A Multi-Objective Optimization Framework for Offshore Wind Farm Design in Deep Water Seas

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 [9–11].

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.

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 $0 m/s$ to $25 m/s$ with a step of $1 m/s$⁠. 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.

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.

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

where N_loc 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.

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.

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].

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 [35–37] to be the most effective and has, therefore, been chosen in this work.

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.

where n_tur is the number of turbines in the plant, n_lines is the number of lines per substructure, h(x, y) is the water depth, and C_anch, C_lines, and C_chain 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.

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 C_intarr 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.

where C_vessel is the charter cost of the installation vessel and V_vessel its average velocity, T_in is the time required for the installation, and n_trips is the required trips number.

being n_fl the number of floaters to carry, set same as n_tur, 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.

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].

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

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.

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 (C_p) and thrust (C_t) coefficient curves as functions of wind speed.

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.

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 $Pm2=0.1$⁠.

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.

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 C_LT. It can immediately be observed that the black dots are clustered on lines with similar C_LT. 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.

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.

where $AEPNO wake$ 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.

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.

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.

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: $6.89 m/s$ for case A and $8.68 m/s$ 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.

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.

• 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).

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