Elsevier

Renewable Energy

Volume 35, Issue 3, March 2010, Pages 685-694
Renewable Energy

Design of wind farm layout for maximum wind energy capture

https://doi.org/10.1016/j.renene.2009.08.019Get rights and content

Abstract

Wind is one of the most promising sources of alternative energy. The construction of wind farms is destined to grow in the U.S., possibly twenty-fold by the year 2030. To maximize the wind energy capture, this paper presents a model for wind turbine placement based on the wind distribution. The model considers wake loss, which can be calculated based on wind turbine locations, and wind direction. Since the turbine layout design is a constrained optimization problem, for ease of solving it, the constraints are transformed into a second objective function. Then a multi-objective evolutionary strategy algorithm is developed to solve the transformed bi-criteria optimization problem, which maximizes the expected energy output, as well as minimizes the constraint violations. The presented model is illustrated with examples as well as an industrial application.

Introduction

The wind energy market is rapidly expanding worldwide [1]. This rapid growth of the wind energy industry has led to cost reduction challenges. There are various ways of reducing the cost of producing wind power: for example, the site selection, site layout design, predictive maintenance, and optimal control system design [13]. The wind farm layout design is an important component of ensuring the profitability of a wind farm project. An inadequate wind farm layout design would lead to lower than expected wind power capture, increased maintenance costs, and so on. Equation (1) captures the cost of energy (COE) [12]COE=CI×FCR+CRAEP+CO&Mwhere CI is the initial capital cost ($) of the wind farm; FCR is the fixed charge rate (%/year); CR is the levelized replacement cost ($/year); CO&M is the cost of maintenance and operations ($/kWh); AEP is the annual energy production (kWh/year)

Note that at the design stage, AEP is the expected (planned) wind energy production. The annual energy production AEP is affected by the turbine availability, i.e., the number of operational hours in a year. Maximizing the AEP is an effective approach for reducing the cost of energy production. In this paper, AEP is improved by optimizing the wind farm layout design, specifically minimizing the wake loss.

Wind farm layout has been addressed in the literature [4], [6], [9], [17]. Grady et al. [4] and Mosetti et al. [9] used a genetic algorithm to minimize a weighted sum of wind energy and turbine costs. The wind farm is divided into a square grid to facilitate the encoding of a 0–1 type solution. Lackner and Elkinton [6] presented a general framework to optimize the offshore wind turbine layout. Details of how to solve the optimization problem are neither discussed in ref. [6], nor any other wind farm layout design tools, such as WASP [18]. Castro Mora et al. [17] also used a genetic algorithm to maximize an economic function, which is related to turbine parameters and locations. Similarly in ref. [17], the wind farm is represented with a square grid. One of the shortcomings of the approach presented in [17] was that wake loss was not considered. In ref. [4], [9], the wind energy calculation is not based on the power curve function, and wind direction was not fully discussed in their optimization models. This paper extends the approach of [6] by developing specific mathematical models to calculate the wake loss based on turbine locations. Solution of the constrained optimization problem is fully discussed with a double-objective evolutionary strategy algorithm, which can be easily extended by considering additional constraints.

Section snippets

Problem formulation and methodology

Modeling the wind farm layout design problem calls for assumptions. However, the assumptions made in this paper are acceptable in industrial applications and they could be modified or even removed, if necessary.

Assumption 1

For a wind farm project, the number of wind turbines N is fixed and known before the farm is constructed. A typical wind farm project has its total capacity goal dictated by various factors, e.g., finances and turbine availability. For example, to achieve 150 MW (Mega Watt) capacity, a

Power curve

Though a power curve P = f(v)usually resembles a sigmoid function, it could be described as a linear function with a tolerable error. For example, model (17) illustrates a linear power curve function, where vcut−in is the cut-in wind speed. If the wind speed is smaller than the cut-in speed, there is no power output because the torque is not sufficient enough to turn the generator. Similarly, if the wind speed is greater than the rated speed and smaller than the cut-out speed vcut−out, the wind

Discretization of wind speed and wind direction

Assume the wind direction is discretized into Nθ + 1 bins of equal width; let θ1,θ2,,θNθ be the dividing points of wind direction with 0°<θ1θ2θNθ<360°,θ0=0°,θNθ+1=360°. Each bin is associated with a relative frequency 0ωi1,i=0,,Nθ. For example ω0 is the frequency of bin [0°,θ1], ωNθ is the frequency of bin [θNθ,360°]. The frequency ωi can be easily estimated from the wind data.

Wind speed is also discretized into Nv + 1 bins; let v1,v2,v3,,vNv be the dividing points of wind speed with vcut

Evolutionary strategy algorithm

The solution of model (4) can be encoded as a vector used by an evolutionary strategy algorithm [14].

The general form of the jth individual in the evolutionary strategy algorithm is defined as (zj,σj), where zj and σj are two vectors with 2N entries, i.e., zj=(x1j,y1j,,xNj,yNj)T,σj=(σ1,xj,σ1,yj,,σN,xj,σN,yj)T.

The pair (x1j,y1j) is the position of the first turbine, while (σ1,xj,σ1,yj) is used to mutate the first turbine's position. The pair (xNj,yNj) is the position of the Nth turbine and(σN,x

Computational study

To illustrate the concepts presented in this paper, numerical examples and an industrial case study are presented. The wind turbines used in this paper have the following parameters: rotor radius is 38.5 (m); cut-in speed is 3.5 (m/s); rated speed is 14 (m/s); rated power is 1500 (kW). For the linear power curve function, λ = 140.86, η = −500. Hub height is 80 (m). The thrust coefficient CT is assumed to be 0.8, the spreading constant κ is assumed to be 0.075 for land cases.

Knowing the cut-in wind

Conclusion

A generic model for optimizing in-land wind farm layout was presented. The optimization model considered wind farm radius and turbine distance constraints. However, other constraints can be easily incorporated in this model. The model maximizes the energy production by placing wind turbines in such a way that the wake loss is minimized.

As the optimization model is nonlinear, and it is hard to derive an analytical solution from the integration part, wind speed and wind direction are discretized

Acknowledgement

The research reported in this paper has been supported by funding from the Iowa Energy Center, Grant No. 07-01.

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