Since information from cycle identification is forwarded to the MPC via parameters, which are varying over the prediction horizon, the total fatigue damage has to be distributed over the prediction horizon, as visualized in Fig. b. Therefore, the damage of each stress cycle is split into two halves, which are allocated to the two contributing stress samples. For example, cycle 4 is formed by samples k=4 and k=8. Their fatigue cost terms therefore are allocated to these samples, as shown by the blocks in Fig. b. This example also shows an important property of the Rainflow algorithm, which identifies cycle 4 even though it is interrupted by the nested cycle 2, as shown in Fig. a. If, for a given stress sample, the complementary stress sample is not controllable (i.e., lies in the residue), all damage is allocated to the given sample. Here, this is the case for cycles 1 and 3, in which all damage is allocated to sample k=2 and k=4, respectively.

Figure a visualizes the generation of the time-varying parameters. Since each stress extremum belongs to one or two stress cycles , one or two stress references are set per extremum. These stress references are considered as optimization or tracking references for the current MPC step. If both stress samples of a cycle lie in the prediction, mean stresses (M2, M4) become the stress references. If the complementary stress sample of a cycle lies in the uncontrollable residue (“mixed cycle”), this complementary stress value (C1, C3) becomes the stress reference for the considered sample in the controllable prediction. However, in many cases, a mixed cycle is crossing the level of the initial stress σ(t0). In this case, the best possible tracking reference is this initial stress value itself (R1, R3) since zero oscillation in the prediction corresponds to zero fatigue cost. A more detailed derivation and explanation can be found in .

The fatigue cost function is defined by an integral over two cost terms, each one representing one potential cycle contribution of a stress sample, i.e., 4Jfatigue,PORFCσ,p‾=1Tcntrl∫t0tend(Jfatigue,c(σ(t),σ‾^ref,c1(t),w‾^c1(t))+Jfatigue,c(σ(t),σ‾^ref,c2(t),w‾^c2(t)))dt[EUR]. The notation (⋅)^ means fixed for one MPC step, while (⋅)‾ means sampled on the control intervals of the prediction horizon. The cost terms are “switched on” by nonzero cycle weights w‾^c1/2(t). Reference stresses σ‾^ref,c1/2(t) and cycle weights w‾^c1/2(t) are collected in the parameter vector, 5p‾=σ‾^ref,c1,σ‾^ref,c2,w‾^c1,w‾^c2, which is defined as piecewise constant over the control intervals of the prediction horizon. The cost of individual cycles is defined by 6Jfatigue,c=w‾^c1/2(t)am|σ(t)-σ‾^ref,c1/2(t)|m, where the fatigue coefficient am and the fatigue exponent m are derived from the damage curve of Eq. ().

Various controller formulations are tested with the National Renewable Energy Laboratory (NREL) 5 MW onshore reference turbine in the aeroelastic simulator OpenFAST. This turbine has a hub height of 110 m and a rotor diameter D=126 m. All mechanical degrees of freedom (DOF) are activated.

All turbulent results in this work are mean values of 12 simulations (each with a different seed) of 600 s length in DLC 1.2 with category A turbulence. For the chosen turbine with a hub height of 110 m and coastal onshore setting, a mean annual wind speed of Vw,mean=7 m s-1 can be assumed . Thus, in the turbulent simulations the probability of wind speed is assumed to follow the Rayleigh distribution 11p(Vw)=π2VwVw,mean2exp⁡-π4VwVw,mean2, as shown in Fig. . The Rayleigh distribution is a variant of the Weibull distribution with the simplification of having only a single parameter .

The model of a pulsed lidar with four beams is employed. The model is implemented in the lidar simulator from sowento GmbH, which generates lidar wind estimates offline, and thus independently from the wind turbine simulation suite . Considered physical effects are the limitation to line-of-sight wind speeds, spatial averaging via a Gaussian range weighting function, discrete scanning, and “unfrozen” wind evolution. Particularly the wind evolution can be parameterized by an exponential decay constant; here, a higher value results in higher variation in the wind during its convection towards the rotor. Finally, the spatially distributed measurements are converted to rotor-effective wind speed by wind field reconstruction.

The MHE and MPC are implemented in the state-of-the-art acados framework , using the interior-point solver HPIPM for the underlying quadratic programs (QPs).

In the following, the performance of five MPC formulations and the baseline conventional controller (CC) from NREL are compared. The MPCs involve the conventional formulation of TTVP (see Sect. ) and the novel formulation of PORFC (see Sect. ). For PORFC, a fatigue exponent of m=2 (see Eq. ) is utilized that, resulting in quadratic cost functions, is particularly suited for quadratic programming. This case is assessed in combination with (named PORFC-2R) and without (named PORFC-2) the use of residue. Additionally, a fatigue exponent of m=5 is also tested (named PORFC-5) because this value is present at low stress amplitudes in the actual S–N curve of the tower material. However, this parameterization in combination with residue (which would be named PORFC-5R) has not led to satisfactory results and thus is not considered further.

Considered performance indicators are revenue (analogous to energy), fatigue cost derived from damage at tower base (based on tower capital expenditures (CAPEX) and a realistic piecewise S–N curve), a simplified definition of profit (revenue minus fatigue cost), pitch travel, and torque travel. It should be noted that these definitions of fatigue cost and profit have various limitations:

A change in tower damage can have further implications, e.g., on maintenance costs, which are not considered here.

Another important remark is that the standard baseline CC is based on a controller design whose major objective has been neither the reduction of fatigue nor the maximization of a profit metric. Therefore, benefits of the MPCs with respect to the CC in regard of these metrics are to be expected and should not be considered as key findings of the present work. This CC is utilized here because it has been used for similar comparisons in previous publications and therefore allows for some cross-comparisons to these other works. The more sophisticated – and thus relevant – comparison controller formulation is the TTVP MPC, which also has been utilized in other publications .

In , the same wind turbine plant model (NREL 5 MW onshore) and lidar simulator are used as in the present work. There, the parameters of the lidar simulator are tuned in order to maximize the measurement coherence bandwidth for the rotor-effective wind speed. In other words, the smallest detectable eddy size is maximized, reaching a value of Deddy,min=1.58D=199 m. Since control of fatigue – and not lidar tuning – is the focus of the present work, the parameters of the lidar simulator in Table  are adopted from . This “default lidar” scenario with a low decay constant of 0.1 is accompanied by a “high decay lidar” scenario, in which the exponential decay constant is increased to 0.4 .

The raw rotor-effective wind speed from the lidar simulator has to be buffered in order to compensate for time delays and filtered to remove uncorrelated high-frequency information. The buffer and filter parameters have to be tuned. In , the tuning of the lidar processing is performed using the reduced wind turbine model as the plant model for performance reasons. However, simulations in have shown that unrealistically high fatigue reduction is possible if the MPC-internal and the plant models are matching. Thus, in the present work, the mid-fidelity OpenFAST model is used for tuning the lidar processing in order to benefit from its more realistic fatigue behavior.

Inspired by , the raw rotor-effective wind speed is buffered by an adaptive buffer time span of 12Tbuffer(t)=Ttravel(t)-0.5Tscan-Tfilter. Here, the traveling time Ttravel(t)=dtravel(t)/Vmean(t) from the closest scanning plane to the rotor is obtained by the traveling distance dtravel and the current mean wind speed Vmean. The traveling distance dtravel(t)=dscan,close+dT(t) is the nominal distance corrected by the current tower tip deflection. The total scan time Tscan=1/fscan is obtained from the scanning rate. The filter delay Tfilter is zero here since a zero-phase filter is utilized.

Since the lidar correlation varies with wind speed , the uncorrelated high-frequency information has to be processed by a low-pass filter, which is adaptive as well. In order to avoid the above-mentioned filter-delay compensation, only zero-phase algorithms have been considered. Particularly, a zero-phase forward–backward infinite impulse response (IIR) filter based on a first-order Butterworth filter (function filtfilt in MATLAB) has been compared to a central moving mean filter (function movmean in MATLAB). For this purpose, the lidar simulator parameterization from Sect.  and a reasonable initial parameterization of the MHE and MPC have been utilized in turbulent simulations. Despite its simplicity, with different MPC formulations and horizon lengths, the moving mean filter has exhibited superior performance and thus has been chosen for the present application. It should be noted that this superiority of the moving mean filter has only been observed for the present lidar and wind turbine configuration and cannot be generalized. Thus, for another configuration, the comparison should be repeated. Beneficially, the moving mean filter only requires tuning of its window length. The empirical formula 13Tmovmean(t)=Deddy,minVmean(t), which is based on the smallest detectable eddy size and the current mean wind speed, has led to a very good adaptive tuning for the present setup.

Due to its nature, the central moving mean filter requires sufficient information from the past and future. Except for the beginning of a simulation, the amount of past information is typically sufficient and even growing in the course of the simulation. In contrast, sufficient future information beyond the prediction horizon is only ensured if the inequality 14Tpred(t)-0.5Tmovmean(t)≥Thoriz holds, where half the moving mean filter length Tmovmean subtracted from the lidar-predicted time Tpred exceeds the MPC horizon length Thoriz. The lidar-predicted time Tpred(t)=dscan,far/Vmean(t) depends on the distance of the farthest scanning plane to the rotor and the current mean wind speed. As shown in Fig. , the inequality typically holds and is only slightly violated at Vmean>22 m s-1.

In order to verify the above adaptation formula, simulations have been executed in which these buffer and filter window lengths are increased or decreased. These results are generated for the “high decay lidar” scenario, in which lidar data quality is lower, and thus correct filtering is more important. The simulations reveal that the above adaptation laws lead already to high revenue and low fatigue cost. As shown for the filter window length in Fig. , for both TTVP and PORFC with scaling factors of 1, a high revenue is maintained while achieving low fatigue cost. Lower scaling factors (shorter filter windows) have a tendency towards higher actuator usage, while higher scaling factors dramatically increase fatigue cost. Consequently, scaling of the buffer and filter parameters is not applied in the present work.

The MHE is set up with an estimation horizon length of 8 s and a sample time of 0.1 s, which are fixed for all MPC configurations.

Practical experience has shown that the goal of the MHE should not be an accurate reconstruction of the unmeasured true plant states of tower tip deflection and velocity, as shown in Fig. . In fact, an accurate reconstruction would contain high frequencies from the plant higher-order degrees of freedom that, not being matched by corresponding degrees of freedom of the MPC-internal model, would spill over and pollute its lower-order degrees of freedom.

Instead, the MHE should provide low-frequency initial states for the low-frequency MPC-internal model (“estimated initial” states). Consequently, the MHE is tuned to estimate state trajectories that best fit the behavior of the reduced model. This is achieved by setting low weights for the tower variables in the weighting matrix Wmeas, as shown in Table . These low weights allow for significant deviations from the measured tower acceleration and the lidar-based references and thus for a greater focus on the reduced model dynamics. For tower deflection and velocity, very low values of 10-4 are chosen since these quantities are not measured. For tower acceleration, an intermediate value of 10-2 is chosen since it is measured, but its trajectory does not need to be tracked carefully.

The estimated tower deflection in Fig.  exhibits a significant steady-state offset from the unmeasured true tower deflection. This can be possibly attributed to the inaccurate system model since this offset also pertains in the MPC prediction, which is based on the same system model.

By the intermediate weights in the weighting matrix Wprev, the current state trajectories are only loosely tied to the previous ones.

In the weighting matrix Wv, significant noise is only permitted for the pitch angle and torque rate in order to enable a close match of pitch and torque estimations with their already accurate measurements.

The controller sample time is set to 0.1 s like in and . The maximum horizon length of Thoriz,max=8 s is chosen based on the findings of , which indicate that a considerable portion of the plant stress cycles will be contained in this prediction horizon. However, since horizon length has a substantial impact on performance and since, moreover, the longest horizon is not always the best, shorter horizons of Thoriz={4;2;1} s will be tested throughout all turbulent studies.

One QP is solved per MPC step. The Hessian matrix is automatically convexified to account for possible numerical issues due to the highly non-standard cost formulation of PORFC. Practical experience has shown that performance is improved if the Newton step length of the QP is reduced from 1 to 0.1. This can be explained by the frequently changing optimization problems, especially for PORFC. In this case the initialization of the QP might not be sufficiently close to the optimum, and full Newton steps could leave the region of validity of the quadratic approximation of the nonlinear program .

As shown in Sect. , all weights in the cost function except for the fatigue weight αfatigue are pre-defined for simplicity. Thus, only the fatigue weight has to be tuned in the following. Tuning is executed at a single reference wind speed of Vw,ref=9 m s-1. This wind speed is chosen since, for the conventional controller CC, the highest profit contribution occurs there, as shown in terms of profit density in Fig. . Profit density represents the incremental contribution to total cumulative profit at a certain wind speed. Other meaningful criteria for a suitable tuning wind speed would be the wind speed at which half of the total cumulative profit is reached or where the highest revenue contribution occurs.

The variation in important key performance indicators (KPIs) with fatigue weights is shown in Fig. . For brevity, only the results for the maximum MPC horizon length of 8 s are shown. All variants of PORFC are able to maintain high revenue levels while decreasing fatigue cost with increasing fatigue weights. In contrast, the revenue of TTVP declines rapidly above a certain fatigue weight. Thus, the tuning of PORFC can be considered as less critical than that of TTVP. Since in most cases torque travel and pitch travel increase with fatigue weight, low fatigue weights should be preferred as long as fatigue cost is not harmed significantly. Following this strategy, the fatigue weights are determined for all controller formulations and horizon lengths, as shown in Table .

As a next step, the optimal tuning weights from Sect.  are fixed, and simulations at different reference wind speeds Vw,ref={5;7;…;25} m s-1 are performed for each controller formulation and prediction horizon length. The simulations result in the Weibull-weighted cumulative KPIs shown in Fig. .

For all MPCs, revenue is at least slightly below the one of the CC. However, especially for PORFC-2R and PORFC-5 with a prediction horizon of Thoriz≥2 s, the revenue losses remain moderate. In contrast, all MPCs with Thoriz≥2 s exhibit substantially lower fatigue cost than the CC. As a rough indication of the combined effect, the changes in revenue and fatigue cost can be assessed in terms of the simplified profit indicator. The highest profit gain is achieved by PORFC-2R at maximum prediction horizon, which surpasses CC by 30 % and the best TTVP by 2.5 %. At least for the present setting, very short horizons of 1 s cannot be recommended since they significantly decrease revenue and greatly increase fatigue.

Over different horizon lengths, PORFC-2R exhibits very stable revenue and fatigue levels. In contrast, for the PORFC formulations without residue (PORFC-2, PORFC-5), a shorter horizon of Thoriz=4 s exhibits higher profit than Thoriz=8 s. This phenomenon may be explained by a higher influence of the prediction errors: due to model errors and wind evolution, the predicted states at the end 4 s <t≤8 s of a long horizon may be affected by large errors. Since PORFC-2 and PORFC-5 have to rely solely on the predictions, their performance may suffer from long horizons.

For all MPCs, the fatigue reduction with respect to CC comes at the price of a higher pitch travel. Here, TTVP exhibits around 6 times the pitch travel of CC. Since in the literature more moderate increases (e.g., by a factor of 2) are reported , further studies on pitch penalization should be conducted. For PORFC, pitch travel is even slightly higher and is somewhat reduced by a reduction in horizon length. Torque travel exhibits different behavior in that several MPC formulations have a lower torque travel than CC.

A look at the profit density and cumulative profit in Fig.  shows that both MPC formulations (TTVP, PORFC) “earn money” very similarly over wind speed. Compared to CC, the MPC fatigue reduction strategies lead to profit benefits at very low and at intermediate wind speeds. With the present tuning, TTVP has a slight extra advantage at very low wind speeds, while PORFC-2R is superior over a broad range of intermediate wind speeds.

The “default lidar” scenario of the previous sections can be considered as very favorable for lidar-assisted control since the wind does not change very much between the lidar measurement planes and the rotor. Thus, the lidar provides a fairly good estimate of the true incoming wind. In order to challenge the MPCs even more, a further assessment is performed for the “high decay lidar” scenario (see Table ).

Despite the significant reduction in lidar signal quality, the profit benefit of the MPCs over CC decreases only slightly, as shown in Fig. . Particularly, the best-performing PORFC-2R (Thoriz=4 s) still surpasses CC by 26 % for the “high decay lidar” scenario, in comparison to the above-mentioned 30 % for the “default lidar” scenario.

The profit benefit of 5.1 % of the best PORFC-2R (Thoriz=4 s) over the best TTVP (Thoriz=8 s) shows that PORFC-2R is particularly strong in handling situations of low lidar data quality. In a direct comparison using the same horizon length (Thoriz=4 s), PORFC-2R is superior by almost 9 %. Just like in the “default lidar” scenario, PORFC-2R exhibits better revenue stability over the horizon lengths Thoriz={8;4;2} s but also exhibits excessive fatigue cost at Thoriz=1 s.

The increasing benefit of PORFC-2R with respect to TTVP with lower lidar data quality conversely suggests decreasing benefits with very high lidar data quality. This hypothesis is actually partially confirmed by the extreme scenario of a perfect wind prediction (without lidar errors). As shown in Fig. , as expected, fatigue cost can be reduced by TTVP and PORFC-2R even further than in the “default lidar” scenario of Sect. . However, relative to TTVP, the advantage of PORFC-2R decreases significantly. For the maximum horizon length, the profit of TTVP even slightly surpasses PORFC-2R by 0.4 %. On the other hand, as revenue and fatigue cost of PORFC-2R are more stable for shorter horizons, this formulation retains a significant advantage there.

All previous scenarios assumed a wind preview. However, to date, lidar systems can still account for a significant portion of the capital and operational expenditures of wind turbines . In order to avoid lidar-related costs or effort, some studies are directed towards predictive control without explicit preview measurement . In this case, the wind prediction over the MPC horizon can for instance be generated via constant extrapolation of the instantaneous wind estimate at the rotor (persistence). This motivates an analysis of how the novel PORFC MPC actually benefits from a predictive preview compared to a persistent preview.

Since the design of a rotor-effective wind speed estimator is out of scope of the present work, a “perfect persistence” scenario is employed and compared to the above “perfect prediction” scenario.

As shown in Fig. , all MPC formulations significantly benefit from prediction (instead of persistence). For all formulations, this is primarily achieved by high fatigue reduction. At the same time, actuator usage is moderately decreased or even increased at some horizon lengths for TTVP but is significantly decreased for PORFC-2R if Thoriz≥2 s.

These results further indicate the technical benefit of lidar-assisted control and motivate further studies comparing the realistic lidar wind preview with a sophisticated wind speed estimator.

Figure shows a situation where the stress prediction at the initial value σ(t0) turns from a mildly rising slope (MPC step 1) to a mildly falling slope (MPC step 2). Consequently, a new stress maximum is formed and added to the “right-hand side” of the residue set at MPC step 2, as shown in Fig. a. In the following steps 3 to 5, the size of the residue set remains constant; only the right-hand-side value is updated by the current initial stress value.

As shown in Fig. b, the change in extrema in the stress prediction over the course of MPC steps leads to frequent changes in the PORFC stress references (see Sect. ) or, more in general, in the PORFC parameter structure (see bars in Fig. b). However, since many of the emerging or vanishing stress cycles are small in amplitude, also their corresponding stress reference values are close to the stress prediction trajectory (compare bars to the solid blue line) and thus have low impact on the overall optimization problem.

Since by nature of MPC the stress trajectory is shifted to the left-hand side with each simulation step, also the PORFC parameter samples are shifted. This becomes even more clear in Fig. , where stress reference 1 is plotted over the prediction horizon and over MPC steps (simulation time). Here, over the course of MPC steps, the stress reference pattern is evolving smoothly towards the beginning of the prediction horizon. While some references emerge within the prediction horizon, many references originate at the end and do not vanish before reaching the beginning of the prediction horizon.