Nonlinear model predictive control of wind turbines using LIDAR

Authors


ABSTRACT

LIDAR systems are able to provide preview information of wind disturbances at various distances in front of wind turbines. This technology paves the way for new control concepts in wind energy such as feedforward control and model predictive control. This paper compares a nonlinear model predictive controller with a baseline controller, showing the advantages of using the wind predictions in the optimization problem to reduce wind turbine extreme and fatigue loads on tower and blades as well as to limit the pitch rates. The wind information is obtained by a detailed simulation of a LIDAR system. The controller design is evaluated and tested in a simulation environment with coherent gusts and a set of turbulent wind fields using a detailed aeroelastic model of the wind turbine over the full operation region. Results show promising load reduction up to 50% for extreme gusts and 30% for lifetime fatigue loads without negative impact on overall energy production. This controller can be considered as an upper bound for other LIDAR assisted controllers that are more suited for real time applications. Copyright © 2012 John Wiley & Sons, Ltd.

1 INTRODUCTION

An important design goal for large wind turbines is to reduce fatigue and extreme loads on support structure and blades by control. This is a challenging task because transients such as gusts represent an unknown disturbance to the control system. Conventional feedback controllers can only provide delayed compensation for such excitations, since the disturbance effects must propagate through the entire wind turbine before showing its effects in the controller outputs. This usually results in additional loads for the wind turbine and requires high actuator rates as discussed later in this work. Those effects can be avoided, if the wind inflow is measured by remote sensing techniques such as LIDAR (LIght Detection and Ranging).

While early work on LIDAR assisted control was done in Harris et al., [1] this field of investigations increased significantly in recent years: Feedforward controllers for load reduction have been proposed in Schlipf and Kühn, [2] Dunne et al. [3] and Schlipf et al. [4] to assist collective pitch control and in Laks et al., [5] Schlipf et al., [6] Dunne et al. [7] and Wang and Johnson [8] to improve individual pitch controllers. Also, the possibility to increase the energy yield by LIDAR assisted yaw or speed control has been investigated in Kragh et al. [9] and Schlipf et al. [10] For model predictive control (MPC), the estimated wind or filtered hub height wind speed can be used to schedule the controller (e.g., see in Bottasso et al. [11] and Kumar and Stol [12]). In Körber and King, [13] it has been shown that a linear model predictive controller can be significantly improved using perfect wind preview during the optimization. This was also used in other publications, such as in Henriksen [14] or Körber and King. [15] But investigations such as Schlipf et al. [16] and Simley et al. [17] account for the fact that the LIDAR system does not provide a perfect preview of the wind. In other implementation of model predictive controllers, [18-20] more realistic wind previews were used.

The present work illustrates that the aforementioned improvement by the knowledge of future wind still exists if imperfect but more realistic nacelle-based LIDAR measurements are combined with nonlinear model predictive control (NMPC), which fits to the nonlinear nature of the variable speed and collective pitch control problem. Major effort has been invested to design an NMPC, which controls the collective pitch angle and the electrical torque simultaneously for the whole operation region of a wind turbine and to estimate the lifetime weighted load reduction.

This paper is organized as follows: Section 2 deals with the modeling of the wind turbine. Subsequently, in Section 3, the different controllers are presented, and Section 4 describes the simulated LIDAR measurements. Simulation results with the full aeroelastic model disturbed by coherent gusts and three-dimensional stochastic wind fields are shown in Section 5. Considerations for real time applications are also discussed in this section. Section 6 concludes the paper with a summary and gives an outlook to future work.

2 MODELING THE WIND TURBINE

The crucial part of a successful NMPC controller design is the modeling of the system to be controlled. In theory, the presented NMPC controller can deal with any nonlinear and time variant model. Nevertheless, the model should be simple enough to allow a reasonable computational time and should be realistic enough to capture the system dynamics that are relevant for the wind turbine control. In this work, the simulations are carried out with an aeroelastic model that is too complex to be included in the NMPC control approach. In addition, real LIDAR system measurements deliver only limited information about the wind and therefore do not justify a higher order model. Here, a reduced model serves as internal model of the NMPC controller, which has to capture relevant effects. An in depth description of both models and a comparison between them are included in this section.

2.1 The aeroelastic model

Simulations are performed with an aeroelastic model of a 5 MW three-bladed pitch-controlled variable-speed wind turbine designed by the National Renewable Energy Laboratory (NREL) as described in Jonkman et al. [21] The following 16 degrees of freedom are enabled in the FAST [22] simulation: first and second flapwise and first edgewise blade modes, first and second fore–aft and side-to-side tower bending modes, drivetrain rotational flexibility, generator and yaw. It is supplemented with a second-order linear model to account for the pitch actuator dynamics (1c), and the pitch rate limit is 8°/s. This sums up to a total of 35 dynamic states of the aeroelastic model (basic data, see Table 1.

Table 1. Specification of the used aeroelastic wind turbine model. [21]
Rated power outputPrated5 MW
Rated rotor speedΩrated12.1 rpm
Rated wind speedvrated11.2 m/s
Cut-in wind speedvin3 m/s
Cut-out wind speedvout25 m/s
Rotor radiusR63 m/s
Rotor diameterD126 m/s
Hub heighthH90 m/s
Hub inertia on low-speed shaftJH115,926 kg/m 2
Blade inertia on low-speed shaftJB11,776,047 kg/m 2
Generator inertia on high-speed shaftJG534.116 kg/m 2
Gear box ratioi1/97
Mass towermT347,460 kg
Mass nacellemN240,000 kg
Mass hubmH56,780 kg
Mass blademB17,740 kg
Natural frequency of first tower fore–aft bendingf00.32 Hz
Structural damping ratiods0.01
Undamped natural frequency of the blade pitch actuatorω2π rad/s
Damping factor of the blade pitch actuatorξ0.70
Optimal tip speed ratioλopt7.55
Peak power coefficientcp,max0.482
Distance from hub to first airfoilrmin9.7 m
Maximum pitch angleθmax90°
Minimum pitch angleθmin
Pitch angle at which the rotor power has doubledθK6.30°

Aeroelastic simulators such as FAST apply the blade element momentum method to compute the impact of the wind field to the turbine model. The calculation has to be carried out iteratively, and therefore, no explicit model equation can be derived to express the impact from each component of the three-dimensional wind field to the wind turbine states. This increases the computational effort in aeroelastic simulations and makes the use of an aeroelastic model unwieldy for NMPC. In addition, current remote sensing methods such as LIDAR are not able to provide a wind field estimate that is comparable with a generic wind field normally used in aeroelastic simulations. In the next section, a reduced turbine model is derived from physical fundamentals, and the wind field is reduced to a disturbance that is measurable with the existing LIDAR technology.

2.2 The reduced nonlinear model

For the nonlinear reduced model of the turbine used by the model-based controller, the disturbance is reduced to the rotor effective wind speed and only three degrees of freedom (see Figure 1).

Figure 1.

Degrees of freedom for the reduced nonlinear model.

The first tower fore–aft bending mode, the rotational motion and the collective pitch actuator are based on Bottasso et al. [23]:

display math(1a)
display math(1b)
display math(1c)

The first equation (1a) models the drivetrain dynamics, where Ω is the rotor speed, Ma is the aerodynamic torque and Mg the electrical generator torque, xT is the tower top fore–aft displacement, θ is the effective collective blade pitch angle and v0 is the rotor effective wind speed. Moreover, i is the gear box ratio, and J is the sum of the moments of inertia about the rotation axis of the rotor hub JH, blades JB and the electric generator JG:

display math(2)

The second equation (1b) describes the tower fore–aft dynamics. Fa is the aerodynamic thrust, and mTe, cT and kT are the tower equivalent modal mass, structural damping and bending stiffness, respectively. These values were calculated according to Gasch et al. [24] and are given in Table 1:

display math(3a)
display math(3b)
display math(3c)

Finally, (1c) is a second-order model of the blade pitch actuator, where θc is the collective blade pitch control input, ω is the undamped natural frequency and ξ is the damping factor.

The nonlinearity in the reduced model is contained in the aerodynamic thrust and torque acting on the rotor with radius R:

display math(4a)
display math(4b)

where ρ is the air density, λ is the tip speed ratio defined as

display math(5)

and cP and cT are the effective power and thrust coefficients, respectively. The cP and cT coefficients are included in the model as a polynomial fit to two-dimensional lookup tables (see Figure 2), which can be obtained from steady state simulation, e.g., with WT_Perf. [25]

Figure 2.

(a) Effective power coefficient. (b) Effective thrust coefficient.

The relative wind speed vrel is defined as a superposition of tower top speed and mean wind speed

display math(6)

and is used to model the aerodynamic damping. The electrical power Pel is calculated by

display math(7)

where η represents the efficiency of the electromechanical energy conversion. With the hub height hH, the tower base fore–aft bending moment MyT is

display math(8)

Equations (1) to (5) can be organized in the usual nonlinear state space form:

display math(9)

where the system states x, the system inputs u, disturbance d and measurable outputs y are

display math(10)

2.3 Model comparison

The quality of the internal model is crucial for NMPC. The internal model of the NMPC control algorithm has to contain all relevant effects of the system to be controlled. Collective pitch and speed control react on rotor speed variation. Models for collective pitch and speed controller need to reproduce the behavior below the rotation frequency of the rotor (the 1P-frequency) well, because wind disturbances within this frequency range are responsible for the main impact on the rotor speed. Capturing the behavior at higher frequencies is less important, because the rotor acts like a filter for wind disturbance above the 1P-frequency. For individual pitch control, a higher correlation is required to capture and reduce the 1P-loads. However, this paper concentrates on collective pitch and speed control, and therefore, following criteria for an adequate model accordance is chosen: the unbiased coherence (see Bendat and Piersol [26] for details) for rotor speed and tower fore–aft bending moment between the aeroelastic and the reduced nonlinear model should be above 0.5 at the 1P-frequency (for rated rotor speed ≈ 0.2 Hz). The value γ2 = 0.5 is used because it is half way between perfectly correlated (γ2 = 1) and fully uncorrelated (γ2 = 0).

The reduced model previously developed demands an effective wind speed that impacts the simplified rotor disk in a way that the full wind field would do. Here, the rotor effective wind speed for the reduced nonlinear model is obtained from the TurbSim [27] wind fields at each time step with a discrete version of the two-dimensional weighting function

display math(11)

where u(r,ϕ) is the longitudinal wind component at the polar coordinates r and ϕ from the turbulent wind fields. The stationary spanwise variation of power extraction ∂cp ∕ ∂r with

display math(12)

can be obtained by modeling tip and root losses following Leishman [28] and Burton et al. [29] Figure 3 shows the used weighting function with cp,max and rmin from Table 1. This curve is an approximation of the aerodynamics and will differ in aeroelastic simulations from the actual values but still covers the effect of tip and root losses compared with a simple average over the rotor disk.

Figure 3.

Used spanwise variation of power extraction in the presence of tip and root losses (solid) and Betz optimal curve without losses (dashed).

With v0, it is possible to compare models directly in closed loop simulations. To emphasize the need for a nonlinear model, a linearized model from FAST extended by the pitch actuator model (1c) is added to the comparison. The linearization point is the corresponding mean wind speed of the used wind field. A baseline controller [21] (basic PI controller for pitch control and a lookup table for torque control) is used for all three models. The aeroelastic model is disturbed by TurbSim wind fields, whereas the linearized FAST model and the reduced nonlinear model are disturbed by the corresponding rotor effective wind speed using (11). Figure 4 shows some results from simulations with mean wind speeds of 8 m/s, 12 m/s and 16 m/s in the time and frequency domain. The reduced nonlinear model and the linearized model show similar behavior compared with the aeroelastic model for 8 m/s and 16 m/s in the time domain. In the coherence plots, it can be seen that frequencies below the 1P-frequency are captured. Especially at wind speeds around rated, the reduced nonlinear model reflects the low frequency behavior of the aeroelastic model significantly better than the linearized model.

Figure 4.

Model comparison: (a), (c), (a) Top, rotor effective wind speed; center and bottom, response rotor speed and tower base fore–aft bending moment for the reduced nonlinear model (black), the linearized FAST model (light gray) and the aeroelastic model (dark gray). (b), (d), (f) Coherence of rotor speed (top) and tower base fore–aft bending moment (bottom) between the aeroelastic model and the reduced nonlinear model (dark gray) and between the aeroelastic model and the linearized FAST model (light gray), from 10 min simulations.

Overall, it can be concluded that the reduced nonlinear model fulfills the design criteria and thus can be used for the proposed NMPC.

3 CONTROLLER DESIGN

At the beginning of this section, the baseline controller is introduced, which controls the wind turbine only via feedback without any wind information. Then, an NMPC using the wind preview information is designed and classified by comparison with existing approaches. Furthermore, a nonlinear estimator for the NMPC is presented.

3.1 The baseline controller

The baseline controller is implemented as described in Jonkman et al. [21] and combines a variable-speed generator torque controller and a collective blade pitch controller.

Both controllers use the generator speed as the input. The speed signal is filtered using a single-pole low-pass filter with a corner frequency of 0.25 Hz, to mitigate high-frequency excitation of the control systems. The goal of the torque controller is to maximize energy yield when operating below rated speed and to regulate power during rated speed operation. The task of the collective blade pitch controller is to regulate the rotor speed when the turbine operates at rated power. The collective blade pitch angle command is computed using a gain-scheduled PI controller on the speed error between the filtered and the rated generator speed. The baseline torque and pitch controllers primarily operate independent of each other. The torque controller continues to regulate power as long as the blade pitch angle remains above a certain threshold. A pitch angle limitation and an anti-windup assure that the collective pitch controller only operates above rated wind conditions.

3.2 Nonlinear model predictive control

Model predictive control is an advanced control tool which predicts the future behavior of the system using an internal model and the current measurements. With this information, the control actions necessary to regulate the plant are computed by solving an optimal control problem over a given time horizon. Part of the solution trajectory for the control inputs are transferred to the system, new measurements are gathered, and the optimal control problem is solved again. Feedback is obtained, since the current state of the turbine is implemented as the initial condition of the optimal control problem. In this subsection, a short overview of MPC is given, and the optimal control problem is derived and then solved.

Model predictive control can be categorized as either linear or nonlinear model. Linear MPC is based on linear models and is successfully applied in several industrial applications since the 1980s, mainly in chemical engineering. [30] However, many real systems have nonlinearities that cannot be neglected. Here, NMPC often yields improved results by considering nonlinear models, objective functions and constraints.

There are several advantages of MPC in general. One is that it can handle multivariable and non-quadratic (different number of inputs and outputs) control tasks naturally: Additional control inputs or outputs will merely increase the number of optimization variables. Another advantage is that it considers actuator and system constraints when solving the optimal control problem. Furthermore, it provides a framework for incorporating a disturbance preview dynamically, and tuning of MPC controllers is carried out intuitively by changing weights of a definable objective function. However, the main advantage of MPC is that it is in mathematical sense an optimal controller. Solving the optimal control problem is not an easy task and several methods exist. A brief overview is provided, and the publications of MPC in wind energy applications that are known to the authors are related to the corresponding method.

Optimal control is the optimization of the operation of a dynamic system, in general described by ordinary differential equations (ODEs). Optimal control problems can be solved by dynamic programming using direct or indirect approaches [31]: dynamic programming uses the Principle of Optimality to transform the problem into partial differential equations, which can be solved only for small state dimensions. Indirect approaches transform the problem into a boundary value problem using Pontryagin's Principle, which can be solved numerically (used in Bottasso et al. [11]).

Direct approaches are typically used for nonlinear problems that transform the infinite-dimensional problem to a finite nonlinear dimensional problem by discretization of the control inputs. The most common way is to use piecewise linear or constant parametrization. Linear MPC then in addition discretizes the ODEs at one single operation point obtaining a pure static optimization problem (used in Körber and King, [13] Laks et al. [5] and Dang et al. [32]). For nonlinear systems, several linear models can be incorporated by switching (used in Henriksen and Poulsen, [33] Kumar and Stol [12] and Soliman et al. [34]) or continuous linearization having one linear model over the entire horizon (used in Körber and King [15]) or even in each stage (used in Henriksen [14] and Soltani et al. [20]). Methods such as Direct Single Shooting and Direct Multiple Shooting keep the ODEs: With Direct Single Shooting (used in Santos [18] and Körber and King [15]), the system simulation (over the entire prediction horizon) and the optimization are performed sequentially. In contrast with Multiple Shooting, optimization and simulation (of multiple stages) are performed simultaneously (used in Dang et al. [35] and in this work). Furthermore, several direct collocation methods exist that have not been used in this study.

Independent of the used method, the basic principle of MPC is illustrated in Figure 5 using piecewise constant parametrization: Future control action is planned to fulfill the control goal, e.g., reference signal tracking, considering a predicted disturbance.

Figure 5.

Principle of nonlinear model predictive control: Over a given time horizon, the control action to a nonlinear system is optimized considering predicted disturbances to fulfill the control goals such as reference signal tracking.

3.3 Definition of the optimal control problem

The considered optimal wind turbine control problem can be described by the following problem: The objective is to find the optimal control trajectory u( · ) which minimizes the cost function JOCP, which is defined as the integral over the time horizon Tf of the objective function F from the actual time t0 to the final time t0 + Tf, with the reduced nonlinear model and the set of constraints H:

display math(13)

The crux of designing the NMPC is to translate the verbal formulation of the control goal to a mathematical formulation of F and H. In wind energy, the overall goal of development can be stated very roughly as ‘minimizing energy production cost’. Such an optimization including the wind turbine design, manufacturing cost and operation can be found in Bottasso et al. [36] However, for the optimal control problem, the wind turbine is already designed, and the optimal control goal can be stated very roughly as ‘maximizing energy production without damage during the lifetime of the turbine’, neglecting secondary requirements, e.g., noise limits. By classic wind turbine control, [29] this is generally carried out by tracking optimal tip speed ratio below a certain wind speed defined as rated wind speed and by limiting rotor speed and power above the rated wind speed. This could be redefined for the NMPC, but special care has to be taken with NMPC characteristics. If, for example, the electrical power is directly maximized for wind speeds below rated wind speed by F = (Pel − Prated)2, the NMPC will slow down the turbine by increasing the generator torque, because this is optimal for the limited view of the NMPC, but evidently not for the overall energy production. [14] In this work, the optimal control problem is based on the classic interpretation of wind turbine control. The used objective function and constraints will be stated and explained here.

The objective function should be quadratic for numerical reasons (see Section 3.4). This requires the weights to be independent of the states x and inputs u but are allowed to be dependent on the external disturbance d. Here, F(x(τ),u(τ),d(τ)) is chosen to

display math(14)

The first line of (14) penalizes deviation from a reference rotor speed Ωref(τ) depicted in Figure 7 and defined as

display math(15)

The rotor speed Ωin (see Table 2) is set to be above 6.4 rpm at which the 3P-frequency f3P coincides with the natural frequency f0 of the first tower fore–aft bending mode. The weight for deviation to the reference rotor speed Q1(v0(τ)) changes the value at rated wind speed:

display math(16)
Table 2. Parameters for the presented nonlinear model predictive controller.
Spinning rotor speedΩspin4 rpm
Cut-in rotor speedΩin8 rpm
Minimum tip speed ratioλmin7.05
Maximum tip speed ratioλmax8.70

In the second line of (14), the tower fore–aft velocity is penalized. Here, the basic idea is to minimize loads on the tower due to variation of MyT; see (8). Using directly math formula would make the NMPC try to minimize xT included in math formula, and finally, the steady state of the system would depend on the weights Q1(v0(τ)) and Q2, which is not reasonable. This shows that special care has to be taken to define the objective function without conflicting the requirements. The weight for the deviation from rated power Q3(v0(τ)) is only active for wind speeds above rated wind speed. This avoids the above-mentioned problem that the NMPC increases the generator torque for short-term energy capture:

display math(17)

The weight for the pitch rate R1(v0(τ)) is designed to penalize the pitch actuator rate. To account for the higher sensitivity of the pitch at higher wind speed (see Figure 6), the static pitch angle over static wind speed θss(vss) is used together with the gain correction factor GK(θ) from Hansen et al., [37] θK from Table 1 and the static weight R1:

display math(18)
Figure 6.

(a) Static pitch angle over static wind speed; (b) normalized weight for the pitch rate.

In contrast to (17), the weight for the pitch angle R3(v0(τ)) is only active for wind speeds below rated wind speed. This means that the NMPC is still able to pitch below rated but is penalized:

display math(19)

The set of constraints H(x(τ),u(τ),d(τ)), which can be organized in the form H(x(τ),u(τ),d(τ)) ≥ 0, is chosen as

display math(20a)
display math(20b)
display math(20c)
display math(20d)
display math(20e)

The constraint (20a) limits the rotor speed to 120% of Ωrated, (20b) limits the pitch angle to its feasible positions, and (20c) and (20d) constrain the pitch angle and the generator torque rate, respectively. The idea behind (20e) and defining Q1a ≪ Q1b is maximizing the power output without having the drawback of optimal- λ tracking: If the tip speed ratio λ is tracked perfectly, the loads on the shaft will increase significantly. [10] Here, λmin(v0(τ)) and λmax(v0(τ)) are chosen such that the power coefficient cp is above cp,min without changing (15); see Figure 7.

Figure 7.

(a) Limits for maximum power output; (b) reference rotor speed and limits.

3.4 Solving the optimal control problem

The optimal control problem is converted by the Direct Multiple Shooting method [30] into a nonlinear program. Here, the control inputs are discretized in K piecewise constant stages (Figure 8). The ODEs of the model are solved numerically on each interval, starting in stage i with the initial values si for all states. The optimization is performed over the set of initial values and the control outputs. Additional constraints are applied to ensure that the states at the end of each stage coincide with the initial conditions of the subsequent stage. This method gives significant improvements over the Direct Single Shooting approach, especially with respect to numerical stability.

Figure 8.

Principle of the direct multiple shooting method: x0 is the initial point from the current measurement, si are the starting points for the nonlinear simulations, which have to coincide with the final point of each simulation by changing the projected control inputs ui over the K control steps up to the final time Tf.

The resulting nonlinear program can be described as follows:

display math(21)

This nonlinear program can be solved iteratively with Sequential Quadratic Programming (SQP). The separation of the optimization problem into multiple stages results in a faster solution. This is caused by the better approximation of the Lagrangian Hessians of the nonlinear problem parts in each stage by low rank updates. [38]

Here, Omuses [38] is used, a front-end to the large-scale SQP-type nonlinear optimization solver HQP. The prediction horizon is chosen to be Tf = 10 s as a compromise between the different preview times of the LIDAR (see Sections 4 and 5.3). The time steps are set equal to the LIDAR update rate ΔtL = 0.2 s, resulting in K = Tf ∕ ΔtL = 50 stages. The differential equations are solved with a fourth-order explicit Runge–Kutta method.

The used control structure is depicted in Figure 9. To avoid resonance cases, notch filters (Butterworth, second order) with stop band at math formula and [0.9f0,1.1f0] for y is used, where f0 is the natural frequency of the tower. The number of control steps applied in a feedforward control to the system after each optimization is chosen to be KF F = 1. This implies that the optimization is repeated with new measurements each 0.2 s to close the control loop. The proof of closed loop stability of a nonlinear and constrained system solved by a model predictive controller is beyond the scope of this work and is quite complicated as JOCP has to be a local Lyapunov function. There are some theoretical approaches [39] and practical recommendations, [40] but the following results will show that there is no stability problem in this case. The NMPC controller needs the full state vector x0 at the start of the optimization horizon. Only the rotor speed Ω, the tower fore–aft acceleration math formula, the pitch angle θ and the pitch rate math formula can be considered as measurable signals. Therefore, an estimator has to be implemented to reconstruct math formula and xT.

Figure 9.

Closed loop nonlinear model predictive control.

3.5 Nonlinear estimator

A nonlinear estimator is used to estimate the tower dynamic states math formula and math formula and consists of a static nonlinear estimation of the aerodynamic thrust and a linear Luenberger estimator. The wind measurement v0 from the simulated LIDAR and the filtered measured outputs θ and Ω are used to estimate the aerodynamic thrust math formula with the nonlinear equation (4b) neglecting the influence of the tower movement by setting math formula m/s. Then, math formula can be used to estimate math formula with the filtered tower acceleration math formula. In contrast to the standard case, [41] the ODE of the estimator is (due to the direct feedthrough)

display math(22)

where the matrixes A, B, C and D are obtained from (1b), using cTe instead of cT to model the missing aerodynamic damping:

display math(23)

and L was designed such that (A − LC) has eigenvalues to the left of the eigenvalues of A in order that the estimator can estimate faster than the system changes.

Compared with an Extended Kalman Filter design commonly used in wind energy research, the proposed estimator is quite simple but showed satisfactory results.

4 SIMULATED LIDAR MEASUREMENTS

The NMPC needs a preview of the rotor effective wind speed over the prediction time Tf = 10 s. In simulations, this could be obtained easily from the TurbSim wind fields using (11).

In reality, nacelle mounted LIDAR systems are capable to scan the incoming wind field. It is possible to extract a rotor effective wind speed from the raw data delivered by such a LIDAR system. The procedure is represented in a LIDAR simulator [42] and described as follows.

Three-dimensional stochastic wind fields used for the aeroelastic simulations are scanned online during the simulations, and depending on the LIDAR system, a wind estimate is obtained. Different LIDAR systems (pulsed or continuous wave) and scanning modes can be simulated with the use of a LIDAR simulator. In this work, a pulsed system with a circle trajectory is used, which is performed within Tt = 2.4 s with 12 focus points in five focus distances, resulting in an update rate of ΔtL = 0.2 s. Dimensions can be found in Table 3 and Figure 10(b).

Table 3. Dimensions of circle trajectory with the rotor diameter D = 126 m.
Index focus distance12345
Distance perpendicular to the rotor plane fi0.50D0.75D1.00D1.25D1.50D
Diameter of circle Di0.33D0.50D0.67D0.83D1.00D
Figure 10.

SWE nacelle-based pulsed LIDAR system (a) in experiment and (b) in simulations.

This trajectory was realized by a real scanning LIDAR system installed on the nacelle of a 5 MW turbine (see Figure 10(a) and Rettenmeier et al. [43]). In the simulation, effects such as collision of the laser beam with the blades, volume measurement and mechanical constraints of the scanner from data of the experiment are considered to obtain realistic measurements. For instance, the same loss of approximately one-third of the points could be observed in both the simulation and the measurements because of the collision with rotating blades. Measurement noise is not considered.

Taylor's frozen turbulence hypothesis assumes that the turbulent wind field is unaffected when approaching the rotor and moving with average wind speed. This hypothesis is used in the simulation of the measurements and for the wind estimation. To evaluate, for example, a measurement in 1D distance during time t of a wind field with mean wind math formula, the wind field is analyzed at math formula.

To account for the volume measurement of the LIDAR system, the wind field is not only analyzed at the focus points but also in the area around the focus point along the laser beam. Here, a Gaussian shape weighting function fL(a) depending on the distance a to the focus point with full width at half maximum of W = 30 m is used (Figure 11), following the considerations of Cariou and Pulsed, [44] Lindelöw [45] and Banakh and Smalikho [46]:

display math(24)

With the weighting function, it is possible to calculate the line-of-sight wind speed of each focus point with fL(a) by

display math(25)

where [lx ly lz]T is the normalized laser beam vector and [u(a) v(a)w(a)]T is the wind vector at the distance a to the focus point.

Figure 11.

Normalized range weighting function fL(a) for the considered LIDAR system.

As LIDAR systems measure only the wind speed in the line-of-sight direction, the three-dimensional wind vector is reconstructed using the assumption of perfect yaw alignment with the wind direction. If the turbine is perfectly aligned with the wind, the estimated lateral and vertical wind components are assumed to be zero, and the longitudinal component math formula can be calculated:

display math(26)

For each distance i, the longitudinal wind component math formula is then averaged over the last trajectory, and the obtained time series of the measurements vi is time-shifted according to Taylor's frozen turbulence hypothesis. The preview time of each measurement plane is the distance fi divided by the mean wind speed math formula, reduced by the time shift because of the running average of half of the trajectory duration Tt; see Figure 12. The farthest right point of each line represents the newest measurement of the corresponding focus distance. The rotor effective wind speed v0(t) is then calculated by

display math(27)

where the weights wi were chosen similar to (11), integrating the math formula over the area associated to each circle, e.g., the area for the first circle begins at r1 = 0 m and ends at the span positions between the first circle at D1 ∕ 2 and the second one at D2 ∕ 2:

display math(28)

The dimensions of the proposed circle trajectory are chosen to have the weights almost equal. This produces an improvement of the short-term estimation, because the measurements of further distances can be stored and used to obtain more information when reaching the nearest distance. If there is no measurement available for the first focus distance, the weighted average is made only over the last four distances and so on. If the requested preview for the NMPC is larger than the possible preview, the last available value is held constant.

Figure 12.

Scope of the wind prediction: The line-of-sight wind speeds are measured at different fixed distances, corrected and averaged over the last trajectory. The resulting preview of the rotor effective wind speed v0(t) is a weighted average over all vi available during time t.

Figure 13 shows the coherence between the rotor effective wind speed extracted from a turbulent wind field using (11) and from simulated LIDAR measurements using (28). A good correlation (γ2 > 0.5; see Section 2.3) can be observed for wave numbers below math formula rad/m. Similar results have been revealed in an investigation, where the LIDAR system was mounted on a 5 MW turbine and performing the aforementioned circle scan. [16, 47, 48] Therefore, the wind preview v0 is filtered by a low-pass filter (second order) with cutoff frequency fcutoff at

display math(29)

The time delay introduced by the filter has to be considered. In reality, filtering will be important to avoid incorrect control action caused by inaccurate predictions. Finally, the filtered v0(t) gives a realistic estimation of the preview provided by a real LIDAR system, because the LIDAR measurements are simulated in detail and experiences from experiments are considered. Furthermore, this section gives an outlook of necessary algorithms needed in real applications.

Figure 13.

Solid: Coherence between the rotor effective wind speed extracted from a turbulent wind field and from simulated LIDAR measurements. Dashed: maximum wavenumber math formula used to filter LIDAR measurements.

5 SIMULATION RESULTS

The most important benefits for the presented control may lie in the reduction of extreme loads in gust events. High load reduction can be also observed for fatigue loads. Both cases will be investigated in this section. For the extreme loads, simulations are carried out first with the reduced nonlinear model assuming perfect measurement. Then, the more realistic reduction with the full aeroelastic model and simulated LIDAR measurements are shown for the extreme and subsequently for the fatigue loads. At the end of this section some considerations regarding the applicability of the NMPC in real time are presented. The NMPC is compared with the baseline controller [21] which has no information of the approaching wind.

5.1 Extreme loads

In the time domain, the different control strategies are compared with their reaction to gusts. Therefore, hub height time series are created with extreme operation gusts according to current standards [49] at vrated + 2m/s = 13.2m/s and vout = 25m/s. At first, the simulations are run with the reduced nonlinear model such that the internal model and the simulation model are identical. Furthermore, the wind speed is directly fed into the NMPC assuming perfect measurements, and the tower states are assumed to be measurable. This is carried out to make results more apparent and to show the effect of different optimization goals: The full NMPC tries to reduce rotor speed variation and tower movement. The reduced NMPC (NMPC red) is designed to reduce only the rotor speed by setting Q2 = 0. Figure 14 compares the pitch angle, generator torque, rotor speed, and tower base fore–aft bending moment for the different controller implementations and for the different wind speeds. For 13.2m/s, the NMPC and the NMPC red are able to minimize the rotor speed deviation. The NMPC red only uses the pitch angle to achieve this goal. The demanded pitch angle is similar to the pitch angle, which can be obtained by analytical nonlinear system inversion, [6] neglecting the influence of the tower movement:

display math(30)
Figure 14.

Simulation with reduced nonlinear model. Wind speed, pitch angle demand, generator torque, rotor speed and tower base fore–aft bending moment for baseline controller (dark gray), the reduced nonlinear model predictive control (NMPC red) for reduction of the rotor speed (light gray) and full nonlinear model predictive control for load reduction (black); ideal feedforward (dotted) from (30).

Differences in the pitch action between the NMPC red and the system inversion in Figure 14 are due to the weight on the pitch rate. This demonstrates that the NMPC red finds the optimal trajectory numerically. The full NMPC additionally uses the generator torque to achieve the minimization of the tower movement and the variation of the rotor speed because of its competence to incorporate multivariable control. For 25m/s, the reactions of the NMPC and the NMPC red are quite similar, although the NMPC induces greater variations in generator torque actuation, which suggests the conclusion that the equilibrium manifolds for the aerodynamic torque (4a) and thrust (4b) are more similar for high wind speeds, and therefore, it is easier for the NMPC to compensate the effect of the wind disturbance to the aerodynamic torque and thrust with the pitch only. The results in Table 4 show that a reduction of up to 57% of the maximum value is possible. In a next step, the same gusts are used to investigate if this load reduction holds for the full aeroelastic model together with the realistic LIDAR simulations and the nonlinear estimator. Here, the NMPC is used with the same parameters as in the detailed fatigue analysis (see Section 5.2). Figure 15 and Table 5 depict that even under more realistic conditions, the load reduction is still significant. Along with the wind speed, the LIDAR estimation is plotted in the top part of Figure 15, which shows the spatial and temporal filtering effects depending on the wind speed of the LIDAR simulation: For low wind speeds, the cutoff frequency (29) increases, and the gust is smoothed by the measurement volume (24) and the applied filter.

Table 4. Maximum values of Figure 14.
 EOG 13.2 m/sEOG 25 m/s
MyT [MNm]ΔΩ [rpm]MyT [MNm]ΔΩ [rpm]
  1. EOG = extreme operation gusts; NMPC = nonlinear model predictive control.

Baseline1321.741002.45
NMPC red810.05350.06
NMPC570.15320.08
NMPC red/baseline [%]623352
NMPC/baseline [%]439323
Figure 15.

Simulation with aeroelastic model. Top: wind speed (black) and LIDAR estimated rotor effective wind speed (gray). Below: pitch angle demand, generator torque, rotor speed and tower base fore–aft bending moment for baseline controller (gray) and nonlinear model predictive control (black).

Table 5. Maximum values of Figure 15.
 EOG 13.2 m/sEOG 25 m/s
MyT [MNm]ΔΩ [rpm]MyT [MNm]ΔΩ [rpm]
  1. EOG = extreme operation gusts; NMPC = nonlinear model predictive control.

Baseline1292.34993.01
NMPC630.57360.29
NMPC/baseline [%]49243610

5.2 Fatigue loads

To estimate the benefit for fatigue load reduction, various simulations with a set of turbulent TurbSim wind fields (see Table 6) are conducted, featuring A-type turbulence intensity according to IEC 61400-1 [49] and a Rayleigh distribution with C = 12m/s.

Table 6. Specifications of the used TurbSim wind fields.
Vertical grid points23
Horizontal grid points23
Time step [s]0.25
Grid height [m]132
Grid width [m]132
Mean wind speed at hub height [m/s]4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24
IEC turbulence characteristicA
Random speeds per mean wind3

The NMPC coupled to the LIDAR simulator and the nonlinear estimator is tuned to have high load reduction on tower and blades together with low pitch activity and slightly improved energy production. A larger power oscillation is tolerated in the partial load region.

In Figure 16, the behavior of both controllers below rated wind conditions is shown. Here, the NMPC has higher generator torque activity but is better in tracking the optimal tip speed ratio λopt (until t ≈ 240 s). Similar to the results in Körber and King [15] and Schlipf et al., [10] the improved λopt tracking does not justify the increase in the shaft loads.

Figure 16.

Simulation with mean wind 6 m/s and turbulence intensity 27%: rotor effective wind speed, generator torque, tip speed ratio, rotor speed and tower base fore–aft bending moment for baseline controller (gray) and nonlinear model predictive control (black).

The NMPC is also able to maintain the rotor speed above Ωin (after t ≈ 240 s), showing the capability of the NMPC in handling system constraints. Therefore, the turbine's rotation keeps more distance from the tower natural frequency, which can be noted in less oscillation in the tower base fore–aft bending moment. Similar improvement could be obtained by changing the 1.5 region of the baseline controller.

Figure 17 depicts the improvement for above rated wind conditions and transitions between both regions. The NMPC can reduce rotor speed variations with less pitch and generator activity. In the transition to the partial load region at t ≈ 310 s and back to full load region at t ≈ 340 s, the tower base fore–aft blade bending moment is decreased significantly in comparison with the baseline controller, showing the benefit of including nonlinearity. Because of this improvement, the energy capture can be slightly improved (see Figure 19). The effect of using LIDAR assisted control is more obvious in the frequency domain (see Figure 18, power spectral density for the simulation shown in Figure 17). The NMPC can significantly reduce the influence of the wind disturbance to rotor speed and to the tower base fore–aft bending moment for low frequencies. The pitch rate is also reduced in this region.

Figure 17.

Simulation with mean wind 16 m/s and turbulence intensity 18%: rotor effective wind speed, pitch angle demand, generator torque, rotor speed and tower base fore–aft bending moment for baseline controller (gray) and nonlinear model predictive control (black).

Figure 18.

Power spectral densities for the 10 min simulation of Figure 17, baseline (gray) and nonlinear model predictive control (black).

Over all simulations, the NMPC stabilizes the system and leads to satisfying control performance: Damage equivalent loads (DEL) are calculated on the basis of a rainflow counting with Wöhler exponent of 4 and 10, typical for steel and composite material. [50] The distribution of the lifetime weighted DEL [20 years with Rayleigh distribution (C = 12m/s), reference number of cycles 2 × 106] of the tower base fore–aft bending moment and the low-speed shaft torque are shown in Figure 19(a) and  19(b), respectively. The loads for the tower are reduced not only for high wind speeds but also for simulations with mean wind speeds of 4 and 6 m/s by limiting the rotor speed. Therefore, less energy capture (see Figure 19(d)) is tolerated for these wind speeds. The loads on the low-speed shaft are only reduced for high wind speeds. For lower wind speeds, the shaft loads are increased because of the improved λopt tracking and limiting the rotor speed at lower wind speeds. The pitch activity (see Figure 19(c)) decreases for all wind bins except for 8 m/s, where the slight increase is beneficial to achieve the optimization criteria.

Figure 19.

Distribution of lifetime weighted damage equivalent loads of (a) the tower base fore–aft bending moment and (b) the low-speed shaft torque, (c) weighted standard deviation for pitch rate and (d) energy production for the baseline controller (gray) and nonlinear model predictive control (black) over the considered bins.

Figure 20 and Table 7 summarize the results for all 33 simulations. For the LIDAR assisted NMPC, the possible reduction of tower and blade DEL can be estimated to approximately 30% and 10%, respectively. The standard deviation of the pitch rate and the rotor speed are decreased to approximately 30% and 10%. Furthermore, the energy production can be increased slightly by 0.30%, but also the standard deviation of the power is increased by approximately 7%.

Figure 20.

Overall improvement for the nonlinear model predictive control with respect to the baseline: Lifetime weighted damage equivalent loads for tower base fore–aft bending moment MyT, out-of-plane blade root bending moment of blade 1 Moop1 and low-speed shaft torque MLSS, lifetime energy production, lifetime weighted standard deviation of pitch rate, rotor speed and electrical power.

Table 7. Overall performance for the nonlinear model predictive control with K = 50 stages, repeating optimization after KFF = 1 stage.
 DEL(MyT) [MNm]DEL(Moop1) [MNm]DEL(MLSS) [MNm]EP [GW h]math formula [ ° ∕ s]σ(Ω) [rpm]σ(Pel) [MW]
  1. DEL = damage equivalent loads; EP = energy production; NMPC = nonlinear model predictive control.

Baseline87.6612.872.89548.420.460.590.52
NMPC62.3511.372.87550.050.320.530.55
NMPC/baseline [%]71.1388.3599.01100.3070.1989.72106.44

5.3 Considerations for real time application

The NMPC previously presented was not capable of running in real time on the author's PC (single core 2.19 GHz). The 10 min simulations need approximately 1 h to run. It should be possible to reduce the computing time, e.g., by the usage of higher clock rate or improvement of the software communication, but the main issues for real time applications will remain.

The first problem of the Direct Multiple Shooting method is that adding the initial states as the optimization variables may produce an intermediary result that is often far away from the optimum if the end states of one stage do not fit with the initial states of the next stage because of infeasible sets of ui and si. The second problem is that the presented approach leads to the iterative solution of a non-convex optimization problem, and thus, there is no guarantee to find the global minimum in the allotted time slot. [31]

It is beyond the scope of this work to present an implementation that is applicable in real time, but some considerations and analysis will be presented in the remainder of this section. Two steps are introduced to reduce the time needed for the optimization below the allotted time: On the one side, the computational effort is reduced, and on the other side, the allotted time is extended.

In the first step, the computational effort to solve the nonlinear program (21) can be decreased by reducing the stages K. Here, this is achieved by a shorter prediction horizon Tf, and the length of each stage is maintained equal to the update rate of the LIDAR system ΔtL. To find an appropriate value for the new prediction horizon, the available preview of the LIDAR system is considered: The preview time of the filtered wind can be calculated by the time TTaylor provided by Taylor's hypothesis at the fifth focus distance reduced by the time shift of the running average and the filter delay (see Section 4):

display math(31)

The delay introduced by the second-order Butterworth filter depends on the frequency fd. Here, fd = 0.1 Hz is used. This is the center of the collective pitch operation domain from 0 Hz to the 1P frequency. The prediction time depending on the mean wind speed math formula is depicted in Figure 21. For the simulations in Sections 5.1 and 5.2, the time horizon Tf = 10 s is chosen as a compromise. For simulations with math formula, the last available value of the available preview is held constant. To avoid this and to lower the computational effort, the new prediction horizon is set to Tf = 5.6 s, the minimum preview available for the considered wind fields, resulting in K = 28 stages.

Figure 21.

Black: Available preview time for the nonlinear model predictive control. Light gray: Used preview time in Sections 5.1 and 5.2. Dark gray: Used preview time in Section 5.3.

In the second step, the allotted time is extended by increasing the number of control steps KF F applied to the system after each optimization. In the previous simulations, KF F = 1 is used. Thus, the optimization has to be finished for real time applications in 0.2 s. Figure 22(a) shows a histogram of the times needed to execute all optimizations of the 33 simulations of Section 5.2. All 99,000 optimization times are above the available time. The 33 simulations are repeated with the same parameters except the new prediction horizon Tf = 5.6 s and KF F = 4. In these simulations, only 24 of the 24,750 optimizations last longer than the allotted time (see Figure 22(a)). Those outliers may be due to the non-real time capable operating system of the author's PC. The values are distributed for higher wind speeds around 0.7 s and for lower wind speeds around 0.4 s, because of the weight R3 on the pitch angle for below rated wind conditions. Table 8 summarizes the results for the modified NMPC RT. Compared with Table 7, the loads on the shaft increases as well as the standard deviation of the rotor speed and the power. The standard deviation of the pitch rate can be further reduced. The new implementation has a small effect on the load reduction on the tower and blades.

Figure 22.

Histogram of the time used by the nonlinear model predictive control to find the optimal solution. (a) With K = 50 stages, repeating optimization after KFF = 1 stage. (b) With K = 28 and KFF = 4. Dashed: allotted time slot.

Table 8. Overall performance for the NMPC RT with K = 28 stages, repeating optimization after KFF = 4 stages.
 DEL(MyT) [MNm]DEL(Moop1) [MNm]DEL(MLSS) [MNm]EP [GWh]math formula [ ° ∕ s]σ(Ω) [rpm]σ(Pel) [MW]
  1. DEL = damage equivalent loads; EP = energy production; NMPC = nonlinear model predictive control.

Baseline87.6612.872.89548.420.460.590.52
NMPC RT62.8211.372.90549.440.290.580.57
NMPC RT/baseline [%]71.6688.34100.34100.1964.1498.56110.11

This analysis shows that the aforementioned restrictions do not have a strong impact on the given implementation. Therefore, the application of the presented approach on real systems is worth considering because a supervisory control could be designed which can switch to the baseline controller for the rare cases in which the solution is not found in the allotted time slot, improving the robustness of the control strategy. A supervisory control could also avoid applying suboptimal solutions of local minima, e.g., if the value of the cost function is far away from the range of previous minima.

6 CONCLUSION AND OUTLOOK

In the presented work, a nonlinear model predictive controller based on LIDAR measurements of the wind field has been presented. The controller was evaluated with simulated measurements of a real LIDAR system and compared with a baseline controller. Promising load reduction on tower and blades and reduction of the pitch activity were achieved by including the information about future disturbance in the optimal control problem of the nonlinear model predictive controller. The benefits of the nonlinear design are especially evident for wind conditions near and above rated wind speed.

Although real time issues are addressed in this work, the NMPC should be considered more as an upper bound for other controllers.

In future work, the presented controller will be compared with a LIDAR-assisted feedforward controller and with the controller of the European UpWind project. [50] Also, a comparison with a linear and multilinear model predictive controller will help to estimate the benefit of this approach and to be closer to real time applications. Furthermore, this approach seems to be promising to control floating wind turbines using wind preview provided by LIDAR and a wave preview provided by buoys or SODAR.

ACKNOWLEDGEMENTS

Part of this research is funded by the German Federal Ministry for the Environment, Nature Conservation and Nuclear Safety (BMU) in the framework of the German joint research project ‘LIDAR II—Development of Nacelle-based LIDAR Technology for Performance Measurement and Control of Wind Turbines’ (FKZ 0325216B). Thanks to Timo Maul for his help with the LIDAR simulator and the very useful discussions.

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