Jensen Wake Model Research Articles

Wind farm layout optimization using a Gaussian-based wake model

The wind farm layout optimization problem has received considerable attention over the past two decades. The objective of this problem is to determine the wind farm layout that maximizes the annual energy generated. The majority of studies that have solved this problem simulated the velocity deficit using the Jensen wake model. However, this model is not in agreement with field measurements and computational fluid dynamics simulations. In this study, an approach to solve the wind farm layout optimization problem based on a Gaussian wake model is proposed. The Gaussian wake model uses an exponential function to evaluate the velocity deficit, in contrast to the Jensen wake model that assumes a uniform velocity profile inside the wake. The proposed approach minimizes the annual cost of energy of a wind farm using a genetic algorithm. The application of the proposed approach yields higher annual generation and a lower computational time for all wind scenarios under study. Under a more complex wind scenario, the improvement was relatively small. This suggests that the use of a more robust wake model in the WFLO problem, does not lead to greater efficiency in real wind cases.

Windfarm Power Optimization Using Yaw Angle Control

In this paper, we extend the Park and Jensen wake model to include the effects of yaw angle rotation and wake deflection of wind turbines. We first perform a numerical study to find the optimal values of induction factor and yaw angle of wind turbines in a single column of a windfarm for achieving the maximum total power with wake effects. This study shows that the maximum power in a single column is achieved by keeping the induction factor at one-third and only changing the yaw angle to deflect the wake. We then propose a dynamic programming formulation to maximize the total power production in a windfarm with a single column of turbines using yaw angle as the control variable. We also extend our expressions to a windfarm with multiple rows and columns of turbines and perform simulations on the 3 $\times$ 3 and 4 $\times$ 4 grid topologies. Our results show that the optimal induction factor for most turbines is quite close to one-third and yaw angle acts as the dominant optimization variable.

Optimizing the number and locations of turbines in a wind farm addressing energy-noise trade-off: A hybrid approach

Micro-siting is an optimal way of placing turbines inside a wind farm while considering various design objectives and constraints. Using a well-established Jensen wake model and ISO-9613-2 noise calculation, this study performs a wind farm layout optimization based on a multi-objective trade-off between minimization of the noise propagation and maximization of the energy generation. A novel hybrid methodology is developed which is a combination of probabilistic real-binary coded multi-objective evolutionary algorithm and a newly proposed deterministic gradient based non-dominated normalized normal constraint method. Based on the Inverted Generational Distance metric, the performance of the proposed method is found to be better than the conventional normalized normal constraint method or the concerned evolutionary method alone. Moreover, in contrast to the previous studies, the generated non-dominated front is capable of providing a trade-off between various alternative energy-noise solutions, along with an additional information about the corresponding turbine numbers and their optimal location coordinates. As a result, the decision maker can choose from different competing wind turbine layouts based on existing noise and other standard regulations.

Prediction of multi-wake problems using an improved Jensen wake model

The improved analytical wake model named as 2D_k Jensen model (which was proposed to overcome some shortcomes in the classical Jensen wake model) is applied and validated in this work for wind turbine multi-wake predictions. Different from the original Jensen model, this newly developed 2D_k Jensen model uses a cosine shape instead of the top-hat shape for the velocity deficit in the wake, and the wake decay rate as a variable that is related to the ambient turbulence as well as the rotor generated turbulence. Coupled with four different multi-wake combination models, the 2D_k Jensen model is assessed through (1) simulating two wakes interaction under full wake and partial wake conditions and (2) predicting the power production in the Horns Rev wind farm for different wake sectors around two different wind directions. Through comparisons with field measurements, results from Large Eddy Simulations (LES) as well as results from other commercial codes, it is found that the predictions obtained with the 2D_k Jensen model exhibit good to excellent agreements with experimental and LES data.

A Continuously Differentiable Turbine Layout Optimization Model for Offshore Wind Farms

This article presents a continuously differentiable modification of the commonly used Jensen wake model. This property is con- served in the wake combination model and, by formulating the turbine's power curve as a set of constraints, also in the computation of a turbine's power output. The resulting objective function, maximizing the total power production, and optimizing the turbine positions, is thus continuously differentiable, and gradient based solution methods can be applied. Numerical experiments are conducted with simulated wind data of six offshore wind farm locations.

Surrogate based wind farm layout optimization using manifold mapping

High computational cost associated with the high fidelity wake models such as RANS or LES serves as a primary bottleneck to perform a direct high fidelity wind farm layout optimization (WFLO) using accurate CFD based wake models. Therefore, a surrogate based multi-fidelity WFLO methodology (SWFLO) is proposed. The surrogate model is built using an SBO method referred as manifold mapping (MM). As a verification, optimization of spacing between two staggered wind turbines was performed using the proposed surrogate based methodology and the performance was compared with that of direct optimization using high fidelity model. Significant reduction in computational cost was achieved using MM: a maximum computational cost reduction of 65%, while arriving at the same optima as that of direct high fidelity optimization. The similarity between the response of models, the number of mapping points and its position, highly influences the computational efficiency of the proposed method. As a proof of concept, realistic WFLO of a small 7-turbine wind farm is performed using the proposed surrogate based methodology. Two variants of Jensen wake model with different decay coefficients were used as the fine and coarse model. The proposed SWFLO method arrived at the same optima as that of the fine model with very less number of fine model simulations.

Multi-Objective Random Search Algorithm for Simultaneously Optimizing Wind Farm Layout and Number of Turbines

A new algorithm for multi-objective wind farm layout optimization is presented. It formulates the wind turbine locations as continuous variables and is capable of optimizing the number of turbines and their locations in the wind farm simultaneously. Two objectives are considered. One is to maximize the total power production, which is calculated by considering the wake effects using the Jensen wake model combined with the local wind distribution. The other is to minimize the total electrical cable length. This length is assumed to be the total length of the minimal spanning tree that connects all turbines and is calculated by using Prim's algorithm. Constraints on wind farm boundary and wind turbine proximity are also considered. An ideal test case shows the proposed algorithm largely outperforms a famous multi-objective genetic algorithm (NSGA-II). In the real test case based on the Horn Rev 1 wind farm, the algorithm also obtains useful Pareto frontiers and provides a wide range of Pareto optimal layouts with different numbers of turbines for a real-life wind farm developer.

Wind farm micro-siting optimization using novel cell membrane approach

Micro-siting aims to determine every wind turbine's position to reduce velocity deficits caused by the wake effect. The Novel CMO (cell membrane optimization) approach is proposed to overcome this weakness. It plays a vital role to utilize more wind resources while the type of wind turbine and the area to build a wind farm have been determined. The work is based on the Jensen wake model, and the hypothetical situations are the same as those used by the former researchers. There are three wind cases: constant speed with one direction, constant speed with variable directions and variable speeds with variable directions. The area of wind farm is assumed to be a plane 2km×2km square. The numbers of the wind turbines is 26, 19 and 15 in three cases respectively. Compared with Gene Algorithm introduced by G. Mosetti, CMO's results are acceptable and the velocity deficit is smaller, which results from that CMO's variables is continuous and can make the most of the area the wind turbines can be placed. Moreover, it performs well to avoid the local optimal solutions by dividing the searching particles into different types which move according to different rules.

Generalized coupled wake boundary layer model: applications and comparisons with field and LES data for two wind farms

We describe a generalization of the Coupled Wake Boundary Layer (CWBL) model for wind-farms that can be used to evaluate the performance of wind-farms under arbitrary wind inflow directions whereas the original CWBL model (Stevens et al., J. Renewable and Sustainable Energy 7, 023115 (2015)) focused on aligned or staggered wind-farms. The generalized CWBL approach combines an analytical Jensen wake model with a "top-down" boundary layer model coupled through an iterative determination of the wake expansion coefficient and an effective wake coverage area for which the velocity at hub-height obtained using both models converges in the "deep-array" portion (fully developed region) of the wind-farm. The approach accounts for the effect of the wind direction by enforcing the coupling for each wind direction. Here we present detailed comparisons of model predictions with LES results and field measurements for the Horns Rev and Nysted wind-farms operating over a wide range of wind inflow directions. Our results demonstrate that two-way coupling between the Jensen wake model and a "top-down" model enables the generalized CWBL model to predict the "deep-array" performance of a wind-farm better than the Jensen wake model alone. The results also show that the new generalization allows us to study a much larger class of wind-farms than the original CWBL model, which increases the utility of the approach for wind-farm designers.

The impact of land use constraints in multi-objective energy-noise wind farm layout optimization

Recently the environmental impact of onshore wind farms is receiving major attention from both governments and wind farm designers. As land is more extensively exploited for wind farms, it is more likely for wind turbines to be in proximity with human dwellings, infrastructure (e.g. roads, transmission lines), and natural habitats (e.g. rivers, lakes, forests). This proximity makes significant portions of land unusable for the designers, introducing a set of land-use constraints. In this study, we conduct a constrained and continuous-variable multi-objective optimization that considers energy and noise as its objective functions, based on Jensen's wake model and the ISO-9613-2 noise standard. A stochastic evolutionary algorithm (NSGA-II) solves the optimization problem, while the land-use constraints are handled with static and dynamic penalty functions. Results of this study illustrate the effect of constraint severity and spatial distribution of unusable land on the trade-off between energy generation and noise production.

On the application of the Jensen wake model using a turbulence‐dependent wake decay coefficient: the Sexbierum case

We present a methodology to process wind turbine wake simulations, which are closely related to the nature of wake observations and the processing of these to generate the so-called wake cases. The method involves averaging a large number of wake simulations over a range of wind directions and partly accounts for the uncertainty in the wind direction assuming that the same follows a Gaussian distribution. Simulations of the single and double wake measurements at the Sexbierum onshore wind farm are performed using a fast engineering wind farm wake model based on the Jensen wake model, a linearized computational fluid dynamics wake model by Fuga and a nonlinear computational fluid dynamics wake model that solves the Reynolds-averaged Navier–Stokes equations with a modified k-ε turbulence model. The best agreement between models and measurements is found using the Jensen-based wake model with the suggested post-processing. We show that the wake decay coefficient of the Jensen wake model must be decreased from the commonly used onshore value of 0.075 to 0.038, when applied to the Sexbierum cases, as wake decay is related to the height, roughness and atmospheric stability and, thus, to turbulence intensity. Based on surface layer relations and assumptions between turbulence intensity and atmospheric stability, we find that at Sexbierum, the atmosphere was probably close to stable, although the stability was not observed. We support these assumptions using detailed meteorological observations from the Høvsøre site in Denmark, which is topographically similar to the Sexbierum region. © 2015 The Authors. Wind Energy published by John Wiley & Sons Ltd.

Wind farm power optimization including flow variability

A model-based optimisation approach is used to investigate the potential gain of wind-farm power with a cooperative control strategy between the wind turbines. Based on the Jensen wake model with the Katic wake superposition rule, the potential gain for the Nysted offshore wind farm is calculated to be 1.4–5.4% for standard choices 0.4 ≥ k ≥ 0.25 of the wake expansion parameter. Wake model fits based on short time intervals of length 15sec ≤ T ≤ 10 min within three months of data reveal a strong wake flow variability, resulting in rather broad distributions for the wake expansion parameter. When an optimized wind-farm control strategy, derived from a fixed wake parameter, is facing this flow variability, the potential gain reduces to 0.3–0.5%. An omnipotent control strategy, which has real-time knowledge of the actual wake flow, would be able to increase the gain in wind-farm power to 4.9%.

Development and validation of a new two-dimensional wake model for wind turbine wakes

A new two-dimensional (2D) wake model is developed and validated in this article to predict the velocity and turbulence distribution in the wake of a wind turbine. Based on the classical Jensen wake model, this model is further employing a cosine shape function to redistribute the spread of the wake deficit in the crosswind direction. Moreover, a variable wake decay rate is proposed to take into account both the ambient turbulence and the rotor generated turbulence, different from a constant wake decay rate used in the Jensen model. The obtained results are compared to field measurements, wind tunnel experiments, and results of an advanced k−ω turbulence model as well as large eddy simulations. From the comparisons, it is found that the proposed new wake model gives a good prediction in terms of both shape and velocity amplitude of the wake deficit, especially in the far wake which is the region of interest for wind farm development projects.

Power output of offshore wind farms in relation to atmospheric stability

Atmospheric stability is known to influence wind farm power output, by affecting power losses due to wakes. This research tries to answer what atmospheric stability does to the power production and how conventional simulations using the Jensen wake model compare and can be improved. Data is used from two offshore wind farms, Egmond aan Zee (OWEZ) and North Hoyle. Stability distributions are determined using metmast data. By combining this data with the production data, the influence of stability on the power output is studied. It is found that very unstable conditions result in higher power output (i.e. smaller wake losses) than near-neutral conditions, and these again show higher power output than during very stable conditions. Differences in normalized power output of 10-20% exist between the very unstable and very stable conditions. Simulations can be improved by adapting the wake decay constant (WDC). Observed WDC values are k ≥ TI, as opposed to the conventional k ≈ 0.5TI. A hypothesis for further research is proposed regarding the influence of vertical turbulence.

Multi-Objective Wind Farm Layout Optimization Considering Energy Generation and Noise Propagation With NSGA-II

Recently, the environmental impact of wind farms has been receiving increasing attention. As land is more extensively exploited for onshore wind farms, they are more likely to be in proximity with human dwellings, increasing the likelihood of a negative health impact. Noise generation and propagation remain an important concern for wind farm's stakeholders, as compliance with mandatory noise limits is an integral part of the permitting process. In contrast to previous work that included noise only as a design constraint, this work presents continuous-location models for layout optimization that take noise and energy as objective functions, in order to fully characterize the design and performance spaces of the wind farm layout optimization (WFLOP) problem. Based on Jensen's wake model and ISO-9613-2 noise calculations, single- and multi-objective genetic algorithms (GAs) are used to solve the optimization problem. Results from this bi-objective optimization model illustrate the trade-off between energy generation and noise production by identifying several key parts of Pareto frontiers. In particular, it was observed that different regions of a Pareto front correspond to markedly different turbine layouts. The implications of noise regulation policy—in terms of the actual noise limit—on the design of wind farms are discussed, particularly in relation to the entire spectrum of design options.

Wind farm layout optimization in complex terrain: A preliminary study on a Gaussian hill

One of the crucial problems for wind farm (WF) development is wind farm layout optimization. It seeks to find the optimal positions of wind turbines (WTs) inside a WF, so as to maximize and/or minimize a single objective or multiple objectives, while satisfying certain constraints. Although this problem for WFs in flat terrain or offshore has been investigated in many studies, it is still a challenging problem for WFs in complex terrain. In this preliminary study, the wind flow conditions of complex terrain without WTs are first obtained from computational fluid dynamics (CFD) simulation, then an adapted Jensen wake model is developed by considering the terrain features and taking the inflow conditions as input. Using this combined method, the wake effects of WF in complex terrain are properly modelled. Besides, a random search (RS) algorithm proposed in previous study is improved by adding some adaptive mechanisms and applied to solve the layout optimization problem of a WF on a Gaussian shape hill. The layout of the WF with a certain number of WTs is optimized to maximize the total power output, which obtained steady improvements over expert guess layouts.

Data Driven Modelling of the Dynamic Wake Between Two Wind Turbines

Wind turbines in a wind farm, influence each other through the wind flow. Downwind turbines are in the wake of upwind turbines and the wind speed experienced at downwind turbines is hence a function of the wind speeds at upwind turbines but also the momentum extracted from the wind by the upwind turbine. This paper establishes flow models relating the wind speeds at turbines in a farm. So far, research in this area has been mainly based on first principles static models and the data driven modelling done has not included the loading of the upwind turbine and its impact on the wind speed downwind. This paper is the first where modern commercial mega watt turbines are used for data driven modelling including the upwind turbine loading by changing power reference. Obtaining the necessary data is difficult and data is therefore limited. A simple dynamic extension to the Jensen wake model is tested without much success. The best model turns out to be non linear with upwind turbine loading and wind speed as inputs. Using a transformation of these inputs it is possible to obtain a linear model and use well proven system identification methods. Finally it is shown that including the upwind wind direction to explain the wake improve the prediction performance.

A fast algorithm based on the submodular property for optimization of wind turbine positioning

In the design of a wind farm, the placement of turbines is an important factor that affects the efficiency and profit, but automatic placement of turbines is still a challenging problem. This study reveals the “submodular” property of the wind turbine positioning problem based on Jensen wake model. Based on this property, a “lazy greedy” algorithm is used to optimize the placement. This method can obtain solutions with theoretical guarantee of quality. It can also estimate the lower bound of the optimal value of the objective function. This method is tested on three types of wind scenarios. Compared to previous research, this algorithm takes much less time, and always gains a better solution. To enlarge the application scope, the wake model is extended to the large scale complex terrain in this study. The present algorithm and some other algorithms are tested in the simulation of the complex terrain. The experimental results demonstrate the present method’s superior performance.