2.1 The PArallelized LES Model (PALM)

In this study, an LES wind farm model is utilized, which employs the PArallelized Large-eddy simulation Model (PALM) version 4.0 (Maronga et al., 2015) coupled with the actuator disc model (ADM) of a wind turbine (Witha et al., 2014). PALM is an open-source LES code, which is developed for simulating atmospheric and oceanic flows and optimized for massively parallel computer architectures. PALM makes use of the Schumann volume averaging approach (Schumann, 1975) and uses central differences to discretize the nonhydrostatic and incompressible Boussinesq approximation of the three-dimensional Navier–Stokes equations on a structured Cartesian grid. For more information about the general capabilities of the PALM, the reader is referred to Maronga et al. (2015). Much more detailed wind turbine models with a more realistic near-wake structure, e.g., the actuator disc model with rotation (ADM-R) and actuator line model (ALM) are also implemented in PALM. However, ADM is computationally efficient and provides a good approximation of the far wake structure, making it suitable for the present study. The static ADM is extended with a first-order lag and a mass-spring-damper system in order to estimate the dominant aerodynamic power response and the first tower fore–aft mode excitations of an individual wind turbine to the wind farm turbulent flow and wakes in a computationally efficient manner.

2.2 The wind turbine model

The individual wind turbines are parameterized with an actuator disc model (ADM) to exert a thrust force on the incoming flow and extract a certain amount of energy from the incoming wind. The wind turbines are modeled in PALM with only 2 degrees of freedom (DoF) as follows:

Equation (1) represents the power response of a wind turbine to the aerodynamic power P_a with the aerodynamic time constant τ, which can be associated to the drive-train dynamics. The axial induction factor a stands for the ratio of the reduced wind velocity at the rotor to the effective wind speed U₀ at a far distance upwind from the rotor disc and can be translated to the practical torque and pitch control inputs of a wind turbine. Equation (2) describes the first tower fore–aft dynamic mode with the tower top fore–aft displacement x_T; the aerodynamic thrust force F_a; and the tower equivalent modal mass $m_{{\mathrm{T}}_{\mathrm{e}}}$, structural damping c_T, and bending stiffness k_T according to Schlipf et al. (2013). Note that the tower top fore–aft displacement x_T is considered positive in the downwind direction.

The aerodynamic thrust force for a single turbine is calculated using the employed ADM in the PALM simulation code as follows (Gasch and Twele, 2011):

where the thrust coefficient C_T is described as a function of the axial induction factor a as

and ρ is the air density and A_d is the swept area of the rotor plane. Note that the axial induction factor is limited to its value for maximum (greedy) power extraction (Gasch and Twele, 2011), i.e., $a\le \mathrm{1}/\mathrm{3}$, to avoid high wake losses and violating the thrust approximation at higher induction factors. The relative wind speed U_rel is defined as a superposition of the effective wind speed and the structural tower top velocity as (Schlipf et al., 2013):

in order to model the aerodynamic damping of the tower fore–aft mode.

Resolving Eq. (5) and considering the induction effect of a rotor disc, the effective wind speed U₀ can be approximated from the measurable axial disc-averaged wind velocity U_d from the PALM code as

and enables us to model the applied aerodynamic thrust force (Eq. 3) acting in the negative direction on the flow (Witha et al., 2014). Then, the aerodynamic power of an individual wind turbine is calculated as

The tower base fore–aft bending moment of an individual wind turbine is approximated as (Schlipf et al., 2013):

where h_H is the hub height. In this investigation wind turbines are operating in the below-rated region, wherein the effective wind speed U₀ is always below the rated wind speed of the simulated wind turbine. It is assumed that the generator torque is fast enough to realize the commanded axial induction factor. Therefore, the actuator dynamics are neglected here.

In Sect. 3.2.2, the tower base fore–aft bending moment is used as a representative load indicator to mitigate the wake-induced global load at the different wind turbines inside the wind farm rather than to determine the actual fatigue load damage of the tower or any other component. Therefore the simplified structural and dynamical model Eqs. (1), (2), and (8) are considered meaningful for our control purpose and applied modeling approach. The ADM turbine model is integrating the turbulence-induced loads over the entire swept area and is neglecting important dynamic load effects, e.g., the rotational sampling of partial gusts and partial wakes. If a more realistic rotor model, actuator line model (ALM) with individual blades, is applied, the modeling of the structural dynamics and the load effect of the turbine should be improved as well.

Figure 1Power response of the simulated ADM compared with the resultant electrical power output of the drive-train model. Both models are subjected to the same wind speeds.

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Figure 2Ramped effective wind speed U₀ and the relative wind velocity U_rel, which takes the tower top fore–aft movement into account according to Eq. (5).

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Figure 1 plots the power response of the simulated wind turbine model Eqs. (1)–(2), which is incorporated later in the PALM code for simulating a wind farm example and used for the controller designs in the following sections. The power response of the model is compared with the resultant electrical power of the first-order drive-train dynamics presented in Schlipf et al. (2013) without considering any electromechanical losses. Table 1 lists the key parameters of the wind turbine models, which are taken from the freely available model of the NREL 5 MW reference wind turbine (Jonkman and Buhl, 2005).

Table 1Specifications of the wind turbine models taken from Jonkman and Buhl (2005).

^∗ Time constant at an effective wind speed U₀=8 m s⁻¹.

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Both models in Fig. 1 are subjected to the same effective wind velocities at the optimal operating point of the NREL 5 MW reference turbine with rotor diameter D=126 m, which is obtained from steady-state power and thrust computations as a function of the effective wind speed, e.g., using blade element momentum (BEM) theory. The aerodynamic time constant τ in Eq. (1) is scheduled as a function of the effective wind speed in order to describe the resultant electrical power of the considered drive-train as a reference. The fluctuations of the calculated aerodynamic power (black curves) originate from tower fore–aft mode excitations because of the step changes in the wind speed U₀. Note that the above analysis on the wind turbine model has not been simulated with PALM as it is not possible to step the wind speeds in the PALM code. Figure 2 compares the ramped effective and the relative velocities, showing the effect of the tower structural dynamic model Eq. (2) on the aerodynamic power and thrust of the extended ADM. The following section focuses on our investigated wind farm example, which is simulated after coupling the presented wind turbine model Eqs. (1)–(2) with PALM (see Level 0 of Fig. 6).

2.3 The case study

A layout of a 3×4 wind farm example with different wake overlaps with downstream wind turbines is considered here (Fig. 3). The wind turbines are spaced 5 rotor diameters (or 5D) in the streamwise direction. In the first and third rows, i.e., turbines $\mathit{\{}\mathrm{1},\mathrm{4},\mathrm{7},\mathrm{10}\mathit{\}}$ and $\mathit{\{}\mathrm{3},\mathrm{6},\mathrm{9},\mathrm{12}\mathit{\}}$, the rotor centers of the downstream turbines are intentionally offset by half a rotor diameter (D) from the centers of their upwind ones, while the wind turbines in the middle row, i.e., turbines $\mathit{\{}\mathrm{2},\mathrm{5},\mathrm{8},\mathrm{11}\mathit{\}}$, are spaced without lateral offset to create full wake interactions. Figure 3 shows the instantaneous field of the u component of the wind at the hub height of the wind turbines. Our convention for the rows (R₁ to R₃) and columns (C₁ to C₄) of wind turbines is also clarified. Table 2 summarizes the key parameters of the PALM simulation setup for the main simulations of the considered wind farm case study with active power control.

Figure 3The layout of the waked 3×4 wind farm model simulated with PALM. R_i and C_j stand for the ith row and the jth column of the simulated wind turbines, respectively.

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A neutral boundary layer (NBL) is simulated with a capping inversion and a mean wind speed of 8 m s⁻¹ at hub height. Under such conditions, large wake losses can be expected during standard operation of the wind farm, i.e., without active power control. A precursor simulation of the atmospheric boundary layer is conducted without any turbines in order to allow for the generation of a fully developed undisturbed turbulent flow field. Then, it is used for the initialization of the main wind farm simulation runs in which a turbulence recycling method is used.

A high turbulence intensity is one of the key drivers for fatigue loading under both ambient and wake conditions (Frandsen, 2007). The longitudinal turbulence intensities at hub height of the first wind turbine are approximately 5 % at the turbine location and 15 % at 5 D distance downstream in the wake center, respectively. These values are calculated after resolving wake structure and its propagation downstream under greedy control as indicators for the assessment of intensified turbulence intensities inside the wake compared with the free-stream flow. Indeed, the wake turbulence is inhomogeneous and varies across and along the wake. Note that the turbulence intensity will indeed be slightly higher, as generally the subgrid-scale turbulent kinetic energy (TKE) will also contribute. Furthermore, the employed ADM in the large-eddy simulation indeed reproduce lower thrust variations due to averaging the local turbulence over the rotor-swept area and especially in the side rows due to wake meandering and partial wake overlaps.

Figure 4Wind farm power (left vertical axis) and deviations of the applied thrusts from their turbine average (right vertical axis) with locally greedy induction factors.

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Figure 5SD of the applied thrust forces on the individual wind turbines operating with the greedy induction factors $a_i=\mathrm{1}/\mathrm{3}$.

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Table 2The key parameters of the PALM simulation setup.

^∗ Sum of the individual wind turbine's power production with the local greedy control setting $a_i=\mathrm{1}/\mathrm{3}$.

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Figure 4 summarizes the total power production and dynamic loading of the considered case study, wherein the wind turbines operate with the locally greedy optimal control settings $a_i=\mathrm{1}/\mathrm{3}$. Note that the wind turbines always operate in the below-rated region, in which the control objective is to maximize energy capture from the incoming wind. The total turbulent wind farm power production and the corresponding time-averaged power have been used as indicators for the wind farm available power. The thrust force which is applied on average on the wind turbines and the root mean square (RMS) of the errors across all wind turbines between the individually applied thrust forces and their turbine-averaged value are employed as indicators for uneven dynamic loading in a waked wind farm. Figure 5 shows the standard deviations (SD) of the time series of the applied thrusts of the individual wind turbines, operating with their locally optimal axial induction factors. Although the downwind turbines are subjected to lower wind velocities compared with the upwind turbines, they experience mainly higher dynamic loading, which is attributed to the intensified turbulence intensity and velocity fluctuations inside the wakes. One important note is that we have focused so far on the dynamic loading caused by wakes and their interactions with the boundary layer. In the following, we demonstrate how the wind turbine control settings, e.g., the axial induction factors, might be used to influence the structural loading of the wind turbines in the presence of wakes with higher turbulence intensity.

3.1 Wind turbine controller design

Each wind turbine has its own feedback controller such that it behaves as a dominant second-order system with a certain frequency ω and a damping ratio ζ to regulate the rate of power production (see Level 1 of Fig. 6). The control objective of the wind turbine control system is defined to locally track the power demand $P_i^{\text{dem}}$, which is commanded by the high-level wind farm controller.

The wind turbine controller is designed based on a linear representation of the aerodynamic power with respect to the control input. Hence, the following mapping is applied to the axial induction factor (Boersma et al., 2016):

One important note is that the power derating control is governed by the induction control Eq. (9). A more comprehensive wind turbine control system would consider the actual torque and pitch controllers. For more details on practical aspects of implementing the torque and pitch control for power derating and their effects on the wind turbine loads (see Aho et al., 2013, 2016).

A proportional-integral-based (PI) control law is defined for APC of the ith wind turbine as

with the local tracking error $e_{i,k}=P_{i,k}^{\text{dem}}-P_{i,k}$ at time instance k. Standard pole placement is utilized for computing the gain-scheduled proportional and integral gains K_P,i and K_I,i as follows.

with the locally rotor-averaged wind velocity $U_{{\mathrm{d}}_i}$ as the scheduling variable. The closed-loop frequency and the damping ratio with the designed wind turbine controllers are set to ω=0.06 Hz and ζ=1, respectively. It is important to note that the gain-scheduling approach is necessary in order to keep the closed-loop response invariant to changes in the aerodynamic control distribution. The axial induction factor is constrained to its value for maximum (greedy) power extraction, i.e., $a_i\le \mathrm{1}/\mathrm{3}$, equivalent to ${\mathit{\beta }}_i\le \mathrm{1}/\mathrm{2}$. Therefore, the PI controller is extended with an antiwindup scheme for resetting the integrator component whenever the control input constraints are activated.

3.2 Wind farm controller design

The wind farm power reference P^ref is distributed among the individual wind turbines on the basis of a power distribution control law, e.g., an open-loop pre-selection (Fleming et al., 2016; van Wingerden et al., 2017) or a closed-loop adjustment (Vali et al., 2018b) of the power set-points. In the current study, we propose an extension to the APC of waked wind farms to actively regulate the distributing set-points, yielding reduced structural loading when the total power production tracks a time-varying power reference, demanded by the transmission system operator (TSO). The proposed control architecture for APC with a CLD is depicted in Fig. 7.

Figure 7Schematic illustration of the proposed closed-loop APC of wind farms. The gray block depicts the main components of the wind farm control (WFC) block in Level 2 of Fig. 6. One feedback loop focuses on APC, while a second feedback loop provides coordinated load distribution (CLD).

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4.1 Active power control of waked wind farms

Figure 9 illustrates the total power productions of the simulated wind farm with the equal power set-points of 1∕12. The demanded power reference by the TSO is perturbed about b=95 % (Fig. 9a) and b=100 % (Fig. 9b) of the time-averaged power production of the locally greedy control. Therefore, high wake-induced power losses of the downstream turbines adversely degrade the performance of the baseline. As demonstrated also in van Wingerden et al. (2017), the feedback controller compensates for the accumulated power losses by demanding more from turbines with higher wind power reserve. The accuracy of the APC approaches is assessed using the RMS of the tracking errors over the illustrated simulation runtime.

Figure 9Total power production of the wind farm with closed-loop APC, compared with the baseline. Both controllers rely on constant power set-points ${\mathit{\alpha }}_i=\mathrm{1}/\mathrm{12}$ for equally distributing the AGC power reference among the wind turbines with indices of $i=\mathrm{1},\mathrm{\dots },\mathrm{12}$. The time-varying power reference parameters Eq. (23) are chosen here as $P_{\text{farm}}^{\text{base}}=\mathrm{12.3}$ MW, b=95 % (a), b=100 % (b), and c=10 %. $P_{\text{farm}}^{\text{base}}$ represents the time-averaged wind farm power with locally greedy induction factors $a_i=\mathrm{1}/\mathrm{3}$ (see Fig. 4).

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Figure 10The axial induction factor trajectories of the individual wind turbines with the baseline and closed-loop APC for the power reference with $P_{\text{farm}}^{\text{base}}=\mathrm{12.3}$ MW, b=95 %, and c=10 %. Both controllers rely on equal power set-points, i.e., ${\mathit{\alpha }}_i=\mathrm{1}/\mathrm{12}$.

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Figure 11The power productions of the individual wind turbines with the baseline and closed-loop APC for the power reference with $P_{\text{farm}}^{\text{base}}=\mathrm{12.3}$ MW, b=95 %, and c=10 %. Both controllers rely on equal power set-points, i.e., ${\mathit{\alpha }}_i=\mathrm{1}/\mathrm{12}$.

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One important note is that the turbulent kinetic energy of the flow plays an important role in the closed-loop APC. For the power reference with b=95 %, the closed-loop APC is able to track a power reference signal 5 % larger than $P_{\text{farm}}^{\text{base}}=\mathrm{12.3}$ MW from 450 to 850 s. However, the total power production suddenly drops due to a turbulence-related strong decrease in the wind speed at about the time instant 880 s. All wind turbines operate with the locally optimal induction factor $a_i=\mathrm{1}/\mathrm{3}$ during this short period of time to capture the kinetic energy from the wind as much as possible. This situation is intensified by demanding the higher power reference with b=100 %, which leads to more frequently switching between the APC and the greedy control. As discussed above, the constraint Eq. (16) has been introduced to distinguish the wind farm tracking errors caused by a temporary lack of available wind farm power from a lack of power due to inefficient wake-induced losses. Indeed, the control signal ΔP^ref adjusts the TSO power reference depending on the controllability of the APC problem. A satisfactory tracking performance is achieved again as soon as the energy content of the wind increases and thus the wind farm available power goes up beyond the TSO power reference.

Figure 10 demonstrates the trajectories of the axial induction factors of the individual wind turbines for following the power reference with b=95 % in the most critical time span up to 1300 s. The closed-loop APC mostly saturates the downwind turbines to the locally optimal operating point $a_i=\mathrm{1}/\mathrm{3}$, which means that the controllability of the problem is limited for high power references. Thus, they are not responsive enough for rejecting the sources of dynamic loading, e.g., a high turbulence intensity inside the wake, through their control inputs. High wake losses are mainly compensated by the upwind turbines (first column) using the total wind farm power feedback. Indeed, high structural loading are expected on the upwind turbines because of their large power variations to compensate for power losses at the waked wind turbines.

Figure 11 plots the corresponding power productions of the individual wind turbines. Both wind farm controllers apply the concept of equal power derating (dashed black curves), which is then adjusted in the closed-loop APC. The upwind turbines 1, 2, and 3 mainly contribute to the wind farm power-tracking performance, as shown in Fig. 9. Moreover, the second row of the wind turbines, containing wind turbines 2, 5, 8, and 11 (see Fig. 3), interact fully with their wakes. Compared with the baseline, higher losses of the fifth turbine are caused by the higher energy extraction of the second turbine. However, the flow is recovered faster with the closed-loop APC when it reaches the eighth wind turbine, which operates at the greedy operating point for both cases (see Fig. 10).

4.2 Active power control with coordinated load distribution law

Applying our new control law, the distributed power set-points are actively adjusted at each time instant to achieve smaller deviations of structural loading of the individual wind turbines from their mean value. In the previous section, it has been discussed that a demand at b=95 % of $P_{\text{farm}}^{\text{base}}=\mathrm{12.3}$ MW limits the flexibility and the APC solution domain relative to a lower demand b. Therefore, we elaborate the performance of APC with CLD with different parameterizations of the power reference Eq. (23) in order to enlarge the domain of APC solution.

Figure 12Total power production of the wind farm with closed-loop APC with coordinated load distribution (CLD). The time-varying power reference parameters Eq. (23) are chosen here as $P_{\text{farm}}^{\text{base}}=\mathrm{12.3}$ MW, b=80 %, and c=10 %. $P_{\text{farm}}^{\text{base}}$ represents the time-averaged wind farm power with locally greedy induction factors $a_i=\mathrm{1}/\mathrm{3}$ (see Fig. 4).

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Figure 12 plots the AGC response and indicates the RMS of the wind farm power-tracking error from 300 to 2100 s for the studied APC approaches, where the demanded power reference from the TSO is perturbed about b=80 % of $P_{\text{farm}}^{\text{base}}=\mathrm{12.3}$ MW with c=10 % of the normalized AGC signal. The tracking error of the baseline (dashed blue curve) is reduced due to the lower wake deficits. However, the wake-induced power losses still degrade the grid stability and this can be addressed using the closed-loop APC (solid red curve). Note that the power set-points are chosen as 1∕12 as before for both cases. The wind farm power-tracking accuracy is further improved when the power set-points are actively adjusted using the proposed CLD control law (dashed green curve).

Figure 13The trajectories of the power set-points with APC with coordinated load distribution (CLD). The baseline and Ref. APC rely on equal power set-points, i.e., ${\mathit{\alpha }}_i=\mathrm{1}/\mathrm{12}$.

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Figure 13 depicts the trajectories of the regulated power set-points. The APC with CLD exploits the flexibility of the wind turbine control inputs in order to find the APC solution that yields a better balance of structural loading for the individual wind turbines. Contrary to the equal set-points of 1∕12, the power set-points of the upstream turbines 1 to 3 are significantly increased because they are operating in the free stream with high power reserves and low turbulence intensity (of approximately 5 %). Therefore, there exists the possibility to increase their contributions toward structural load reduction of the downwind turbines operating inside wakes with higher turbulence intensity (of approximately 15 %). Furthermore, the set-points of the saturated wind turbines are adjusted according to Eq. (22), which causes the wind farm controller to demand these waked wind turbines to produce about their locally available wind power.

Figures 14 and 15 reveal the impact of the CLD law on the wind farm power tracking and the dynamic loading, respectively. The first figure illustrates the control signal ΔP^ref, which actively regulates the wind turbine power demands in order to compensate for the total wake-induced power losses. Contrary to the reference case (red curve), APC with CLD demands significantly smaller corrections via the overall power demand, and the power set-points are adjusted more using the local power and structural load measurements of the individual wind turbines. This addresses the accuracy improvement of the wind farm power tracking, compared with the reference closed-loop APC. Note that the APC and CLD loops are activated at time instants 200 and 300 s, respectively. Each distributing signal α_i,k is initialized with an equal power set-point of 1∕12, as illustrated in Fig. 13.

Figure 14The control signal ΔP^ref for adjustment of the wind turbine power demands.

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Figure 15RMS errors between the applied thrust of the individual wind turbines and their mean value.

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Figure 15 shows the pattern of dynamic loading of the individual wind turbines for the studied APC approaches. At each time instant, the RMS of the errors across all wind turbines between the applied thrust forces and their mean value is calculated. The SD of the applied thrust forces on all wind turbines from 300 to 2100 s is also denoted in the legend for each approach. Although the reference APC improves the accuracy of the tracking performance by almost 74 % (see Fig. 12), the loading pattern remains similar to the baseline (see solid red and dashed blue curves in Fig. 15) due to the same selection of the power set-points. The reader is referred to Vali et al. (2018b) for analyses on the dependency of the thrust distribution patterns on different stationary selections of the power set-points. The increase in the SD of the exerted thrust forces is related to the compensation of wake-induced power losses using feedback. As demonstrated, APC with CLD is capable of regulating the wind turbine power productions with at least the same quality of AGC response of the Ref. APC without CLD. The RMS of the defined thrust-based error over time (see green line in Fig. 15) and the associated SD of the exerted thrust forces (see the legend in Fig. 15) are significantly reduced, meaning that the wind turbines are loaded more evenly.

Figure 16The standard deviation (SD) of the applied thrust forces on the individual wind turbines for the different APC approaches. R and C stand for the row and column of wind turbines, respectively, for our simulated wind farm (see Fig. 3). The power reference Eq. (23) is parameterized as $P_{\text{farm}}^{\text{base}}=\mathrm{12.3}$ MW, b=80 %, and c=10 %. The Ref. APC without CLD (red bars) and the APC with CLD (green bars) lead to almost the same accuracy of the wind farm power tracking (cf. Fig. 12).

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Figure 16 compares the SD of the applied thrust forces on the individual wind turbines from 300 to 2100 s. APC with CLD (green bars) results in a reduction in deviations of the thrust forces of the downwind turbines operating inside the wakes, compared with the baseline and Ref. APC. In total, the aggregated structural loading is remarkably lowered by slight load transfer to the upwind turbines, i.e., the first column (C₁). Such an increase in the dynamic loading of these turbines is not critical since they are operating in a low ambient turbulence anyway.

Figure 17The adjusted power demands (a, b, c) and the corresponding trajectories of the axial induction factors (d, e, f) of the last column (C₄) of turbines in the wind farm. Both APC approaches provide almost the same quality of AGC response (see Fig. 12). The power reference Eq. (23) is parameterized as $P_{\text{farm}}^{\text{base}}=\mathrm{12.3}$ MW, b=80 %, and c=10 %.

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Figure 17 provides a closer view on the trend of the proposed CLD law for the reduction of wake-induced structural loads, e.g., by looking at the performance of the wind turbines 10 to 12 in the last wind farm column, compared with the Ref. APC without CLD. The adjusted power demands with the APC law Eq. (13) are actively derated over time using feedback from the local power and structural load measurements. It should be noted that the wind farm power-tracking performance is kept unchanged through uprating the wind turbines that are subjected to lower external turbulent excitations, particularly the upwind turbines (see Fig. 13). Indeed, limiting the power demands increases the controllability of the waked wind turbines. As shown, the axial induction factors are saturated less often and hence have more freedom (see green curves) to reject the intensified fluctuations due to the turbulence, which is the main source of the wake-induced structural loading on the downstream turbines.

4.3 Performance analyses

This section evaluates the applicability of the proposed APC with CLD from some practical criteria, e.g., the APC accuracy, the extreme loading, and the aggregated fatigue loading. The wind turbine structural fatigue loading is among the key factors in the design of wind turbines (IEC, 2005). The fatigue load analysis is achieved by comparing the load spectra obtained from rainflow counting of the time series of the stresses with a characteristic curve of the design resistance of the component under investigation, the so-called S–N or Wöhler curve. This study only provides a qualitative assessment of the corresponding damage equivalent load (DEL) of the tower base fore–aft bending moment as a descriptor for structural fatigue loading in wind farms. As shown in Sect. 2.2, it is straightforward to extend the ADM in the PALM simulation code with the tower structural dynamics Eq. (2). Other fatigue load sources, e.g., quasi-periodic load disturbances due to wakes partially overlapping the rotor-swept area, will be more representative for a comprehensive fatigue load analysis when they are included inside the simulation model. The corresponding DELs are computed using the 30 min time series (from 300 to 2100 s) of the tower base fore–aft bending moment. The inverse slope of the S–N curve is considered m=4, which is commonly used for steel components like the tower.

Table 4Performance assessment of the studied APC approaches with PALM for different power reference parameterizations. The time-varying power references (Eq. 23) are parameterized as $P_{\text{farm}}^{\text{base}}=\mathrm{12.3}$ MW, b=80 %, 85 %, 90 %, 95 %, respectively, and c=10 %. All quantities are computed for the 30 min time series (from 300 to 2100 s) of the wind farm operation.

^∗ The quantity is calculated using the applied thrust forces or tower base load measurements of all wind turbines.

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Table 4 summarizes the performance results of the studied APC approaches. Each case is evaluated with four AGC-based power references (Eq. 23) with different parameterizations of b in order to investigate the effect of wind power reserves and thus different sizes of the APC solution domains. The accuracy of APC is evaluated first with the RMS of the wind farm power-tracking error. Then, the SD of the applied thrust forces on all wind turbines are outlined as well. The possible impacts on the structural loading of the wind farm are analyzed using the maximum amplitude and the corresponding DELs of the fore–aft tower base bending moment of all 12 wind turbines, which are plotted in Fig. 18 for all case studies. We introduce the mean DEL and the SD of the corresponding DEL as two indicators for analyzing the wind farm fatigue loading as a whole and the variation in fatigue load distributions among the wind turbines, respectively. The local quantity is of special interest since particular turbine situations with higher DELs are not balanced out by other periods with lower DELs due to the strong nonlinearity of the fatigue damage on the stress amplitude.

Figure 18Corresponding DELs of the tower base fore–aft bending moment with the APC approaches for different levels of AGC power demand (Eq. 23), parameterized as $P_{\text{farm}}^{\text{base}}=\mathrm{12.3}$ MW, b=80 %, 85 %, 90 %, 95 %, respectively, and c=10 %. R and C are referring to the turbine positions inside the simulated wind farm (see Fig. 3).

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Figures 19 and 20 provide a comparative insight into the overall performances for achieving the defined control objectives at the wind farm level. The RMS of the wind farm power-tracking error (Fig. 19), the mean DEL (Fig. 20a), and the standard deviation (Fig. 20b) of the corresponding DELs are normalized with respect to the baseline case with the power reference level b=80 %. The bar graphs reveal the relative performance of the studied APC approaches with different levels of the TSO power reference and the corresponding wake effects. In general, higher power demands create stronger wake effects downstream, e.g., larger wind velocity deficits and fluctuations, which lead to accumulated power-tracking errors and higher structural fatigue loading, respectively.

Figure 19Normalized root mean square (RMS) of wind farm power-tracking error for different levels of AGC power reference (Eq. 23), parameterized as $P_{\text{farm}}^{\text{base}}=\mathrm{12.3}$ MW, b=80 %, 85 %, 90 %, 95 %, respectively, and c=10 %. The baseline case with power demand level b=80 % is considered the reference.

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Figure 20Normalized mean (a) and standard deviation (b) of the corresponding DELs of the tower base fore–aft bending moment for different levels of AGC power reference (Eq. 23), parameterized as $P_{\text{farm}}^{\text{base}}=\mathrm{12.3}$ MW, b=80 %, 85 %, 90 %, 95 %, respectively, and c=10 %. The baseline case with power demand level b=80 % is considered the reference.

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