2.1. Wind Generation System and Model

Systems interact beyond the individual capacities of their elements. The science of systems, which studies these interactions, postulates universal principles, emphasizing the systemic nature that allows a system to be composed of parts of others while maintaining its totality [13, 14]. The functional and structural hierarchy is another systemic characteristic. Universal properties include totality, composition, and internal and external organization. These systems, composed of basic units such as elements or parts, interact and exchange information, with their behavior emerging from these interactions [13]. Models are simplified representations of interactions that seek to approximate the behavior of the real system [15]. They should be simpler but detailed enough to generate valid values comparable to the real system [16]. Validation occurs when the same set of inputs generates the same results in both the system and the model. Various representations are possible depending on the designer’s experience and the purpose of the model [14].

2.2. Turbine–Generator Coupling

Various electrical generator technologies are employed for wind power generation, each with distinct advantages and disadvantages [21]. Induction generators operate at higher speeds, while permanent magnet generators function at lower speeds with high torque. Switched reluctance generators offer a broad operating speed range. Because the turbine rotor speed ω_t is much lower than the synchronous generator speed, gearboxes are often used to increase the transmission ratio. The commonly used models include the two-mass model illustrated in Figure 3, adapted from Hansen and Butterfield [22].

In Figure 3, T_t represents the wind turbine torque; T_g is the generator torque; J_t is the turbine inertia; J_g is the generator inertia; ω_t and ω_g are the rotational speeds of the wind turbine and the generator, respectively; K_s is the shaft stiffness; and D_s is the damping constant. The gearbox increases the rotational transmission ratio, converting ω_t to a higher speed suitable for the operation of the generator. However, this transmission process incurs power losses [23].

2.3. Modeling of the Switched Reluctance Generator

In Figure 4, when the actuation of g_(θ) from the asymmetric half-bridge (AHB) converter occurs at positions where the inductance of the coils L increases (represented by the red curve), the torque is positive, and the machine operates as a motor. In contrast, when the actuation of g_(θ) from the AHB converter occurs at positions where the inductance of the coils L is decreasing (represented by the blue curve), the torque is negative, and the machine operates as a generator.

The inherent requirement to determine the position of the rotor makes the SRG drive complex. Since the 1970s, advances in power electronics and microprocessors have made it possible to optimize the performance of SRGs [25]. There are two ways to excite the SRG: (i) with an independent fixed voltage source or (ii) with self-excited operation [26]. The converter commonly used to drive the SRG, regardless of the excitation strategy, is the AHB [3]. Figure 5(a), adapted from Araújo et al. [3], illustrates this converter to drive the SRG with an independent excitation source (SRGIE). Each phase of the converter, composed of two series-connected switches Q with the coil and two diodes D, allows individual current control, enabling the magnetization of one phase while the other demagnetizes. Figures 5(b) and 5(c) illustrate the operating cycle of one phase.

In the SRGIE, the excitation of the AHB converter keeps the switches Q₁ and Q₂ on, connecting the voltage V_e to the phase coil. The current flows from the positive terminal of the source V_e through the switch Q₁, the coil, the switch Q₂, and finally through the negative terminal of V_e. In the subsequent generation stage, the switches Q₁ and Q₂ are turned off, allowing the electromotive force generated in the coil to keep the current flowing through the diodes D₁ and D₂. Thus, there are two driving stages for each phase. After excitation, freewheeling occurs, an intermediate stage when the current circulates through the switch Q₂ and the diode D₁. During this phase, the concatenated flux remains constant, but the current continues to increase due to the electromotive force of the coil, producing zero voltage [27].

In the case of SRGSE, self-excitation cannot be generated directly, but it is possible to add permanent magnets to the stator to generate magnetic flux for self-excitation [28]. However, this strategy has disadvantages, such as (i) increased manufacturing costs, (ii) dependence on speed to produce sufficient flux, and (iii) complications in the manufacturing process [26]. Another approach is to add capacitors in series or parallel to the phase winding and load, using the AHB converter. Figure 6(a), adapted from Touati et al. [26], illustrates the schematic of the AHB converter for the SRGSE drive with a capacitor in parallel with the load. Unlike SRGIE, the excitation bus is the same as the one receiving the generated energy. In this process, initial capacitor charging does not require an external source, as the capacitor starts to excite the phases [29]. Figures 6(b) and 6(c) detail the excitation and generation stages. The capacitor current I_cap is divided into two parts: one for excitation I_phase and the other to power the load I_load [9]. After initiating excitation, the phase current increases as a result of the excitation current and the action of the electromotive force. When the switches Q₁ and Q₂ open, the electromotive force allows the current to circulate through the diodes D₁ and D₂ [9].

3.1. Turbine Model

Turbine modeling is an important step in the proposed wind power generation system. The methodology involves defining input parameters and calculating variables such as the power coefficient C_P, the mechanical power of the turbine P_t, and the torque of the turbine T_t. Input constants, such as the diameter of the rotor A, the angle of motion β, and the density of the air ρ, are important for the precision of the model, together with variables such as the wind speed v_v and the rotation speed of the turbine ω_t. In calculating C_P shown in Equation (4), which influences η and turbine efficiency, variations in v_v, β, and ω_g are considered. Although the theoretical maximum value of C_P is 0.59, Betz limit, real turbines typically exhibit lower values, defined as $$.

Subsequently, P_t in Equation (2) and the turbine torque T_t in Equation (6) are calculated based on the value of C_P, considering the variable v_v. To ensure precise analyses of SRG performance in wind power generation, real data for v_v are collected. Since P_t depends on ω_t, the maximum power point tracking (MPPT) method is implemented to determine the optimal tip-speed ratio λ^∗ = v_v × ω_g and optimize η.

3.2. Maximum Power Point Tracking Proposed Approach

Given that the radius of the turbine R_t remains constant, the value of ω_t can be determined based on the knowledge of v_v. Taking into account the variation in v_v over time, finding the value of λ^∗ allows for the identification of ω_t that achieves $$ and consequently the maximum power P_t that can be extracted, as expressed in Equations (2) and (6).

3.3. Coupling Turbine–Generator

The coupling between the wind turbine and the SRG is modeled to establish the relationship between the wind speed v_v, the wind turbine torque, and the generator torque. Given that wind turbines operate at low speeds, a gearbox is commonly used to elevate this speed to values within the generator’s operating range. However, the SRG is uniquely capable of functioning at low rotational speeds, which distinguishes it from other generators used in wind power generation, such as induction generators. This characteristic implies that the gear ratio G for the SRG may be lower than that of other machines. In some cases, this ratio may not even be required, allowing the wind turbine and the generator to operate at the same speed.

The gearbox model requires knowledge of the inertia values J_t, stiffness K_s, and shaft damping constant D_s. Accurately representing these parameters poses challenges, especially given the SRG’s low operating speeds. This characteristic might eliminate the need for a gearbox in some cases. Therefore, the optimal value of G for the proposed model is evaluated based on the nominal speed of the SRG.

3.4. Switched Reluctance Generator Design

In this design approach, our objective is to extract the actual mechanical and electrical parameters of the SRG through simulations. The SRG is configured as 6 × 4, with its external dimensions shown in Figures 8(a) and 8(b). Our housing design allocates 0.003178 m³ space to accommodate the laminated core. This 6 × 4 configuration allows the SRG to operate at higher speeds, facilitated by a six-switch power converter, making the control more cost-effective. However, the reduction in stator pole count results in more oscillatory torque production by the SRG. Figure 8(c) illustrates the geometric parameters, which are calculated to achieve the desired output parameters.

3.5. Implementation of the Proposed Generator Model

Several important steps are involved in the SRG simulation process, including implementing the mathematical model, defining the inductance profile, specifying the switching logic of the AHB converter, and determining the construction parameters (both electrical and mechanical). Figure 9 outlines the flowchart of this process, showcasing two distinct excitation strategies: SRGIE and SRGSE. Despite their differences in excitation methods, both strategies utilize the same machine model. Once the machine model is established, the simulation process begins by acquiring electrical parameters such as the resistance of the stator coil R and mechanical parameters such as the viscous friction coefficient D_A and the moment of inertia J_g. These parameters are integral in computing the state matrix, as given by Equation (12).

The AHB converter, illustrated in Figure 5(a) for SRGIE and Figure 6(a) for SRGSE, is fundamental for regulating instantaneous phase currents i based on coil voltages v. Its selection is driven by its capability to control individual phase currents, allowing for simultaneous magnetization and demagnetization of phases. The switching logic block, as illustrated in Figure 9, utilizes the instantaneous rotor position θ obtained from the angular position sensor. This position is compared with the angles θ_on and θ_off of the AHB converter switches to generate the switching logic. Each phase of the AHB converter comprises two electronic switches: insulated gate bipolar transistors (IGBT) labeled Q₁ to Q₆ and fast recovery diodes (FRDs) labeled D₁ to D₆. The results are thoroughly evaluated to validate the proposed model, with simulations carried out across varying excitation voltages V_e, angles θ_on and θ_off of the AHB converter switches, and generator rotation speeds ω_g.

3.6. Analysis of the Wind Power Generation System

Once real wind speed data are collected and wind turbine, coupling, SRGIE, and SRGSE models are built, the complete model of the wind power generation system is obtained. The flowchart guiding the construction of this model is illustrated in Figure 10, highlighted in the gray region. For each value of v_v, the wind turbine block computes ω_t and the developed torque T_t. The coupling block connects the wind turbine on the low-speed side with the generator on the high-speed side. The SRG block receives inputs including ω_g, torque T_g from the generator, V_e from the AHB converter, and P_s. Once the model is assembled, simulations are executed to observe the behavior of SRGIE and SRGSE for different inputs v_v, which facilitates performance evaluation with real data.

3.7. Tracking Process

The optimization process manipulates variables such as V_e, angles θ_on, and θ_off to maximize P_s and η. Our methodology involves fixing V_e and θ_on while tracking the values of θ_off that produce the maximum power throughout the operating speed range of SRGIE and SRGSE. After defining the value of V_e, simulations are performed to determine the optimal value of θ_on considering P_s and η. Once this is determined, the θ_off tracking block is introduced into the model, as illustrated in Figure 10. This block receives ω_g from the coupling block and searches for the optimal value of θ_off. This search takes into account the P_e, P_s, and P_mec curves, along with the η curve, analyzing them with varying values of θ_off and ω_g. The proposed methodology for θ_off follow-up is illustrated in Figure 11.

At the end of this process, a surface of power and efficiency η is generated, covering the entire range of θ_off and ω_g. This surface allows for pinpointing the values of θ_off that yield the maximum power and efficiency for each speed analyzed. However, the goal extends beyond the identification of these optimal values for specific speeds. The aim is to derive a function that can perform this calculation at any speed. This function can then be incorporated into the simulation within the θ_off tracking block, as illustrated in Figure 11, thus generating the θ_off reference for the AHB converter.

3.8. Applied Optimization Process

Control of the SRG output voltage V_s is implemented alongside the θ_off tracker using a PID controller integrated in the simulation. The manipulated variables to achieve V_s control are (i) V_e, (ii) θ_on, and (iii) θ_off [3]. They evaluated these manipulated variables for V_s control of the SRG. Initially, they searched for the value of θ_on that contributes to increasing P_s and η and reducing the torque oscillation T_R. Under these conditions, the conduction window is kept at θ_D = 30°, and the value of θ_on is decreased until the established conditions are met. Following this procedure, the authors applied the optimization process to the analyzed variables. With the optimized values, they performed a sensitivity analysis of the parameters [13], obtaining sensitivity indices [3]. They found θ_on = −10°to be the optimal value, which is considered fixed.

The normalized minimization of the error area, represented by the integral of the absolute error, the overshoot represented by V_s(t_M) − V_s(∞), and the steady-state error represented by $$, is defined in Equation (18). With optimized gains from the controller, simulations can be performed applying disturbances to the system to verify the performance in controlling V_s.

3.9. Wind Power Generation System Validation

Computational experiments are conducted to analyze and validate the proposed wind power generation model, providing information for decision-making on the use of the SRG. Initially, these experiments involve testing with various values of wind speed v_v, starting with constant v_v and then introducing fluctuations to assess the system’s response to sudden changes over short periods. Given the variable and dynamic nature of wind, which directly influences power output P_s, it is necessary for the SRG to demonstrate its ability to generate power over a broad spectrum of v_v values.

Data for these experiments are sourced from databases that contain information from meteorological stations or wind atlases, providing information on the wind potential of specific regions. Based on these data, locations are selected as potential sites for wind turbine installation. It is important to note that adjustments may be necessary for wind speed data, as measurement heights can vary among meteorological stations. After adapting the data to the relevant heights for analysis, the regions are chosen to encompass low, medium, and high profiles v_v.

4.1. Wind Turbine Simulation

Simulations were performed using a computational model derived from expressions shown in Equations (1), (2), (3), (4), (5), and (6) to analyze the behavior of the wind turbine. The radius of the turbine, R_t, is set at 1.6 m. At a wind speed of v_v = 10 m/s, the generated mechanical power, P_mec, reaches 2, 000 W, which is sufficient to meet the power requirements for the switched reluctance generator (SRG). To achieve the maximum power coefficient $$, the pitch angle of the blade is set to β = 0°. The simulations cover wind speeds ranging from 4 to 10 m/s. Table 1 outlines the characteristics of the wind turbine, while Figure 12 illustrates the behavior of the turbine power P_t, at various wind speeds. It is observed that, as v_v increases, P_t also increases. However, the influence on power extraction is not solely dependent on v_v. Specific values of ω_t allow optimal power extraction in each v_v, as illustrated by the ascending red curve in Figure 12.

4.2. Search for the Maximum Power Point and Turbine–Generator Coupling

To implement the maximum power point tracking algorithm (MPPT), the pitch angle of the blade, β, is first evaluated. This angle represents the deviation between the turbine blades and the line perpendicular to its axis. When β = 0°, the blades align precisely with the direction of the wind. As β increases, the blades tilt in relation to the direction of the wind. Keeping β = 0°, the value of $$ is determined. With a constant β, the power coefficient, C_P, varies as a function of λ. The goal is to find the specific value of λ that yields the maximum $$. To achieve this, the curve C_P in relation to λ is analyzed, as illustrated in Figure 13.

The optimal value of λ^∗ = 7.8 was determined, corresponding to $$. The angular velocity, ω_t, for each wind speed, v_v, was then calculated using the expression in Equation (13). With the MPPT algorithm in place, simulations were conducted on the wind turbine. The simulation covered wind speeds of v_v = [7,  10,  6] m/s, with a 3-s interval. This abrupt variation in v_v mirrors real-world wind turbine scenarios. Figure 14 illustrates the response of the turbine to sudden changes in v_v. During a total simulation time of t = 10 s, at t = 3 s, v_v increases from 7 to 10 m/s, resulting in instantaneous increases in both turbine power P_t and angular velocity ω_t. Similarly, at t = 6 s, v_v decreases from 10 to 6 m/s, leading to reductions in both P_t and ω_t. Thus, the model effectively responds to changes in wind speed.

In the speed ranges where wind turbines typically operate, which are considered low compared to conventional generators, the ideal value of the speed multiplication ratio, G, was established as G = 2. This choice aligns with the switched reluctance generator’s (SRG) ability to function effectively at low rotational speeds. One of the objectives of this study was to evaluate the performance of the SRG in low-speed operations, and selecting G = 2 ensured the lowest feasible value capable of generating energy.

4.3. Machine Parameter Identification

Tests were carried out using the handmade prototype of the switched reluctance generator (SRG) to obtain fundamental parameters for simulation. During the construction of the phase coils, the conductors were extended by equipping the terminals with cables and connecting connector pins to each coil. The resulting phase resistance was 3.25 Ω.

The accuracy of the SRG model depends on the representation of inductances. To achieve this, parametric regression is employed to determine the optimal inductance surface. In the surface analysis of experimental inductance, data were collected at 108 points corresponding to the angular positions of 0°, 6°, 11°, 16°, 22°, 27°, 33°, 39°, and 45° (the position of total misalignment). For each angular position, the coil received a current ranging from 0 A to 6 A, at intervals of 0.5 A. By applying parametric regression, the expression L(θ, i) was derived, and the experimental inductance surface was obtained. This surface significantly contributes to the accuracy of the model while also reducing the simulation time. Refer to Figure 15 for a visualization of the experimental inductance surface of the SRG.

Figures 16(a) and 16(b) provide detailed views of the custom generator prototype. To determine the viscous friction and inertia values, dynamic tests were performed applying varying torque levels and measuring the resulting accelerations. The equation of motion T_g = J_g · α_g + D_A · ω_g was used, where α_g represents the angular acceleration of the SRG. Table 2 lists the key electrical, mechanical, and geometric parameters of the constructed SRG.

4.4. Simulation of the Switched Reluctance Generator

During the excitation phase, a voltage of approximately 100 V is reached. After IGBTs are turned off, the voltage reverses and reaches approximately 216 V during the generation phase. In this phase, the diodes of the AHB converter are reversed, directing the voltage to the load and maintaining the same current direction, as illustrated in Figure 5. The IGBT command pulse is applied when the inductance is decreasing, allowing the machine to operate as a generator. Specifically, the conduction window occurs between the positions θ_on = 0° (maximum inductance) and θ_off = 30° (minimum inductance), resulting in a conduction window of θ_D = 30°. This configuration, known in the literature as ideal for 6 × 4 SRG, ensures efficient operation.

Figure 17 shows the current and voltage behavior in Phase A, with peak values of i_a = 15.76 A and v_a = 97.12 V. During the excitation phase, v_a reaches 212 V, while in the generation phase, the machine exhibits resilient performance. The power characteristics P_mec, P_e, and P_s are illustrated in Figure 18(a). It is observed that, when an input power is approximately P_e = 400 W, the machine achieves mechanical power P_mec ≈ 600 W and shaft power P_s ≈ 559WP_s ≈ 559 W, both exceeding the power consumed by the excitation source. Finally, Figure 18(b) presents the efficiency curve of the simulated SRG, obtained using Equation (17).

4.5. Validation of the Wind Turbine and Switched Reluctance Generator Models

In this study, the models were validated by comparing the simulation results with the experimental data and the simulations of other studies. The focus was specifically on models representing the horizontal axis wind turbine (HAWT) and the switched reluctance generator (SRG). The HAWT model, widely recognized in the literature, has been extensively employed in various studies. Its expression for power calculation, defined in Equation (2), originates directly from physical principles. This expression establishes the relationship between the conversion of kinetic wind energy to mechanical energy and subsequent generation of electrical energy [17]. The precision of this model depends on the precise definitions of the parameters and the calculation of the power coefficient C_P. The methods for computing C_P rely on the parameters β and λ. The study carried out by Castillo et al. [33] analyzed existing methods, highlighting that the approaches proposed by Ovando et al. [34] and Guo et al. [35] effectively represent the coefficients using expressions Equations (4) and (5).

These methods undergo rigorous validation through simulations and experimental tests. The same approach to calculate the power coefficient C_P is used consistently in various studies, including those of [36, 37, 38]. Our wind turbine simulations show that increasing the parameter β leads to a reduction in both the total power P_t and the coefficient C_P. Furthermore, this investigation identifies an optimal λ value that maximizes power output. These findings align with the existing literature.

The computational model for the switched reluctance generator (SRG) was developed based on the expressions in Equations (7) to (12). The SRG parameters were experimentally determined and are summarized in Table 2. The accuracy of the model critically depends on how the inductances are represented. In this study, the inductance representation model proposed by Araújo et al. [3] and da Cunha Reis [16], which uses parametric regression, was adopted. The experimental inductance surface, Figure 15, was then compared with the simulated continuous inductance surface (Figure 19), further validating our SRG model.

The SRG was simulated with V_e = 100 V and a rotational speed of ω_g = 600 rpm. The voltage, current, voltage pulse, and inductance curves in Figure 17 can be compared with those obtained in previous studies, such as those of Araújo et al. [3] and Barros et al. [7]. Typically, all studies initiate machine excitation with θ_on = 0° and θ_off = 30°, marking the beginning of the generation stage. During this phase, the machine achieves an output voltage greater than the input. In our work, the SRG was reached 216 V during the generation phase with an input voltage of V_e = 100 V, consistent with the results reported in the literature. It is observed that in all simulations, the shaft power P_s exceeded the electrical power P_e, as illustrated in Figure 18(a).

4.6. Simulation of the Wind Generation System

Following the implementation and validation of the wind turbine, coupling and SRG models, each model was successfully interconnected to create a comprehensive wind generation system model, as shown in Figure 10. This integrated model allowed the assessment of SRG performance under real wind conditions, including abrupt changes. In Table 3, the specific parameters adopted for this simulation are presented, while the remaining parameters for the wind turbine and the SRG remain consistent with those listed in Tables 1 and 2.

To perform simulations with real v_v data, information was obtained from the Weather Spark Platform. This platform offers detailed information on current and historical weather conditions, including comprehensive analyses of various meteorological parameters such as wind speed. Users can explore weather data for specific locations and download relevant information [39]. Using these collected data, the wind energy generation was estimated at each chosen location, allowing the evaluation of the performance of the SRG under different wind speed scenarios. The Weather Spark Platform provides parameters such as the annual average wind speed $$ at various heights, as well as additional measures such as the annual diurnal profile and typical daily variation. Specifically, the arithmetic average of v_v recorded at a consistent time throughout the year, at a height of 10 m, defines each hourly value. By analyzing latitude and longitude data for specific locations, the behavior of wind speed over extended periods, such as 12 months, can be studied. In selecting simulation locations, regions with varying wind speed profiles were considered, including classifications of low, medium, and high wind generation potential. Table 4 displays the extrapolated $$ for a height of 100 m for the three chosen locations.

From the selected locations, daily data series for each city were extracted. Subsequently, the monthly averages for each city were calculated and simulated over 12 months. Figure 20 illustrates the wind speed profiles in the three regions, reflecting the three speed profiles analyzed. The operation of the MPPT algorithm involves increasing ω_g to match the increase in v_v. However, this leads to a reduction in the excitation time of the generator. As a consequence, P_e decreases, resulting in a corresponding reduction in P_mec. To improve generation efficiency, it is fundamental that P_e increases as ω_g grows. To achieve this, the tracking of θ_off is implemented. This strategy involves keeping the values of θ_on and V_e fixed while determining the optimal value of θ_off that maximizes the generation.

4.7. θ_off Tracking Strategy for SRGIE

In the θ_off tracking strategy, the value θ_on = −10°, as indicated by Araújo et al. [3], was adopted. This choice ensures efficient generation, high efficiency η, and reduced torque oscillation. With a fixed input voltage of V_e = 100 V, the goal was to optimize the system. Simulations were carried out in the range 100 rpm ≤ ω_g ≤ 2, 000 rpm and −5° ≤θ_off ≤ 30°, as illustrated in Figure 11. By analyzing the power and efficiency curves η, the optimal θ_off that maximizes the shaft power P_s can be identified. However, since the SRG relies on an external excitation source, the excitation power P_e, which represents the energy supplied to the excitation system, was also evaluated. Unlike traditional MPPT algorithms, the proposed optimization aims to find the value θ_off for each ω_g, resulting in the highest achievable P_s and η.

In Figure 21(a), P_e is almost negligible for small angles θ_off, particularly in the range −5° to 10°. This behavior arises because the fixed θ_on = −10° results in a narrow conduction window θ_D. Consequently, each SRG coil experiences brief excitation periods. Furthermore, P_e remains below 2,000 W for all speeds when θ_off < 13°. Beyond this value, P_e reaches its peak when the generation rotation speed ω_g = 100 rpm and θ_off = 30°. Similarly, P_mec, as shown in Figure 21(b), is also influenced by values of θ_off below 15°. Although there is an increase in P_mec as θ_off increases, this effect is observed only until ω_g ≈ 1, 200 rpm, after which P_mec starts to decrease.

In Figure 22(a), increasing the values of θ_off leads to higher P_s. However, the optimal values of θ_off vary for each ω_g, requiring dynamic adjustment of θ_off to achieve maximum utilization of the generation. In addition to P_mec, P_e, and P_s, efficiency η is also considered, as shown in Figure 22(b). The efficiency η is observed to increase with greater ω_g. The goal of our optimization process is to find the θ_off angles that achieve the highest possible P_s and η for each generator operating speed. This approach considers tradeoffs: the search for values θ_off that maximize P_s can lead to a high P_e, resulting in η ≈ 0 and making the operation unfeasible. In contrast, focusing solely on the θ_off value that maximizes η could result in P_s ≈ 0, rendering the operation impractical. By analyzing the power surfaces in Figures 21((b) and 22((a) with respect to η, shown in Figure 22(b), the angles θ_off that strike the optimal balance between P_s and η can be determined.

From Figures 23(a), 23(b), 23(c), and 23(d), the relationship between shaft power P_s, efficiency η, and θ_off for 100 rpm ≤ ω_g ≤ 2, 000 rpm is explored. To aid in visualization, P_s is normalized relative to its base value of 2,626 W. Identifying the θ_off values that maximize both P_s and η for each ω_g, the optimal θ_off for each analyzed speed range can be determined. However, the ultimate goal is to derive a mathematical expression that relates any ω_g to θ_off. To achieve this, 20 θ_off values were initially collected, and an interpolation was performed to obtain 400 data points. Using these interpolated values, a ninth degree polynomial was derived using the least squares method. The resulting polynomial θ_off(ω_g) is given by the expression in Equation (24):

Through the expression shown in Equation (24), the optimal values of θ_off throughout the range of ω_g are determined, as illustrated in Figure 24. These values correspond to different wind speed profiles v_v from various locations. Consequently, the values of θ_off that maximize system operation are obtained. The expression in Equation (24) is then incorporated into the simulation of the wind generation system, with the aim of finding the ideal θ_off values and consequently maintaining the optimal λ ratio, resulting in the extraction of maximum P_mec from the wind turbine.

The simulations were carried out using wind speed profiles corresponding to three regions, as listed in Table 4. In the simulation without θ_off tracking, a minimum value of P_e = 543 W was observed at Location 3, while the maximum value of P_e = 1, 303 W was recorded at Location 1. In simulation with θ_off tracking, the minimum value of P_e = 768 W occurred at Location 3, while the maximum value reached P_e = 1, 134 W at Location 1. The θ_off tracker resulted in an increase of 35.27% in P_e at Location 2 and 21.96% at Location 3, which have higher wind speeds. Regarding P_s, in the simulation without the tracker, P_s = 1, 268 W was obtained at Location 3, with the maximum value of P_s = 1, 724 W recorded at Location 1. In the simulation with the tracker, the minimum value of P_s = 1, 551 W occurred at Location 1, while the maximum value reached P_s = 1, 701 W at Location 3. Figures 25(a), 25(b), 25(c), 25(d), 25(e), and 25(f) present P_s and η of the SRGIE with and without θ_off tracking.

In the simulation without the θ_off tracker, the maximum value of P_s was achieved. However, analyzing the complete simulation, a reduction of 67.11% was observed in the variation $$ with the θ_off tracker. The θ_off tracker demonstrated efficiency in optimizing generation and ensuring maximum utilization. Furthermore, in the simulation with the tracker, the algorithm adjusts θ_D to prolong the machine excitation time, adjusting to the increase in v_v. The influence of v_v is also evident in P_mec, as the θ_off tracker alters ω_g. Consequently, in the tracker simulation, P_mec is higher in regions with elevated v_v. Regarding efficiency η, values of approximately 59% ≤ η ≤ 75.6% were recorded in the simulation without tracking and 61% ≤ η ≤ 72% in the simulation with tracking, representing the minimum and maximum values, respectively. Although there was no significant increase in η, the variation was reduced from Δ_η = 16.6% without tracking to Δ_η = 11% with tracking.

Table 5 provides the averages of powers P_e, P_mec, P_s, and η under conditions both without and with θ_off tracking. An increase in P_s is observed for all three locations when θ_off tracking is implemented. Although the simulation without θ_off tracking showed a higher overall η, the values of η remained above 65% even with tracking. The θ_off values selected during the tracking resulted in a significant increase in P_s for the three locations analyzed and helped to reduce the oscillatory effect by widening the conduction window.

In Location 1, which has an average wind speed of 5.4 m/s (Table 4), the system without optimization generated 1,522 W, equivalent to ≈1.5 kW. However, by applying the optimization process using the proposed method, energy generation increased to ≈1.6 kW. These results demonstrate the positive impact of the θ_off tracking technique on the performance of the SRG, resulting in significant improvements in efficiency and the amount of energy generated, even at relatively low wind speeds. Furthermore, in Location 3, with an average wind speed of 8.4 m/s (Table 4), the optimization process increased the output power from 1,286 W without optimization to ≈1.65 kW, as shown in Table 5. These data indicate the efficiency of optimization under different wind conditions, maximizing energy generation.

4.8. Output Voltage Control of the Generator

To implement the control of the output voltage V_s of the SRG in conjunction with the θ_off tracker, a proportional integral derivative (PID) controller was incorporated into the simulation. The value of θ_on was kept constant at −10°, and the excitation voltage V_e was chosen as the variable manipulated. The PID gains were fine-tuned using the Nelder–Mead simplex algorithm. The reference voltage $$ was set at 180 V and the initial parameters, obtained from prior knowledge of the plant, were k_p = 3°/V, k_i = 5°/V, and k_d = 0.1°/V. The optimization process resulted in gains of k_p = 2.3889°/V, k_i = 6.6075°/V, and k_d = 0.0752°/V, with a deviation of IAE = 20.2185 V. Figures 26(a) and 26(b) present the results of the simulations for v_v = [7, 10, 6] m/s, at intervals of 3 s.

By controlling V_s through V_e, consistent behavior was observed throughout the simulation, with V_s closely tracking the reference voltage $$. At time t = 3 s, the generator increases ω_g to follow the change in v_v, which goes from 7 to 10 m/s, and the same happens at t = 6 s. Despite the variation in ω_g, the PID controller kept V_s practically constant. Figures 26(c) and 26(d) show the results of P_s and η with V_s control. Even with variation in ω_g, P_s remains at approximately 1,000 W, and η shows no significant variations, ranging from η = 68.14% to η = 72.89%. After simulations with variable speed, a simulation was performed varying the reference value, starting with $$ and changing to $$ at t = 3 s, followed by another change to $$ at t = 6 s. Figure 26(e) shows the results of V_s and $$. In the simulation with variation in $$ and ω_g, the PID controller with optimized gains keeps V_s constant. Furthermore, the PID controller operates in conjunction with the θ_off tracker.

4.9. Self-Excited Switched Reluctance Generator

In simulating the self-excited switched reluctance generator (SRGSE), a model similar to the SRGIE is adopted, with the addition of a capacitor for initial excitation. It is observed that there is no segregation in the connection circuit between the excitation and generation stages, as illustrated in Figure 6(a). The capacitor, initially charged by an external source such as a battery or solar energy, facilitates the initial excitation of the machine. Subsequently, once the voltage surpasses that of the capacitor, the diode D₇ becomes reverse biased, which moves the generator into self-excited mode. To avoid interference from the excitation source, an on–off switch is incorporated, programmed to open after a duration of 0.5 s.

The specifications of the capacitor include a capacitance of 5 mF and an initial charge of 50 V. Keeping a constant speed of ω_g = 600 rpm, the firing window is restricted to 0°≤θ_D ≤ 30°. For simulating the self-excited switched reluctance generator (SRGSE), the same model was employed as the switched reluctance generator with initial excitation (SRGIE). However, in the SRGSE, a capacitor was introduced to facilitate the initial excitation of the machine. Unlike SRGIE, there is no separation between the connection circuits for the excitation and generation stages.

4.10. Analysis with Different Load Values

Initially, a simulation was performed using the same value of R_load = 33 Ω as in the SRGIE model. The resulting output voltage is illustrated in Figure 27(a). Upon analysis of Figure 27(a), it was observed that the SRGSE model did not generate energy with R_load = 33 Ω, as V_s remained below the initial V_e, which led to no energy generation. To comprehensively assess the performance of SRGSE at different values of R_load, simulations were carried out within the range of 10 Ω ≤ R_load ≤ 120 Ω, with intervals of 10 Ω. Furthermore, the speed varied within the range of 100 rpm ≤ ω_g ≤ 2, 000 rpm, with intervals of 100 rpm. The initial charge voltage of the capacitor was maintained at 50 V to ensure excitation throughout the range of ω_g. From the results of these simulations, the values of P_s were calculated for all combinations of R_load and ω_g. The surface plot generated based on the values of P_mec, R_load, and ω_g is presented in Figure 27(b). In Figure 27(b), it is observed that P_s increases as the values of R_load and ω_g increase.

When the resistance is below 60 Ω, the machine reaches a maximum of P_s = 1, 000 W throughout the analyzed range of ω_g. In the range between 60 and 120 Ω, there is a significant increase in P_s, which escalates with increasing ω_g, eventually exceeding the values of P_s = 4 kW. However, considering the limitation of the actual machine of 2.6 kW, a load value of 100 Ω was selected. This choice allows P_s to approximate 2.6 kW at ω_g ≈ 1, 500 rpm. In Figure 28(a), which presents the output voltage of this simulation, the progression can be observed from an initial V_e = 50 V to peak values of V_e ≈ 140 V in the SRGSE can be observed. The waveform shows moments of capacitor charging and discharging. Furthermore, Figure 28(b) illustrates the performance metrics of P_mec, P_e, and P_s for the SRGSE. Following the initial excitation of the capacitor, P_e remains negligible. Under these conditions, P_mec ≈ 1, 200  W and P_s ≈ 184 W were recorded for the SRGSE.

The inductance curves of the three phases of the machine were analyzed using the same method as in the SRGIE. In Figure 29(a), the behavior of voltage V_e, current i, command pulse, and inductance L of one of the phases of the SRGSE is presented. During the phase excitation stage, V_e was approximately 138 V. After the switches were turned off during the generation stage, V_e reached approximately 232 V, with the maximum current i_p (peak) reaching values close to 25 A. Figure 29(b) displays the efficiency η, calculated considering P_mec and P_s, as there is no external excitation. With a value of η ≈ 18%, it is relatively low compared to SRGIE, which achieved η > 50%. However, it is important to note that in this configuration, no external excitation source was used, offering a significant advantage for wind power generation applications.

4.11. θ_off Tracking Strategy for SRGSE

The tracking of θ_off for SRGSE followed the same procedure as for SRGIE. The parameters were set as θ_on = −10°, V_e = 50 V for the initial capacitor charge, and R_load = 100 Ω. To optimize the system, simulations were performed covering 100 rpm ≤ ω_g ≤ 2, 000 rpm and −5° ≤θ_off ≤ 30°. Figure 30 illustrates the surface that represents the relationship between P_s, ω_g, and θ_off. In this simulation, the power P_s exceeded the nominal electrical power of the machine. Hence, P_s = 2.62 kW was defined as the maximum observed in the red plane in Figure 30.

Incorporating Equation (25) in the simulation of the wind power generation system enables us to track the optimal value of θ_off in the same way as in the SRGIE. This tracking of θ_off is executed for each wind speed profile illustrated in Figure 20. Figures 32a), 32(b), 32(c), 32(d), 32(e), and 32(ff) present the corresponding P_s for these profiles, with and without θ_off tracking. For comparison purposes, Figure 25 for SRGIE and Figure 32 for SRGSE present P_s and η associated with the same profiles, with and without θ_off tracking. It was observed that P_s with θ_off tracking performed better in both cases, in the SRGIE and in the SRGSE, highlighting the efficacy of the tracker in conjunction with optimized parameters. It is observed that, although the SRGSE θ_off tracking aims to maximize P_s, the tracking, in addition to achieving this maximization, also enhances η. In contrast, in the SRGIE, while there was a maximum of power, there was no significant improvement in η. This underscores the dual benefits of the tracker in the SRGSE scenario, optimizing both power output and efficiency.

4.12. Discussion

The amount of electrical energy generated from wind power is intrinsically related to wind speed v_v. Although wind farms are often located in regions with higher wind availability, it is possible to harness electrical energy even at low wind speeds, provided that these winds remain consistent and that the generator used can operate effectively at low speeds. This study builds on previous research involving the switched reluctance generator (SRG), which functions as both a motor and a generator. Specifically, an SRG with a geometry of 6 × 4 was designed and built, its performance was evaluated, and a novel technique to determine the inductance profile was introduced. This enhancement improves the efficiency of the model and, consequently, the overall system.

Our innovative evaluation of the SRG for wind power generation integrates wind turbine modeling, the θ_off tracking technique, and optimization of PID controller gains for both the SRGIE and SRGSE models. The primary objective is to analyze the performance of the SRG under low wind speed conditions. The results presented here are derived from laboratory experiments and simulations, all validated experimentally. The wind turbine was sized to match the built and validated SRG, and the SRG simulations were compared with results from other research studies [3, 4, 6, 7, 8, 16].

As the wind speed v_v increased, the generation of electrical energy decreased. This phenomenon occurred because increasing the generator speed ω_g to match the MPPT of the turbine reduced the machine’s excitation time. To optimize performance across the entire speed range, the θ_off tracker was implemented, dynamically adjusting the value of θ_off to v_v and consequently ω_g varied. The θ_off tracking methodology was adapted for both the SRGIE and SRGSE models. For SRGIE, both the shaft power P_s and the efficiency η were analyzed. In the case of SRGSE, which does not rely on an external excitation source, the value of θ_off was selected that provided the highest P_s. To evaluate the performance of the θ_off tracker, simulations were performed using real v_v data for both SRGIE and SRGSE models. The results demonstrated the efficiency of the θ_off tracker in effectively maximizing generation in both system configurations.

In the SRGIE model, the voltage output control V_s was implemented using an optimized PID controller. Remarkably, this controller operates in parallel with the θ_off tracker, allowing efficient control of the SRGIE’s V_s while simultaneously enhancing energy generation. Meanwhile, in the SRGSE model, simulations were performed by varying the load resistance. Our results confirmed the observations made by Bernardeli et al. [9], showing that P_s increases as the load value increases. Simulations using real wind speed data v_v allowed us to evaluate the performance of SRGIE and SRGSE in three regions. In Location 1, with v_v ≈ 5 m/s, the SRGIE achieved P_s ≈ 1, 548 W, and SRGSE achieved P_s ≈ 480 W. In Location 3, with v_v ≈ 8.4 m/s, the SRGIE obtained P_s ≈ 1, 655 W, and the SRGSE obtained P_s ≈ 2, 442 W.

Both the SRGIE and SRGSE models demonstrated the ability to generate electrical energy across different v_v profiles. The SRGIE is observed to perform better under lower v_v conditions, while the SRGSE excelled at higher speeds. However, determining which model is superior remains challenging, as each involves distinct parameter definitions and employs different methodologies for the tracking of θ_off. It should be noted that all results were obtained using a custom SRG prototype (handmade SRG prototype). Consequently, on a commercial machine with optimized parameters, performance is expected to surpass that found in this study.