Realistic Optimization of Parallelogram-Shaped Offshore Wind Farms Considering Continuously Distributed Wind Resources

energies Article Realistic Optimization of Parallelogram-Shaped Offshore Wind Farms Considering Continuously Distributed Wind Resources Angel G. Gonzalez-Rodriguez 1,*,†,‡ , Javier Serrano-González 2,‡ , Manuel Burgos-Payán 2,‡ and Jesús Manuel Riquelme-Santos 2,‡   Citation: Gonzalez-Rodriguez, A.G.; Serrano-Gonzalez, J.; Burgos-Payan, M., Riquelme-Santos, J.M. Realistic Optimization of Parallelogram-Shape Offshore Wind Farms Considering Continuously Distributed Wind Resources. Energies 2021,14, 2895. https://doi.org/10.3390/en14102895 Academic Editor: Galih Bangga Received: 21 April 2021 Accepted: 12 May 2021 Published: 17 May 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Department of Electronic Technology and Automation, University of Jaen, 23071 Jaen, Spain 2 Department of Electrical Engineering, University of Seville, 41004 Seville, Spain; [email protected] (J.S.-G.); [email protected] (M.B.-P.); [email protected] (J.M.R.-S.) *Correspondence: [email protected] † Current address: Department of Electronic Technology and Automation, University of Jaen, A3-442 Jaen, Spain. ‡ These authors contributed equally to this work. Abstract: Offshore wind power plants are becoming a realistic option for the renewable production of electricity. As an improvement tool to the profitability of OWFs, this work presents the first complete non-genetic (and non-binary) evolutionary algorithm to optimize the location, size and layout of a parallelogram-shaped offshore wind farm, as the arrangement that is becoming an standard for offshore wind farms. It has been tested in the HR-I site. Most relevant economic data influencing the investment profitability have been taken into account. In addition, the paper introduces a new approach to offshore wind farm optimization based on a continuous behaviour of varying wind conditions, which allows a more realistic estimation of the energy produced. The proposed optimization approach has been tested based on the available information from HR-I. Obtained solutions present similar values to the actual offshore wind farm in terms of investment and annual energy produced, but differs with respect to the optimal orientation and profitability. The contributions of this paper are: it details the first method to interpolate a continuous distribution of wind rose and Weibull parameters; it presents the first algorithm to obtain a realistic optimal solution to the location+sizing+micro-siting problem for regular arrangements; it is prepared to work with the most complete set of economic, bathymetric, and wind data. Keywords: offshore wind farms; non-genetic evolutionary algorithm; macro-siting; layout optimization; parallelogram-shaped wind farms; continuous distribution; investment costs 1. Introduction Offshore wind technology is becoming established as a realistic option for the production of electricity from renewable sources. The accumulated power capacity at the end of 2019 was 28.3 GW, of which 22 GW corresponds to Europe [ 1 ]. This represents only 0.3% of global energy production, albeit swith a growing role in the world electricity market. The knowledge acquired in the development of previous power plants, and the determination of the sources and severity of the contingencies that delay the development of the project, make offshore wind farm (OWF) projects an investment that is less and less risky and therefore more attractive to potential developers. However, it is yet one of the most expensive technologies in terms of cost-per-MW installed, and a thorough profitability analysis is necessary before considering the realization of an offshore wind farm. In this analysis, the site conditions must be taken into account in order to refine the cost of the investment and deduce the best arrangement of the turbines to maximize the production of electrical energy and minimize the investment. Energies 2021,14, 2895. https://doi.org/10.3390/en14102895 https://www.mdpi.com/journal/energies Energies 2021,14, 2895 2 of 20 1.1. Genetic and Non-Genetic Evolutive Algorithms There are numerous papers that propose an algorithm to solve the micro-siting problem and obtain the optimal arrangement of the turbines in order to maximize the annual energy production (AEP). Serrano in [2] and Herbert-Acero in [3], among others, conduct a deep review of the literature about the wind farm optimization problem (WFOP). It should be noted that the vast majority of previous works that have tackled the problem of optimal micro-positioning of wind turbines have resorted to meta-heuristic optimization techniques to solve the problem. This is because it is a problem with multiple optimal solutions (convexity), and it is not possible to describe its objective function in a completely analytical manner, which makes the problem non-derivable, thus preventing the use of classical gradient-based optimization techniques. Kirchner-Bossi [ 4 ] uses two GA parametrizations to determine the position of the turbines that optimize the energy produced, assuming that the turbines are restricted to the area occupied by Horns Rev I (HR-I) and for Princess Amalia. Gao et al. [ 5 ] presented a multi-population genetic algorithm for a case study optimization applied to an offshore wind farm in Hong Kong. A similar study was also performed by Pookpunt and Ongsakul [6] , in this case for an offshore wind farm in Thailand, by means of a binary particle swarm algorithm. Chowdhury et al. [ 7 ] introduced a particle swarm algorithm to optimize the location of wind turbines and considered different wind conditions over the study area. Furthermore, a particle swarm optimization algorithm was proposed by Hou et al. [8] to address the design of large offshore wind farms. Salcedo et al. [ 9 ] relied on a novel coral reef algorithm with the objective of maximizing the output of offshore wind projects. Feng in [ 10 ] uses a random search algorithm to maximize the power production in HR1. A hybrid method based on a genetic algorithm (GA) combined with local search was proposed by Mittal et al. [ 11 ]. A newly reported mutation operator was developed by Mayo with the objective of maximizing energy yield [ 12 ]. In a similar way, but referring to large wave farm consisting of fully submerged wave energy converters, Neshat et al. [ 13 ] use different approaches to evolutionary algorithms to maximize the harnessed power. Five of the approaches fall into the group of continuous methods, like the one used in the present work, and four of them in that of the discrete methods, of which GAs are its greatest exponent. After reviewing these works and other similar ones, we found that they are mostly based on two approaches to model the WFOP: • Defining a discrete computational domain in the form of a grid in which the turbines can be located in the centre of each cell. This is the most popular method. For example, Kunakote et al. in [ 14 ] conduct a comparative performance of twelve metaheuristics to optimize the layout of a wind farm after discretizing the studied area into 10 × 10 grids. Binary reasoning is applied to discriminate whether each cell includes a turbine or not. • Assuming a predefined number of wind turbines in the wind farm and optimizing the value of the two geographic coordinates of each wind turbine. These approaches generally result in power plants with very irregular layouts. This characteristic is not desirable in real OWFs due to the problems that it originates in navigation and the access to turbines for maintenance and repair. Despite this, little work has been done to limit solutions to OWFs with evenly distributed turbines [ 15 – 18 ], which is a solution that is increasingly being adopted by designers of real OWFs. Among them, parallelogram-shaped wind farms are a particular and frequent case. In cases with a very limited concession area where the optimal solution will hardly have a regular shape, Serrano et al. in [ 15 ] uses a search algorithm for a parallelogram-shape solution to determine the distance between turbines, orientation and angle of the parallelogram that maximize energy production. Neubert in [ 16 ] uses a deterministic algorithm to investigate a discrete number of cases with the motivation of achieving an appealing visual impact and presumably reduce the infrastructure cost for cables. Feng in [ 17 ] uses a parallelogram shape solution in a first stage of the WFOP to Energies 2021,14, 2895 3 of 20 deduce the best orientation and shape of the OWF, before performing the complete layout optimization. Finally, Stanley in [ 18 ] uses five parameters to define the solutions bounded in a concession area with the benefit of reducing the computational expense. For other renewable sources, a continuous exploration of the position and orientation of a solar array is presented in [19]. 1.1.1. Contribution 1. Non-Genetic Evolutionary Algorithm Searching in a Vast Concession Area By exploiting the complexity reduction derived from considering parallelogram-shape layout solutions (with evenly distributed turbines), this work introduces a complete WFOP that searches the optimal values for position, size and layout of the OWF in order to maximize the project profitability. Contrary to existing works based on a GA, the proposed method can search within a vast concession area without compromising the computation time. Such a complete search spectrum is only possible if the idea of using GAs to independently position each turbine is abandoned. Instead, this work proposes a non-genetic evolutionary algorithm (NGEA) that works with eight continuous variables regarding the centre coordinates, number of rows, number of turbines per row, distances between rows and between turbines in a row, row orientation and parallelogram angle. As a relative drawback, the turbine locations cannot freely be located in the OWF area and are forced to be uniformly distributed, thus reducing the spectrum of possible candidates to the optimum. However, optimizations made with and without the restriction of uniformly distributed turbines show a very small difference in the resulting net energy yield, less than 1% [4,16,17]. 1.1.2. Contribution 2. Analysis of the Optimal Orientation for Rows and Columns The proposed method has been tested in the popular HR-I site (see Figure 1). This OWF has been used in many research works as a testing bench to prove the validity of a proposed optimization algorithm, or as a method to evaluate the wake losses. For the first objective, the output of most of the tested algorithms was a layout with turbines irregularly distributed along the concession area, and hence no useful information resulted about the preferential directions of the turbine alignments. Figure 1. Nautical chart including depths, coast and forbidden zones for HR-I site. However, few research papers have been found discussing whether the choice of orientation and inclination of the concession area was appropriate. Gerdes in [ 20 ] stated that the turbine spacing was chosen as 560 m both between rows and along a row (equivalent to seven rotor diameter of the upwind turbine ( D )), based on a wide wind direction distribution that revealed no clear preferable orientation and demanded an equal-distance spacing for both wind farm axes. The report [ 21 ] exposed that the modeled wake losses varied by only 0.11% over the full range of grid rotation, which was relatively unimportant in determining the total plant losses. The present work analyzes the optimal values for orientation and inclination for a parallelogram-shaped OWF and compares the result to the actual layout. An additional advantage of using uniformly distributed turbines is that two of the variables defining the optimum, orientation, and inclination provides the designer with information about Energies 2021,14, 2895 4 of 20 the preferred directions to align the turbines with the aim of being used in future for the selection of the concession area that is most attractive to developers. If the concession area is large enough, albeit with limitations in the number of turbines or MW to be installed, the algorithm also finds the optimal solution by completely defining its location, distance, orientation, inclination and layout (distribution in rows and columns). 1.2. Wake Loss Model Regarding the models used for the evaluation of the energy produced, Archer et al. in [ 22 ] reviews and compares six popular analytical wake loss models. Among them, the vast majority of published works use the Jensen model [ 23 ] (also named as the PARK model) to calculate the wake effect at distances in excess of three turbine [ 24 ] diameters. This is the analytical model used in commercial software packages such as WAsP, WindPro and Meteodyn to evaluate the production of wind farms. In addition to the analytical models, there are others based on computational fluid dynamics (CFD) that allow a more accurate assessment of the effect of wakes. Walker et al. in [ 25 ] benchmarks the performance of four commonly used wake models against production data from five offshore wind farms. Nevertheless, the very high computational cost required makes their applicability to the optimal wind turbine micro-siting problem unfeasible in practical terms. On the other hand, due to unavailability of the complete wind series or difficulty in their treatment, wind roses or Weibull parameters distributed in few sectors are generally used. The consideration of a discrete wind rose leads to the fact that the wake effect for incoming wind directions close to the transition zone between two sectors is not properly evaluated, which in turn causes discontinuities and unrealistic estimates of the produced energy. This aspect is even more important in the case of large offshore wind farms based on parallelogram-shaped layouts, since the errors made by concentrating all the probability on the bisector of a wind rose sector are amplified due to the high degree of symmetry, being successively reproduced for all the rows or columns. Contribution 3. Continuous Interpolation of Sectorized Wind Data For this reason, it is necessary to transform a sectorized wind rose into a continuous probability distribution, and analogously for the Weibull parameters. In the reviewed examples [ 26 ], the estimated distribution matches the available wind rose fairly well. However, instead of looking for a high match, a more realistic distribution will be estimated that yields the same mean value or the same cumulative probability at each sector. To sum up, the literature review shows the existence of two gaps in the current body of knowledge on the problem of optimal micro-siting of OWFs. On the one hand, there are no previous works published that deal efficiently with the problem of optimal micro-siting of offshore wind farms based on parallelogram-shape layout, which is also one of the most common types of configurations. On the other hand, all previous works on optimal wind farm design have considered a discrete wind rose (usually composed of 8, 12 or 16 sectors), which can lead to errors in the evaluation of the wake effect with respect to a realistic situation in which the wind direction has a continuous evolution. Therefore, the present work introduces two significant improvements by developing an efficient optimization approach for parallelogram-shaped offshore wind farm designs and by introducing a new realistic model for evaluating the energy produced based on the continuous evolution of the wind resource. The remaining of the paper is structured as follows: • Section 2lists the expressions for two of the most popular economic indicators used to assess the profitability of a investment, presents a brief but complete description of the Park model to evaluate the wind speed deficit, introduces a new method to interpolate a continuous distribution of the wind rose and the Weibull parameters when data for a reduced number of sectors are given, and presents the basis of the NGEA used to solve the WFOP. • Section 3presents and discusses the results obtained after testing it in the HR-I site; Energies 2021,14, 2895 5 of 20 • Finally last Section exposes the conclusions extracted from the work. 2. Materials and Methods 2.1. Economic Indicators Rodrigues in [ 27 ] explains different economic functions to evaluate the profitability of a prospective investment over time. Among them, the levelized cost of energy (LCOE), the internal rate of return (IRR), the discounted payback time, the return on investment, or the benefit to cost ratio are used as measurements of the operation risk and the investment quality. On the other hand, the net present value (NPV) is suitable to choose between mutually exclusive options for OWF layouts and hence should be used as objective function in optimization algorithms, although limiting the maximum investment to a certain value. Since NPV is the difference between the present value of a company’s cash inflows and the present value of cash outflows over a given time period, it can be expressed as: NPV = T ∑ k=1 AEP pkWh(1+∆pkWh)k−1−OPEX(1+∆OPEX)k−1 (1+r)k−CAPEX −Dec (1) where T is the utility life-time, AEP is the annual net energy production (after discounting the total losses due to wake interferences and in the electrical infrastructure), pkwh and ∆pkWh are the energy price and its annual increment, OPEX and ∆OPEX are the operation and maintenance cost and its increment, CAPEX is the total investment cost and Dec is the on-time decommissioning cost after deducing the residual price of the plant. Finally, r is the discount rate, obtained from the nominal interest rate riand the inflation in f as r=1+ri 1+in f −1. (2) The annuity factor (a) can be defined as a= T ∑ k=1 1 (1+r)k=1−1 (1+r)T r(3) and LCOE can be expressed as LCOE =CAPEX/a+OPEX AEP . (4) In some research works, the discount rate is assumed to be similar to the weighted average cost of capital (WACC) and slightly higher to adjust for risk. WACC is more difficult to obtain [ 28 ], although reference values are given in [ 29 ] for Denmark, oscillating between 7.4% in 2001 and 8.9% in 2013. In this work, ri = 9.4% and in f = 1.5% have been used, yielding r= 7.78%. The main drawbacks of NPV are: • A positive value is indicative of a profitable investment but does not give an idea of its quality since, in general, NPV will increase with the OWF size. It cannot be used to compare two OWFs with different sizes, locations or bid prices. • The value of LCOE is strongly influenced by the discount rate, while the value of IRR is influenced by the sale price of the energy produced. The calculation of NPV is determined by both data, and the variation in one of them will not only produce an erroneous value in the objective function but will also distort the selection between individuals. For example, if a too low energy price or a too high discount rate have been taken, then the best solution found will tend to reduce the distances between turbines more than what is ideal. A value of 58 € /MWh can be assigned to the energy price, as the sum of the feed-in tariff and the market price (at least during 10 years) [ 20 , 30 ]. Nevertheless, with this energy Energies 2021,14, 2895 6 of 20 price, profitability is doubtful. As a consequence, LCOE has been chosen instead of NPV or IRR as a objective function, since this indicator does not depend on the energy price. 2.2. Annual Energy Production (AEP) The annual energy produced by a wind power plant can be obtained by adding together the energy, Ek , produced by each of the Nt turbines, for every wind direction ϕ and for every wind speed u. Throughout a year (Ty=8766h), AEP can be expressed as: AEP = Nt ∑ k=1 Ek=Ty Nt ∑ k=1Zucut−out ucut−in Z2π 0Pc(νk)f r(ϕ,u)dudϕ(5) where ucut−in is the cut-in speed, ucut−out is the cut-out speed, Pc is the electrical energy at a certain wind speed given by the turbine power curve and fr(ϕ , u) is the density of probability at speed ν and wind direction ϕ . In this expression, the power produced by a turbine k of the array depends of the effective speed of the air flow that reaches the k -th turbine, νk . Due to the wake effect, this effective speed is smaller than the free-flow air speed u . There are different methods to estimate the wind speed deficit caused by the wakes. The simplest and most popular wake model used by researchers in WFOP is the PARK model [ 31 ]. It was first proposed by Jensen and Katic [ 23 ], and starting from a diameter of the wake ( Dw ) equal to the rotor diameter of the upwind turbine ( D ), it assumed a expansion of the wake behind the upwind wind turbine linear with the projection of the distance between turbines onto the wind direction (s) (see Figure 2) Dw=D+2kws(6) The proportionality factor is the wake decay constant ( kw ), which can be expressed as a function of the tower hub height (h) and the surface roughness length (z0) kw=0.5 ln(h)−ln(z0). (7) For the first evaluations, a value of 0.0525 was used, obtained from z0= 0.005 (as usual for offshore sites) and h= 70. However, the resulting park efficiency for the HR-I layout yielded 91.5%, higher than the observed one (89% in [ 32 ]). In fact, the value obtained from (7) should be adjusted to take into account the turbulence intensity TI , considering that the lower the stability, the higher the kw [ 33 ]. When TI is not available, authors fit kw so that the results obtained from the wake loss model match the observed values, or more frequently, they use kw=0.04 for offshore sites [34]. This is the value used in this work. According to the momentum theory, the wake expansion produces a wind speed deficit δthat depends on the upwind thrust coefficient (Ct) through δ(s) = v∞−v v∞=1−p1−CtD Dw2 (8) where v is the wake velocity at the downstream position and v∞ is the undisturbed wind speed. In the case of turbines that are perturbed by more than one upwind turbines, the effect of the wakes must be added to obtain the effective wind speed deficit. Different methods of aggregation have been suggested [ 31 ], and among them, the preferred one is the root sum of squares δ2 agg = N ∑ i=1 δ2 i(9) which obtains the total wind speed deficit at any turbine as the root mean square of the deficits at downstream place due to the single wake from the N upstream turbines. In the Energies 2021,14, 2895 7 of 20 work of Tian et al. [ 35 ], this combination method is the one that exhibits the best match with the observed power data. The previous expression can be easily modified to account for the situation in which the downwind turbine is only partially affected by the wake. Figure 2illustrates this situation that allows the calculation of the effective surface of the downwind rotor affected by the upstream weak ( Ae f f ) once the projection s and the transversal separation between turbines (y) are known. Ae f f =     0 if y>Rw+Rt πR2 tif y<Rw−Rt R2 wγw−sin(2γw) 2+R2 tγt−sin(2γt) 2otherwise (10) with γw=cos−1R2 w−R2 t+y2 2y Rw γt=cos−1R2 t−R2 w+y2 2y Rt(11) and expression (9) for the wind speed deficit turns into δ2 agg = N ∑ i=1 δ2 iAe f f i(12) where Ae f f i is the surface of the partial wake of the upwind turbine i affecting the downwind turbine under study. Downwind turbine Wake radius y Rw Downwind turbine Upwind turbine y s Rt t w Aeff v v v (1 − 𝛿 (s)) Figure 2. Downwind turbine partially covered by an upwind stream. 2.3. Continuous Distribution of Magnitudes Given by the Wind Rose and Averaged Values Inspecting (5) makes it clear that accurate data for the density of probability fr(u , ϕ) are mandatory for an adequate AEP evaluation. Complete wind data sets are frequently unavailable to researchers or for a first site assessment. It is more common to have such data as values collected and summarized in a wind rose or as a set of mean values for each of the ns sectors under study. In the case of HR-I, several sources can be found with different values, although they share the existence of a prevailing wind coming from NW. This work will use the ones obtained from [36], represented in Table 1, resulting in the wind rose of Figure 3. Table 1. Values for probability, and Weibull parameters (scale factor Aat 62 m and shape factor K) for every sector. Sector N NNE NEE E EES ESS S SSW SWW W WWN WNN freq (%) 3.8 4.3 5.5 8.3 8.7 6.7 8.4 10.5 11.4 12.2 13.9 6.1 WeibA (m/s) 8.71 9.36 9.29 10.27 10.89 10.49 10.94 11.23 11.93 11.94 12.17 10.31 WeibK2.08 2.22 2.41 2.37 2.51 2.75 2.61 2.51 2.33 2.35 2.58 2.01 Energies 2021,14, 2895 8 of 20 It is necessary to adjust the scale parameter A from the measurement height ( zref =62 m ) to the hub height ( zhub = 70 m) due to the wind shear effect. The relationship between scale factors, and in general between wind speeds, at different heights is given by A=Are f ln(zhub)−ln(z0) ln(zre f )−ln(z0)(13) As previously mentioned, the usual value of z0 for offshore sites is 0.005, which is also consistent with the wind profiles presented in [36]. 3% 6% 9% 12% 15% 0% EW N S Wind Speeds in m/s WS 25 19 WS < 25 13 WS < 19 8 WS < 13 3 WS < 8 Figure 3. Wind rose obtained from [ 36 ]. Only values between cut-in speed and cut-out speed are represented. Logically, these data do not present a continuous evolution, and in general there appear abrupt changes in the edge from one sector to another. This unrealistic distribution prevents a proper evaluation of produced energy, leading to inconsistent situations and solutions that are far from the real one. It is therefore necessary to obtain a distribution that, although estimated, is more realistic and consistent with the continuous distributions of parameters and probabilities that are assumed for the site. Feng [ 26 ] proposed a way to obtain a continuous distribution starting from cubic spline interpolations. The result effectively describes an evolution that passes through the bisectors of the ns sectors, but in general it will give rise to distributions that do not have the same mean value (in the case of the Weibull parameters), or the same cumulative probability (in the case of the wind rose) as the starting data. Instead, this article proposes a way to obtain a continuous distribution from the averaged data for the different sectors, while respecting the aforementioned concerns. In the following, we will assume that the average value for a certain magnitude x is available for the ns sectors. At the end of this section, we will recover the Weibull parameters as this magnitude. For simplicity, we define a sector sias si=2π ns (i−1),2π ns i(14) although the explanation can be easily extended to s i=2π ns (i−1.5),2π ns (i−0.5) in order to centre the first sector at the origin (0 deg ). The average value for a sector i can be expressed as: xj=ns 2πZ2π ns j 2π ns (j−1)xdϕj∈N,j≤ns (15) where xcan be considered as unavailable and hence unknown. Energies 2021,14, 2895 9 of 20 Let us define the uniform knot vector, and a vector of ns +1 ordinates t:={0, t1, . . . , tns}with ti=i2π ns i∈ {0, 1, . . . ns}(16) y:={0, y1, . . . , yns}with yi=2π ns i ∑ j=0 xj(17) where for convenience, we can consider or define ¯ y0=¯ x0:=0. A cubic spline S(ϕ) can be obtained as a piecewise function defined from a series of polynomials Pi:[ti,ti+1)→R(18) passing through the ns +1 points (ti,yi), i.e., satisfying S(ti) = Si2π ns =2π ns i ∑ j=0 xj(19) On the other hand, since S(t) is a cubic (i.e., smooth) spline, its derivative exits, is finite and continuous ∀ϕ∈[0, 2π]. By calling ˜ x(ϕ):=dS(ϕ) dϕ(20) it yields that, along a sector i, the mean value for ˜ xis ˜ xi=ns 2πZ2π ns i 2π ns (i−1)˜ xidϕ=ns 2π(S(ti)−S(ti−1))= i ∑ j=0 xj− i−1 ∑ j=0 xj=xi(21) with the result that that ˜ xis a coherent estimation of the unknown x. Consequently, for a certain magnitude x , by finding the derivative of the cubic spline of the sum of its average values, it is possible to obtain a continuous distribution of this magnitude while keeping its mean along a sector. The procedure is: • Calculate the accumulated sum of the average values from (17). • Obtain the cubic spline Ssatisfying (19). • An estimation of xcan be found as the derivative of S. This procedure can be applied to Weibull parameters, starting from their mean values for each of the sectors. Figure 4shows the original discrete values for the Weibull parameters, together with their obtained continuous distributions. 030 60 90 120 150 180 210 240 270 300 330 0 5 10 Weibull A parameter 030 60 90 120 150 180 210 240 270 300 330 0 1 2 3 Weibull K parameter Figure 4. Original discrete values for the Weibull parameters, and their corresponding continuous distributions. The case of the wind rose is conceptually different, because the available data are the probability PSi that the wind comes from sector i , and the goal is to obtain a continuous Energies 2021,14, 2895 16 of 20 presented, with an average of 2100 s per optimization search for long distances between turbines and 2700 s for normal separations. 0 50 100 150 # gen 59 59.5 60 60.5 0 500 1000 1500 2000 2500 Time (s) dr, dtr [5D, 20D] 0 50 100 150 200 250 # gen 60.8 60.9 61 61.1 61.2 61.3 61.4 0 1000 2000 3000 4000 5000 6000 Time (s) dr, dtr = 7D Figure 14. Evolution of the fitness and the simulation time during the search process. Figure 15 represents the evolutions of the most frequent orientations for the rows and columns. The column orientation is obtained by subtracting the orientation and the inclination (see Figure 9). The preferred alignment directions are around 60 deg and 150 deg , trying to avoid the wind directions with more energy deficit (90 deg and 170 deg from Figure 11). 0 50 100 150 200 250 Generation 0 50 100 150 200 Orientation for rows or columns dr, dtr [5D, 20D] 0 50 100 150 200 250 Generation 0 50 100 150 200 Orientation for rows or columns dr, dtr = 7D Figure 15. Preferred orientations for rows (o) and columns (*) of the best individual at each generation. Results for the five optimization searches are represented with different colors. The plot at the left represents the values when no separation restriction is imposed to the separation between rows and columns. In the right-hand plot, a separation of 7D is imposed. Finally, Table 3lists the decision variables of the final optimum for both types of searches and compares them with the one of the actual HR-I OWF, together with the breakdown of estimated costs. Table 3. Configuration and cost breakdown for the actual layout, the optimal solution assuming 7D as dr and dtr , and the optimal solution. Actual Optim Optim Actual Optim Optim Layout d=7Dd∈[5D, 20D]Layout1d=7D1d∈[5D, 20D]1 dtr (D)7 7 5.35 Turbines 154.8 154.8 154.8 dr(D)7 7 19.98 Foundations 48.18 49.08 48.51 ntr 8 5 16 Electrical Offshore 58.04 60.04 58.04 nr10 16 5 Electrical Onshore 15.88 15.88 15.88 θ( deg) 173 71.05 62.5 Others 16.60 16.60 16.60 φ( deg) 83 91.16 99.97 OPEX 14.24 14.47 14.97 AEP (GWh) 712.47 723.55 748.60 LCOE (€/MWh) 61.26 60.95 59.34 Elect. losses (GWh) 9.83 8.53 16.35 IRR 13.78% 13.99% 14.41% Wake losses (GWh) 76.81 64.35 40.68 NPV (M€)72.93 77.47 86.09 1Estimated costs. Energies 2021,14, 2895 17 of 20 With regard to the actual layout for HR-I, the estimated park efficiency is 90.27% ( 712.47 712.47+76.81 ), reasonably close to the observed value (89%). Costs and prices were obtained from [ 40 ]. Starting from these, the estimated cost for the actual layout is EUR 293.5 M, which means 1.83 M EUR/MW. The real project cost, obtained from [ 20 ], was EUR 278 M. The third column corresponds to the optimization result when distances between turbines are fixed and equal to 7 D between rows, and between turbines in a row. Optimum orientations for rows and columns are different from the actual ones and similar to the central plot of Figure 9. However, the main difference lies in the number of rows, which in this case increases to 16, trying to orientate as much turbines as possible in the less frequent direction (71 deg together with 251 deg ). The AEP improvement is 1.56% ( 723.55−712.47 712.47 ). Finally, the column Optim [ 5 D ,20 D] refers to the optimization result when distances are bounded between 5 and 20 D . In this case, the solution results in the maximum distance (20 D ) between the five rows, and reduced distances (5.35 D ) between turbines in a row, which are aligned along less frequent wind directions. The AEP improvement is 5.09% (748.60−712.47 712.47 ). With regard to the economic indicators, the investment cost hardly varies from one solution to another. An exception is the significant reduction in the collection MV electrical infrastructure for the Optim 7 D , due to grouping of sixteen turbines in the same row, for which the cable section can be optimized. For any of the solutions, the values for LCOE indicate that the price of the energy should be sufficiently higher than 60 EUR/MWh to make this project attractive to developers. 4. Conclusions and Discussion There are numerous research works carried out on the energy losses suffered by the wake effect, especially relative to the HR-I OWF. Genetic algorithms also abound that obtain the layout that maximizes the return on investment, although none takes into account all of the factors that affect the profitability of the project. In addition, they give rise to configurations that are not desirable due to their lack of uniformity in the location of the turbines. In this work the result of a complete optimization algorithm is presented: • that takes into account all of the factors (or at least more than any other paper found) required for the economic evaluation; • that fully defines the position, size and configuration of the optimal OWF with regularly distributed turbines and shaped like a parallelogram; • that, alternatively, indicates what should be the orientation and inclination of the concession area to make it more attractive to potential developers of wind farms. With the fixed electricity production feed-in tariff provided for grid connections at that moment equal to 0.058 EUR/kWh [ 20 , 30 ], the project is not profitable except for very low interest rates. In the case of uncertain values for the energy price, the most suitable indicator to evaluate the project profitability is the LCOE. From the exploration of the results obtained with a wide concession area, a tendency is observed for the algorithm to search for very dispersed OWFs, with little exploitation of the available surface. If, to reduce the occupied area, the distance between turbines is limited to, for example, seven diameters (as is the case with HR-I), a solution search algorithm with regular patterns offers the optimal orientation, inclination and layout. In this case, and assuming the rows and columns configurations of (4 × 20, 5 × 16, 8 × 10, 10 ×8 , 16 ×5, 20 × 4), an exhaustive search only needs 180 × 90 × 6 ≃ 90,000 evaluations to calculate the optimal configuration, so that the use of a random search algorithm is meaningless. With regard to the orientation and inclination, results show that optimal solutions tend to avoid the more frequent wind direction given by the wind rose. However, if the algorithm also has to find the exact location, orientation, inclination and optimal separation distances between turbines (respecting a more or less wide concession area), the number of possibilities increases to the extent of requiring an efficient algorithm that can handle these variables. Under these requirements, a traditional GA Energies 2021,14, 2895 18 of 20 is not appropriate. Instead, an NGEA that operates with real values has been used to determine the decision variables that define a parallelogram-shaped OWF. The inspection of the results shows that other regular layouts were possible with a better site exploitation. In order to have a confident estimation of wind resources, a novel method of interpolating the data given for a reduced number of sectors is presented. Consequently, the discontinuous values describing the wind direction probability and Weibull parameters are interpolated along 360 deg using cubic splines, with the concern of maintaining the accumulated probability (in the case of the wind rose) or the average value (in the case of the Weibull parameters) at each sector. Obtained results show significant differences between the optimal orientation values when the algorithms uses continuous distributions for the wind resources and when it uses sectorized data. Additional contributions from this project are: • discussion on the optimal orientation of the concession area for HR-I, • recommendation to reduce the range of speeds to be studied when evaluating the orientation and inclination of the OWF during the optimization algorithm execution, • possibility to deduce the optimal orientation and inclination replacing the original layout into a scaled version of the OWF, with half of the rows and half of the turbines per row, • the evaluation of a quincunx configuration, which does not entail an increase in the energy produced, mainly if the position of the turbines must be limited to a certain concession area. Author Contributions: Conceptualization, A.G.G.-R.; Data curation, A.G.G.-R. and J.S.-G.; Formal analysis, A.G.G.-R.; Funding acquisition, M.B.-P. and J.M.R.-S.; Investigation, A.G.G.-R.; Methodology, A.G.G.-R., M.B.-P. and J.M.R.-S.; Project administration, M.B.-P. and J.M.R.-S.; Software, A.G.G.-R.; Validation, J.S.-G.; Visualization, J.S.-G.; Writing—original draft, A.G.G.-R., J.S.-G., M.B.-P.; Writing— review and editing, M.B.-P., J.S.-G. and J.M.R.-S. All authors have read and agreed to the published version of the manuscript. Funding: This work was partially funded by the Spanish MEC (Ministerio de Economía y Competitividad (Ministry of Economy and Competitiveness)), co-funded by the European Commission (ERDF - European Regional Development Fund) under grant ENE2016-77650-R, the CYTED Network Program under grant 718RT0564, and by the CERVERA research programme of CDTI (Industrial and Technological Development Centre of Spain) under the research Project HySGrid+ (CER-20191019). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. Glossary OPEX operational expenditure CAPEX capital expenditure NPV net present value IRR internal rate of return LCOE levelized cost of energy T wind farm life time aannuity factor AEP annual energy production pkwh price per kWh in f annual inflation rdiscount rate rinominal interest rate Dec decommissioning cost WACC weighted average cost of capital Energies 2021,14, 2895 19 of 20 ns number of sectors nrnumber of rows ntr number of turbines per row drdistance between rows, in diameters dtr distance between turbines in a row, in diameters θorientation of an array with respect to North, in deg, sense CW φparallelogram angle; equal to 90 deg for rectangle z0surface roughness length Ascale factor of the Weibull distribution Kshape factor of the Weibull distribution δwind speed deficit at a downwind turbine kwwake decay constant htower hub height Drotor diameter of the upwind turbine Dwdiameter of the wake Ctthrust coefficient sprojection of the distance between turbines onto the wind direction ytransversal separation between turbines Ae f f effective surface of the downwind rotor affected by the upstream weak GA genetic algorithm OWF offshore wind farm WFOP wind farm optimization problem NGEA non-genetic evolutionary algorithm HR-I Horns Rev I References 1. 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