A passively self-adjusting floating wind farm layout to increase the energy production: a sensitivity analysis

Offshore wind farm layout design techniques and methodologies currently focus on fixed bottom offshore wind turbines. As more floating wind farm (FWFs) are planned, new methodologies for FWF layout design and optimization are required to consider the different attributes between fixed bottom and floating offshore wind turbines (FOWTs). One main difference is the ability of FOWTs to move in the horizontal plane (surge and sway motions). In our work in [1], we showed that the motions of the FOWTs in the horizontal plane represent an opportunity which can be used to increase the FWF’s annual energy production (AEP). We can passively control the motions of the FOWTs in a FWF according to the incoming wind direction, moving the downwind turbines out of the wake of upwind ones. Since the horizontal motions of a FOWT are governed by the mooring system design (MSD) attached to it, this passive control can be done by designing a customized mooring system (MS) for every FOWT in the farm. In this work we build on our work in [1], as we carry out a sensitivity analysis to check the effect of the farm size and the wind rose on the methodology we introduced. The results show that the percentage increase of energy production due to passively relocating the FOWT is sensitive to the FWF size. Very small FWFs will gain less energy by relocating the FOWTs. Moreover, we show that even with a more uni-directional wind rose relocating the FOWTs in a FWF remains beneficial.


Introduction
Wake losses occur in a wind farm as the upwind turbines extract energy out of the free flow, decreasing the velocity of the downwind flow and increasing its turbulence.The bigger the wind farm the higher the wake losses as the flow loses more of its energy as it travels through the farm.To mitigate the wake losses in a fixed bottom wind farm, we optimize the wind farm layout to locate the turbines out of the wakes of each other as much as possible.Moreover, wake deflection can be used to decrease wake interactions and the wake losses for the downwind turbines.However, these techniques do not account for the differences between the conventional fixed bottom wind farms and FWFs.One of the main differences comes from the ability of a FOWT to move in the horizontal plane (surge and sway).
On our quest to reach a carbon neutral energy transition, we need to build FWFs at windy sites in deep waters.Moreover, we will need to cluster the FOWTs in bigger wind farms to benefit from these attractive sites as much as possible.Therefore, we need to come up with new methods for FWFs layout design and optimization to decrease wake losses.One method to decrease wake losses in FWFs is to take advantage of the ability of FOWTs to move in the horizontal plane and relocate the downwind turbine out of the wake of the upwind ones.Our work in [1] investigated this idea and introduced a new method for FWFs layout design.The work showed that passively relocating the FOWTs according to the inflow direction, increases the FWF's energy production.Since the horizontal motion of a FOWT is governed by the MS attached to it, as the MS allows the FOWT to draft freely only within a defined area called the watch circle.Therefore, we designed a customised MSD for each of the FOWTs inside the farm to passively relocate the FOWTs with an objective to increase the farm's energy production.The FOWTs in the FWF will relocate their position only due to the MSD, there is no active control involved in this method.Hence in this method, only the mooring lines forces are responsible of relocating the wind turbines.
The mooring lines are responsible for the station keeping of FOWTs, as they provide stiffness in the surge, sway and yaw degrees of freedom (DoFs).In the absence of a MS, FOWTs will float freely following the wind direction.The mooring lines tensions counteract the horizontal forces due to the aerodynamic thrust forces and wave forces.Moreover, the tensions in the lines also counteract the yaw moment from the rotor aerodynamics.When the wind and wave forces are applied on a FOWT, it will displace in the surge, and sway DoFs until equilibrium is achieved between all the forces acting on the FOWT in the horizontal direction and the horizontal components of the mooring lines tensions.The tension in the mooring lines depends on the design parameters of each line.Changing the design parameters leads to a different line tension and hence a different displacement of the FOWT.Therefore, by attaching a customised MSD to each FOWT in the FWF we can regulate the FOWTs displacements within the FWF and decrease the wake losses.
In this work, we perform a sensitivity analysis for the method we previously presented in [1].We want to test the performance of the methodology for different farm sizes and different wind roses.Our goal is to answer two questions.First, what is the effect of the farm size on the energy gain due to relocating the FOWTs?Second, what is the effect of the wind rose on relocating the FOWT?Do we still gain energy through relocating the FOWTs for more uni-directional wind roses?Answering these questions will give us a better understanding of the potential of relocating the FOWTs in a FWF, and understand the limitations of the methodology.
The paper is organised as follows, we start with a quick introduction for the methodology.Then we show the layouts and wind roses we used in the sensitivity analysis.Afterwards, we will go through each step of the method and show the results at each step of the process.Finally, we summarise our results and last but not least we present our conclusions from this study.

FWF layout design methodology
The novel FWF layout design method is summarised in Figure 1.The method is composed of five main steps.First, we optimise the baseline layout using the existing state of the art layout optimization methods.Afterwards, we optimize the FWF layout for each wind direction separately, which means we will have a different layout for each wind direction.From this step we can save the optimal displacement done by each FOWT in the FWF to increase the farm's energy production.In parallel, steps three and four are carried on to create a MS database.The database will save the watch circle of each MSD, which will show us the displacement a FOWT will make in every wind direction when attached to each MS in the database.Finally, in the fifth step we find a MS from the database that can match the required motions of the floater we obtained in the second step.The method represents a first step towards integrating the MSD as part of the FWF layout design.The results we obtain are not meant to present the optimum solution for the FWF layout.We will go through the details of each of these steps, in each of the following sections and will show the results obtained at each step.

Sensitivity analysis parameters
In this section we will present the two input parameters for the sensitivity analysis, which is the main goal of this paper.First, Figures 2 and 3 introduce the wind roses used in the study.Figure 2, shows the wind rose used within the IEA task 37 [2].The wind rose is multi-directional as only the west to east wind direction is slightly more dominant than the south east to north west wind direction.Figure 3, shows the alpha ventus wind rose [3] which is more uni-directional with the wind mostly blowing from southwest to northwest sector.Throughout the paper for both wind roses we assume the mean wind speed is constant and equal to 10m/s.This assumption makes it easier to compare the effect of the wind directionality on our method without including the effects of the Weibull distribution of the wind speed at each wind direction.The value 10m/s is chosen as it is very close to the rated wind speed of the IEA 15 MW reference wind turbine [4] that we will use in this analysis.Since we are trying to decrease the wake losses a value just below the rated wind speed is needed, because the wake losses above the rated wind speed do not decrease the power production of the FOWTs.For the floating platform, the Activefloat semi-submersible floater will be coupled to the IEA 15 MW reference model [5].
The second parameter in the sensitivity analysis is the wind farm size.We created four baseline layouts with two different boundary shapes and sizes as shown in Figure 4. We have two layouts with a circular boundary shape with 7 and 37 turbines, with energy densities of 23W/m 2 and 14W/m 2 respectively.Moreover, we have two layouts with square shaped boundaries with 9 and 25 turbines with energy densities of 23W/m 2 and 16W/m 2 respectively.The energy densities of the two small and the two large FWF lay within the same range.The two boundary shapes show how each boundary shape affects the performance of the method we are presenting for FWF layout design.Through the analysis we want to know if it is beneficial to relocate the turbines in small FWFs of few turbines, or in bigger FWFs.In the following steps, we will apply our method to each layout with each wind rose.Each of the following sections will describe one of the steps shown in Figure 1.Then we will show the energy gain due to relocating the wind

Conventional fixed bottom wind farm layout optimization
The first step in our method is to optimize the baseline layout in a conventional layout optimization process.This is not needed if the layout was optimized beforehand.This step is important to be able to quantify that the energy gain we will calculate later on comes solely from passively relocating the turbines and not because the starting layout is so far from the global optimum.Our optimization objective is to increase the farm's energy production.We have two optimization constraints, first the turbines must be within the farm's boundaries.Second, the Figure 5. Conventionally optimized bottom fixed wind farm layouts for the IEA wind rose minimum distance between two turbines cannot be less than 2D, where D is the turbine's rotor diameter.The results of the optimized layouts with the IEA wind rose can be seen in Figure 5, while the optimized layouts with the alpha ventus wind rose can be seen in Figure 6.These optimized wind farm layouts (OWFL), are the baseline layouts for the rest of the study.The energy gain will be calculated relative to the energy produced by these OWFLs and not to the initial baseline layout designs.
The optimization was carried on with a gradient based optimizer called SNOPT [6], which showed good results for layout optimization in [2].We used FLORIS v3 [7] to provide the Gaussian wake model that we used for the velocity deficit calculation.The ambient turbulence intensity used was 6%.Table 1 shows the wake losses of each OWFL and each wind rose.After the conventional fixed bottom layout optimization, the wake losses are smaller for smaller wind farms compared to the losses in larger wind farms.This is true for both wind roses.This illustrates that for a smaller wind farm it is easier for the optimization algorithm to find a layout with low wake interactions and hence small losses.

Wind farm layout optimization separately for each wind direction
The next step is to optimize the OWFL layout separately for each wind direction to increase the energy production of the farm for each wind direction.For this optimization problem, we had two constraints.First, our turbines can only move perpendicular to the wind direction.Second, the maximum displacement a turbine can make is 0.5D.The maximum displacement cannot be more than 0.5D because of the minimum distance we used to obtain the OWFL in section 4. We chose the minimum distance between two FOWTs to be 2D, therefore if two FOWTs beside each other displace more than 0.5D this will lead to them colliding.The reasoning for choosing these two constraints is explained further in details in [1].This means that for every wind direction our optimization gives a slightly different layout.The layout we obtained for every wind direction gives the maximum energy our wind farm can produce if we are allowed to relocate the FOWTs a distance up to 0.5D in the crosswind direction.Since this step represents the targeted goal we want to achieve by relocating the wind turbines we call these layouts targeted layouts.An example of the targeted layout at one wind direction is shown in Figure 7.The figure shows how relocating the turbines decreases the wake losses within a FWF.The figure only represents the solution for one wind direction, and this step is repeated for every wind direction in the wind rose.The gain we can achieve if we can fulfil the targeted layouts for the IEA wind rose as well as the alpha ventus wind rose is shown in Table 2.The gains shown in Table 2 are calculated assuming the wind speed is constant for all wind direction and equal to 10m/s.These targeted layouts energy gain is the maximum gain we can achieve by relocating the FOWTs in each of the OWFLs.
Comparing the wake losses in Table 1 to the energy gain in Table 2 shows that for smaller wind farms, with 7 and 9 turbines, achieving the targeted layouts will make the wake losses close to zero.While for bigger farms even if we achieve the targeted layouts we can only decrease the wake losses.This is clear as for smaller wind farms relocating the wind turbines is easier as Figure 7. On the left the OWFL and on the right the targeted layout for the 9 turbines layout with the IEA wind rose for southeast to northwest wind direction the wake interaction happens between two or three turbines maximum, while for a bigger wind farm the wake interactions will happen between more turbines.
Analysing the effect of the wind rose on relocating the FOWTs we can see that the smaller layouts are performing better for the IEA wind rose, while the bigger layouts are performing better for the alpha ventus wind rose.In Table 1 the OWFLs of the smaller wind farms have smaller losses for the alpha ventus wind rose which is mostly uni-directional, while they have higher wake losses for a multi-directional wind rose like the IEA wind rose.This is reflected directly on the targeted maximum gain we can achieve by relocating the wind turbines, because the goal of relocating the wind turbine is to avoid the wake that the conventional fixed bottom layout optimization cannot achieve.The bigger the wake losses in the OWFL the higher the potential of relocating the FOWTs in a FWF.
Realistically, we cannot relocate our turbines to achieve the targeted layout for each wind direction.The reason is that these targeted layouts are created assuming each FOWT in the FWF can move freely for each wind direction.In reality, this is not the case as the movement of a FOWT is governed by the MSD attached to it and can only move within its watch circle.Therefore, we can achieve the targeted layouts for some wind directions but not all wind directions.In the following steps, our goal is to find a MSD for each FOWT in the farm to increase the overall energy production and reach as close as possible to the targeted energy gain value.

Mooring systems database
This section presents steps three and four from Figure 1.In these two steps we define the design space for our MSDs and create a database to store watch circles.To create the database we will follow the same method we presented in [8], where our design space will be divided into fixed design parameters and variable design parameters.The fixed parameters include: • The number of mooring lines is fixed to three lines in each MS design.
• The water depth is fixed to 200m.
• All mooring lines are made of steel chains.
While the variable parameters include: • The mooring lines diameter with values of 0.06m, and 0.12m.
• The lines headings with every possible line headings combination with a step of 10 • .
• The anchor radii with values 3D, 4D, and 5D.The line lengths were defined as functions of the anchor radii following the same approach as in [8], and for each anchor radius three line lengths were assigned.
Using these parameters to create the MS database led to a total of 419904 MSDs, but only 6309 MSDs were saved in the database.The reason is that for any MSD to be saved in the database it needed to fulfill the following criteria: • No vertical forces on the anchors were allowed.
• The maximum horizontal displacement in the wind direction is 240m.
• The maximum accepted platform yaw angle is 10 To create the database we used MoorPy [9] which is a quasi-static tool capable of calculating the mooring lines forces as well as the platform's position.The inputs to MoorPy were the hydrodynamic property of the Activefloat platform, and a force vector representing the aerodynamic thrust of the IEA 15 MW wind turbine at wind speed of 10m/s.

Customised MS design for each FOWT in the FWF
After obtaining the targeted layouts and creating the MSD database, our goal is to find a MSD from the database with a watch circle as close as possible to the targeted layouts displacements of each FOWT to increase the FWF's energy production.There is no MSD that can achieve the same displacements our targeted designs require for all wind directions.Therefore, for some wind directions we achieve the targeted energy gain when the MSD's watch circle displacement is equal to the targeted displacements.On the other hand, we will lose energy in other wind directions where the MS' watch circle displacement is not equal to the value of the targeted displacement.Hence the goal here is to find a MSD for each FOWT which leads to higher energy gains than the energy losses.We achieve this goal through two steps.First, we narrow down the design space of MSDs further.We use the method we introduced in [1] to narrow down the design space from the 6309 MSDs in the database to a few hundreds designs for each FOWT.The details of how we do this are explained in [1] and are out of the scope of this paper.The second step is that we use brute force optimization to iterate over these MSDs until the energy gain of the entire FWF converges.Once the energy of the FWF converges to a fixed value, the iteration stops.At the end of these two steps, we have a customised MSD for each FOWT in our FWF.

Results
After choosing a customised MSD for each of FOWTs in the OWFLS, we calculated the energy production of each layout with each wind rose assuming a constant wind speed of 10m/s.The energy gain of the layouts with the customised MSD attached to each FOWT in each FWF is shown in Table 3.We can see that passively relocating the FOWTs leads to energy gain for all cases, but the values are different according to the farm size and the wind rose.In order to analyse the results, we compared the final energy gain percentages to the targeted energy gain percentages of each FWF size with both wind roses.From Table 2 we expect, that the final energy gain percentage of the FWF with 9 FOWTs to be the highest followed by the energy gain percentages of the FWFs with 7, 25, and 37 FOWTs respectively.However, the final energy gain percentages do not follow this order.As expected, the layout with 9 FOWTs has the highest energy gain for both wind roses.On the other hand, the final energy gain of FWF with the 7 FOWT is lower than what we expected from its targeted layout energy gain for both wind roses.The energy gain of the FWF with 25 FOWTs is as expected for both wind roses when compared with the targeted energy gains.Finally, the percentage of the energy gain of the FWF with the 37 FOWTs is the lowest as expected.In order to understand why the final gain percentages differ from the targeted gain percentages, we show the final energy gain as a percentage of the targeted energy gain in Table 4.While analysing this comparison we have to keep in mind that the targeted gains are the maximum gains we can achieve by relocating the FOWTs.While the final gain is what our FOWTs can realistically achieve when attached to the customised MS following the method we presented in the paper.Therefore we do not expect the final gain to achieve the targeted gain, our goal is to be as close as possible to the targeted gain.Table 4 shows that for five FWF layouts and wind rose combinations the final layout achieves more than 60% of the maximum energy gain we can achieve by relocating the FOWTs.While for three cases, the gain is around 30% of the targeted gain.The three cases with the lower gains are the smallest wind farm with 7 FOWTs for both wind roses, and the largest layout of 37 FOWTs for the alpha ventus wind rose.
For the smallest wind farm of 7 FOWTs, the small energy gain is due to the low wake losses of the OWFL shown in Table 1.These low losses are due to the small farm size leading to minimum wake interactions within the farm.As the wake losses are small, it is hard to achieve the maximum possible gain we can achieve by relocating the turbine.The reason is that the OWFL is close to the point where the wake losses cannot be reduced further by the FOWTs motion.This shows that as the farm gets smaller relocating the FOWTs will become less beneficial as the wake losses are already minimal.
For the FWF with 37 FOWTs with the alpha ventus wind rose there can be two reasons why the energy gain achieved is small compared to the maximum possible energy gain.First, the motions required by the targeted layout is impossible to achieve by any of the MSDs we have in our database.The second reason is that the method we introduce narrowed down our design space more than it should in step five which is described in section 7. We believe that the low percentages of the final energy gain is due to a mixture of these two reasons.This shows that the method we are presenting is limited by the MS design space.Moreover, it shows that better methods are required for customizing the MSD attached to each FOWT, and that the method we present is just a first step.We expect that enhancing the methodology will lead to higher energy gain for bigger FWFs in the future.
Although the higher gains for the FWF with 9 FOWTs make it look more attractive, we have to keep in mind that in Table 3 we are comparing energy gain percentages and not absolute values.If we look at the absolute values, the energy gains for a the FWF with 25 FOWTs are higher.Therefore from the economical point of view, a smaller gain of a larger wind farm still means higher profit.From this perspective, the results show that as the FWF becomes smaller relocating the FOWTs will become less beneficial, while this will not be the case as the FWF gets bigger.The bigger the FWF the more wake losses and the bigger the opportunity we can benefit from the relocation of the FOWTs.

Conclusion
In our work done in [1], we presented a new method for FWF layout design.The goal was to benefit from the ability of a FOWT to move in the horizontal plane and passively relocate the downwind turbines out of the wake of the upwind turbines.In this paper we conducted a sensitivity analysis to understand the effect of the farm size and the wind rose on the the method from [1].The goal was to understand how these parameters affect the method's performance and the energy gain due to relocating the FOWTs in a FWF.We used two wind roses in this study, the almost uni-directional wind rose from alpha ventus site, and the multi-directional wind rose used within the IEA task 37.For the farm sizes, we used four farm sizes with two boundary shapes.The farm sizes included two small FWFs with 7, and 9 FOWTs, and two larger FWFs with 25, and 37 FOWTs.We applied our method on each FWF size using the two wind roses and calculated the energy gain in each case.
Analysing the maximum gain we can obtain by relocating the FOWTs in Table 2, we see that the maximum energy gain is higher if we relocate FOWTs in a small FWF in a site with a multi-directional wind rose.While the maximum energy gain of relocating FOWTs in a larger wind farm is higher at a uni-directional site.Moreover, the maximum gain in the smaller FWFs can totally compensate the wake losses if achieved.While in large FWF, passively relocating the FOWTs can compensate around 20% of the wake losses.
After customising the MSDs of each FOWT in each of the FWF, we achieved between 73% and 31% of the maximum possible energy gain.This shows that passively relocating the FOWTs will lead to energy gain for all FWFs sizes and for different wind roses.However, the energy gain achieved varies according to the wind rose and the farm size.For a very small FWF, the conventional wind farm layout optimization brings the layout design to a point close to the global minimum wake losses.Therefore, for a very small FWF with minimum wake interactions relocating the FOWTs will only lead to a small energy gain.Hence, we can conclude that as the FWF becomes smaller the benefit of relocating the FOWTs decreases.On the other hand, as the FWF becomes bigger the wake losses increases, and so does the potential of relocating the FOWTs in the farm.From the results, it is also clear that the performance of the current methodology is limited by the design space of the MSD database.Further enhancements are needed for the method for large wind farms.Finally, we can see that relocating the FOWTs in a FWF has a big potential and should be considered while designing the layout of a FWF.As long as every FOWT relocates its position according to the inflow direction, we should benefit from this movement to increase the FWF's energy production instead of neglecting the effect of the FOWT's motions.

Figure 6 .
Figure 6.Conventionally optimized bottom fixed wind farm layouts for the alpha ventus wind rose

Table 1 .
Wake losses in the OWFL assuming constant wind speed of 10m/s

Table 2 .
Energy gains of the targeted layouts compared to the OWFLs assuming constant wind speed of 10m/s

Table 3 .
Energy gains of the final layouts assuming constant wind speed of 10m/s Number of Turbines IEA 37 wind rose Alpha ventus wind rose

Table 4 .
Percentage of the final gain compared to the targeted gain Number of Turbines IEA 37 wind rose Alpha ventus wind rose