The share of wind energy in the electricity market is rapidly increasing . Offshore wind farms impose fewer geographical and social constraints than onshore wind farms, which leads to larger design spaces. The performance of an offshore wind farm indicates how efficient the system is at achieving its main objective . Examining operational wind farms, a development of farm layouts over time can be recognised. Earlier wind farms show regular patterns such as the wind farms Horns Rev 1 (2002) and Prinses Amalia (2008) . Newer and larger wind farms show more variation in patterns such as the wind farms Horns Rev 2 (2009) and Rødsand (2010) and partial irregularity such as Anholt (2013) and many more. A number of desktop optimisation studies even suggest fully irregular wind farms such as Research Layout 1 and Research Layout 2 , which are obtained from existing research as shown in Fig. .

investigated the preliminary design of the layout, electrical collection system and support structures of an offshore wind farm using two different optimisation approaches. The sequential approach neglects the interaction between the three selected performance characteristics, while precisely this is taken into account in the multidisciplinary design analysis and optimisation (MDAO) approach. The two approaches were used to optimise the total-system levelised cost of energy (LCOE) of a regular and an irregular farm layout. The study focussed on the comparison of the two approaches and their interaction effects, not comparing the performance of the different geometric patterns. used a multi-objective genetic algorithm (GA) to maximise the wind farm efficiency and minimise its cost applying real wind conditions. Investigating one regular and three irregular layouts with identical total geographical area, the comparison suggested that irregular geometric patterns may perform better than regular layouts, yet no final conclusion was drawn in the study. The goal of maximising energy extraction while minimising cost was also pursued by , who compared regular and irregular wind farm layouts. The resulting power, capacity factor and efficiency were all higher for the irregular layout, although it should be noted that a constant wind speed and constant wind direction were considered. For general validity of this conclusion, the different layout options also need to be investigated at variable wind speeds and with variable wind directions.

Three observations can be made for comparisons of wind farms with regular and irregular layouts conducted to date. First, the performance indicators for wind farms are not well-defined in the literature. An overview of all possible indicators and how they affect the overall performance is lacking. Also, the degree to which these indicators are influenced by the geometry of the wind farm has not been investigated.

Second, the effects of optimised wind farm layouts on all performance indicators either are unknown or have only been partially investigated in the literature. The existing studies which include regular and irregular wind farm layouts focus on the performance of the optimisation tools. The aim of these optimisation studies is not to compare the overall performance of the regular and irregular wind farms.

Third, an optimisation of an existing regular wind farm pattern inherently leads to an increase in irregularity, as shown by optimisation studies . This is only logical given the enormous design space of irregular patterns with many local optima compared to regular patterns. A particle swarm optimisation (PSO) or genetic algorithm (GA) is, for example, unlikely to find a regular pattern. Based on the nature of optimisation algorithms, many studies are naturally biased towards irregular wind farm layouts.

The objective of this paper is thus to quantify the effect of regular and irregular offshore wind farm layouts on selected performance indicators by means of a comparative case study using state-of-the-art simulation tools and models. The paper is based on the graduation project of the first author and is structured as follows. In Sect. , the selection of performance indicators is described. In Sect. , the power production of the different layouts and, in Sect. , the wake-induced tower fatigue of the different layouts are assessed. In Sect. , the general applicability of the results is investigated with a sensitivity analysis of the performance indicators for the Borssele wind farm. Conclusions are presented in Sect. .

All performance indicators by which a wind farm has been assessed so far were inventorised using, among others, the works of and . The indicators were divided into three levels with the third level containing sub-categories, as shown in Fig. . To incorporate changes in energy price, instead of using the LCOE as an overarching key performance indicator, profit (positive net present value) was selected .

To direct the research, a multi-criteria decision analysis was performed based on these criteria: (1) affected by wind farm layout, (2) feasibility to research, (3) site independence and (4) technical nature. This resulted in the selection of five sub-indicators of performance, which were grouped to represent “power performance” (yield/wake losses, predictability and value on the electricity market) and “wake-induced tower fatigue” (wind turbine cost, component replacement cost).

To assess the performance of these indicator groups for the different layout categories, the regular and irregular farm layouts depicted in Fig.  were selected from the work of .

For both layouts, the International Energy Agency (IEA) Wind Task 37 reference wind turbine with 10 MW rated power and a rotor diameter of 190.8 m was used . The degree of irregularity was quantified mathematically with (1) the sum of the distance to surrounding turbines within a radius of 10 rotor diameters (10 RD) for each turbine and (2) the minimum inter-turbine spacing. The number of turbines with the same unique sum of distances to their surrounding turbines with a tolerance of 0.001 RD was 66 for the regular wind farm layout and only 5 for the irregular farm layout. The minimum inter-turbine spacing was 2.73 RD for the irregular wind farm layout compared to 8.88 RD for the regular wind farm.

The following two sections will present the analyses of performance indicator groups 1 and 2, respectively.

The annual energy production (AEP) was analysed in windPRO, computing the absolute difference between regular and irregular wind farm layouts. The wind climate imported to windPRO was obtained from . The data were extrapolated to hub height using the power-law wind profile with a power exponent α of 0.08. The windPRO calculation shows a higher AEP of approximately 0.66 % for the irregular wind farm layout, corresponding to approximately EUR 700 000 according to the average European Power Exchange (EPEX) price in the Netherlands between 2007 and 2020. The individual turbine performance shows that the difference in AEP is not caused by the outliers (best-performing and worst-performing turbines) but by the average-performing turbines in the wind farm. Relating the performance and positions shows that the distribution of the lower-performing turbines is more evenly spread for the regular wind farm than for the irregular wind farm, as is apparent from Fig. .

Figure  shows the persistence of power in relation to wind direction, i.e. the extent to which power production varies with wind direction, measured in degrees, for a certain wind speed. The maximum power drop is quantified as the maximum uninterrupted decrease in power for an increase or decrease in wind direction. This maximum power drop decreases by 73.7 % for the irregular wind farm compared to the regular wind farm layout. The orientation of turbine rows in the wind farm with a regular layout is driving for the angle at which the power drops occur as well as for their magnitude. More turbines in a row correspond to a larger power drop. It is expected that a wind farm layout with a higher persistence in relation to wind direction will lead to a decrease in prediction errors. This will likely result in lower imbalance costs. A rough estimation based on historical imbalance cost data shows that the imbalance cost would amount to approximately 1.6 % of the AEP revenue. The difference between the imbalance cost of the regular and irregular wind farm would then become visible within this 1.6 %.

The analysis for persistence in relation to wind direction was executed using a constant mean wind speed of 9.5 m s-1. This simplification likely overestimates the power drops as a function of wind direction because the time-dependent change in wake losses in the wind farm was not included. Additionally, the wind speed is just below the rated power, which means that the wake losses play a significant role. For higher wind speeds (15 m s-1 and above), the effect of wake losses is reduced or disappears.

Based on the analysis and assumptions in this section, the irregular wind farm layout performs better for all three sub-indicators of performance analysed: the energy yield, predictability and value in the electricity market. The importance and value of persistence in relation to wind direction will increase as the impacts of wind power on power system operation increase with the very large growth foreseen in the coming decades. The indicator predictability and value in the electricity market do require a much more extensive analysis for proper quantification.

The layout of a wind farm and the wind environment determine to what degree downstream turbines are affected by the wakes of upstream turbines. Especially for offshore farms, these wake effects drive wind turbine fatigue loading . With low terrain roughness and low ambient turbulence intensities, the effect of wakes is higher than in onshore wind farms. Multiple studies confirm that one of the fundamental parameters determining the wall thickness of the tower design is fatigue . Therefore, the layout of the farm will affect the cost of the towers.

Interestingly, while monopile foundation designs are optimised for individual locations within an offshore wind farm, typically only a single tower design is applied based on the turbine location with the highest turbulence intensity. Therefore a single high-turbulence turbine location within a project impacts the wall thickness of all towers of that project, which implies a high-cost multiplication factor.

The effective turbulence intensity was quantified using the Frandsen model, which implies that the structural load ranges vary linearly with the turbulence intensity . The model was used to determine the damage-equivalent bending moment

The damage-equivalent load is a load with constant amplitude and fixed frequency causing the same damage as the actual variation in loads over a lifetime.

The sensitivity analysis presented in this section serves as a method to predict the outcome of results for the following:

annual energy production,

Based on the work done in this paper, it is found that irregular wind farm layouts outperform regular layouts regarding energy production, as overall wake losses are reduced. In the performed case study, an overall yield increase of 0.66 % was found for the chosen layout pair. A notable finding is that irregular layouts also increase the persistence in relation to wind direction, which means that the power output is less sensitive to fluctuations in wind direction. For the case study done, the maximum power drop of the irregular layout is roughly one-third of the maximum power drop observed in the regular layout. This characteristic of irregular layouts improves the predictability of the power output, reducing the impact of wind forecasting errors in power system operation and potentially increasing the value of power in the electricity market. The actual benefit of an irregular layout compared to a regular one, ceteris paribus, will vary between the actual layouts compared.

A drawback of irregular layouts is that the turbulence intensity for some turbines may be higher. Using the Frandsen model, the irregular layout is found to generate 14 % to 24 % higher worst-case wake-induced turbulence levels. This results in higher fatigue loads, increases in tower wall thickness and therefore higher steel costs. Since typically only one wind turbine tower design is used per project, this effect may be significant at the project level. By increasing the minimum spacing, the worst-case turbulence intensity is reduced by 10.4 %, bringing the worst-case wake-induced turbulence within the range of the regular layout. This conclusion has general validity, since irregular layouts inherently have at least a few turbines positioned with limited spacing. Although the use of irregular wind farm patterns increases energy yield, improving the performance of the wind farm as a whole requires caution with regard to inter-turbine spacing to limit the negative effect of increased fatigue loading.