Exploring the Potential of Hybrid Excitation Synchronous Generators in Wind Energy: A Comprehensive Analysis and Overview

Abstract

Due to the unpredictable nature of the wind, uncertainty in the characteristics of wind electrical conversion systems (WECSs), and inefficient management tactics, wind turbines have historically had operational inefficiencies. In order to overcome these drawbacks, the hybrid excitation synchronous generator (HESG), an alternative to traditional generators, is presented in this study along with the suggestion to use robust regulators to regulate HESGs. This research begins with a thorough review of the literature on generators often seen in modern wind systems. Next, a simulation platform that merges a WECS with a HESG tied to an isolated load is built using the MATLAB Simulink environment. Pitch angle control investigation shows a new experimental approach to determine the link between turbine output and the reference pitch angle. Furthermore, an evaluation of the mechanical stability of the WECS is conducted by a comparison of the performance of a H∞ controller and a CRONE controller. The simulation results demonstrate the efficiency of the CRONE controller in reducing mechanical vibrations in the WECS. By reducing vibrations, the proposed control technique enhances the overall performance and efficiency of the wind turbine system. The field is extended by the demonstration of how HESGs and reliable control systems can enhance wind turbine performance while eliminating inherent limitations.

1. Introduction

Faced with the limits, high prices, pollution issues of fossil fuels, and the desire to lower carbon dioxide emissions and protect nature, the use of renewable energies such as wind power, solar, geothermal, etc., is now necessary. Wind energy is one of the most profitable of these options [1]. Wind turbines (WTs) do not affect water or land during production and do not release greenhouse gases (GHGs), despite their damaging manufacturing and dismantling processes [2]. The application of this technology is fast growing in Asia, the European Union (EU), and the United States. The Global Wind Energy Council (GWECS) reported a 12.4% increase in wind power installed capacity to 837 GW in 2021, up from 2020 [3].

According to Wind Europe, WECSs will supply up to 35% of the EU’s electricity demands by 2030 [4]. By 2030, the United Nations intends to satisfy 20% of its electricity demand for wind energy [5]. Asia represents the greatest wind energy market. China, for instance, plans to construct 200 GW of wind power by 2020, 400 GW by 2030, and 1000 GW by 2050 [6].

The significance of wind energy’s potential and its competitive cost compared to other forms of RE have substantially facilitated the growth of this industry [7]. This explains why researchers are interested in the topic of wind energy.

Many grid-connected WECS setups have been studied in the literature. Variable-speed WECSs with dual-fed asynchronous generators and multistage gearboxes continue to be the most prevalent systems on the market today. This is supported by the fact that the power consumed by an electrical power converter is 25–30% of the capacity of the generator, making this approach economically viable [8]. A reduced size of convertors and thereby filters implies cost savings, which explains why the double-fed generator is the favored option for grid-connected WECSs [9]. Additionally, a converter can compensate for the reactive power, ensuring a smooth connection to the grid. In light of increasing dependability requirements, permanent magnet and direct drive synchronous generators are becoming increasingly attractive alternatives [10].

In this study, we will enumerate the various types of wind generators and their architectures. Subsequently, we introduce the double-excited synchronous generator. Then, the use of an HESG in a wind conversion system is presented as an attractive alternative to conventional systems. While various studies have examined the concept and structural features of this novel generator, few, if any, have tested its efficiency in a wind conversion system. The key objective here is to develop an efficient and trustworthy overall control strategy for wind energy hybrid generators. Permanent magnets and direct current coils are used to create the excitation flux in this generator. This specificity enhances the degree of freedom in the WCS design since the excitation winding current may be used as a control parameter. The rotational speed is tracked by changing the excitation winding current. Furthermore, this control requires only a DC/DC converter, which consumes far less power than a more typical architecture.

This paper is divided into two parts: the first part reviews the literature regarding various wind generator types, and the second part investigates the HESG potential in WECSs. In this paper, the authors present the detailed mathematical modeling of a WECS based on an HESG. Step-by-step procedure for controlling the system: both pitch control and speed control are presented. The mechanical stability performances of a CRONE and H∞ regulators are compared in the last part of this paper.

2. Reviewing the Literature Regarding Various Wind Generator Types

The speed of a wind energy conversion system (WECS) can be either fixed or variable. In the absence of a power electronics interface, fixed-speed wind turbines typically utilize a squirrel-cage induction generator (SCIG) or an asynchronous wound-rotor generator (AWRG) [11]. Variable-speed wind turbines are managed by a partial/full-scale power converter to control the electricity flow, offering a wide range of generators and power converters for selection. Asynchronous and synchronous machines are commonly used generators for these wind turbines.

Table 1. Commercial wind turbines and their generators.

Manufacturer	Model	Rated Power (MW)	Generator Type
Vestas	V150-4.2 MW	4.2	Permanent Magnet Synchronous Generator (PMSG)
Siemens Gamesa	SG 5.0-145	5.0	Doubly Fed Induction Generator (DFIG)
GE Renewable Energy	Haliade-X	12	Direct Drive Synchronous Generator
Enercon	E-126 EP3	4.0	Synchronous Generator with Direct Drive
Nordex	N149/4.0-4.5	4.5	Doubly Fed Induction Generator (DFIG)

gives a short review of some of the main commercial wind turbines on the market, as well as their respective generator types, allowing for a better grasp of the technical landscape.

Choosing the correct generator is critical for gathering wind energy at different speeds, particularly at low wind speeds that necessitate a highly efficient conversion system. In reference [12], the selection of an electric generator for a self-contained turbine is examined. A comparison between induction and synchronous generators revealed that a permanent magnet synchronous generator (PMSG) is the most suitable for standalone operation. Numerous articles [13] have highlighted IGs as mature equipment for standalone WECS applications, with [14,15] specifically advocating the SCIG for its simplicity, durability, brushless nature, and cost-effectiveness. However, considerations about the generator’s efficiency may diminish its appeal. Research in [16] assessed the losses, output powers, and efficiency of a WECS employing a SCAG, a double power supply, and a PMSG depending on wind velocity. The data show that the asynchronous cage generator has the lowest efficiency, especially at low rotating speeds.

In summary, this paragraph outlines the most common generators for standalone WECSs, including the Wound Rotor Asynchronous Generator (WRAG), Squirrel Cage Asynchronous Generator (SCAG), Double-Fed Asynchronous Generator (DFAG), Wound Rotor Synchronous Generator (WRSG), Brushless Dual-Supply Asynchronous Generator (BDSAG), Brushless Dual-Supply Variable Reluctance Generator (BDSVRG), Permanent Magnet Asynchronous Generator (PMAG), Variable Reluctance Generator (VRG), and Permanent Magnet Synchronous Generator (PMSG).

2.1. Wound Rotor Asynchronous Generator (WRAG)

In a WECS utilizing a WRAG, as depicted in Figure 1, the inclusion of a starter is essential for mitigating the starting inrush current. Additionally, for this configuration, adjusting the reactive power becomes a requirement.

The independent nature of the WRAG allows easy manipulation to generate stable voltages with consistent amplitude and frequency. Due to its limited speed fluctuation range, the WRAG has traditionally been applied in fixed-speed wind systems rather than variable-speed systems [17].

The WRAG’s stator is connected to the electrical network, and the rotor winding is in series with a variable resistance. Variable-speed operation is possible by adjusting the energy produced by the WRAG rotor. However, it is vital to note that the external resistor wastes the power. Should there be a desire to expand the dynamic speed range, the slip will increase, leading to higher power drawn by the rotor and a reduction in the efficiency of the WRAG due to the enlarged size of the resistance.

2.2. Squirrel Cage Asynchronous Generator (SCAG)

The SCAG is recognized as the smallest, most cost-effective, and most durable conventional induction generator (IG) [18]. As a well-established generator in the WECS field, SCAG (depicted in Figure 2) has been the focus of numerous research endeavors, particularly concerning emulator design, the development of new power converters, the exploration of control schemes, the implementation of power-up techniques, and investigations into self-excitation for off-grid applications [19]. The SCAG is known for its simplicity, durability, brushless nature, and cost-effectiveness.

However, there are certain drawbacks associated with SCAG [8]:

• The range of speed variation is limited, and the generator can only operate at speeds surpassing the synchronization speed.
• This type of wind turbine necessitates a three-stage gearbox in the transmission, with the multipliers constituting a substantial portion of the nacelle’s bulk, making them both bulky and expensive [8].
• Operating at a constant speed leads to immediate electromechanical torque variations with changes in the wind speed, resulting in high mechanical stresses and eventual system fatigue. Additionally, adjusting the turbine speed based on the wind speed to maximize the aerodynamic efficiency is not possible.

2.3. Double-Fed Asynchronous Generation (DFAG)

A WECS employing a DFAG, as illustrated in Figure 3, is configured with the stator connected to the network, while the rotor is linked to the point of common coupling (PCC) through a partial-scale power electronic converter. Power flow is unidirectional through the stator, and the flow through the rotor depends on the generator’s operational mode. Below the synchronous speed, electricity is provided to the rotor; beyond this speed, the rotor generates electricity.

The electric power converter regulates the rotor’s frequency. This configuration enables speeds that are more than 30% higher than the synchronous speed, allowing for operation across a wide range.

As mentioned earlier, the key advantage of DFAG lies in the reduced power requirement of its power converter (25 to 30 percent of the generator’s capacity). However, DFAG has several drawbacks [11]:

• A multiplier is still necessary in the transmission because the speed range for a DFAG far exceeds the typical turbine rates of 10 to 25 rpm.
• Neglecting routine maintenance on the slide ring system may lead to machine malfunctions and electrical losses.

2.4. Wound Rotor Synchronous Generator (WRSG)

WCSs utilizing a WRSG, as depicted in Figure 4, necessitate DC excitation. This excitation can be provided through a brushless exciter with a power converter and auxiliary AC generator or an external DC power supply via brushes and slip rings, as proposed in [20]. The authors of [20] see this strategy as a viable alternative for supplying demand in isolated loads. Ref. [21] investigated several regulation algorithms for managing stator voltage in an off-grid WRSG. This generator eliminates the need for permanent magnets (PMs), resulting in cost savings [22].

2.5. Brushless Dual-Supply Asynchronous Generator (BDSAG) and Brushless Dual-Supply Variable Reluctance Generator (BDSVRG)

The BDSAG is constructed by combining two wound-rotor asynchronous machines, with the first dedicated to control and the second dedicated to power generation. In Figure 5, the BDSAG features two sets of stator windings: power winding (PW) and control winding (CW). Although BDSAG offers advantages similar to those of DFAG, it is bulkier and has more complex assembly and control requirements for the same rated power.

Despite these drawbacks, BDSAG remains an attractive solution for grid-connected installations (Figure 5), especially for offshore applications, where it is reliable and nearly maintenance-free [23]. Although BDSAG has resolved commutator issues, its size and complexity may make it unsuitable for smaller wind farms. Similarly, the BDSVRG, while more efficient, faces similar challenges.

The BDSVRG introduces a distinctive design featuring a reluctance rotor instead of a wound rotor, as seen in BDSAG [24]. Despite its efficiency and reliability in comparison to the BDSAG, the BDSVRG rotor design is intricate, and its size is larger. Recent advancements in reluctant rotor design, however, may enhance its appeal [25].

2.6. Permanent Magnet Asynchronous Generator (PMAG)

The voltage regulation capabilities of an IG are limited because it relies on magnetizing current from an excitation source [26], resulting in a reduced power factor and efficiency [27]. The use of a PMAG can improve the power factor, voltage control, and efficiency [27]. Although the stator of a PMAG is similar to that of a conventional IG, its rotor design is distinct, comprising two components: a permanent magnet rotor and a squirrel cage rotor. The squirrel cage rotor is partially energized by the magnet rotor, reducing the need for reactive power from an external source and consequently lowering the capacitance in an MPAG-based wind system compared to systems based on the WRAG. Furthermore, the AGPM can operate without a multiplier. PMAG-based WECSs have two available configurations, as depicted in Figure 6.

As a relatively new technology, PMAG can serve as a direct-drive generator in grid-connected [27] or isolated [28] applications, offering an improved power factor and enhanced overall performance.

2.7. Variable Reluctance Generator (VRG)

A VRG is a structurally simple and robust device commonly constructed from steel sheets. Its stator consists of poles with concentrated windings around them, while the rotor is made up of salient poles without windings or permanent magnets (PMs) [29]. Typically, a half-bridge converter powers the machine [30]. As a durable, reliable, and cost-effective device with relatively straightforward control, VRGs have shown great potential as direct-drive machines in both off-grid WECSs [31] and grid-connected wind systems [32] (Figure 7).

Despite proposals for the use of VRGs in WECSs dating back to the 1990s, their performance evaluation has been limited to simulations and a few research studies, with no actual deployment.

2.8. Permanent Magnet Synchronous Generator (PMSG)

A WECS incorporating a PMSG, as illustrated in Figure 8, has the advantage of eliminating the need for a gearbox. Given the self-exciting nature of the PMSG, a three-phase rectifier can be employed as a machine-side converter. To regulate the generator shaft’s speed, a chopper is necessary. One noteworthy advantage of PMSG-based WECSs is the ability to use a diode rectifier without relying on self-exciting capacitors, making them a cost-effective and flexible alternative in wind turbine design [33].

The presence of permanent magnets in a PMSG eliminates the need for a magnetizing current to maintain a constant flux in the air gap. Consequently, only the stator current is responsible for generating torque, allowing for operation with a high-power factor and increased efficiency.

However, the interaction between the stator slots and the rotor magnets in a PMSG produces a detrimental torque known as cogging torque. This torque can lead to oscillations in shaft speed, vibrations, and noise in the machine, especially at low speeds, significantly impacting the overall production of a wind turbine [34]. Additionally, PMSG has its drawbacks:

• The high cost of permanent magnets.
• Manufacturing difficulties.
• Demagnetization of permanent magnets at high temperatures.

Finally, the effective voltage for a PMSG can be expressed as in Equation (1) [34]:

$$V=p\Omega \sqrt{L_s{I}_s^2+{\varphi }_{ext}^2-2{\varphi }_{ext}{L}_s{I}_s\sin \psi }$$

(1)

Stator voltage (V) is determined by the following factors: rotor mechanical speed (Ω), number of pole pairs ($p$), flux generated by the magnets in the stator winding (${\varphi }_{ext}$), angle (ψ) in radians between the electromotive force (EMF) and the current (load angle), stator cycle inductance ($L_s$), and stator current ($I_s$).

According to Equation (1), one can conclude that the speed or tension may be controlled by acting on ${\varphi }_{ext}$. However, as this flux is continuous, power electronics are employed to deflux electricity. This approach forms the foundation of the hybrid excitation synchronous generator (HESG) described and utilized in this research, as further explanations are provided in the next paragraph.

3. HESG Potential in WECSs

Although there is no consensus on the optimal design for future WTs, synchronous machines appear to be the trend. Permanent magnets or wrapped rotors are utilized in these devices.

The use of permanent magnets, as opposed to rotor windings, can provide considerable benefits, such as increased efficiency owing to the elimination of copper loss, increased power density, and quicker reaction due to the low electromechanical time constant. However, the principal flux produced by the permanent magnet cannot be adjusted, which is the primary downside of this machine type. The double-excited synchronous generator is an intriguing alternative to standard synchronous generators.

A number of works have proposed the use of HESM in generator mode in applications such as marine [35,36], small hydroelectric power plants [36], and aeronautics [37,38]. In [35], the performance of the HESM compared to that of the PMSM was proven via finite element analysis (FEA). For the application under consideration (marine diesel generator), the generator voltage must be maintained within ±10% of its nominal value, and the short-circuit current must be at least three times the nominal current for at least two seconds; this second condition is a major problem for PMSM.

In [36], the authors compared the performance of a dual-excitation synchronous machine to that of a wound-excitation SM and to that of the PMSM. They showed that, compared to the wound-excitation SM, HESM had much greater efficiency for a hydroelectric plant. The stator copper losses are slightly greater (due to the length of the HESM stator), but the rotor copper losses are much lower, which shows the efficiency of the hybrid generator compared to the wound excitation MS. For a unit power factor, the efficiencies of the PMSM and HESM are comparable.

According to [38], the HESG is a viable alternative to the three-stage synchronous machine often employed in commercial and military aviation power systems [39] due to its better power density, simpler structure, and smaller size.

The application of HESGs in WECSs has attracted increasing attention in the last several years. The benefits of winding field excitation and permanent magnets are combined in HESG generators, which produce a highly efficient generator that can produce large torque at low speeds. As a result, HESGs are now a viable option to WECSs, especially for applications that need high efficiency and little maintenance. Numerous studies have been conducted on the use of HESGs in WECSs, with many focusing on the design, control, and optimization of HESGs for optimal performance. The GREYC team in France [40] proposed the use of an HESG in this field by connecting it to the grid through a three-stage power electronics system (AC/DC/AC converter). Another study conducted at the L2EP laboratory in France highlighted the mass and energy efficiency of this generator for high -power applications, particularly wind power, compared to that of the Salient Pole Synchronous Generator (SPSG) [34]. The authors investigated the influence of several HESG design factors on WECS performance, including stator and rotor size, winding configurations, and magnetic material qualities. In their work, Ref. [41] devised a control approach for a HESG-based WECS that optimizes generator torque and speed to optimum power production.

Other studies have focused on the integration of HESGs into WECSs and the potential benefits of using HESGs in different WECS applications. For example, Ref. [42] proposed and modeled a 1.5 MW grid-connected WCES based on an HESG. The authors analyzed the performance of an HESG-based WECS in both grid-connected and standalone modes and demonstrated that the HESG system was highly efficient and reliable in both scenarios.

In a study conducted by [43], the application of HESGs in offshore wind farms was examined. The findings indicated that HESGs have the potential to substantially decrease generator size and weight, leading to enhanced overall system efficiency. Additionally, Ref. [44] introduced a novel HESG design featuring a dual-stator configuration, aiming to enhance efficiency and lower costs.

HESG are designed to balance weight and size, making them lighter and more compact than direct drive synchronous generators while providing comparable or higher performance. This makes HESGs easier to transport and install, which lowers logistical hurdles and costs. Their efficiency is extremely high, frequently equaling or exceeding that of Permanent Magnet Synchronous Generators (PMSG) and direct drive systems, since their design reduces energy losses, which is critical for increasing the energy output of Wind Energy Conversion Systems (WECS). Although HESGs may be more expensive than certain alternatives, such as Doubly Fed Induction Generators (DFIGs), they are more cost-effective in the long run due to fewer maintenance requirements and improved energy output efficiency.

Reduced operational expenses and increased HESG efficiency can result in considerable cost savings during the wind turbine’s lifespan. Furthermore, HESGs often have fewer moving parts than DFIGs, resulting in less wear and tear and cheaper maintenance expenses. Their strong design improves dependability and extends the wind turbine’s operating life. Furthermore, HESGs retain high efficiency even in varied wind conditions, making them ideal for a variety of geographical areas with changing wind patterns.

Despite these developments, incorporating HESGs into WECSs remains hard. These include issues with HESG control, cost-effectiveness, and system stability. This research aims to contribute to the field by providing a comprehensive review of HESG technology’s potential applications in WECSs, as well as highlighting its distinct strengths and drawbacks.

To summarize, the research on HESGs in WECSs has demonstrated that this technology has the potential to increase system performance and efficiency. This work contributes to current knowledge by providing a comprehensive overview of HESG technology and its potential applications in WECSs.

3.1. Model of the WCS Based on an HESG

A WCS transforms wind energy into electricity, comprising essential components such as the turbine, gearbox, and generator. The typical configuration of a WCS utilizing a standalone HESG is depicted in Figure 9.

3.1.1. Dynamic Model of a Wind Turbine

With the existence of an aerodynamic torque $C_t$, the gearbox, which links the turbine and the generator, aligns the turbine rotation speed ${\mathrm{\Omega }}_t$ with that of the generator ${\mathrm{\Omega }}_g$.

$$C_t=0.5·C_p(\lambda ,\beta )·\rho ·S·V_w^3/{\Omega }_t$$

(2)

The wind velocity $\left(V_w\right)$, air density $\left(\rho \right)$, and surface area swept $\left(s\right)$ by turbine blades with a radius of $\left(R_p\right)$ are factors in turbine operation. The turbine performance coefficient $\left(C_p\right)$ is determined by the pitch angle $\left(\beta \right)$ and the tip speed ratio $\left(\lambda \right)$. The value of parameter $\lambda$ is defined as follows:

$$\lambda ={\Omega }_t·R_p/V_w$$

(3)

3.1.2. Gearbox Dynamic Model

This study employs a two-mass mechanical model (depicted in Figure 10) to describe the mechanical behavior.

Equations (4) and (5) in the slow shaft reference provide the relevant formulations.

$$J_t\frac{d{\Omega }_t}{dt}=C_t-D_{ls}({\Omega }_t-\frac{{\Omega }_g}{m_p})-K_{ls}({\theta }_t-\frac{{\theta }_g}{m_p})-K_t{\Omega }_t$$

(4)

$$J_g\frac{d{\Omega }_g}{dt}=C_{em}-\frac{D_{ls}}{m_p}({\Omega }_t-\frac{{\Omega }_g}{m_p})-\frac{K_{ls}}{m_p}({\theta }_t-\frac{{\theta }_g}{m_p})-K_g{\Omega }_g$$

(5)

${\theta }_g$ and ${\theta }_t$ indicate the angular displacements of the turbine and generator, respectively. $J_g$ and $J_t$ denote the inertias of the generator and turbine, respectively. $K_g$ and $K_{ls}$ are the viscous friction coefficients for the generator and slow shaft, respectively, while $D_{ls}$ is the torsion coefficient of the slow shaft.

3.1.3. HESG Model

To consider the impact of harmonics, the modeling of HESG incorporates the findings presented in reference [42]. Specifically for phase “a”, the stator inductance is represented by (6), the mutual inductance is expressed as shown in (7), and the flux is given by (8). To express the stator inductances, an 18th-order Fourier series expansion is utilized.

$$L=L_{s_0}+{\displaystyle \sum _{h=1}^9{L}_{s_{2h}}\cos (2hp\theta -{\zeta }_h)}$$

(6)

$$M=M_{s_0}+{\displaystyle \sum _{h=1}^9{M}_{s_{2h}}\cos (2hp\theta -{\zeta }_h)}$$

(7)

$${\varphi }_e={\displaystyle \sum _{h=0}^8{\varphi }_{a_{(2h+1)}}\cos (p\theta (2h+1)-{\zeta }_h)}$$

(8)

With $L_{s_0}=(L_d+L_q)/3$, $M_{s_0}=-L_{s_0}/3$

The inductances $L_d$ and $L_q$ represent the inductance values along the $d$ and $q$ axes, respectively. The flux ${\varphi }_a$ is generated by the magnets in the armature coils, while ${\varphi }_e$ is generated by the magnets in the excitation coils. The variable $p$ corresponds to the number of pole pairs.

Subsequently, the HESG is represented in the Concordia frame following the expressions presented in Equations (9) through (13) where:

• ϕ_ae: Excitation flux produced by the magnets
• I_e: Excitation current
• [ϕ]: Stator flux vector
• [I]: Stator current vector
• [V]: Stator voltage vector
• [ϕ_e3]: Excitation flux produced by the magnets in the stator coils
• [M_e]: Stator-excitation mutual vector
• L_e: Excitation inductance
• [L]: Stator inductance matrix
• R_s: Armature resistance
• R_e: Excitation coil resistance
• L_s0: Stator self-inductance
• M_s0: Mutual inductance between the inductor and the armature
• φ_a: Flux created by the magnets on a stator winding
• φ_ae: Flux created by the magnets on an excitation winding
• L_s: Stator cyclic inductance
• Fluxes:

$$\left[{\varphi }_2\right]=\left[L_2\right]\left[I_2\right]+\left[M_{e2}\right]I_e+\left[{\varphi }_{e2}\right]$$

(9)

$${\varphi }_e={\left[M_{e2}\right]}^t\left[I_2\right]+L_e{I}_e+{\varphi }_{ae}$$

(10)

• Voltages:

$$\left[V_2\right]=R_s\left[I_2\right]+d\left[{\varphi }_2\right]/dt$$

(11)

$$V_e=R_e{I}_e+d{\varphi }_e/dt$$

(12)

• Electromagnetic torque:

$$\left[C_{em}\right]=\frac{1}{2}{\left[I_2\right]}^t\frac{d\left[L_2\right]}{d\theta }\left[I_2\right]+{\left[I_2\right]}^t\frac{d\left[M_{e2}\right]}{d\theta }{I}_e+{\left[I_2\right]}^t\frac{d\left[{\varphi }_{e2}\right]}{d\theta }$$

(13)

where:

$$\begin{array}{l}{\left[{\varphi }_2\right]}^t=\left[{\varphi }_{\alpha }{\varphi }_{\beta }\right]\\ {\left[{\varphi }_{e2}\right]}^t=\left[{\varphi }_{e\alpha }{\varphi }_{e\beta }\right]\end{array}\begin{array}{l}{\left[V_2\right]}^t=\left[V_{\alpha }{V}_{\beta }\right]\\ {\left[I_2\right]}^t=\left[I_{\alpha }{I}_{\beta }\right]\end{array}\left[L_2\right]=\left[\begin{array}{cc}{L}_{\alpha }& M_{\alpha \beta }\\ M_{\alpha \beta }& L_{\beta }\end{array}\right]\begin{array}{c}\\ {\left[M_{e2}\right]}^t=\left[M_{e\alpha }{M}_{e\beta }\right]\end{array}$$

3.1.4. Converter and Load Modeling

A DC/DC converter is utilized to regulate the DC coils of the HESG, while a resistive load is linked to the WCS through a full bridge rectifier.

The HESG we use is a parallel HESG. We must be able to increase and decrease the excitation flux ϕ_e. To increase ϕ_e, I_e must be positive, and to decrease it, I_e must be negative. For this purpose, the excitation circuit of the HESG used in our application is powered by a four-quadrant chopper controlled by the current.

The converters are modeled using SimPowerSystem tools, which enables the consideration of commutation effects and facilitates the testing of controllers in a realistic environment.

The implementation of the resistive load $R_c$ is also carried out using SimPowerSystem blocks. A specific resistance value of 15 Ω is chosen for this purpose, as indicated in reference [45].

Table 2. Parameters of the WCS.

Parameter	Nominal Value (Variation Range)	Parameter	Nominal Value (Variation Range)
ρ (kg/m3)	1.2	Pn (kW)	3
Kt (kg·m2·s−1)	0.055 (±75%)	P	6
Jt (kg·m2)	3.6	Me (mH)	1.1 (80–100%)
Rp (m)	1.5	Ld (mH)	5 (50–100%)
mp	5	Lq (mH)	9.2 (50–100%)
Dls	0.8 (±25%)	Le (mH)	46 (50–100%)
Kls	160 (+48%/−34%)	Re (Ω)	3 (100–150%)
Jg (kg·m²)	0.015	Rs (Ω)	0.87 (100–150%)
ϕa (mWb)	66 (80–100%)	EDC (V)	50
G0	10	VPM (V)	5
Vd (m/s)	3	Rc (Ω)	15
Vn (m/s)	15	Vm (m/s)	24

provides a summary of the nominal parameter values for the WCSs along with their corresponding ranges of variation.

3.2. Control System

3.2.1. Pitch Angle Control

The blade rotation model and the control loops are depicted in Figure 11. This illustration demonstrates that pitch regulator modeling involves three steps: generating the reference pitch angle denoted as ${\beta }_{ref}$, regulating the pitch angle $\beta$, and finally regulating the speed of variation of this angle. Three controllers are employed: $K_v\left(s\right)$ for speed, $K_{p\beta }\left(s\right)$ for position, and $K_p\left(s\right)$ for power. This section will delve into the synthesis of these three controllers [46,47].

-

Speed controller:

For safety reasons and considering the stress experienced by each of the blades, the rate of change of the pitch angle must be limited to ±10°/s. A PI controller is typically employed to adjust the rate of change of this angle, consisting of two parameters, $K_v$ and $T_v$, which are calculated to achieve a dynamic response of 10 ms, which is deemed appropriate for regulating the rotation speed of the pitch angle. $T_v$ is determined through pole compensation. The $K_v$ gain is computed by approximating the velocity loop behavior as a second-order system. Using the relationship ${\omega }_0·t_r=f\left(\xi \right)$ that relates the damping factor $\xi$ to the system bandwidth ${\omega }_0$ and its response time, ${\omega }_0$ can be set. For a damping factor of approximately 0.6, ${\omega }_0$ equals 500 rad/s. The final regulator is expressed by the following equation:

$$\hspace{1em}{K}_v(s)=K_v\frac{1+T_v·s}{T_v·s}=50·\frac{1+0.1·s}{0.1·s}$$

(14)

-

Position controller:

Given the presence of an integrator in the direct chain (14), two types of controllers may be used for position regulation: proportional or proportional-integral controllers. In this example, we examine a proportional controller ($K_{p\beta }$). The response time of the outer position loop should be at least 10 times greater than that of the inner velocity loop. The value of $K_{p\beta }$ is selected to achieve a second-order closed-loop transfer function (TF) (Figure 12) with a response time of 0.1 s. The damping coefficient is set at 0.7. For this value, the product ω0·tr is minimized, and the overshoot does not exceed 5%. Consequently, the bandwidth is set at 30 rad/s. Ultimately, the determined value for $K_{p\beta }$ is 29 s⁻¹.

-

Generation of the reference angle:

The wind conversion system is regarded as a single unit where the only variables of interest are the reference pitch angle ${\beta }_{ref}$ and the wind turbine power $P_t$. Identification is performed on the complete model of the wind conversion system for different values of the pitch angle ${\beta }_{ref}$ (Figure 13).

As illustrated in Figure 13, the input of the model is ${\beta }_{ref}$, and the output is $P_t$. Five operating points were selected within zone III ($v_w$ = 12 m/s, $v_w$ = 15 m/s, $v_w$ = 18 m/s, $v_w$ = 21 m/s, and $v_w$ = 23.5 m/s).

Figure 14 shows the Bode plots of the identified local TFs.

The provided TF represents the selected model, which is representative of average characteristics, specifically for a wind speed of 18 m/s.

$$\hspace{1em}H(s)=\frac{P_t(s)}{\beta (s)}=\frac{-1393·(9.97·s+1)}{1+13.5·s}\hspace{1em}$$

(15)

3.2.2. Regulation of the WCS Speed

The objective in wind turbine control is to achieve accurate reference tracking and minimize dynamic error. In the case of an HESG for maximum power extraction, the optimal rotation speed of the turbine can be controlled by adjusting the excitation current of the generator. This section focuses on designing two robust controllers, a H∞ controller and a second-generation CRONE controller, which can handle uncertainties in system parameters and external disturbances such as sudden changes in wind conditions and deliver satisfactory performance in all situations when operating in a closed-loop mode. Figure 15 shows a closed-loop system, with $K_{\mathsf{\Omega }}\left(s\right)$ representing the velocity controller to be synthesized.

As an HESG is highly nonlinear, an experimental identification is performed to drive an equivalent plant model. Three operating points in the MPPT zone are considered in the identification process. Figure 16 shows the Bode diagrams of the local TFs ${\Omega }_g(s)/i_{eref}(s)$ obtained from the linearization technique.

The TF of the average model (corresponding to v_w = 6.5 m/s) is given by:

$$G(s)=\hspace{1em}\frac{{\Omega }_g(s)}{i_{eref}(s)}=\frac{G_n}{T_{n}s+1}=\frac{-19.5}{3.28s+1}$$

(16)

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H∞ control strategy:

The Normalized Coprime Factors method is the focus of the current work. It is necessary to express the model under control using its prime elements. The normalized left coprime factorization of the nominal model, denoted by G(s), is as follows:

$$G(s)={\Omega }_g(s)/i_{eref}(s)=M^{-1}(s)·N(s)$$

(17)

In this context, $N\left(s\right)$ and $M\left(s\right)$ refer to stable TFs.

The NCF method provides a controller that stabilizes both the nominal model $G\left(s\right)$ and any model within the set of perturbed models, which includes additive uncertainties on $N\left(s\right)$ and $M\left(s\right)$ and is expressed as:

$$G_{\epsilon }=\left\{\begin{array}{l}{G}_p={\left(M+{\Delta }_M\right)}^{-1}\left(N+{\Delta }_N\right)\hspace{1em}\hspace{1em}\\ \hspace{1em}{\Vert \left[\begin{array}{cc}{\Delta }_N& {\Delta }_M\end{array}\right]\Vert }_{\infty }\le \epsilon \end{array}\right\}$$

(18)

In this context, the symbol $\epsilon$ represents the magnitude of the uncertainty that the plant can tolerate without causing destabilization.

The goal is to find a unifying controller, K(s), that stabilizes all the plants described by (19) by resolving the following equation:

$$\underset{K \mathrm{stabilising}}{\inf }{\Vert \left(\begin{array}{c}K\\ I\end{array}\right){\left(I+GK\right)}^{-1}{M}^{-1}\Vert }_{\infty }={\gamma }_{\min }={\epsilon }_{\max }^{-1}$$

(19)

In this particular context, the symbol ${\epsilon }_{max}$ refers to the maximum allowable stability margin.

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CRONE Control strategy:

CRONE control, which stands for non-integer order robust control (abbreviated in French), is a robust control methodology based on a frequency approach. In this approach, the corrected open-loop TF has a non-integer (fractional) order, either real or complex, allowing for the definition of an optimal open-loop TF based on parameters such as overshoot, speed, and precision. The CRONE control system was separated into three generations. The first generation entails keeping the controller’s phase constant around the appropriate open-loop crossover frequency. The second generation is employed when the nominal model’s gain and transitional frequencies vary. The third generation is useful when the model’s frequency response includes uncertainty other than gain and phase shifts.

According to the Bode plots shown in Figure 8, the second generation appears to be a suitable choice. It involves determining the open-loop TF $\beta \left(s\right)$ defined by (20) for the nominal plant state, which guarantees the desired specifications [48].

$$\begin{array}{l}\beta (s)=K_{CRONE}(s)·G(s)\\ =K_u{\left((1+\frac{s}{w_l})/(\frac{s}{w_l})\right)}^{n_1}·{\left((1+\frac{s}{w_h})/(\frac{s}{w_l})\right)}^n·\left(1/{\left(1+\frac{s}{w_h}\right)}^{n_h}\right)\end{array}$$

(20)

where

• $G\left(s\right)$: plant model;
• K_CRONE(s):velocity controller;
• K_u: constant ensuring unity gain at the desired frequency;
• $w_0$: desired frequency;
• $w_h$: transitional high frequencies;
• $w_l$: transitional low frequencies;
• $n_h$: order at high frequencies;
• $n_l$: the order at low frequencies;
• $n$: order around the crossover frequency.

4. Results and Simulations

The simulation is carried out under stochastic wind circumstances with an average velocity of 17 m/s (Figure 17). The H∞ controller maintains the generator’s rotating speed at 239 rd/s ± 1 rd/s, resulting in a 0.4% oscillation (Figure 18). In contrast, the CRONE controller limits oscillation to a maximum of 0.18 rd/s (Figure 19). This large reduction in oscillation reduces mechanical stress on the turbine shaft and dampens vibrations on the slow side of the shaft, increasing system lifetime.

To quantify the advantages, the turbine’s power output is limited to its ideal value of 3000 W (Figure 20) by changing the blade pitch angle (β). Although this change somewhat reduces aerodynamic efficiency (Figure 21), it ensures the system operates within safe mechanical limits, thereby extending the turbine’s operational lifespan.

Both controllers play critical roles in wind system stability, particularly at high wind speeds. For example, at a wind speed of 13.5 m/s, the reference excitation current calculated from open-loop experiments is 3.6 A, allowing the generator to maintain a constant speed of 239 rd/s. Figure 22 depicts the excitation current profiles, with the constant profile in blue and the speed regulator output in red, which averages 3.6 A.

Figure 22, Figure 23 and Figure 24 show the excitation current, generator speed, and torsional torque variation, respectively, versus time. Injecting a steady excitation current (blue profile in Figure 22) causes generator speed oscillations (Figure 23, blue profile), which result in damaging torque oscillations (Figure 24, blue profile). Such oscillations can result in mechanical fatigue and structural damage. However, creating an excitation current that oscillates around an appropriate average value (red profile in Figure 22) guarantees functioning at the desired speed while reducing mechanical vibrations.

Figure 23 and Figure 24 show that the developed controllers, notably the CRONE controller, successfully inject an excitation current to compensate for rotational speed fluctuations and therefore stabilize the system. At specific operating points, the controllers’ robust correction greatly minimizes vibrations, increasing the overall efficiency and dependability of the WECS.

The CRONE controller is very useful since it not only lowers mechanical oscillations but also improves the WECS’ efficiency. The CRONE controller ensures the turbine runs smoothly and effectively by maintaining appropriate rotational speeds and minimizing mechanical stresses, resulting in less wear and tear and potentially cheaper maintenance costs. This efficiency improvement, while not immediately measured in power production, transfers into enhanced long-term performance and dependability of the wind energy system.

5. Conclusions

By examining the impacts of wind stochasticity, control strategies, and WECS parameter uncertainty, this work successfully addressed the long-standing problem of wind turbine inefficiency. We evaluated many generator types often seen in contemporary wind systems after carefully reviewing the body of research, which enabled us to propose the hybrid excitation synchronous generator as a potential substitute for conventional generators. Furthermore, we provided a robust control strategy for HESGs that uses powerful regulators to reduce inefficiencies.

To assess the mechanical stability of the WECS, an integrated simulation platform using MATLAB Simulink R2020b has been developed. The simulation results clearly demonstrate the effectiveness of the CRONE controller when compared to the H∞ controller in reducing mechanical vibrations within the WECS. The wind turbine system’s overall performance and efficiency are greatly improved by this vibration reduction.

Further research and experimentation in this field are encouraged to refine and validate the proposed approaches and technologies for real-world implementation. By continuously refining and enhancing wind energy systems, we can accelerate the adoption of renewable energy and contribute to a greener and more sustainable future.