Typically, an optimization of the control settings for a WPP requires 200–1000 power evaluations for each mean wind speed and direction. To calculate a proper AEP metric, we use 23 speeds and 360 directions; i.e. 1.6–8.2 million flow field computations are needed to optimize the control settings for a given WPP layout. Obviously, this puts excessively high demands on the computational speed of the flow solver.

The linear CFD RANS solver, Fuga (Ott et al., 2011), is extremely fast, has previously compared well with full-scale measurements (Peña et al., 2018; van der Laan et al., 2019), and is thus considered ideal for this task. The governing Navier–Stokes equations, neglecting the Coriolis forcing, are consistently linearized using a formal perturbation expansion and subsequently retaining only the first-order perturbation terms. Thus, mass conservation is identically satisfied; momentum conservation is satisfied to first order; and the resulting WPP fields are divergence free, as they should be for an assumed incompressible flow. The resulting equations are in turn conveniently formulated and solved in a mixed-spectral domain for efficiency reasons. The velocity perturbation around a single WT in the physical domain is derived from Fourier components of the mixed-spectral solution using a fast inverse Fourier integral transform and stored in a system consisting of both general and WT-specific look-up tables, which facilitates the extreme computational speed of the solver. Because of the linearity of the model, wakes from multiple upstream WTs can consistently be superimposed to construct the flow field further downstream. From an efficiency perspective, this is a big advantage.

The WTs are modelled as actuator discs, which in general can be vertically inhomogeneous but are often assumed to be uniform in wake studies. The actuator discs embedded in the flow field represent the rotor drag forces, which in turn are responsible for the creation of rotor downstream wakes. The specifications of the individual actuator discs are based on detailed aerodynamic models of the WPP rotors as accounted for in Sect. 2.2. The WPP wind field, impinging on an arbitrary WT in the WPP, depends on the ambient wind field and wakes from relevant upstream WTs linearly superimposed.

The inflow conditions, i.e. mean wind speed and direction, are assumed to be horizontally homogeneous over the spatial extend of the WPP. More specifically, neutral atmospheric boundary conditions are assumed, meaning that a logarithmic mean wind shear profile applies. The characteristics of the shear profile is thus in turn defined by a terrain roughness length and the friction velocity u*. For neutral atmospheric conditions, Monin–Obukhov scaling dictates the standard deviation of the velocity fluctuations to be invariant through the atmospheric boundary layer and proportional to the friction velocity. The turbulence inflow is thus expressed in terms of the same input parameters as the mean wind shear field. For the Lillgrund site, a roughness length of z0=0.03 m is used, which results in an inflow turbulence intensity of approximately 12 %. This is at the high end for an offshore location and relates to the proximity of the Lillgrund site to urban areas. No attempt was made to link the roughness length to inflow wind speed, because site measurements have shown only marginal variations in the turbulence intensity with wind speeds below the rated wind speed.

For each wind direction, the local wind speed, i.e. ambient wind speed minus the sum of deficits from upstream turbines, the power production, and the thrust coefficient as well as the wake deficits at downstream WT positions are evaluated starting with the most upstream WT position and continuing in the downstream order.

As mentioned, we consider detailed aerodynamic rotor performance expressed in terms of power- and thrust coefficients as fully satisfactory for WT AEP simulations.

Initially, the power and thrust coefficients of the rotor are modelled using HAWCStab2 – a linearized aero-servo-elastic code designed for stability analysis and steady-state simulation of WTs (Hansen et al., 2017). HAWCStab2 relies on an extended formulation of the traditional blade element momentum (BEM) approach (Madsen et al., 2007), and consequently detailed geometric and aerodynamic input is required, e.g. the blade planform and twist distribution as well as blade aerodynamic properties in terms of aerodynamic coefficients over the blade length. In the present application, HAWCStab2 uses a fully flexible WT model formulation to account for the equilibrium-static wind-speed-dependent deflections of the WT main components and thus the potential effects on the WT thrust and power performance.

For traditional layout optimization without WPP control, the WPP production is implicitly based on WTs running at maximum Cp. For the present application, which aims at system-optimal design, the aerodynamic modelling includes a WT derating feature, which links to a unique set of the tip speed ratio and collective pitch angle, (αCt,λCt). Consequently, the aerodynamic WT model must facilitate computation of Ct and Cp conditioned on these design variables. Assuming zero yaw error, the tip speed ratio, λ, is defined as 1λ≡RΩU, where R is the rotor radius, Ω denotes the rotor speed, and U is the hub-height mean wind speed.

The conditional dimensionless rotor thrust and power coefficients are defined as 2Ct(U|α,λ)≡TWT(U|α,λ)12ρAU2 and 3Cp(U|α,λ)≡PWT(U|α,λ)12ρAU3, respectively, where TWT is the rotor thrust force; PWT is WT power production; ρ is the air density; and A is the rotor area, which depends on both the rotor tilt (θt) and the blade-coning (θc) angles as 4A=π(Rcos⁡θccos⁡θt)2. In this context, PWT and TWT are obtained from HAWCStab2 simulations of the Siemens SWT-2.3-93 WT, which operates at the Lillgrund WPP; see Sect. 4. The steady-state power and thrust have been simulated for a range of collective pitch and rotor speed settings in a uniform flow field of 8 m s-1. In principle such steady-state parameter sweep simulations must be performed for all relevant mean wind speeds to account for the steady-state blade deflections. However, assuming that these deflections have only a minor effect on the steady-state power and thrust performance of the WT in question, then one mean wind speed suffices. This is justified under the assumption that the thrust scales with U2 and the thrust coefficient is normalized with U2, whereas the power scales with U3 and the power coefficient is normalized with U3.

Note from Eq. (1) that for a fixed wind speed, a variation in rotor speed corresponds to a variation in λ. Thus, from the above-described simulation outputs, the power and thrust coefficients are easily calculated as a function of the tip speed ratio and the collective pitch via Eqs. (1)–(4); see Fig. 1.

The results shown in Fig. 1 can be used for the entire range of mean wind speeds requested for the system optimization; see Eq. (1). This is convenient from a computational point of view and thus consolidates the status of the tip speed ratio as a design variable as an appropriate choice.

Another important computational simplification is, as previously mentioned, the reduction from two control design variables per WT to one control design variable per WT. This reduction is based on the previously mentioned findings by Vitulli et al. (2019) showing that optimal derating is obtained by selecting the unique set of design variables (αCt,λCt), which, for a given derating (i.e. power production reduction), corresponds to the smallest possible thrust. This condition, which is also intuitively clear, provides a unique relationship between αCt and λCt and justifies the reduction in design space to one control variable per WT, conditioned on ambient mean wind direction and mean wind speed.

As a consequence of the control design space collapse, a specific derating factor corresponds to a deterministic path through the original (α, λ) design space, where the points on this path correspond to certain mean wind speeds. Note that these paths are constrained by the minimum and maximum rotor speed limits as well as the maximum power limit; see Fig. 2.

The last step needed to prepare for an efficient optimization procedure is to transform the above-described aerodynamic rotor computations into a surrogate model, which maps mean hub wind speed and the requested derating factor into a power production coefficient and a thrust coefficient conditioned on the operational settings, i.e. Cp(U|α,λ) and Ct(U|α,λ). The surrogate thereby establishes the link between the derating settings, to be specified by the control optimizer, and the characteristics of the uniformly loaded actuator discs needed by the flow solver. Note that controller-specific constraints such as the tower exclusion zone and smooth transition between regions as well as controller implementation issues are not taken into account. At present, it has not been found essential to model the actuator discs as vertically inhomogeneous, although this is possible within the framework.

The objective function defined for the present optimization platform is the AEP of the WPP. Financial costs of the internal WPP power grid, access roads, foundation, etc. are not considered, which in turn means that the positions of the individual WPP WTs are only constrained by the minimum allowable distance to the nearest neighbouring WT and the line of demarcation defining the permissible WPP area. Considering two rotor diameters (2 D) to be the minimum realistic WT interspacing distance, this minimum spacing constraint has been selected for all show cases presented in this paper. The permissible WPP area for the Lillgrund case is the stylized convex shape of the Lillgrund reef. Finally, we have incorporated two additional constraints associated with the operational conditions of the Lillgrund Siemens SWT-2.3-93 WT used in all cases: Ωϵ [9 rpm, 16 rpm] and αϵ [-2∘, 90∘].

In each iteration of the optimization procedure, the objective function – in this case the AEP performance metric – must be computed. Computational efficiency is in particular needed for the present CFD-based approach, and maximum efficiency is assured through implementation of the “shortcuts” described in Sect. 2.2.

For a given layout (i.e. associated with a given iterative step in the layout optimization process), the WPP AEP, PAEP, is estimated from 5PAEP=T∑i=1N∫0∘360∘∫UinUoutPi(U|θ)fU(U|θ)fθ(θ)dUdθ, in which U denotes the undisturbed ambient hub-height mean wind speed and Pi (U|θ) is the production (in watts) of the ith WT at ambient hub-height mean wind speed, U, and associated operating conditions dictated by the internal WPP flow field. fU (U|θ) is the ambient hub-height mean wind speed pdf, conditioned on the ambient mean wind direction (i.e. often a two-parameter Weibull distribution), and fθ (θ) is the ambient mean wind direction pdf. Assuming SI units, T is the number of seconds corresponding to 1 year and N is the predefined number of WTs within the WPP considered.

In practice, Eq. (5) is discretized to facilitate evaluation of the involved integrals. In the succeeding case studies, a directional discretization of 1∘ was used combined with an ambient mean wind speed discretization of 1 m s-1.

Overall, there are three common ways to design the WPP system optimization. The most elaborate of these is to design the fully integrated approach by involving all design variables simultaneously – the one-step approach. The layout-optimization-related design variables amount to two (i.e. the WT position in a Cartesian coordinate system) per WT. The WPP control optimization, conditioned on ambient mean wind direction and mean wind speed, requires, utilizing the design space collapse described in Sect. 2.2, one design variable per WT. However, because the AEP computation requires all wind directions and all wind speeds to be accounted for, the control-related design variables amount to NdNs per WT. Here Nd is the number of ambient inflow directions and Ns is the number of ambient mean wind speeds considered in the discrete version of Eq. (5). Thus, in total the number of design variables amounts to N(2+NdNs). This is clearly infeasible within the present framework – even when utilizing a high-performance computing cluster.

An alternative and more efficient strategy for a fully integrated system optimization is a two-step nested approach, in which, for each optimization step, first the layout is advanced and then, based on this iteration of the layout, the associated optimal control schedule, conditioned on ambient mean wind speed and direction, is determined. Merging the sequentially determined WPP layout and associated optimal control schedule, the AEP estimate, associated with the actual iterative step, can be evaluated. The associated workflow is illustrated in Fig. 3.

Both the one-step optimization strategy and the two-step nested optimization strategy are fully integrated strategies, which eventually will lead to the same result.

If the optimal system design is separable, in the sense that only a weak coupling exists between the layout and the WPP control optimization, the problem can be significantly simplified. This will be quantified in Sects. 3.3 and 4.1 using two demonstration cases. The significant reduction in computational complexity is obtained taking a two-step sequential approach by approximating a weak system coupling with no system coupling. The sequential workflow, in which the conventional “greedy” individual WT control settings are used for the WPP layout optimization, is succeeded by an optimization of the WPP control scheduling conditioned on both ambient mean wind speed and direction. Thereby, the greedy WT control settings are replaced by optimized “collaborative” WT settings to the benefit of the WPP AEP. The workflow associated with this sequential strategy is shown in Fig. 4.

The merger of these two optimization steps makes up the optimized system design and is in essence a sequential application of the TopFarm2 (Pedersen et al., 2019a) layout platform and the open-loop WPP control scheduling platform described by Vitulli et al. (2019).

The layout is optimized using a combination of random-search and gradient-based (SLSQP – sequential least-squares programming) optimization. The random-search algorithm, described by Feng and Shen (2015), does not get stuck at local optima and is consequently suitable to finding a good global solution, while the gradient-based optimizer, applied subsequently, is used to trim the random-search solution to the nearest optima. In this setup, the gradients are approximated by a finite-difference approach. For a complex non-convex optimization problem, a global optimum cannot theoretically be ensured, but running numerous random sequences converging to almost identical results gives confidence in the result being close to the global optimum.

The WPP control scheduling optimization problem has in general only a few local optima and can therefore easily be solved by the gradient-based optimizer using gradients computed via finite difference. This control optimization is, however, rather time-consuming, as the WT control settings must be optimized for all 360×23 combinations of wind directions and wind speeds; see Table 1. These combinations are, fortunately, independent, and the workflow is therefore suitable for parallel computation. For the current study, a parallel workflow utilizing 360 CPUs (i.e. corresponding to a 1∘ mean wind direction resolution) has been set up, where each CPU optimizes all WT control settings for one wind direction. Table 1 gives an idea of the computational resources needed for the case studies described in Sects. 3 and 4.

The sanity check of the control optimization is performed on a simple example consisting of three WTs in a row, separated by 4 D and with a uniform inflow of 10 m s-1 aligned with the row; see Fig. 5.

Figure 6 shows the power produced by the three WTs as a function of the derating of the two upstream WTs. In panel a, it is seen that the power for the most upstream WT, WT1, only depends on its own derating setting. The power of WT2, on the other hand, depends on the derating of both itself and of WT1 (panel b). Finally, it is seen that WT3, obviously, produces the most if both WT1 and WT2 are derated 100 % (panel c).

The total power produced by the three WTs is seen in Fig. 7, and it appears that the total power can be increased by 4.01 % if WT1 is derated by 7 % and WT2 is derated by 5 %.

A sanity check of the layout optimizer is also performed on the three-WT row. In this case, the position of WT2 is allowed to vary between 2 and 6 D behind WT1. Figure 8 shows the individual relative power production of the three WTs as well as the total power production as a function of the position of WT2 in a uniform flow of 10 m s-1 aligned with the row. As expected, WT1 is unaffected by the position of WT2, while the power production of WT2 increases with the distance to WT1 and vice versa for the power of WT3. Finally, the total power production is seen to increase slightly when WT2 is moved downstream.

For other wind speeds, however, the picture is quite different, as seen in Fig. 9. The optimal position thereby depends on the wind speed distribution, which links to the dependence of Cp and Ct on wind speed with the hub-height mean wind speed. Plotting the relative AEP computed using the Weibull distribution associated with westerly winds at the Lillgrund wind farm (c.f. the wind rose shown in Fig. 12) reveals that the optimal spacing, under these conditions, is very close to 4 D.

The performance of the integrated layout and control optimization is illustrated in Fig. 10. The blue line indicates the relative AEP of the three WTs as a function of the position of WT2 in the case that all WTs are operated with greedy settings (i.e. no derating). This is the base case. The optimal position of WT2 is found to be 3.96 D downstream of WT1. Applying layout-dependent optimal derating of WT1 and WT2 (orange curve) sequentially increases the AEP of the initial layout by 2.221 %. Finally, the AEP is seen to increase only infinitesimally (i.e. increasing from +2.221 % to +2.226 %) when applying integrated two-step nested system optimization. For the investigated simplistic case, this result indicates a very weak system coupling between WPP layout and control optimization.

This case study basically consists of one of the three Lillgrund WWP rows with eight WTs, meaning that the WT interspacing in the base case is 3.3 D (see Fig. 11) and that the WTs are Siemens SWT-2.3-93. The wind climate is fictitious, as only an along-row inflow direction is considered, which ensures the largest possible mutual WT wake interactions. Within this framework we have, without loss of generality, assumed Weibull-distributed mean wind speeds corresponding to the 270∘ site condition (truncated, however, to the relevant wind speed regime [3 m s-1, 25 m s-1]) although the “true” inflow direction associated with this row is 300∘.

With the purpose of investigating the strength of the system coupling, we have optimized the following: (1) the WPP layout, (2) the WPP control, (3) the integrated WPP layout and WPP control based on the two-step sequential approach (see Sect. 2.4), and (4) the integrated WPP layout and WPP control based on the two-step nested approach (see Sect. 2.4). Based on a pre-investigation of optimizers, where the random-search approach was compared to the SLSQP gradient-based optimization algorithm, the latter was found to be clearly superior and was consequently used in this study.

The results of the investigation are, together with the base case 0, summarized in Table 2.

The base case, case 0, represents the existing layout with the conventional greedy control of the individual WTs. Case 1 represents the base case layout with the WPP control optimized. The associated increase in AEP, relative to the base case, is significant and amounts to 8.0 %. The close spacing in the Lillgrund WPP case (3.3 D) is comparable with the WT interspacing (2.3–3.1 D) in the Goole Fields WPP investigated in van der Hoek et al. (2019), where an increase of 5.6 % for a row of five WTs was predicted and an increase of 3.3 % was realized in the accompanying full-scale study. As noted in the discussion of the results in this paper, the k-ε turbulence closure of the CFD model, which was used for the tuning of the derating settings, supposedly makes the CFD predictions underestimate wake effects for closely spaced WTs, whereby the used pitch settings are likely to be suboptimal. Furthermore, only the first WT in the investigated row was derated, their two-step pitch-offset derating strategy was suboptimal, and finally the derating potential increases with the number of WTs. This, paired with the fact that full-scale studies will always suffer from imperfect inflow (due to, for example, wind direction variability within the 10 min recording sequences) and WT operational conditions (such as moderate yaw errors), makes us believe that the case 1 result is fairly consistent with the results presented in van der Hoek et al. (2019). In case 2 the WT applies the greedy control, and the WT row layout is optimized. The increase in AEP, relative to the base case, amounts to 1.4 %, which is considerably less than achieved in case 1. Compared to the three-WT case in Fig. 10, the AEP increase achieved by layout optimization in this case is much more pronounced because the number of design variables has increased from one to six. It is seen that the distances between the two most upstream and the three most downstream WTs are smaller than in the base case. This allows larger spacing and thereby production of the middle turbines, which, in this case, results in an increase in the AEP of the whole row. Obviously, this strategy is not possible with only three WTs. Case 3 represents one of two system optimization approaches. Here we assume that the system optimization is separable and consequently can be performed by first optimizing the layout and subsequently the WPP control. The combined effect is an increase in AEP amounting to 9.1 %, which is significant and exceeds what was obtained by only optimizing the WPP control (i.e. case 1). In the second and last system optimization strategy, case 4, the integrated two-step nested approach is taken. Although the approach is more complex and time-consuming (around 540 times slower) than the case 3 strategy, the outcome is not significantly improved (see Table 2).

In conclusion, we have shown that the strength of the system coupling between WPP layout and WPP control optimization is only marginal for the considered eight-WT case study characterized by “heavy” mutual WT wake interactions.

This case study comprises the entire Lillgrund WPP, and it ultimately aims to quantify the potential of an integrated system optimization of WPP layout and WPP control.

In analogy with Sect. 4.1, we will investigate a variety of WPP layout and WPP control optimization strategies. The control optimization schedule and the associated results appear in Table 3.

The investigated cases are analogous to the cases investigated for the eight-WT case in Sect. 4.1. As for case 1 we see a considerably drop in performance increase compared to the eight-WT situation, which is due to the persistently more severe mutual WT wake interactions in the fictitious eight-WT situation compared to the full Lillgrund WPP, where WT wake interactions for some inflow directions are limited (see Fig. 14). With less wake interaction, less potential for WPP control follows intuitively. Case 2 represents an isolated WPP layout optimization retaining the greedy individual WT control performance. The associated increase in AEP performance amounts to 2.8 % – or more than double that of the WPP control optimization, case 1. The last case, case 3, represents a system optimization approach. Based on the investigations performed in both Sects. 3.3 and 4.1, we assume that the system optimization is separable in the sense described in Sect. 4.1. The rationale justifying this assumption is that the system coupling between WPP layout and WPP control optimization was shown to be marginal in the eight-WT case, in which the overall WT wake interaction, over all inflow directions, is significantly more pronounced than for the full Lillgrund case. Taking the sequential approach, the combined Lillgrund WPP optimization results in an AEP improvement of 4.0 %, which is significantly more than each of the individual layout and WPP control optimization approaches. Finally, it should be noted that, although possible, the two-step nested approach will require horrendous CPU resources and even on a cluster take on the order of a few months to conduct.

The layout resulting from the Lillgrund WPP system optimization is shown in Fig. 13 together with the baseline layout.

From a pure production perspective, it makes sense to locate WTs densely at the boundary of the “admitted area” for the WPP, because it intuitively will reduce the WT wake interactions. Notable is also that the individual WT deratings for the shown example, except for one row, are considerably less than for the baseline case.

The results for all the investigated optimization strategies are summarized in Figs. 14 and 15. Figure 14 shows the increase in AEP conditioned on the inflow mean wind direction. As expected, the AEP gains vary with the wind direction with huge increases, up to 50 %, for the optimized layout for the wind directions that are parallel to the rows of the original layout, i.e. 120∘ / 300∘, 42∘ / 222∘, and 0∘ / 180∘. These increases, however, are almost balanced out by the decrease in other directions resulting in the average increases of the 2.8 % and 4 % that are reported in Table 3.

In Fig. 15, the AEP gains are shown as a function of the mean inflow wind speed. The largest increases are seen below 10–11 m s-1 where all WTs operate below rated power. At higher wind speeds, the WPP production wake losses decrease, as more and more WTs reach rated power, thus eventually completely eliminating any WPP control potential.