With lower turbulence and less rigorous restrictions on noise levels, offshore wind farms provide favourable conditions for the development of high-tip-speed wind turbines. In this study, the multi-objective optimization is presented for a 5MW wind turbine design and the effects of high tip speed on power output, cost and noise are analysed. In order to improve the convergence and efficiency of optimization, a novel type of gradient-based multi-objective evolutionary algorithm is proposed based on uniform decomposition and differential evolution. Optimization examples of the wind turbines indicate that the new algorithm can obtain uniformly distributed optimal solutions and this algorithm outperforms the conventional evolutionary algorithms in convergence and optimization efficiency. For the 5MW wind turbines designed, increasing the tip speed can greatly reduce the cost of energy (COE). When the tip speed increases from 80m/s to 100m/s, under the same annual energy production, the COE decreases by 3.2% in a class I wind farm and by 5.1% in a class III one, respectively, while the sound pressure level increases by a maximum of 4.4dB with the class III wind farm case.

This paper presents a computational method for solving stochastic Ito-Volterra integral equations. First, Haar wavelets and their properties are employed to derive a general procedure for forming the stochastic operational matrix of Haar wavelets. Then, application of this stochastic operational matrix for solving stochastic Ito-Volterra integral equations is explained. The convergence and error analysis of the proposed method are investigated. Finally, the efficiency of the presented method is confirmed by some examples.

The stability of the Nyström method for the Sherman-Lauricella equation on contours with corner points cj, j = 0, 1, …, m relies on the invertibility of certain operators belonging to an algebra of Toeplitz operators. The operators do not depend on the shape of the contour, but on the opening angle θj of the corresponding corner cj and on parameters of the approximation method mentioned. They have a complicated structure and there is no analytic tool to verify their invertibility. To study this problem, the original Nyström method is applied to the Sherman-Lauricella equation on a special model contour that has only one corner point with varying opening angle In the interval (0.1π, 1.9π), it is found that there are 8 values of θj where the invertibility of the operator may fail, so the corresponding original Nyström method on any contour with corner points of such magnitude cannot be stable and requires modification.