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WIND ENERGY Wind Energ. (2016) Published online in Wiley Online Library (wileyonlinelibrary.com) RESEARCH ARTICLE Combining economic and fluid dynamic models to determine the optimal spacing in very

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WIND ENERGY Wind Energ. (2016) Published online in Wiley Online Library (wileyonlinelibrary.com) RESEARCH ARTICLE Combining economic and fluid dynamic models to determine the optimal spacing in very large wind farms Richard J. A. M. Stevens 1,4, Benjamin F. Hobbs 2, Andrés Ramos 3 and Charles Meneveau 4 1 Department of Physics, Mesa+ Institute and J. M. Burgers Centre for Fluid Dynamics, University of Twente, 7500 AE Enschede, The Netherlands 2 Department of Geography and Environmental Engineering and the Environment, Energy, Sustainability and Health Institute, Johns Hopkins University, Baltimore, Maryland , USA 3 ICAI School of Engineering, Institute for Research in Technology, Comillas Pontifical University, C/ Santa Cruz de Marcenado 26, Madrid, Spain 4 Department of Mechanical Engineering and Center for Environmental and Applied Fluid Mechanics, Johns Hopkins University, Baltimore, Maryland 21218, USA ABSTRACT 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, (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 The Authors. Wind Energy Published by John Wiley & Sons, Ltd. KEYWORDS wind farm; engineering economics; fluid dynamic models; coupled wake boundary layer model; optimal turbine spacing; wind farm design; turbine wakes; renewable energy Correspondence Richard J. A. M. Stevens, Department of Physics, Mesa+ Institute and J. M. Burgers Centre for Fluid Dynamics, University of Twente, 7500 AE Enschede, The Netherlands. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. Received 18 February 2016; Revised 24 June 2016; Accepted 6 July INTRODUCTION Wind farms are becoming increasingly larger. For instance, the Alta and Roscoe wind farms in Texas have approximately 600 turbines. Therefore, it is important to better understand the influence of wake effects on the optimal turbine spacing in wind farms with hundreds or thousands of turbines. 1 For smaller wind farms, the majority of the turbines can be placed such that wake effects are rather limited. For the design of such wind farms, the industry uses site-specific, detailed optimization 2016 The Authors. Wind Energy Published by John Wiley & Sons, Ltd. Combining economic and fluid dynamic wind farm models R. J. A. M. Stevens et al. calculations for wind turbine placement based on wake type models. 2 5 Such calculations aim to place the turbines such that wake effects are limited with respect to the main incoming wind direction at the site under consideration. There are also academic studies that use wake models to optimize the placement of a limited number of turbines on a given land area using Monte Carlo simulations, 6 genetic algorithms 7 or evolutionary algorithms. 8, 9 For an overview on this, we refer to the review by Herbert Acero et al. 10 The typical wind turbine spacing that is used in actual wind farms nowadays is 6 10D, whered is the turbine diameter. It is known that these wake type models do not explicitly capture the effect of the interaction between the atmosphere and very large wind farms These effects are better described by top-down type models, which are based on momentum analysis and horizontal averaging. These models give a vertical profile of the average velocity profile by assuming the existence of two logarithmic velocity regions, one above the turbine hub-height and the other below The mean velocity at hub-height is used to predict the wind farm performance as function of wind farm design parameters. The top-down model approach allows one to analytically calculate the wind farm performance for very large wind farms. Meyers and Meneveau 24 used this approach to analyze the optimal spacing in the limit of very large wind farms by accounting for the turbine and land costs and found an optimal spacing of 12 15D, which is significantly larger than the value found in actual wind farms. Later work by Stevens 23 showed that the predicted optimal spacing is much closer to the values found in actual wind farms when the smaller number of turbines (and thus smaller contribution of wake effects) in these farms are taken into account. The purpose of this paper is to generalize the Meyers and Meneveau 24 analysis for very large farms by including effects of cable and maintenance costs in addition to land and turbine costs upon optimal wind turbine spacing under various economic criteria. Figure 1 shows in schematic fashion the general trade-offs that we analyze. The horizontal axis is the spacing between turbines while the vertical axis expresses the levelized cost per MWh of energy production, defined as the present worth (at an assumed interest rate) over the wind farm lifetime of the farm s capital as well as operations and maintenance (O&M) costs, divided by the present worth of energy (MWh) produced over the farm s life. The cost is normalized so that 1.0 is the hypothetical levelized cost of a single wind turbine whose output is unaffected by wakes from other turbines and for which there are no capital costs beyond the turbine and its associated structure. A real wind farm with multiple turbines will have a levelized cost greater than this level because there will be some wake effects (decreasing the denominator) as well as additional costs for land acquisition (perhaps in the form of offshore leases), cables between turbines and associated electrical resistance losses, and roads (in the case of onshore wind farms). The downward sloping curve (turbine capital and O&M cost) shows how wakes affect the levelized turbine, structure and O&M costs; at a small spacing, wake effects significantly shrink the MWh production (and thus the denominator of levelized costs). That curve approaches the 1.0 asymptote for large spacings. Meanwhile, the upward sloping curve (quadratic and linear cost) represents the levelized cost of land, cables and roads, which approaches zero at the origin and is convex (bending upwards) if land costs are important, because they are a quadratic function of spacing. The sum of these two cost curves gives the overall levelized cost of power production, which will be U-shaped and will have a minimum at the cost-optimal spacing. The spacing that minimizes the cost of wind depends on shapes and levels of both the capital and O&M cost curve and the quadratic/linear cost curve. In this paper, we use the coupled wake boundary layer (CWBL) model of Stevens et al. 13, 14, 25 to account for wake effects in the capital/o&m cost curve, and then combine it with an improved methodology for estimating the cost curve for very large farms. Because the CWBL model is an analytical model, it allows a very fast evaluation of the wind farms performance. Important benefits of the CWBL model compared with the Calaf et al. 21 model are (i) that the CWBL model Figure 1. Schematic of analysis of optimal spacing in a large wind farms, showing trade-off between wake losses (reflected in higher capital and O&M costs for lower values of turbine spacing) and land, cable and other spacing-related expenditures (reflected in increased quadratic plus linear costs as a function of spacing). R. J. A. M. Stevens et al. Combining economic and fluid dynamic wind farm models is able to predict the effect of the turbine layout on the performance of very large wind farms and (ii) that the CWBL model captures the entrance effects. Thus, the use of the CWBL model allows us to analyze the optimal spacing of wind farms in more detail than the Calaf et al. 21 model. This CWBL approach can be used for wind farms of varying size. As wake effects on the optimal spacing are most pronounced in very large fully developed wind farms, we focus on that case here. Apriori, it is difficult to specify a very sharp definition of the fully developed regime of large wind farms because it depends on case-by-case arrangements. For the purpose of mean power optimization as carried out in this paper, we take the view that fully developed means that the turbine power becomes nearly constant with downstream distance. Common experience such as in Horns Rev (80 turbines) shows that at the end of the wind farm turbines is exposed to fully developed wind conditions. Therefore, we expect fully developed wind conditions to become relevant for wind farms with several hundred turbines. The improved costing methodology used in this paper considers cost categories that are quadratic (e.g., area-related costs arising from leasing or occupying land) and linear (e.g., the expense of inter-turbine cabling, electrical resistance losses and roads) related to the inter-turbine spacing. Based on an order of magnitude estimation of the quadratic and linear costs, we argue in this paper that linear costs are significant and are especially important for offshore wind farms. In addition, to account for the effect of both quadratic and linear cost components in this paper, we also present versions of the model that address other issues not considered by Meyers and Meneveau. 24 One version accounts for the effect of spacing-affected turbulence upon cost-optimal spacing, because closely spaced turbines are likely to experience increased turbulence-related fatigue and damage. Another version recognizes that areas that are particularly profitable for wind development can be limited, implying that the objective of design is not to minimize cost of production but to maximize profit per unit area. In Section 2, we start with the definition of the quadratic and linear cost components and the dimensionless cost parameters that are important for very large wind farms. In Section 3, we introduce the CWBL model 13, 25 that will be used to estimate the power and turbulence intensity for different wind farm designs. In Section 4, the modeling approach introduced by Meyers and Meneveau 24 is extended to include linear costs (e.g., the costs of cables, roads and resistance losses that scale linearly with the distance between turbines) while in Section 5 the optimal turbine spacing is determined by optimizing the profit per unit area of the wind farms instead of the normalized power per unit cost. In Section 6, the effect of the maintenance costs, based on the predicted turbulence intensity by the CWBL model, is evaluated. Subsequently, we will use the CWBL model to estimate the effect of the wind farms layout on the optimal turbine spacing in very large wind farms in Section 7. Conclusions are given in Section DEFINITION OF DIMENSIONLESS PARAMETERS In this paper, we will study the effect of different cost factors on the optimal turbine distance S D sd,whered is the turbine diameter, and s the dimensionless turbine distance that will be used in the remainder of this paper. For convenience, we introduce the following dimensionless parameters to analyze the effect of the main economic influences on optimal turbine spacing: D Cost quadratic Cost turbine =D 2, ˇ D Cost linear Cost turbine =D, D Rev 1, D Main 1 (1) Cost turbine Cost turbine where all the cost factors have been normalized with respect to the turbine costs and the turbine diameter when appropriate. Cost turbine is defined as the present worth (at some discount rate) of the capital and O&M cost of a single turbine, excluding all costs that depend on distance between turbines, such as cabling, land leasing and roads. Cost quadratic and Cost linear are, respectively, the present worth of area and length dependent capital and other costs that increase with turbine spacing. Quadratic costs include for example costs arising from leasing or occupying land while linear costs include items such as inter-turbine cabling, electrical resistance losses and roads. Rev 1 is the present worth of expected revenue of a single wind turbine that does not experience wake effects. Main 1 are the maintenance costs over lifetime, again for a single turbine without wake effects, the estimate should indicate for which turbulence intensity this estimate was obtained, such Table I. Reference estimates of reasonable dimensionless cost parameters for wind farms, see details in the Appendix. (quadratic) ˇ (linear) (revenue) (maintenance) Onshore Offshore The cost parameters (quadratic), ˇ (linear), (revenue) and (maintenance) are normalized with the turbine costs, see equation (1). Combining economic and fluid dynamic wind farm models R. J. A. M. Stevens et al. that this can be accounted for in the calculations. Lifetime revenue is defined as being net of the expense of interconnecting the farm to the main power grid, because that cost is not included in Cost turbine. In Sections 4 7, we assess the optimal spacing and profitability for wide ranges of the linear (ˇ) and quadratic () turbine cost parameters, in acknowledgement of the large uncertainty in their values for particular locations. In the Appendix, we present some order of magnitude estimates for the dimensionless cost parameters for onshore and offshore wind farms, see Table I, to direct the reader s attention to more practically relevant regions of the parameter space. The parameter is meant to include all sources of revenues, including tax incentives and renewable energy credits. The precise value is not based on actual experience but upon an expectation that constructed wind farms will be profitable. Therefore, we assume the same for both onshore and offshore facilities under the assumption that public policy and market conditions are such that either are somewhat but not greatly profitable for development. Thus, although costs are higher for offshore facilities, we anticipate that revenues will also be higher, so that the ratio of revenues to costs for financially viable wind farms will somewhat but not greatly above 1.0. For instance, offshore farms are given more subsidies in the UK than onshore farms. 26 To assure the aforementioned assumption of wind farm profitability, we have selected D 1.5 for the onshore and offshore reference cases. 3. INPUT PREDICTIONS FROM THE CWBL MODEL In the CWBL model, the top-down approach by Calaf et al. 21 is used to calculate the performance of the wind turbines in the fully developed region of the wind farm. In the CWBL model, the ratio of the mean velocity to the reference incoming velocity at hub-height in the fully develop region is hui.z h / hu 0 i.z h / D ln ıh =z 0,lo ln zh ln 1 C D # b 1 zh ln (2) ı H =z 0,hi z 0,hi 2z h z 0,lo Here, ı H indicates the height of the atmospheric boundary layer in the fully developed region of the wind farm, b D w =.1 C w /, w 28p C T =.8w f s x s y /, z 0,hi denotes the roughness length of the wind farm, which is evaluated in the model according to z 0,hi D z h.1 C R / b exp Œq C lnœ.z h =z 0,lo /.1 R / b / 2 1=2,whereR D D=.2z h /, q D C T =.8w f s x s y 2 /, hu 0 i.z h / D u = ln z h =z 0,lo and wf indicates the effective wake area coverage, which is obtained from the two-way coupling with the wake model part of the CWBL model, and is equal to the ratio of wake area divided by total area. 25 Because of the two-way coupling between the wake and the top-down model, w f depends on parameters such as the streamwise distance between the turbines s x, the relative positioning of the turbines and the wake coefficient in the fully developed region of the wind farm k w,1. As we focus on the optimal spacing for very large wind farms, we assume that the average power output of the wind turbine is the same as the turbine power output in the fully developed region of the wind farms. The power ratio P 1 =P 1 is given by the ratio of cubed mean velocity at hub-height with wind turbines compared with the reference case without wind farms: P 1 hui.zh / 3.s x, s y, layout,.../ D (3) P 1 hu 0 i.z h / Figure 2. The velocity field, obtained with the CWBL model, in the fully developed part of a very large wind farm with a spacing of s D 10.5 with an (a) aligned, (b) staggered and (c) full wake coverage limit layout. The black lines indicate the turbine positions. The wake coverage area w f is defined as the percentage of the area in the fully developed regime of the wind farm where u 0.95u Here, w f D 0.35 (aligned), w f D 0.60 (staggered) and w f D 1.00 (full wake coverage limit). R. J. A. M. Stevens et al. Combining economic and fluid dynamic wind farm models In earlier work, we showed 13, 25 that the CWBL model gives improved predictions for the power output in the fully developed region compared with the top-down model introduced by Calaf et al. 21 or stand-alone wake models. 3 In addition, the CWBL model is able to predict the difference between the performance of different wind farms geometries. Here, we specifically focus on aligned, staggered and full wake coverage limit layouts. Figure 2 shows a visualization of these different wind farm layouts. The full wake coverage limit is defined as a wind farm configuration for which the wake coverage area w f D 1 in the CWBL model. In the full wake coverage limit layout, the performance of the wind farm is better than for aligned or staggered wind farms, because the distance between directly upstream turbines is larger, by placing turbines such that turbines on the next downstream row are just outside the wake. For s 7, the staggered wind farm layout still has a wake area coverage w f D 1, and therefore, the full wake coverage limit only outperforms the staggered layout for s 7. This effects is discussed in more detail in the work of Stevens et al. 27 We note that most calculations presented in this paper, except in Section 7 where the effect of the wind farms layout is discussed, are based on the power estimates for the full wake coverage limit layout. 4. OPTIMAL TURBINE SPACING OF VERY LARGE WIND FARMS WITH CABLE COST In order to investigate the robustness of the results from Meyers and Meneveau, 24 we include, in addition to the quadratic (land) costs, a linear cost component defining the total cost per turbine as Cost D Cost turbine C.sD/Cost linear C.sD/ 2 Cost quadratic (4) Following the approach by Meyers and Meneveau, 24 this leads to the following normalized power per unit cost P.s x, s y, layout,.../ D P 1 Cost D P 1 Cost turbine C.sD/Cost linear C.sD/ 2 Cost quadratic P 1 P 1 1 D Cost turbine P 1 1 C ˇs C s 2 (5) where P 1 D P 1.s x, s y, layout,.../ is the aver

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