Maximum thermodynamic power coefficient of a wind turbine

According to the centenary Betz‐Joukowsky law, the power extracted from a wind turbine in open flow cannot exceed 16/27 of the wind transported kinetic energy rate. This limit is usually interpreted as an absolute theoretical upper bound for the power coefficient of all wind turbines, but it was derived in the special case of incompressible fluids. Following the same steps of Betz classical derivation, we model the turbine as an actuator disk in a one dimensional fluid flow but consider the general case of a compressible reversible fluid, such as air. In doing so, we are obliged to use not only the laws of mechanics but also and explicitly the laws of thermodynamics. We show that the power coefficient depends on the inlet wind Mach number , and that its maximum value exceeds the Betz‐Joukowsky limit. We have developed a series expansion for the maximum power coefficient in powers of the Mach number that unifies all the cases (compressible and incompressible) in the same simple expression: .     

Uncertainty quantification of infinite aligned wind farm performance using non‐intrusive polynomial chaos and a distributed roughness model

Uncertainty of wind farm parameters can have a significant effect on wind farm power output. Knowledge of the uncertainty‐produced stochastic distribution of the entire wind farm power output and the corresponding uncertainty propagation mechanisms is very important for evaluating the uncertainty effects on the wind farm performance during wind farm planning stage and providing insights on improving the performance of the existing wind farms. In this work, the propagation of uncertainties from surface roughness and induction factor in infinite aligned wind farms modeled by a modified distributed roughness model is investigated using non‐intrusive polynomial chaos. Stochastic analysis of surface roughness indicates that 30% uncertainty can propagate such that there is up a 8% uncertainty in the power output of the wind farm by affecting the uncertainty in the position of the individual wind turbines in the vertical boundary layer profile and uncertainty in vertical momentum fluxes which replenish energy in the wake in large wind farms. Induction factor uncertainty of the wind turbines can also have a significant effect on power output. Not only does its uncertainty substantially affect the vertical boundary layer profile, but the uncertainty in turbine wake growth which affects how neighboring turbine wakes interact. We found that optimal power output in terms of reduction of uncertainty closely correlates with the Betz limit and is dependent on the mean induction factor. Copyright © 2016 John Wiley & Sons, Ltd.     

Performances of ideal wind turbine

If there is an ideal wind turbine, its performances will be the pursuit goals for designing the actual wind turbine. In this paper, the wind turbine that has the maximum efficiency is defined as ideal wind turbine, which has three main features: lift-drag ratio is infinite, it has enough number blades so that the blade tip and root losses can be ignored, and its blades are limited in width. Using blade element theory, the differential equations of power, torque, lift and thrust of blade element were derived, and the expressions of power, torque, lift and thrust coefficients of the ideal wind turbine were gained by integrating along the blade span. Research shows that the power, torque and lift coefficients of the ideal wind turbine are functions of tip-speed ratio. When the lift-drag ratio and the tip-speed ratio is approaching infinity, power coefficient of the ideal wind turbine is close to the Betz limit; The torque limit is 0.401 when the tip-speed ratio equals about 0.635; The Lift limit is 0.578 when the tip-speed ratio equals about 0.714; The thrust coefficient is 8/9, which is unrelated with tip-speed ratio. For any wind turbine which tip-speed ratio is less than 10, the power coefficient is unlikely to exceed 0.585, for any high-speed wind turbine which tip-speed ratio is greater than 6, the torque coefficient in steady state is unlikely to exceed 0.1, and the lift coefficient is unlikely to exceed 0.2.     

基于偏航尾流模型的风电场功率协同优化研究

为减小风电场尾流效应的影响,提升风电场整体发电量,提出一种基于偏航尾流模型的风电场功率协同优化方法。首先建立风电场偏航尾流模型,该模型包括用于计算单机组尾流速度分布的Jensen-Gaussian尾流模型、尾流偏转模型及多机组尾流叠加模型,对各机组风轮前来流风速进行求解;再根据来流风速计算风电场输出功率,并以风电场整体输出功率最大为优化目标,利用拟牛顿算法协同优化各机组轴向诱导因子和偏航角度。以4行4列方形布置的16台NREL-5 MW风电机组为对象进行仿真研究。结果表明,所提出的基于偏航尾流模型的风电场功率协同优化方法能显著提升风电场整体输出功率。     

Tidal farm analysis using an analytical model for the flow velocity prediction in the wake of a tidal turbine with small diameter to depth ratio

A tidal turbine is a device converting hydrodynamic power into electrical power. Lately, more and more projects have been developed in order to optimize the productivity of this kind of energy. In such research with industrial interest, under the impact of the wake effect on the output power, the analysis of a tidal farm layout is regarded as the first priority. Simple approaches such as those developed for wind farms could be used in tidal turbine arrangement optimization. These methodologies can be improved by taking into account the turbulence in tidal farms and tidal turbines' mechanical characteristics. The goal of this work is to propose a predictive analytical model to estimate the tidal speed in the far wake of tidal turbines with small diameter to depth ratio (20% here). It is a first step prior to integrate the wake model in a tidal farm layout optimization algorithm. The wake model development is achieved reanalyzing the far wake's equations used in wind farm applications. A turbine represented by an Actuator Disc (AD) in conjunction with a Computational Fluid Dynamics (CFD) numerical model is used as a reference for this purpose. The CFD-AD model has been validated with experimental results from literature. The novelty of the present work consists in expressing the far wake's radius expansion as a function of the ambient turbulence and the thrust coefficient. The proposed equation is used in conjunction with the Jensen's model in a manner that the velocities downstream a tidal turbine can be estimated. The velocity distribution in the far wake of a single turbine obtained by the proposed model is in good agreement with the CFD numerical model. As a matter of fact, the model provides satisfactory accuracy in the cases of two parks: one with five aligned turbines and one with ten staggered turbines.     

An experimental investigation on the wake interferences among wind turbines sited in aligned and staggered wind farms

An experimental investigation was conducted for a better understanding of the wake interferences among wind turbines sited in wind farms with different turbine layout designs. Two different types of inflows were generated in an atmospheric boundary layer wind tunnel to simulate the different incoming surface winds over typical onshore and offshore wind farms. In addition to quantifying the power outputs and dynamic wind loads acting on the model turbines, the characteristics of the wake flows inside the wind farms were also examined quantitatively. After adding turbines staggered between the first 2 rows of an aligned wind farm to increase the turbine number density in the wind farm, the added staggered turbines did not show a significant effect on the aeromechanical performance of the downstream turbines for the offshore case. However, for the onshore case, while the upstream staggered turbines have a beneficial effect on the power outputs of the downstream turbines, the fatigue loads acting on the downstream turbines were also found to increase considerably due to the wake effects induced by the upstream turbines. With the same turbine number density and same inflow characteristics, the wind turbines were found to be able to generate much more power when they are arranged in a staggered layout than those in an aligned layout. In addition, the characteristics of the dynamic wind loads acting on the wind turbines sited in the aligned layout, including the fluctuation amplitudes and power spectrum, were found to be significantly different from those with staggered layout.     

Grounding and the influence of the total pressure losses in ducted wind turbines

Ducted wind turbines are known to exceed the Betz limit for the power that may be extracted from free currents per unit cross‐sectional area of their rotors. The present study points to another advantage: the fact that the turbine rotor (and the associated subassemblies) need not be lifted up on the air and rotated to align with the wind. They can function just as effectively by installing them on the ground and sucking air inside it. The analysis evaluates the efficiency of a ducted wind turbine plant, exhausting into the wake of a bluff body, under both ideal and realistic component performance. The results indicate that there exists a strong need to develop diffusers with a very low loss coefficient (e.g. by employing suction or transpiration techniques). Under such conditions and by employing the above exhaust assist techniques, the ducted wind turbine concept may deliver efficiencies more than an order of magnitude larger than those of the conventional plants. Copyright © 2009 John Wiley & Sons, Ltd.     

Non‐uniform velocity distribution effect on the Betz–Joukowsky limit

A simple analytic correction is derived for the maximum efficiency of an ideal wind turbine rotor, the Betz–Joukowsky limit. The analytic correction accounts for the effect that the non‐uniform atmospheric boundary layer velocity distribution has on the Betz–Joukowsky derivation. The maximum power coefficient predicted by using the atmospheric boundary layer velocity profile is slightly higher than that predicted by using a uniform velocity distribution. The application of the correction to a 100 m rotor diameter at 80 m hub height in a neutrally stratified boundary layer flow predicts a maximum power increase of 1–2%, depending on the approach terrain. Copyright © 2012 John Wiley & Sons, Ltd.     

风电场采用健康系数与变下垂输出系统参与电网调频研究

针对风电场参与电网调频导致备用功率冗余的问题,建立了风电场基于健康系数的备用功率分配系统与变下垂系数的备用功率输出的系统。为了分配风机备用功率与控制风机备用功率输出,采用了ARMA风功率预测法对风电场5台风机的历史功率数据进行处理,得到风机的能量密度和功率预测准度,再通过模糊逻辑系统求得风机健康系数,并按照健康系数进行备用功率分配。当频率响应系统检测到频率误差后,控制风机变下垂功率输出系统进行功率补偿,稳定电网频率。仿真结果表明,基于健康系数的功率分配系统与变下垂功率输出系统增加了风电参与电网调频的能力,减少了系统备用功率的冗余。     

Large eddy simulations of the effect of vertical staggering in large wind farms

In order to study the effect of vertical staggering in large wind farms, large eddy simulations (LES) of large wind farms with a regular turbine layout aligned with the given wind direction were conducted. In the simulations, we varied the hub heights of consecutive downstream rows to create vertically staggered wind farms. We analysed the effect of streamwise and spanwise turbine spacing, the wind farm layout, the turbine rotor diameter, and hub height difference between consecutive downstream turbine rows on the average power output. We find that vertical staggering significantly increases the power production in the entrance region of large wind farms and is more effective when the streamwise turbine spacing and turbine diameter are smaller. Surprisingly, vertical staggering does not significantly improve the power production in the fully developed regime of the wind farm. The reason is that the downward vertical kinetic energy flux, which brings high velocity fluid from above the wind farm towards the hub height plane, does not increase due to vertical staggering. Thus, the shorter wind turbines are effectively sheltered from the atmospheric flow above the wind farm that supplies the energy, which limits the benefit of vertical staggering. In some cases, a vertically staggered wind farm even produced less power than the corresponding non vertically staggered reference wind farm. In such cases, the production of shorter turbines is significantly negatively impacted while the production of the taller turbine is only increased marginally.     

Application of a LES technique to characterize the wake deflection of a wind turbine in yaw

When a wind turbine works in yaw, the wake intensity and the power production of the turbine become slightly smaller and a deflection of the wake is induced. Therefore, a good understanding of this effect would allow an active control of the yaw angle of upstream turbines to steer the wake away from downstream machines, reducing its effect on them. In wind farms where interaction between turbines is significant, it is of interest to maximize the power output from the wind farm as a whole and to reduce fatigue loads on downstream turbines due to the increase of turbulence intensity in wakes. A large eddy simulation model with particular wind boundary conditions has been used recently to simulate and characterize the turbulence generated by the presence of a wind turbine and its evolution downstream the machine. The simplified turbine is placed within an environment in which relevant flow properties like wind speed profile, turbulence intensity and the anisotropy of turbulence are found to be similar to the ones of the neutral atmosphere. In this work, the model is used to characterize the wake deflection for a range of yaw angles and thrust coefficients of the turbine. The results are compared with experimental data obtained by other authors with a particle image velocimetry technique from wind tunnel experiments. Also, a comparison with simple analytical correlations is carried out. Copyright © 2009 John Wiley & Sons, Ltd.     

基于CFD的二维垂直轴风力机性能计算

采用计算流体力学方法(CFD)针对垂直轴风力发电机,开展简化的二维绕流特性研究。首先,基于开放型转子和增强型转子,研究网格节点数和壁面y+、计算时间步长和湍流模型等的变化对计算结果的影响,对计算模型和方法进行确认。随后,计算分析增强型垂直轴风力机与开放型垂直轴风力机的特性。结果表明,与开放性垂直轴风力发电机相比,增强型垂直轴风力发电机的功率系数和转矩系数有明显增加,且达到最大值的位置向叶尖速比增大的方向移动。然后对增强型垂直轴风力机发电机在不同来流风速下进行计算,发现增强型垂直轴风力发电机的转子转矩随来流风速增加,而转矩系数和功率系数与来流风速无关。最后,针对定子叶片在不同的方向开展计算研究。结果表明,定子叶片在不同方向时,增强型垂直轴风力机的转子转矩不同,且转矩到达峰值的位置也不同;在当前3个方向角中,叶片处于0°方向角时风力机具有最高的转矩系数,即具有最佳的功率系数。     

Combining economic and fluid dynamic models to determine the optimal spacing in very large wind farms

Wind turbine spacing is an important design parameter for wind farms. Placing turbines too close together reduces their power extraction because of wake effects and increases maintenance costs because of unsteady loading. Conversely, placing them further apart increases land and cabling costs, as well as electrical resistance losses. The asymptotic limit of very large wind farms in which the flow conditions can be considered ‘fully developed’ provides a useful framework for studying general trends in optimal layouts as a function of dimensionless cost parameters. Earlier analytical work by Meyers and Meneveau (Wind Energy 15, 305–317 (2012)) revealed that in the limit of very large wind farms, the optimal turbine spacing accounting for the turbine and land costs is significantly larger than the value found in typical existing wind farms. Here, we generalize the analysis to include effects of cable and maintenance costs upon optimal wind turbine spacing in very large wind farms under various economic criteria. For marginally profitable wind farms, minimum cost and maximum profit turbine spacings coincide. Assuming linear‐based and area‐based costs that are representative of either offshore or onshore sites we obtain for very large wind farms spacings that tend to be appreciably greater than occurring in actual farms confirming earlier results but now including cabling costs. However, we show later that if wind farms are highly profitable then optimization of the profit per unit area leads to tighter optimal spacings than would be implied by cost minimization. In addition, we investigate the influence of the type of wind farm layout. © 2016 The Authors Wind Energy Published by John Wiley & Sons Ltd     

Evaluation of the effects of turbulence model enhancements on wind turbine wake predictions

The modelling of wind turbine wakes is investigated in this paper using a Navier–Stokes solver employing the k–ω turbulence model appropriately modified for atmospheric flows. It is common knowledge that even single‐wind turbine wake predictions with computational fluid dynamic methods underestimate the near wake deficit, directly contributing to the overestimation of the power of the downstream turbines. For a single‐wind turbine, alternative modelling enhancements under neutral and stable atmospheric conditions are tested in this paper to account for and eventually correct the turbulence overestimation that is responsible for the faster flow recovery that appears in the numerical predictions. Their effect on the power predictions is evaluated with comparison with existing wake measurements. A second issue addressed in this paper concerns multi‐wake predictions in wind farms, where the estimation of the reference wind speed that is required for the thrust calculation of a turbine located in the wake(s) of other turbines is not obvious. This is overcome by utilizing an induction factor‐based concept: According to it, the definition of the induction factor and its relationship with the thrust coefficient are employed to provide an average wind speed value across the rotor disk for the estimation of the axial force. Application is made on the case of five wind turbines in a row. Copyright © 2010 John Wiley & Sons, Ltd.     

Testing basic performance of a very small wind turbine designed for multi-purposes

A very small wind turbine system for multi-purposes was developed and its performance was reported in this paper. The rotor diameter of the turbine is 500 mm. The tests of the energy output, turbine speed, power coefficient, and torque of turbine were carried out for a wide rage of free stream velocity. The flow around the wind turbine and the influence of the turbulence were investigated with a particle image velocimetry. Experimentally obtained power coefficient was 0.4 in maximum and 0.36 in the rated running condition, respectively. The tip speed ratio corresponding to the optimum driving condition was 2.7. Comparing with the other commercial turbines, the performance was excellent at a slow turbine speed. By the flow visualization and PIV measurement around the wind turbine, the approaching flow velocity and the accelerated flow field passing the blade tip was obtained. It was confirmed that the actual flow passed through the blades was about 20% slower than the ideal flow. Tip vortex shed from the blade tip was also visualized clearly.     

Experimental investigation of wake effects on wind turbine performance

The wake interference effect on the performance of a downstream wind turbine was investigated experimentally. Two similar model turbines with the same rotor diameter were used. The effects on the performance of the downstream turbine of the distance of separation between the turbines and the amount of power extracted from the upstream turbine were studied. The effects of these parameters on the total power output from the turbines were also estimated. The reduction in the maximum power coefficient of the downstream turbine is strongly dependent on the distance between the turbines and the operating condition of the upstream turbine. Depending on the distance of separation and blade pitch angle, the loss in power from the downstream turbine varies from about 20 to 46% compared to the power output from an unobstructed single turbine operating at its designed conditions. By operating the upstream turbine slightly outside this optimum setting or yawing the upstream turbine, the power output from the downstream turbine was significantly improved. This study shows that the total power output could be increased by installing an upstream turbine which extracts less power than the following turbines. By operating the upstream turbine in yawed condition, the gain in total power output from the two turbines could be increased by about 12%.     

海上风电场功率提升和疲劳平衡综合优化控制

针对海上风电场,综合功率提升和疲劳平衡分配的优化目标,提出一种以天为优化周期的优化策略。在电网高负荷时段,基于Jensen尾流模型,以轴向诱导因子为优化变量,风电场整场功率最大为目标,运用随机粒子群算法进行风功率利用提升优化控制;在电网低负荷时段,基于风电机组综合疲劳系数计算方法,以机组轴向诱导因子为优化变量,应用尾流计算模型调整轴向诱导因子来满足电网限功率指令,以机组疲劳系数标准差最小为目标,采用粒子群算法寻优进行疲劳平衡优化。以某海上风电场进行算例分析,结果表明该优化策略在一天的优化周期内可较好地实现风电场功率提升和疲劳平衡的综合优化。     

An experimental investigation on the effect of individual turbine control on wind farm dynamics

Individual wind turbines in a wind farm typically operate to maximize their performance with no consideration of the impact of wake effects on downstream turbines. There is potential to increase power and reduce structural loads within a wind farm by properly coordinating the turbines. To effectively design and analyze coordinated wind turbine controllers requires control‐oriented turbine wake models of sufficient accuracy. This paper focuses on constructing such a model from experiments. The experiments were conducted to better understand the wake interaction and impact on voltage production in a three‐turbine array. The upstream turbine operating condition was modulated in time, and the dynamic impact on the downstream turbine was recorded through the voltage output time signal. The flow dynamics observed in the experiments were used to improve a static wake model often used in the literature for wind farm control. These experiments were performed in the atmospheric boundary layer wind tunnel at the Saint Anthony Falls Laboratory at the University of Minnesota using particle image velocimetry for flow field analysis and turbine voltage modulation to capture the physical evolution in addition to the dynamics of turbine wake interactions. Copyright © 2015 John Wiley & Sons, Ltd.     

Wind power distributions: A review of their applications

This paper presents the features of wind power distributions that have been analytically obtained from wind distribution functions. Simple equations establishing a relationship between mean power density and wind speed have been obtained for a given location and wind turbine (WT). Four different concepts relating wind power distribution functions are shown: the power transported by the wind; the theoretical maximum convertible power from it according to the Betz’ law; the maximum convertible power from the wind considering more realistic limits that will be explained; finally an even more approximate limit to the maximum power obtained from a wind turbine, considering its parameters. Similarly, four different equations are obtained establishing relationships between the mean power density and the mean wind speed. These equations are very simple and very useful when discarding locations for wind turbine installation.     

The potential of cross-stream aligned sub-arrays to increase tidal turbine efficiency

A theoretical model is proposed for a row of sub-arrays of tidal turbines aligned in a cross-stream fashion across part of a wide channel. This model builds on previous work investigating the behaviour of a single partial row array that split the problem into two flow scales; device and channel. In the present work, three flow scales are proposed: device, sub-array and channel flow, allowing the mass, energy and momentum conservation balance to be assessed separately at each scale. The power potential of a row of sub-arrays with varying blockage ratios at each flow scale is investigated, and it is found that increasing device local blockage has the greatest potential to increase power yield. It is also found that, for such a single row tidal farm with a sufficient number of devices in a very wide channel, splitting the long fence array into multiple smaller co-linear sub-fences can increase the overall energy extraction potential. A new maximum power coefficient is found in infinitely wide flow, increasing from the Lanchester-Betz limit of 0.593 for turbines in unblocked flow, past the partial row array limit of 0.798, to a new limiting value of 0.865 for a row of multiple sub-arrays.