Fuzzy PID Individual Pitch Control with Effective Wind Speed Estimation for Offshore Floating Wind Turbines

Abstract

Individual Pitch Control (IPC) is a crucial mechanism for mitigating asymmetric loads in offshore floating wind turbines (OFWTs). Conventional IPC systems face significant limitations in wind speed estimation accuracy and control strategy robustness, leading to load fluctuations and power degradation. To address these challenges, this study proposes a novel IPC system incorporating an innovative effective wind speed estimation method and a fuzzy PID control strategy. The wind speed estimation is achieved using polynomial fitting of the tip speed ratio and pitch angle. The fuzzy PID control strategy for IPC employs variable control gains calculated based on wind speed, azimuth angle, and blade root loads. To verify the performance of the proposed control system, it is compared against the baseline control system implemented in the OpenFAST software v1.0.0 by a case study of the NREL 5MW OFWT. Results demonstrate that the proposed system has high accuracy in wind speed estimation and maintains rated power output while reducing blade flapwise and pitching moments. Notably, the proposed EWSE has a 53.1% improvement in median error and a 19.23% improvement in data error threshold compared with a reference EWSE. Under strong turbulent conditions (15% turbulence intensity), the proposed system achieves a reduction of 17.9% in flapwise moment and 12.9% in pitching moment compared with a baseline controller.

1. Introduction

The increased geometry of offshore floating wind turbine (OFWT) blades has led to significantly higher asymmetrical loads, which in turn shorten the service life of the OFWTs [1,2,3,4]. This negative impact can be mitigated through the development of an individual pitch control (IPC) system [5,6,7]. There are two key issues to develop such an IPC system: the obtainment of wind speed data and the development of a control strategy.

For the first key issue, the conventional method for measuring wind speed resorts to a single-point measurement of wind speed sensors [8,9]. However, the single-point measurement cannot accurately obtain the effective wind speed of the rotor due to uneven wind speed distribution, which leads to wrong pitch actions [10,11,12]. Thus, several effective wind speed estimation (EWSE) methods were proposed [13]. The EWSE methods are roughly classified into four groups (see

Table 1. EWSE methods comparison.

Method	Calculation Speed	Accuracy
Statistical law-based	medium	low
Kalman filter-based	medium	medium
Neural network-based	low	high
The present study	high	medium

) [14]. The first group is based on statistical law [15,16]. This kind of method focuses on analyzing the probability distribution of wind speed, which suffers from low computational efficiency. The second group is based on Kalman filters [17,18]. This kind of method focuses on linearizing the whole system model, which exhibits poor performance under an unsteady wind field. The third group is based on neural networks [19]. This kind of method focuses on sampling data to develop a real-time training model, which is hard to implement in practical applications. The fourth group is based on polynomial fitting [20,21]. Such methods typically concentrate on polynomial fitting of wind speed and pitch angle at rated rotor speed, offering rapid computational efficiency, strong feasibility, and excellent performance in unsteady wind fields. However, the fluctuations of rotor speed were seldom considered in these previous studies, which have been shown to reduce estimation accuracy [22]. From this perspective, an EWSE method based on polynomial fitting considering the fluctuation of rotor speed is proposed in the present study to enhance the estimation accuracy.

For the second key issue, the related investigations can be traced back to Bossanyi’s studies [23,24], where IPC systems were realized by single-variable control strategies. However, these single-variable IPC strategies cannot directly manage the coupling relationships between azimuth angle and loads in complex OFWT systems, which often leads to poor control performance [25]. To enhance the control performance of OFWTs, some researchers have adopted advanced multivariable IPC strategies [26,27,28]. For instance, Zhang and Plestan proposed an adaptive second-order sliding mode control strategy to realize an IPC system in an OFWT, which was proven to achieve superior control performance. However, under highly turbulent environments, these IPC strategies often exhibit undesirable control performance. The reason is that these control strategies are subject to poor robustness. As an alternative solution, the fuzzy PID control strategy offers strong robustness [29,30,31], making it a promising choice for IPC systems.

Based on the aforementioned literature review and discussions, the present study presents a novel IPC system that offers distinct innovations and practical application value through two primary contributions: an effective wind speed estimation (EWSE) method and a fuzzy PID control strategy. Specifically, an EWSE method is proposed to address the limitations of conventional approaches by introducing a rotor speed fluctuation compensation mechanism with a fifth-order polynomial fitting of the tip speed ratio (TSR) and pitch angle. Furthermore, a real-time adaptive fuzzy PID control strategy for IPC is developed for the IPC system with high robustness, which dynamically adjusts gains based on Mamdani fuzzy reasoning and the centroid method. Then, numerical simulations are carried out to verify the performance and practical value of the proposed IPC system in the OpenFAST software. These contributions will provide tangible benefits for both IPC system development and the safe operation assurance of the OFWTs.

The rest of this paper is organized as follows: The EWSE method is introduced in Section 2. The fuzzy PID control strategy for IPC is described in Section 3. The proposed IPC system is verified in Section 4. Finally, the conclusions are presented in Section 5.

2. EWSE Method

The IPC system (illustrated in Figure 1) includes three components: the EWSE method, the collective pitch control (CPC) unit based on PID, and the fuzzy PID control strategy (described in Section 3). As the first component, the EWSE method is essential for the accuracy of the input data. In this section, an EWSE method is constructed by polynomial fitting, which considers the fluctuation of rotor speed to provide a precise wind speed signal.

The wind power P passing through the swept area of the rotor can be calculated as follows [32]:

$$P={\displaystyle \frac{1}{2}}\rho \mathsf{\pi }{R}^2{V}^3$$

(1)

where ρ, R, and V are the density of air, the rotor radius, and the wind speed, respectively.

The power P_r of OFWTs can be calculated as follows:

$$P_{\mathrm{r}}=C_P\left(\beta ,\lambda \right)\times P$$

(2)

where C_P(β, λ) is the power coefficient; β is the pitch angle; λ is the TSR and is calculated using the following:

$$\lambda ={\displaystyle \frac{{\omega }_{\mathrm{r}}R}{V}}$$

(3)

where ω_r is the rotor angular speed.

By combining Equations (1)–(3), one can get the following equation:

$$C_P(\beta , \lambda )={\displaystyle \frac{2{\lambda }^3{P}_{\mathrm{r}}}{\rho \pi R^5{\omega }_{\mathrm{r}}^3}}$$

(4)

Generally, the C_P(β,λ) can be given as follows [33]:

$$\left\{\begin{array}{c}{C}_P(\beta , \lambda )=c_0({\displaystyle \frac{c_1}{\hat{\lambda }}}-c_2\beta -c_3{\beta }^{c_4}-c_5)e^{-{\displaystyle \frac{c_6}{\hat{\lambda }}}}+c_7\lambda \\ {\displaystyle \frac{1}{\hat{\lambda }}}={\displaystyle \frac{1}{\lambda +d_0\beta +d_1}}-{\displaystyle \frac{d_2}{{\beta }^3+1}}\end{array}\right.$$

(5)

where c₀, c₁, …, c₇ and d₀, d₁, d₂ are dimensionless coefficients, which can be obtained by controlling the pitch angle in various wind speeds above the rated value [34].

By combining Equations (4) and (5), λ can be regarded as the zero point of a function f(λ). f(λ) can be drawn as follows:

$$f(\lambda )=c_0({\displaystyle \frac{c_1}{\hat{\lambda }}}-c_2\beta -c_3{\beta }^{c_4}-c_5)e^{-{\displaystyle \frac{c_6}{\hat{\lambda }}}}+c_7\lambda -{\displaystyle \frac{2{\lambda }^3{P}_{\mathrm{r}}}{\rho \mathsf{\pi }{R}^5{\omega }_r^3}}$$

(6)

A bisection method is adopted to calculate the root λ in Equation (6). Considering the fluctuation of rotor speed, the initial interval of the bisection procedure is defined as follows:

$$\left\{\begin{array}{c}{\lambda }_0\in (\lambda (\beta )-c, (\lambda (\beta )+c){\displaystyle \frac{{\omega }_r}{{\omega }_{rated}}}), {\omega }_r\ge {\omega }_{rated}\\ {\lambda }_0\in ((\lambda (\beta )-c){\displaystyle \frac{{\omega }_r}{{\omega }_{rated}}}, \lambda (\beta )+c), {\omega }_r<{\omega }_{rated}\end{array}\right.$$

(7)

where ω_rated is the rated rotor angular speed; c is a correction coefficient; the initial value λ(β) can be calculated as the follows:

$$\lambda (\beta )=b_0+b_1\beta +b_2{\beta }^2+\cdot \cdot \cdot +b_n{\beta }^n$$

(8)

where b₀, b₁, …, b_n are dimensionless coefficients.

Then, the real-time inflow wind speed V_in is calculated as follows:

$$V_{\mathrm{in}}={\displaystyle \frac{{\omega }_{\mathrm{r}}R}{\lambda }}$$

(9)

Finally, considering the effect of platform motion, the effective wind speed V_E can be calculated as follows [35]:

$$V_{\mathrm{E}}=V_{\mathrm{in}}-V_{\mathrm{motion}}$$

(10)

where V_motion is the speed of platform motion in the inflow wind speed direction, with the center of the rotor as the reference point.

3. Control Strategy Design

3.1. Multi-Blade Coordinate Transformation

The asymmetric loads vary periodically with the azimuth angle of the blades, which brings difficulties to the development of the IPC strategy. To address this issue, the multi-blade coordinate (MBC) transformation is adopted to convert the loads into a time-invariant linear system as follows:

$$\left[\begin{array}{c}{M}_{\mathrm{tilt}}\\ M_{\mathrm{yaw}}\end{array}\right]={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}\sin {\alpha }_1& \sin ({\alpha }_1+{\displaystyle \frac{2}{3}}\mathsf{\pi })& \sin ({\alpha }_1+{\displaystyle \frac{4}{3}}\mathsf{\pi })\\ \cos {\alpha }_1& \cos ({\alpha }_1+{\displaystyle \frac{2}{3}}\mathsf{\pi })& \cos ({\alpha }_1+{\displaystyle \frac{4}{3}}\mathsf{\pi })\end{array}\right]\left[\begin{array}{c}{M}_1\\ M_2\\ M_3\end{array}\right]$$

(11)

where M_tilt and M_yaw are the tilting moment and the yawing moment, respectively; M_i (i = 1, 2, 3) is the root bending moments of the blade in the rotating coordinate system; α₁ is the azimuth angle of blade 1.

The moments M_tilt and M_yaw are fed into the fuzzy PID controller unit (described in Section 3.2) to obtain the pitch angle increments β_tilt and β_yaw. Then, an inverse MBC transformation is adopted to calculate the individual pitch angle differential value as follows:

$$\left[\begin{array}{l}\Delta {\beta }_1\\ \Delta {\beta }_2\\ \Delta {\beta }_3\end{array}\right]=\left[\begin{array}{cc}\begin{array}{c}\sin {\alpha }_1\\ \sin ({\alpha }_1+{\displaystyle \frac{2}{3}}\mathsf{\pi })\\ \sin ({\alpha }_1+{\displaystyle \frac{4}{3}}\mathsf{\pi })\end{array}& \begin{array}{c}\cos {\alpha }_1\\ \cos ({\alpha }_1+{\displaystyle \frac{2}{3}}\mathsf{\pi })\\ \cos ({\alpha }_1+{\displaystyle \frac{4}{3}}\mathsf{\pi })\end{array}\end{array}\right]\left[\begin{array}{l}{\beta }_{tilt}\\ {\beta }_{yaw}\end{array}\right]$$

(12)

where ∆β_i (i = 1, 2, 3) is the individual pitch angle differential value of the blade.

3.2. Fuzzy PID Control Strategy for IPC

The mathematical expression of the fuzzy PID control strategy for the IPC shown in Figure 2 is as follows:

$$U(t)=K_{\mathrm{P}}e(t)+K_{\mathrm{D}}{\displaystyle \frac{de(t)}{dt}}+K_{\mathrm{I}}{\displaystyle {\int }_0^{t}e(\tau )}d\tau$$

(13)

where e(t) is the input error; K_P is the proportional gain; K_I is the integral gain; and K_D is the differential gain.

Fuzzification aims to transform input parameters into fuzzy subsets and membership functions for fuzzy reasoning by the membership value method. In this study, the input parameter X(t) is [M_tilt, M_yaw]^T and the output parameter U(t) is [β_tilt, β_yaw]^T. The domains of dimensionless error e and its gradient ec are both set as [−6, 6]. The fuzzy subsets E and EC are both set as {PB, PM, PS, NS, NM, NB, ZE}, where PB, PM, PS, NS, NM, NB, and ZE denote positive big, positive medium, positive small, negative small, negative medium, negative big, and zero, respectively. Meanwhile, the fuzzy domains of the output parameter are all set as [−3, 3]. The triangle-type membership function is adopted for PID calculation, as shown in Figure 3. And the fuzzy rules are shown in

Table 2. Fuzzy rules for calculating K_P/K_I/K_D.

KP/KI/KD		EC
		PB	PM	PS	ZE	NS	NM	NB
E	PB	PB/NS/PM	PB/NS/PS	PM/NM/ZE	NM/NM/NS	NS/NB/NM	ZE/NB/NM	ZE/NB/NB
	PM	PB/ZE/PM	PM/ZE/PM	PS/NS/PS	NM/NS/ZE	NS/NM/NS	ZE/NM/NM	PS/NB/NM
	PS	PM/PS/PB	PM/PS/PM	PS/ZE/PM	NB/ZE/PS	ZE/ZE/ZE	PS/NS/NS	PS/NS/NM
	ZE	PM/PB/PB	PS/PM/PB	ZE/PS/PM	ZE/ZE/ZE	ZE/PS/PM	PS/PM/PB	PM/PB/PB
	NS	PS/NS/NM	PS/NS/NS	NB/ZE/PS	NB/ZE/ZE	PS/ZE/ZE	PM/PS/PM	PM/PS/PB
	NM	PS/NB/NB	ZE/NM/NM	NS/NM/NM	NM/NS/NS	PS/NS/ZE	PM/ZE/PS	PB/ZE/PM
	NB	ZE/NB/NB	ZE/NB/NB	NS/NB/NM	NM/NM/NM	PM/NM/NM	PB/NS/ZE	PB/NS/PS

1. When E is PB and EC is PB, then K_P is PB, K_I is NS, K_D is PS, and others are similar. 2. PB, PM, PS, NS, NM, NB, and ZE denote positive big, positive medium, positive small, negative small, negative medium, negative big, and zero, respectively.

.

The Mamdani direct reasoning method is chosen to obtain the outputs fuzzy subsets as follows:

$$\alpha =({\mu }_e(i)\wedge C_{\mathrm{out}}(i))\wedge ({\mu }_{ec}(i)\wedge C_{\mathrm{out}}(i))$$

(14)

where α is the fuzzy subset of output parameter when the ith fuzzy rule is activated; μ_e(i) and μ_ec(i) are membership functions of e and ec obtained from Figure 3a under the ith fuzzy rule; C_out(i) is the fuzzy subset of output parameters obtained from Figure 3b under the ith fuzzy rule.

Finally, a clarification on the fuzzy is performed by the center of area method as follows:

$$K_{\mathrm{out}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^k\alpha (i)D_{\mathrm{out}}(i)}}{{\displaystyle \sum _{i=1}^k\alpha (i)}}}$$

(15)

where k is the element number of α. D_out is the output value corresponding to the fuzzy subset of the output parameter; K_out is the output of the fuzzy control strategy, that is, K_P, K_I, and K_D of the PID control strategy.

4. Case Study

The NREL 5 MW OFWT is taken as the reference OFWT with the basic parameters shown in

Table 3. The basic parameters of the NREL 5MW OFWT.

Parameters	Value
Rated power	5 MW
Rotor diameter	126 m
Rated wind speed	11.4 m/s
Rated rotor speed	12.1 rpm
Generator efficiency	94.4%
Cut-out wind speed	25 m/s

(more specifications can be found in References [36,37]).

4.1. Verification of the EWSE Method

A fifth-order polynomial is selected to fit the λ-β curve, balancing the reduction of overfitting with the need for precision. The optimal pitch angles for rated power at various inflow wind speeds are simulated by the AeroDyn module of the OpenFAST software. In the simulation, the inflow wind speed is set within the range of [11.4 m/s, 25 m/s], with an increment of 1 m/s as shown in

Table 4. The optimal pitch angles for rated power.

Inflow Wind Speed (m/s)	λ	β (rad)
11.4	7.002	0
12.0	6.652	0.065
13.0	6.141	0.112
14.0	5.702	0.149
15.0	5.322	0.180
16.0	4.989	0.208
17.0	4.696	0.234
18.0	4.435	0.258
19.0	4.201	0.280
20.0	3.991	0.302
21.0	3.801	0.323
22.0	3.629	0.345
23.0	3.471	0.367
24.0	3.326	0.387

. Then the coefficient b₀, b₁…, b_n in Equation (8) is obtained and listed in

Table 5. The coefficients of λ-β polynominals.

b0	b1	b2	b3	b4	b5
7.003	−0.772	−91.790	324.190	−489.409	294.210

.

To verify the performance of the proposed EWSE method (EWSE-current) under varying wind speeds, comparative numerical simulations are conducted between the conventional EWSE method used in OpenFAST software (EWSE-ref) and the EWSE-current. Simultaneously, the simulation configurations incorporate a linearly ramped inflow wind speed profile from 15 m/s to 20 m/s over a 300 s duration, while platform motion speed is randomly varied within the range of [−0.5 m/s, 0.5 m/s] to simulate the realistic dynamic conditions.

Figure 4 presents the comparative results of estimation accuracy between the estimations and the actual wind speed. As demonstrated in Figure 4a, the EWSE-current exhibits superior estimation performance in terms of accuracy and consistency. The quantitative error analysis in Figure 4b reveals that the median error of EWSE-current is 0.15 m/s, representing a 53.1% improvement over the EWSE-ref median error of 0.32 m/s. Notably, the EWSE-current demonstrates enhanced precision with only 1.63% of data points exceeding the 0.4 m/s error threshold, compared to 19.23% for EWSE-ref. These results collectively demonstrate the enhanced reliability of the proposed EWSE for wind speed tracking in dynamic operating scenarios.

4.2. Performance of Fuzzy PID Control Strategy for IPC

To comprehensively evaluate the control performance of the fuzzy PID control strategy with the proposed EWSE for IPC, the comparative simulations are conducted against the baseline controller (BC) by OpenFAST under extreme turbulent conditions compliant with IEC 61400-1 standards [38]. The average wind speed value of the numerical simulations is set as 16 m/s to ensure the OFWT operates in the above-rated region (Figure 5), and the six-degree-of-freedom platform motion speed fluctuates randomly between −0.5 m/s and 0.5 m/s.

The pitch control strategy is configured with operational constraints reflecting practical engineering requirements: a pitch angle range of [0, π/2] with a maximum rate limit of 0.21 rad/s. Simulation results demonstrate that the proposed fuzzy PID control strategy with the proposed EWSE maintains pitch rate within safe operational margins, with the maximum observed rate of 0.098 rad/s (46.7% of the design limit). Compared to the proposed control strategy with the EWSE-ref, the proposed control strategy with the proposed EWSE method has a reduction in pitch fluctuations, as evidenced in Figure 6a. Figure 6b reveals that the fuzzy PID control strategy achieves similar energy capture to the BC for IPC. Notably, compared to the control strategy with the EWSE-ref, the control strategy with the proposed EWSE demonstrates superior power stability (standard deviation: 113.91 kW vs. 121.96 kW).

Figure 7a shows that both the proposed fuzzy PID with EWSE-ref and the proposed fuzzy PID with the EWSE-current effectively reduce flapwise moment fluctuations compared to BC for IPC and a Model Predictive Control (MPC) for IPC. Results show that the BC for IPC has a better stability than the MPC for IPC and a poorer stability than the proposed fuzzy PID controller. Specifically, the standard deviations of BC for IPC and MPC for IPC are 1770.34 kN·m and 2841.83 kN·m, respectively, while the proposed fuzzy PID with EWSE-ref and the proposed fuzzy PID with the EWSE-current achieve reductions of 13.0% and 17.9% (compared with BC for IPC), respectively. Additionally, in the time range of 40 s to 180 s, the maximum flapwise moment is reduced from 8299.63 kN·m (BC for IPC) and 8050.35 kN·m (MPC for IPC) to 8020.26 kN·m (the proposed fuzzy PID with EWSE-ref) and 7356.81 kN·m (the proposed fuzzy PID with the EWSE-current), which exhibits lower peak values. Figure 7b shows the proposed fuzzy PID with the EWSE-current achieves the most stable performance in reducing the fluctuations of pitching moment. The standard deviations of the pitching moment for BC for IPC and MPC for IPC are 29.63 kN·m and 35.80 kN·m, respectively. Compared with BC for IPC, the proposed fuzzy PID with EWSE-ref reduces this value by 5.3%, while the proposed fuzzy PID with the EWSE-current further decreases it by 7.6%.

To further investigate the control efficacy of the proposed control strategy with the proposed EWSE method for IPC, numerical simulations are conducted against the BC across turbulence intensities spanning 0% to 20%.

As shown in Figure 8, compared with the BC for IPC, the proposed fuzzy PID control strategy significantly reduces the fluctuation of the moments over these cases. Notably, the proposed control strategy with both EWSE methods achieves similar performance in mitigating moment fluctuations in low turbulence intensities (0% to 5%). While in high turbulence intensity environments (turbulence intensity greater than 5%), compared with the proposed control strategy with the EWSE-ref, the proposed control strategy with the proposed EWSE has more precise wind speed estimation for IPC to achieve superior control performance (especially when turbulence intensity equals 15%). It is important to note that turbulence intensity exacerbates the fluctuation of the moments, which is a challenge for the robustness of the control strategy. Compared with the BC for IPC, the proposed control strategy can calculate the control gains (K_P, K_I, K_D) in real time to adjust the pitch angle. This capability provides strong robustness for the IPC system in high-turbulence environments. Consequently, in practical engineering applications, this strategy can be employed in the design of IPC systems to reduce loads and regulate power generation at rated levels, which are crucial for the service life and reliability of OFWTs.

5. Conclusions

This study presents an IPC system incorporating an innovative EWSE method and a fuzzy PID control strategy for IPC. The main conclusions are as follows:

(1) An EWSE method is developed by using polynomial fitting to guide the pitch control. A power coefficient error function is added to enhance the estimation accuracy. Simulation results demonstrate that the proposed EWSE method achieves superior estimation performance, with relative errors between the estimated and actual wind speed ranging from −3.0% to 2.9%. Comparative simulations reveal that the proposed method significantly outperforms conventional EWSE approaches, reducing the median estimation error by 53.1% (from 0.32 m/s to 0.15 m/s). Furthermore, the proposed method exhibits enhanced robustness, with only 1.63% of data points exceeding the 0.4 m/s error threshold, compared to 19.23% for the conventional method. Notably, the method proves highly effective in supporting the IPC system to maintain optimal control performance, even under high turbulence intensity conditions.

(2) A novel fuzzy PID control strategy for IPC is proposed to enhance the mitigation of asymmetrical loads. This strategy employs the Mamdani reasoning method and the centroid method for real-time PID gain calculation, enabling adaptive and precise control. Simulation results demonstrate that, compared to the BC for IPC, the proposed strategy with the proposed EWSE method significantly reduces flapwise moment fluctuations by 17.9% and pitching moment fluctuations by 12.9% when turbulence intensity equals 15%, while maintaining stable generator performance without noticeable degradation. Furthermore, the effectiveness of the proposed control strategy is validated across wind field cases with varying turbulence intensities (ranging from 0% to 20%). The results consistently show superior load mitigation performance, particularly under high turbulence intensities, where the control system achieves more precise wind speed estimation and enhanced control efficacy compared to the BC for IPC.

In conclusion, the proposed IPC system can effectively enhance wind speed estimation accuracy and reduce the moments of blades while maintaining the power output at the rated value.

Future work will focus on two aspects. First, additional influential factors, such as variation in air density and blade elastic deformation, should be integrated into the control system to further enhance the accuracy of the EWSE. Second, the proposed control system should be validated through wind tunnel tests or scaled OFWT model experiments.