Abstract

Floating offshore wind energy will play a key role in the clean energy transition. The number of large-scale wind farm projects is growing in regions like Northern Europe, the East Coast of the U.S., and the Mediterranean Sea. Offshore wind farms face fewer layout constraints, as they can be situated in vast, open sea areas. Turbines are often arranged in simple layouts, such as grid patterns, but this can cause significant annual energy production (AEP) losses due to wake–rotor interaction. Increasing spacing can mitigate this effect, but it may not always be feasible due to marine space limitations or higher costs for cabling and maintenance. This paper introduces a multi-objective wind farm optimization framework using a non sorted genetic algorithm (NSGA II) to minimize costs and maximize AEP. The method is applied to two case studies in the Mediterranean Sea, assuming 15 MW wind turbines. Case A is located off the coast of Civitavecchia, and case B in the Gulf of Squillace. AEP evaluation is performed with the open-source library FLOw Redirection and Induction in Steady-State (floris), while optimization is done using pymoo. In case A, the layout characterized by the lowest levelized cost of energy (LCOE) features 16 turbines, achieving an AEP of 709 GWh, an LCOE of 112.06 €/MWh, and wake losses of 2.6%. Meanwhile, in case B, the layout with the lowest LCOE consists of 19 turbines, achieving an AEP of 1140 GWh, an LCOE of 80.82 €/MWh, and wake losses of 4%.

1 Introduction

Offshore wind technologies have a central role in the context of energy transition toward renewable sources, as also confirmed by the plans of the European Commission for the installation of 111 GW of offshore renewable energy by 2030 [1], almost doubling the previous target of 60 GW [2]. The drivers of the steady growth in the production of energy from offshore sources and, in particular, from wind are to be found in the opportunity to exploit a vast reservoir of potential energy with reduced visual or acoustic constraints, thus allowing the installation of larger wind turbines in high cost-effective multi-GW wind farms. In fact, considering that installation costs are such a high portion of total capital expenditure costs, installing fewer but larger turbines is more cost-effective. These considerations are further supported by the increasing trend in the average size of offshore turbines, which has grown from 2 MW in 2002 to 8 MW in 2022 and is expected to exceed 14 MW by 2027 [3].

In the described scenario, the design of the wind farm layout plays a crucial role. On the one hand, there is a need to install the turbines in confined marine spaces to limit maintenance and cabling costs. On the other hand, a reduced installation distance leads to larger energy losses due to wake–rotor interaction. In actual design of offshore wind farms, wind turbines are positioned based on empirical knowledge at a distance ranging from seven to twelve diameters in the direction of the prevailing wind, and from three to five diameters in the orthogonal direction to the prevailing wind [4]. In such regular layout, the loss caused by wake effect is estimated to be around 15% of the annual energy production (AEP) that would be otherwise achieved in the same wind farm if each wind turbine was considered as isolated entity [5]. These intuitive arrangements are now being challenged by more complex layouts to reduce the wake–rotor interaction [6]. However, finding such layouts present both the need to solve complex optimization problem and to calculate the effects of wake-induced velocity reduction.

The second aspect can be achieved with computational fluid dynamics (CFD) based tools, but although these tools provide high accuracy, they are too computationally demanding to be applied to optimization problems. For this reason, the use of simplified analytical models for the wake effect is usually adopted in layout optimization problems, which allow for faster calculations while still providing sufficient accuracy. These models are constantly improving and evolving; however, they share some similarities. Specifically, they all originate from the one proposed by Jensen [7], and in general they model and quantify the downstream flow velocity deficit as a function of the axial thrust on the rotor. Jensen's model assumes a linear expansion of the wake described by a parameterized coefficient. The multizone model enhances Jensen's model by identifying multiple wake zones characterized by different expansion coefficients [8]. More recent models are proposing a representation of the velocity deficit as a Gaussian distribution in the spanwise direction [911].

The approaches used in various layout optimization methods differ in terms of the objective functions to be maximized or minimized, the constrains applied, the adopted wake and cost models, and the chosen optimization algorithms [12]. The most commonly used objective functions include AEP [13], total power [14], levelized cost of energy (LCOE) [15], net present value [16], and linear combinations of the previous [17]. Some of them are also characterized by multi-objective optimizations: AEP and efficiency [18]; cost per unit power and efficiency [19]; AEP; Euclidean distance between turbines and layout perimeter [20]. Numerous algorithms have been proposed in literature, such as genetic algorithms [17], particle swarms [21], algorithms based on combinations of mixed integer linear programing models and heuristic methods [22], and topology based algorithms [6].

This study aims to introduce a multi-objective layout optimization methodology to maximize AEP and minimize total costs over the lifetime of the wind farm. For this purpose, the multi-objective non sorted genetic algorithm NSGA II [23] is used, as implemented in the pymoo library [24]. Each evaluated individual represents a wind farm, defined by a different layout and total number of turbines. The layouts are obtained by discretizing the selected area into a Cartesian grid of possible locations for the wind turbines. Imposed constraints are the outer perimeter, a minimum distance between the rotors, as well as both a minimum and maximum number turbines. For each feasible individual, AEP and total cost (capital expenditures (CAPEX) and operating and maintenance expenditures (OPEX)) during the life cycle of the wind farm are evaluated as objective functions. The proposed cost model takes into account factors such as seabed depth, distance from the shore, and the offshore substation's location. The FLOw Redirection and Induction in Steady-State (floris) library [25] is used to model the farms and evaluate the wake losses.

Section 2 presents the applied methodology, the selected tools, and models. In Sec. 3, the case studies are identified with a strong focus on the definition of the wind resource and the bathymetry in the selected areas. In addition, plant characteristics such as turbine type, position of the offshore substation, length and number of cables for energy transport, and distance from port are explored. In Sec. 4, the results are presented, while Sec. 5 provides the conclusions and final remarks.

2 Problem Description and Methodology

The study aims to propose a multi-objective and multiconstrained layout optimization problem resolution methodology. The possibility of arranging wind farm turbines within a fixed external perimeter is investigated to maximize energy output while simultaneously minimizing total costs. The latter is the sum of OPEX and CAPEX, which vary based on water depth, length of electrical cables, and other site characteristics. The problem is approached by considering solutions with a variable number of turbines within a specified range. Additionally, a minimum distance between rotors is established. The selected area is associated with a bathymetric profile and a wind rose. Anemometric and bathymetric data are made available through the Global Wind Atlas [26]. Seabed depth is obtained through the acquisition of a vector image available from the aforementioned source. With respect to the hub height of the rotor, data on annual average wind speed, wind direction frequency, and characteristic parameters of the Weibull distribution for each direction are provided. The wind direction frequency is presented through a rose diagram divided into 12 sections, each with an amplitude of 30 deg. For each direction, the values of the Weibull's scale parameter A and the shape parameter k are known, allowing the reconstruction of the wind speed frequency curve. After obtaining the 12 curves, the process involves considering wind velocities in the range from 0m/s to 25m/s with a step of 1m/s. Multiplying the frequency of each speed by the frequency of the respective wind direction results in a joint frequency distribution of wind speed and direction f(u,θ). Section 2.1 outlines the optimization algorithm, while Sec. 2.2 describes the modeling of wind farm performance and Sec. 2.3 presents the cost models.

2.1 Optimization Algorithm.

Given the complexity of the optimization problem under investigation, the nondeterministic genetic algorithm NSGA II implemented in the open-source library pymoo [24] is used.

In this study, the layout optimization problem is addressed by discretizing the spatial domain of the wind farm: given an external perimeter, the enclosed area is divided into a grid of possible turbine locations. Each individual, and therefore each layout, is described with a binary array in which a 1 value represents the presence of a turbine at the corresponding index position. Conversely, a 0 value represents the absence. The array is defined starting from one at the bottom-left vertex of the area perimeter and progressing toward the right as the index increases. Upon completing a row, the position shifts to the one above, as described in Fig. 1 where numbers on the left side represent the index and numbers on the right side represent the presence or the absence of turbines. The obtained grid is superimposed onto a vector image representing the bathymetry map of the selected area. In this way, the respective seabed depth is associated with each possible position. The objective functions of the problem, evaluated for each individual, are
(1)
where C is the sum of CAPEX and OPEX, and it is calculated for the entire lifetime of the wind farm (LT). Since higher AEP is desired, the minimization algorithm requires a negative sign in the corresponding objective function. The constraints imposed on the problem are assumed as
(2)

where NMAX and NMIN are, respectively, the maximum and minimum number of turbines; ntur is the number of turbines in each farm; dist(i,j) represents the Euclidean distance matrix between turbines i and j; and Dmin is a minimum distance between turbines to allow for proper floater anchoring and safe operations.

Fig. 1
Binary array representation of layout
Fig. 1
Binary array representation of layout
Close modal
Each individual in the initial population is created randomly assigning to each element of the binary array a value of either 0 or 1. In order to obtain an initial population composed by mostly meaningful individuals, a probability for the presence of a turbine in the ith location is imposed, equal to
(3)

where Nloc is the number of possible locations.

The previously described multi-objective problem falls within the category of Pareto optimization, since it lacks a specific hierarchical criterion between the two objective functions. This approach aims to identify a set of nondominated individuals within the two-dimensional objective space. An individual is considered nondominated, or Pareto-optimal, if there is no other individual that outperforms it in both objective function values. In addition, considering the constraints exposed in Eq. (2), X1 is considered dominant over X2 if [27]:

  • (i) X1 is feasible while X2 is not.

  • (ii) X1 and X2 are both infeasible and X1 dominates in objective space.

  • (iii) X1 and X2 are both feasible and X1 dominates in objective space.

Moreover, when comparing two nondominated individuals, the one located in a region of lower density ρ in the objective space is preferred [23]. Referring to Fig. 2, the density ρ of the region in which individual i is located is evaluated as
(4)
Fig. 2
Objective space representation
Fig. 2
Objective space representation
Close modal
The density is set to zero for the two extreme individuals in the objective space (i.e., individuals corresponding to the minima of the objective functions), as it would be not possible to apply the definition of ρ as it is for those two values. Since individuals in lower density regions are preferred, this choice ensures a wide Pareto front while preserving diversity of individuals (i.e., penalizing individuals that are clustering together and preserving well distantiated individuals). These criteria, implemented through the tournament selection operator [23], allow to identify the best performing individuals within each generation. Subsequently, these selected individuals are randomly paired to generate the offspring using crossover and mutation operators [28]. The former switch elements of the binary arrays of parent individuals, providing two new individuals from each parental pair. With the latter, the individuals experience a random mutation of specific bits. The mutation operator is defined by a two-step sequence bitflip mutation defined by
(5)

where Pm1 is defined as the probability for an individual to mutate, while Pm2 represents the mutation probability of each ith gene of the mutating individual (with Pm1>Pm2). The top-performing individuals, matching the original population size, are finally selected from the combined pool of parents and offspring to constitute the next generation.

To evaluate the convergence, the hypervolume (HV) indicator value is calculated for each generation. In a two-objectives minimization problem, HV measures the area bounded by the approximated front of dominant individuals and the segments corresponding to the coordinates of an arbitrary reference point Q>(1,1) in the normalized objective space [29]. Figure 3 provides an example of HV assuming Q=(1.2,1.2) and considering four dominant individuals.

Fig. 3
Hypervolume indicator in the two-objectives case
Fig. 3
Hypervolume indicator in the two-objectives case
Close modal

2.2 Wind Farm Modeling.

Calculation of the velocity field and, therefore, of energy output is carried out using the floris library. This is an open-source library to model wind farms using reduced-order wake models. In particular, floris evaluates the velocity of the flow at the turbines on a grid of points arranged on a vertical plane immediately upstream the rotor. This approach, combined with semi-empirical and analytical wake models, allows a fast computation of the power of the wind farm, avoiding expensive CFD simulations. Several works in literature provide a validation of the power estimation made by floris through comparison with real data and numerical simulations. In particular, in works [8] and [11], results obtained with floris are compared with scada data and with results by CFD simulation made with sowfa [30].

In this study, the Gauss curl hybrid wake model, as presented in Refs. [9] and [10], is selected. The model relies on the assumption of a self-similar Gaussian wake, with a shape characterized by the parameters provided below. The condition of self-similarity is achieved at the end of the near wake region, corresponding to a distance x0 from the rotor. The ratio between the wind speed at the point with coordinates (x, y, z) and the undisturbed flow speed is given by the following expression:
(6)
where zh is the hub height, Cd is the velocity deficit at the center of the wake, δ is the wake deflection, and σy and σz are, respectively, the wake width in y and z directions. The velocity deficit Cd is defined as follows:
(7)
where Ct is the thrust coefficient. The subscript 0 is assigned to quantities calculated at the point of self-similarity. The wake widths can be expressed as
(8)
where
(9)
in which the yaw offset γ is assumed to be zero, uR represents the velocity of the flow immediately after passing the rotor, and u represents the velocity of the undisturbed wind. The two parameters ky and kz are assumed to be equal and are determined from the turbulence intensity TI
(10)
The values of the parameters are assumed to be ka=0.38 and kb=0.004. The distance from the turbine at which the wake reaches self-similarity is normalized with respect to the rotor diameter
(11)

with α and β typically fixed parameters set to 0.58 and 0.077, respectively.

The wake deflection is determined as a combination of yaw misalignment and rotor–boundary layer interaction, the former being set to zero, the latter obtained from an empirical relation [8].

Moreover, an additional turbulence component that accounts for turbines operations is quantified by the model proposed in Refs. [31] and [32] and added to the ambient turbulence intensity.

The velocity field corresponding to the wind farm is ultimately obtained by superimposing the wake effects calculated for each individual wind turbine. In literature, several methods have been proposed for this purpose [33], including geometric sum, linear sum, energy balance, and quadratic sum. The latter, proposed by Katic et al. [34], has been shown in various studies [3537] to be the most effective and has, therefore, been chosen in this work.

2.3 Costs Model.

To assess the value of the second objective function, an economic analysis has been carried out. According to Martinez and Iglesias [38], CAPEX can be defined as the composition of development & consent (D&C), turbine and floater, mooring and anchoring system, transmission infrastructure (including interarray cables, export cables, and the substations), and the installation costs. Decommissioning costs should be offset by the returns from selling scrap materials. Assuming this balance, the net expense of decommissioning is nullified, as proposed by Cavazzi and Dutton [39].

Table 1 outlines the considered D&C and turbines costs. In particular, D&C comprises multiple preliminary surveys, including seabed and environmental surveys, with a primary focus on development and consent services. Regarding the substructure, in this work, a semisubmersible floater is supporting the turbine.

Table 1

CAPEX values: D&C and turbine

ParameterValueMeasurement unitReference
D&C210k€/MW[38]
Turbine12M€/turbine[40]
Substructure10M€/floater[40]
ParameterValueMeasurement unitReference
D&C210k€/MW[38]
Turbine12M€/turbine[40]
Substructure10M€/floater[40]
Regarding the mooring system, mooring lines connected to drag embedment anchors are considered, and their cost reads as
(12)

where ntur is the number of turbines in the plant, nlines is the number of lines per substructure, h(x, y) is the water depth, and Canch, Clines, and Cchain are the cost of the anchor, the lines, and the chain, respectively. The terms in the round brackets represent the length of the single line, and they derive from a calculation based on 500 m of line for the initial 100 m of water depth. Subsequently, an additional 150 m of line is considered for every succeeding 100 m, with an extra 60 m included. Furthermore, at the end of each line, there is an additional 50 m of chain. The values of the components of the mooring systems are reported in Table 2.

Table 2

Mooring components costs

ParameterValueMeasurement unitReference
nlines4[38]
Canch123k€/anchor[38]
Clines48€/m[38]
Cchain270€/m[38]
ParameterValueMeasurement unitReference
nlines4[38]
Canch123k€/anchor[38]
Clines48€/m[38]
Cchain270€/m[38]
The transmission costs can be expressed as
(13)

where d(x, y) is the distance from the shore, nex_cab,nsub_off, and nsub_on are the number of export cables, offshore substation, and onshore substation, Cex_cab,Csub_off,Csub_on, and Cintarr are the costs of export cables, offshore and onshore substations, and interarray cables, and lint_arr is the length of the interarray cables. Specifically, nex_cab is obtained considering an export cable every 330 MW of installed rated power.

Additionally, according to Ref. [38], an high voltage alternating current transmission system is considered when the substation is within 20–60 km from the onshore station, while the transmission is high voltage direct current for further plants.

Adopted values of the transmission systems are listed in Table 3.

Table 3

Transmission components costs

ParameterValueUnitReference
Cex_cab(AC)2.336M€/km[38]
Cex_cab(DC)1.168M€/km[38]
Csub_off(AC)39M€[38]
Csub_off(DC)142.75M€[38]
Csub_on84.35M€[38]
Cint_arr303.5k€/km[38]
ParameterValueUnitReference
Cex_cab(AC)2.336M€/km[38]
Cex_cab(DC)1.168M€/km[38]
Csub_off(AC)39M€[38]
Csub_off(DC)142.75M€[38]
Csub_on84.35M€[38]
Cint_arr303.5k€/km[38]
For the installation of the turbine and substructure, different strategies are available depending on which components are (pre)assembled onshore. In a general form, these costs can be written, according to Ref. [38], as
(14)

where Cvessel is the charter cost of the installation vessel and Vvessel its average velocity, Tin is the time required for the installation, and ntrips is the required trips number.

Depending on the chosen strategy, the first term in the brackets can be expanded. In this work, tower and floater are considered as pre-assembled onshore and carried on site by an AHTS (anchor handling tug supply) vessel, while the turbines are carried by a PSV (platform supply vessel). Therefore, Eq. (14) can be rewritten as
(15)
where ntrips_fl and ntrips_tur represent the number of trips required to transport floaters and turbines, respectively. These two values are defined as
(16)

being nfl the number of floaters to carry, set same as ntur, and nfl_per_trip and ntur_per_trip the number of floaters and turbines that can be carried per trip.

The adopted quantities are listed in Table 4.

Table 4

Installation components costs

ParameterValueUnitReference
Cvessel0.012M€/h[39]
VAHTS10km/h[41]
nfl_per_trip2[41]
VPSV61.7km/h[41]
ntur_per_trip3[41]
Tin48h[38]
ParameterValueUnitReference
Cvessel0.012M€/h[39]
VAHTS10km/h[41]
nfl_per_trip2[41]
VPSV61.7km/h[41]
ntur_per_trip3[41]
Tin48h[38]

Finally, the last component to be analyzed is relative to the operation and maintenance of the wind farm. Here, OPEX is composed of two elements, specifically fixed costs related to repairs, port facilities, and equipment and variable costs associated with traveling expenses. Due to a limited number of active offshore floating plants available as reference, values of 138 k€/MW/yr for the former and 40 €/MW/km/yr for the latter are assumed [38].

Annual energy production and costs are often combined in the LCOE, defined as the price at which the electricity produced by the farm has to be sold to reach the break-even point [38] with the ratio
(17)

where CAPEX and OPEX are evaluated each year i for LT = 25 yr of the plant lifetime, considering a discount rate r =0.05.

3 Case Studies

The described methodology is applied to two different virtual case studies in the Mediterranean Sea, namely, case A and case B. Case A is located off the coast of Civitavecchia, Lazio, while case B is located in the Gulf of Squillace, off the coast of Catanzaro, Calabria. For both sites, a square area of 20D × 20D is identified.

Figure 4 provides representations of the bathymetry of the two cases and their corresponding wind roses. In the same figure, the positions of the offshore substations are also shown. The main characteristics of the sites are summarized in Table 5.

Fig. 4
Bathymetry maps and wind roses for case A (top) and case B (bottom)
Fig. 4
Bathymetry maps and wind roses for case A (top) and case B (bottom)
Close modal
Table 5

Site characteristics comparison

AB
Most frequent wind direction150 deg, 30 deg300 deg
Mean wind speed (at hub height)6.89 m/s8.68 m/s
Ambient turbulence intensity6%6%
Depth range100–250 m600–1000 m
Shore distance20 km15 km
Port distance20 km90 km
AB
Most frequent wind direction150 deg, 30 deg300 deg
Mean wind speed (at hub height)6.89 m/s8.68 m/s
Ambient turbulence intensity6%6%
Depth range100–250 m600–1000 m
Shore distance20 km15 km
Port distance20 km90 km

The ambient turbulence intensity is assumed to be 6%, a value that falls in the range of TI reported in Ref. [42] and derived from the FINO1 experiment [43]. Case B presents an irregular bathymetric profile and deeper seabeds compared to case A. Wind resource of case B is characterized by higher intensity and a single prevailing direction. By contrast, in case A, wind blows more frequently in two, almost orthogonal directions. The nearest ports for installation operations are assumed to be those of Civitavecchia for case A and Crotone for case B.

The 15 MW reference turbine from the International Energy Agency (IEA) [44] is selected for both cases. Its main characteristics are shown in Table 6, while Fig. 5 shows power (Cp) and thrust (Ct) coefficient curves as functions of wind speed.

Fig. 5
Cp and Ct curves of the IEA 15 MW wind turbine
Fig. 5
Cp and Ct curves of the IEA 15 MW wind turbine
Close modal
Table 6

International Energy Agency 15 MW turbine specifications

ParameterValue
Rated power15 MW
Diameter240 m
Hub height150 m
Cut-in wind speed3 m/s
Cutoff wind speed25 m/s
Rated wind speed10.59 m/s
Expected lifespan25 yr
ParameterValue
Rated power15 MW
Diameter240 m
Hub height150 m
Cut-in wind speed3 m/s
Cutoff wind speed25 m/s
Rated wind speed10.59 m/s
Expected lifespan25 yr

Both 5 × 5 km areas are discretized by a Cartesian grid of possible locations, with cells of 250 m side. This resulted in 441 possible locations for the turbines. The selected constraints of the optimization problem defined in Sec. 2 are reported in Table 7.

Table 7

Optimization constraints

ParameterValue
NMAX30
NMIN5
Distmin3 ·D
ParameterValue
NMAX30
NMIN5
Distmin3 ·D

In both cases, an initial population of 600 individuals evolves for 1000 generations. Finally, regarding mutation, the following parameters are set: Pm1=0.5 and Pm2=0.1.

4 Results

In this section, results obtained for the two sites are presented. At first, a sensitivity study on the initial population is performed on a reduced version of case A (12 × 12 grid), showing no variations in the resulting Pareto front when generating different initial populations randomly. Individuals with minimum LCOE are also consistently the same across multiple runs.

Figure 6 shows, for both cases, the trend of the HV as the number of evaluations of individuals increases. The indicator is calculated considering the reference point Q = (1.2, 1.2). For each generation, consisting of 600 individuals, the front of nondominated individuals is computed, and the value of the indicator defined in Sec. 2.1 is displayed. The curve exhibits a region with a steep slope until reaching 100,000 function evaluations, after which it assumes an asymptotic behavior, converging around the value of 0.95. This graph confirms the algorithm's capability, with a fixed population size of 600 individuals, to achieve convergence in fewer than 1000 generations, as used in the present study.

Fig. 6
Hypervolume convergence
Fig. 6
Hypervolume convergence
Close modal

Figure 7 presents for both case studies the entire population of the final generation in the objective space. Each black dot represents a nondominated individual. On the horizontal axis of the graph, there is the AEP (with inverted sign), while the vertical axis reports the CLT. It can immediately be observed that the black dots are clustered on lines with similar CLT. Each of these clusters corresponds to the set of individuals having the same number of turbines. The red line of Fig. 7 highlights the envelope of individuals of these clusters with lower total costs. The blue line, on the other hand, highlights simultaneously individuals with the highest AEP and the lowest LCOE for each isoturbine count cluster. This behavior is due to the limited impact of the cost components, recalling Eqs. (12) and (13), that are variable even with a fixed number of turbines. In particular, the length of internal cables and seabed depth are not significantly affecting the cost related objective function. Values of the AEP, on the other hand, due to wake losses, have shown significant variability even among individuals with the same number of turbines. These considerations explain how, for each cluster having the same number of turbines, individuals with higher AEP coincide with those having lower LCOE.

Fig. 7
Pareto front of dominant individuals: (a) case A and (b) case B
Fig. 7
Pareto front of dominant individuals: (a) case A and (b) case B
Close modal

Furthermore, Fig. 7 highlights the three points of interest: (i) the Pareto-optimal (purple cross), (ii) the maximum AEP (red circle), and (iii) the minimum LCOE (green square). The first is defined as the point with the minimum Euclidean distance from the origin in the normalized objective space. These individuals represent an interesting subset to focus on (i) the final result of the proposed multi-objective optimization procedure, (ii) the individual of maximum possible energy production according to the model and constraints, and (iii) the individual that would be optimal taking the LCOE as single cost function of the optimization.

Figure 8 shows the LCOE and wake losses over the number of turbines and the same three point of interest defined above, associated with the entire population of the final generation. Wake losses are defined as
(18)

where AEPNOwake represents the energy production that would be obtained considering all turbines as isolated. The blue line shows, for each number of turbines, the individuals with the lowest LCOE and wake losses. These two lines collect the same individuals. In both cases, the slope of wake losses exhibits an increasing trend due to the saturation of available space. Indeed, for a limited number of turbines, configurations capable of preventing wakes from impacting the rotors are feasible. However, as the number of turbines increases, achieving this becomes progressively more difficult and the slope of the curve increases. Moreover, case B has a unique predominant wind direction, which allows for better turbine placement and results in lower losses compared to case A.

Fig. 8
LCOE and wake losses for different turbines number for population of the last generation: (a) case A and (b) case B
Fig. 8
LCOE and wake losses for different turbines number for population of the last generation: (a) case A and (b) case B
Close modal

On the other hand, economical performances of both cases are described by an initially decreasing trend of the LCOE until reaching a point beyond which wake losses assume an excessively high impact. The left side of the plot shows that increasing the number of turbines allows for better amortization of costs related to wind farm infrastructures. However, beyond a certain number of rotors, their operation becomes so compromised that a further increase of turbines installation is no longer economically sustainable. Moreover, the LCOE graphs clearly show the discontinuity corresponding to the increased number of necessary export cables. In fact, as explained in Sec. 2.3, the use of a second cable is deemed necessary once the nominal power reached 330 MW. Case B, despite being characterized by much deeper waters and a greater distance from the nearest port, shows lower values of LCOE with respect to case A. In fact, the higher wind resource availability for case B results in larger AEP increase for a lesser increase in costs, which is also indicating a better coupling between this site and the chosen wind turbine model. Furthermore, in case B, due to the unique prevalent wind direction, LCOE exhibits a decreasing trend up to a higher number of turbines compared to case A.

Tables 8 and 9 report the main characteristics of the wind farm layouts with the lowest LCOE, the higher AEP, and the minimum Euclidean distance from the origin, for both cases. It can be observed how the Pareto-optimal individuals are very close but not equal to the individuals that minimize LCOE index. In case A, the Pareto optimality tends to a configuration with higher installed power than in the case of minimal LCOE, preferring an increase in energy to a reduction in costs. The opposite trend is observed in case B where, despite a slightly lower LCOE associated with a project with 19 turbines, the Pareto-optimal shows a better option selecting 17 turbines (preferring a reduction of overall costs instead of an increase in total AEP). Considering that the interest in the point with the minimum distance from the origin arises from the assumption that the two objective functions (LCOE and AEP) are of equal importance, this point represents, by definition, an individual in a median position on the Pareto front. On the other hand, the number of turbines for the individual with the lowest LCOE depends, as explained previously, on the site-dependent benefit of installing a larger number of turbines. For this reason, we observe that, in case B (characterized by higher wind intensity and a single predominant wind direction), the individual with the lowest LCOE has more turbines than the point with the minimum distance from the origin, whereas in case A, this is not true.

Table 8

Case A minimum LCOE and maximum AEP individuals

Opt ParetoMin LCOEMax AEP
Turbines number171629
LCOE (€/MWh)112.19112.06122.72
AEP (GWh)7487091154
Wake losses (%)3.92.612.5
Opt ParetoMin LCOEMax AEP
Turbines number171629
LCOE (€/MWh)112.19112.06122.72
AEP (GWh)7487091154
Wake losses (%)3.92.612.5
Table 9

Case B minimum LCOE and maximum AEP individuals

Opt ParetoMin LCOEMax AEP
Turbines number171930
LCOE (€/MWh)80.9680.8285
AEP (GWh)102711401698
Wake losses (%)349.7
Opt ParetoMin LCOEMax AEP
Turbines number171930
LCOE (€/MWh)80.9680.8285
AEP (GWh)102711401698
Wake losses (%)349.7

Figures 9 and 10 show the layout, the power output of each turbine, and the velocity field in the wind farm, for the wind farm layouts corresponding to the Pareto-optimum, minimum LCOE, and maximum AEP, respectively. The plots are generated only for the most frequent wind conditions of each site, in order to have a summary view on the most probable operating state of these wind farms.

Fig. 9
Pareto-optimal (left column), minimum LCOE (center), and maximum AEP (right column) individuals: (a) opt. Pareto turbines power, (b) minimum LCOE turbines power, (c) maximum AEP turbines power, (d) opt. Pareto wind farm velocity field, (e) minimum LCOE wind farm velocity field, and (f) maximum AEP wind farm velocity field
Fig. 9
Pareto-optimal (left column), minimum LCOE (center), and maximum AEP (right column) individuals: (a) opt. Pareto turbines power, (b) minimum LCOE turbines power, (c) maximum AEP turbines power, (d) opt. Pareto wind farm velocity field, (e) minimum LCOE wind farm velocity field, and (f) maximum AEP wind farm velocity field
Close modal
Fig. 10
Case B Pareto-optimal (left column), minimum LCOE (center), and maximum AEP (right column) individuals: (a) opt. Pareto turbines power, (b) minimum LCOE turbines power, (c) maximum AEP turbines power, (d) opt. Pareto wind farm velocity field, (e) minimum LCOE wind farm velocity field, and (f) maximum AEP wind farm velocity field
Fig. 10
Case B Pareto-optimal (left column), minimum LCOE (center), and maximum AEP (right column) individuals: (a) opt. Pareto turbines power, (b) minimum LCOE turbines power, (c) maximum AEP turbines power, (d) opt. Pareto wind farm velocity field, (e) minimum LCOE wind farm velocity field, and (f) maximum AEP wind farm velocity field
Close modal

In particular, for case A, the most frequent wind direction is 150 deg and it occurs with a frequency of 14%, while for case B the most frequent wind direction is 300 deg and it occurs with a frequency of 30%. The considered wind speeds are equal to the average wind speed at each site: 6.89m/s for case A and 8.68m/s for case B. All figures presents the spatial coordinates expressed in rotor diameters (D = 240 m). The plots show how individuals of maximum AEP exhibit a high number of turbines operating in waked conditions. All layouts show clearly the trend to place the turbines toward the two sides of the domain exposed to the prevailing wind. Furthermore, the wake effect is avoided by controlling the turbine density in the center of the domain. Moreover, in both cases, optimal and minimum LCOE individuals have almost the same (or larger) number of turbines that works at the maximum power with respect to the individual with maximum AEP. The preceding considerations justify the need to identify the number of turbines capable of maximizing the energy potential of the site without incurring in efficiency reductions that would compromise the economic sustainability of the project. Comparing Pareto-optimal with the minimum LCOE individual in case A, no significant differences are recorded, while in case B, the Pareto-optimal solution tends to a layout with two turbines less, having one turbine less working at maximum power, but fewer turbines working under strong wakes.

Finally, to evaluate the impact of uncertainty in the wake model, an additional run of case A is performed using the well-established Jensen wake model. This yields a minimum LCOE value of 113.12 €/MWh, with a deviation of only 0.83% compared to the result obtained with the Gauss curl hybrid model. This outcome suggests that the total uncertainty is primarily due to the cumulative effect of all assumptions embedded in the model used for this study, rather than from the wake loss prediction alone.

5 Conclusions

This study presents a methodology for multi-objective optimization in a floating offshore wind farm layout optimization problem, using NSGA II genetic algorithm where each tested individual is representing a different farm with its own layout and number of turbines. The problem is defined using the python multi-objective optimization library pymoo by defining two objectives, namely, the maximization of AEP and the minimization of total costs. The two-dimensional objective space is explored by varying the number of turbines within a specified range. Constraints are imposed by specifying maximum and minimum numbers of turbines and their minimum distance. Total costs, subdivided in CAPEX and OPEX, are evaluated according to literature, while AEP (including wake losses) is calculated using the library floris. The algorithm proceeds by selecting individuals compliant with constraints and nondominated in the objective space for survival, leading to the final generation. This methodology is applied to two test cases in the Mediterranean Sea, case A, located offshore Civitavecchia, and case B, in the Gulf of Squillace. Case B is characterized by a unique predominant wind direction, higher average wind speed, and greater water depth compared to case A. The results suggest that case B is significantly more cost-effective in terms of LCOE. For case A, the individual with the lowest LCOE is characterized by having 16 turbines, while in case B, it is characterized by 19 turbines. From these results, it can be inferred that:

  • For the cost model used, the depth of the seabed has a negligible impact on costs.

  • The presence of a single wind direction allows for the placement of a greater number of turbines without them operating under disadvantageous conditions.

Observing the values of the functions in the objective space, it can be deduced that, with an equal number of turbines for both cases, costs show little variation. An element of uncertainty may derive from the current interarray cost model, which could be refined in future studies. The comparison, in both test cases, between layouts with maximum AEP and minimum LCOE demonstrates how, as the number of turbines increases beyond the site's saturation, the wake losses become so high that they do not justify the installation of additional rotors. Indeed, the wake losses exhibit an increasing slope trend as the number of turbines increases, gradually assuming a greater significance.

Funding Data

  • Ministry of University and Research (MUR) as part of the European Union program NextGenerationEU, PNRR-M4C2-PE0000021 “NEST—Network 4 Energy Sustainable Transition” in Spoke 2 “Energy Harvesting and Off-shore renewables” (Funder ID: 10.13039/501100003407).

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

References

1.
European Commission
,
2023
, “
Delivering on the EU Offshore Renewable Energy Ambitions
,” Communication from the Commission to the European Parliament, the Council, the European Economic and Social Committee and the Committee of the Regions, Bruxelles, Belgium, Report No. COM(2023) 668 final.
2.
European Commission
,
2020
, “
An EU Strategy to Harness the Potential of Offshore Renewable Energy for a Climate Neutral Future
,” Communication from the Commission to the European Parliament, the Council, the European Economic and Social Committee and the Committee of the Regions, Bruxelles, Belgium, Report No. COM(2020) 741 final.
3.
Musial
,
W.
,
Spitsen
,
P.
,
Duffy
,
P.
,
Beiter
,
P.
,
Marquis
,
M.
,
Hammond
,
R.
, and
Shields
,
M.
,
2022
, “
Offshore Wind Market Report: 2022 Edition
,” National Renewable Energy Lab. (NREL), Golden, CO, Report No.
DOE/GO-102022-5765
.https://www.energy.gov/eere/wind/articles/offshore-wind-market-report-2022-edition
4.
Barthelmie
,
R. J.
,
Larsen
,
G.
,
Frandsen
,
S.
,
Folkerts
,
L.
,
Rados
,
K.
,
Pryor
,
S.
,
Lange
,
B.
, and
Schepers
,
G.
,
2006
, “
Comparison of Wake Model Simulations With Offshore Wind Turbine Wake Profiles Measured by Sodar
,”
J. Atmos. Oceanic Technol.
,
23
(
7
), pp.
888
901
.10.1175/JTECH1886.1
5.
Hou
,
P.
,
Zhu
,
J.
,
Ma
,
K.
,
Yang
,
G.
,
Hu
,
W.
, and
Chen
,
Z.
,
2019
, “
A Review of Offshore Wind Farm Layout Optimization and Electrical System Design Methods
,”
J. Mod. Power Syst. Clean Energy
,
7
(
5
), pp.
975
986
.10.1007/s40565-019-0550-5
6.
Pollini
,
N.
,
2022
, “
Topology Optimization of Wind Farm Layouts
,”
Renewable Energy
,
195
, pp.
1015
1027
.10.1016/j.renene.2022.06.019
7.
Jensen
,
N. O.
,
1983
,
A Note on Wind Generator Interaction
, Vol.
2411
,
Citeseer
, Risø National Laboratory, Roskilde, Denmark.
8.
Gebraad
,
P. M.
,
Teeuwisse
,
F. W.
,
Van Wingerden
,
J.
,
Fleming
,
P. A.
,
Ruben
,
S. D.
,
Marden
,
J. R.
, and
Pao
,
L. Y.
,
2016
, “
Wind Plant Power Optimization Through Yaw Control Using a Parametric Model for Wake Effects—A CFD Simulation Study
,”
Wind Energy
,
19
(
1
), pp.
95
114
.10.1002/we.1822
9.
Annoni
,
J.
,
Fleming
,
P.
,
Scholbrock
,
A.
,
Roadman
,
J.
,
Dana
,
S.
,
Adcock
,
C.
,
Porte-Agel
,
F.
,
Raach
,
S.
,
Haizmann
,
F.
, and
Schlipf
,
D.
,
2018
, “
Analysis of Control-Oriented Wake Modeling Tools Using Lidar Field Results
,”
Wind Energy Sci.
,
3
(
2
), pp.
819
831
.10.5194/wes-3-819-2018
10.
Bastankhah
,
M.
, and
Porté-Agel
,
F.
,
2016
, “
Experimental and Theoretical Study of Wind Turbine Wakes in Yawed Conditions
,”
J. Fluid Mech.
,
806
, pp.
506
541
.10.1017/jfm.2016.595
11.
Bay
,
C. J.
,
Fleming
,
P.
,
Doekemeijer
,
B.
,
King
,
J.
,
Churchfield
,
M.
, and
Mudafort
,
R.
,
2023
, “
Addressing Deep Array Effects and Impacts to Wake Steering With the Cumulative-Curl Wake Model
,”
Wind Energy Sci.
,
8
(
3
), pp.
401
419
.10.5194/wes-8-401-2023
12.
Tesauro
,
A.
,
Réthoré
,
P.-E.
, and
Larsen
,
G. C.
,
2012
, “
State of the Art of Wind Farm Optimization
,”
Proceedings of EWEA
, Copenhagen, Denmark, Apr. 16–19, pp.
1
11
.https://backend.orbit.dtu.dk/ws/portalfiles/portal/7990594/State_of_the_Art_of_Wind_Farm_Optimization.pdf
13.
Gebraad
,
P.
,
Thomas
,
J. J.
,
Ning
,
A.
,
Fleming
,
P.
, and
Dykes
,
K.
,
2017
, “
Maximization of the Annual Energy Production of Wind Power Plants by Optimization of Layout and Yaw-Based Wake Control
,”
Wind Energy
,
20
(
1
), pp.
97
107
.10.1002/we.1993
14.
do Couto
,
T. G.
,
Farias
,
B.
,
Diniz
,
A.
, and
de Morais
,
M. V. G.
,
2013
, “
Optimization of Wind Farm Layout Using Genetic Algorithm
,”
Tenth World Congress on Structural and Multidisciplinary Optimization
, Orlando, FL, May 19–24, pp.
1
10
.
15.
Grady
,
S.
,
Hussaini
,
M.
, and
Abdullah
,
M. M.
,
2005
, “
Placement of Wind Turbines Using Genetic Algorithms
,”
Renewable Energy
,
30
(
2
), pp.
259
270
.10.1016/j.renene.2004.05.007
16.
González
,
J. S.
,
Rodríguez
,
A. G.
,
Mora
,
J. C.
,
Santos
,
J. R.
, and
Payán
,
M. B.
,
2009
, “
A New Tool for Wind Farm Optimal Design
,”
IEEE Bucharest PowerTech
, Bucharest, Romania,
June 28–July 2
, pp.
1
7
.10.1109/PTC.2009.5281977
17.
Mosetti
,
G.
,
Poloni
,
C.
, and
Diviacco
,
B.
,
1994
, “
Optimization of Wind Turbine Positioning in Large Windfarms by Means of a Genetic Algorithm
,”
J. Wind Eng. Ind. Aerodyn.
,
51
(
1
), pp.
105
116
.10.1016/0167-6105(94)90080-9
18.
Rodrigues
,
S.
,
Bauer
,
P.
, and
Bosman
,
P. A.
,
2016
, “
Multi-Objective Optimization of Wind Farm Layouts—Complexity, Constraint Handling and Scalability
,”
Renewable Sustainable Energy Rev.
,
65
, pp.
587
609
.10.1016/j.rser.2016.07.021
19.
Chen
,
Y.
,
Li
,
H.
,
He
,
B.
,
Wang
,
P.
, and
Jin
,
K.
,
2015
, “
Multi-Objective Genetic Algorithm Based Innovative Wind Farm Layout Optimization Method
,”
Energy Convers. Manage.
,
105
, pp.
1318
1327
.10.1016/j.enconman.2015.09.011
20.
Li
,
W.
,
Özcan
,
E.
, and
John
,
R.
,
2017
, “
Multi-Objective Evolutionary Algorithms and Hyper-Heuristics for Wind Farm Layout Optimisation
,”
Renewable Energy
,
105
, pp.
473
482
.10.1016/j.renene.2016.12.022
21.
Wan
,
C.
,
Wang
,
J.
,
Yang
,
G.
,
Gu
,
H.
, and
Zhang
,
X.
,
2012
, “
Wind Farm Micro-Siting by Gaussian Particle Swarm Optimization With Local Search Strategy
,”
Renewable Energy
,
48
, pp.
276
286
.10.1016/j.renene.2012.04.052
22.
Fischetti
,
M.
, and
Monaci
,
M.
,
2016
, “
Proximity Search Heuristics for Wind Farm Optimal Layout
,”
J. Heuristics
,
22
(
4
), pp.
459
474
.10.1007/s10732-015-9283-4
23.
Deb
,
K.
,
Pratap
,
A.
,
Agarwal
,
S.
, and
Meyarivan
,
T.
,
2002
, “
A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II
,”
IEEE Trans. Evol. Comput.
,
6
(
2
), pp.
182
197
.10.1109/4235.996017
24.
Blank
,
J.
, and
Deb
,
K.
,
2020
, “
Pymoo: Multi-Objective Optimization in Python
,”
IEEE Access
,
8
, pp.
89497
89509
.10.1109/ACCESS.2020.2990567
25.
NREL
,
2023
, “
FLORIS. Version 3.5
,” NREL, Golden, CO.
26.
Davis, N. N., Badger, J., Hahmann, A. N., Hansen, B. O., Mortensen, N. G., Kelly, M., et al., 2023, “The Global Wind Atlas: A High-Resolution Dataset of Climatologies and Associated Web-Based Application,”
Bull. Am. Meteorol. Soc.
, 104(8), pp. E1507–E1525.10.1175/BAMS-D-21-0075.1
27.
Kukkonen
,
S.
, and
Lampinen
,
J.
,
2005
, “
GDE3: The Third Evolution Step of Generalized Differential Evolution
,”
IEEE Congress on Evolutionary Computation
, Edinburgh, UK,
Sept. 2–5
, pp.
443
450
.10.1109/CEC.2005.1554717
28.
Deb
,
K.
,
Sindhya
,
K.
, and
Okabe
,
T.
,
2007
, “
Self-Adaptive Simulated Binary Crossover for Real-Parameter Optimization
,”
Proceedings of the Ninth Annual Conference on Genetic and Evolutionary Computation
, London, UK, July 7–11, pp.
1187
1194
.10.1145/1276958.1277190
29.
Fonseca
,
C. M.
,
Paquete
,
L.
, and
López-Ibánez
,
M.
,
2006
, “
An Improved Dimension-Sweep Algorithm for the Hypervolume Indicator
,”
IEEE International Conference on Evolutionary Computation
, Vancouver, BC, Canada,
July 16–21
, pp.
1157
1163
.10.1109/CEC.2006.1688440
30.
Churchfield
,
M.
, and
Lee
,
S.
,
2012
, “
NWTC Design Codes-SOWFA
,” NREL, Golden, CO, accessed Dec. 18, 2024, https://www.nrel.gov/wind/nwtc/sowfa.html
31.
Niayifar
,
A.
, and
Porté-Agel
,
F.
,
2016
, “
Analytical Modeling of Wind Farms: A New Approach for Power Prediction
,”
Energies
,
9
(
9
), p.
741
.10.3390/en9090741
32.
Crespo
,
A.
, and
Hernández
,
J.
,
1996
, “
Turbulence Characteristics in Wind-Turbine Wakes
,”
J. Wind Eng. Ind. Aerodyn.
,
61
(
1
), pp.
71
85
.10.1016/0167-6105(95)00033-X
33.
Göçmen
,
T.
,
Van der Laan
,
P.
,
Réthoré
,
P.-E.
,
Diaz
,
A. P.
,
Larsen
,
G. C.
, and
Ott
,
S.
,
2016
, “
Wind Turbine Wake Models Developed at the Technical University of Denmark: A Review
,”
Renewable Sustainable Energy Rev.
,
60
, pp.
752
769
.10.1016/j.rser.2016.01.113
34.
Katic
,
I.
,
Højstrup
,
J.
, and
Jensen
,
N. O.
,
1986
, “
A Simple Model for Cluster Efficiency
,”
European Wind Energy Association Conference and Exhibition
,
Rome, Italy
, Oct. 7–9, pp.
407
410
.https://backend.orbit.dtu.dk/ws/portalfiles/portal/106427419/A_Simple_Model_for_Cluster_Efficiency_EWEC_86_.pdf
35.
Habenicht
,
G.
,
2011
, “
Offshore Wake Modelling
,”
Presentation at Renewable UK Offshore Wind
Conference and Wxhibition, Liverpool, UK, June 29–30, pp.
1
8
.
36.
Ti
,
Z.
,
Deng
,
X. W.
, and
Zhang
,
M.
,
2021
, “
Artificial Neural Networks Based Wake Model for Power Prediction of Wind Farm
,”
Renewable Energy
,
172
, pp.
618
631
.10.1016/j.renene.2021.03.030
37.
Yang
,
K.
,
Deng
,
X.
,
Ti
,
Z.
,
Yang
,
S.
,
Huang
,
S.
, and
Wang
,
Y.
,
2023
, “
A Data-Driven Layout Optimization Framework of Large-Scale Wind Farms Based on Machine Learning
,”
Renewable Energy
,
218
, p.
119240
.10.1016/j.renene.2023.119240
38.
Martinez
,
A.
, and
Iglesias
,
G.
,
2022
, “
Mapping of the Levelised Cost of Energy for Floating Offshore Wind in the European Atlantic
,”
Renewable Sustainable Energy Rev.
,
154
, p.
111889
.10.1016/j.rser.2021.111889
39.
Cavazzi
,
S.
, and
Dutton
,
A.
,
2016
, “
An Offshore Wind Energy Geographic Information System (OWE-GIS) for Assessment of the UK's Offshore Wind Energy Potential
,”
Renewable Energy
,
87
, pp.
212
228
.10.1016/j.renene.2015.09.021
40.
Ministero dell'ambiente e della sicurezza energetica, 2024, “Definizione contenuti SIA,” Ministero dell'ambiente e della sicurezza energetica, Rome, Italy, accessed Dec. 18, 2024, https://va.mite.gov.it/it-it/procedure/viaelenco/1/9
41.
Myhr
,
A.
,
Bjerkseter
,
C.
,
Ågotnes
,
A.
, and
Nygaard
,
T. A.
,
2014
, “
Levelised Cost of Energy for Offshore Floating Wind Turbines in a Life Cycle Perspective
,”
Renewable Energy
,
66
, pp.
714
728
.10.1016/j.renene.2014.01.017
42.
Emeis
,
S.
,
2014
, “
Current Issues in Wind Energy Meteorology
,”
Meteorol. Appl.
,
21
(
4
), pp.
803
819
.10.1002/met.1472
43.
Berge
,
E.
,
Byrkjedal
,
O.
,
Ydersbond
,
Y.
,
Kindler
,
D.
, and
Kjeller Vindteknikk
,
A.
,
2009
, “
Modelling of Offshore Wind Resources. Comparison of a Meso-Scale Model and Measurements From FINO 1 and North Sea Oil Rigs
,”
EWEC 2009
, Marseille, France, Mar. 16–19, pp.
1
8
.https://www.enecafe.com/interdomain/idlidar/paper/2009/offshore%20mesosclale%20EWEC2009.pdf
44.
Gaertner
,
E.
,
Rinker
,
J.
,
Sethuraman
,
L.
,
Zahle
,
F.
,
Anderson
,
B.
,
Barter
,
G. E.
,
Abbas
,
N. J.
, et al.,
2020
, “
IEA Wind TCP Task 37: Definition of the IEA 15-Megawatt Offshore Reference Wind Turbine
,” National Renewable Energy Lab. (NREL), Golden, CO, Report No.
NREL/TP-5000-75698
.https://www.nrel.gov/docs/fy20osti/75698.pdf