Turbine repositioning sustains a net yaw angle θ misalignment to generate lateral thrust forces, displacing the floating platform to a new equilibrium (Fig. b). The net yaw angle θ quantifies the misalignment between the rotor plane and the incoming wind direction. It is defined as the sum of the nacelle yaw angle γ (°) and the platform yaw angle θz (°), minus the inflow direction ∠v∞ (Fig. ): 3θ=γ+θz-∠v∞. By shifting the turbine laterally to the displaced equilibrium, repositioning can reduce the overlap between upstream wakes and downstream rotors, thereby enhancing energy capture and mitigating wake losses.

Despite its potential, the repositioning capability of each turbine in a farm may be limited depending on wind conditions. For the symmetric three-line mooring system in Fig. a, assume that the wind blows from left to right and the net yaw angle is set to θ=0. The turbine drifts downwind to an equilibrium where the wind-aligned mooring line becomes load-bearing and the other two lines slacken; consequently, crosswind motion encounters relatively little resistance and substantial repositioning becomes possible. By contrast, when the wind blows from right to left (Fig. b), the two right-side mooring lines remain taut and jointly oppose the motion, which increases restoring stiffness and severely restricts lateral displacement, reducing the effectiveness of repositioning. This directional dependence motivates augmenting repositioning with additional strategies, including wake steering, power derating, and dynamic wake mixing.

When platform mobility is limited, the same net yaw angle θ can be used to deflect the wake laterally without displacing the platform (Fig. c). This deliberate misalignment steers the wake away from the downstream turbine, improving its inflow conditions. Power derating constrains turbines to prescribed power caps Pset, thereby reducing wake deficits and alleviating downstream impacts (Fig. d). Dynamic wake mixing (Fig. d) enhances turbulent diffusion to accelerate wake recovery. In this study, wake mixing is realized using FLORIS's Helix mode (periodic individual-blade pitch actuation), with the wake-mixing intensity parameterized by the mean-to-peak excitation amplitude βamp .

While each strategy provides distinct benefits, their combined application enables more flexible and effective farm-wide operation across diverse wind conditions. By integrating turbine repositioning, wake steering, power derating, and dynamic wake mixing within a unified optimization framework, the controller can adaptively select and coordinate strategies based on wind conditions and control objectives. This improves robustness in the practical sense that the controller can still mitigate wakes and meet farm-level goals even when one strategy is less effective. For example, when platform mobility is high, repositioning can be prioritized, whereas in constrained orientations, wake steering and dynamic wake mixing can take over.

The objective function J(⋅) in Eq. () is defined as 5J(U)=Pref-∑i=1NPi2+w∑i=1N‖v∞-vi‖2. This objective function balances competing control objectives through weighted penalties. The first term enforces power-related objectives: for power maximization, the wind farm's reference power Pref can be set to a sufficiently high (possibly unattainable) upper-bound target, such as the sum of rated turbine powers, whereas for power regulation it corresponds to a prescribed farm-level target. Deviations of the actual output ∑i=1NPi from this reference are penalized accordingly. The second term promotes wake mitigation by penalizing discrepancies between the free-stream wind velocity v∞ and the local velocity vi at the rotor of turbine i. Collectively, these terms enable the controller to satisfy power objectives while simultaneously reducing wake interactions. The power-tracking term and the wake-mitigation term in Eq. () have different units and typical magnitudes; therefore, the constant weight w is a key design parameter. It is to be selected via trial and error to balance the relative influence of the two terms.

The turbine powers P, inflow velocities V, and thrust coefficients Ct in Eq. () are predicted using FLORIS , an engineering wake model developed by the US National Renewable Energy Laboratory, which takes as inputs the stacked control vector U, turbine locations X, and free-stream wind condition v∞. FLORIS is computationally efficient, making it well suited for embedding in the proposed optimization-based controller. The wake-model components used in FLORIS, the implementation of the wind farm control methods in the model, and relevant modeling limitations are summarized next.

In this study, wake effects in FLORIS are represented by four submodels, each used with the default parameter values provided by FLORIS :

Velocity deficit model. For the wake velocity deficit, we use the empirical Gaussian wake model, which assumes a Gaussian distribution of the velocity deficit .

The wind farm control methods are implemented in FLORIS through the control inputs U as follows:

Wake steering. Wake steering is implemented by prescribing yaw misalignment commands θi, which modify turbine power/thrust and the resulting wake deflection in a steady-state sense.

It should be noted that FLORIS has not yet been extensively validated for cases in which a turbine is simultaneously yaw-misaligned and operated in either derated mode or active wake-mixing mode. In our optimization framework, we nevertheless allow yaw misalignment to be applied concurrently with either power derating or Helix wake mixing for each turbine. Under such combined operation, FLORIS predicts steady-state changes in turbine power/thrust, wake deflection, and wake recovery; however, the corresponding results should be interpreted with this validation limitation in mind. Moreover, when wake steering and power derating/Helix wake mixing are applied simultaneously, yaw-induced power loss is not captured (as observed in our simulations) in the FLORIS prediction; therefore, to avoid overestimating power capture under yaw misalignment, we apply a yaw-power loss correction for yawed turbines, Pi←Pi(cos⁡θi)pP, with a parameter value pP=1.88 . For yaw-only wake-steering/turbine-repositioning cases, we do not apply this additional correction to avoid double counting.

The optimization problem incorporates both equilibrium and operational constraints in Eqs. () and (). First, each turbine must satisfy static force equilibrium between the aerodynamic thrust and the restoring forces of the mooring system to maintain its steady-state position. The aerodynamic thrust force on turbine i is given by 6Faero,i:=12ρA‖vi‖Ct,ivi, where ρ (kgm-3) is the air density, A (m2) is the rotor swept area, vi (ms-1) is the inflow velocity at the rotor plane, and Ct,i (–) is the thrust coefficient. In addition, all control inputs are constrained by minimum and maximum bounds, umin⁡ and umax⁡, which ensure ui remains within physically realizable and safe operating ranges.

The wake model and the force equilibrium in Eq. () are solved simultaneously using an iterative procedure. Given the free-stream wind velocity v∞ and candidate control inputs U, an initial guess of the turbine positions X is provided to FLORIS to compute the local velocities V, thrust coefficients Ct, and turbine powers P. The aerodynamic thrust forces Faero,i are then computed from V and Ct using Eq. (). Next, for each turbine, the mooring restoring force Fmoor,i required to satisfy the equilibrium constraint Eq. () is determined, and a precomputed look-up table generated from a quasi-static mooring model  is used to update the corresponding turbine positions X. The updated positions are fed back into FLORIS to recompute V, Ct, and P, and the iteration continues until the turbine positions converge. Once convergence is achieved, the converged P and V are used to evaluate the objective function in Eq. (). Overall, the optimization problem formulated in Eq. () is nonlinear and nonconvex because of the complex wake interactions and spatial constraints, which can be solved using a global optimization solver. The next section presents simulation results that demonstrate the performance of the proposed controller under a wide range of wind conditions.

The proposed integrated controller, which combines turbine repositioning, wake steering, and dynamic wake mixing, is benchmarked against (i) a combined repositioning + wake steering case, (ii) wake steering only, (iii) dynamic wake mixing only, and (iv) greedy control. Here, greedy control refers to the baseline operation in which each turbine independently maximizes its own power production without coordination across the farm. A “repositioning-only” case is not considered because, in this study, repositioning is achieved through net yaw offsets that modify the aerodynamic thrust direction to induce crosswind drift; the same yaw offsets inevitably steer the wake, so repositioning cannot be decoupled from wake steering. For the wake-steering-only and wake-mixing-only benchmarks, all turbines are held at their neutral (no-wind) positions. Power derating is excluded from this comparison because FLORIS simulations consistently show that it produces lower total output than greedy control across all wind speeds and inter-turbine spacings.

Figure  shows the maximum achievable farm power for steady inflow wind speeds of 4, 6, 8, and 10 ms-1 as the inflow direction is swept over the full 0–360° range (without wind-rose probability weighting). Wind direction is defined clockwise from the top of the wind farm (0° at the top). All simulations assume an ambient turbulence intensity of 0.06, a power-law vertical shear exponent of 0.12, and zero wind veer. Ocean wave effects are neglected. Each plot compares the different strategies, with the numbers displayed inside each plot indicating the wind farm power in megawatts. Results at higher wind speeds are omitted because even greedy control alone delivers rated farm power despite wake overlaps.

Under greedy control, farm power exhibits a strong dependence on inflow direction, with pronounced deficits at directions that align multiple turbines with the wind and thus intensify wake overlap (seen as deep “notches” in the black curve). In these wake-dominated sectors, the integrated controller yields the largest benefits by filling in the troughs and recovering several megawatts at 10 ms-1, with similarly scaled gains at lower wind speeds. In contrast, for directions associated with weak wake interactions, the strategy curves nearly collapse, indicating that all approaches converge when there is little wake loss to mitigate.

A comparison of the benchmark strategies in Fig.  further highlights the value of coordination. In all four subplots (a–d), the integrated controller (blue line) delivers the highest power across directions by adaptively combining control mechanisms. The wake steering + repositioning case (orange line) generally remains close to the integrated control envelope, indicating that combining crosswind mobility with wake deflection captures much of the attainable improvement for this layout. Wake steering (green line) and dynamic wake mixing (red line) each provide clear improvements over greedy control (black line) while largely preserving the same direction-dependent performance pattern.

To summarize overall performance across directions, Fig.  reports the average maximum farm power over all wind directions for inflow speeds between 4–10 ms-1, expressed as the percentage change relative to greedy control. Several trends are evident. First, all strategies yield positive mean gains, indicating that wake-mitigation actions provide net benefits when averaged over the full 0–360° direction sweep. Second, the fully integrated controller achieves the largest improvement at every wind speed, with the highest gain at 4 ms-1 (up to 6.5%) and gains remaining around 3.9% at 10 ms-1.

At the same time, as discussed in Sect. , the simultaneous use of yaw-based control and Helix wake mixing is more complex than the standalone application of either method, and this combined operation has not been extensively validated in the present FLORIS model. Therefore, the fully integrated case may provide optimistic quantitative estimates of the achievable performance gains. To further assess the main conclusions, Fig.  also includes a restricted integrated-control case that provides a conservative lower-bound benchmark for the fully integrated study. In this restricted case, for each wind direction, all turbines in the farm are constrained to use the same control method, either yaw-based control (turbine repositioning/wake steering) or Helix wake mixing. The chosen method may vary with wind direction, but the two methods are not used together within a given wind-direction case. Even under this restriction, excluding the simultaneous same-turbine use of yaw-based control and Helix wake mixing does not eliminate the benefit of integrated control, although the resulting gains are smaller than those of the fully integrated case. These results indicate that the main qualitative benefit of the proposed integrated framework is preserved.

Among the remaining strategies, wake steering + repositioning provides the next-highest gains, followed by wake steering, and then dynamic wake mixing. The separation between wake steering + repositioning and wake steering alone highlights the additional value of exploiting crosswind mobility to reduce wake overlap beyond what can be achieved through wake deflection alone.

To further illustrate how the integrated approach coordinates different strategies, Fig.  presents two representative power maximization cases at an inflow speed of 10 ms-1. Panels (a)–(c) correspond to wind from the left, and panels (d)–(f) correspond to wind from the lower left. For each inflow direction, panels (a) and (d) show the hub-height wind-speed field under greedy control, and panels (b) and (e) show the corresponding field under integrated control. In the left-inflow case, the farm power increases from 29.8 MW under greedy control to 33.7 MW under integrated control (+13.1 %); in the lower-left case, it increases from 26.0 to 27.6 MW (+6.2 %). Color denotes wind speed (red: near free-stream; white/blue: wake deficits), and the numbers annotated near each turbine report the corresponding local turbine inflow velocity. Panels (c) and (f) summarize the integrated-control solution with per-turbine bar plots of net yaw angle, Helix excitation amplitude, and crosswind displacement. The net yaw angle and Helix excitation amplitude are normalized by their maximum allowable magnitudes (20 and 5°), and the crosswind displacement is normalized by 120 m, chosen as a representative large lateral repositioning. Bars close to zero indicate negligible utilization of that strategy for a given turbine, whereas larger magnitudes indicate greater control effort. For yaw and displacement, bars extending above (below) zero correspond to positive (negative) values, indicating opposite yaw offsets and crosswind shift directions.

In Fig. a–c, where the wind originates from the left, turbines have substantial crosswind mobility. Consequently, turbine repositioning is prioritized to mitigate wake effects and enhance overall farm power. For example, to reduce wake interactions among T1, T3, and T5, T1 applies a negative net yaw angle to shift downward, T3 remains almost at the same position, and T5 applies a positive net yaw angle to shift upward. As a result, all three turbines gain access to near-free-stream wind and generate higher overall power. Similarly, wake interactions between T2 and T7, as well as between T8 and T10, are alleviated by repositioning the turbines in opposite crosswind directions. T9 experiences a mild wake influence; although the annotated inflow speed is 10 ms-1, this value is rounded, and the true inflow is slightly below the free-stream speed. Accordingly, T4 and T9 exhibit small opposite crosswind shifts (as seen in the small orange bars in panel c) to further reduce wake overlap. Meanwhile, T6 is neither waked nor a wake source for downstream turbines and therefore remains unchanged.

When the wind originates from the lower left (Fig. d–f), crosswind mobility is limited, and wake steering together with dynamic wake mixing become the dominant strategies. To mitigate wake interactions among T1, T2, T4, T6, and T8, T1 and T2 rely on a combination of wake steering and dynamic wake mixing, while T4 and T6 employ wake steering alone. T8, being the most downstream turbine, makes only a small adjustment. In addition, T5 applies dynamic wake mixing accompanied by a small yaw offset to enhance the inflow to T10; T10 itself applies no control action. The remaining T3, T7, and T9 are unaffected by wakes and therefore remain unchanged.

Overall, these cases highlight how the integrated controller adaptively coordinates strategies to maximize performance under varying wind conditions.

This case study examines farm-level power regulation with an added wake-overlap penalty, where the integrated controller coordinates turbine repositioning, wake steering, and power derating. The constant weight in Eq. () is set to w=800. Results are reported for a below-rated inflow speed of 10 ms-1 under two representative inflow directions, with corresponding farm-level power references (Pref=32 MW for inflow from the left and Pref=27 MW for inflow from the lower left); see Fig. . In both cases, the controller closely tracks the prescribed reference. The figure format matches the power maximization cases, except that panels (c) and (f) replace the Helix excitation amplitude with the normalized derating level (5MW-Pset,i)/5MW.

For inflow from the left (Fig. a–c), greedy control produces clear wake deficits for T3, T5, T7, and T10, whose annotated inflows drop to 8.1, 8.9, 8.6, and 8.9 ms-1, respectively, while the remaining turbines stay at the 10 ms-1 free stream (panel a). Under integrated control (panel b), these deficits are visibly reduced: T3, T5, and T10 recover to 10 ms-1, and T7 increases to 9.9 ms-1. Panel (c) indicates that wake mitigation is achieved primarily through crosswind repositioning, while farm-level power tracking is enforced through distributed per-turbine derating and net-yaw offsets. Notably, T4, T6, and T9, located in free-stream inflow and with minimal downstream influence, also apply large yaw offsets, reflecting yaw's dual role: it generates lateral repositioning forces and reduces the turbine's own power, thereby decreasing wake overlap while simultaneously contributing to regulation.

For inflow from the lower left (Fig. d–f), the aligned wakes under greedy control produce pronounced deficits at T2, T4, T6, T8, and T10, with annotated inflows of 7.2, 7.3, 7.8, 8, and 8.8 ms-1, respectively (panel d). Under integrated control, the inflow to these waked turbines increases to 8.1, 8, 8.6, 8.5, and 9.2 ms-1 (panel e). Consistent with the small orange bars in panel (f), crosswind repositioning is limited for this inflow direction; consequently, the controller relies on wake steering together with per-turbine derating to simultaneously reduce wake overlap and regulate farm-level power.

As discussed in Sect. , when yaw-based control and power derating are applied simultaneously to the same turbine in the present study, the power of the yawed turbine is evaluated by applying the correction Pi←Pi(cos⁡θi)pP to the derated power, where pP=1.88 . However, this representation remains approximate. Recent studies indicate that, for a yawed turbine, the power depends on the yaw angle, the thrust coefficient, and the control strategy governing the rotor, rather than being described by a single constant cosine-law exponent . Therefore, the combined effect of yaw misalignment and power derating is not fully captured by the present model.

To further assess the impact of this modeling limitation, we considered a restricted benchmark using the mixed operation model in FLORIS, in which each turbine was allowed to use either yaw-based control or power derating, but not both simultaneously. For the two representative cases considered here, the restricted benchmark still achieved the farm-level power-regulation objective while mitigating wake overlap, with results similar to those of the unrestricted integrated case. The detailed results are therefore omitted here for brevity.

Overall, these cases show that the controller maintains reliable farm-level tracking via distributed derating and yaw-induced power reduction while adaptively using wake steering and turbine repositioning to reduce wake overlap.