Pitching compensation system to improve floating offshore wind turbine performance

Abstract

In this paper the inﬂuence of the wave oscillation on the behavior of ﬂoating off-shore wind turbine is analyzed. Aerodynamic analysis of forces on wind turbine blade has been carried out as a function of the wind turbine tilt angle due to wave oscillation. This analysis has resulted in a theoretical model based on the variation of the angle of attack which allows the characterization of the turbine under the effect of the oscillation of the sea surface. The results obtained show the inﬂuence of the oscillation in a particular case and its impact on the generated power. Subsequently, a so-called “pitching compensation” system that allows eliminating the effect produced by the variation in the angle of attack and, therefore, minimizing the effect of oscillation on the power generated, is proposed.

Keywords

Floating off-shore wind turbine wave oscillation pitch and attack angle wind turbine performance pitching compensation system

Introduction

Off-shore wind farms are, among the renewable sources, one of the most promising due to the large areas where they can be installed without interfering with human activities. The installations of marine wind farms, however, have been suffering from a serious problem since they have been traditionally located in shallow sea areas where the wind turbine mast can be anchored on the seabed to provide better stability. Unfortunately, there are no many areas where this condition takes place in the nearby of the coastline. The solution of this problem has been the development of the floating wind turbines. In the last years some floating wind farm projects have arisen (Buljan, 2020; Durakovic, 2019; Hywind Scotland, n.d.; Repsol, n.d.) with innovative techniques like the double wind turbine platform (W2Power, n.d.).

The continuous increase of floating off-shore wind farms in the last years is due to a better use of the wind resource in the marine environment, the lack of social opposition to this type of installations and the feasibility of implementing wind turbines in deep sea areas. Nevertheless, the design of floating wind turbines is more complex and its vulnerability increases due to the wave movement, thus influencing the wind energy generation and limiting the reliability of the wind turbine (Li et al., 2020, 2021; Liu et al., 2021; Moghaddam et al., 2020; Okpokparoro and Sriramula, 2021; Zhang et al., 2016).

Among the problems a floating off-shore wind turbine (FOWT) shows due to its configuration, the effects of wave movement onto the performance are of high relevance (Xu et al., 2019; Yue et al., 2020) because of the continuous misalignment of wind turbine mast to the wind direction. This situation not only may affect the efficiency of wind energy conversion, but also increase the wind turbine degradation due to additional loads (Chen and Basu, 2018; Dai et al., 2018; Park et al., 2020; Wen et al., 2020a, 2020b, 2020c; Xu et al., 2019; Zhang et al., 2019).

Floating off-shore wind farms are, in opposition to conventional off-shore wind farms, much cheaper since the wind turbine does not require to be anchored to the seabed. However, a FOWT requires a roll compensation mechanism that makes the design more complex. Four solutions have arisen to avoid a rigid anchoring system to the seabed, Semi-submersible, Spar, Tension Leg Platform and Barge, all of them including a type of connecting line to the seabed (Collu and Borg, 2016; Floating Windturbine, n.d.; Liu et al., 2016; Ma et al., 2019; Odijie et al., 2017). These systems avoid an excessive rolling of the wind turbine structure, but they are subject to the wave movement anyway.

Although many studies have already been devoted to analyze the influence of waves onto the performance of the FOWT, none of them has been focused on a compensation mechanism to make the FOWT operating in similar conditions to a steady off-shore wind turbine (OWT). By doing this, a simpler analysis of the aerodynamic conditions can be applied to the FOWT response to wave movement. This mechanism should maintain the aerodynamic surface the FOWT shows to the incoming wind by modifying the aerodynamic profile of the rotor blade with the wave movement.

Wave movement

Wave formation on the sea surface is mainly due to the wind action generating the characteristic sinusoidal pattern. Depending on wind conditions the sea surface profile is smooth or rough. Roughness depends on how much the wind is blowing creating turbulences onto the sea surface; these turbulences are directly related to the own wind turbulences. The sea surface roughness alters the wind speed value, thus the wind power onto the wind turbine rotor, what causes a reduction in the energy generation; therefore, the mutual interaction between wind and sea surface should be characterized to determine how much the wave movement affects the wind energy. Wind turbulence reduces its intensity over the sea level because of the lower roughness (Figure 1).

Figure 1.

Wind turbulence dependence on type of terrain.

Wind turbulence can be defined mathematically in terms of sea surface roughness as (Burton et al., 2011):

z_{o} (\bar{U}) = \frac{A_{c}}{g} {(\frac{κ \bar{U}}{\ln (z_{hub} / z_{o} (\bar{U}))})}^{2}

(1)

Where A_c is the Charnock constant, $\bar{U}$ the average wind speed, z_hub the position of the wind turbine hub over the sea level, z_o the sea surface roughness and κ is the von Karman (Slotta, 1962) constant whose typical value is 0.4. The Charnot constant values are in the range from 0.011 for open sea to 0.034 for the nearby coastline (Charnock, 1864).

Sea surface is the result of the overlapping of a great number of regular waves of different amplitude and period moving at a specific velocity. The wave group can be described in terms of its amplitude through a frequency spectrum. The main parameters are the wave height, H, defined as the average height of the third of tallest waves in a specific period of time, and the frequency corresponding to the peak of spectral density, f_p. The wave height is what it can be seen while the area covered by the spectral function matches the variance of the surface elevation, σ_η². We can use the Pierson-Moskowitz spectrum (Teixeira and Murilo, 2015) that is based on the fact that the significant wave height is four times the standard elevation of the wave surface, which is true for a Rayleigh distribution; mathematically:

S_{η} (f) = \frac{5}{16} \frac{H_{s}^{2}}{f} {(\frac{f_{p}}{f})}^{4} \exp [- \frac{5}{4} {(\frac{f_{p}}{f})}^{4}]

(2)

Sea surface state

We are going to divide the state of the sea surface in two, calm sea and rough sea. Calm sea can be defined in terms of a stochastic model based on a Pierson-Moskowitz spectrum; for calm sea the height of a representative wave should be taken as the expected value for a given wind speed, using the local database. Besides, the period of the spectral peak, T_p, the inverse of the frequency corresponding to the maximum of the frequency spectrum value, and the mean zero-crossing period, T_z, can be related through the equation:

T_{p} = T_{z} \sqrt{\frac{11 + γ}{5 + γ}}

(3)

Where the coefficient γ can be obtained from:

γ = \exp (5.75 - 1.15 T_{p} / H_{s}^{0.5})

(4)

The rough states of the sea, in which the wind turbine continues operating, are distinguished from the normal states only at the significant wave height. For each mean value of the velocity U, the significant wave height H_s(U) should be taken at a value such that its recurrence period in combination with the mean velocity of the wind is 50 years, based on the extrapolation of the database of the tested area.

For rough sea, the severe wave height, H_SWH(U) is the expected height of the tallest wave. Considering a Rayleigh distribution, the probability for a wave to overpass H is given by:

P (h > H) = \exp [- 0.5 {(\frac{H / 2}{σ_{n}})}^{2}] = \exp [- 2 {(\frac{H}{H_{s}})}^{2}]

(5)

Environmental outline

One of the most representative cases for wind energy generation is the 50 years returning combination of the wind speed together with the wave height. From these two parameters we can define an environmental outline that allows a probabilistic study. In a general way, the probability of not overpassing the average wind speed and wave height at the same time can be expressed as:

P = F (H_{s} | \bar{U}) F (\bar{U})

(6)

Where F is the cumulative distribution function.

For a returning period of 50 years, the probability adopts the value

P = (1 - 1 / N)

(7)

Where N is the number of states of the sea in the 50 years period, whose value is N = (50 × 365 × 24/3) = 146,000, considering every state last for 3 hours.

The environmental outline can be obtained converting the wind speed and wave height into two non-correlated variables, U₁ and U₂ whose distribution is normalized.

F (\bar{U}) = Φ (U_{1}); F (H_{s} | \bar{U}) = Φ (U_{2})

(8)

Where:

Φ (U) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{U} \exp (- u^{2} / 2) du

(9)

Therefore:

P (U_{1}, U_{2}) = \frac{1}{\sqrt{2 π}} \exp (- \frac{U_{1}^{2} + U_{2}^{2}}{2})

(10)

It can be observed that the probability defined in equation (10) is axially symmetric respect to the origin of the coordinates U₁ and U₂. A graph representation has been pictured in Figure 2.

Figure 2.

Graph representation of the environmental outline.

Now considering the returning average wind speed for the year N, we have:

F (\bar{U}) = 1 - 1 / N \to U_{1} = Φ^{- 1} (1 - 1 / N)

(11)

In the defined environmental outline the probability that the wind speed exceeds the returning value for the year N is represented by the probability of finding the point U₁, U₂ on the right side of the ABC line. In the same way, all points placed at equal distance from the axis U₂ have the same probability that the wave height exceeds the returning value for the year N.

Wind forces on the rotor blade

The three main forces acting on the rotor blade can be mathematically defined in the following way:

Lift force : F_{L} = \frac{1}{2} C_{L} ρ {AU}_{rel}^{2}

(12a)

Thrust force : F_{T} = B P_{2} S_{N}

(12b)

Drag force : F_{D} = \frac{1}{2} C_{D} ρ A {(U - Ω r)}^{2}

(12c)

Where C_L and C_D are the lift and drag coefficients, A is the wind turbine rotor area, ρ the air density, U and U_rel the absolute and relative wind speed, Ω the angular speed, r the radius of the rotor, B the number of blades, S_N the rotor area perpendicular to the wind direction, and P₂ the wind pressure onto the rotor blades. Figure 3 shows the drawing of the acting forces onto the wind turbine rotor blade.

Figure 3.

Schematic representation of the acting forces onto the wind turbine rotor blade.

Wave effects

The main effect of the waves onto the wind turbine is changing the effective area of the rotor due to a change in the relative direction of the wind onto the wind rotor plane; the effective area can be defined as:

A (γ) = A_{o} \vec{r} \cdot \vec{v} = A_{o} \cos γ

(13)

Where γ is the angle of the relative direction of wind.

The aerodynamic effect of the wind onto the rotor blade depends on γ, but also on the rotor blade profile and on the chord line; this makes difficult to find a simple mathematical relation to define the aerodynamic effect dependence on the γ angle. Nevertheless, this can be made using the projection effect in the measurements obtained at different inclinations of the rotor (Figure 4).

Figure 4.

Vector representation of the wave effect onto the wind turbine (a and b): vertical; (c and d): tilted at γ = −10º.

Figure 4 shows the schematic representation of the vector analysis on the wind rotor plane; black line represents the torsion plane, dark blue arrow the relative wind direction to the horizontal plane, yellow arrow the angular speed direction, light blue arrow the relative wind direction to the vertical plane, and red arrow the real relative wind direction. α represents the angle of attack. It can be observed that α changes with the relative position of the wind turbine respect to the vertical plane. To obtain α, the following expression has been used:

\cos α = \frac{{\vec{U}}_{rel} \cdot R_{γ} \cdot {\hat{P}}_{T}}{| {\vec{U}}_{rel} |}

(14)

Where Rγ is the rotational operator for γ = −10° and P_T is the edge of the torsion plane.

Marine surface characterization

The wave shape can be described through the mathematical expression:

Y (x, t) = - \frac{H}{2} \sin (\frac{2 π}{T} t - \frac{2 π}{λ} x + ψ)

(15)

Where T is the period, λ the wave length and ψ the initial phase at t = 0.

Assuming the tilt angle of the wind turbine and wave are equal, the wave lag is zero, and setting up x = 0, we can establish:

\tan γ = \frac{\partial Y}{\partial x} = π \frac{H}{λ} \cos (\frac{2 π}{T} t)

(16)

The two key parameters in equation (16) move in the range T [10–60] and H/λ [0.02–0.5]. This last parameter corresponds to the interval of waves from 1 m high and 250 wave length to 25 m high and 50 m wave length. The time evolution of the tilt angle can be obtained from equation (16) as:

\frac{d γ}{dt} = - \frac{\frac{2 π^{2} H}{λ T} \sin (\frac{2 π}{T} t)}{1 + {[\frac{π H}{λ} \cos (\frac{2 π}{T} t)]}^{2}}

(17)

Whose maximum value is attained at:

{(d γ / dt)}_{max} = 2 π^{2} H / λ T

(18)

Now representing the tilt angle evolution as a function of H/λ (Figure 5):

Figure 5.

Evolution of wind turbine tilt angle.

To maintain the relative wind direction respect to the rotor blade the pitch angle must be changed as the wind turbine angle does; this forces to include a continuous pitching variation that requires a specific control unit. Considering the wind turbine does not stand for tilt angles over 7°, we can establish the following condition:

H / λ = \tan 7 ° / π

(19)

and taking into account equation (17) the maximum angular speed is given by:

ω_{max} = 0.7718 / T

(20)

this expression allows to determine the maximum lineal speed of the wind rotor provided the rotor blade length is known.

Power generation

Power generation in a wind turbine strongly depends on the lift force, which has been defined in equation (12a). Assuming the rotor area is fixed and for a given wind speed, neglecting changes in air density, the lift force only depends on the lift coefficient that depends on the attack angle (Figure 6). The variation of the lift coefficient with the angle of attack can be correlated to a fifth degree polynomial function of the type:

C_{L} = 2.973 x 1 0^{- 6} α^{5} - 1.378 x 10^{- 4} α^{4} + 1.753 x 10^{- 3} α^{3} + 4.748 x 10^{- 3} α^{2} + 0.026 α + 0.511

(21)

Figure 6.

Lift coefficient vs. angle of attack (Haque et al., 2015).

That shows a regression coefficient of R² = 0.9893

The value of C_L for the critical attack angle (α_o = 13.2°) is C_L = 1.067

On the other hand, mechanic power generation can be related to the lift coefficient through the expression:

P = \frac{1}{2} C_{L} ρ A^{2} U_{rel}^{2} r ω

(22)

Electric power generation can be defined from mechanical power as:

P_{el} = P_{N} (C_{L} / C_{LC})

(23)

Where P_N is the nominal wind turbine power and C_LC is the maximum lift coefficient.

Angle of attack

To calculate the angle of attack is necessary to determine the relative wind speed vector and the chord line vector, both depending on time and distance to the hub; since the relative wind speed is given by the difference between the wind speed and the rotor blade linear velocity:

\begin{matrix} {\vec{U}}_{rel} = \vec{U} - \vec{v_{p}} (r, t) = R_{γ} (t) R_{ω} (t) [\begin{matrix} \to & \to \\ \hat{ω} & r \end{matrix}] = \\ = (\begin{matrix} \cos γ & - \sin γ & 0 \\ \sin γ & \cos γ & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix} 1 & 0 & 0 \\ 0 & \cos (ω t) & \sin (ω t) \\ 0 & - \sin (ω t) & \cos (ω t) \end{matrix}) (\begin{matrix} 0 \\ - ω r \sin (ω t) \\ - ω r \cos (ω t) \end{matrix}) = \\ = (\begin{matrix} U - ω r \sin γ \sin (ω t) \\ ω r \cos γ \sin (ω t) \\ ω r \cos (ω t) \end{matrix}) \end{matrix}

(24)

Where R_γ(t) and R_ω(t) are the rotation operator in the Z and YZ plane.

The chord line vector depends on the pitch angle, which for the optimum position of the wind rotor can be expressed as:

θ_{opt} (r) = φ (r) - α_{o}

(25)

Where α_o is the critical attack angle, and φ(r) is the incidence angle of wind, given by:

φ (r) = \arctan (U / ω r)

(26)

Considering the pitch angle remains constant along the rotor blade length:

〈 φ 〉 = \frac{\int_{0}^{L} \arctan (U / ω r) dr}{L}

(27)

Equation (25) can be transformed into:

θ_{opt} (r) = 〈 φ 〉 - α_{o}

(28)

The chord line direction can, therefore, be defined as:

\begin{matrix} {\vec{L}}_{C} (t) = R_{γ} (t) R_{ω} (t) (- \vec{k} - \tan (θ_{opt}) \vec{i}) = \\ (\begin{matrix} - \tan (θ_{opt}) \cos γ + \sin (ω t) \sin γ \\ - \tan (θ_{opt}) \sin γ - \sin (ω t) \cos γ - \cos^{2} (ω t) [(1 - \cos γ) \sin (ω t) - \tan (θ_{opt}) \sin γ] \\ - \cos (ω t) + (\sin (2 ω t) / 2) [(1 - \cos γ) \sin (ω t) - \tan (θ_{opt}) \sin γ] \end{matrix}) \end{matrix}

(29a)

Taking into account the projection over the rotor plane, equation (29a) reduces to:

{\vec{L}}_{C} (t) = (\begin{matrix} - \tan (θ_{opt}) \cos γ + \sin (ω t) \sin γ \\ - \tan (θ_{opt}) \sin γ - \sin (ω t) \cos γ \\ - \cos (ω t) \end{matrix})

(29b)

The angle of attack can be defined, then, as:

α (r, t) = π - \arccos (\frac{{\vec{L}}_{C} (t) \cdot {\vec{U}}_{rel} (r, t)}{| {\vec{L}}_{C} (t) | \cdot | {\vec{U}}_{rel} (r, t) |})

(30)

Since the oscillation time of the wind turbine and rotational time of the wind rotor are different, equation (30) should be transformed into:

α (r, t, t') = π - \arccos (\frac{{\vec{L}}_{C} (t, t') \cdot {\vec{U}}_{rel} (r, t, t')}{| {\vec{L}}_{C} (t, t') | \cdot | {\vec{U}}_{rel} (r, t, t') |})

(31)

Where t and t’ account for the oscillation and rotational time.

Now averaging the angle of attack:

〈 α (t) 〉 = \frac{1}{π L / ω} \int_{0}^{2 π / ω} \int_{0}^{L} [π - \arccos (\frac{{\vec{L}}_{C} (t, t') \cdot {\vec{U}}_{rel} (r, t, t')}{| {\vec{L}}_{C} (t, t') | \cdot | {\vec{U}}_{rel} (r, t, t') |})] drdt'

(32a)

Equation (32a) tends to overestimate the value of the attack angle due to the lack of symmetry along the turning of the rotor; to avoid this effect the integration has been divided in two, each one corresponding to a half cycle; therefore:

\begin{matrix} {〈 α (t) 〉}_{1} = \frac{1}{π L / ω} \int_{0}^{π / ω} \int_{0}^{L} [π - \arccos (\frac{{\vec{L}}_{C} (t, t') \cdot {\vec{U}}_{rel} (r, t, t')}{| {\vec{L}}_{C} (t, t') | \cdot | {\vec{U}}_{rel} (r, t, t') |})] drdt' \\ {〈 α (t) 〉}_{2} = \frac{1}{π L / ω} \int_{π / ω}^{2 π / 2 ω} \int_{0}^{L} [π - \arccos (\frac{{\vec{L}}_{C} (t, t') \cdot {\vec{U}}_{rel} (r, t, t')}{| {\vec{L}}_{C} (t, t') | \cdot | {\vec{U}}_{rel} (r, t, t') |})] drdt' \end{matrix}

(32b)

Since equation (32b) is of great complexity, a numerical method has been applied using Maple 18 from Maplesoft and the Monte Carlo integration method with a tolerance of 0.0005

Lift coefficient

The average lift coefficient can be defined as:

C_{L} (〈 α (t) 〉) = \frac{C_{L} ({〈 α (t) 〉}_{1}) + C_{L} ({〈 α (t) 〉}_{2})}{2}

(33)

Where the C_L(α) function has been defined in equation (21).

To solve equation (32b) we have developed a numerical simulation using the following data (Table 1):

Table 1.

Data base for the numerical simulation of the attack angle.

U	7 m/s	P_N	4.2 MW	< φ >	21.0608°
ω	−0.810811 rad/s	L	73.7 m	θ_opt	7.85584°
T_ω	7.7178484 seconds	I_P	14,000 kg·m²	T	0.1 second

T and T_ω are the wave and turbine rotor period, and I_p the inertia momentum of the rotor blade.

FOWT power

The power generated by a floating off-shore wind turbine can be expressed as:

\begin{matrix} P_{FOWT} = \frac{1}{2} C_{p} ρ A_{o} U^{3} \cos γ = \frac{1}{2} C_{p} ρ A_{o} U^{3} \cos {\arctan [π \frac{H}{λ} \cos (\frac{2 π}{T} t)]} = \\ = \frac{1}{2} \frac{C_{p} ρ A_{o} U^{3}}{\sqrt{1 + {[π \frac{H}{λ} \cos (\frac{2 π}{T} t)]}^{2}}} \end{matrix}

(34)

An expression that allows to determine the FOWT power generation as a function of the wave parameters.

Combining equations (21), (24), and (34), we have calculated the power generation as a function of the wave oscillation time. The results are shown in Figure 7.

Figure 7.

Simulated values of power generation by FOWT.

The analysis of the data from Figure 7 shows the maximum power occurs at zero tilt, while the minimum happens at the maximum tilt, in perfect agreement with expected results.

Using the aforementioned values, and converting power into energy, we obtain for T = 10, ξ =41.718 MJ = 11.588 kWh. When comparing this value with a conventional on-land wind turbine of same characteristics, we obtain 42 MJ, what means that the FOWT power generation is 0.671% lower. This agreement proves the validity of the simulation model.

Using a pitching compensation system that maintains the attack angle constant, and reproducing the calculations for α constant, we have obtained:

Using values from Figure 8, we obtain a global energy of 41.843 MJ, or 11.623 kWh, a 0.3% higher than in the case of no pitching compensation.

Figure 8.

Simulated values of power generation by FOWT with pitching compensation system.

Assuming the wave and average wind speed are constant over a 1 hour period, we can extrapolate to obtain the increase in energy using the pitching compensation system. A simple calculation gives:

Δ ξ_{1 h} = (ξ_{c} - ξ_{o}) (t / T) = (0.035 kWh) (3600 / 10) = 12.6 kWh

(35)

And normalizing this value per unit of power, it results:

Δ ρ_{ξ} = \frac{(ξ_{c} - ξ_{o}) (t / T)}{P_{N}} = \frac{12.6 kWh}{4.2 MW} = 3 \frac{kWh}{MW}

(36)

The so presented values are valid for the specific simulated test, but can be applied to other situation simply applying the corresponding values. The problem, however, is only partially addressed, since there are effects that cannot be included due to the extreme complexity of the equations.

Analysis of feasibility

The pitching compensation system requires extra energy to operate continuously; this energy must fulfill the condition of requiring less power than generated by the compensation mechanism, otherwise the energy balance is negative. Provided this condition is fulfilled, the use of a pitching compensation system in a continuous way may cause a severe degradation of the mechanism, resulting in an early replacement, what in the marine environment might represent a serious problem; therefore, the feasibility of installing a pitching compensation mechanism should be carefully evaluated taking into consideration not only the extra energy generation, but the maintenance requirements and the cost of replacement. Anyway, this is a promising solution to increase the power and energy generation in FOWT if the pitching compensation system is reliable and cheap.

Conclusions

A theoretical model to analyze the performance of a FOWT subject to wave oscillation has been developed.

A mathematical function to evaluate the influence of the wave oscillation onto the FOWT performance and power and energy generation has been obtained from the analysis of the effects the wave oscillation has onto the wind attack angle.

A pitching compensation system is proposed to optimize the performance of the FOWT; this system maintains the attack angle constant at its optimum value. The analysis has proved the feasibility of correcting the pitch angle to optimize the attack angle, thus the power and energy generated by the wind turbine.

The simulated power and energy generation results have been compared to the ones obtained from a non-compensated pitch angle, showing an improvement of 0.3%. Although this value looks of little significance, it may represent a considerable extra energy in large FOWT.

Tested simulation has produced an increase of 3 kWh/MW that may represent a global amount of 1.5 MWh in a marine wind farm of medium size.

The study concludes that the continuous use of pitching compensation mechanisms may cause a degradation in the system forcing to its early replacement, what is marine turbine may represent a problem.

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

Carlos Armenta-Déu

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