Simplified support structure design for multi rotor wind turbine systems

In this study different multi rotor wind turbine systems (MRSs) are designed in such a way that the space frame, forming the connection between rotor nacelle assemblies (RNAs) and tower, is modeled as an ideal truss work. To dimension the tube diameters and wall thicknesses, a simplified load case is used with an adjusted safety factor for loads. This simplified approach allows fast computations of a large variety of different support structure designs. By variation of rotor number, space frame topology, space frame depth and the positioning of yaw bearings, it is possible to gain an understanding of the optimal 5 MRS design. As such, the simplified approach is a preliminary step helping to choose a good design parameter combination for a more detailed and comprehensive analysis.

In the revised paper this will be added as a remark in the outlook.
(3) Author's changes in manuscript: In line 194-196 (marked-up version) a remark was added to clarify the approach to find the minima in this study: "These minima are determined manually with no incorporation into an overarching mathematical optimization approach of the dimensioning procedure." In line 255-257 (marked-up version) a remark was added to indicate the possible need for a formal optimization approach: "With increasing complexity in future works due to a growing number of designs and design parameters, a framing of the dimensioning process as part of a formal optimization approach needs to be considered."

Introduction
In times of ever growing wind turbines and their components, the industry is facing new challenges in manufacture and transportation, as well as loads and strength. A multi rotor wind turbine system (MRS) could overcome the obstacles of this growth 10 trend.
Studies from the INNWIND project showed the potential of a 20 MW MRS with 45 rotors to reduce the levelized cost of energy (LCoE) compared to a power equivalent single rotor (SR) (Jamieson et al., 2017). Recent results from Vestas' four rotor MRS demonstrator revealed advantages in aerodynamic efficiency and in wake recovery when compared to a SR (van der Laan et al., 2019).

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Assuming a MRS with an overall capacity of 20 MW, due to the resulting lower rotor nacelle assembly (RNA) masses, based on the square cube law, and the load averaging effect (Jamieson et al., 2017), it should be more suitable to build up a MRS with a high number of small rotors, rather than a small number of large rotors. This can be categorized as a multi digit MRS (MD-MRS). To allow original equipment manufacturers (OEMs) to move towards MRSs by using their existing turbine portfolio, a medium term solution might be the use of few rotors. This single digit MRS (SD-MRS) would be built up of three 20 to nine rotors in the megawatt range. In the long term, a MRS with a high number of rotors using a newly developed small RNA in the kilowatt range seems favorable.
The DTU 10 MW research wind turbine (Bak et al., 2013) is used in this study as a basis for down-and upscaling. This includes downscaling to the size of the rotors used for the SD-MRS (set to 2 MW, 4 MW or 8 MW), downscaling to the size designs that have been ✿✿✿ are ✿✿✿✿✿ being analyzed are in the range of an overall capacity of 14 MW to 28 MW. This simplified approach allows fast computations of a large variety of different support structure designs. By variation of rotor number, space frame topology, space frame depth and the positioning of yaw bearings, it is possible to gain an understanding of the optimal MRS design. As such, the simplified approach is a preliminary step helping to choose a good design parameter combination for a more detailed and comprehensive analysis.
2 Simplified support structure design 45 An overall capacity in the 20 MW range is assumed for the MRS. Regardless of the number of rotors, or rather the distinction of SD-MRS or MD-MRS, rotor data like the masses and diameters of rotors are needed. The basis for this rotor data is the DTU 10 MW research turbine (Bak et al., 2013) which is scaled down for the rotors of the MRS, as well as scaled up for a SR with an equal overall capacity.
Scaling is done under the assumption of similarity rules for wind turbines and a constant tip speed ratio (Jamieson, 2018).

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The rotor diameters for the scaled turbines D rotor,i are calculated with the rotor diameter of the DTU rotor D DTU : The masses of rotors and nacelles are scaled via: with the scaling exponent n. Due to the different influences on masses like new and lighter materials, better light weight design 55 and higher experience in manufacturing, scaling is somewhat a critical task, especially for blades. Inter-and extrapolated scaling trend lines also dependent ✿✿✿✿✿✿✿ depends on the considered data. In (Jamieson, 2018) this leads in one analysis to a scale exponent slightly above two. On the other hand, due to larger blade lengths, self weight bending could become a design driver, resulting in a higher exponent than three. Fundamentally blade mass is scaling in a cubic way.
The wind industry almost exclusively applies upscaling, due to the growth trend of turbines for a higher energy yield. For 60 the MRS downscaling is of importance. Downscaling with an exponent around two seems not be suitable. Then the gain from new materials, technology and experience would be discarded and would result in a heavier and not modern blade.
An upscaling exponent of n up = 2.6 is set for blade and nacelle masses for the large SRs implying technological improvement. For downscaling an exponent of n down = 3 is set, assuming a scaled state of the art small turbine without any new future improvements.

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The simplified support structure design is described on the example of a SD-MRS with three rotors, as seen in Figure 1. An MRS support structure is composed of a tower and a space frame, connecting the RNAs among themselves and with the tower.
The space frame consists of tubular steel connections.
In the INNWIND project (Jamieson et al., 2017) the spacing between rotors D spacing was set to the rotor diameter plus five percent:D spacing = 1.05 · D rotor . Their simulations resulted in an increase of both thrust and power generation in comparison to 70 a single rotor. A change to D spacing = 1.025 · D rotor resulted in no change to the mean value for thrust and power production.
Here in this study a D spacing = 1.03 · D rotor is set.
All chosen layouts are based on equilateral triangles, so the vertical distance between rotor rows results to: The height of the first row results from a set blade tip ground clearance of 22 m.

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Tower and space frame are connected through yaw bearings, in this study always with a fixed/loose bearing combination.
The MRS should be able to follow the wind via a global yaw system. Meaning that the whole space frame should be able to align itself perpendicular to the actual wind direction, rather than each rotor itself. How the yaw bearings would be connected The vertical positions of the fixed and loose bearing, characterized with h depth,1 and h depth,2 is ✿✿✿ are varied to investigate the influence on the cost and to find the optimal design regarding the cost. The fixed bearing is in this study always on the tower top, so the position also dictates the tower height, see Figure 2. In the designs with three rotor rows, the parameter h depth,3 is needed. Another geometric design parameter that is varied is the depth of the space frame.

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To dimension both tower and space frame a simplified design load case is defined: maximum thrust force (at steady rated wind speed) on all rotors simultaneous. Because of wind shear, turbulent wind, gusts and the pitching behavior this is an artificial and somehow unlikely case, but for this study it represents a worst case regarding the ultimate loads.
The MRSs are designed in such a way that the space frame is modeled as an ideal truss work. To determine the member forces of the space frame, a finite element analysis (FEA) carried out via ANSYS APDL is used. The space frame members are 90 modeled with bar elements. Bar elements have one local degree of freedom (DoF) per element node, the axial displacement, resulting in three global DoFs per node. The corresponding reaction force to the local axial displacement is the local axial force.
The FEA requires initial diameters and wall thicknesses of the space frame members to determine and use the stiffness matrix.
Since the space frame is modeled as an ideal truss work the FEA solution of interest, the axial member force, is independent of the initial cross section parameters.

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The use of bar elements implies that there are no other DoFs and reaction forces in the nodes, apart from the axial ones. In reality the connections between the space frame elements and the rotors would be welded or bolted. Therefore, ✿ shear forces as well as bending and torsional moments would occur in the nodes. This could be modeled in the FEA via beam elements with six DOFs and reaction forces per node. With the use of beam elements, the FEA solution would depend on the initial cross section and the dimensioning process would be an iterative one for all elements/members and all space frame design configurations.

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The differences between bar and beam element solutions was investigated and deemed neglectable for this preliminary study.

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The thrust forces F t on the rotors are calculated via the c T = 0.827 value of the DTU turbine at rated wind speed v rated = 11.4 m s (Bak et al., 2013), the scaled rotor diameter D rotor and ρ air = 1.225 kg m 3 (IEC 61400-1 Ed.4): This is still under the assumption of an unchanged tip speed ratio λ due to scaling as well as unchanged c P -λ and c T -λ curves.

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Also acting on the support structure are the gravitational forces of the RNAs with applied partial safety factor γ f,gravity = 1.35.
The partial safety factor for the thrust forces is set to γ f,thrust = 1.5 instead to the suggested value of 1.35 according to (IEC 61400-1 Ed.4). This is due to the simplicity of the design and to compensate for the simplified load case. Both kind ✿✿✿✿ kinds ✿ of forces are modeled as external forces acting on the rotor nodes of the space frame in the FEA.
The yaw bearings are modeled as boundary conditions with their respective DoFs fixed bearing disables all three spatial translations, the lower loose bearing has one DoF in the tower height direction. In Figure   2 a loose bearing is seen in the back behind the fixed bearing. This is required for the FEA simulation to run, otherwise the model wouldn't be kinematically determined. The space frame could still rotate and this would result in a singular reduced stiffness matrix.
The dead load of the space frame and the drag forces of both space frame and tower are neglected. This is due to the fact, 115 that both tower and space frame are going to be dimensioned and an iterative process is to be avoided.
The space frame members are first dimensioned against ultimate strength with applied safety factor for material γ m = 1.1 and an assumption of a thin-walled tube: the wall thickness t is much smaller than the diameter D: t ≪ D.
A ratio for the wall thickness to diameter is defined r t = t D . This is set to r t,b = 1 120 for the space frame bars and r t,t = 1 250 for the tower. For the cross section follows: For both space frame and tower a construction steel with a yield strength of σ yield = 355 MPa is used. Based on σ = F A and the FEA based axial forces F bar,i , including both safety factors for loads, the bar diameters D i can be calculated now: If necessary, in case of a compression state in the member element, a redimensioning against stability (Euler's critical load) with applied safety factor for buckling γ m,buckling = 1.2 is required. Since both ends of the members are free to rotate in theory, the column effective length factor l k is set to l k = l i , the whole length of each space frame bar element i. Euler's critical load is defined as: with Young's modulus E = 2.1e5 MPa and area moment of inertia for bending I b . Again, like with the cross section A, a simplification for thin-walled bars ✿✿✿✿ tubes can be used (Wriggers et al., 2007): In case of a positive member axial force, the member is in a tension state and stability is of no concern. A negative axial force means a compression state. If the difference N crit − F bar is negative, the bar diameter can be dimensioned with: The conclusive bar diameter is set to the maximum of D i and D i,buckling .
The tower diameters and wall thicknesses are determined by the tower bending reaction moment. The bearing reaction forces thin-walled tubes can be used: The tower diameter D tower follows to: Since there are no bending moments  (Fraunhofer ISE, 2018) and selected free available turbine data. The resulting factors can be seen in Table 1. The calculated cost factor values for tower, rotor and nacelle are in good accordance to (Jamieson et al., 2017). The assumption for the space frame cost factor is taken from their study.
Each SD-MRS design is simulated for each design parameter combination of h depth,i and the depth of the space frame.
The energy yield of the MRS is determined based on a ✿✿ an ✿✿✿✿✿✿✿✿ assumed Rayleigh wind speed distribution with a mean wind speed  With the cost and the annual energy production (AEP) of the designs the SCoE can be calculated: with an assumed wind turbine lifetime of n l = 25 years.

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To compare and normalize the SCoE values, a power equivalent single rotor SR is designed with the same assumptions as the SD-MRS (blade tip clearance, loads, etc.).

SD-MRS
To keep the design space somehow limited and to reflect currently available turbines on the market, the rotors for the SD-MRS are set to a single capacity of 2 MW, 4 MW or 8 MW. Table 2   Possible rotor numbers are set to 3 (Tri rotor), 5 (Penta rotor), 7 (Hepta rotor) or 9 (Ennea rotor) rotors. An even numbered SD-MRS would result in a cantilever design with or without steel ropes to reduce loads. These cantilever designs and therefore even numbered SD-MRSs are not considered in this study. The uneven numbered SD-MRS layouts are all designed with 175 equilateral triangles which results in the highest packing density of the rotor area. The disadvantage of design no. 1 is the higher tower base moment, resulting in higher cost.
-Penta rotor: 5 · 4 MW resulting in 20 MW. The five rotors can also be arranged in an increasing or decreasing order.
Two version are designed for increasing and decreasing order each, with the same rotor layout, but a slightly different 185 arrangement of the bars.
-Hepta rotor: 7 · 2 MW resulting in 14 MW or 7 · 4 MW resulting in 28 MW. Additional to the increasing or decreasing order a circular/hexagonal arrangement is possible.
-Ennea rotor: 9 · 2 MW resulting in 18 MW. Two or three rows of rotors are possible, both with increasing and decreasing order.

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This results in an overall number of 18 designs for the SD-MRS, as seen in Figure 3.
The design parameters shown earlier, h depth and depth are varied via unit less ratios and the cost are calculated. Since the variation of these design parameters doesn't change the heights of the rotors, there is no influence of the variation on the energy yield of each design. To find the minimum SCoE  This is due to the relative small fraction of tower and space frame mass on the overall mass and therefore cost. Design driver for the SD-MRS are the RNA masses, they benefit from smaller rotors based on cubic scaling.   Table 3.
215 The design parameters to be varied are again the depth of the structure and the fixed bearing position and therefore the tower height. Instead of a quasi-continuous variation of the fixed bearing position over the height, three discrete positions are investigated. Position 1 in the second row at the bottom, position 2 in the middle of the space frame and position 3 at the top.
All three variants have the loose bearing in the first space frame row.   should not be to build up a MRS with a high number of rotors and a small overall capacity. It just seems that way, since only a fixed number of rotors was investigated for the MD-MRS. The investigation with a variable, high number of rotors is missing and seems to be the next step. The Tri 240 rotor shows the least potential to reduce cost, based on the relative large rotors and therefore cost.

Conclusions and outlook
The aim of this study was to develop a simplified method for preliminary calculations of masses and therefore cost for multi rotor wind turbine systems. The simplifications in the dimensioning process were used to avoid iterations for convergence and to allow a fast way to investigate a variety of designs and design parameters.

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Several SD-MRS designs were designed, simulated and optimized regarding the cost. The Members can change from a tension to the compression state or the other way around. Stability seems to have a big influence since many space frame members needed to be redimensioned, when in the compression state.

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The SD-MRS designs with small single rotors showed the highest potential to reduce cost. One particular MD-MRS design with 45 rotors was also investigated and showed an optimal depth to width ratio for the space frame of 10%-13%. A fixed bearing position and therefore a tower height in the middle of the space frame was most promising. Overall the MD-MRS was showing more potential than the SD-MRS to reduce cost.
Next steps include more MD-MRS designs with variable number of rotors. ✿ After that, one or two promising designs will be analyzed in detail, regarding design load cases according to (IEC 61400-1
Author contributions. SSt performed all simulations, wrote the pre-and post-processing code, as well this paper. PDa and MTa helped