Wind farm cooperative control for optimal power generation

Abstract

The power generation of a wind farm depends on the efficiency of the individual wind turbines of the farm. In large wind farms, wind turbines usually affect each other aerodynamically at some specific wind directions. Previous studies suggest that a way to maximize the power generation of these wind farms is to reduce the generation of the first rows wind turbines to allow the next rows to generate more power (coordinated case). Yet, other studies indicate that the maximum generation of the wind farm is reached when every wind turbine works at its individual maximum power coefficient C_Pmax (individual case). This article studies this paradigm and proposes a practical method to evaluate when the wind farm needs to be controlled according to the individual or the coordinated case. The discussion is based on basic principles, numerical computations, and wind tunnel experiments.

Keywords

Wind farm control optimal power generation wake effect blade element momentum theory experimental validation

Introduction

Wind farm (WF) control strategies and wind turbine (WT) coordination methods have attracted many research efforts in the last few years. Commercial WTs in a WF are commonly operated individually at their maximum power coefficients C_Pmax. However, this solution does not take into account the potential aerodynamic effects produced on every WT by other neighboring WTs. According to some theoretical calculations, previous studies have mentioned that the power generation of a WF could be optimized by reducing the rated power (derating) of the upper stream WTs (see Bitar and Seiler, 2013; Gebraad et al., 2013; Johnson and Thomas, 2009; Schepers and Van Der Pijl, 2007). In these cases, the WF operator may apply the following operating methods to increase the wind velocity that reaches the downstream WTs (see Campagnolo et al., 2016; Zalkind and Pao, 2016):

Derate the upstream WTs by modifying the generator torque, which allows more wind flow to pass to the downstream WTs (same C_P/λ curve at a lower working point, being C_P the power coefficient and λ the tip-speed ratio, λ = Ω R_i/V_i);

Derate the upstream WTs by modifying the pitch angle, which also allows more wind flow to pass to the downstream WTs (change to a lower C_P/λ curve);

Modify the yaw angle of upstream WTs to direct the wake effect to the downstream WTs differently.

These three operating methods (1–3) are potential solutions for the so-called coordinated control strategy, which assumes that the maximum generation of the WF is reached when the generation of the WTs of the front rows is properly reduced to allow the next rows to generate more power. However, it could also be possible that the maximum generation of the WF is reached when every WT works at its individual maximum power coefficient C_Pmax, which is the so-called individual control strategy. Both strategies, individual and coordinated, are currently under discussion by the international wind energy community, with different opinions and proposals in the recent literature (see Bitar and Seiler, 2013; Campagnolo et al., 2016; Gebraad et al., 2013; Johnson and Thomas, 2009; Schepers and Van Der Pijl, 2007; Zalkind and Pao, 2016). In this article, we study this problem and propose a methodology to evaluate whether the WF has to be controlled following the individual or the coordinated strategy. The discussions and results are based on three complementary aspects: WT basic principles, numerical computations, and wind tunnel experiments.

Section “WF model analysis” starts with a simple model, based on the ideal rotor disk theory for WT power capture and the Park model to characterize the aerodynamic wake interaction among the WTs in the WF. With several examples, the discussion shows how to apply these basic principles to understand the main differences between the individual and coordinated strategies. Based on these principles, section “Individual/coordinated case selection” develops a new methodology to evaluate when the WF should be controlled according to the individual or the coordinated case. Moving forward, section “Numerical Validation with Blade Element Momentum Theory” expands the discussion and study a more realistic scenario by applying the Blade Element Momentum (BEM) theory. The validation of the methodology is carried out numerically with five different rotor configurations. Afterwards, section “Wind tunnel experimental results” validates the method experimentally, using a fully instrumented wind tunnel and several lab-scale WT prototypes. Finally, section “Conclusion” summarizes the main results and presents suggestions for future research.

WF model analysis

The first-principles WF model used in this article and two illustrative case studies are presented in this section. The first case study considers the simple situation of two identical WTs, arranged in a straight line perfectly aligned with the wind direction, with a given distance between the two WTs. The second case study extends the discussion by adding a third WT behind the second WT, in the same straight line in the wind direction, at the same distance from the second WT. The following subsections present the main principles, power capture discussions, and the two case studies.

Wake effect model

A number of mathematical models that describe the effect of a wake in a WF have been developed in the last few decades. One of them, originally developed by Jensen and Katic, is the Park wake model (see Kattic et al., 1986). The model is based on basic principles, easy to use, and gives a first approach to understand the problem.

As one of the main assumptions, the Park model considers the wind velocity in the wake as the linear distribution, where the velocity V_i at the location x_i is given by equation (1)

V_{i} (x_{i}, a_{i - 1}) = V_{i - 1} (1 - δ V (x_{i}, a_{i - 1}))

(1)

where “i-1” and “i” denote the upstream and downstream WTs, respectively; V_i_-1 is the wind velocity at the upstream turbine; V_i is the wind velocity at the downstream turbine after passing an upstream turbine; x_i is the distance between the two WTs “i” and “i-1,” and a_i_-1 is the axial induction factor of the turbine i-1. In addition, δV is the relative velocity factor defined as

δ V = (1 - \sqrt{1 - C_{T i - 1}}) {(\frac{R_{i - 1}}{R_{i - 1} + k x_{i}})}^{2}

(2)

where k is a constant that expresses the wake expansion at a distance x_i from the upstream turbine, R_i_-1 is the rotor radius of the “i-1” turbine, and C_Ti_-1 is the thrust coefficient of the rotor of the “i-1” turbine. Typically, k = 0.04 for the first row or upstream WTs, and k = 0.08 for the downstream turbines (see Gonzalez-Longatt et al., 2012; Sϕrensen et al., 2008).

The Park model has enough accuracy when the distance between the WTs is larger than 6D (six rotor diameters) (see Annoni et al., 2014). The equations (1) and (2) are used in this section to calculate the downstream WT wind velocities V_i in the simulations.

Ideal rotor disk model

Based on the well-known ideal rotor disk assumption utilized to derive the Betz limit, the non-dimensional power coefficient C_P and the thrust coefficient C_T can be ideally expressed as functions of the axial induction factor a, so that (see Burton et al., 2008; Garcia-Sanz and Houpis, 2012)

C_{P} = 4 a {(1 - a)}^{2}

(3)

C_{T} = 4 a (1 - a)

(4)

As seen in equations (1) to (4), the axial induction factor a affects the WT power coefficient C_P, the thrust coefficient C_T, and the downstream wind speed V_i in the WF numerical model. Consequently, the axial induction factor a plays a central role in the control strategy for the optimization of the WF power generation. The power captured by the “i” WT can be expressed as

P_{i} (a_{i}) = \frac{1}{2} ρ A_{i} C_{P} (a_{i}) V_{i}^{3} (x_{i}, a_{i - 1})

(5)

where ρ is the air density, A_i = π R_i² is the turbine rotor swept area, and R_i the rotor radius (see Garcia-Sanz and Houpis, 2012).

Case study I (2-WT)

In the two WTs case (first case study, i = 2), WT₁ denotes the upstream WT, WT₂ the downstream WT and a₁ the axial induction factor for WT₁. In order to compare the results of this section with the wind tunnel experiments presented at the end of the article, we select the lab-scale WT prototype rotor radius for both WTs, R = R₁ = R₂ = 0.145 m. The total mechanical power P_mec of the WF at the shaft of the WTs can be expressed as

P_{m e c} (a_{1}) = \frac{1}{2} ρ π R^{2} (V_{1}^{3} C_{P 1} (a_{1}) + V_{2}^{3} C_{P 2 \max})

(6)

where we consider that WT₂ works at its individual maximum C_Pmax₂ and WT₁ at a C_P₁ that can be either C_Pmax₁ (individual case) or less than C_Pmax₁ (coordinated case). Now, considering the distance between the WTs proportional to the rotor radius, that is, x₁ = n R, and substituting equations (1) and (2) into the equations (5) and (6), we obtain

P_{m e c} (a_{1}) = \frac{1}{2} ρ π R^{2} V_{1}^{3} (C_{P 1} (a_{1}) + f {(a_{1})}^{3} C_{P 2 \max})

(7)

where f (a₁) is

f (a_{1}) = 1 - 2 a_{1} {(\frac{R}{R + 2 k x_{1}})}^{2} = 1 - 2 a_{1} {(\frac{1}{1 + 2 k n})}^{2}

(8)

Incidentally, it is interesting to see that the free upstream wind speed V₁ and the WT blade radius R do not affect the factor f (a₁), and then they are not necessary for the selection of the individual/coordinated strategy to optimize the WF power generation. Actually, f (a₁) is just a function of a₁, k, and the distance between turbines in terms of number or radius n, being k and n constant numbers.

Using equations (1) to (6), Figure 1 shows the total mechanical power output of the WF when the distance between the WTs is eight rotor diameters (8D). The power coefficient of the first turbine C_P₁ is changed from 0 to the Betz limit = 16/27 = 0.593, and the power coefficient of the second turbine is kept at the ideal maximum value C_P₂ = C_PmaxIdeal = 0.593 (Betz limit). As we can see, the maximum power output of the WF is achieved when the first turbine operates at C_P₁ = 0.5625, with a₁ = 0.238. This corresponds to the coordinated case and shows a ~4% power increase in comparison with the case when both WTs operate at the individual maximum power coefficient, C_P₁ = C_P₂ = C_PmaxIdeal = 0.593 (Betz limit), or individual case. The MATLAB code for this example is included in the Appendix 1.

Figure 1.

2-WT case study. Mechanical power P_mec of two wind turbines using ideal disk theory. Maximum WF power is obtained at C_P₁ = 0.5625 and C_P₂ = C_PmaxIdeal = 0.593 (coordinated case).

Case study II (3-WT)

In the three WTs case (second case study, i = 3), the principles introduced in the previous subsections give the results shown in Figure 2. The distance between the first and the second WT is 8D, as well as the distance between the second and the third WT. The maximum total mechanical power output of the 3-WT WF is obtained when the first WT operates at C_P₁ = 0.5120, the second WT operates at C_P₂ = 0.5695, and the third one operates at ideal maximum value C_P₃ = C_PmaxIdeal = 0.593 (Betz limit).

Figure 2.

3-WT case study. Mechanical power P_mec of three wind turbines using ideal disk theory. Maximum WF power is obtained at C_P₁ = 0.5120, C_P₂ = 0.5695, and C_P₃ = C_PmaxIdeal = 0.593 (coordinated case).

This also corresponds to the coordinated case, and shows a ~9% power increase in comparison with the case when the three WTs operate at the maximum individual power coefficient, C_P₁ = C_P₂ = C_P₃ = C_PmaxIdeal = 0.593 (Betz limit), or individual case.

In conclusion, according to the basic principles given by the ideal rotor disk theory and Park model, the optimal power generation of a WF is achieved by derating properly the upstream WTs, which is the coordinated case.

Individual/coordinated case selection

In the previous section, we just found that the coordinated case strategy maximizes the power generation of the WF. However, some recent publications have proposed the opposite solution, suggesting the individual case as the strategy to maximize the power generation of the WF (see for instance Campagnolo et al., 2016).

This controversy compelled us to rethink the previous results and look for additional parameters to understand the problem better. As we looked deeper into the physics, we found that the solution of this paradigm is not unique but depends on some properties of the WTs and WF. They are introduced in this section.

At first step, we multiply the power coefficient C_P in equation (3) by a gain G_cp and the thrust coefficient C_T in equation (4) by a gain G_ct. That is to say

C_{P s c a l e d} = 4 a {(1 - a)}^{2} G_{c p}

(9)

C_{T s c a l e d} = 4 a (1 - a) G_{c t}

(10)

These two new gains, G_cp and G_ct, are selected in the ranges [1–δ₁ ⩽ G_cp ⩽ 1], |δ₁| < 1, and [1–δ₃ ⩽ G_ct ⩽ 1 +δ₄], |δ₃| < 1, |δ₄| < 1 in order to see the effect of small variations of the power and thrust coefficients around the nominal values given by equations (3) and (4). Note that the maximum value for the first gain is G_cp = 1, which represents the Betz limit for the power coefficient C_P. On the other hand, the second gain can be G_ct ⩾ 1 or G_ct ⩽ 1, as in practice C_T can take values below and above the value given by equation (4).

Using equations (6) to (10) and taking into account the numerical results for the 2-WT case study and the gains [0.8 ⩽ G_cp ⩽ 1], [0.4 ⩽ G_ct ⩽ 1.4], the total power generated by the WF is now calculated as a function of the power coefficient C_P of the first turbine, as shown in Figure 3. The MATLAB code for these calculations is included in the Appendix 1. Note that the case (G_cp = 1, G_ct = 1) is the one presented previously in Figure 1.

Figure 3.

2-WT case study. Optimal power point as a function of the gains G_cp and G_ct in C_P and C_T, and at a distance 8D. The solid curves show the power of the wind farm for each C_P of the first WT with G_cp = 1 and with 0.4 ⩽ G_ct ⩽ 1.4. The solid curves for G_cp = 0.8 and 0.9 are not plotted. The three asterisk curves (for G_cp = 0.8, 0.9, and 1) show the maximum power generation at each G_ct case (0.4 ⩽ G_ct ⩽ 1.4).

Note also that for almost every G_cp gain with a large G_ct gain the maximum power generation of the WF is not at C_P₁ = C_Pmax but at a lower value—see the asterisk curves. In addition, the variation of G_ct affects more significantly the WF optimal point. Actually, as G_ct becomes smaller, the WF optimal point shifts to the right, reaching C_P₁ = C_Pmax, as shown in Figure 3. This means that in these right-curve cases the WF has to be operated according to the individual case control strategy. As a result, we can say that:

The individual WT power coefficients C_P alone do not provide enough information to decide which strategy, individual or coordinated case, is the best to optimize the power generation of the WF.

This decision mainly depends on how the upstream WT slows down the wind velocity at a certain downstream distance, and this fact is better represented by the rotor’s thrust coefficient C_T.

The wind velocity in the wake area is recovering as the distance between the WTs increases—see equations (1) and (2). Then, the distance (nD) between the WTs is another key factor to select the WF control strategy.

With these remarks in mind, we present in the following paragraphs a practical methodology to facilitate the selection of the WF control strategy between the individual and coordinated cases.

First, we normalize equation (9) dividing it by the Betz limit, C_{P. Betz} = 0.593 and multiplying it by the actual rotor power coefficient C_Pmax. This C_Pmax coefficient can be obtained from the C_P/λ curves provided by the blade manufacturer, by specific WT experiments (see section “Wind tunnel experimental results”) or by using appropriate blade design computer programs (QBlade). In addition, the value of the gain G_ct in equation (10) is selected according to the distance between the turbines. Hence, the C_Pref and C_Tref expressions are defined as

C_{P r e f} = 4 a {(1 - a)}^{2} \frac{C_{P m a x}}{C_{P . B e t z}}

(11)

C_{T r e f} = C_{T s c a l e d} = 4 a (1 - a) G_{c t}

(12)

Second, we represent the reference curves “C_Tref/C_Pref” taking C_Pref as the horizontal axis and C_Tref as the vertical axis. Figure 4 shows an example where the WT has an actual maximum power coefficient of C_Pmax = 0.4. The functions represented in the figure are the “C_Tref/C_Pref” reference curves for that C_Pmax and for distances 6D, 8D, 10D, and 12D (D = rotor diameter)—see equations (11) and (12). The MATLAB code for these calculations is included in the Appendix 1.

Figure 4.

C_Tref/C_Pref reference curves for WTs at distances 6D, 8D, 10D, and 12D.

Note that, as discussed in the previous section, the Park model gives enough accuracy in cases where the distance between the WTs in the wind direction is larger than 6D (n = 6 or six rotor diameters) (see also Yang and Sotiropoulos, 2013). In addition, we have seen that in cases where that distance is larger than 12D (n = 12 or twelve rotor diameters) the wind speed recovers to values higher than the 85% of the upstream velocity, meaning that the aerodynamic effect of neighbor WTs is negligible.

As a result and depending on the distance nD between the WTs in the wind direction, the methodology gives practical results in the following scenarios:

n > 12: The individual case is always recommended.

6 ⩽ n ⩽ 12: The method is valid to select between the individual and coordinated cases.

0 < n < 6: If the method suggests the individual case for these short distances, it can be inferred that the individual case is the appropriate one for n ⩾ 6. However, if the method suggests the coordinated case, no specific answer can be confirmed for these short distances (0 < n < 6).

The following paragraphs present the analysis for the distances between 6D and 12D or lower and upper boundaries. Table 1 shows the G_ct gains selected individually for each distance (6D, 8D, 10D, and 12D). They are chosen so that the coordinated case produces at least m = 5% more energy than the individual case.

Table 1.

G_ct gains for different distances between the WTs.

Distance	6D	8D	10D	12D
G_ct	0.9	1	1.05	1.1

The factor m is the ratio between the energy produced by the coordinated case over the energy produced by the individual case. This ratio is selected so that it justifies economically the increase of complexity of the coordinated case over the individual case. In Table 1, we selected m = 5% for all cases. In addition, the case defined in Table 1 by “Distance = 8D and G_ct = 1” is based on the results presented in Figure 3, with m = 5%.

Once we have the “C_Tref/C_Pref” reference curves—see for instance Figure 4 or Figures 7 to 9, we plot in the same figure the actual “C_T/C_P” curve of the WT rotor. This actual “C_T/C_P” curve can be obtained from the information provided by the blade manufacturer, by specific WT experiments (see section “Wind tunnel experimental results”) or by using appropriate blade design computer programs based on the BEM theory (see section “Numerical Validation with Blade Element Momentum Theory”) or other advanced aerodynamic theories (see Burton et al., 2008; Garcia-Sanz and Houpis, 2012; Hansen, 2008; QBlade). Figures 7 to 9 show some results.

Now, by comparing the slopes of the “C_Tref/C_Pref” reference curves and the actual “C_T/C_P” curve, we can select the control strategy (individual or coordinated) for the WF. Specifically, as shown in Figure 4, if the WT rotor’s C_T/C_P curve presents a slope smaller than the lower boundary, or 6D reference curve, then the WF will be operated according to the individual control strategy if the distance between the WTs is greater than 6D.

On the contrary, if the rotor’s C_T/C_P curve of the WT presents a slope in the middle of the upper (12D) and lower (6D) boundary curves—see Figure 4 again, the WF might run as individual case or coordinated case, depending on the distance between the WTs.

Finally, if the rotor’s C_T/C_P curve of the WT presents a slope greater than the upper boundary curve (12D)—see Figure 4, the WF will achieve the maximum power using the coordinated control strategy in cases where the distance between the WTs is less than 12D.

For simplicity, we can approach the reference curves in Figure 4 as first-order functions “C_T = A C_P + B” that pass through the points (0, 0) and (C_Pmax, C_T (C_Pmax)). As these lines pass through the origin, or B = 0, we only need to evaluate the parameter A to select the WF control strategy. In the example of Figure 4, with C_Pmax = 0.4, if the C_T/C_P curve of the WT has a slope A less than 1.92, then the WF will be operated as individual case (in cases where the distance between the WTs is greater than 6D). If the slope A is in the range of 1.92–2.35, the WF can be operated as individual case or coordinated case, depending on the distance between the WTs. Finally, if the slope A is greater than 2.35, the WF will follow the coordinated case in cases when the distance between the WTs is less than 12D.

To validate this practical methodology, the following section presents five different rotor configurations. The validations are carried out with numerical simulations and wind tunnel experiments.

Numerical validation with BEM theory

In order to validate numerically the methodology introduced in the last section, this new section applies the well-known BEM theory (see Burton et al., 2008; Garcia-Sanz and Houpis, 2012; Hansen, 2008) and the blade design packages XFOIL and QBlade.

As discussed in the previous section, the WT rotor coefficients C_P and C_T depend on the rotor configuration, blade geometry, and airfoils. Five different WT rotors are designed here following the BEM theory and using the same reference wind velocity condition, V_wind = 8 m/s. The following paragraphs describe the details.

BEM theory

According to the BEM theory, the lift (F_L) and drag (F_D) forces per unit of length applied by the wind to the perpendicular sections of the blade, from the root to the tip of the blade, are calculated as

F_{L} = \frac{1}{2} ρ V_{r e}^{2} C_{L} c

(13)

F_{D} = \frac{1}{2} ρ V_{r e}^{2} C_{D} c

(14)

where c is the chord of the blade at each section, and C_L and C_D the lift and drag coefficients, respectively. We divide each blade into 100 sections, from the root to the tip of the blade. V_re is the relative wind velocity that affects any given section, located at a distance r from the central axis of the rotor

V_{r e} (r) = \sqrt{{[V_{w i n d} (1 - a)]}^{2} + {[Ω r (1 + a^{'})]}^{2}}

(15)

where, a is the axial induction factor, a′ is the tangential induction factor, V_wind is the upstream undisturbed wind velocity, and Ω is the rotor velocity. The algorithm to calculate both a and a′ is defined in Hansen (2008). The normal (F_n) and tangential (F_t) forces to the rotor plane are described respectively as follows

F_{n} = F_{L} \cos (ϕ) + F_{D} \sin (ϕ)

(16)

F_{t} = F_{L} \sin (ϕ) - F_{D} \cos (ϕ)

(17)

where φ is the sum of the angle of attack, blade twist, and blade pitch angles, with $ϕ = atan [(V_{w i n d} (1 - a)) / (Ω r (1 + a^{'}))]$ . In addition, the normal (C_n) and tangential (C_t) thrust coefficients are defined as

C_{n} = \frac{F_{n}}{0.5 ρ V_{r e}^{2} c}

(18)

C_{t} = \frac{F_{t}}{0.5 ρ V_{r e}^{2} c}

(19)

and

C_{n} = C_{L} \cos (ϕ) + C_{D} \sin (ϕ)

(20)

C_{t} = C_{L} \sin (ϕ) - C_{D} \cos (ϕ)

(21)

F_n and F_t are forces per unit of length. On each section of length dr (in the root-tip direction) the normal force and torque at a distance r from the central rotor axis are respectively

d N = \frac{1}{2} ρ N_{b} \frac{{[V_{w i n d} (1 - a)]}^{2}}{\sin^{2} ϕ} c C_{n} d r

(22)

d T = \frac{1}{2} ρ N_{b} \frac{V_{w i n d} (1 - a) Ω r (1 + a^{'})}{\sin ϕ \cos ϕ} c C_{t} r d r

(23)

where N_b is the number of blades. Taking the integrals from the root (r_hub) to the tip of the blade (rotor radius R), the power (C_P), and thrust (C_T) coefficients are respectively

C_{P} = \frac{P_{m e c}}{P_{w i n d}} = \frac{\int_{r_{h u b}}^{R} Ω d T}{0.5 ρ π R^{2} V_{w i n d}^{3}}

(24)

C_{T} = \frac{\int_{r_{h u b}}^{R} Ω d N}{0.5 ρ π R^{2} V_{w i n d}^{2}}

(25)

In addition, we apply two well-known corrections for the BEM computation. First, the Prandtl’s tip loss factor, which corrects the assumption of an infinite number of blades. Second, the Glauert correction for the high values of a, which is an empirical relation between the induction factor (when it is greater than 0.4) and the thrust coefficient (see Garcia-Sanz and Houpis, 2012).

Method validation with BEM theory

This section analyses several WT rotors using the BEM theory presented in the previous subsection, equations (13) to (25). The study includes the airfoils: S1223 (R = 0.145 m and R = 5 m), S834E (R = 10 m), SG6051 (R = 20 m), and AH 93-W-300 (R = 30 m). The corresponding C_P and C_T curves are shown in Figures 5 and 6. The lab-scale S1223 rotor (R₁) presents a C_Pmax = 0.22; the 5 m S1223 rotor (R₂) has a C_Pmax = 0.49; the 10 m S843E rotor (R₃) has a C_Pmax = 0.38; the 20 m AH 93-W-300 rotor (R₄) has a C_Pmax = 0.47; and the 30 m FX 77-W-270 S rotor (R₅) has a C_Pmax = 0.47—see Table 2.

Figure 5.

Numerical computation of C_P/λ curves using BEM theory.

Figure 6.

Numerical computation of C_T/λ curves using BEM theory.

Table 2.

Rotor blade characteristics.

Rotor name	R ₁	R ₂	R ₃	R ₄	R ₅
Airfoil	S1223 lab-scale	S1223	S834E	AH 93-W-300	FX 77-W-20 S
C_Pmax	0.22	0.49	0.38	0.47	0.47

Using the C_P and C_T curves shown in Figures 5 and 6, we plot in Figure 7 the R₁ rotor C_T/C_P curve along with their reference curves. As the R₁ rotor C_T/C_P curve (A = 0.54) is less than the 6D lower boundary curve (A = 3.61), then the WF with R₁ rotors has to be operated according to the individual case strategy when the distance between the WTs is greater than 6D.

Figure 7.

Study for R₁ rotors with C_Pmax = 0.22. The solid line is the R₁ rotor C_T/C_P curve. Dashed-lines are the reference curves at different distances.

In the same way, Figure 8 shows the R₂ rotor C_T/C_P curve along with their reference curves. Now, the R₂ rotor C_T/C_P curve slope (A = 1.84) is slightly larger than the lower boundary (A = 1.60) and smaller than the upper boundary (A = 1.95). Hence, the WF based on this rotor has to operate as coordinated case when the distance is less than 6D, and as individual case when the distance is greater than 12D. In the range of 6D–12D, the WF may operate as individual case or coordinated case.

Figure 8.

Study for R₂ rotors with C_pmax = 0.49. The solid line is the R₂ rotor C_T/C_P curve. Dashed-lines are the reference curves at different distances.

In addition, Figure 9 shows the R₄ rotor C_T/C_P curve along with their reference curves. Now, the R₄ rotor C_T/C_P curve slope (A = 1.52) is smaller than the lower boundary (A = 1.69) and the upper boundary (A = 2.07). Hence, the WF based on this rotor has to operate as individual case when the distance between the WTs is greater than 6D.

Figure 9.

Study for R₄ rotor with C_pmax = 0.47. The solid line is the R₄ rotor C_T/C_P curve. Dashed-lines are the reference curves at different distances.

Finally, the blade with the airfoil S834E (R₃) and FX 77-W-20 S (R₅) also have less slope compared to their references. Then, the WF based of these WTs have to be operated as individual case in cases when the distance between the WTs is greater than 6D.

Wind tunnel experimental results

For a better understanding and a more complete validation of the proposed methodology, this section presents a collection of experiments with lab-scale WT prototypes at the fully instrumented wind tunnel at the Control and Energy System Center (CESC), Case Western Reserve University (CWRU).

The experiments were conducted with two identical lab-scale WTs, each one with a 3-blade S1223 R₁ rotor. Both WTs were placed in a straight line perfectly aligned with the wind direction in the middle of the wind tunnel—see Figure 10.

Figure 10.

Experiments with two identical lab-scale WTs with R₁ rotors in the CWRU-CESC wind tunnel.

Figure 11 shows the experimental results of the power coefficient curve (C_P/λ) of one of the S1223 R₁ WTs working alone in the middle of the wind tunnel. The experiments apply five different undisturbed upstream wind velocities: V_∞ = 6.1, 7.1, 7.6, 8.0, and 8.35 m/s. In order to obtain the C_P/λ curves at each wind speed, the electrical current of the generator, which is proportional to the antagonistic WT torque, was controlled at 20 different working points, being each point the average of 1-min data, with 600 samples per minute—see five C_P/λ curves in Figure 11. The maximum aerodynamic power coefficient obtained for all cases is C_Pmax = 0.22—see Figure 11 (dashed line), which is consistent with the numerical results obtained in the previous section for the same rotor R₁—see Figure 5 and Table 2.

Figure 11.

Experimental results with the R₁ S1223 lab-scale WT prototype. C_P/λ curves at five different wind speeds. The red dash line is the maximum C_P. Each C_P/λ curve has 20 points, each point is the average of 1-min data or 600 samples.

Once we obtain the individual C_P/λ curve for the WT, we conducted the WF experiments with two WTs arranged at a relative distance of 3.3D in the middle of the wind tunnel, as shown in Figure 10.

The undisturbed upstream wind velocity was set at V_∞ = 8 m/s. During the experiment, the first WT tracked a pre-defined angular velocity Ω that covers the complete range of operation.

At the same time, the second WT was controlled with a Maximum Power Point Tracking (MPPT) strategy, according to a classical Power Signal Feedback (PSF) method, to maintain the maximum R₁ rotor power coefficient at C_P₂ = C_Pmax = 0.22 at every wind speed (see Garcia-Sanz and Houpis, 2012; Annoni et al., 2014; Wang et al., 2016). As defined in this strategy, the optimum target power P_opt was

P_{o p t} = K_{o p t} Ω^{3}

(26)

where Ω is the rotor speed in rad/sec measured in real-time, and K_opt is the coefficient calculated for the R₁ rotor, which is

K_{o p t} = \frac{0.5 π C_{P m a x} ρ R^{5}}{λ_{o p t}^{3}} = 9.8514 \times 10^{- 7}

(27)

Figure 12 shows the mechanical power (power at the input of the drive train) of both WTs (WT₁ and WT₂) versus the tip-speed ratio (λ) of the upstream WT (WT₁). The power of the downstream WT (WT₂) slightly changes as we change the generator electric current (torque control) of the upstream WT (WT₁).

Figure 12.

Wind tunnel experimental results: mechanical power of the wind turbines WT₁ and WT₂ with a 3.3D distance.

By adding up the two mechanical power curves of the two WTs, the maximum WF power generation is achieved when the first WT operates at C_P₁ = C_Pmax = 0.22, which is the individual case—see Figure 12. These results are consistent with the numerical calculations for the same rotor R₁ and the 2-WT case study presented in previous sections.

Also, as discussed previously, if the method suggests to select the individual case for this n = 3.3 (3.3D) short distance, it can be inferred that the individual case is the appropriate one for the distances n ⩾ 6. However, no specific answer can be confirmed for distances n < 6.

Conclusion

Some studies suggested that to maximize the power generation of a WF, the energy extraction of the turbines of the first rows have to be reduced (derated) to allow the next rows to generate more power (coordinated case). However, other studies indicated that the maximum generation of the WF is reached when every WT works at its maximum power coefficient C_Pmax (individual case).

This article analyzed this paradigm and proposed a practical methodology to evaluate when a WF should be operated according to the individual case or the coordinated case. The methodology introduces a simple graphical analysis based on a set of power coefficient/thrust coefficient (C_T/C_P) reference curves. The article discussed and validated the methodology by using three complementary steps: (1) studies based on basic principles, (2) numerical computations with the Park model and BEM theory for different rotor configurations, and (3) wind tunnel experiments with lab-scale WTs.

Footnotes

Appendix

Appendix 1

The MATLAB code to generate the main results of the article is included next.

CP = []; CT = []; PowerWF = [];
V1 = 8; % wind speed of first wind turbine (m/s)
R = 0.145; % rotor radius (m)
D = 2*R; % rotor diameter (m)
x = [6*D 8*D 10*D 12*D]; % distance between WTs (m)
ro = 1.225; % air density (kg/m3)
CpBetz = 0.593; % Betz limit
CpMax = 0.4; % Maximum Cp of WT
k = 0.04; % Wake expansion constant, for Eq.(2)
Gcp = 1; % Gcp Gain
Gct = [0.9 1 1.05 1.1]; % Gct Gain
% Gct = 1; CpMax = CpBetz; x = 8*D; % Remove “%” for Figure 1.
% Gct = [0.4:0.1:1.4]; CpMax = CpBetz; x = 8*D + 0*(1:1: length(Gct)]; % Remove “%” for Figure 3.
jjmax = length(Gct);
a = [0:0.01:0.33]; % axial induction factor
iimax = length(a);
for jj = 1:1: jjmax % Gct Gain
for ii = 1:1: iimax % axial induction factor
CP(ii, jj) = 4*a(ii)*(1-a(ii))^2*(CpMax/CpBetz); % Eq.(11)
CT(ii, jj) = 4*a(ii)*(1-a(ii))*Gct(jj); % Eq.(12)
DV = (1-(1-CT(ii, jj))^0.5)*(R/(R + k*x(jj)))^2; % Eq.(2)
V2(ii, jj) = V1*(1-DV); % Second wind turbine wind speed, Eq.(1)
CP1 = CP(ii, jj);
CP2 = CpBetz;
PowerWF(ii, jj) = 0.5*ro*(pi*R^2)*(V1^3*CP1 + V2(ii, jj)^3*CP2*Gcp); % Wind Farm Power, Eq.(6)
end
MaxP(jj) = max(PowerWF(:, jj));
CP_MaxP(jj) = CP(find(PowerWF(:, jj) == MaxP(jj)),jj);
end
figure; % For Figures 1 and 3
plot(CP, PowerWF); hold on; plot(CP_MaxP,MaxP,’*r’); grid on;
xlabel(‘Cp for the first wind turbine,’’FontSize,’12);
ylabel(‘Power of Wind farm,’’FontSize,’12);
figure; % For Figure 4
plot(CP, CT); grid on;
xlabel(‘CpRef,’’FontSize,’12); ylabel(‘CtRef,’’FontSize,’12);

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

Fa Wang

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