The paper studies the applicability of the IEC 61400-1 ed.3, 2005 International Standard of wind turbine minimum design requirements in the case of an onshore Darrieus VAWT and compares the results of basic Design Load Cases (DLCs) with those of a 3-bladed HAWT. The study is based on aeroelastic computations using the HAWC2 aero-servo-elastic code A 2-bladed 5 MW VAWT rotor is used based on a modified version of the DeepWind rotor For the HAWT simulations the NREL 3-bladed 5 MW reference wind turbine model is utilized Various DLCs are examined including normal power production, emergency shut down and parked situations, from cut-in to cut-out and extreme wind conditions. The ultimate and 1 Hz equivalent fatigue loads of the blade root and turbine base bottom are extracted and compared in order to give an insight of the load levels between the two concepts. According to the analysis the IEC 61400-1 ed.3 can be used to a large extent with proper interpretation of the DLCs and choice of parameters such as the hub-height. In addition, the design drivers for the VAWT appear to differ from the ones of the HAWT. Normal operation results in the highest tower bottom and blade root loads for the VAWT, where parked under storm situation (DLC 6.2) and extreme operating gust (DLC 2.3) are more severe for the HAWT. Turbine base bottom and blade root edgewise fatigue loads are much higher for the VAWT compared to the HAWT. The interpretation and simulation of DLC 6.2 for the VAWT lead to blade instabilities, while extreme wind shear and extreme wind direction change are not critical in terms of loading of the VAWT structure. Finally, the extreme operating gust wind condition simulations revealed that the emerging loads depend on the combination of the rotor orientation and the time stamp that the frontal passage of gust goes through the rotor plane.
(a) Aerodynamic loads
(i) Fundamentals of wind turbine aerodynamics
The purpose of a wind turbine is to extract kinetic energy from the wind flowing through it. The earliest aerodynamic model for a wind turbine is generally attributed to Betz , based on one-dimensional momentum flow theory developed by Rankine  and Froude . This model consists of representing the wind turbine as a permeable actuator disc through which the flow of fluid experiences a loss of energy. The resulting reduction in fluid flow velocity and pressure drop, Δp, downwind of the actuator disc results in a thrust force, T, acting on the actuator disc 2.1 where A is the area of the disc. Through the application of the Bernoulli, mass conservation and momentum conservation equations, (2.1) is transformed into 2.2 where ρ is the fluid density, is the free-stream velocity and a is the axial induction factor that relates the free-stream velocity to the fluid velocity passing through the actuator disc. An equivalent drag or thrust coefficient, CT, can be defined as 2.3
Furthermore, the power extracted by the actuator disc from the fluid is given by 2.4
The above is then integrated differently with blade element theory to predict blade and rotor forces for HAWTs and VAWTs. The different approaches used are summarized below.
(ii) Horizontal axis wind turbine momentum model
First formulated by Glauert  in the investigation of propellers, blade element theory involves the discretization of the blades into a number of aerodynamically independent elements that are treated as two-dimensional aerofoils. Local aerodynamic blade forces can be found based on local flow conditions, and the global aerodynamic forces are found through the integral of local blade forces along all blade lengths. Coupling this theory with the momentum flow theory by assuming the loss of momentum/pressure across the turbine equals the work done on the blade elements by the fluid flowing through the turbine, the retardation of the wind through the rotor may be obtained. Subsequent work expands on this to consider a number of different aspects and corrections related to HAWTs, including dynamic stall, tip and hub losses, high loading and skewed wake .
(iii) Vertical axis wind turbine momentum model
A number of increasingly detailed momentum models for VAWTs have been developed over the years [12–14], with the most elaborate variant formulated by Paraschiviou  and known as the double-multiple streamtube model. Briefly, the VAWT is decomposed into two regions: the upwind half-cycle and the downwind half-cycle and are mathematically represented as tandem actuator discs. Furthermore, each region is divided horizontally and vertically into a number of aerodynamically independent streamtubes. By applying blade element momentum theory to each streamtube, the induction factors as a function of azimuthal position, and subsequently local blade forces, can be evaluated.
(iv) Typical rotor loads
The aerodynamic forces produced by HAWTs and VAWTs are markedly different in all six d.f. For identical steady wind conditions, VAWT aerodynamic forces are highly oscillatory while HAWT aerodynamic forces are fairly constant. To illustrate this difference, sample time series of the rotor thrust forces generated by the NREL 5 MW reference HAWT  and the NOVA 5 MW VAWT  at the respective rated wind speeds are presented in figures 2 and 3.
VAWT and HAWT rotor thrust forces at rated wind speeds obtained using FloVAWT  and FAST , respectively. (Online version in colour.)
VAWT and HAWT rotor inclining moments at rated wind speeds obtained using FloVAWT  and FAST , respectively. (Online version in colour.)
While the HAWT thrust forces contain small oscillations at the (n⋅p)HAWT frequency (n=number of blades, p=rotation frequency), the VAWT thrust forces vary from approximately zero to maximum at a frequency equal to the (n⋅p)VAWT frequency. These trends are also present for the aerodynamic pitch inclining moments and shaft torque. As highlighted by figure 3 there is a substantial difference in the inclining moments, which is due to the much lower line of action of the VAWT as compared to the HAWT (illustrated in figure 4).
Front view schematic of HAWT and VAWT, with the centre of thrust pressure, CT, indicated. Note that the height of the VAWT CT varies as the turbine rotates, with maximum and minimum values indicated. (Online version in colour.)
Another significant difference between HAWTs and VAWTs is that VAWTs generate significant oscillatory aerodynamic forces in all six d.f. while HAWTs generate relatively steady aerodynamic forces in d.f. relating to thrust, pitch overturning moment and roll overturning moment. This aspect is of particular importance to floating wind turbines as it may have a significant impact on the design, cost and feasibility of such devices, as will be further elaborated in §3.
(b) Static stability
The basic theory of hydrostatics of floating bodies is well known (e.g. ). It is, however, appropriate to start from the prime principles and apply them to the specific case of a floating offshore wind turbine (FOWT).
In general, for a partially submerged body of arbitrary shape, subject to arbitrary large inclination angles, it is necessary to adopt a fundamental approach, which is the integration of the pressure distribution on the submerged area, in order to estimate all the necessary hydrostatic characteristics.
However, in order to develop a deeper understanding of the static stability of floating wind turbine systems, the following simplifying hypotheses are considered in the present work: the fluid in which the body is immersed is at rest, the body is always in equilibrium and therefore the amount of submerged volume is constant during the (quasi-static) rotation, and the angle of inclination of the body is small (small angle approximation). This is usually called ‘initial stability’ analysis. The analysis will also be restricted to the pitch rotational d.f. (rotation about the y axis), but can be easily extended to roll rotational displacements.
(i) Reference points
Given the floating body in figure 5, the following definitions are given:
— Axis system. An orthogonal axis system is defined, with x aligned with the direction of the wind, z perpendicular to x and vertical upward and the origin coincident with F (therefore z=0 at waterline level).
— Centre of buoyancy (B). The geometric centroid of the submerged volume of a body through which the total buoyancy may be assumed to act.
— Centre of flotation (F). The geometric centroid of the area of the waterplane of any waterline. A waterline is the intersection line of the free water surface with the moulded surface of the body.
— Centre of gravity (G). The centre through which all the weights constituting the system may be assumed to act.
— Centre of mooring line action (MLA). The intersection of the line of action of the horizontal component of the mooring force with the z axis is defined as the reference point of the mooring line action.
— Centre of pressure of environmental forces (CP(env)). The environmental forces acting on the FOWT system will be: aerodynamic forces, hydrodynamic forces and current forces. If an equilibrium state is considered (no waves, only constant wind and current forces), the centre of pressure of environmental forces is defined as the point on which the sum of the environmental forces (Fenv) act.
Diagram of forces and moments acting on a floating wind turbine system, longitudinal plane (pitch degree of freedom). (Online version in colour.)
(ii) Inclining moment
Referring to figure 5, the sum of the environmental forces Fenv is counteracted by the horizontal component of the mooring system force (F=Fenv). The inclining moment (in the x–z plane) MI is the moment resulting from these two forces, and can be estimated as Fenv multiplied by the vertical distance between CP(env) and the point where Fenv is counteracted, CMLA, or 2.5
(iii) Restoring moment
The moments counteracting the inclining moment, whose sum is the restoring moment, can depend on three system characteristics: geometrical, inertial (mass and G) and in the case of tensioned mooring systems (e.g. tension leg platform (TLP)), the mooring system. The initial position of B and the second moment of the waterplane area (linked to the movement of B when the platform is inclined) are the characteristics determining the geometric contribution to the restoring moment.
For freely floating bodies (such as ships), where FB is equal to mg, all the contributions are summarized in the parameter called the metacentric height, GM (see equation (2.14)).
For an offshore floating wind turbine, FB can be higher than mg, due to the downward force of the mooring system. It is then preferred to classify the stabilization mechanisms in: a term taking into account the waterplane area contribution (geometric), a term taking into account the G position and the initialB position (geometric-inertial) and a term related to the (possible) contribution of the mooring system (mooring). In the following, this last approach is illustrated, and the equivalence with the classic approach is demonstrated for the particular case of freely floating bodies (equation (2.14)).
Restoring moment due to the waterplane area. If a rotation around the y-axis is imposed on the body, part of the initial waterplane area where x is positive will be submerged, while the part where x is negative will emerge. This means that on the first side there will be an additional submerged volume, while on the other side there will be an equal (in quantity) amount of volume no longer submerged. This will create a restoring couple that can be estimated by integrating the contribution of the change of submerged volume over the waterplane area (figure 5) 2.6 where Ix is the second moment of area of the initial waterplane area (within the approximation of small inclination, the waterplane area does not change) with respect to the x axis.
Restoring moment due to B and G positions. Since the change of submerged volume and, consequently, the change of position of B is taken into account through equation (2.6), B can be considered fixed in the initial position for this approach, and the moments generated with respect to the origin by the buoyancy force, FB, and the weight force, FW, can be taken into account separately through 2.7
Restoring moment due to the mooring system. There are several mooring systems that have been adopted by the offshore floating wind industry. In general, its contribution to the total stiffness of the FOWT system may be represented by a 6×6 matrix, as it can generate counteracting forces in all d.f., and there can be coupled terms. For the present 1 d.f. analysis, it is assumed that the restoring moment in pitch is only proportional to the rotational displacement in pitch (decoupled from the other d.f.) and can be estimated as 2.8
Total restoring moment in pitch. Considering the three contributions to the restoring moment in equations (2.6)–(2.8), the total restoring moment is 2.92.10
If the small angle of inclination approximation is adopted, and , and from equation (2.10), the total stiffness in pitch C55 can be defined as 2.112.12
Freely floating body case. If the body is freely floating and the contribution of the mooring system to the restoring force can be considered negligible, FB=m⋅g, and (2.12) can be simplified to 2.132.14 where GM is the longitudinal metacentric height, or distance between CG and M, the metacentre. If GM is positive, i.e. M above CG, then the floating body is stable.
N.B. Mooring system contributing both to inclining and restoring moment. Keeping in mind that the pitch d.f. is analysed here, the mooring system is contributing to the inclining moment through the horizontal force equal to Fenv, but (in the case of tensioned mooring platform) can also contribute to the restoring moment through a restoring couple (e.g. for a TLP system, the difference between the vertical components of the tension in the cables provides a restoring moment).
(iv) Static stability requirement and classification of floating offshore wind turbine
Maximum angle of inclination and minimum stiffness. The aerodynamic performance of floating HAWT and VAWT systems operating with their axes of rotation not parallel (HAWT) or perpendicular (VAWT) to the wind direction is still a relatively unexplored research field [20–23]. According to Zambrano , a maximum mean pitch/roll angle of 5° combined with ±15° of dynamic amplitude should be imposed, in order not to substantially compromise the performance of the wind turbine, but it is only a guideline. In any case, it is clear that a maximum angle () of inclination should be imposed in the design phase as one of the requirements for the floating support structure or, given an inclining moment, a minimum total stiffness should be provided by the floating support structure. According to equation (2.12), the minimum total stiffness is 2.15
Classification based on main stability mechanism. Referring to (2.10), the mechanism through which a FOWT system fulfils condition (2.15) is often used to classify FOWT systems . The system is defined as waterplane-stabilized if the main contribution to the total MR is MR,WP (e.g. Dutch Trifloater , WindFloat , NOVA semi-submersible ), as ballast-stabilized if the main contribution is given by MR,CG (e.g. Hywind SPAR by Statoil ), or as mooring-stabilized if the main contribution comes from MR,moor (e.g. MIT TLP , BlueH by BlueH ). This is usually represented by the ‘stability triangle’, where the sum of the three contributions (each one going from contributing 0 to 100% of the total stiffness) amounts to the total stiffness of the system .
It is important to note that usually the contribution of the CG position is represented as positive or zero, while this is valid only if 2.16
If the body is freely floating, the buoyancy-to-weight ratio is equal to one, and therefore, if the CG is below CB, the contribution of this term to the restoring moment is positive, while if CG is above CB the contribution is negative. This is an important consideration for FOWTs, since even if the mass of the wind turbine system (rotor plus nacelle plus tower) is usually smaller than the mass of the support structure, the CG of the wind turbine can be very high and push the total CG above the CB. If the body is not freely floating, as for example for a taut or TLP system (see below), the buoyancy-to-weight ratio can be greater than one and therefore, to have MR,CG>0, the CG position has to be lower than the buoyancy-to-weight ratio multiplied by the CB position.
In figure 6, the modified stability triangle concept is represented, taking into account the potential negative contribution of the restoring term due to the CG position, called ‘ballast stabilized’. (N.B. In this figure, the negative contribution goes to −25% for the sake of simplicity, but can be higher in modulus.)
Stability triangle. The three corners of the triangle represent the three stabilization mechanisms exploited by floating wind turbine systems (see §2b(iii),(iv)). (Online version in colour.)
(c) Dynamic response
(i) Dynamic response of floating bodies
The six d.f. dynamic response of a free-floating body can be represented by a set of coupled linear second-order differential equations in the complex-frequency domain, based on linear hydrodynamic theory assuming small amplitude displacements as defined, for example, by Faltinsen  and Fossen  2.17 where x is the displacement complex vector, M is the body inertia matrix, A(ω) is the hydrodynamic frequency-dependent added mass matrix, Bhyd(ω) is the hydrodynamic frequency-dependent damping matrix, C is the restoring stiffness matrix and τexc is the excitation force. τexc is a combination of all external forces, i.e. wave excitation forces, current forces and wind turbine forces. Wave excitation forces and current forces may be computed through various numerical and semi-empirical approaches . Likewise, wind turbine forces may be computed from a selection of engineering and higher order models . For this set of coupled equations of motion, the eigenvalue problem can be solved to obtain the natural frequencies and modes of the floating body.
(ii) Considerations for floating wind turbines
For the case of a floating wind turbine, the excitation force in (2.17) derives both from incident wave excitations and aerodynamic excitations from the turbine, and the total restoring stiffness matrix consists of hydrostatic stiffness, Chyd, and mooring restoring stiffness, Cmoor, given by 2.18
The mooring system also contributes to the total system inertia, and in preliminary analyses this contribution is combined with the FOWT inertia matrix. In certain severe operating conditions, the structural and hydrodynamic damping of mooring lines can also have a significant effect on global platform response, although it has not been included here. On this basis, the eigenvalue problem to deduce the six d.f. natural frequencies and coupled vibration modes of the FOWT is given by 2.19
The identification of floating wind turbine natural frequencies allows designers to avoid frequency ranges where incident wave forces are significant by appropriately modifying the floating wind turbine characteristics. Equation (2.17) allows for the investigation of the dynamic response of a floating wind turbine for steady-state sinusoidal motions only. The analysis of the transient behaviour of a floating wind turbine would require the time-domain variant of (2.17) that considers free-surface memory effects, as formulated by Cummins  and further developed by Oglivie . This time-domain model also allows for the inclusion of the inherent coupling between aerodynamic and hydrodynamic system response, which would otherwise be impractical to consider in frequency-domain models.