Drivetrain fatigue strength characteristics of model-predictive control for wind turbines

2.1 Relative assessment of fatigue strength

The relative assessment of the fatigue strength is based on two different concepts—the DEL and the damage sum D. The calculation of a DEL was originally derived, to compare damaging of differing load signals.¹⁸ It is independent of a specific component geometry, purely based on load signals and applies only general material properties. D is stress-based and correlated to the scalar utilization factor a_BK, which is defined by the FKM guideline.^{16, 17} The FKM guideline describes, among others, industrially used methods for fatigue strength verification of variable amplitude nominal stress signals for rod-shaped components. D is strongly dependent on the cross-section of a component. But a relative comparison of two D resulting from two different load signals acting on the same component allows to eliminate this dependency. Within this paper, only torsional stresses and loads are considered to determine D and the DEL.

The calculation of D is based on S-N curves, which relates the allowable number of cycles to failure to an alternating, that is, mean-free, stress amplitude. Equivalently, a DEL is based on M-N curves,^{18, 20} which relates the allowable number of cycles to failure to an alternating load amplitude. A time domain analysis based on turbulent full-field wind speeds results in variable amplitude spectra with variable means stress and load. In order to map them to mean-free spectra, at first, a classification of both signals into stress and load amplitudes along with its corresponding number of cycles and mean stress and mean load values is done by the rainflow counting method. This results in two sets of 3-tupels and . Second, the amplitudes are mapped to damage-equivalent alternating stress and load amplitudes. To this end, each element of is scaled by a mean stress factor.

Definition 2. ((mean stress factor and damage-equivalent alternating stress amplitude[¹⁷]).)For some i ∈ {1,…,N_τ} consider a 3-tupel . The corresponding mean stress factor K_AK is determined based on the amplitude τ_a and its corresponding mean value τ_m. For torsional stresses from variable amplitude and variable mean stress signals, it is given by

(1)

where R denotes the stress ratio

(2)

M_τ is called the mean stress sensitivity and here given for stainless steel. R_m denotes the tensile strength and depends on the material. The damage-equivalent alternating stress amplitude is given by

(3)

This results in a damage-equivalent set of 3-tupels . Equivalently, a Goodman-correction is applied to map a variable amplitude and mean load classification into a set of damage-equivalent alternating load amplitudes.

Definition 3. ((Goodman-correction[¹⁸]).)The Goodman-correction is determined based on the amplitude M_a and its corresponding mean value M_m. The corrected mean-free amplitude is given by

(4)

where M_mean denotes a fixed mean load and M_ult a design ultimate load.

This results in a damage-equivalent set of 3-tupels . The utilization factor a_BK and its corresponding damage sum D are determined by the component variable amplitude fatigue strength.

Definition 4. ((component variable amplitude fatigue strength[¹⁷]).)The component variable amplitude fatigue strength T_BK is given by

(5)

where T_AK=K_AKT_WK denotes the amplitude of component fatigue limit. The mean stress factor K_AK is equal to 1.0 for signals without a mean stress.¹⁷ The component fatigue limit for completely reversed stress T_WK is independent of the stress signal and will be eliminated in a relative comparison. For a variable amplitude stress spectrum, the variable amplitude fatigue strength factor K_BK is calculated by the elementary version of Miner's rule.

(6)

where , N_D=10⁶ denotes the number of cycles at the knee point and k denotes the slope of the component amplitude S-N curve for shear stresses of non-welded stainless steel.

Definition 5. ((cyclic degree of utilization and damage sum[¹⁷]).)The cyclic degree of utilization of rod-shaped components a_BK and its corresponding damage sum are defined by

(7)

(8)

Remark 1..Please note that

(9)

In the later, only the relation of two damage sums D_NMPC and D_Reference will be analyzed for the same rod-shape machine element. Both stress signals will be obtained from the same multibody simulation (MBS) model. As the component fatigue limit for completely reversed stress T_WK is independent from the actual stress signal τ(t),¹⁷ it is not necessary to be determined.

To map the alternating amplitudes of to a scalar value DEL, the hypothesis about linear accumulation of damage is consequently used.¹⁸

Definition 6. ((DEL[¹⁸]).)The DEL for a variable amplitude alternating load spectrum is given by

(10)

f^eq denotes the DEL-frequency, for example. 1 Hz, and T=t_f−t₀ the time domain of the load signal M_x(t). k denotes the slope of the M-N curve.

2.2 Optimal control

NMPC describes an iterative strategy in which an OCP is solved in each step. An OCP of ordinary differential equations (ODEs) is defined as an optimization problem, where functions of one independent variable—for example, the time t – is determined such that they without loss of generality (w.l.o.g.) minimize a given target function. Optimal control theory provides first- and second-order necessary conditions for optimality. In this section, these necessary conditions are summarized up to that level, which is needed to formulate an NMPC scheme based on indirect methods.

In the following functions, whose derivative is not explicitly occurring in the problem formulation, are called control variables. They are queued up in the vector where defines its number. The optimization includes conditions like ODEs. In contrast to a control variable, each function, whose derivative is defined by this ODE, is called a state variable. They are queued up in the vector , where defines its number. OCPs may also involve inequality constraints and terminal conditions. The following defines the most general OCP used within this paper. W.l.o.g. is assumed to be autonomous.

Problem 1..Let denote the following OCP.

(11)

subject to

(12)

with 1 and , ∀i∈{1,…,p} and ∀j∈{1,…,q} with k ≥ 2 sufficiently high. is called initial value, g(x,u) control constraints and h(x) state constraints.

Definition 7..Let the following sets be defined:

(13)

(14)

Definition 8. ((Hamiltonian and augmented Hamiltonian).)The Hamiltonian and the augmented Hamiltonian are defined by

(15)

(16)

Definition 9. ((order of state constraint).)The order of a state constraint h_j(x) is defined by the smallest number such that there exists at least one i ∈ {1,…,m} for which the following holds

(17)

Definition 10. ((solution structure).)Let (x^⋆, u^⋆) be a solution of . The sets of the active control constraints, the almost active control constraints, and the active state constraints are defined by

(18)

(19)

(20)

Moreover, the sets are defined by

(21)

(22)

If , then is called a solution structure.

If does not contain state constraints, then the necessary conditions of first order are given by the following theorem.

In case of general state constraints, a complete theory is not yet available. Necessary conditions in case of one scalar state constraint and no control constraints are provided in Jacobson et al.²² In case of multiple state constraints, necessary conditions are described in Maurer.²³ A survey of the necessary conditions for state constrained problems is given in Hartl et al.¹⁵ For a problem with a regular Hamiltonian and first-order state constraints, the adjoint variables λ(t) and the Lagrangian multipliers associated to the control constraints μ_g(t) are continuous.²⁴

4.1 Mathematical model of the wind turbine

In this section, two models of the wind turbine with different levels of fidelity are used. The first one is an MBS model based on the definition of the NREL-5 MW reference wind turbine.²⁸ It is also provided with the MBS software Simpack 2 and was extended by a standard appropriate planetary gear box by the IMM 3 of the Technische Universität Dresden. It consists of two planetary stages with four and three planets and the outer rings fixed to the gear box housing. A spur gear stage is located between the second planetary stage and the clutch (cf. Figure 3). To account for small torsional flexible deformations in the drivetrain, the main shaft, the two planetary carriers, the intermediate, and the high speed shaft are modeled flexible by standard modal linear model order reduction techniques for finite element models. This MBS model serves as the system to be controlled and analyzed. To predict the dominant dynamic characteristics within the NMPC approach, a nonlinear model of reduced order (NMOR-model) is derived from the MBS model.

An important characteristic of the NMOR is the nonlinear dependency of the aerodynamic torque M_aero on the rotor speed ω, the scalar-valued rotor effective wind speed V_Wind(t), and the collective pitch angle β. M_aero is often approximated by^{9-11, 29, 30}

(28)

where C_p denotes the coefficient of power, ρ the air density, and R the nominal rotor radius. C_p is a dimensionless quantity describing the mean efficiency of converting the aerodynamic power into the rotor power. It is nonconstant and limited to the theoretical maximum of 59.2 % by Betz's law.³¹ C_p(λ,β) is often given as an array of data derived for a set of tip speed ratios and pitch angles β. The array entries are determined by quasi-static equilibria of a wind turbine MBS model. The only MBS velocity states, which are not vanishing in terms of these equilibria, are all the rotor speed related states. The nominal rotor speed is given by λ, for which the C_p-table is created. As λ also depends on the wind speed, a nominal V_Wind needs to be specified to obtain C_p(λ,β) in terms of λ and β.

In Figure 4, C_p(λ,β) is plotted versus λ for different uniform wind speeds and β=0°. The maximum value for is round about 49.48%, and for , it is round about 46.83%, which makes a difference of round about 2.6% in the estimated efficiency. When including conditions into the OCP to consider drivetrain fatigue strength, it turned out that a C_p dependent on λ and β only does not capture the dynamics sufficiently. Alternatively, C_p(ω,V_Wind(t),β) is considered to be dependent on the rotor speed ω, the rotor effective wind speed V_Wind(t), and the collective pitch angle β.

(29)

where J denotes the total axial moment of inertia, i_GBX the total gear box ratio, and η the generator efficiency. For convenience this equation is reformulated.

(30)

(31)

A tensor product of Chebyshev polynomials is used to interpolate the C_p-data.^{32, 33} The rotor speed related polynomials are of order 10, the wind speed related ones are of order 8, and the pitch related polynomials are of order 6. These orders were found by a sensitivity analysis. In Figure 5, the rotor speed of the NMOR model is plotted versus the rotor speed of the MBS model. As it can be seen, the NMOR model captures the dominant dynamics of the rotor speed sufficiently.

As the rates of the generator torque and the pitch angle are limited, both are states, whereas its time derivatives are control variables.

(32)

NMPC requires an objective to be optimized. For the wind turbine, the generated electrical energy is maximized.

(33)

The maximum and minimum values of x₂ and x₃ result in first order state constraints as well as the limit on the maximum of generated power. The maximum rotor speed is a second order state constraint. NMPC is applied to control complex systems, which generally contain model simplifications and state measurement deviations. Even if a hard fulfillment of a second-order state constraint can be found in one step of NMPC, these uncertainties may lead to initial conditions such that no solution of the OCP exists any more in the next step of NMPC. The application of terminal integral inequality conditions relaxes these second-order state constraints, such that violations up to an a priori specified amount are accepted in each NMPC step.²⁷ This technique is exemplary described by limiting the rotor speed to its maximum value: x₁(t)−x_1,max ≤ 0 ∀t∈[t₀,t_f]. A mean tolerated violation is specified by the definition of two limits x_1,max,a and x_1,max,b with x_1,max,a<x_1,max,b. The following condition is added to the OCP.

(34)

In the later, x_1,max,a=10.1 rpm and x_1,max,b=12.1 rpm was chosen. This leads to small violations of the maximum rotor speed of 12.1 rpm, which are accepted by the reference controller in the same way. An equivalent term with the according conditions is added regarding the maximum generator power. P_max,a=4 MW and P_max,b=5 MW are the corresponding limits.

4.2 Regularization and fatigue strength consideration

So far, the formulation of the OCP still leads to a nonregular Hamiltonian with discontinuous optimal control.²⁷ A regularization is mandatory from the engineering point of view.¹¹ It can be chosen such that the control is continuous, but the final value of the objective differs from the nonregular solution within an acceptable range.^{27, 34} It is not unique but can be formulated in several different ways. Here, they are chosen such that the damage is limited. As a natural choice the generator torque might be prevented to oscillate with high frequencies by the following term.

(35)

where c_reg>0 is chosen a priori. The more c_reg gets close to zero, the more the nonregular solution is approximated. If one determines c_reg in Equation (35) such that the damage is kept at the level of a reference controller,²⁸ the energy production performance of NMPC drops significantly far bellow the reference controller. As NMPC is also numerically more expensive, this makes NMPC noncompetitive. The damage sum is dominantly influenced by the corresponding (half-) cycle of the maximum stress amplitude, but the regularization generally prevents an oscillation of the generator torque. To reduce the maximum stress amplitude, the coefficient c_reg in Equation (35) needs to be chosen that large, such that the energy extraction efficiency is also reduced in ranges, which have small influence on the fatigue strength. More sophisticated conditions are necessary to take the fatigue strength into account.

The damage is determined based on the Rainflow counting method. It is a characteristic of the signals on times scales a lot larger than prediction horizons, which can be handled in an NMPC scheme. The fatigue strength cannot be taken into account directly within the prediction, but needs to be considered by conditions on the already passed signal. The following regularization was applied successfully.

(36)

where denotes the mean value of u₁ from the preceding NMPC step and x_2,max,last is the maximum and x_2,min,last is the minimum generator torque applied to the system within the last 300 s.

These terms will regularize the Hamiltonian with respect to u₁. A second regularization term needs to be added to ensure a continuous optimal control of u₂. For the pitch angle, the natural choice was found to be sufficient. This results in the following complete objective:

(37)

4.3 Simulation results

The presented indirect NMPC scheme was applied to six sample load cases. Each of it is a standard A-classified full-field turbulent wind speed with a mean of 11 and was created with the NREL software TurbSim.³⁵ Their configurations are given in Table 1. The following discussion is limited to this mean value, to allow a reasonable discussion of the results. The presented approach can easily be extended to a broad range of mean wind speeds, which is shown in Schwarz and Schulz.³⁶

The regularization coefficients and allow to further configure the indirect NMPC. Three different settings are determined.

1. Purely maximize the extracted amount of energy without explicitly constraining the damage of drivetrain components.
2. Maximize the extracted amount of energy while keeping the damage sum at the level of the reference controller.
3. Further reduce the damage sum while keeping the energy performance at the level of the reference controller.

Configuration 1: The energy extraction performance of these three settings is analyzed in relation to the reference controller and illustrated in the first plot of Figure 6. The largest increase of 0.6% to 1.32% can be obtained, when damaging is not explicitly considered. Its prediction capability allows NMPC to keep the rotor speed close to its optimal value, which strongly depends on the wind speed, cf. Figure 4. This requires the generator torque signal to oscillate around its optimal value,²⁷ cf. fourth plot of Figure 6. It illustrates the maximum load amplitude, the mean load value (position of the box), and the standard deviations of the load signal (vertical size of the box). For all the six load cases, both the maximum torque amplitude and the standard deviation of the torque signal is increased (black vs. red bars). Thus, the amplitudes are increased significantly for low and high frequencies. At the same time, the D is raised by round about 147% to 269%, cf. second plot of Figure 6. A similar result can be observed for the DEL, which is increased by 8.9% and 19.3%, cf. third plot in Figure 6.

Configuration 2: Larger costs of construction and operation to run NMPC for a wind turbine may not be compensated by this increase of extracted energy, cf. configuration 1. The nature of NMPC allows to explicitly consider the damage, while still obtaining a control signal maximizing the energy. Thus, and are chosen by a parameter variation such that D and DEL are kept at the level of the reference controller, cf. blue bars in Figure 6. The extracted energy can still be raised by 0.4% to 1.0%.

Configuration 3: The prediction capability of NMPC can also be used to focus on the reduction of D and DEL. The maximization of the energy extraction is still the objective of NMPC. But the damage is penalized such that the amount of extracted energy is kept at the level of the reference controller. This allows to reduce D by 23.8% to 48.3%. Equivalently, the DEL is decreased by 1% to 9%. The advantage of NMPC with this configuration will be the reduction of the maintenance rate as well as the extension of the life time.

In Figure 7, the generator torque is plotted in the time domain in the left column for all six load cases. The red curves of NMPC configuration 1 show large maximum load amplitudes, whereas the corresponding amplitudes of the blue and black curves are almost coincident. The green curves of NMPC configuration 3 illustrate significant smaller maximum load amplitudes. In the right column, the power spectral density (PSD) of the joint torque between the main shaft and the first planetary carrier is plotted on a double logarithmic scale. The amplitude peaks at 0.6, 1.2, 1.6, 2.4, and 2.9 Hz all can be identified with eigenmodes of the MBS system, which have strong torsional character. It can be seen that the amplitudes below the resonance at 0.6 Hz are reduced by all NMPC configurations. For frequencies above 0.6 Hz, NMPC configuration 1 shows the largest amplitudes, whereas the NMPC configuration 3 still have the smallest amplitudes beside of the resonance frequencies. These resonances are currently not taken into account of NMPC and show further potential, especially when combining the approach with a linear state space model of the drivetrain like it is done in Schulz.⁴

The optimal control in each NMPC step was determined within a mean time period of 5 ms and a maximum of 51 ms.