# Optimize Rotating Wind Energy Rotor Blades Using the Adjoint Approach
## 1 Introduction  
The generated electric power from wind turbines increases linearly with the swept rotor area. Therefore, wind turbines grow larger and larger. Otherwise, limits in maximal mountable rotor weight and constraints coming from material science restrict the growth of rotor blades. Further limiting factors are alternating wind loads, gusts, turbulences and rotational forces. As wind turbines are normally designed for a life time of 20-25 years, the estimated total amount of rotations is about \(150\times 10^{6}\), which leads to \(5\times 10^{8}\)-\(10\times 10^{8}\) load changes. Periodic loads due to the vertical wind velocity gradient and the tower shadow, each one of them proportional to the blade rotation frequency, add up to this [1]. Added to those load changes due to the rotational movement are the wind loads, which were shown to be highly intermittent [2]. At the same time, most favorable sites for wind turbines with little turbulence intensity and large average wind speeds are already exploited, such that either repowering or worse sites are chosen in order to increase the generated wind power [3]. For slow average wind speeds, slender blades are favorable, which react even on small wind velocities due to a large aerodynamic efficiency. This enforces the trend towards long, slender blades. Such blades are prone to elastic deformations and even more influenced by wind load changes. In return, this might lead to critical fatigue of the material, both from blades, but also from bearings and support structures. However, flexibility and light structures offer the possibility of passive control. The pitch systems of wind turbine blades are too slow to react on highly intermittent load fluctuations within the wind field, the use of structural bend twist coupling, which reacts instantly on aerodynamic loads, is a promising concept to reduce the amount of load changes and therefore of fatigue by a pre-designed deformation of the blade [4]. Especially adaptive changes in the twist of theblade can reduce the angle of attack and thus lead to a significant load reduction, which results in a longer life time or less cumbersome maintenance of the wind turbine blades and therefore in the end in lower specific costs of energy [5; 6]. Based on the actual development in rotor design, we present in this work an optimization tool for the aerodynamics of rotating rotor blades within the open-source toolbox for numerical simulations OpenFOAM. This tool is tested for the optimization of the twist distribution in order to reduce the overall thrust force. Gradient-based optimization offers on the one hand the large advantage of a high-quality search direction for optima. On the other hand, only local optima are found. Thus, this method is suitable if there is not more than one optimum or if the initial point is not too far away from the global optimum. In our case, the aerodynamics of wind turbine rotor blades can be considered as sophisticated, so that the mentioned disadvantage of the gradient-based method can be neglected, while the advantage of a high-quality search direction is utilized. The continuous adjoint approach is used for the computation of gradients. In contrast to other methods, the adjoint approach is marked by the independence of the computational cost on the amount of design parameters, thus no full simulation per each single design parameter is required [7]. This qualifies the adjoint approach for the combination with computationally expensive Computational Fluid Dynamics (CFD). The optimization is based on the solution of steady-state flow equations of the current blade geometry. Obtained gradients are then evaluated to find the optimum. The implementation of the continuous adjoint equations in OpenFOAM was firstly done by [PERSON] [8] for ducted flows[PERSON] Several publications show the applicability of the adjoint approach in combination with CFD, both for external and ducted flows [9; 10; 11; 12].