Analysis of Wind Farm Productivity Taking Wake Loss into Account: Case Study

Abstract

Due to the growing demand for green energy, there is a shortage of land available for the location of wind farms. Therefore, the distances between turbines are being reduced, and the power of the turbines is being increased. This results in increased wake loss. The article describes a study of the impact of wind speed deficit and loss of wind turbine output due to wake loss on the decrease in energy efficiency of a wind farm. Two proposed wind farms, where the maximum number of turbines are located, were analyzed. The facilities were designed for implementation in Central Europe. The basic costs of construction and operation of the wind farms (WFs) were estimated. Based on the results of wind measurements and the performance characteristics of wind turbines, the productivity of the WFs was determined. The impact of removing individual turbines with the largest wake losses from the wind farm on the economic outcome of the project was studied. Evaluation criteria were proposed to quantify losses, which can serve as a benchmark for evaluating other wind farms. It was found that the higher the turbine’s power rating, the faster the payback resulting from the wake losses of a single turbine.

1. Introduction

The negative effects of climate change are so evident that it is a priority to focus on the goals of the 2030 Agenda for Sustainable Development [1]. The strategic solution is to make the elimination of the dependence of countries’ economic development on fossil fuel energy while transitioning to renewable energy solutions (RES). The global energy mix is shifting from fossil fuels to greater use of renewable energy sources [2]. The global energy transition has to do with maximizing renewable electricity while respecting the environment [3]. The introduction of economic (market) mechanisms in the energy sector has meant that energy sources with the lowest cost of production and environmental compatibility have the best chance of development. One source that meets the above criteria is wind turbines. Every year, new wind turbines are put into operation around the world. The total installed wind power capacity in Europe is expected to exceed 450 GW by 2030. In the European Union, for example, 6.4 GW of new wind capacity was installed in the first half of 2024 [4]. This is due to the decarbonization strategies for energy sources adopted by European Union countries [5].

Maximizing turbine output in the limited space of wind farms is becoming possible due to the development of large-scale turbines [6]. The distance between turbines in wind farm systems is also decreasing [7]. The above behaviors exacerbate the interference of the aerodynamic trace between turbines, which causes serious wake losses and negatively affects the economic benefits of wind farm operation [8]. A turbine extracting energy from the wind creates a trace behind it. If another turbine is operating in the vicinity of the turbine, the output power of that turbine is reduced compared to a single turbine operating in the open. The reduction in turbine output power is typically 5–15% per year, depending on the wind distribution, the characteristics of the wind turbine, and the layout of the wind equipment on the wind farm [9]. Turbines operating in the trace are not only subjected to reduced wind speeds but also increased dynamic loading. This is due to the increased turbulence induced by the turbines causing the trace [10]. The increased turbulence must be taken into account when selecting the appropriate class of turbine. Initially, in the second half of the 20th century, onshore wind turbines set up on farms were spaced 5 to 7 rotor diameters apart. Today, due to in-depth studies of the aerodynamic trace effect and the effects of turbulence, wind farm turbines are spaced a minimum of 3 rotor diameters apart. The large size of the latest wind turbines and their densification in wind farms highlight the effect of aerodynamic trace on power output, leading to a reduction in the economic benefits of wind farms [11,12].

When designing wind farms (WFs) in large-scale areas free of human settlements and devoid of valuable natural habitats, we can freely shape the placement of turbines [13]. The same is true in offshore areas [14]. In areas with numerous scattered residential developments, such as in Europe, the location of turbines on WFs is subject to a number of restrictions. Noise emissions have the greatest impact on the distance of wind turbines from residential developments [15,16]. In addition, there are also natural and infrastructural constraints (e.g., Natura2000 areas, forests, roads, and overhead power transmission networks). Adequate legal distances must be maintained from the mentioned areas [15]. The only way to increase the installed capacity of WFs then remains to condense the distribution of turbines and maximize their power rating.

There are many ways to model an aerodynamic trace. A simple model of a single trace is the Jensen model [17]. It is based on the assumption of a linearly expanding trace diameter. The Eddy viscosity model uses an axisymmetric formulation of the time-averaged Navier–Sokes equations with the closure of eddy viscosity in the turbine trace [18]. The Larsen model is a semi-analytical model derived from asymptotic expressions of the Prandtl equations for a turbulent boundary layer with rotational symmetry [19,20]. TurbOPark is a version of the Jensen model [17], which uses turbulence intensity (TI) to locally set the trace fading parameter [21]. Most trace models are single trace models. Thus, to get a useful result for wind farms with multiple turbines, single traces must be combined into a cumulative effect. This is accomplished through exclusively empirical means, using various trace combination models that take into account: the sum of squares of velocity deficits, energy balance, geometric sum, and linear superposition [22,23]. Currently, research on the footprint model is advanced and widely used [24,25,26,27].

Trace models require internal parameters as inputs. These parameters take into account terrain type and wind conditions [26,28]. Input parameters for a trace model can be turbulence intensity and roughness length. Usually, such parameters can be assumed to depend on the roughness class of the terrain.

Table 1. Estimated selected parameters of the trace model depending on the type of terrain [29].

Terrain Type	Roughness Class	Roughness Length	Wake Decay Constant	Ambient Turbulence at 50 m	Comments
Offshore water areas	0.0	0.0002	0.040	0.06 ÷ 0.08	Water reservoirs.
Mixed water and land	0.5	0.0024	0.052	0.07 ÷ 0.10	Also applies to very smooth terrain.
Very open farmland	1.0	0.0300	0.063	0.10 ÷ 0.13	Scattered buildings. Smooth hills.
Open farmland	1.5	0.0550	0.075	0.11 ÷ 0.15	Some buildings. Crossing hedges 8 m high and 1250 m apart.
Mixed farmland	2.0	0.1000	0.083	0.12 ÷ 0.16	Some buildings. Crossing hedges 8 m high and 800 m apart.
Trees and farmland	2.5	0.2000	0.092	0.13 ÷ 0.18	Enclosed appearance. Dense vegetation with 8 m hedges 250 m apart.
Forests and villages	3.0	0.4000	0.100	0.15 ÷ 0.21	Villages, small towns, and enclosed farmland. Numerous wood and forest areas.
Large towns and cities	3.5	0.8000	0.108	0.17 ÷ 0.24	Large towns, cities with extended built-up areas.
Large built-up cities	4.0	1.6000	0.117	0.21 ÷ 0.29	Large cities with high buildings.

lists examples of estimated trace model parameters, depending on the type of terrain. The parameters are estimates and should be corrected by taking field measurements.

The article describes a study of the effect of wind speed deficit and loss of wind turbine output due to wake losses on the decrease in energy efficiency of a wind farm. At the same time, it is shown that the loss of energy efficiency is economically compensated for during the 25-year life of the WF.

The aim of the research is to find an answer to the question: is there an economic justification for abandoning individual turbines in a wind farm with reduced productivity due to wake losses?

The proposed evaluation criteria, which allow for the quantification of losses, can serve as a benchmark for the evaluation of other wind farms and inspire further research.

2. Materials and Methods

Two wind farm sites, A and B, were analyzed. The facilities were designed for implementation in Central Europe. Based on the results of the wind measurements and the productivity characteristics of the available wind turbines, the investor selected the equipment. Farm A consists of 34 V136–3.45 MW turbines with a tower height of 131 m. Farm B plans to install 32 V162–6.2 MW turbines with a tower height of 119 m. Two locations with significantly different wind conditions were selected. Due to the equipment manufacturer’s requirements, the turbine towers were arranged, maintaining a minimum spacing of 3 rotor diameters between them. In the cases analyzed, the minimum distances between the turbine towers at Farm A and B were 408 m and 486 m, respectively.

The wind farms analyzed are actual projects that are currently in the design phase. They have a strictly fixed location for wind turbines resulting from the following constraints:

• provision of local spatial development plans,
• distance from the nearest residential development,
• possibility of signing lease agreements with landowners,
• distance from infrastructure (roads, overhead power lines, gas pipelines, etc.),
• distance from valuable forms of nature conservation.

Taking into account the above limitations, the maximum number of turbines on each WF was located, with no possibility of adjusting their location. The distribution of turbines on each WF is shown in Figure 1 and Figure 2.

Due to the need to protect investor-sensitive data, energy productivity analyses were conducted using virtual mast data based on ERA5 Gaussian grid data updated on 31 August 2024. ERA5 is a climate analysis dataset developed by the Copernicus Climate Change Service (C3S) with raw data processed and provided by the European Center for Medium-Range Weather Forecasts (ECMWF). Wind data from the ERA5 database is accepted by banks to provide financial support for investments in wind parks. Professional companies dealing with wind measurement campaigns use ERA5 data for monthly correlations of actual results when determining long-term (multi-year) corrections. It should be mentioned that there are many simulation methods for forecasting wind conditions in the field. An innovative way is to use an artificial neural network (ANN) to predict wind speed. The authors of many publications found high result consistency of such forecasts with measurement [30,31,32]. However, the most reliable way to determine wind conditions is long-term field measurement.

The Copernicus DEM GLO-30 digital terrain surface model was used in the calculations, taking into account landforms, including buildings, infrastructure, and vegetation. The resolution of the data is 30 m. The Copernicus DEM model data were acquired as part of the TanDEM-X mission between 2011 and 2015. The datasets were released for use in 2019 and will be stored until 2026 [33]. The acquired data were correlated with European CORINE Land Cover data containing up-to-date information on land cover and land use throughout Europe [34].

Wake loss was calculated using commercial specialized software. The software used for the wind farm aggregates individual traces into a combined result using empirical combination rules.

Based on the results of the wind measurements and the productivity characteristics of the wind turbines, the productivity of the wind farm (WF) was evaluated. Knowing the productivity characteristics of the power plant, the value of power was estimated for wind speeds falling into successive class intervals of 0.5 m/s. In the next step, the energy generated by the power plant over one year was calculated. Calculations were performed in each class interval. Summing up the component energy from all intervals, the total energy generated over one year by the power plant was obtained [35,36].

The basic factors affecting the productivity of a wind turbine are the available wind power and the associated wind speed distribution at the location under study. Having the results of the average wind speed at the tested heights, we can use the two-parameter Weibull function to determine the mathematical distribution of wind speed. Despite the availability of other mathematical models of wind speed distribution over time, the applied two-parameter Weibull function is the most commonly used in wind power [37].

The two-parameter Weibull probability density function is expressed by:

$$p\left(v\right)=\left({\displaystyle \frac{k}{A}}\right){\left({\displaystyle \frac{v}{A}}\right)}^{k-1}{e}^{-{\left({\displaystyle \frac{v}{A}}\right)}^k},$$

(1)

where v, k, and A > 0.

The parameters k and A appearing in Formula (1) are the shape and scale factors, respectively. The coefficients k and A affect the density of the wind speed distribution. The dimensionless parameter k has values from 1 for variable wind speeds to 3 for stabilized speeds. Parameter A was calculated on the basis of the following relationship:

$$A={\displaystyle \frac{v}{\mathsf{\Gamma }\left(1+\frac{1}{k}\right)}}$$

(2)

Having the given wind speed distributions at the measured heights, the Weibull distribution was fitted to them. Figure 3 and Figure 4 show the power curves of the wind turbines selected for analysis.

Table 2. Turbine power generated in successive wind speed class ranges.

Wind Speed	Power V136–3.45	Power V162–6.2
m/s	kW	kW
3.0	49	34
3.5	127	150
4.0	224	292
4.5	339	467
5.0	480	676
5.5	651	927
6.0	857	1229
6.5	1099	1584
7.0	1382	2000
7.5	1705	2476
8.0	2067	3017
8.5	2460	3626
9.0	2849	4284
9.5	3174	4917
10.0	3369	5483
10.5	3434	5882
11.0	3449	6114
11.5	3450	6176
12.0	3450	6197
12.5	3450	6200
13.0	3450	6200
13.5	3450	6200
14.0	3450	6200
14.5	3450	6200
15.0	3450	6200
15.5	3450	6200
16.0	3450	6200
16.5	3450	6200
17.0	3450	6186
17.5	3450	6077
18.0	3450	5853
18.5	3450	5590
19.0	3450	5348
19.5	3450	5095
20.0	3450	4825
20.5	3450	4538
21.0	3450	4251
21.5	3450	3954
22.0	3450	3664
22.5	3450	3367
23.0	0 *	3064
23.5	0 *	2763
24.0	0 *	2451

* The turbine is cut off if the wind speed exceeds 22.5 m/s.

summarizes the productivity of the analyzed wind turbines in successive wind speed class intervals of 0.5 m/s.

3. Analysis and Results

3.1. Energy Analysis of Farm A

Considering the assumptions presented in Section 2, wind data for Farm A were obtained, and energy analyses were performed. Average wind speeds and directionality (wind rose) at 131 m above ground level are illustrated in Figure 5.

Knowing the wind speed and directionality at a height of 131 m, the wind speed distribution was generated using a two-parameter Weibull function. It was determined that the shape parameter is k = 2.06, the scale parameter A = 7.11 m/s, and the average wind speed is 6.3 m/s. Based on the wind rose, the directional distribution of productivity was determined for the selected wind turbine type (energy rose). The results are shown graphically in Figure 6 and Figure 7.

Taking into account the Weibull distribution (Figure 6) and the power curves of the selected turbines (Figure 3), Figure 7 shows the distribution of annual turbine productivity in each sector of the wind rose. The wake losses are also shown.

Table 3. Annual productivity of Farm A.

Turbine Number	Annual Energy	Wake Loss	Annual Energy	Wake Loss
	MWh/y	%	MWh/y	%
1	2	3	4	5
A01	10,324.1	7.3	10,331.7	7.2
A02	10,210.2	8.3	10,212.4	8.3
A03	10,432.6	6.3	10,435.2	6.2
A04	10,378.6	6.7	10,381.8	6.7
A05	10,499.8	5.7	10,502.1	5.6
A06	10,161.4	8.7	10,165.9	8.7
A07	10,201.3	8.4	10,210.9	8.3
A08	10,891.4	2.1	10,893.2	2.1
A09	9721.5	12.6	9762.6	12.3
A10	9363.6	15.8	9452.0	15.0
A11	10,835.3	2.5	10,838.7	2.5
A12	9010.7	19	9385.3	15.6
A13	9054.6	18.6	9120.9	18.0
A14	9054.0	18.6	9160.7	17.6
A15	9470.7	14.8	9519.2	14.4
A16	10,093.9	9.2	10,106.0	9.1
A17	8571.4	22.9	0.0	0.0
A18	9032.7	18.8	9084.9	18.3
A19	9582.5	13.8	9610.6	13.5
A20	8851.2	20.4	8986.6	19.2
A21	10,579.2	4.8	10,586.4	4.8
A22	9012.0	18.9	9105.3	18.1
A23	10,036.1	9.7	10,044.6	9.6
A24	9417.8	15.2	9439.0	15.0
A25	8707.2	21.6	8824.1	20.6
A26	9001.0	19.1	9152.9	17.7
A27	9843.1	11.4	9858.6	11.3
A28	10,261.3	7.6	10,271.4	7.6
A29	9243.1	16.8	9290.4	16.4
A30	8880.5	20.1	8942.6	19.5
A31	9334.4	16	9398.5	15.4
A32	9891.5	11	9914.6	10.8
A33	9383.2	15.5	9417.4	15.2
A34	10,116.5	8.9	10,145.9	8.6
Mean wake loss WF	-	12.9	-	12.1
Total annual energy WF	329,448.4		322,552.4

(columns 2 and 3) summarizes numerically the annual WF productivity and losses resulting from the aerodynamic footprint. It was determined that, according to the wind rose, the greatest wake losses occurred in the two wind directions SSW and NNW. Based on the analysis of Table 3, Figure 1 and Figure 7, wind turbine A17 was selected for removal from the wind park. Again, a productivity analysis was conducted for the 33 wind turbines at location A, and wake losses were determined. The results are summarized in Table 3 (columns 4 and 5).

3.2. Energy Analysis of Farm B

Considering the assumptions presented in Section 2, wind data for Farm B were obtained, and energy analyses were performed. The average wind speeds and directionality at a height of 119 m above ground level are illustrated in Figure 8.

Knowing the wind speed and directionality at a height of 119 m, the wind speed distribution was generated using a two-parameter Weibull function. It was determined that the shape parameter is k = 2.54, the scale parameter A = 7.02 m/s, and the average wind speed is 6.4 m/s. Based on the wind rose, the directional distribution of productivity was determined for the selected wind turbine type (energy rose). The results are shown graphically in Figure 9 and Figure 10.

Taking into account the Weibull distribution (Figure 9) and the power curves of the selected turbines (Figure 4), Figure 10 shows the distribution of annual turbine productivity in each sector of the wind rose. The wake losses are also shown.

Table 4. Annual productivity of Farm B.

Turbine Number	Annual Energy	Wake Loss	Annual Energy	Wake Loss
	MWh/y	%	MWh/y	%
1	2	3	4	5
B01	17,295.4	5.4	17,303.0	5.4
B02	16,965.7	7.2	16,971.9	7.2
B03	17,259.1	5.6	17,267.6	5.6
B04	16,851.6	7.9	16,865.3	7.8
B05	16,629.7	9.1	16,642.8	9.0
B06	16,650.0	9.0	16,664.0	8.9
B07	17,005.6	7.0	17,021.6	6.9
B08	16,862.9	7.8	16,878.0	7.7
B09	16,542.9	9.5	16,570.3	9.4
B10	15,931.8	12.9	15,969.1	12.0
B11	16,170.6	11.7	16,208.1	11.0
B12	15,491.9	15.4	15,540.6	15.0
B13	17,014.6	7.1	17,036.6	7.0
B14	15,470.2	15.5	15,534.4	15.0
B15	16,101.8	12.1	16,133.2	11.0
B16	15,496.5	15.4	15,551.9	15.0
B17	15,884.4	13.2	15,973.0	12.0
B18	17,050.0	6.9	17,085.4	6.7
B19	16,156.3	11.7	16,204.1	11.0
B20	15,608.2	14.7	16,109.6	11.0
B21	15,799.3	13.7	16,473.9	10.0
B22	16,250.5	11.1	16,403.8	10.0
B23	15,463.4	15.5	0.0	0.0
B24	16,678.4	8.8	16,720.2	8.5
B25	16,337.7	10.7	16,615.8	9.2
B26	16,666.0	8.9	16,793.4	8.2
B27	15,985.6	12.6	16,043.3	12.0
B28	16,300.6	10.9	16,368.8	10.0
B29	17,257.1	5.7	17,338.5	5.3
B30	17,062.1	6.7	17,086.9	6.6
B31	16,593.7	9.2	16,621.2	9.1
B32	16,841.1	7.9	16,869.9	7.7
Mean wake loss WF	-	10.2	-	9.6
Total annual energy WF	525,674.7	-	512,866.2	-

(columns 2 and 3) summarizes the annual WF productivity and wake losses numerically. It was determined that, according to the wind rose, the highest wake losses occur in wind direction W. Based on the analysis of Table 4, Figure 2 and Figure 10, wind turbine B23 was selected for removal from the wind park. Again, a productivity analysis was conducted for the 31 wind turbines at location B, and wake losses were determined. The results are summarized in Table 4 (columns 4 and 5).

3.3. Cost Analysis

It was assumed that the wind farms would be operated for a period of 25 years. The basic costs of construction and operation of the two wind farms were estimated. Common costs, independent of the number of installed devices, such as measuring wind conditions and designing the spacing of turbines on the wind farm, were omitted. Costs for disposal, dismantling of the equipment, and land reclamation, which will occur after the operation period, were not included. A summary of costs in Euro (€) is shown in

Table 5. Selected construction and operating costs of wind turbines in 2024.

Cost Type	V162	V136
	€	€
Annual land lease	34,000	21,000
Construction of the foundation including connection to the internal power grid	280,000	280,000
Maintenance yard	82,000	82,000
Access road 500 m	81,000	81,000
Purchase, transportation, and installation of the turbine	6,000,000	4,000,000
Annual operation (maintenance)	60,000	50,000

. Valuations of individual costs were estimated on the basis of cost estimates of currently implemented wind farms in Poland.

It is difficult to forecast how the price of energy extracted from WFs will develop in future years. The price depends on many political and economic factors. A constant value of EUR 110 per MWh was assumed for the analysis, which is close to current valuations. Based on the analyses in Section 3.1 and Section 3.2, the reduced annual production of Farm A and Farm B resulting from the removal of one turbine at each WF was estimated. The differences are summarized in

Table 6. Annual production of wind farms.

Farm A—V136		Farm B—V162
MWh/y		MWh/y
34 Turbines	33 Turbines	32 Turbines	31 Turbines
329,448.4	322,552.4	525,674.7	512,866.2
Productivity difference = 6896.0		Productivity difference = 12,808.5
Profit difference = 758,560 €		Profit difference = 1,408,935 €

.

Using the data in Table 5, the construction cost of the wind turbines was determined, taking into account the construction of the foundations, access roads, and maintenance yards, as well as the purchase of the devices. An amount of EUR 4,443,000 was obtained for the V136 turbines (Farm A), while an amount of EUR 6,443,000 was obtained for the V162 turbines. The above figures were divided by the annual profit difference caused by the removal of one turbine from each WF, together with the annual cost reduction of leasing the land and maintenance costs of the wind turbines. On this basis, an estimate was made of the payback period it would take to recoup the investment had it not been decided to remove one turbine from each WF. For the V136 turbine (Farm A), the payback period is about 6.5 years, and for the V162 turbine (Farm B), the payback will occur in less than 5 years. This means that for the next 18.5 and 20 years, respectively, the turbines will be profitable and generate clean electricity. The above calculations were carried out for specific types of wind turbines in a well-defined location. They certainly cannot be directly applied to other wind farms. However, the result obtained indicates that there is no economic justification for removing turbines due to increased wake loss.

It should be kept in mind that the actual WF productivity will be reduced to about 5% due to turbine shutdowns at maximum winds due to wake loss-induced turbulence. The manufacturer always adjusts the turbine control software individually for each wind farm. With small distances between towers, the software shuts down individual turbines at wind speeds above 20 m/s.

4. Conclusions

Two proposed wind farms were analyzed where the maximum number of turbines was located, taking into account the limitations described in Section 2. Evaluation criteria were proposed to quantify losses, which can serve as a benchmark for evaluating other wind farms.

It was found that due to wake losses, WF productivity was lower at Farm A by 12.9% and at Farm B by 10.2%. The maximum wake losses of individual turbines at Farm A and Farm B were 22.9% and 15.5%, respectively. The energy efficiency of the WFs was 28.8% (Farm A) and 27.2% (Farm B). When one unit (A17 and B23) was removed at each WF, the wake losses of Farm A and Farm B dropped slightly to 12.1% and 9.6%. In contrast, the maximum wake losses of individual turbines on Farm A and Farm B decreased to 20.6% and 15.0%, respectively. The WF energy efficiency increased slightly to 29.1% (Farm A) and 27.4% (Farm B).

The primary purpose of building WFs is to harvest green energy. If the scenario of reducing a single unit on each of the analyzed farms was realized, less renewable energy would be acquired in total. Over a period of 25 years, the difference in green energy generated at Farm A is 172,400 MWh/25 y, and at Farm B is 320,212.5 MWh/25 y.

The answer to the question: is there an economic justification for the removal of turbines with reduced productivity on a wind farm resulting from aerodynamic flux losses? is negative. It has been shown that the loss of energy efficiency is economically compensated for over the 25-year life of the WF.

The study shows that the payback period for the 3.45 MW V136 turbine (Farm A) is about 6.5 years, while the payback period for the 6.2 MW V162 turbine (Farm B) is shorter, at less than 5 years. It can therefore be projected that the higher the turbine’s power rating, the faster the payback resulting from the wake loss of a single turbine will occur. It should be remembered that the calculations are for specific WF locations and installed equipment. This means that the result is affected by local wind conditions and the type of turbines installed. In other locations, the payback period will be determined by these factors.

The results and conclusions of the study do not suggest that the wake losses of turbine rotors are insignificant. It was only pointed out that the construction of wind farms on which the wake losses are high has economic and environmental justification.

Due to the limited availability of land for WF siting, reducing the distance between turbines will continue. Further research should focus on advanced wind turbine control techniques and shaping the design of towers and rotor blades to minimize the turbulence caused by wake loss.