The impact of wind direction on wind farm power output calculation considering the wake effects of wind turbines

Abstract

The wind farm is a collection of all wind turbine generators situated at a particular distance. The wind speed and wind direction play an important role in wind farm power calculation. The power curve of the wind farm is not simply the summation of the wind turbine’s power curve. It is complex due to the intermittent nature of wind speed and its direction. The power curve is obtained from the data taking the wind speed and wind power at that speed. There are many logistic functions used in the literature to analyze the wind power curve that helps to calculate wind farm power output and energy generated. In this paper, the 3-parameter deterministic process (3P-DP) method is used for wind power curve calculation. The wake effect is analyzed by Jensen’s model with wind speed and wind direction. The wind farm power is obtained from the new proposed formula and compared with the already existing one. The results are verified from real data obtained from the literature.

Keywords

Wake effect windfarm wind turbine power curve parameter logistic function Jensen model

Introduction

To study the electric power supplied by the wind turbine (WT) or a wind farm (WF) based on wind speed the power curve is used. Manufacturers obtain this curve by taking readings at every 0.5 m/s of wind speed and corresponding power generated (Iec, 2019). The graph provided by manufacturers is helpful to find output power at a given speed and hub height (Vestas, n.d.). However, when dealing with big data, the graph is not sufficient for analysis of the power curve so the mathematical expression is required. The mathematical model uses pairs of point readings to frame piecewise continuous functions. Due to the specific shape of the power curve, there are several models are proposed to analyze it. In the piecewise model linear (Khalfallah and Koliub, 2007), quadratic (Albadi and El-Saadany, 2009), cubic (Jangamshetti and Rau, 1999), least-square (Thapar et al., 2011), and spline model (Villanueva et al., 2012) are mostly used. However, they show a lack of continuity of slope and error near rated wind speed.

The single continuous function model resolves these issues as it is continuous, derivable, and more accurate at rated wind speed (Feijoo and Cidras, 2000; Lydia et al., 2013). Among the single continuous function model, 4-parameters and 5-parameters logistic functions are most widely used. These models required the use of parameters so some procedure is described to calculate them. Lydia et al. (2013) proposed the 4-parameters logistic function model using an optimization process for Vestas V80 WT. The suggested method did not give always a good result as it depends on the objective function.

Villanueva and Feijóo (2016) describe the alternative method to obtain parameters of the 4-parameter logistic function. Three continuous models, that is, 4P-DP, 4P-DS, 3P-DP models for power curve improvement are discussed. In which, 3-parameters logistic function using a deterministic process (3-P DP) model is the most suitable as it involves a lesser number of parameters that can be easily obtained from the WT manufacturer. Either the model is piecewise or continuous, it is defined for all possible values of wind speed. The interval of the output power curve is defined between the cut in (2–5 m/s) and cut out speeds (20–30 m/s) of wind.

The mathematical representation of the WT power curve includes many functions. In the case of the WF power curve, it is not simply the summation of all the individual WT power curves. The representation of the power curve in the WF is more complex due to wind speed variation experience by different WTs. A review of the deterministic WT power curve is discussed in Villanueva and Feijóo (2020). The factors involved in the WT power curve are based on models, the number of parameters, ease of use, and accuracy. It is suggested that the methods are selected based on the objective taken, that is, minimizing the number of parameters, ease of use, relationship with WT parameters, and priority of the accuracy. It is recommended that the 5-parameter logistic functions (PLE) method should be used for large parameters and the 3PLE method is used when the minimum number of parameters are needed for consideration.

In Villanueva and Feijóo (2018), a comparison of different PLE for the WT power curve is presented. The results discuss the 5 PLE give better results than 6 PLE as consideration of 6 parameters of the turbine is difficult. 3 PLE is better than 4 PLE in terms of results and parameter consideration. If X is the number of parameters between three and six, XPL and Gompertz (GPTZ) function should not be used for power curve determination. Fitting of WT power curve using Weibull cumulative distribution function (CDF) expression by deducing shape and scale parameters of WT generators have been discussed in Bokde et al. (2018). The results are compared with the WT power curve obtains from many logistic functions. The proposed algorithm achieves accurate results when three or four parameters are used in the curve fitting of the WT power curve.

It is important to consider minimum area (due to the economy of land) while the WF construction planning. However, minimizing the area leads to a decrease in distance between WTs. As the distance decreases, the shadow effect of WT comes into the picture. Upstream WT extracts energy and reduces the wind speed for downstream WT causing a turbulent flow. Due to this, the second WT would not run optimally and produce less energy. This mutual shadowing phenomenon between the neighboring WT is called the wake effect or array effect. Moskalenko et al. (2010) reviews some models which include Lissaman’s model, Larsen’s model, Jensen’s model, and Ainslie’s model. It also describes the importance of the shadowing effect of WT considering the wind direction and wind speed during WF planning.

Among the available wake model, Jensen’s wake model is simpler and require less computational time. In Jensen’s wake model, it is assumed that the wake flow is linear, the wind profile for total wake is rectangular and behind the rotor, the near-wake effect is negligible. In this model, the entrainment constant “k,” controlled the wake flow (Jensen, 1983). Modeling and simulation of the WF under wake effect is discussed in Hwang et al. (2015). The effect of wind speed time delay on the power output of the WT generator is also discussed. The arrival time delay is proportional to the WT distance. The wake is modeled in the real-time digital simulator for analysis and the effect of wake effect on WF power output is shown. The wind direction and operation condition of a WT generator is ignored during all simulation process.

A Monte Carlo simulation-based adequacy analysis of WF considering the wind speed, wake effect, location, and reliability of WT generator is discussed in Han et al. (2011). The optimal location of the WT is selected based on the interpolation method. The power output of the WF is more when WT’s are optimally located. It can be concluded that the wake effect has a significant impact on the power output of WT’s in WF when it is not optimally located. An Artificial Neural Network (ANN) based approached for WT power output prediction is proposed in Bilal et al. (2018). The spatial climate data are collected from four sites on the northwest coast of Senegal to identify the parameters of ANN. It is observed that wind output power depends on the site location and climate data. The power output is close to real observed data when wind direction is considered with wind speed data.

González-Longatt et al. (2012) discussed the static and dynamic behavior of a WT. Upstream WT has a shadowing effect on downstream WT due to changes in the wind direction. Multiple wake effects, wind direction, and wind speed delays on downstream WT are also discussed. The efficiency of WF depends on the geometry of WT based on wind direction considering the wake effects. Mostly, WF is planned in such a way that when there is a need to increase energy output then more WT can be added to it. For such planning, the wake effect is an important parameter to consider. From an economic point of view, neglecting the wake effect can lead to overestimation of power output. WF power curve using probabilistic analytical method under the wake effect is proposed in Shi et al. (2014).

Three-dimensional (3D) gaussian wake model for the WF layout optimization problem is discussed in Tao et al. (2020). In comparison to 1D and 2 D wake models, the 3D wake model is more efficient to address the optimization problem of WF layout. Uniform and nonuniform WF optimization process with maximization of the power output as the objective function is considered. The Mixed-Discrete Particle Swarm Optimization (MDPSO) algorithm result shows nonuniform WF layout generates more power output as compared to uniform WF layout power. When the WT is under maintenance, it does not generate power. The downstream WT will not experience the wake effect by the turbine under maintenance. The output power gets affected by the maintenance of WT’s in the WF. The optimization scheduling model considering wake effect and wind direction for maintenance of offshore WF is discussed in Ge et al. (2020). The objective function is considered to maximize power generation and minimize maintenance costs. It is observed that wind direction and maintenance both have a significant influence on the wake effect on the downstream WT.

Feijoo and Villanueva (2017a) discusses the 3P-DP logistic function model of the WT power curve using the Jensen wake effect model where the wind direction is not considered. The aggregation-based model based on the coherence matrix of wind speed in a WF has been discussed in Rudion et al. (2007). Based on the matrix information, WF is reduced and this reduced structure is used for dynamic simulation. The input of this matrix is wind speed and wind direction. Feijoo and Villanueva (2017b) gives the idea about continuous expression for WF power output curve calculation using the logistic function (4P-DP, 4P-DS) model. The paper suggests including the wind direction in the power curve calculation for more accurate results.

A new mathematical model for WF power calculation considering multiple wake effects is discussed in Ulku and Alabas-Uslu (2019). The paper discussed the optimal location of WT’s in WF increases the net power generated. Some other methods for improving WF power output calculation are discussed in the literature, that is, global sensitivity analysis methods (Carta et al., 2020), cooperative control method for power output optimization (Deljouyi et al., 2021), active power setpoint method under the influence by varying the meteorological and operational parameters (Díaz et al., 2020), thrust coefficient control strategy (Meng et al., 2020), probabilistic estimation model of power curve (Yun and Hur, 2021), the impact of mountain waves on wind power generation (Draxl et al., 2021), and new multiple wake model for prediction of turbulence intensity (Qian and Ishihara, 2021) for power maximization, etc.

The impact of wind direction on the wake effect will give a more accurate power calculation of the WF. Although many literatures have discussed the wake model and power curve model in which very few include wind direction. The 3P-DP model is not considered with the wind direction for WF power calculation so, in this paper, Jensen’s wake effect model of WT using Shi et al. model for power curve as well as a 3P-DP model for parameters calculation is considered. The wind speed, as well as direction under the shadow effect for the output power of individual WT, is also considered. The proposed method proposes a new formula for WF power output calculation considering the wind direction which was not considered in Feijoo and Villanueva (2017a).

Wind turbine power curve

The power curve is obtained by plotting wind speed and corresponding power output data (Villanueva and Feijóo, 2016). The real-time data of 1 year (for the year 2019, 8.2649°N, 77.5668°E) has been used to plot the power curve as shown in Figure 1. The inflection wind speed is 9.039 m/s and the corresponding WT power output is 1.047 MW. The mathematical formula expression to obtain generated power with the wind speed is given by equation (1)

P (v) = P_{r} \frac{1 + (\frac{2 P_{i p}}{P_{r}} - 1) e^{2 s (v_{i p} - v) / (P_{r} - P_{i p})}}{1 + e^{2 s (v_{i p} - v) / (P_{r} - P_{i p})}}

(1)

where

$v_{ip}$ = Wind speed at the inflection point on the power curve

$P_{ip}$ = Power at the inflection point on the curve

$P_{r} =$ Rated power of WT

Figure 1.

Power Curve of Vestas V80 2MW (for the year 2019, 8.2649°N, 77.5668°E) (Feijoo and Villanueva, 2017b).

Equation (1) can also be written with the four parameters (a, m, n, and $τ$ ) logistic model using deterministic process (4P-DP) (Villanueva and Feijóo, 2016) as

P (v) = a \frac{1 + m e^{- \frac{u}{τ}}}{1 + n e^{- \frac{u}{τ}}}

(2)

where

a = P_{r}

n = e^{2 s v_{ip} / (P_{r} - P_{ip})}

m = (\frac{2 P_{ip}}{P_{r}} - 1) n

τ = \frac{(P_{r} - P_{ip})}{2 s}

Now taking some assumption (2 $P_{ip}$ = $P_{r}$ ) for converting equation (1) to 3 parameters deterministic process (3P-DP) model can be given by equation (3).

P (v) = \frac{P_{r}}{1 + e^{4 s (v_{ip} - v) / P_{r}}}

(3)

Wind wake effect considering wind direction

In a WF, the wake effect is evoked due to the mutual shadowing phenomena. The incoming wind speed to the downstream WT’s that are placed in the wind shadow of upstream and other WT’s, decreases (Jensen, 1983). Jensen’s model is presented in Figure 2

Figure 2.

Wake effect model by Jensen (1983).

The velocity over the upstream turbine which is a function of thrust coefficient is given by equations (4) and (5) respectively. Thrust coefficient varies according to the wind speed (Moskalenko et al., 2010).

V_{wt} = V_{n} \sqrt{1 - C_{T}}

(4)

C_{T} = \frac{2 F_{T}}{ρ π R_{wt}^{2} V_{n}} m

(5)

where

$V_{n}$ = Natural wind speed (m/s)

$V_{wt}$ = Speed of wind just passing through the turbine (m/s)

$C_{T}$ = Thrust coefficient of WT

$F_{T}$ = Thrust force

$ρ$ = Air density

Due to the wake effect the wake radius and the speed of wind for the next turbine change. The wake radius at distance x from the rotor plane of upwind WT is given in equation (6). The entrainment constant “k” in equation (7) depends on the offshore (k = 0.075) and onshore (k = 0.04) surrounding sites of the turbine (Feijoo and Villanueva, 2017a). It also describes the wake expansion, that is higher value of k- means the wake cone is wider and the wind speed to the downstream turbine will be less deficit and vice versa.

R_{wake} (x) = k x + R_{wt}

(6)

k = \tan \propto = \frac{k_{wake} (σ_{wake} + σ_{0})}{V_{wake} (x)}

(7)

where,

$R_{wt}$ = Radius of WT rotor (m)

$k$ = Wake attenuation coefficient or entrainment coefficient or wake decay constant

$k_{wake}$ = Wake constant

$σ_{wake}$ = Mean square deviation of turbulence generated by WTs

$σ_{0}$ = Mean square deviation of natural turbulence

The single wake model and multiple wake model have been discussed in González-Longatt et al. (2012). The idea is inspired by González-Longatt et al. (2012) as a single wake model will be assumed when the upstream WT wake shadowing area overlaps downstream WT area and for the rest of the case, it can be considered as a multiple wake effect with partial shadowing.

V_{wake} (x) = {\begin{matrix} V_{n} [1 - (1 - \sqrt{1 - C_{T}}) {(\frac{R_{wt}}{R_{wake} (x)})}^{2}] & 0^{°} < θ < 20^{°} \\ V_{n} [1 - \sum_{1}^{nupWT} [(1 - \sqrt{1 - C_{T}}) {(\frac{R_{wt}}{R_{wake} (x)})}^{2} \frac{A_{shadow}}{A_{wt}}]] & 20^{°} < θ < 70^{°} \\ V_{n} [1 - (1 - \sqrt{1 - C_{T}}) {(\frac{R_{wt}}{R_{wake} (x)})}^{2}] & 70^{°} < θ < 90^{°} \end{matrix}}

(8)

The wind speed due to the wake effect and wind direction on downstream WT can be calculated by equation (8). It is assumed that when the wind direction is between $0^{°} < θ < 20^{°}$ & $70^{°} < θ < 90^{°}$ the shadow area of upstream WT is equal to the rotor swept area downstream WT, that is full shadowing case. For wind direction $20^{°} < θ < 70^{°}$ , the multiple upstream WTs wake shadow occur. The total single wake effect is the sum of all the upstream WTs shadowing effects on downstream WT.

$nupWT$ = number of upstream WTs for particular downstream WT as shown in Figure 3.

Figure 3.

Multiple Wake effect model of upstream WT on Downstream WT (Partial and full shadow case) (González-Longatt et al., 2012).

The shadow area can be calculated by equation (9) (González-Longatt et al., 2012).

\begin{matrix} A_{shadow} = [R_{wake} (x)]^{2} co s^{- 1} \frac{d^{2} + {[R_{wake} (x)]}^{2} - {[R_{wt}]}^{2}}{2 d R_{wake} (x)} + [R_{wt}]^{2} co s^{- 1} \frac{d^{2} + {[R_{wt}]}^{2} - {[R_{wake} (x)]}^{2}}{2 d R_{wt}} \\ - \frac{1}{2} \sqrt{(- d + R_{wake} (x) + R_{wt}) (d + R_{wake} (x) - R_{wt}) (d - R_{wake} (x) + R_{wt}) (d + R_{wake} (x)} + R_{wt)} \end{matrix}

(9)

Equation (9) can be written as

\begin{matrix} A_{shadow} = [R_{wake} (x)]^{2} co s^{- 1} \frac{d^{2} + {[R_{wake} (x)]}^{2} - {[R_{wt}]}^{2}}{2 d R_{wake} (x)} + [R_{wt}]^{2} co s^{- 1} \frac{d^{2} + {[R_{wt}]}^{2} - {[R_{wake} (x)]}^{2}}{2 d R_{wt}} \\ - \frac{1}{2} \sqrt{({(R_{wake} (x) + R_{wt})}^{2} - d^{2}) (d^{2} - {(R_{wake} (x) - R_{wt})}^{2})} \end{matrix}

(10)

Aggregation approach for WF power calculation

The aggregation model concept is used in Rudion et al. (2007). The coherence index is used to find the same input wind speed turbine to convert it into an equivalent model. In the case of WF, the turbines have the same wake coefficient consider into one column (shadow groups are made based on coherence matrix). The wind direction is taken as south to north initially then it rotates clockwise direction as shown in Figure 4.

Figure 4.

wind farm shadowing pattern (Moskalenko et al., 2010).

The output power of WF considering the wind speed, wind direction, and shadow effect given by equation (11). WT’s experiencing the same incoming wind speed for operation due to wake effect is considered into one shadow group (the number of turbines in one shadow group can be considered as the same number of WTs in one column).

The first group of turbines experiencing normal wind speed is shown by the red color group. The second group is given by green color experience same wind speed due to the neighbor red color WT wake effect. Similarly, group three with lighter green experience the same incoming reduce wind speed due to the wake effect of the green WTs group and so on. The last WT will experience lower input wind speed according to wind direction given by lighter color. The concept of direction is taken from Rudion et al. (2007) in which the number of the WT of one group has been selected based on the coherence matrix.

The WF power output will be equal to the sum of all the shadow group WTs power generated (red + green + light green +…). So, equation (3) can be modified and given by equation (11). For the first quadrant (let wind direction vary between $0^{°} < θ < 90^{°},$ the total power from the WF can be calculated by equation (11). Similarly, other quadrant wind direction can be calculated.

P_{WF} = {\begin{matrix} \sum_{i = 1}^{l} \frac{R * P_{r}}{1 + e^{4 s (v_{ip} - V_{wake} {(x)}_{i}) / P_{r}}} & 0^{°} < θ < 20^{°} \\ \sum_{i = 1}^{l} \frac{(R + C + 1 - 2 i) P_{r}}{1 + e^{4 s (v_{ip} - V_{wake} {(x)}_{i}) / P_{r}}} & 20^{°} < θ < 70^{°} \\ \sum_{i = 1}^{l} \frac{C * P_{r}}{1 + e^{4 s (v_{ip} - V_{wake} {(x)}_{i}) / P_{r}}} & 70^{°} < θ < 90^{°} \end{matrix}}

(11)

where

R = number of rows

C = number of columns

$l$ = Number of rows or column either of which one has lesser values

$V_{wake} {(x)}_{i}$ = Speed of wind velocity considering the wake effect in shadow group

$P_{WF}$ = Total power of WF considers wind direction and wake effect

When row and column will be different in the WF, it is clear from equation (11) that for the $0^{°} < θ < 20^{°}$ only, row (R) will be considered. For $20^{°} < θ < 90^{°},$ the row and column (R + C) both will be considered and for $70^{°} < θ < 90^{°},$ column (C) will be considered as the first row.

Numerical analysis and simulation results

The deficit in the wind speed due to the wake effect is shown in Figure 4. The WF power depends on the wind speed experienced by WT’s placed at a particular distance in the farm. For verification of the proposed model, the result obtained is compared with the data as published in Elliott (1991). It is found that the result is almost the same as discussed in the literature which verifies the proposed formula gives accurate results. In this paper, two cases for different “ $θ$ ” angels are considered to include wind direction for wind power calculation. The parameters and data used in the simulation are given in Table 1.

Table 1.

Simulation parameters.

Rated Power	P_r	2 MW (Feijoo and Villanueva, 2017b)
Power at an inflection point on the curve	P_ip	1.0972 MW (Feijoo and Villanueva, 2017b), 1.0472 MW from figure
Wind speed at the inflection point of the curve	$v_{ip}$	9.37 m/s (Feijoo and Villanueva, 2017b), 9.039 m/s from figure
The slope of power curve	s	0.3137514 MW. s/m (Feijoo and Villanueva, 2017b)
Natural wind speed (m/s)	$V_{n}$	According to Figure 5
Thrust coefficient of wind turbine	$C_{T}$	Based on Moskalenko et al. (2010), 0.2 (Feijoo and Villanueva, 2017a)
The entrainment constant	k	offshore (k = 0.075) (Feijoo and Villanueva, 2017a) and onshore (k = 0.04)
Three parameter data	a, m, n, $τ$	a = 2000, m = 0, n = 3,570,702, $τ$ =1.5936 (Villanueva and Feijóo, 2016)
Wind direction	$θ$	210°–220° According to Moskalenko et al. (2010)
Distance between WTs	x	400 m (Feijoo and Villanueva, 2017a)
The radius of the wind turbine rotor	$R_{wt}$	40 m (Feijoo and Villanueva, 2017a)
Wake radius from upstream WTs	$R_{wake} (x)$	70 m(case 1), 72 $\sqrt{2}$ m (case 2)

The steps involve in the simulation is as follow

Collection of real-time WT data, that is, wind speed and corresponding power output

Obtain the power curve according to WT data using the 3P-DP model

Calculation of inflection wind speed point and corresponding WT power output

Consider WF with turbine placed in row and column

Consider Jensen’s wake effect model for simulation

Calculation of wake velocity at each turbine with wind speed and wind direction

Calculation of WF power due to wake effect and wind direction

Comparison of WF power with already existing expression and proposed mathematical expression.

Case 1. When wind direction between $0^{°} < θ < 20^{°}$ or $70^{°} < θ < 90^{°}$

For numerical analysis, row × column (5 × 5) structure has been taken. The number of rows and the number of columns are the same in this structure. All the WT has a 2 MW capacity. For the first case, the total shadowing case is taken, that is shadow area of upstream WT will be equal to the downstream WT rotor swept area. The real-time data of wind for 24 hours of 50 MW (5 × 5 × 2 MW) installed capacity WT is considered. The wind speed data and its variation of 24 hours are shown in Figure 5. Table 2 gives the WF output power for Case 1. For the case 1 direction is assumed between the range 0° < θ < 20° or 70° < θ < 90° with varying wind speed according to data obtained as shown in Figure 5.

Figure 5.

24 Hours wind speed variation.

Table 2.

WF power output for 24 hours when wind direction between $0^{°} < θ < 20^{°}$ or $70^{°} < θ < 90^{°}$ .

Date time	PwF (kW) (Feijoo and Villanueva, 2017a)	PwF (kW)_prop	Difference in kW
19-07-2019 00:30	40,482.50	40,166.02	316.48
19-07-2019 01:30	41,264.38	40,944.37	320.01
19-07-2019 02:30	41,568.39	41,247.74	320.65
19-07-2019 03:30	41,128.42	40,808.83	319.59
19-07-2019 04:30	40,286.89	39,971.69	315.20
19-07-2019 05:30	40,116.14	39,802.18	313.96
19-07-2019 06:30	38,663.79	38,364.54	299.25
19-07-2019 07:30	40,143.98	39,829.81	314.17
19-07-2019 08:30	43,275.78	42,960.37	315.41
19-07-2019 09:30	45,301.34	45,016.49	284.85
19-07-2019 10:30	46,391.00	46,137.11	253.89
19-07-2019 11:30	46,909.92	46,675.34	234.58
19-07-2019 12:30	47,244.85	47,024.56	220.29
19-07-2019 13:30	47,229.56	47,008.59	220.98
19-07-2019 14:30	47,354.08	47,138.79	215.29
19-07-2019 15:30	47,533.05	47,326.33	206.72
19-07-2019 16:30	46,774.78	46,534.85	239.93
19-07-2019 17:30	43,925.77	43,616.90	308.87
19-07-2019 18:30	39,585.55	39,276.13	309.42
19-07-2019 19:30	36,835.24	36,563.33	271.91
19-07-2019 20:30	35,835.08	35,581.34	253.74
19-07-2019 21:30	35,305.97	35,062.58	243.39
19-07-2019 22:30	35,733.76	35,481.97	251.79
19-07-2019 23:30	34,703.38	34,472.30	231.08

The proposed method modified the formula by including wind direction so the comparison between power output calculation by the method used in Feijoo and Villanueva (2017a), that is, equation (3) and the proposed method that is, equation (11) has been shown in Figure 6. The direction of wind plays a significant role as the power calculated by the proposed method is more accurate. The table shows the difference in power in kW as it is very accurate to calculate the exact power output.

Figure 6.

WF power output comparison when wind direction between $0^{°} < θ < 20^{°}$ or $70^{°} < θ < 90^{°}$ .

Case 2. When wind direction between $20^{°} < θ < 70^{°}$

Each WT in downstream in the green group experience a wake effect from neighboring two red WTs. Let each turbine in the red group having a 10% shadow effect on downstream WT, the total shadow experience by green group WT will be 20%. So, wake velocity can be calculated by taking $\frac{A_{shadow}}{A_{wt}}$ = 0.2. The distance between WTs will increase by $\sqrt{2}$ times as diagonal distance will be considered now. For case 2 direction is assumed between the range 20° < θ < 70° with varying wind speed according to data obtained as shown in Figure 5. The comparison of WF power is shown in Figure 7.

Figure 7.

WF power comparison when wind direction between $20^{°} < θ < 70^{°}$ .

Table 3 shows the 24 hours power output calculation of WF. The total WTs power is added considering the wind direction and wind speed. In the case of partial shadowing, the output power is more as compare to the first case (full shadow) because in the case of full shadowing the wind speed reduces more due to the wake effect of upstream WTs. The difference is in MW in the case of partial shadowing so it is required to take wind direction with wake effect for WF power output calculation.

Table 3.

WF power output for 24 hours when wind direction between $20^{°} < θ < 70^{°}$ .

PwF (Feijoo and Villanueva, 2017a) (MW)	PwF_prop (MW)	Difference (MW)
40.4825	43.44319	2.960690
41.26438	44.05083	2.786449
41.56839	44.28432	2.715937
41.12842	43.94590	2.817481
40.28689	43.28958	3.002695
40.11614	43.15499	3.038845
38.66379	41.99087	3.327077
40.14398	43.17697	3.032984
43.27578	45.56630	2.290515
45.30134	47.01922	1.717875
46.39100	47.76755	1.376558
46.90992	48.11487	1.204950
47.24485	48.33568	1.090823
47.22956	48.32566	1.096093
47.35408	48.40708	1.053004
47.53305	48.52342	0.990374
46.77478	48.02502	1.250238
43.92577	46.04084	2.115066
39.58555	42.73368	3.148128
36.83524	40.47686	3.641627
35.83508	39.62624	3.791162
35.30597	39.16984	3.863876
35.73376	39.53919	3.805427
34.70338	38.64470	3.941318

Conclusion

While WF construction planning, it is important to consider the optimal area due to the economy of the land. However, minimizing the area leads to a decrease in distance between WTs. As the distance decreases the shadow effect of WT comes into the picture. Upstream WT extracts energy and reduces the wind speed for downstream WT causes a turbulent flow. Due to this, the second WT would not run optimally and produce less energy. The power output of WF is affected by the wake effect of WT due to wind speed and wind direction. In this paper, a new equation has been suggested to calculate total WF power considering the 3P-DP method for parameter control of power curve, with the impact of wind speed and direction on wake effects of WT. The comparison of WF power calculated from the already existing equation and the proposed equation is also discussed. The proposed mathematical expression gives a more accurate WF power calculation that reduces the chance of calculating over-estimation of WF energy. From an economical point of view, it calculates and shows more realistic results. For future work, yaw control and pitch control of wind turbines for WF power enhancement can be considered.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Aishvarya Narain

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The impact of wind direction on wind farm power output calculation considering the wake effects of wind turbines

Abstract

Keywords

Introduction

Wind turbine power curve

Wind wake effect considering wind direction

Aggregation approach for WF power calculation

Numerical analysis and simulation results

Case 1. When wind direction between 0 ° < θ < 20 ° or 70 ° < θ < 90 °

Case 2. When wind direction between 20 ° < θ < 70 °

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Case 1. When wind direction between $0^{°} < θ < 20^{°}$ or $70^{°} < θ < 90^{°}$

Case 2. When wind direction between $20^{°} < θ < 70^{°}$