Open AccessArticle

Investigation into the Yaw Control of a Twin-Rotor 10 MW Wind Turbine

Amira Elkodama

A. Abdellatif

S. Shaaban

Mostafa A. Rushdi

Shigeo Yoshida

^3,4,* and

Amr Ismaiel

^1,*

Faculty of Engineering and Technology, Future University in Egypt (FUE), 5th Settlement, New Cairo 11835, Egypt

Mechanical Engineering Department, Arab Academy for Science Technology and Maritime Transport, Cairo 11799, Egypt

Research Institute for Applied Mechanics (RIAM), Kyushu University, Fukuoka 816-8580, Japan

⁴

Institute of Ocean Energy (IOES), Saga University, Honjo-Machi, Saga 840-8502, Japan

Authors to whom correspondence should be addressed.

Appl. Sci. 2024, 14(21), 9810; https://doi.org/10.3390/app14219810

Submission received: 14 September 2024 / Revised: 13 October 2024 / Accepted: 19 October 2024 / Published: 27 October 2024

(This article belongs to the Section Energy Science and Technology)

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Browse Figures

Figure 1
Proposed TRWT configuration. "> Figure 2
Betz tube. "> Figure 3
WT mechanical system. "> Figure 4
Simplified DC motor closed-loop TF. "> Figure 5
DC motor with SMC and PID. "> Figure 6
Cp vs. TSR curve. "> Figure 7
Cp and λ Simulink results. "> Figure 8
Vw vs. Cp curve. "> Figure 9
Verification of Vw vs. mechanical power per rotor. "> Figure 10
Verification of Vw vs. thrust force per rotor. "> Figure 11
Twin-rotor mechanical outputs. "> Figure 12
Response of SMC vs. PID. "> Figure 13
SMC response. "> Figure 14
System’s chattering signal before applying SMC. "> Figure 15
Sliding mode surface control. "> Figure 16
Chattering signals before and after applying SMC control. ">

Versions Notes

Abstract

Multi-rotor system (MRS) wind turbines can provide a competitive alternative to large-scale wind turbines due to their significant advantages in reducing capital, transportation, and operating costs. The main challenges of MRS wind turbines include the complexity of the supporting structure, mathematical modeling of the aerodynamic interaction between the rotors, and the yaw control mechanism. In this work, MATLAB 2018b/Simulink^® software was used to model and simulate a twin-rotor wind turbine (TRWT), and an NREL 5 MW wind turbine was used to verify the model outputs. Different random signals of wind velocities and directions were used as inputs to each rotor to generate different thrust loads, inducing twisting moments on the main tower. A yaw controller system was adapted to ensure that the turbine constantly faced the wind to maximize the power output. A DC motor was used as the mechanism’s actuator. The goal was to achieve a compromise between aligning the rotors with the wind direction and reducing the torque induced on the main tower. A comparison between linear and nonlinear controllers was performed to test the yaw system actuator’s response at different wind speeds and directions. Sliding mode control (SMC) was chosen, as it was suitable for the nonlinearity of the system, and its performance showed a faster response compared with the PID controller, with a settling time of 0.17 sec and a very low overshoot. The controller used the transfer function of the motor to generate a sliding surface. The dynamic responses of the controlled angle are shown and discussed. The controller showed promising results, with a suitable response and low chattering signals.

Keywords:

multi-rotor systems; renewable energy; sliding mode control; wind energy; yaw control

1. Introduction

In recent decades, the consumption of non-renewable energy sources, such as fossil fuel, coal, petroleum, and natural gas, has increased rapidly due to population growth and technological innovations. Non-renewables are limited in their sources and have a significant negative impact on the environment due to greenhouse gas (GHG) emissions from burning fossil fuels and natural gas, which is reflected in climate change and global warming [1,2]. According to the Intergovernmental Panel on Climate Change (IPCC), CO₂ emissions have increased by 1.9% in the past three decades alone, and in 2007, a 40–110% increase in CO₂ emissions by 2030 was predicted [3]. Consequently, there have been great efforts to reduce GHG emissions by searching for alternative sources of energy production that are sustainable and environmentally friendly, in order to avoid the potential dangers of global warming and climate change, which are threats to the whole world. One of the most effective ways of eliminating fossil fuel usage is shifting toward using renewable energy (RE) sources [4], which have minimal negative impacts on the environment, as they are sustainable and clean [5]. Renewable energy sources, including biomass energy [6], tidal energy [7,8], solar energy [9,10,11], and wind energy [12,13], can be safe and economically feasible alternatives for generating power on a megawatt scale.

Currently, solar and wind energy sources are considered to be the cleanest, cheapest, and fastest-growing RE sources in the world, and they are expected to play a dominant role in the transition to fully RE systems that cover global energy needs [14,15]. A long time ago, it was concluded that 45% solar PV and 55% wind power would be the optimum balance for a 100% RE system [16]. Wind energy is considered to be one of the core pillars of RE sources, as it is the least polluting, most sustainable, efficient, and cost-competitive energy source compared with other sources [17]. The year 2022 was recorded as the third best year for wind power production, with a capacity of 78 GW added, resulting in a global cumulative wind power capacity of 906 GW, representing a year-over-year (YOY) growth of 9%. According to the 2023 Global Wind Energy Report (GWEC), it is expected that an additional capacity of 143 GW will be installed by the end of 2030 [18]. Therefore, wind energy is becoming an object of great research interest, and researchers are competing to find new sources and develop existing ones.

In order to benefit from wind energy and utilize it to its full potential, new and innovative wind energy systems have been developed [19], including some novel unconventional technologies such as airborne wind energy devices like giant kites [20], aerial vehicles [21], and wings [22], which are tethered to the ground and fly high in the air where the wind is stronger and more consistent.

Wind turbines (WTs) are conventional devices for harvesting energy from flowing wind, and they have the highest efficiency. Horizontal-axis wind turbines (HAWTs) are complex nonlinear systems with a single rotor consisting of three fiberglass blades that come in different sizes based on capacity. According to the general rule of WT power, the amount of energy produced is directly proportional to the cube of the wind speed and the swept area, which means that WTs of enormous sizes are required to produce more wind energy [23]. HAWTs such as the Haliade-X WT, which has a 220 m rotor diameter and a 107 m blade length, can typically produce up to 14 MW [24]. However, many major problems are associated with components of enormous size [25]. These problems include but are not limited to the design complexity due to stresses on longer blades and an increased rotor weight, which also increases the gearbox weight. These problems result in transportation, installation, and maintenance cost increases, with transportation accounting for about 10% of the capital cost of a WT, equivalent to USD millions [26,27].

In the 1930s, the concept of multi-rotor wind turbines (MRWTs) was first introduced by Honnef as an alternative solution to the complications of single-rotor wind turbines (SRWTs) [28]. MRWTs replace the large single rotor with multiple small rotors on the same support structure. MRWTs with “n” number of rotors showed a reduction in mass by a ratio of 1/

\sqrt{n}

compared with SRWTs with the same capacity [29]. Such mass reduction results in the use of less material, and hence, a reduction in production, transportation, installation, and maintenance costs.

Several studies have been performed on MRWT designs, considering mass and cost analyses, power performance enhancement, and aeroelasticity. Jamieson compared a 20 MW SRWT with two MRWT configurations: 4-rotor × 5 MW and 45-rotor × 444 kW. The results showed cost reductions of 20% and 30% for the 4-rotor and 45-rotor models, respectively [25]. Mate also demonstrated a 37% mass reduction and a 25% cost reduction for a five-rotor WT compared with a 5 MW SRWT [30]. Elkodama et al. reported reductions in mass of 25.6%, 16.9%, and 22.5% for twin-, tri-, and quad-rotor configurations, respectively, compared with an SRWT with a 5 MW capacity [31]. In light of this power production enhancement, an experimental study showed that two- and three-diffuser-augmented rotor configurations increased the power produced by 5% and 9% per rotor, respectively, compared with an SRWT [32]. In addition, a simulation study was performed on seven 2 MW rotors, using CFD and vortex models, and the results showed an increase in power of 3% compared with seven non-interacting single rotors [33]. Aeroelastic simulations performed by Ismaiel and Yoshida proved the advantage of MRWTs with regard to power production compared with SRWTs, along with the improved dynamic behavior of the support structure [34]. However, it was found that the twisting moment generated by the difference in thrust forces in each rotor could lead to structural issues and yaw misalignment.

Problem Statement and Contribution

Multi-rotor systems (MRSs) face many challenges, including the complexity of the support structure and the yaw control, so they are still being researched and have not yet been put into use commercially. The yaw system is responsible for directing the WT rotors to the direction of wind flow in order to extract the maximum power and to prevent failure of the WT due to misalignment with the flowing wind. Yaw control techniques have been developed over the years to optimize WTs’ produced power, as yaw misalignment significantly affects their power performance [35]. The challenges of MRWTs’ yaw systems include not only directing the rotors toward the wind flow but also overcoming changes in the thrust between the different rotors aligned at different distances from each other and from the main support tower, due to the nonlinearity of the flowing wind. This latter challenge was first observed in work carried out by Ismaiel and Yoshida [34], when only a slight phase change that occurred in the azimuth angle of the two rotors generated a twisting moment on the tower top. Therefore, a thorough investigation is required to develop and apply a yaw control algorithm to overcome this phenomenon.

This work presents the design and simulation of a yaw control system that manages to pivot a coplanar twin-rotor WT (TRWT) toward the wind while ensuring optimal output energy production. It also takes into consideration the impact of different thrust levels between the rotors. Due to the WT’s nonlinear behavior, SMC is used to control the yaw mechanism actuators, which are responsible for directing the WT’s rotors toward the wind flow. An NREL 5 MW WT rotor was used for the twin-rotor WT configuration, for its data availability [36]. Table 1 shows the main specifications of the NREL 5 MW rotor, and Figure 1 shows the proposed TRWT configuration.

In Figure 1, the proposed configuration of the TRWT includes two NREL 5 MW rotors placed on top of a T-shaped supporting structure. The rotors are connected by side booms such that the tips of the rotors are separated by a distance of 5% of the rotor diameter, to reduce the aerodynamic interaction between them. The main contribution of this work is to control the side booms under the effect of the thrust difference between the two rotors, which mainly operate under different wind conditions.

This manuscript is organized as follows: Section 2 shows the methodology followed for modeling the twin-rotor WT. Section 3 shows the yaw control formulation and mechanism. Section 4 shows the main results of the control mechanism, and the system stability analysis and response. Finally, Section 5 presents the main findings of this paper.

2. Twin-Rotor WT Model

This section presents the mechanical power and thrust calculation of the WT to design a yaw control for a twin-rotor WT that overcomes the thrust force difference between the two rotors, which affects the system’s performance. The MATLAB Simulink^® tool was used to model and simulate the overall NREL 5 MW WT system’s behavior.

WT devices convert the kinetic energy in flowing wind into electrical energy through the rotor’s shaft, which is connected to a generator, converting rotational mechanical power into electrical power. The mechanical output power of WTs is affected by several factors, as shown in Equation (1) [23]:

P_{m} = \frac{1}{2} ρ A {V_{w}}^{3} C_{p} (λ, β)

(1)

where P_m is the WT’s output mechanical power,

V_{w}

is the wind velocity,

ρ

is the air density, A is the rotor’s swept area, and

C_{p}

is the rotor’s power coefficient or efficiency, which is the ratio between the actual obtained power and the maximum power available in the wind.

The maximum value of

C_{p}

represents the maximum power the WT can produce. Theoretically, the maximum value of

C_{p}

, called the Betz limit, is 0.59, which indicates that the WT produces 59% of the available power in the flowing wind.

A WT’s power coefficient varies with the tip speed ratio (TSR) of the turbine (

λ)

and the blades’ pitch angles (β), as expressed in Equation (2) [37]:

C_{p} (λ, β) = 0.5176 \cdot (116 \frac{1}{λ_{i}} - 0.4 β - 5) e^{\frac{21}{λ_{i}}} + 0.0068 λ

(2)

where

\frac{1}{λ_{i}} = \frac{1}{λ + 0.08 β} - \frac{0.035}{β^{3} + 1}

(3)

and

λ = \frac{ω_{r} R R}{V_{w}}

(4)

where β is the blade pitch angle,

ω_{r}

is the rotational speed of the rotor, and RR is the rotor’s radius. Regarding the thrust load applied to the rotor plane from the flowing wind, the rotor is considered to be a permeable ideal disc that reduces the flowing air from V₀ before the rotor to V₁ at the rotor disc and to V₂ beyond the rotor, which is called the wake velocity, as shown in Figure 2.

Bernoulli’s equation can be applied to calculate the pressure drop, thus calculating the dynamic axial force applied to the rotor center. Assuming frictionless and incompressible flow and no external forces acting on the fluid, Equation (5) represents Bernoulli’s equation:

\frac{P_{0}}{ρ_{0}} + g z_{1} + \frac{V_{0}^{2}}{2} = \frac{P_{2}}{ρ_{2}} + g z_{2} + \frac{V_{2}^{2}}{2}

(5)

where g is gravity, z denotes the tube height, and

ρ

is the fluid density. Since the flow is at the same height, the fluid density does not change, and the wind velocity before the rotor plane is greater than that beyond the rotor (V₀ > V₁), so the pressure drop along the Betz tube can be calculated according to Equation (6):

∆ P_{m a x} = \frac{1}{2} ρ {V_{0}}^{2}

(6)

Therefore, the maximum axial force acting on the rotor plane can be calculated using Equation (7):

F_{A} = \frac{1}{2} ρ A {V_{0}}^{2}

(7)

Regarding the HAWT, the fraction of the maximum axial force that the turbine actually experiences is the axial thrust force coefficient (C_t). Thus, the thrust force (F_T) is the maximum axial force centered on the rotor’s rotational axis. Equation (8) represents the F_T calculation considering V₀ = V_w [38]:

F_{T} = F_{A} \times C_{t} = \frac{1}{2} ρ A {V_{w}}^{2} C_{t}

(8)

Due to the decrease in the wind velocity between the free stream and the rotor’s plane, the axial induction factor (a_Axial) is calculated as follows:

a_{A x i a l} = \frac{V_{0} - V_{1}}{V_{0}}

(9)

The power coefficient (C_P) and thrust force coefficient (C_t) can be calculated in terms of the axial induction factor, as follows:

C_{t} = \frac{F_{T}}{\frac{1}{2} ρ A {V_{w}}^{2}} = 4 a_{A x i a l} (1 - a_{A x i a l})

(10)

C_{P} = \frac{P_{m}}{\frac{1}{2} ρ A {V_{w}}^{3}} = 4 a_{A x i a l} {(1 - a_{A x i a l})}^{2}

(11)

To calculate C_t, and thus calculate the axial thrust force, the induction factor (a_Axial) should be calculated first. As seen in Equation (11), the induction factor can be calculated from C_P, so the output result of Equation (2) is used to calculate a_Axial. According to the Betz limit, the value of the induction factor at the maximum power extraction at C_P = 0.59 is a_Axial = 1/3; thus, at the maximum C_t = 1, a_Axial = ½, so the induction factor a_Axial ranges from 0 to 0.5.

However, to calculate the mechanical power, mechanical torque, and thrust force, the input parameters that need to be assigned are the pitch angle, which is assumed to be zero, the rotor’s radius for calculating the area of the turbine rotor, the flowing wind velocity, and the rotor rotational speed. Figure 3 shows the overall mechanical system of a WT on Simulink.

The concept of the twin-rotor WT is obtained by duplicating the rotor’s mechanical model but with only one signal for the input wind velocity. Each rotor should produce 5 MW of output power, while the wind velocity input is considered a variable vector due to the random nature of the wind.

3. WT Yaw Mechanism

The active (forced) yaw method is an automatic control method that uses electrical or hydraulic motors meshed with electrical drives, sets of gears, and bearings to align the rotor–nacelle assembly toward the wind flow direction [39]. It is mainly used in medium and large wind turbines, as it can track the maximum power according to the wind direction measurement feedback. In this work, the TRWT yaw mechanism included multiple, separately excited DC motors that took the same input signal to rotate the boom carrying the two WT rotors into alignment with the flowing wind direction.

3.1. Electrical Motor Mathematical Model

DC electrical motors that provide feedback signal control are called servomotors; they respond quickly and have precise angular positions. The back EMF (V_e) induced in the armature circuit by the permanent magnet can be calculated as follows [40]:

V_{e} = K_{e} \cdot ω = K_{e} \cdot \dot{θ}

(12)

The produced torque from the motor is directly proportional to the motor current, as follows:

T = K_{t} \cdot i (t)

(13)

Assuming no mechanical or electromechanical losses, the mechanical and electrical power are equal, as follows:

T \cdot ω = V_{e} \cdot i (t)

(14)

∴ K_{t} \cdot i (t) \cdot ω = K_{e} \cdot ω \cdot i (t) \to K_{t} = K_{e} = K

(15)

By applying Kirchhoff’s law voltage,

V = V_{e} + R \cdot i (t) + L \cdot \frac{d i (t)}{d t}

(16)

the following can be substituted into Equation (12):

V = K \cdot ω + R \cdot i (t) + L \cdot \frac{d i (t)}{d t}

(17)

Laplace transform is applied to obtain the armature current in the S-domain:

V (s) - K \cdot ω (s) = (L s + R) \cdot I (s)

(18)

∴ I (s) = V (s) - K \cdot ω (s) \frac{1}{(L s + R)}

(19)

To calculate the motor’s torque, Newton’s second law can be applied as follows:

T = J_{m} \cdot \ddot{θ} + b \cdot \dot{θ} + T_{L o a d}

(20)

Laplace transform is applied as follows:

T (s) = J_{m} \cdot s^{2} \cdot θ (s) + B \cdot s \cdot θ (s) + T_{L o a d}

(21)

∴ θ (s) = (T (s) - T_{L o a d}) \frac{1}{(J s + B)} \frac{1}{s}

(22)

The input signal to the DC motor mode is volt V(s), and the output is the angular position θ(s). Hence, the open loop transfer function of the DC motor with no load is as follows:

\frac{θ (s)}{V (s)} = \frac{K}{s ((L s + R) (J s + B) \cdot K^{2})}

(23)

For angular position control of the DC motor, a closed loop with feedback and a controller law are required. Figure 4 shows a simplified block diagram model. The motor’s response was tested by applying step input.

As mentioned above, the motor used in this work was a separately excited DC motor, selected due to its high starting torque and variable speed drive capability. Four separately excited DC motors with a gear ratio of 1:60 were used to overcome the load torque resulting from the different thrust levels on the tower top and to control the position of the WT’s side booms. The motors were selected with capacity to cover the torque required to overcome the twisting moment induced by the difference in thrust. Table 2 shows the DC motor specifications [39].

3.2. Sliding Mode Control

Due to the significant nonlinearity of the yaw mechanism of this work, SMC was used to address the nonlinear tracking problem and maintain the system’s state within the control limits. Two steps were carried out to obtain SMC. The first step was to create the sliding surface on which the system’s behavior was restricted, to obtain the desired response. Hence, the system’s state dynamic variables were constrained to satisfy a set of differential equations with lower degrees of freedom, defining the so-called switching surface. The second step was to obtain switch feedback gains to help the system’s state follow the sliding surface [41].

The sliding surface for the SMC is defined as follows:

s = {(\frac{d e (t)}{d t} + C \cdot e (t))}^{n - 1}

(24)

where C is a strictly positive constant that represents the bandwidth of the motor’s system, e is the error signal, and n is the system’s TF degree order. This work used a DC motor with a second-degree TF, so by substituting the angle error and n = 2 into Equation (24), the sliding surface is given as follows:

s = \frac{d θ_{e} (t)}{d t} + C \cdot θ_{e}

(25)

The control signal that the system can maintain and which can reach the sliding surface is subject to the following:

s \cdot \dot{s} > 0

(26)

u = K \cdot s g n (s)

(27)

where the discontinuous controller output signal is obtained using a sign function, and the variable is the value of the instantaneous sliding surface, while K is a positive constant. The sign function is defined as follows:

s = \{\begin{matrix} - 1, s < 0 \\ + 1, s \geq 0 \end{matrix}

(28)

However, the sign function leads to the chattering phenomenon, which may harm the motor’s system. Using a pseudo function instead of the sign function can solve this chattering problem, as follows:

u = K \cdot \frac{s}{|s| + δ}

(29)

where δ is a positive tuning parameter to reduce the chattering around the sliding surface. Selecting the proper value for δ is crucial; if it is too small, the chattering problem will not be solved, and if it is too big, it will be difficult to reach the reference value.

PID controllers are known to be a robust solution for controlling motor speeds, and they are also easy to implement. Thus, to ensure the motor’s performance, a comparison was made between a PID controller and the SMC algorithm for controlling the DC motor system, and their performances were tested. The PID parameters were automatically tuned via the parameter tuner in the Simulink PID controller block. Figure 5 shows the applications of both SMC and PID to the DC motor system with a constant input signal.

The simulation results of the WT’s system model and sliding mode control model are presented and discussed in detail in the Results and Discussion section.

4. Results and Discussion

4.1. WT Mechanical Model

A Simulink mechanical model of a WT was designed and tested to operate in region 2 (transition region) of a utility-scale wind turbine. For the NREL 5 MW rotor, the wind speed in this region ranges from 8 m/s to 11.4 m/s. At low wind speeds under 8 m/s, torque control is needed to maximize the power output. Above 11.4 m/s, region pitch control is needed to keep the power output at the rated value of 5 MW, which was not in this work’s scope. The WT’s power coefficient calculation model was first tested by plotting a curve representing the Cp values with respect to λ at different pitch angles, as illustrated in Figure 6, and comparing it with the original Cp-λ curve of the NREL 5 MW WT.

As this work focused on region 2 in the operation of a WT, the values of the power coefficient varied with wind speeds from 8 m/s to 11.4 m/s. According to the NREL definition report, at the rated wind speed of 11.4 m/s, the rotor rotational speed (ω_r) is 12.1 rpm, so applying these values to the mechanical model resulted in an optimal power coefficient factor (C_p) = 0.4514 and TSR (λ) = 7.002, as shown in Figure 7. The results of the power coefficient for different wind speeds are shown in Figure 8.

The mechanical power simulation showed an increase in the produced mechanical power over wind speed until it reached a rated power of 5.108 × 10⁶ W per rotor.

The results shown were verified by the reference NREL 5 MW wind turbine, with a maximum error of −5%. Accordingly, the axial thrust force was calculated and verified with a maximum error of −14.2% compared with the reference turbine at a wind speed of 8 m/s. Figure 9 shows the variation in the mechanical power per rotor with the wind speed in region 2. Figure 10 shows the verification of the axial thrust force.

4.2. WT Yaw Mechanism Model

As mentioned above, the wind changes dramatically over time and between locations. In this work, the distance between the center of the two rotors was 1.05 times the rotor diameter, which was equivalent to 134.4 m. This very large distance caused differences in the wind speed and direction between the rotors. These changes resulted in differences in the thrust force between the two rotors, with the potential to cause a large twisting moment and destroy the WT. Applying different random wind speeds, up to the rotor’s rated speed, and wind directions to each rotor resulted in different power and thrust outcomes. Applying different thrust forces to each rotor resulted in a side-boom twisting moment on the tower top where the yaw actuator was installed. The multiple-yaw DC motors were designed to control and optimize this difference, thus enhancing the output power. Figure 11 shows the resulting mechanical parameter outputs due to the variations in the wind speed and direction on each of the WT’s rotors.

4.3. Stability Analysis of the System Dynamic Model

To apply a position control law to the system Simulink model, first, the system state space model had to be obtained. The MATLAB system identification toolbox was used to capture the system dynamics using nonlinear ARX model estimators. The resultant system state space model and its state space matrices are shown in Equation (30):

A = [\begin{matrix} - 0.554 & - 0.056 \\ 0.0625 & 0 \end{matrix}], B = [\begin{matrix} 128 \\ 0 \end{matrix}], C = [\begin{matrix} 186.8 & 2.232 \end{matrix}] a n d D = [0]

(30)

Then, the Lyapunov direct method was used to determine the system’s stability. The Lyapunov equation is shown in Equation (31):

- Q = A^{T} P + P A

(31)

where Q is the positive definite matrix, A is the system matrix, and P is the solution matrix.

To determine the system’s stability, the eigenvalues of the P matrix and A matrix were calculated and found to be

[\begin{matrix} 0.904 \\ 0336.886 \end{matrix}] a n d [\begin{matrix} - 0.547 \\ - 0.006 \end{matrix}]

, respectively. This means that the P matrix is a positive definite matrix and the eigenvalues of the A matrix have negative real parts. Thus, the system is asymptotically stable.

4.4. Application of Position Control on the System Model

As stated in the previous section, yaw position control can be applied to the system as planned. To rotate the WT’s boom carrying the two rotor–nacelle assemblies toward the wind flow direction, the separately excited DC motor parameters were fine-tuned to obtain a high torque, and a gearbox with a gear ratio of 1:60 was used. In order to reach the total torque required for rotating the WT’s boom to a specific angular position, four DC motors with specifications designed for angular position control were simulated. The system’s TF and parameters were first tested for a step input signal to verify its output response and torque as an open-loop system before applying any control techniques. The system did not reach the required input, which meant that a control method was required for the system.

PID and SMC controllers were applied to the motors’ transfer function. A comparison of the SMC and PID controllers with fine-tuned parameters is shown in Table 3. The simulation results of the step reference input signals for SMC and PID, respectively, are illustrated in Figure 12 and show that the system response with the SMC technique had very low overshoot compared with that of the PID controller. Additionally, the rise and settling times of SMC were much better and faster than those of PID.

The detailed system response results are shown in Table 4. Compared with the PID controller, the SMC controller had a better and faster response, which was more suitable for the purpose of this work.

SMC was applied to the TF of this designed system by changing the SMC parameters to construct a sliding surface. The system’s dynamic state on this sliding surface was restricted due to the changes in the random input angle signals. SMC has three different parameters. “C” is the bandwidth constant of the system, which is multiplied by the input angle and added to the differentiation of this input angle signal in order to obtain the sliding surface that the dynamic variables will follow; “u” is a control law based on the sliding surface obtained and tuned through the parameter “δ”, which forces the controlled signal to smoothly chatter around the reference signal. The parameter values were fine-tuned until the response was accepted at C = 10, K = 300, and δ = 50. The SMC signal with respect to the reference input signal is shown in Figure 13.

It can be observed that the SMC forced the system’s signal to slide over the sliding surface, following the desired output angular position. This action led to a reduction in yaw misalignment; hence, the rotors faced upwind while counteracting the induced twisting moment on the tower top. Accordingly, the power of the turbine was maximized for the different flow conditions at each rotor.

Figure 14 shows the significant chattering of the input signal on the sliding surface due to its sudden variation. This chattering represents the deviation of the system’s dynamic state from the desired state before applying the SMC. Figure 15 shows the enhancement in the chattering signals after applying the SMC control law, which significantly reduced it to follow the sliding surface with fewer fluctuations. However, SMC showed better responses in tracking the dynamic state of the angular position control of the system.

Figure 16 shows the chattering in the signals before and after applying SMC. The SMC-controlled system showed very low chattering compared with the signal chattering. The signal appears as a line, since the order of magnitude was less than a factor of 10⁶. This shows the significant improvement in the system provided by SMC, and thus, its suitability in this case.

5. Conclusions

This paper presents a detailed mathematical model for the mechanical system of a WT, with equations applied in Simulink to calculate the mechanical outputs, such as the mechanical power and thrust force, and study the system’s performance. The MATLAB Simulink^® tool was used to model and simulate the overall mechanical system, and the NREL 5 MW WT was used as a reference to verify the system’s behavior. This model is the core contribution of this work and was used to show the results of the proposed yaw control mechanism configuration.

A twin-rotor model was adopted as a case study to design and simulate the yaw control mechanism, which is one of the main challenges of MRS turbines. The mechanical power of the simulated model was verified, and the thrust force was calculated and verified, as this is a crucial factor affecting the yaw action of MRS turbines. The proposed yaw mechanism considers the side booms carrying the two rotors as one singular moving part. Two different vectors for the wind speed were used as inputs to the model for each of the two rotors, with random values for the wind speed and direction. The thrust force was calculated for each rotor, and the twisting moment on the tower top resulting from the difference in thrust force was used as a determination factor for the yaw mechanism’s action.

A DC motor was used as an actuator for the yaw control action. The controller parameters were determined through fine-tuning until the desired torque was achieved. The SMC algorithm was used to slide over the transfer function of the system. The response curve of the controller state was very smooth, and the settling time was as little as 0.6 s. The sliding mode controller also improved the chattering in the signal significantly along the sliding surface. The yaw control mechanism played a major role in compensating for the torque induced by the difference in the thrust force between the two rotors.

Although SMC resulted in a good response, the proposed control in its current form was unable to optimize the yaw misalignment in relation to the torque induced on the main tower. In the current configuration, where the side booms move as one part, it seems impossible to compromise between the wind flow direction and the thrust difference between the rotors.

It is recommended that future work should also consider rotating the rotors around a vertical axis to be able to include the advantages of both yaw mechanisms—the rotors and the side booms. It is also suggested that different control algorithms should be used and optimized to obtain better results for the yaw mechanism of multi-rotor wind turbines.

Author Contributions

Conceptualization, A.I., A.A., S.Y. and S.S.; methodology, A.E., A.I., S.Y. and A.A.; software, A.E., A.I., A.A. and M.A.R.; validation, A.A., M.A.R. and A.I.; resources, A.A. and M.A.R.; writing—original draft preparation, A.E.; writing—review and editing, A.A., S.S., S.Y. and A.I.; visualization, A.E. and M.A.R.; supervision, A.A., S.S., S.Y. and A.I. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

List of Abbreviations

Symbol	Definition
A	Swept area of the rotor (m²)
a_Axial	Axial induction factor (--)
B	Viscous damping constant (N.m/rad/s)
C_P	Power coefficient (--)
C_t	Thrust force coefficient (--)
F_T	Thrust force (kN)
J	Inertia constant (kg.m²)
K_t	Torque constant (--)
K_v	Back EMF constant (--)
L	Armature inductance (H)
P_m	Mechanical power (kW)
R	Armature resistance (Ω)
RR	Rotor’s radius (m)
T	Torque (kN.m)
V₀	Wind speed before the rotor (m/s)
V₁	Wind speed at the rotor disc (m/s)
V₂	Wind speed beyond the rotor (m/s)
V_e	Back electromotive force (EMF) (volts)
V_w	Wind velocity (m/s)
z	Betz tube height (m)
β	Blade pitch angle (°)
ω_r	Rotational speed of the rotor (rad/s)
$λ$	Tip speed ratio (--)
$ρ$	Air density (kg/m³)

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Figure 1. Proposed TRWT configuration.

Figure 2. Betz tube.

Figure 3. WT mechanical system.

Figure 4. Simplified DC motor closed-loop TF.

Figure 5. DC motor with SMC and PID.

Figure 6. C_p vs. TSR curve.

Figure 7. C_p and λ Simulink results.

Figure 8. V_w vs. C_p curve.

Figure 9. Verification of V_w vs. mechanical power per rotor.

Figure 10. Verification of V_w vs. thrust force per rotor.

Figure 11. Twin-rotor mechanical outputs.

Figure 12. Response of SMC vs. PID.

Figure 13. SMC response.

Figure 14. System’s chattering signal before applying SMC.

Figure 15. Sliding mode surface control.

Figure 16. Chattering signals before and after applying SMC control.

Table 1. NREL 5 MW rotor specifications [36].

Property	Value
Power Rating	5 MW
Rotor Configuration	Upwind, 3 blades
Rotor Diameter	126 m
Hub Height	90 m
Cut-In Wind Speed	3 m/s
Rated Wind Speed	11.4 m/s
Cut-Out Wind Speed	25 m/s
Rotor Mass	110,000 kg
Nacelle Mass	240,000 kg

Table 2. Separately excited DC motor specs. [39].

Parameter	Value
DC Supply Voltage (V)	220 V
DC Motor Capacity (P)	3 HP
Armature Resistance (R)	0.6 Ω
Armature Inductance (L)	0.008 H
Inertia Constant (J)	0.011 Kg.m²
Viscous Damping Constant (B)	0.004 Nm/rad/s
Torque Constant (K_t)	0.55
Back EMF Constant (K_v)	0.55

Table 3. SMC and PID tuned parameters.

Controller	Tuned Parameters
PID	K_P	K_I	K_D
PID	1.10	50	0.07
SMC	C	K	δ
SMC	60	600	5

Table 4. SMC vs. PID response results.

Controller	System Response Parameters
Controller	Rise Time (ms)	Settling Time (s)	Overshoot %	Undershoot %
PID	46.438	0.867	30.921	1.990
SMC	35.465	0.170	0.802	1.999

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MDPI and ACS Style

Elkodama, A.; Abdellatif, A.; Shaaban, S.; Rushdi, M.A.; Yoshida, S.; Ismaiel, A. Investigation into the Yaw Control of a Twin-Rotor 10 MW Wind Turbine. Appl. Sci. 2024, 14, 9810. https://doi.org/10.3390/app14219810

AMA Style

Elkodama A, Abdellatif A, Shaaban S, Rushdi MA, Yoshida S, Ismaiel A. Investigation into the Yaw Control of a Twin-Rotor 10 MW Wind Turbine. Applied Sciences. 2024; 14(21):9810. https://doi.org/10.3390/app14219810

Chicago/Turabian Style

Elkodama, Amira, A. Abdellatif, S. Shaaban, Mostafa A. Rushdi, Shigeo Yoshida, and Amr Ismaiel. 2024. "Investigation into the Yaw Control of a Twin-Rotor 10 MW Wind Turbine" Applied Sciences 14, no. 21: 9810. https://doi.org/10.3390/app14219810

APA Style

Elkodama, A., Abdellatif, A., Shaaban, S., Rushdi, M. A., Yoshida, S., & Ismaiel, A. (2024). Investigation into the Yaw Control of a Twin-Rotor 10 MW Wind Turbine. Applied Sciences, 14(21), 9810. https://doi.org/10.3390/app14219810

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