Efficiency Improvement of Standalone and Grid-Connected Wind Turbines via Stator Current Optimization Control of a PM Synchronous Generator

Variable-speed wind turbine systems using permanent magnet synchronous generators (PMSGs) have been widely studied due to their high performance in variable-frequency applications. PMSG provide high efficiency by combining electromagnetic and reactive torque, with wide-range operation using flux-attenuation technology. In this study [1], a wind generator system using an internal PMSG is investigated. PMSG are widely used in many applications as high-performance variable speed drives. In particular, PMSG can offer a highly efficient drive by using reactive torque in addition to magnet torque and provide constant power operation over a wide range using the magnetic flux attenuation method [2]. The optimization of the electromagnetic structure and the control method of the PMSG have attracted significant research interest in recent years. Several methods of controlling the PMSG exist, such as vector control at a given vector current value along the axis (i_d = 0), maximum torque per ampere (MTPA), maximum speed per voltage control, loss model control [3-6]. The optimal power search algorithm proposed in this literature [7] maintains peak power points on the power curve. In this research [8], an algorithm for maintaining maximum power based on a neural network is proposed, and in this study [9], an algorithm for maintaining maximum power based on a backpropagation neural network is proposed. However, the control of the PMSG of the surface type is simpler than that of the internal PMSG [10, 11]. Moreover, in many published works, efficiency optimization of PMSG systems has been largely overlooked. In paper [12], the magnetic saturation effect is considered, and the inductance of the internal PMSG along the q axis varies depending on the stator current along the same axis. In addition, in the internal PMSG, the stator current components on both the d and q axis contribute to the developed torque. As a result, the nonlinearity of the system becomes severe if the internal PMSG operates in the flux attenuation region, where the stator current component along the d axis i_d = 0. This nonlinearity complicates the direct application of the linear system theory. To solve this problem, non-linear control schemes have been developed to improve the performance of the internal FGM [13].

The minimization of losses in the PMSM core through the suitable design of magnets and grooves, as well as the selection of the number of poles, was investigated in studies [14, 15]. They highlight that in addition to the need for standalone efficiency optimization, the area of connection between wind energy conversion systems (WECS) and the electrical grid introduces new technical requirements, such as power quality, grid synchronization and the regulation of a stable DC-link voltage. The grid-connected inverter is required to transfer active power to the utility grid in the most efficient manner while maintaining a unity power factor, low harmonic distortion, and appropriate responses to grid disturbances [16, 17]. The total harmonic distortion (THD) of grid currents must satisfy international standards such as IEEE 519, which specifies that the THD should not exceed 5% for grid-connected renewable energy system [18]. In this study [19], the authors proposed a maximum power output control method for a variable speed wind turbine system using the PMSG but without closed-loop speed control. Optimal stator current commands on the d and q axes were obtained depending on the generator speed to achieve loss minimization and maximum power maintenance mode. However, since there was no closed-loop speed control, the maximum power point could not be accurately maintained when system parameters changed during actual operation. In addition, the effect of magnetic saturation on the performance, characteristics and control of the PMSM was not considered. In order to achieve optimal efficiency and dynamic response of the CMS, magnetic saturation must be considered when designing the CMS control system for torque and current.

When determining the loss in the generator, the equivalent resistance of the core loss is assumed to be constant [20]. In addition, to avoid DC-bus voltage stability and power quality issues with the grid, the DC-link voltage must not decrease excessively during rapid variations in wind speed [19]. However, for PMSG-based wind turbines many studies have concentrated on maximizing power extraction, but few works have combined this objective with minimizing system losses while also satisfying grid connection requirements. This gap is highly important because flux-weakening control, like many efficiency optimization techniques, can impact the quality of the current waveform and harmonic performance [20]. Thus, it is necessary to validate efficiency-enhancing method under grid-connected conditions to demonstrate its practicality.

To fill this research gap, in contrast to existing studies that mainly focus on the stand-alone mode of proposed respective individual models, the current study evaluates their performance under realistic grid-connected conditions against key metrics including power-tracking accuracy, DC-link voltage stability (total imbalance), grid current synchronization (frequency mismatch), and THD. However, this assumption makes it difficult to accurately determine core loss, as the equivalent resistance varies with rotor speed. To address these challenges, this study examines the effect of losses in the inverter-electrical machine system operating within a wind turbine on the stator current component d-axis. A method is proposed to determine the optimal value of the current component d for energy efficiency, taking into account inverter losses. The current is adjusted according to the rotor rotation speed using the maximum power generation mode linked to wind speed. A wind turbine model is built in Simulink to implement the proposed method and maximize efficiency under both standalone and grid-connected conditions [21]. A comprehensive study was also conducted on the effect of inverter selection on machine losses. More than one inverter was selected to determine its effect. The selection was made by comparing the losses and energy consumption between silicon IGBT and SiC MOSFET chips of different generations. The main conclusion, on which our study was based, was that improving the performance of the electric vehicle does not depend solely on selecting the latest inverter technologies, but the interaction between the inverter, the type of motor, and the entire system must be taken into account [22]. Many recent studies have addressed improving efficiency by taking into account the losses between the inverter and the motor, as in total loss reduction (TLMM), which uses a single method in inverter switching to reduce losses in the generator via the inverter, where the results were compared on one side with a well-known method, the MTPA conventional method [23]. A recent study presented a comprehensive, systematic approach to evaluating the performance of electric vehicles by combining finite element analysis of an interior permanent magnet synchronous motor (IPMSM) with inverter power electronics modeling to assess efficiency and total combined power losses under realistic driving cycles. The results demonstrated that clustering techniques significantly reduced computation time while maintaining high accuracy [24]. A study was conducted to improve the performance of electric vehicles using a multi-motor, multi-gear drive system. The efficiency of both the motor and the inverter (combiner) was analyzed together due to their direct impact on overall vehicle efficiency. A multi-objective optimization model was proposed to refine the drive system design by considering variables such as core length, revolutions per minute, torque distribution, and gear ratios to enhance efficiency and dynamic performance. To reduce computational complexity, an artificial neural network was employed to accelerate the optimization process. The results demonstrate that relying on the integrated efficiency (motor + inverter) achieves better performance than relying solely on motor efficiency. Furthermore, the artificial neural network model significantly reduces computation time while maintaining accurate results [25].

Stator Current Optimization Control method with the current component d (OTC) is applied according to the power loss criterion, taking into account losses in the inverter-electrical machine system. The dependence of the efficiency of the PMSG, the inverter, and the overall wind turbine on rotor speed is also studied in both standalone and grid-connected modes. The efficiency results are compared between conventional vector control and the stator current component optimization method, demonstrating improved performance when using the proposed method. The proposed system was modeled using MATLAB/Simulink version (R2018 b) on a personal computer protected by an Intel Core i7 processor and 16 GB of RAM to run Windows 10 programs.

The turbine is always operated at its maximum power output. The turbine speed is calculated for each wind speed while maintaining a specific speed, called the optimum turbine speed, from which the maximum power output is derived. This optimum speed is calculated using the tip speed ratio (λ_opt), which is the rotational speed multiplied by the turbine radius, and the wind speed (v_w). The optimal turbine speed can also be calculated as follows:

$\omega_{o p t}=\frac{\lambda_{o p t} \cdot v_w}{\mathrm{R}}$  (9)

where, $\omega_{\text {opt}}$: The rotational optimal speed of turbine; $\lambda_{\text {opt}}$: Optimal value of the tip speed ratio. The rotational speed of the wind turbines is also controlled and regulated using the generator's directional control system, by directly controlling the torque on the generator shaft through calculating the optimal electrical load value, which can be calculated using the equation below:

$\omega_r=\frac{1}{J_{t o t}} \int\left(T_t / i-T_e\right)$  (10)

A variation in the generator rotor rotational speed causes a corresponding variation in the turbine rotational speed according to the gearbox transmission ratio $i$, where the turbine speed is expressed as $\omega_t=\frac{\omega_r}{i}$. Consequently, the turbine rotational speed $\omega_t$ can be represented as follows:

$\omega_t=\frac{1}{i \cdot J_{t o t}} \int\left(T_t / i-T_e\right)$ (11)

In synchronous machines with permanent magnets with an implicit pole rotor, the inductances along the $d$ and $q$ axes are assumed to be equal to: $L_q=L_{d} . J_{\text {tot}}$ - total of moment of inertia, $T_e$ - Electromagnetic torque of PMSG, in this case, the expression for torque can be written as follows:

$T_e=1.5 p\left[\psi_r i_q\right]$   (12)

Thus, there is a linear relationship between the electromagnetic moment of the PMSG and the current on the $q$-axis, and the value $c_t=1.5 \, \mathrm{p} \psi_r$ is called the torque coefficient. The transformation of the coordinate system (Clark and Park transformations) for the transition from a three-phase system (abc) to a two-phase fixed coordinate system ($\alpha, \beta$ system) and a rotating d-q coordinate system is expressed as follows.

$\left\{\begin{array}{l}i_\alpha=i_a, i_\beta=\frac{i_b-i_c}{\sqrt{3}} \\ i_d=i_\alpha \cos (\theta)+i_\beta \sin (\theta) \\ i_q=i_\beta \cos (\theta)-i_\alpha \sin (\theta)\end{array}\right.$  (13)

Also, the inverse transformation of coordinate systems, known as the Clarke-Park inverse transformation, is performed for the purpose of transitioning from a d-q rotary coordinate system to a two-phase ($\alpha, \beta$) fixed coordinate system and a three-phase (abc) coordinate system, as follows [29].

$\left\{\begin{array}{l}V_\alpha=V_d \cos (\theta)-V_q \sin (\theta) \\ V_\beta=V_\beta \cos (\theta)+V_d \sin (\theta) \\ V_a=V_\alpha, V_b=\frac{\sqrt{3}}{2} V_\alpha-\frac{1}{2} V_\beta \\ V_c=-\frac{1}{2} V_\alpha+\frac{\sqrt{3}}{2} V_\beta\end{array}\right.$  (14)

Electromagnetic losses $P_{E M}$ in fixed magnet generators (PMSG) consist of a combination of magnetic losses in the steel $\left(P_{f e}\right)$ and electrical losses in the form of copper losses $\left(P_{cu}\right)$. As shown in the equation below:

$P_{E M}=P_{c u}+P_{f e}$ (25)

Copper losses, iron losses and mechanical losses due to friction of the generator can also be calculated using the equations shown below. These equations provide a quantitative estimation of the main loss components in the generator, allowing the contribution of each loss mechanism to the total power loss to be evaluated.

$\left\{\begin{array}{l}P_{c u}=\frac{3}{2} R_s\left[\begin{array}{l}\left(I_{o d}-\frac{\omega L_q I_{o q}}{R_c}\right)^2+ \\ \left(I_{o q}+\frac{\omega\left(\psi_r+L_d I_{o d}\right)}{R_c}\right)^2\end{array}\right] \\ P_{f e}=\frac{3}{2} \frac{\omega^2}{R_c}\left[\binom{\left(L_q I_{o q}\right)^2+}{\left(\psi_r+L_q I_{o d}\right)^2}\right] \\ P_{\text {mech }}=B_m \cdot \omega_r^2\end{array}\right.$ (26)

where, $P_{\text {mech }}$-mechanical losses in the PMSG; $I_{\text {od,}} I_{\text {oq}}$ - are the magnetizing currents on the $d$ and $q$ axes; $L_d, L_q-$ inductances of the stator windings along $d, q$ axes; $R_s$-is the stator resistance; $R_c$ - equivalent core-loss resistance; $\omega$ - is the electrical speed; $\omega_{r}$- is the mechanical rotor speed; $\psi_r$ the rotor flux linkage; $B_m$ - coefficient of viscous friction; $T_{\mathrm{e}}-$ electromagnetic torque of PMSG.

Using Eq. (26), it is possible to express the electrical losses specific to electric generators as a function of $\omega, I_{o d}$, and $I_{o q}$. At steady state, the electromagnetic losses of a PMSG also depend on the value of $I_{o d}$. For a PMSG with a non-salient rotor, $L_d=L_q$. The optimal value of the current component along the $d$-axis can also be determined by setting the first derivative of the electrical losses with respect to $I_{o d}$ to zero, to determine the optimal stator current $\left(I_{o d}\right)$ according to the minimum electrical losses' criterion for a PMSG.

$\begin{aligned} & \frac{d P_{E M}}{d I_{o d}}=\psi_r^3 \cdot\left(\frac{3}{2} \cdot \frac{P}{2}\right)^2 \cdot I_{o d}\left(R_s R_c^2+\omega^2 L_d^2 R_s+\omega^2 L_d^2 R_c\right)+\psi_r^4 \cdot\left(\frac{3}{2} \cdot \frac{P}{2}\right)^2 L_d\left(R_s+R_c\right) \omega^2 \\ & n p u \frac{d P_{E M}\left(I_{o d}, I_{o q}, \omega\right)}{d I_{o d}}=0 \Rightarrow I_{o d}=-\frac{\psi_r\left(R_s+R_c\right) \omega^2 L_d}{R_s R^2 c+\omega^2 L_d^2\left(R_s+R_c\right)}\end{aligned}$      (27)

The negative d-axis current (I_d) is represented by a weakening of the rotor magnetic flux of the generator, which reduces the total flux along the d-axis and allows the generator to operate at higher speeds compared to the case where I_d = 0. This process also reduces the flux coupling resulting from the weak flux, which reduces iron losses. The optimum value of the stator current component along the q-axis (I_oq) to achieve the maximum electromagnetic torque of the PMSG generator can be determined as follows:

$I_{o q}^*=\frac{\left(R_s R^3{ }_c+\omega^2 L_d^2\left(R_s R_c+R^2{ }_c\right)\right) I_q-\psi_r \omega\left(1+L_d^2\left(R_s+R_c\right)\right)}{R_s R^3{ }_c+\omega^2 L_d^2\left(R_s R_c+R^2{ }_c\right)}$ (28)

where, I_d, I_q – d, q axis currents.

Inverter losses (VSI) can be divided into dynamic switching losses and static conduction losses, and can be calculated using the following equations [14, 31, 32, 35, 36]:

$\left\{\begin{array}{l}P_{S . L}=\frac{6}{\pi} \cdot f_S \cdot\left(E_{o n, I}+E_{o f f, I}+E_{\mathrm{off}, \mathrm{D}}\right) \cdot \frac{V_{d c}}{V_{\text {ref. }}} \cdot \frac{I_L}{I_{\text {ref. }}} \\ P_{\text {cond }, I}=\frac{V_{C E, 0}}{2 \pi} \cdot I_L \cdot\left(1+\frac{M \cdot \pi}{4} \cos (\phi)\right)+ \\ \frac{r_{C E, 0}}{2 \pi} \cdot\left(\frac{\pi}{4}+M\left(\frac{2}{3} \cdot \cos (\phi)\right)\right) \\ P_{\text {cond }, \mathrm{D}}=\frac{V_{F, 0}}{2 \pi} \cdot I_L \cdot\left(1-\frac{M \cdot \pi}{4} \cos (\phi)\right)+ \\ \frac{r_{C E, 0}}{2 \pi} \cdot\left(\frac{\pi}{4}-M\left(\frac{2}{3} \cdot \cos (\phi)\right)\right) \\ P_{V S I}=P_{S . L}+P_{\text {cond }, I}+P_{\text {cond }, \mathrm{D}}\end{array}\right.$  (29)

where, f_s is the VSI switching frequency, $V_{d c}$ is the DC link voltage, I_L is the peak amplitude of the sinusoidal AC line current, E_on,I and E_off,I are the IGBT turn-on and turn-off energies, E_off,D is the power diode turn-off energy due to reverse recovery current, P_cond,I represents the transistor’s static losses, and P_cond,D represents the diode’s static losses. Because inverter losses are influenced by the stator current, selecting the calculated optimal current value will directly impact these losses. Taking this into account, it is essential to establish how the inverter losses vary. The sum of losses in both the inverter and the PMSG is referred to as the combined losses. The relationship between the total losses and the stator current along the d-axis (I_od) can be determined using Eqs. (25) and (29):

$P_{\text {comb }}=P_{E M}+P_{V S I}$   (30)

Eq. (5) allows us to determine how the total losses in the inverter-machine system depend on the Iod current, taking into account the electromagnetic losses in the machine (P_cu + P_fe). The minimum total loss function is achieved when the following condition is met:

$\begin{aligned} & P_{\text {comb }}=P_{E M}+P_{V S I} \\ & P_{\text {comb }}= \\ & \left(\begin{array}{l}\left.\frac{3}{2} R_S\left[\left(I_{\text {od }}-\frac{\omega L_q I_{\text {oq }}}{R_c}\right)^2+\left(I_{\text {oq }}+\frac{\omega\left(\psi_r+L_d I_{o d}\right)}{R_c}\right)^2\right]+\frac{3}{2} \frac{\omega^2}{R_c}\left[\binom{\left(L_q I_{o q}\right)^2}{+\left(\psi_r+L_q I_{o d}\right)^2}\right]\right)+ \\ \left(\begin{array}{l}\frac{6}{\pi} \cdot f_s \cdot\left(E_{o n, I}+E_{o f f, I}+E_{\text {off, } \mathrm{D}}\right) \cdot \frac{V_{d c}}{V_{\text {ref. }}} \cdot \frac{I_L}{I_{\text {ref. }}}+\frac{V_{C E, 0}}{2 \pi} \cdot I_L \cdot\left(1+\frac{M \cdot \pi}{4} \cos (\phi)\right.\end{array}\right) \\ +\frac{r_{C E, 0}}{2 \pi} \cdot\left(\frac{\pi}{4}+M\left(\frac{2}{3} \cdot \cos (\phi)\right)\right)+\frac{V_{F, 0}}{2 \pi} \cdot I_L \cdot\left(1-\frac{M \cdot \pi}{4} \cos (\phi)\right)+ \\ \frac{r_{C E, 0}}{2 \pi} \cdot\left(\frac{\pi}{4}-M\left(\frac{2}{3} \cdot \cos (\phi)\right)\right.\end{array}\right)\end{aligned}$   (31)

The rate of change of electromagnetic losses in the machine with respect to the current component along the d-axis can be calculated.

$\begin{aligned} & P_{E M}=\frac{3}{2} R_s\left[\left(I_{o d}-\frac{\omega L_q I_{o q}}{R_c}\right)^2+\left(I_{o q}+\frac{\omega\left(\psi_r+L_d I_{o d}\right)}{R_c}\right)^2\right]+\frac{3}{2} \frac{\omega^2}{R_c}\left[\binom{\left(L_q I_{o q}\right)^2}{+\left(\psi_r+L_q I_{o d}\right)^2}\right] \\ & \frac{d P_{E M}}{d I_{o d}}=3 R_s\left[\left(I_{o d}-\frac{\omega L_q I_{o q}}{R_c}\right)+\frac{I_{o q} L_d \omega}{R_c}+\left(\frac{L_d \omega^2}{R_c} \frac{\psi_r}{R_c}+\frac{L_d \omega^2}{R_c} \frac{L_d I_{o d}}{R_c}\right)\right] \\ & +3 \frac{\omega^2}{R_c}\left[\left(\psi_r+L_q I_{o d}\right) L_q\right]=\frac{d P_{E M}}{d I_{o d}}= \\ & \left(3 R_s+3 R_s \frac{L_d^2 \omega^2}{R_c^2}+3 \frac{\omega^2}{R_c} L_q L_q\right) I_{o d}+3 R_s \frac{\psi_r L_d \omega^2}{R_c^2}+3 \frac{\omega^2}{R_c} L_q \psi_r\end{aligned}$    (32)

$\frac{d P_{V S I}}{d I_{o d}}=\frac{I_{o d}}{\sqrt{I_{o d}^2+I_{o q}^2}}\left[\begin{array}{l}\frac{6}{\pi} \cdot f_s \cdot\left(\begin{array}{l}E_{o n, I}+ \\ E_{o f f, I} \\ +E_{o f f, D}\end{array} \frac{V_{d c}}{I_{r e f .} V_{r e f .}} \frac{V_{C E, 0}}{2 \pi}\left(1+\frac{M \cdot \pi}{4} \cos (\phi)\right)\right) \\ +\frac{V_{F, 0}}{2 \pi} \cdot\left(1-\frac{M \cdot \pi}{4} \cos (\phi)\right)\end{array}\right]$ (33)

In the same way, and based on Eq. (4), the rate of change of electrical losses in the inverter with respect to the current component along the d-axis is calculated.

The rate of change of total losses depends on the calculated value of the current I_od, taking into account the effect of rotor field attenuation, according to the following formula.

$\frac{d P_{c o m b}}{d I_{o d}}=\frac{d P_{E M}+d P_{V S I}}{d I_{o d}}$ (34)

The algorithm of calculating the total efficiency, which combines all parts of this system including the proposed control method, can be seen in Figure 1.

Figure 1. Algorithm of improving the total efficiency

The overall efficiency calculation algorithm, which integrates all components of this system, including the proposed control method, is shown in Figure 1 below. This diagram illustrates the integrated methodology for modeling and optimizing the efficiency of a wind energy conversion system (WECS) based on a PMSG with a voltage source transformer (VSI). It considers system losses in all components and ensures no losses are neglected. The process begins by determining the incoming wind speed (v_w) and constructing a complete wind turbine model using MATLAB Simulink. This model relies on dynamic and geometric parameters of the turbine, such as blade radius, through-air density, and power factor, which represents the turbine's efficiency according to Betzi’s law. The PMSG model is then developed using standardized data and its electrical and mechanical parameters, based on real data from a well-known global brand of motors. Finally, key operational variables such as the axial currents (d and q) and angular velocity are measured. This work incorporates gearbox losses and their significant impact on the accuracy of the results. It also models the VSI transformer, defining control parameters and connection and switching losses. During the optimization phase, copper and iron losses in the PMSG are reduced by adjusting the d-axis current, while VSI performance is improved by assessing power losses and determining optimal operating values. Finally, the efficiencies of both the PMSG and VSI are calculated separately, and the results are combined to obtain the overall system efficiency, enabling a comprehensive evaluation and effective optimization of the entire wind power system. The control diagram of a PMSG with classical vector control without using the rotor field weakening method (Field oriented control (FOC) when I_od = 0) and using the rotor field weakening method (OTC when I_od ˂ 0) (stator current optimization method of a PMSG with vector control) with grid side, is shown in Figure 2.

Figure 2. Block diagram of standalone and grid control systems with maximum power point tracking (MPPT) based Stator Current Optimization Control method (OTC), classical vector control field oriented control (FOC)

Figure 2 illustrates the integrated operating architecture and control circuitry for the turbine and generator, as well as the grid control, of a wind turbine conversion system based on a PMSG connected to the electrical grid via a back-to-back AC-AC converter to achieve high operational efficiency. The wind turbine extracts kinetic energy from the wind and converts it into mechanical energy, which is then transferred through a gearbox to the generator. The generator produces electrical power with varying voltage and frequency depending on the wind speed. The generator side employs maximum power point tracking (MPPT) and two methods of current optimization—the classic FOC method and the OTC method—to ensure maximum power extraction while minimizing electrical losses through precise regulation of the shaft currents (dq) and to determine the optimal method for maximizing efficiency. The generated power is then stabilized via a DC-link before the grid side regulates the DC-link voltage, controls the active and reactive power, and synchronizes with the grid using PLL technology. This integrated structure ensures improved overall system efficiency, reduced losses in the generator and transformers, and high stability and reliability in pumping power to the grid under various operating conditions.

The power is stored in the DC-Link capacitor to provide voltage stability, and the current then passes to the grid-side converter (GSC), which converts the DC voltage back into a three-phase AC that is synchronized with the grid using a DC-AC voltage source converter and spatial pulse width modulation (SVPWM) technology. The system has active and reactive power control (P and Q) through PI and PLL control panels for phase synchronization, with LCL filters to ensure the quality of the voltage output to the 300 V electrical grid at 50/60 Hz, enabling bi-directional power flow between the generator and the grid. This scheme is implemented using the visual modeling environment for dynamic systems Simulink and is shown in Figure 3.

Figure 3. Model of an autonomous wind energy conversion system based on permanent magnet synchronous generator (PMSG) with Stator Current Optimization Control method (OTC) method

The scheme implements a model of an autonomous wind energy conversion system on the basis of a PMSG with vector control based on SVPWM, taking into account the power losses and efficiency of the gearbox and the mechanical losses of the PMSG, power losses and the VSI losses. The maximum power tracking mode operates in the range of specified wind speed values (from 5 to 15 m/s) and gear ratio i = 3.

7.1 Limitations of the present study

Although promising results were obtained by the proposed OTC method, several limitations should be discussed:

Low Power Wind Turbine: A small wind turbine was used for the study (6 kW rated power, Scirocco E5. 6-6). Considering a multi-megawatt utility-scale wind turbine packs an entirely different generator design, inverter topology and control complexity—multiplied tenfold—the 2.5% efficiency boost seen here at low speeds may not directly translate to MW-class machines. Because the analysis was based on a single wind turbine grid-connected capacity system application, the model did not explicitly account for inter-turbine interactions within the wind farm, such as wake effects, grid voltage variations, and power sharing.

Assumed Wind Speed Profile: A step change in constant wind speeds. Effective dynamic behavior of the MPPT algorithm cannot be expected as realistic turbulent wind profiles (e.g., von Kármán or Kaimal spectra) were not implemented.

Assumed Constant Temperature: All simulations were performed at a constant temperature (25 ℃). In practice, stator resistance and magnet flux linkage are temperature-dependent, which leads to a variation of copper losses and efficiency.

Idealized Grid Conditions: These conditions modeled the grid as an ideal voltage source. It did not account for grid non-idealities, including voltage sags, harmonics, frequency deviations and fault conditions.

Lack of Experimental Validation: Results are only based on simulation (MATLAB/Simulink). Experimental confirmation on a hardware test bench is needed to validate the OTC method in practice.

Loss Models: Mechanical losses (friction, windage) and gearbox losses were modeled with simplified empirical equations. Better approximations of these losses (i.e. tooth friction, oil churning losses) in the model could make it less conservative.

Neglect Magnetic Saturation: The PMSG model with constant inductances (L_d = L_q). Effects due to magnetic saturation, important at high currents were ignored.

7.2 Future work recommendations

Future research should address the following based on the limitations we identified above:

• Scope of Work: Validate the OTC method on medium (50–100 kW) and large-scale (1-5 MW) wind turbine systems.
• Wind Farm Simulation: Generalize to mult-turbine case and add wake modeling and control coordination.
• Turbulent Wind Speed: Use the Kaimal spectrum in wind to assess MPPT performance.
• Thermal Modeling: Use temperature-based resistance and flux linkage to evaluate efficiency at various operating temperatures.
• Grid Fault Ride-Through: Assess OTC performance with grid faults (Voltage sags, frequency deviations, harmonic distortion).
• Validation through experiments: Develop a test bench at laboratory scale (with motor driven PMSG, 2-3 kW) to experimentally validate the simulation results.
• Advanced Loss Models: Make loss models more accurate including magnetic saturation, eddy currents in magnets and gearbox dynamics.
• Real-Time Implementation: The OTC algorithm is implemented on DSP, or FPGA to evaluate real time power computational requirements.
• ML Use: Utilize reinforcement learning or a neural network for real-time adaptive tuning of the optimal d-axis current given changing conditions.
• Economic analysis: Conduct a cost-benefit analysis to help quantify the return on investment related to the use of OTC in commercial wind turbines.