## 1 INTRODUCTION

Wind conditions for wind turbines located inside wind farms are significantly different than outside farms. The main effects are that the average wind speed is decreased, and the turbulence inside the wind farm is increased. The result of this is a reduction in power production together with increased load levels. Several people have worked on the topic, and recent measurements of wake effects have previously been reported in Vølund, [1] Frandsen *et al.*, [2] Christiansen and Hasager, [3] Cleve *et al.*, [4] Barthelmie *et al.* [5] and Barthelmie and Jensen. [6] Many engineering models have been developed in the past. Regarding these models, the main focus has been to enable prediction of power production, [7, 8] whereas only few models address the loading issues [9, 10] and none of these models encompass both production and load aspects. The most well-known model for prediction of fatigue load levels is the one reported by Frandsen, [10] which is also the wake model included in the IEC61400-1 standard for load simulation of wind turbines. This model is basically a method to scale the intensity of the inflow turbulence according to the park configuration and the load component in question. This model is mainly based on full-scale load measurements observed from the Vindeby offshore wind farm in Denmark, where the turbines are located with a distance of approximately eight diameters. Remarkably, none of the previous mentioned models (with slight exception of Ainslie [8]) account for wake meandering even though this phenomenon was addressed by Baker and Stel [11] and Ainslie [8] in the 1980s. In order to enable simulation of the combined effect of both power reduction and load increase in a wind farm, a completely different type of model is needed. Different computational fluid dynamics (CFD) methods have been demonstrated, where especially actuator line simulations seem promising. [12] Also large eddy simulations have provided new insight in the complexity of wind farm wakes. [13, 14] Unfortunately, these methods are all very computationally demanding with simulation times of days or more on high performance computers and therefore not usable in typical load analysis for wind turbine design.

Another type of engineering model capable of predicting the loads on turbines inside wind farms is the dynamic wake meander (DWM) model, which is a more detailed model of the flow field behind the upstream turbine. This method uses the wind speed deficit of the upstream turbine together with a meandering process in order to simulate the incoming flow field of the downstream turbine and thereby enabling detailed analysis of both production and loading through aeroelastic computations. The meandering process causes an intermittent appearance of the flow field with periods of full, half or no wake situation—varying from time to time driven by the large-scale natural turbulence. This more correctly implemented physical process has previously been validated by load, inflow and wake measurements [15-18] and is further verified by this paper. It has also previously been seen that there are important differences in the turbine loading depending on the type of model chosen. [19, 20]

### 1.1 The HAWC2 model

The HAWC2 code is an aeroelastic model intended for calculating wind turbine response in time domain. [21] The core of the code was mainly developed within the years 2003–2007 in the Aeroelastic Design Research Program at Risø, National Laboratory, Denmark, but is continuously being updated and improved.

The structural part of the code is based on a multi-body formulation as described in Shabana [22] using the floating frame of reference method. In this formulation, the wind turbine main structures are subdivided into a number of bodies, where each body is an assembly of Timoshenko beam elements. Each body includes its own coordinate system with the calculation of internal inertia loads when this coordinate system is moved in space; hence, large rotation and translation of the body motion are accounted for. Inside a body, the formulation is linear, assuming small deflections and rotations. This means that a blade modeled as a single body will not include the same nonlinear geometric effects related to large deflections of a blade divided into several bodies. The bodies, representing the mechanical parts of the turbine, are connected by kinematic constraints. The constraints are formulated as algebraic equations, which impose limitations of the bodies’ motion. Examples of such constraints are a fixed connection from a structural node to a global point (e.g. tower bottom clamping), a fixed coupling of the relative motion (e.g. fixed pitch, yaw), a frictionless bearing and a bearing where the rotation angle is controlled by the user. It may be worth to notice that also for the last constraint where the rotation is specified externally, inertial forces related to this movement are accounted for in the response. External forces are placed on the structure in the deformed state, which is especially important for pitch loads and twist of the blades, and since large rotations are handled by a proper subdivision of bodies, the code is suited for calculations on very flexible turbines subjected to, e.g. large blade deflections. The structural model is general, but in its simplest form, a turbine is modeled using one body for the tower, one for the nacelle and one for each blade.

The aerodynamic part of the code is based on the blade element momentum (BEM) theory, but extended from the classic approach to handle dynamic inflow, dynamic stall, skew inflow, shear effects on the induction and effects from large deflections. One example is the effect of large flapwise blade deflections causing a change in the effective rotor diameter and that the blade forces are no longer perpendicular to the rotor plane. This reduces the thrust on the rotor and thereby changes the induced velocities and vice versa. The dynamic stall model [23] is a modified Beddoes–Leishmann [24] model that includes the effects of shed vorticity from the trailing edge (Theodorsen theory [25]), as well as the effects of stall separation lag caused by an instationary trailing edge separation point. These effects are important in relation not only mainly to flutter analysis, but also generally to calculate loads and stability of blades with very low torsion stiffness. The induced velocities are calculated on the basis of the local inflow velocities causing different inductions in the upper and lower parts of the rotor, as in the case of a large wind shear. [26] In order to illustrate the capability of this BEM implementation in the case of a wake situation with a turbine operating in a steady half wake from an upstream turbine, a phenomenological study was conducted in both HAWC2 and the CFD code EllipSys3D, [27] which generally is considered state-of-the-art within CFD simulation of flow around wind turbines. A very fine agreement was obtained, as seen in Figure 1.

The wind conditions are divided into deterministic and stochastic contributions. The deterministic wind is mean wind velocity, wind steps, ramps, special gust events and special shears, including the possibility for fully user-defined shears. The stochastic turbulent wind is generated using the Mann model, [28] which is a non-isotropic full three-dimensional correlated turbulence flow field model. Tower shadow effects are included using a potential flow method.

Control of the turbine is performed through one or more dynamic link libraries and is therefore not part of the HAWC2 core. The reason for this is that each turbine is equipped with its own individual controller, which is normally kept confidential by the manufacturer.

The calculation time, which is obtained using a Newmark-beta solution scheme together with Newton–Raphson iterations within each time step, is approximately a factor of 1–2 slower than real time on a 3 GHz CPU.