Flatness-based Feedforward Controller for Perfect Wind Preview

For the considered 5 MW turbine and baseline controller, the highest static aerodynamic thrust and thus the largest tower deflection is at the rated wind speedvrated = 11.3 m/s, where the CPC starts to pitch the blades, see Figure 8.1. This results in high dynamic loading in the transition region, where the wind changes from below to above rated wind speed. The feedforward (FF) controller described in Chapter 6 uses the derivative of the static pitch curve and the wind speed measurementv_0L when the turbulence hits the turbine. It compensates the effect of wind speed changes to the rotor speed in (3.3a), neglecting the effects of the ISC. The benefit for the industrial application is that only a simple update to common baseline controllers is necessary.

At high wind speeds, holding the aerodynamic torque M_a constant results in smaller changes in the aerodynamic thrust Fa and thus leads to tower load reduction (see Figure 6.5). But the FF control is only able to react if the PI controller is not saturated. Furthermore, close to vrated, holdingMa constant results in high fluctuations of Fa due to the nonlinearities in cP

and c_T. Limiting the feedforward pitch rate close to v_rated is a heuristic solution. Designing a feedforward controller to hold F_a constant would cause contradicting control goals in the feedback and feedforward controllers.

A more direct solution is the NMPC, where a time series of the wind speed preview and a nonlinear model including the tower motion can be used to calculate repetitively an optimal trajectory for both the collective pitch command θc and the generator torque MG while also taking the actuator and state constraints into account [29]. But solving the optimal control problem is computationally intensive and there is no guarantee to find the global minimum in the allotted time slot. Furthermore, the tower states x_T and ˙x_T have to be estimated, because only the acceleration signal ¨x_T is available on standard wind turbines. The work in [30] concentrates on the transition region and proposes the combination of offline calculation based on dynamic programming and an online Trajectory Tracking Controller (TTC). Using a model of the rotor motion, higher load reduction compared to the FF without the need of state estimation and the computational effort of the NMPC can be achieved. Both NMPC and TTC may be the best solution in the future, but they have the drawback that they replace the baseline feedback controller. Thus, they are less attractive to turbine manufacturers. Because of safety concerns, the industrial state-of-the-art extends the baseline controller by various additional control loops such as tower feedback damping and individual pitch control [101].

8.1 Controller Design 163

v₀ [m/s]

xT,ss[m]θss[deg]Ωss[rpm]

v_rated

0 2 4 6 8 10 12 14 16 18 20 22 24

−0.1 0 0.1 0.2 0.3 0.4 0 5 10 15 20 25 02 468 1012 14

Figure 8.1: Steady-state values of the 5 MW reference wind turbine.

The proposed TEQUILA combines the advantages of the feedforward controller (update on existing feedback), the NMPC (considering tower deflection and wind preview over a time hori-zon, multivariable, considering constraints) and TTC (tracking of trajectories without extensive computational effort). It consists of a flatness-based feedforward controller and a trajectory planning algorithm and can be combined with a conventional feedback controller.

The Flat Wind Turbine

Flatness is a system property introduced by [102] that extends the concept of controllability from linear to nonlinear systems. A system is flat if a – not necessarily physical – so-called flat output exists such that all system states and inputs can be explicitly expressed in terms of the flat output and a finite number of its derivatives. This property can be used to plan an input trajectory on a nonlinear system based on the flat output.

The wind turbine model (3.3)-(3.11) with the states Ω, xT, and θ and the control inputs MG

and θc is a flat system with the flat outputs xT and Ω. Thus, all states and the system inputs can be expressed by the desired trajectories xT,d and Ωd, and their derivatives, which will be subsequently explained in detail. The disturbance v₀ can be considered as a time-varying

−100 1020

30 0 5 10 15

−2

−10 1 2

θ [deg] λ[-]

cT[-]

−1.5

−1

−0.5 0 0.5 1 1.5 c_T[-]

5 0 15 10

−1 0

1 2

−100 10 20 30

λ[-]

c_T [-]

θ[deg]

−10 0 10 20 30 θ[deg]

Figure 8.2: Considered area of the thrust coefficient (left) and its inverse regarding the pitch angle (right). Black lines are the steady values. Discretization of figures is reduced for better illustration.

parameter.

Using (3.10) and (3.11), the desired tip speed ratio λ_d and the desired relative wind speed are λ_d= ΩdR

v_rel,d, (8.1a)

v_rel,d= (v0−x˙_T,d). (8.1b)

Based on (3.3b) and (3.9b) the desired thrust coefficient is c_T,d= 2Fa,d

ρπR²v_rel,d² , (8.2a)

with F_a,d =m_eTx¨_T,d+c_eTx˙_T,d+k_eT(xT,d−x_0T). (8.2b) The inverse θ(λ, cT) of the three-dimensional look-up table c_T(λ, θ) has been calculated, see Figure 8.2. Since a bijective relationship is necessary for the inversion (only one value of θ for each set of λand c_T), only a sub-set of the look-up table forc_T is used. The bounds have been chosen such that the variation from the steady states under normal operation for wind speeds from 4 m/s to 30 m/s is covered.

With the inverse θ(λ, cT), one obtains the desired pitch angle

θ_d =θ(λd, c_T,d). (8.3)

Due to the time delay T_B in (3.1), the future desired pitch angle is needed to calculate the desired demanded pitch angle θ_c,d, which is the first system input:

θ_c,d(t) = θ_d(t+T_B). (8.4)

8.1 Controller Design 165

PP ¹_s ¹_s

Ωss

exp(−T₂s) v₀

SC M_a,d,M˙_a,d

¨Ω_d ˙Ω_d Ω_d

¨Ω_min/max ˙Ω_min/max

Figure 8.3: Online trajectory planning for the rotor motion.

Finally, the desired generator torque MG,d – the second system input – can be obtained using (3.3a) and (3.9a):

M_a,d = 1

2ρπR³c_P(λd, θ_d)

λd v²_rel,d, (8.5a)

M_G,d=i_GB M_a,d−J˙Ωd

. (8.5b)

This confirms that the wind turbine model is flat with respect to the outputs Ω and x_T. There are unique trajectories for M_G,d and θ_d based on the desired trajectory for the tower and rotor states and the given wind speed v₀. Continuously differentiable trajectories for x_T,d,x˙_T,d,x¨_T,d,Ωd, ˙Ωd are designed in the next section to provide smooth control.

Trajectory Planning

Flatness-based control is often used for trajectory planning to transfer a system from one equilibrium point to another. For wind turbine control, an online trajectory planning of the flat output is necessary to account for the changes in wind conditions. In this work, the trajectory planning is based on a second-order system for the rotor motion and a third-order system for the tower motion following [103].

For the trajectory of the rotor motion (see Figure 8.3), the rotor effective wind speedv0L is first delayed by the buffer time T₂ and then transferred to a rotor speed set point with the static function Ωss (see Figure 8.1). The rotor motion is then shaped by the Pole Placement (PP).

By choosing the two poles p₂₁ and p₂₂, the second derivative of the rotor speed is

¨Ωd= (p21p₂₂)Ωss+ (p21+p₂₂) ˙Ωd−(p21p₂₂)Ωd. (8.6) In the block System Constraint (SC), the limits ¨Ωmin/max and ˙Ωmin/max are calculated online based on M_a,d and its derivative from (8.5a) such that constraints for the generator torque

PP ¹_s ¹_s ¹_s x_T,ss

exp(−T₃s) v₀

SC F_a,d,max

...x_T,d x¨_T,d x˙_T,d x_T,d

¨ x_T,max

Figure 8.4: Online trajectory planning for the tower motion.

MG,max/min and its rate ˙MG,max/min are not violated:

˙Ωmin/max= M_a,d

J − M_G,max/min

iGBJ , (8.7a)

¨Ωmin/max= M˙a,d

J − M˙_G,max/min

i_GBJ . (8.7b)

The desired rotor motion trajectories ˙Ωd and Ωd are then obtained by applying the constraints and successive integration following [104]. The rotor dynamics are designed for low frequency tracking of the desired speed. The buffer time T₂ is set such that the rotor speed changes with the arrival of the wind field at the rotor.

The desired tower trajectories are planned in a similar way by choosing a buffer time T₃ and the poles p₃₁, p₃₂, and p₃₃, see Figure 8.4. With the poles and the steady tower position x_T,ss, the third derivative of the tower top displacement is

...x_T,d =−(p₃₁p₃₂p₃₃)x_T,ss+ (p₃₁+p₃₂+p₃₃)¨x_T,d−(p₃₁p₃₂+p₃₁p₃₃+p₃₂p₃₃) ˙x_T,d+ (p₃₁p₃₂p₃₃)x_T,d. (8.8) The maximum allowed thrustF_a,d,max to stay above the minimum pitch angleθ_min is calculated and used to limit the tower acceleration:

F_a,d,max = 1

2ρπR²c_T(λd, θ_min)v_rel,d² , (8.9a)

¨

x_T,max = F_a,d,max−c_eT x˙_T,d−k_eT (xT,d−x_0T)

meT . (8.9b)

The maximum pitch angle and pitch rate could be considered similarly. This was omitted here, because the exceedance of these constraints have been determined to be very unlikely.

8.1 Controller Design 167

WT CPC

ISC ISC

TEQUILA v₀

ΩG,rated

M_G

ΩG

−

θ_FB θ_c

M_G,d θ_c,d

∆ΩG,d

∆M_G,d

−

Figure 8.5: Closed loop of the combined feedforward-feedback controller.

Combined Feedforward-Feedback Controller

There are two main issues when combining the TEQUILA controller with the feedback controller as illustrated in Figure 8.5. First, all controllers need to have the same control objective. The common CPC has a constant reference value ΩG,rated, while the desired generator speed

ΩG,d= Ωd

iGB (8.10)

falls below this value for wind speeds below v_rated. This issue is solved by adding the deviation from the rated generator speed to the reference signal:

∆ΩG,d = ΩG,d−ΩG,rated. (8.11)

The desired generator speed could be used as in [100] as well. However, using ∆ΩG,d gives the possibility to set this signal simply to zero if no wind speed preview is available.

The second issue is that the desired generator torque M_G,d cannot be simply applied to the nonlinear state feedback of the ISC. Thus, the desired generator speed ΩG,d, the desired pitch angleθ_d, and an additional ISC are used to calculate a desired generator torque update ∆MG,d. Furthermore, the generator torque in the transition fromM_G,2.5in region 2.5 andM_G,3in region 3 of both ISCs is linearly interpolated using the switching signalσ_R3as proposed in [94] to avoid large differences if both ISCs are not switching simultaneously:

M_G =σ_R3M_G,3+ (1−σ_R3)MG,2.5 with σ_R3 =

( 0, θ ≤θfine

1, θ > θfine (8.12) Here, θ_fine is the minimum blade pitch angle for ensuring region 3 torque. Additionally, σ_R3 is filtered by a first-order low pass filter with a time constant of 5 s.

Im Dokument Lidar-assisted control concepts for wind turbines (Seite 180-186)