amplitude S and the number of load cycles N that would cause failure. This forms the basis for 70 the mathematical relation for fatigue analysis in wind turbines components expressed as where s denotes the stress range amplitude, m the Wöhler exponent (typically 3 for steel materials like the tower and 10 for composites like the blade (Ragan and Manuel, 2007)). The material parameter of fatigue damage at failure K e.g., ultimate tensile strength is related to the number of load cycles N .

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Wind turbine components are designed for a service lifetime of at least 20 years according to the international electrotechnical commission (IEC) standard, with these structural components facing roughly between 10 8 and 10 9 fatigue load cycles (Ziegler et al., 2018). The component lifetime is typically arrived at using the projected number of fatigue cycles and average wind conditions it will encounter in its lifetime. Additionally, the IEC standard specifies that a wind turbine component should be designed to maintain its structural integrity in case it experiences 50 year extreme wind events during its lifetime.

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Fatigue damage in components can be assessed using linear damage accumulation theory based on Miner's rule or nonlinear damage accumulation theories (Yuan et al., 2014). Due to its simplicity, Miner's rule (Miner, 1945) is widely used. Wind speed variability induces varying-amplitude load spectrum on wind turbine components. To use Miner's rule, the complex spectrum of varying load is often transformed using rain-flow counting (RFC) algorithm first proposed by (Matsuishi and Endo, 1968), into simple uniform loading, from which stress range histograms can be extracted and used to assess the accumulated damage. By combining RFC and Miner's rule, damage accumulation D k is calculated as where k denotes the total number of related stress range histograms, d i the incremental damage at the i th stress range histogram,

Online rain-flow counting
Most standard RFC algorithms generate equivalent load cycles from complex load spectra by pairing local minima and maxima points using 3-point counting rule. Therefore, the entire load history is needed beforehand for the equivalent cycles to be generated. This process is computationally inefficient because the algorithm has to process all the stored loading data. Therefore, as implementation of lifetime control. In this paper, the online damage evaluation algorithm (Musallam and Johnson, 2012), is adopted for evaluating the accumulated damage in rotor blades and tower. This information is then used to adapt the lifetime controller to guaranty a predefined service life of the wind turbine components.

Control strategy for load mitigation and speed regulation
In this paper, a Robust Disturbance Accommodation Controller (RDAC) (Do and Söffker, 2021), proposed for rotor speed 110 regulation and mitigation of tower fore-aft bending moments is extended to include an adaptive Independent Pitch Controller (aIPC), which is used as a dynamic lifetime controller for reducing blade flap-wise bending moments in a wind turbine operating in the above-rated wind speed region. In this section the description of the reference wind turbine (RWT) is outlined.
Additionally, the description of the adaptive robust observer-based controller, which is adapted for lifetime control, is summarized.  Table 1. It is a 3-bladed, upwind, horizontal axis wind turbine, having 24 Degrees of Freedom (DoFs) describing its flexibility.
However, a few DoFs are enabled to obtain a reduced order linear time-invariant (LTI) models used for controller design.
where M denotes the mass matrix containing inertia and mass components and f is a nonlinear function of the enabled DoFs q and their first derivativeq, as well as the control input u, the disturbance input u d , and time t. The nonlinear model Eq. (3) 125 available in FAST is linearized about an operating point in the above-rated region. By enabling the DoFs, which capture the most important wind turbine dynamics of interest, and specifying the operating point defined by a constant hub height wind speed, pitch angle, and rotor speed, linearization is carried out numerically in FAST yielding periodic (azimuth dependent) matrices of LTI models.

Controller for load mitigation and speed regulation 130
An adaptive robust observer-based controller, which in combination with an online damage evaluation model used for lifetime control of wind turbine components, is briefly outlined.

Robust disturbance accommodating controller
The RDAC controller, proposed in previous work (Do and Söffker, 2021), is briefly outlined for principal understanding.
where A, B, B d , C denote the state-space system, ubthe control input, which is the collective pitch angle, x the states, d the wind disturbance, and y the measured outputs, which include rotor speed and tower-top fore-aft bending moment.
The model Eq. (4) is augmented with a pitch actuator model, which accounts for the slow pitch actuator dynamics. To counteract wind disturbance effects, the model is extended with an assumed step disturbance waveform (Wright, 2004;Wright 145 and Fingersh, 2008), which approximates sudden uniform rotor effective wind velocity fluctuations. To meet the rotor speed regulation objective with zero steady-state tracking error, the model is further extended with a partial integral action.
To ensure closed-loop system stability, robustness and optimality, a mixed-sensitivity H ∞ norm of the closed loop transfer function is used as a cost function to optimize the disturbance accommodating controller(DAC) parameters including observer gain L x , state controller K x disturbance rejection controller K d , and the integral gain K i in a single step. The mixed sensitivity H ∞ optimization problem is formulated as where R * denotes the optimized controller, R a set of controllers R that stabilize the plant. The weighting functions W 1 , W 2 , and W 3 are introduced to ensure desired robust performance while S, RS, and T denote the related sensitivity, control effort, and complementary sensitivity functions, respectively. The problem to find an optimal RDAC controller RDAC * is formulated where RDAC denotes a set of controllers RDAC that stabilize the generalized plant P , and G zd is the transfer function from the exogenous inputs d to the controlled outputs z.
Nonsmooth H ∞ synthesis proposed in (Apkarian and Noll, 2006), used for problems with structural and stability constraints 160 is applied to find an optimal controller RDAC * with robust gains L and K for tower load mitigation and rotor speed regulation. It is implemented in MATLAB using hinfStruct command (Apkarian and Noll, 2017). In Fig. 2 application of the RDAC controller to the 1.5 MW NREL RWT is shown. An actuator transfer function is included in the generalized plant P, to account for the blade pitch actuator dynamics. Hub height wind disturbance d excites the wind turbine dynamics in above rated operation. Measurement outputs including rotor speed ω and tower fore-aft bending moment ζ are fed to the RDAC controller, 165 which generates a collective pitch angle β as a control signal for regulating rotor speed at the rated value and for reducing tower fore-aft bending moment oscillations. The RDAC controller is robust against modeling errors and wind disturbances.
The desired trade-off between robust stability and performance is achieved by choosing suitable weighting functions W 11 , W 12 , and W 2 . To effect rotor speed response and ensure robustness against wind disturbances, W 11 is designed as an inverted low-pass filter. To reduce the first mode of tower fore-aft oscillation, W 12 is designed as an inverted notch filter centered at 170 2.56 rad/s. To reduce controller activity at high frequencies thereby increasing robustness, W 2 is chosen as high-pass filter.

Wind
Both objectives of rotor speed regulation and tower load reduction for wind turbines operating in above-rated wind speed region are met while ensuring robustness against modeling errors and wind disturbances. However, RDAC * is only valid within its design operating point and suffers performance deterioration outside this envelop. Additionally, its control input signal is a collective pitch angle, hence cannot be applied for reducing blade oscillations due to vertical wind shear, which can 175 only be achieved through IPC control.

Adaptive independent pitch controller
This controller is desired to counteract periodic aerodynamic loading of the rotor blades due to vertical wind shear. It is designed to reduce 1P (0.333 Hz) blade flap-wise oscillations and is adaptive to change in the operating point due to horizontal wind speed fluctuations. Five IPC controllers, each designed to be operational over a particular wind speed bin in the above-rated 180 wind speed region, together with a switching mechanism based on the incoming wind speed are used to realize aIPC. The linear models, used for designing respective IPC controllers are extracted from the nonlinear wind turbine model Eq.
(1) at different operating points as shown in Table 2.  (Bir, 2010) is used to transform blade dynamics from the rotating to the nonrotating frame. The MBC transformed reduced order models are then averaged to obtain a weakly periodic LTI model described in state-space form aṡ 190 where A, B, B d , C denote the state-space system, u = [∆β 1 ∆β 2 ∆β 3 ] T denotes the perturbed independent pitch angles, and d the wind disturbance. The measurements y, which include the blade root flap-wise bending moment for each blade are assumed to be distorted with noise v.
Using linear quadratic gaussian (LQG) control method, Eq. (7) is used to design an observer-based controller. The full-state feedback controller K is designed using linear quadratic regulator (LQR) technique by minimizing the cost function while solving the algebraic Riccati equation (ARE) A T P +P A−P BR −1 B T P +Q = 0, assuming (A, B) is fully controllable.
Here Q and R denote the state and control input weighting matrices respectively, whose elements are tuned to achieve the desired dynamic response with respect to blade load mitigation and rotor speed regulation, while P is the solution to the ARE. To implement optimal full-state feedback control u = Kx using estimated statesx, a Kalman state estimator is used to 200 design the observer gain L by minimizing the state estimation covariance error E((x −x)(x −x) T ), while solving the ARE Here, Q f and R f are process disturbance and measurement noise covariance matrices, respectively, while P f is the solution to the ARE. The five IPC controllers are designed following this procedure, each at a predefined operating point, to cover the entire range 210 of operation in the above-rated regime. A switching mechanism is then implemented to activate each controller at a predefined operating range based on the prevailing wind speed.
4 Control of wind turbine lifetime: An illustrative example using the 1.5 MW NREL reference wind turbine To control the lifetime consumption in wind turbine blades, the adaptive robust observer-based controller (RDAC+aIPC), implemented using two control loops is combined with an online damage evaluation model as shown in Fig. 4. A wind profile 215 excites the wind turbine dynamics in the above-rated regime. The RDAC controller (Do and Söffker, 2021), which is robust against modeling errors generates the primary CPC signal for rotor speed regulation and tower load mitigation, while aIPC is used as the lifetime controller to dynamically control the damage accumulation of the rotor blades. The IPC angles are perturbed about the CPC signal from RDAC, forming the control input u to the wind turbine.
where T k denotes the current time step while D d denotes the accumulated damage at the design lifetime. At every time step 225 T k , the estimated RUL can be calculated as Based on the threshold evaluation of L e , the load mitigation level in the respective IPC controllers is controlled by selecting the appropriate gains L and K every 10 seconds, which is the time interval chosen for lifetime threshold evaluation. For illustrative purposes a lifetime of 600 seconds is chosen. Three threshold levels are set such that if L e is below the lower  Figure 6a shows that with lifetime control, the blade flap-wise bending moment reduces, with a 11.26 % reduction in standard 245 deviation being achieved. Additionally, there is significant reduction in the accumulated damage as shown in Fig. 6b.
Performance of the adaptive lifetime control strategy in mitigating tower loads is also evaluated. As illustrated in Fig. 7a, significant reduction in tower fore-aft oscillation is observed, with the standard deviation reducing by 16.08 %. A reduction in tower damage accumulation can be seen in Fig. 7b. This shows that lifetime control of blades, which reduces 1P fatigue loads, leads to reduced damage accumulation in tower due to 3P fatigue loads.

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Despite the adaptive lifetime controller achieving improved performance in reducing damage accumulation in both rotor blade and tower, this does not compromise the speed/power regulation performance. To illustrate this, the rotor speed and  In this paper, a prognostics-based adaptive control strategy for lifetime control of wind turbines is presented. A robust disturbance accommodating controller (RDAC) designed using mixed sensitivity H ∞ control, is used as the primary controller for mitigating tower loads and regulating rotor speed using a CPC-signal. On the other hand, aIPC controller designed using LQG control method is used as a lifetime controller. The gains of each of its five IPC controllers are adapted based on the state of 260 health of the rotor blades obtained using an online damage evaluation model to strike a compromise between lifetime control through load mitigation and speed regulation.  Through simulation using a 1.5 MW wind turbine model, it is demonstrated that the adaptive lifetime control strategy controls the damage accumulation in the blades to guaranty a given damage limit at the desired lifetime. Reduction in accumulated damage in the tower is also realized. This can potentially be used for optimizing maintenance scheduling in wind farms by 265 synchronizing ageing of wind turbine components, hence reducing O&M costs, and increasing operational reliability. This improvement is realized without compromise in the speed/power regulation performance. However, the result is achieved based on a slightly increased pitch actuator duty cycle, which can potentially increase fatigue loading in the pitch actuator system components. In the future, adaptive lifetime control based on nonlinear damage accumulation models will be considered. Additionally, use of new concepts for state of health indicators such as change in modal parameters for structural health monitoring 270 will be explored.
Code availability. Code is not publicy available and can not be shared. Competing interests. The authors declare that they have no conflict of interest.