Utilizing WFSim to Investigate the Impact of Optimal Wind Farm Layout and Inter-Field Wake on Average Power

Abstract

This paper presents a comprehensive study on optimizing wind farm efficiency by controlling wake effects using the WFSim dynamic simulation model. Focusing on five key factors—yaw wind turbine position, yaw angle, wind farm spacing, longitudinal wind turbine spacing, and yaw rate—we qualitatively analyze their individual and combined impact on the wind farm’s wake behavior and mechanical load. Through a quantitative approach using the orthogonal test method, we assess each factor’s influence on the farm’s overall power output. The findings prioritize the following factors in terms of their effect on power output: yaw wind turbine position, yaw angle, wind farm spacing, longitudinal spacing, and yaw rate. Most significantly, this study identifies optimal working conditions for maximizing the wind farm’s average power output. These conditions include a wind turbine longitudinal spacing of 7.0D, a wind farm spacing of 15.0D, a yaw angle of 30°, and a yaw rate of 0.0122 rad/s, with the first and second rows of turbines in a yaw state. Under these optimized conditions, the wind farm’s average power output is enhanced to 35.19 MW, marking an increase of 2.86 MW compared to the farm’s original configuration. Additionally, this paper offers an analysis of wake deflection under these optimal conditions, providing valuable insights for the design and management of more efficient wind farms.

1. Introduction

Wind energy is a renewable and clean energy source, crucial for sustainable development [1,2]. Currently, wind energy contributes approximately 6% of global electricity generation, and this figure is expected to rise significantly in the coming decades. Increasing the development rate of novel energy resources, such as wind energy, can substantially reduce carbon emissions and help avert an energy crisis [3,4,5]. Projections indicate that by 2050, wind energy will account for between 25% and 33% of the world’s total electricity, making it one of the primary sources of power generation globally [6,7]. According to GWEC (Global Wind Energy Council), in 2022 alone, the global installed wind power capacity reached over 743 GW, with an annual growth rate of around 10%. These numbers highlight the rapid adoption and critical role of wind energy in the future energy landscape.

The wake effect of the wind turbine will reduce the incoming wind speed of the downstream wind turbine and increase the burden of the downstream wind turbine, resulting in a decrease in wind farm power generation and a decrease in the lifespan of the downstream wind turbine [8,9]. Consequently, calculating the wake effect is one of the essential aspects of analyzing wind farm power output [8,10]. Changes in wind farm configuration [11,12,13] and yaw control strategy [14,15,16] can effectively reduce the influence of wake to maximize the power output of wind farms [8]. Nygaard et al. [17] and Hansen et al. [18] investigated the disturbance’s impact between two adjacent Danish offshore wind farms. Nygaard conducted a study to examine the effect of an upstream wind farms on the wind speed reduction experienced by a downstream wind farm. The findings revealed that the wake loss caused by the upstream wind farm primarily affected the initial rows of wind turbines in the downstream wind farm, increasing the disparity in wind speeds between the two locations. Hansen studied the wind speed loss at various distances and determined that the most significant area of wind speed loss occurred 5–10D downstream of the wind farm entrance. Ma et al. [15] investigated the yaw configuration of wind turbines and derived the optimal yaw angle formula for the joint control of maximal wind farm power generation. As wind energy continues to develop, it may become more common to obtain wind resources through wind farm clusters [19,20].

Currently, the general wind farm control and prediction models consist of Simulator fOr Wind Farm Applications (SOWFA) [21], UTD (UT Dallas) Wind Farm (UTDWF) [22], and the PArallel LES (Large eddy simulation) Model (PALM) [23,24] as well. Typically, the calculation time for these three-dimensional, high-fidelity models can take several days [25,26]. Ainslie [27,28] was the first to propose the two-dimensional low-fidelity dynamic wind farm prediction model (only capturing specified disturbances and wind turbine characteristics located in the horizontal plane of the hub height in the wind farm). If we consider the two-dimensional low-fidelity model issue [29,30], such as by using 2D-CFD (Computational Fluid Dynamics) [31] to reduce calculation time, calculation accuracy will be compromised. In recent years, new wake prediction models such as the 2D_k Jensen model [32], the WFSim (WindFarmSimulator) [25] model, and the engineering wake calculation model introduced proposed by Guo et al. [33], have emerged. The WFSim model is a two-dimensional medium-fidelity model that balances the benefits and drawbacks of the low-fidelity model and the high-fidelity model. As a result, it has the advantages of faster computation speed and higher accuracy. It is based on the simplified LES model and simplified Navier–Stokes equations. Its calculation considers the axial induction actuation and yaw state of wind turbines, as well as the effects of turbine-induced turbulence and spatially and temporally changing inflow profiles [34,35]. The WFSim model has been widely utilized in wind farm research because its features make it simpler to evaluate the wake effect of wind farms. Studying the wake effect of wind farms is made simpler because of the features of the WFSim model. The WFSim model has been extensively used in research on wind farms. Zhao et al. [36] used the model to study the maximization of wind farm power generation through cooperative control under the wake effect. Chen et al. [37] used the model to validate their proposed new control-oriented WF (Wind Farm) reduced-order dynamic model and a deep learning-assisted MPC (Model Predictive Control) algorithm. Doekemeijer et al. [38] created a state estimator called WindFarmWatch (WFObs) based on WFS (Wind Farms). These findings demonstrate that the WFSim model has established itself as one of the most trustworthy models for wind farm simulation studies.

With proper optimization of operational settings, reduced turbulence intensity [39] can increasing wind turbine power. Howland [40] studied wind farm yaw control set-point optimization with model parameter uncertainty. Moreover, with the development of artificial intelligence, more and more AI (Artificial Intelligence) technology will be implemented in the energy field and wind farms [41,42,43]. For instance, neural networks have been used to learn the dynamic WF reduced-order model (ROM) to capture the central dynamics [37], and machine learning has been used to identify the mean wake velocity and turbulence intensity of onshore wind turbines [44]. These technologies can significantly help control wind farms, which can perform better in many tasks [45].

This paper investigates the average power output of wind farms under different working conditions based on the WFSim model to study the influence of other factors on wind turbine wake. After that, we use the orthogonal test method to find the optimal working condition combination that maximizes the average power output of wind farms. In Section 2, we validate the WFSim model and describe the model’s parameter settings and experimental parameters. Reflective results of the experiments will be presented in Section 3. In Section 4, we will analyze the wake and wind speed of the wind farm under optimal operating conditions. In Section 5, we will present our conclusions and discuss our findings.

2. Establishment and Verification of Wake Model of Wind Farms

2.1. Mathematical Model Analysis Method

The purpose of this paper is to use the WFSim simulation model to examine the impact of individual factor variations on the average power output of a wind farm, as well as the combined influence of multiple factors on power generation. In addition, it will investigate how the disturbance produced by an upstream wind farm affects the average power output of a downstream wind farm. Since WFSim is a mesoscale two-dimensional simulation model, this paper does not explore scenarios in the vertical direction. Instead, it focuses on investigating wind speed and flow field changes in the cross-section at the height of the wind turbine hub. A generalized power calculation formula is used in WFSim, i.e., Equation (1). In this equation, ρ represents the atmospheric density, A represents the rotor area, C_p represents the power coefficient, and u represents the incoming wind speed of the wind turbine.

$$\mathrm{P}={\displaystyle \frac{1}{2}}\rho A{C}_{\mathrm{p}}{u}^3$$

(1)

Subsequently, u is optimized to U based on the actual arrival of the wind turbine after yawing, as detailed in Equation (2), where U denotes the optimized wind turbine incoming wind speed and γ denotes the wind turbine yaw angle.

$$U=[u\cos \left(\gamma \right){]}^3$$

(2)

Finally, accumulation is introduced to sum up the power of all wind turbines and calculate the whole power output of the wind power plant. The formula for calculating the final power output of the wind farm in question is detailed in Equation (3). In this equation, $\aleph$ denotes the total number of wind turbines, and i denotes the serial number of the wind turbine [25].

$$\mathrm{P}=\sum _{\mathrm{i}=1}^{\aleph }{\displaystyle \frac{1}{2}}\rho A{C}_{{\mathrm{P}}_{\mathrm{i}}}[U_{\mathrm{i}}\cos \left({\gamma }_{\mathrm{i}}\right){]}^3$$

(3)

The natural variation range of ρ is minimal for existing technologies, and the model of the wind turbine determines A. Consequently, this simulation analysis concentrates predominantly on variables that affect the incoming wind speed of wind turbines. Adjusting the wind farm layout and modifying the yaw settings of the wind turbines are the primary strategies discussed in this paper for changing the incoming wind speed. This paper seeks to investigate the influence of wind turbine longitudinal spacing, wind farm spacing, yaw angle, yaw rate, and yaw wind turbine position on the average power output of a wind farm, taking into account both individual and combined changes. The objective is to identify the optimal combination that maximizes the wind farm’s power output. For this experiment, a 3 × 3 configuration comprising two wind farms, each with nine NREL5MW wind turbine models, was chosen. The total simulation time is 998 s and the 2D dimensions of the computational domain are 6600 m and 3095 m.

Table 1. WFSim simulation parameter configuration.

Parameter	Value	Parameter	Value
Area size (km2)	6.600 × 3.095	Rotor diameter of wind turbine, D (m)	126.4
Number of meshes	100 × 50	Wind turbine layout	3 × 3
Unit size (m2)	66 × 61.9	Atmospheric density, ρ (kg/m3)	1.2
Longitudinal incoming wind speed, ub (m/s)	12	Turbulence model	d = 1000 d′ = 140 m ls = 0.05

lists the specific parameters, including hub height, rotor area, and atmospheric density. To guarantee precise data measurement and reduce experimental errors, the yaw procedure utilized in this experiment was static yaw [46], thereby averting variances in total yaw time between various combinations. It depicted the layout of wind turbines in Figure 1. In Figure 1, M1 and M2 denote wind farm numbers, L1 denotes the line, i.e., the first column, and T1–T18 denote wind turbine numbers.

2.2. Model Verification

2.2.1. Power Prediction Verification of the WFSim Model

In this section, we compare the wind speed–power curve data of the NREL 5 MW wind turbine [47] with the power data derived via WFSim simulations to validate the veracity of the experimental data obtained through WFSim. We define the error calculation method as depicted in Equation (4) to quantify the difference between simulation and experimental results.

$$\delta =\left|\right(P_{\mathrm{e}}-P_{\mathrm{s}})/P_{\mathrm{s}}|\times 100\%$$

(4)

In this equation, $\delta$ represents the error, P_e represents the power data obtained from the experiment, and P_s represents the power data obtained via simulation. The power fitting is shown in Figure 2. The analysis of Figure 2 shows that the two data have a high degree of fit.

When the incoming wind speed ranges from 3 to 7 m/s, the data obtained through WFSim shows a slightly higher value than the experimental data, with noticeable discrepancies. Conversely, when the incoming wind speed falls within the 8–12 m/s range, the WFSim data are slightly lower than the experimental data, exhibiting an average error of approximately 2.86%. This wind speed range proves to be the most suitable for simulation analysis using WFSim. Notably, at an incoming wind speed of 12 m/s, the error decreases to 1.52%, representing a minor deviation between the WFSim results and the experimental data.

In summary, the simulation of wind turbine power using WFSim has achieved satisfactory results overall. It demonstrates the capability of WFSim to provide approximate simulations of the wake generated by wind turbines. Moreover, since the wind speed of 12 m/s corresponds to the rated wind speed of the selected wind turbine and exhibits minimal error when compared to the experimental data, it will be utilized for simulation purposes in future research.

2.2.2. Wind Speed Deficit Verification of the WFSim Model

This section employs the WFSim model to validate wind speed reduction resulting from wake turbulence. The wind farm example consists of two wind turbines. The model parameters associated with this example are presented in

Table 2. WFSim simulation parameter configuration in wind speed deficit verification.

Parameter	Value	Parameter	Value
Area size (km2)	1.686 × 0.500	Rotor diameter of wind turbine, D (m)	126.4
Number of meshes	40 × 20	Wind turbine layout	2 × 1
Unit size (m2)	42.15 × 25	Atmospheric density, ρ (kg/m3)	1.2
Longitudinal incoming wind speed, ub (m/s)	12	Turbulence model	d = 1000 d′ = 140 m ls = 0.05

, while the corresponding wind speed loss is depicted in Figure 3. As can be seen in Figure 3, the wind speed at the hub height has a wind speed deficit when the wind passes through the wind turbines at different locations. When the incoming wind passes through the first wind turbine, its wind speed decreases from 12 m/s to about 8 m/s, and then starts to recover slowly at the downstream 3D position of the wind turbine, and when it passes through the downstream wind turbine position, its incoming wind speed decreases to about 6 m/s, and then slowly recovers again at the downstream 3D position of the wind turbine.

2.2.3. Wake Verification of the WFSim Model

This section compares the flow field trails obtained from the WFSim model and the SOWFA dataset supplied by van Wingerden et al. [25]. The results of this comparison are depicted in Figure 4. Based on the analysis of Figure 4, it is evident that the WFSim model, characterized by its medium fidelity, disregards the typical fluctuations in the non-tail area during computing. Consequently, this results in the comparatively diminished accuracy of SOWFA-derived contrails. However, both types of contrails exhibit similar levels of straightness. Consequently, the utilization of WFSim as a means to investigate contrails demonstrates a notable level of precision.

2.2.4. Wake Deflection Verification of the WFSim Model

Many low-fidelity models do not have good wake deflection accuracy, resulting in their inability to guarantee good computational results even after the wind turbine has yawed. This section aims to evaluate the accuracy of WFSim’s wake deflection by performing yaw control using a single wind turbine model. The model parameters utilized in this section are presented in

Table 3. WFSim simulation parameter configuration in wake deflection verification.

Parameter	Value	Parameter	Value
Area size (km2)	1.686 × 0.800	Rotor diameter of wind turbine, D (m)	126.4
Number of meshes	40 × 20	Yaw angle (°)	35
Unit size (m2)	42.150 × 40	Atmospheric density, ρ (kg/m3)	1.2
Longitudinal incoming wind speed, ub (m/s)	12	Turbulence model	d = 1000 d′ = 140 m ls = 0.05

, while the wake deflections acquired from the simulation are depicted in Figure 5.

In Figure 5, the black line at the center of the wake represents the theoretical wake center, and the outer black line represents the theoretical wake edge [48,49]. Figure 5 shows that the simulation results approximate the theoretical centerline in a steady state. This indicates that WFSim can be used to simulate wake–yaw control.

3. Results and Discussion

3.1. Influence of Single-Factor Change on Average Power Output of Wind Farm

3.1.1. Parameter Settings When Studying the Influence of Single Factors

In this section, we aim to provide a detailed analysis of the impact of single-factor changes on the average power output of wind farms. To facilitate this study, we will explain the parameters involved in the research process and outline the specific parameters under different operating conditions in

Table 4. Single-factor independent variable change and its parameter.

Change the Arrangement of Wind Farms				Change the Yaw Setting
Longitudinal Spacing xxt		Wind Farm Spacing xwf		Yaw Angle γ, (°)		Yaw Rate ω, (rad/s)		Yaw Wind Turbine n
Working conditions	Value	Working conditions	Value	Working conditions	Value	Working conditions	Value	Working conditions	Value
Case 1-1	4.5D	Case 2-1	9D	Case 3-1	5	Case 4-1	0.007	Case 5-1	R1 + R2
Case 1-2	5.0D	Case 2-2	10D	Case 3-2	10	Case 4-2	0.009	Case 5-2	R1 + R4
Case 1-3	5.5D	Case 2-3	11D	Case 3-3	15	Case 4-3	0.011	Case 5-3	R1 + R5
Case 1-4	6.0D	Case 2-4	12D	Case 3-4	20	Case 4-4	0.012	Case 5-4	R2 + R4
Case 1-5	6.5D	Case 2-5	13D	Case 3-5	25	Case 4-5	0.014	Case 5-5	R2 + R5
Case 1-6	7.0D	Case 2-6	14D	Case 3-6	30	Case 4-6	0.016	Case 5-6	R3 + R4
Case 1-7	7.5D	Case 2-7	15D	Case 3-7	35	Case 4-7	0.018	Case 5-7	R3 + R5

. There are five key influencing factors: wind turbine longitudinal spacing, x_xt; wind farm spacing, x_wf; yaw angle, γ; yaw rate, ω; and yaw wind turbine position, n. The baseline parameters are set as follows: x_xt = 6.0D, x_wf = 12.0D, γ = 20 °, ω = 0.0122 rad/s, and n = R1 + R4.

When studying the impact of a single variable on the wind farm’s power output, we only alter the value of the specific variable while keeping the remaining variables at their default baseline parameters. For a more accurate examination of the influence of x_xt and x_wf changes on the average power output of wind farms, the orientation of wind turbines will remain fixed. However, for the other three factors, as the impact of non-yaw states on the average power output of wind farms is minimal, we will focus on studying them under yaw conditions.

By conducting this analysis, we aim to gain precise insights into how individual factors impact the average power output of wind farms, contributing to a comprehensive understanding of wind farm optimization.

3.1.2. Influence of Wind Farm Layout on Average Power Output of Wind Farm

This section primarily focuses on investigating the influence of wind farm layout changes on the average power output of wind farms, explicitly examining two key factors: x_xt and x_wf.

Figure 6 illustrates the simulation results for varying the longitudinal spacing of wind turbines. As shown in Figure 6a, the average power output of the wind farms shows an upward trend as the spacing increases. In Cases 1-1 through 1-3, the increase in average power output is marginal. However, in Case 1-4, the average power output rises significantly, surpassing that in Case 1-3 by 0.65%. Case 1-6 achieves the maximum average power output at 31.43 MW, representing a 2.02% increase over that in Case 1-1.

Figure 6b reveals the impact of different spacings on wake patterns. Increasing the longitudinal spacing between wind turbines enhances the wind speed at the end of the wake, elevating the incoming wind speed for downstream turbines and improving the average power output of the entire wind farm. However, when the spacing exceeds a certain critical value (e.g., 7D), the increased turbulence intensity in the wake region can reduce downstream wind speed and affect the turbines’ power output and operational stability. While turbulent mixing can partially restore wind speed by blending low-speed wake with surrounding high-speed airflow, excessive turbulence can offset these gains.

Figure 7 depicts the simulation results for varying the distance between wind fields. As depicted in Figure 7a, the average power output gradually increases as the distance between wind farms increases. In Case 2-5, the rate of increase in power output decreases, followed by a continuous significant increase until Case 2-7. In Case 2-7, the average power output of the wind farms reaches an optimum level of 31.92 MW, a 5.46% improvement over Case 2-1.

In addition, as depicted in Figure 7b, the effect of increasing the distance between wind farms and increasing the longitudinal spacing between wind turbines on the average power output of wind farms is comparable. Increasing the distance between wind farms increases the wind speed at the M1 wind power facilities’ wake end. As a result, they enhance the incoming wind speed at the M2 wind farm, leading to an increase in the average electricity output of the wind farms. However, changing the distance between wind farms has a more stable effect on average power than changing the distance between wind turbines.

3.1.3. Influence of Wind Farm Yaw Setting on Average Power Output

This section investigates the impact of wind farm yaw setting changes on the average power output of the wind farm. It considers three factors: yaw angle, γ; yaw rate, ω; and yaw wind turbine position, n.

Figure 8 displays the simulation results for varying the yaw angle of the wind turbine. As shown in Figure 8a, the average power output consistently increases as the yaw angle increases. In Case 3-1, the average power output of the wind farm is 31.07 MW. In Case 3-6, the average power output increases to 33.26 MW, a 7.06% increase compared to that in Case 3-1. Different yaw angles result in varying degrees of wake deflection, as shown in Figure 8b. Increasing the yaw angle separates the wake generated by the upstream wind turbine from the wake generated by the downstream wind turbine, mitigating the wake effect on the downstream turbine. It increases the incoming wind speed for the turbine downstream, thereby increasing the average power output. Notably, larger yaw angles incur power losses in the yaw wind turbine itself and can harm the duration of the turbine. The overall increase in power output can be attributed to the improvement in the downstream turbine’s power, which compensates for the yaw wind turbine’s power loss. The data in Figure 8a indicate that as the yaw angle increases to 30°, the average power rises until it reaches its maximum. Increasing the yaw angle beyond 35° causes power loss in the yaw wind turbine, and the increased power output of the downstream turbine cannot entirely compensate for the loss. In conclusion, increasing the yaw angle increases the average power output of the wind farm. However, a yaw angle of 30° approaches the limit for maximizing power output gains through yaw angle adjustments.

Figure 9 displays the simulation results for varying the yaw rate of the wind turbine. As depicted in Figure 9a, the average power output decreases gradually as the yaw rate increases. However, the alterations to the data are relatively minor. Case 4-7’s average power output is only 0.49% lower than Case 4-1’s, with a 0.16 MW variance. This test entails a static yaw, in which the wind turbine remains deflected until the conclusion of the simulation after a single yaw. The deflection process typically lasts only tens of seconds, resulting in a negligible effect on the average power. Figure 9b demonstrates that the wake patterns at different yaw rates are similar. However, the yaw rate can substantially affect the average power output in natural dynamic yaw processes.

Figure 10 displays the simulation results for altering the position of the transverse wind turbine. As the yaw wind turbine is progressively moved downstream from Case 5-1 to Case 5-3, the average power output of the wind farm gradually decreases, as shown in Figure 10a in Case 5-4 to Case 5-7. However, the average power output decreases significantly relative to that in Case 5-1 to Case 5-3 and varies within a specific range. Notably, in Case 5-7, the average power output of the wind farm decreases by 7.97%, from Case 5-1 to 31.38 MW. These results indicate that as the yaw wind turbine is progressively moved downstream, the yaw effect diminishes, gradually decreasing average power output. The yaw disturbance generated by the upstream wind turbine influences all downstream wind turbines, increasing the average output of the wind farm.

As shown in Figure 10b, as the yaw wind turbine is transferred downstream, the number of wind turbines affected by the yaw disturbance decreases, resulting in less of an impact on the wind turbines. As a result, the increase in average power output becomes less significant. In addition, due to power losses in the yaw wind turbine, the yaw of the downstream wind turbine may not wholly compensate for the loss in the yaw wind turbine, resulting in a decrease in the field’s average power output. In conclusion, the yaw of the upstream wind turbine enhances the average power output of the wind farm more effectively.

3.2. Influence of Multi-Factor Change on the Average Power Output of Wind Farm

In the preceding section, we analyzed the individual impact of five variables on average power output. In practice, however, the influence of multiple factors can significantly impact the average output of wind farms. Consequently, the purpose of this section is to investigate the optimal combination of operational conditions that can maximize the average power output of wind farms, taking into account the interaction of various factors. Due to the large number of involved factors and working conditions, we employ the orthogonal test method to determine their impact on the average power output of wind farms. The orthogonal test method is a design technique utilized in multi-factor and multi-level research. It entails selecting representative parameters from various factors to perform combined experiments. This method enables us to determine the optimal parameter settings with the shortest simulation time and the lowest operational cost [50,51].

3.2.1. Results of Orthogonal Test

In this section, the average power of the wind farm will be measured and analyzed using the orthogonal test method. Six influencing factors, namely x_xt, x_wf, γ, ω, n, and the additional empty columns for variance analysis, will be considered. These factors will be denoted by letters A to F, while different levels will be represented by numbers 1 to 7. The specific parameter settings for the orthogonal test are provided in

Table 5. Parameter setting of orthogonal test.

No.	A (xxt)	B (xwf)	C (γ) (°)	D (ω) (rad/s)	E (n)
1	4.5D	9.0D	5	0.00698	R1 R2
2	5.0D	10.0D	10	0.00873	R1 R4
3	5.5D	11.0D	15	0.0105	R1 R5
4	6.0D	12.0D	20	0.0122	R2 R4
5	6.5D	13.0D	25	0.014	R2 R5
6	7.0D	14.0D	30	0.0157	R3 R4
7	7.5D	15.0D	35	0.0175	R3 R5

.

We refer to the orthogonal table with eight factors and seven levels for this test, resulting in 49 unique experimental conditions. This number represents the minimum requirement for conducting a comprehensive and systematic analysis. The average power of the wind farm obtained from these tests is presented in

Table 6. Results of orthogonal test.

Test No.	A	B	C	D	E	F	Average Power (MW)
1	1	1	1	1	1	1	30.100
2	1	2	3	4	5	6	30.693
3	1	3	5	7	2	4	32.942
4	1	4	7	3	6	2	31.670
5	1	5	2	6	3	7	31.438
6	1	6	4	2	7	5	31.691
7	1	7	6	5	4	3	32.926
8	2	1	7	6	5	4	30.533
9	2	2	2	2	2	2	30.639
10	2	3	4	5	6	7	31.095
11	2	4	6	1	3	5	32.988
12	2	5	1	4	7	3	31.168
13	2	6	3	7	4	1	31.886
14	2	7	5	3	1	6	35.140
15	3	1	6	4	2	7	32.659
16	3	2	1	7	6	5	30.287
17	3	3	3	3	3	3	31.499
18	3	4	5	6	7	1	31.377
19	3	5	7	2	4	6	32.416
20	3	6	2	5	1	4	32.211
21	3	7	4	1	5	2	31.978
22	4	1	5	2	6	3	30.667
23	4	2	7	5	3	1	32.821
24	4	3	2	1	7	6	30.823
25	4	4	4	4	4	4	31.837
26	4	5	6	7	1	2	34.703
27	4	6	1	3	5	7	31.443
28	4	7	3	6	2	5	30.188
29	5	1	4	7	3	6	31.755
30	5	2	6	3	7	4	31.209
31	5	3	1	6	4	2	30.981
32	5	4	3	2	1	7	33.104
33	5	5	5	5	5	5	31.983
34	5	6	7	1	2	3	33.760
35	5	7	2	4	6	1	32.281
36	6	1	3	5	7	2	30.909
37	6	2	5	1	4	7	32.123
38	6	3	7	4	1	5	34.225
39	6	4	2	7	5	3	31.434
40	6	5	4	3	2	1	33.114
41	6	6	6	6	6	6	32.988
42	6	7	1	2	3	4	32.275
43	7	1	2	3	4	5	30.957
44	7	2	4	6	1	3	33.625
45	7	3	6	2	5	1	31.909
46	7	4	1	5	2	6	31.329
47	7	5	3	1	6	4	32.119
48	7	6	5	4	3	2	33.145
49	7	7	7	7	7	7	32.438

. The research team then performs a range analysis based on the data in Table 6 to determine the impact of multiple factors on the average power. The difference between the optimal parameter level and the lowest power level is denoted by R. Additionally, K₁ to K₇ represent the sum of the average power for levels 1 to 7. In contrast, k₁ to k₇ represent each level’s average power. The results of the range analysis can be found in

Table 7. Range analysis of orthogonal test.

Parameters	A	B	C	D	E	F
K1	221.460	217.580	217.583	223.890	233.108	223.487
K2	223.449	221.397	219.782	222.701	224.631	224.025
K3	222.427	223.473	220.398	225.032	225.921	225.079
K4	222.482	223.739	225.095	226.007	223.126	223.126
K5	225.073	226.941	227.376	223.273	219.973	222.319
K6	227.068	227.123	229.382	221.130	221.106	225.142
K7	225.520	227.226	227.862	225.446	219.615	224.301
k1	31.637	31.083	31.083	31.984	33.301	31.927
k2	31.921	31.628	31.397	31.814	32.090	32.004
k3	31.775	31.925	31.485	32.147	32.274	32.154
k4	31.783	31.963	32.156	32.287	31.875	31.875
k5	32.153	32.420	32.482	31.896	31.425	31.760
k6	32.438	32.446	32.769	31.590	31.587	32.163
k7	32.217	32.461	32.552	32.207	31.374	32.043
R	0.801	1.378	1.685	0.697	1.928	0.403
Order	E > C > B > A > D > F
Optimal combination	A6B7C6D4E1

The green font indicates the data with the smallest k value in each group, the red font indicates the data with the largest k value in each group.

.

3.2.2. Analysis of Range and Variance

According to the orthogonal test theory, the value of R can indicate the influence of a specific factor on the experimental results. Therefore, Table 7 reveals the order of influence on the average power of the wind farm as E, C, B, A, D, and F (empty column). The optimal combination yielding the highest average power is A₆B₇C₆D₄E₁, which aligns closely with the results obtained from the single-factor analysis. Additionally, although the R-value of the G column (error column) is small but close to the D column, it suggests that besides the six factors considered in the test, the average power of the wind farm is also influenced by some other important factors, albeit to a lesser extent. After simulating the theoretically optimal combination of operating conditions, it is determined that the average power of the wind farm reaches 35.19 MW, surpassing the 35.14 MW obtained in the 14th group of orthogonal tests. This demonstrates that the maximum average power output of the wind farm can be achieved when the operating conditions are set to A₆B₇C₆D₄E_1.

Analyzing Figure 11, it can be observed that as x_xt, x_wf, and γ increase, and as the yaw wind turbine position, n, gradually approaches the upstream, the average power output also increases. Considering the comprehensive comparison of these four influencing factors, it is evident that the influence of x_xt on the average power output of the wind turbine is relatively weak. On the other hand, adjusting x_wf, γ, and n can lead to more significant changes in the average power output of the wind turbine, thus facilitating the optimization of wind turbine capacity. It is worth noting that the change in ω does not correlate significantly with the wind turbine’s average power output.

A variance analysis was conducted to further evaluate the influence of various factors on the average power output. The analysis results are presented in

Table 8. Analysis of variance via orthogonal test.

Source	Sum of Squares	Degree of Freedom	Mean Square	F	p	Significance
A	3.482	6	0.580	3.823	0.063695
B	11.015	6	1.836	12.097	0.003957	++
C	18.339	6	3.057	20.141	0.000985	++
D	2.515	6	0.419	2.762	0.120869
E	18.798	6	3.133	20.644	0.000919	++
e	6.911	6	0.152

++ indicates that the parameter has a significant effect on the data.

. The p-values for columns B, C, and E in Table 8 are all less than 0.05, indicating statistical significance. The experiment reveals that selecting optimal operating conditions increases the average power output of the entire wind farm. For example, increasing the yaw angle of the wind turbine or positioning the yaw wind turbine closer to the upstream can result in favorable outcomes. On the other hand, the impact of other influencing factors on the average power output of wind farms is negligible. Choosing parameter settings close to the optimal level is prudent when considering objective conditions such as cost and location.

4. Wake Analysis and Influence of Yaw Operation on Inter-Field Wake and Downstream Wind Farm Wake under Optimal Conditions

4.1. The Influence of the Yaw of the Upstream Wind Farm on the Downstream Wind Farm

This section examines the inflow velocity of the optimal working condition combination determined by the orthogonal test. Figure 12 depicts the comparison between the yaw wake and the non-yaw wake under optimal working conditions, as well as the wind speed of the wind turbine. Inferred from a comparison of Figure 12a,c, the yaw of the upstream wind turbine effectively deflects the disturbance away from the front wind turbine of the downstream wind farm. Under non-yaw conditions, the average power output is 32.33 MW, which is 8.14% less than the average power output under yaw conditions, which is 35.19 MW. Figure 12b reveals that the yaw of the upstream wind turbine in the M1 wind farm increases the average wind speed by 2.40 m/s in the M1 wind farm, 0.97 m/s in the M2 wind farm, and 1.69 m/s for the entire wind farm. However, due to the yawed state of the wind turbine, it cannot wholly capture the inflow speed, resulting in an increase of 1.34 m/s in the actual inflow speed across the entire field.

4.2. The Influence of the Yaw Operation on Inter-Field Wake

Wake velocity is an essential variable in wind turbine and wind farm research. The wind speed data of the wind farm are processed, generating a U–velocity curve. This trajectory is then superimposed on the WFsim-simulated wind farm cloud image, as depicted in Figure 12a,b. Behind the three rows of wind turbines, the red dashed line represents the average wind speed ratio at the lowest wind speed. Along the longitudinal axis, wind speed data are collected at distances of 2541 m, 3531 m, 4521 m, 5511 m, and 6039 m.

After passing through the wind plane of the third row of wind turbines, the inflow experiences a decrease in wind speed, followed by a progressive increase over distance. Before yaw, the ratio of wind speed along the hub center (wake center) at 2541 m and 3531 m increases from 0.21 to 0.34, respectively. The percentage of wind speed at 2541 m and 3531 m rises from 0.42 to 0.49 due to yaw. Due to the yaw of the wind turbine, the discharge deflects, resulting in a considerable increase in the wind speed ratio between 0.55 and 0.87 at 2541 m and 3531 m along the hub center. Yaw can increase the wind speed between wind turbines.

According to the law of conservation of energy, as described by (1), the larger the yaw angle (0 < γ < 90°), the less energy a wind turbine obtains from the wind, resulting in the wake retaining more power after passing through the wind turbine. Consequently, the wind velocity increases. In addition, the yaw operation causes the wake to deviate from its original position, thereby averting or minimizing the wake’s effect on downstream wind turbines. As a result, the downstream wind turbines experience a quicker wake speed, resulting in less speed loss behind the downstream turbines than in situations without yaw.

It is evident from Figure 12b that the downstream wind speed decreases anteriorly to the yaw due to the disturbance generated by the wind farm above. However, through effective yaw operation of the upstream wind farm’s wind turbines, the wake is deflected from its original position, thereby averting or minimizing its impact on the downstream wind turbines. As depicted in Figure 12c, following the yaw operation, the wake conditions for the fourth row of wind turbines considerably improve, with the average wind speed ratio increasing from 0.19 to 0.40.

As longitudinal distance increases, the influence of precession on wake enhancement diminishes until it has minimal effect on the sixth row. We observe values of 0.26 and 0.21 for the wind speed ratios of the fifth row with and without yaw, respectively. Similarly, the final row’s wind speed ratios with and without yaw are 0.205 and 0.202, respectively. Compared to the first few rows, the disparity between the wind speed ratios before and after yaw in the fifth row is much smaller, and the wind speed ratio of the sixth row is very close. On this basis, we can infer that the disturbance effects from upstream wind farms are concentrated predominantly in the first few rows of wind turbines in downstream wind farms.

5. Conclusions

Several factors influence a wind turbine’s power output. Using the WFSim simulation model, this paper investigates the effect of wind turbine longitudinal spacing, wind farm spacing, yaw angle, yaw rate, and yaw wind turbine position on the average power output of a wind farm. A six-factor, seven-level orthogonal table is designed using an orthogonal test method to determine the optimal combination of working conditions for the wind farm and to analyze the disturbance and wind speed under these optimal conditions. The following are the main findings:

(1)

In descending order of importance, the yaw wind turbine position (n), yaw angle (γ), wind farm spacing (x_wf), wind turbine longitudinal spacing (x_xt), and yaw rate (ω) influence the average power output of the wind farm.

(2)

The optimal working conditions for maximizing the average power output of the wind farm are achieved when the wind turbine longitudinal spacing (x_xt) is 7.0D, the wind farm spacing (x_wf) is 15.0D, the yaw angle (γ) is 30°, and the yaw rate (ω) is 0.0122 rad/s. The first and second rows of wind turbines are in a yaw state. Under these conditions, the average power output of the wind farm is 35.19 MW, an increase of 2.86 MW compared to the original configuration of the wind farm, and the incoming wind velocity at the wind farm rises by 1.341 m/s.

(3)

The average power output of the wind farm under the optimal conditions derived from the orthogonal test is 35.19 MW, which is 5.43 MW higher than the 29.76 MW under the worst conditions, indicating that the selection of appropriate yaw and layout settings can significantly increase the average power of the wind farm.

(4)

The effects of the upstream wind farm’s trail predominantly influence the first few rows of wind turbines in the downstream wind farm.

In addition, the test results indicate that, in addition to the five factors considered in the orthogonal test, additional significant factors influence the average power output of the wind farm. We will continue to investigate these additional factors and their impact on the average power output of the wind farm in future work.