2.1. Control Techniques in the Literature

The proportional-integral-derivative (PID) controller is a classical control system known for its cost-effectiveness and simplicity. It consists of three main components, each serving a specific function: the proportional term (P) improves the system’s response to rapid changes in error; the integral term (I) reduces the steady-state error; and the derivative term (D) enhances the controller’s reaction to the rate of change of the error. PID controllers have demonstrated their effectiveness in FSWT applications. However, their performance in VSWTs is less satisfactory due to the system’s inherent nonlinearity. To address this limitation, researchers have begun exploring new techniques to enhance the performance of classical PID controllers in VSWT systems [13, 14].

One of the advanced techniques developed to enhance PID controller performance is the fractional-order PID (FOPID) controller, introduced in 1999. The FOPID controller offers several advantages, such as improved closed-loop response and reduced sensitivity to changes in system parameters. These characteristics make FOPID controllers particularly promising for complex systems like VSWTs. However, despite its benefits, the FOPID controller faces a significant challenge: it is difficult to design in the time domain. This complexity in design remains a key factor limiting its widespread adoption and implementation [15]. The sliding mode controller (SMC) is another advanced control technique used to enhance blade angle control in wind turbines. SMCs can be categorized into three main types: first-order, second-order, and third-order SMCs (FOSMC, SOSMC, and TOSMC, respectively). Among these, the TOSMC demonstrates particularly robust performance. This robustness is attributed to its ability to operate with minimal frequency variations, resulting in a smoother system response [16]. Another advanced variant of the PID controller is the PID-acceleration (PIDA) controller, which was first used in 2020. This controller enhances the conventional PID design by incorporating a double-derivative branch. The inclusion of this acceleration term improves the controller’s transient response in the frequency domain, allowing for faster and more accurate adjustments. As a result, the PIDA controller offers better performance in systems that require quick and precise control actions, such as VSWTs [17]. The summary of the previous literature of control strategies is shown in Table 1.

2.2. Optimization Algorithms in the Literature

The harmony search algorithm (HSA) was an optimization technique, introduced by Geem [18] in 2001. It is a meta-heuristic algorithm inspired by the improvization process of jazz musicians. Another technique is the whale optimization algorithm (WOA), which mimics the bubble-net hunting behavior of humpback whales. First proposed by Mirjalili and Lewis [19] in 2016, the WOA has gained significant attention in the field of optimization due to its simplicity, flexibility, and strong global search capability. Comparatively, HSA has demonstrated superior performance in enhancing system stability, providing smoother responses than WOA. Another robust optimization technique is the turbulent flow water–based optimization algorithm, which has also proven its superiority over WOA in terms of stability and convergence efficiency [20]. In addition to these optimization algorithms, model predictive control (MPC) represents a fundamentally different approach to controller strategy. This method predicts the future behavior of the system and optimizes control inputs over a limited time horizon, continuously updating the control actions based on new measurements and predictions [21]. The summary of the previous optimization literature is shown in Table 2.

This paper examines the effectiveness of two optimization algorithms when applied to the blade angle PID controller: the exponential distribution optimizer (EDO) algorithm and the HSA. We evaluate these algorithms under three distinct conditions: fixed wind speed, practical (variable) wind speed, and faulty conditions. The superior optimization algorithm, as determined by our analysis, will then be utilized to compare the performance of three different controllers: PID controller, proportional double derivative (PD²) controller, and API controller. This comparative study aims to identify the most effective control strategy for blade angle regulation in wind turbine systems, potentially improving power extraction efficiency and system reliability across various operational scenarios.

The remainder of this paper is organized as follows: Section 3 details the implementation of the DFIG wind turbine model in MATLAB/Simulink, which serves as the foundation for the study. Section 4 introduces the EDO algorithm, explaining its mathematical formulation and application in optimizing blade angle control. Sections 5 and 6 describe the PD² and API controllers, highlighting their design and advantages over traditional PID controllers. Section 7 presents the simulation results, comparing the performance of the HSA and EDO in tuning PID controllers under various operating conditions, including normal operation and fault scenarios. Section 8 extends the analysis by evaluating the performance of the PID, PD², and API controllers optimized using the EDO algorithm. Finally, Section 9 concludes the paper, summarizing the key findings, discussing their implications, and suggesting future research directions.

7.1. First Case: PID Controller, Normal Operation, Without Fault, and Winter Wind Speed for a Random Day With HSA and EDO Optimization

Figure 9 illustrates the wind speed variation over time for a randomly selected winter day, showing a rapid decrease from 19.6 to 9.5 m/s within 2 s. This wind speed profile is applied to the model presented in Figure 3. In this section, the impact of this wind variation on the power curve performance is analyzed using an optimized blade angle controller. The results obtained using the EDO algorithm are compared with those from the HSA algorithm. For EDO optimization, the trial-and-error parameter search range was set to ±60% of the corresponding values reported for the same wind profile using the HSA method, as detailed in [2]. The values of K_p, K_i, and K_d are shown in Table 3.

Figure 10 compares the pitch angle responses of PID controllers optimized using the HSA and EDO techniques. The EDO–based controller demonstrates superior performance, as reflected in its higher pitch angle values and faster dynamic response. When the wind speed exceeds the rated value, the EDO-optimized controller adjusts the pitch angle more rapidly, enabling the system to return the captured energy to the rated level in less time than the HSA–based controller. Conversely, as shown in Figure 9, when the wind speed decreases between 10.5 and 12 s, the pitch angle value also decreases under the EDO-optimized controller, as illustrated in Figure 10. This reduction in pitch angle facilitates greater energy capture, allowing the system to reach its rated power more quickly.

Figure 11 presents a comparison of the extracted active power performance curves using PID controllers optimized with the HSA and EDO techniques. Both optimization methods effectively prevent the active power from dropping below −1 p.u.; however, the EDO–based PID controller exhibits significantly lower oscillations. Specifically, the EDO–based optimization achieves a maximum undershoot of −0.46 p.u., compared to −0.83 p.u. for the HSA–based approach, indicating improved system protection. Although both controllers ultimately reach the rated power, the EDO-optimized PID controller demonstrates a shorter settling time, contributing to enhanced system stability and faster recovery following wind speed variations. Table 4 shows a summary of optimization comparison for Case Study 1. EDO optimization consistently achieves faster settling, better standard deviation and lower overshoot than HSA optimization.

7.2. Second Case: PID Controller, Normal Operation, Without Fault, and Variable Wind Speed With HSA and EDO Optimization

In this section, a variable speed profile is applied to the system in the model presented in Figure 3 under normal operating conditions (no fault).

Figure 12 illustrates the wind speed profile, which varies significantly over time. The wind speed drops below the rated speed between 27 and 41 s, resulting in a reduction in the energy extracted by the blades. It exceeds the rated speed during two intervals: from 7.26 to 13.9 s and from 46 to 59.9 s. For the EDO optimization, the trial-and-error parameter ranges were set to ±60% of the results obtained using the HSA method for the same wind profile, as reported in [28]. The values of K_p, K_i, and K_d are shown in Table 5.

Figure 13 compares the pitch angle curves of PID controllers optimized using EDO and HSA techniques. During the intervals from 7.26 to 13.9 s and 46 to 59.9 s, the wind speed shown in Figure 12 exceeds the rated speed. Both optimization methods respond to this, but the EDO–based PID demonstrates superior performance compared to the HSA–based PID. This is evident in Figure 13, where the pitch angle for the EDO-optimized PID is higher than that of the HSA-optimized PID between 16 and 20 s. The increased pitch angle allows the system to maintain the rated wind speed value for a longer duration after exceeding the rated threshold, indicating improved system stability. Conversely, both methods reduce the pitch angle to zero between 27 and 41 s to compensate for the significant drop in wind speed, thereby maximizing energy capture during this period.

Figure 14 compares the extracted active power curves of PID controllers optimized using HSA and EDO techniques. The superior performance of the EDO–based PID controller over the HSA–based counterpart is particularly evident between 16 and 20 s, which aligns with the pitch angle behavior discussed earlier. During this period, although the wind speed exceeds the rated value, the EDO–based controller successfully maintains the active power at the rated level, resulting in reduced oscillations and a smoother response compared to the HSA–based controller. Both techniques exhibit similar performance between 27 and 41 s, when the wind speed is low, effectively maximizing the extractable energy under these conditions. Table 6 shows a summary of optimization comparison for Case Study 2, which proved the superiority of the EDO optimization over HSA optimization.

7.3. Third Case: PID Pitch Controller, Fault Condition, Line to Ground Fault at The Middle of Transmission Line, and Constant Wind Speed With HSA and EDO Optimization

In this section, a fixed wind speed profile of 15 m/s is applied to the system modeled in Figure 3. The system operates under fault conditions, specifically a line-to-ground fault at the middle of the transmission line. The fault occurs from 20 to 25 s. Figure 15 illustrates the wind speed profile, which remains constant at 15 m/s throughout the simulation. For the EDO optimization, the trial-and-error ranges were set to ±60% of the results presented for the same wind profile using HSA, as reported in [28]. The values of K_p, K_i, and K_d are shown in Table 7.

Figure 16 shows the pitch angle curves for PID controllers optimized using EDO–based and HSA–based methods. When a fault occurs between 20 and 25 s, both optimization techniques increase the pitch angle, enabling the wind turbine to remain operational for a longer duration. This response supports wind farm operators in managing system faults while complying with grid code requirements. During this interval, the pitch angle values achieved by the EDO–based PID controller are slightly higher than those of the HSA–based controller, contributing to marginally improved system stability.

Figure 17 compares the extracted active power performance curves of HSA–based and EDO–based PID controllers. The EDO–based PID controller reaches its rated value of 1 p.u. at 30.8 s, overcoming blade inertia and mechanical losses more rapidly. In contrast, the HSA–based PID controller achieves the rated value at 31.4 s. This indicates that the system utilizing the EDO–based controller recovers more quickly from fault conditions than the HSA–based system.

Table 8 presents a summary of the optimization performance comparison for Case Study 3, highlighting the superiority of EDO optimization over HSA optimization.

7.4. Fourth Case: PID Pitch Controller, Fault Condition, Line to Ground Fault at The Middle of Transmission Line, and Variable Wind Speed With HSA and EDO Optimization

In this section, a variable wind speed profile of 15 m/s is applied to the system model shown in Figure 3. A line-to-ground fault occurs at the midpoint of the transmission line, causing the system to experience a fault condition. This fault lasts from 20 to 25 s. For the EDO, the optimization ranges used in the trial-and-error process are set to ±60% of the results presented for the same wind profile using the HSA, as described in [28]. The values of K_p, K_i, and K_d are presented in Table 9.

Figure 18 depicts the wind speed profile, which features an overrated region between 7.26 and 13.9 s and 46 and 59.9 s, as well as an underrated region between 27 and 41 s. The underrated region results in a transition to cut-off mode due to low extracted power.

Figure 19 illustrates that when a fault occurs from 20 to 25 s, both optimization techniques (EDO and HSA) increase the pitch angle. However, the EDO optimization maintains a higher pitch angle value for an extended period after the fault occurrence. Specifically, the EDO optimization sustains an elevated pitch angle until 34.6 s, while the HSA optimization’s pitch angle returns to zero at 31.4 s.

Figure 20 demonstrates the effect of maintaining the pitch angle on the active power curves. The power curve optimized with EDO exhibits lower oscillations while recovering to its rated value compared to the HSA-optimized curve after fault clearance. This is attributed to the prolonged maintenance of the pitch angle value, as mentioned earlier. However, it is important to note that the both power curves drop to zero during the fault period. The system regains efficient operation after the fault, as the wind speed becomes underrated from 27 to 41 s. Consequently, the pitch angle decreases to zero after recovery from the fault disturbance, allowing for increased energy extraction from the system. Both optimizations settle the rated value at same time.

Table 10 presents a summary of the optimization performance comparison for Case Study 4, highlighting the superiority of EDO optimization over HSA optimization after the fault clearance. Comparing these different case studies, the EDO optimization technique outperforms HSA due to faster response to wind speed variations.

8.1. First Case: Normal Operation, Without Fault, and Winter Wind Speed for a Random Day

Figure 21 illustrates the wind speed over time for a random winter day, showing speed variations from 19.6 to 9.5 m/s within 2 s. This wind speed profile is applied to the model in Figure 3. Table 11 presents the parameter values for the three controllers tuned by EDO optimization.

Figure 22 illustrates the pitch angle curves of the three controllers tuned using EDO optimization. Among them, the API controller demonstrates a smoother pitch angle response during wind speed variations compared to the PD² and PID controllers. This distinction is further supported by Figure 23, which shows the extracted active power curves for the three EDO-optimized controllers.

The API controller’s extracted power curve exhibits fewer ripples compared to those of the PD² and PID controllers. It reaches the rated power value more quickly at 5.3 s and maintains it consistently despite wind speed variations. In contrast, both the PD² and PID controllers achieve the rated power level at 8.7 s, indicating a slower response. Additionally, their power curves display noticeable oscillations. The PD² controller experiences a maximum undershoot of −0.37 p.u. at 7.73 s, while the PID controller exhibits a deeper maximum undershoot of −0.47 p.u. at 7.76 s, suggesting that the PD² controller performs better than the PID in terms of stability. This comparison highlights the superior stability of the API controller, characterized by minimal oscillations and faster response. Furthermore, it confirms that the PD² controller offers improved performance over the PID controller. The API controller’s advantages over the PD² and PID controllers are due to their time-varying gains K_P and K_i, which provide quick responses to wind profile variations. The time variations of gains K_P and K_i are shown in Figures 24 and 25, respectively.

Table 12 presents a summary of the optimization performance comparison for Case Study 1, highlighting the superiority of API controller over PID and PD² controllers.

8.2. Second Case: Normal Operation, Without Fault, and Second Wind Speed Profile

Figure 26 illustrates the wind speed profile for the second normal case study. The wind speed curve starts at a minimum value of 7.3 m/s, increases to a peak of 19.6 m/s at 6 s, and then, decreases again to 7.5 m/s. This wind speed profile is applied to the model presented in Figure 3. Table 13 presents the parameter values for the three controllers tuned by EDO optimization.

Figure 27 displays the pitch angle curves of the three controllers for this second normal case study. The API controller’s pitch angle response exhibits a smoother curve compared to the PD² and PID controllers throughout the wind speed variations.

Figure 28 presents the extracted active power curves of the three EDO-optimized controllers for the second normal case study. These curves reflect the influence of each controller’s pitch angle response. The API controller exhibits the least oscillation in its power curve, maintaining stability despite wind speed variations. In contrast, the power curves of both the PD² and PID controllers show noticeable ripples and oscillations. The API controller reaches the rated power value fastest, settling at 5.3 s and maintaining this value consistently. The PD² controller achieves the rated power at 8.77 s, while the PID controller reaches it at 8.9 s. In terms of undershoot performance, the PD² controller experiences a peak undershoot of −0.53 p.u. with relatively low ripple, whereas the PID controller shows a larger undershoot peak of −0.79 p.u.

This comparison highlights the superior stability and faster response of the API controller’s power curve compared to those of the PD² and PID controllers. It also indicates that the PD² controller outperforms the PID controller in terms of active power tracking and stability. The API controller’s advantages stem from its time-varying gains K_P and K_i, which enable quick responses to wind profile variations. The time variation of gains K_P and K_i is are shown in Figures 29 and 30, respectively.

Table 14 presents a summary of the optimization performance comparison for Case Study 2, highlighting the superiority of API controller over PID and PD² controller.

8.3. Third Case: Normal Operation, Without Fault, and Third Wind Speed Profile

Figure 31 illustrates the wind speed over time for the third normal case. This wind speed profile was applied to the model shown in Figure 3. The parameters for the three controllers, optimized using EDO, are presented in Table 15.

Figure 32 displays the pitch angle curves of these three controllers for the third normal case study. The API controller exhibits a smoother pitch angle response compared to the PD² and PID controllers.

Figure 33 shows the extracted active power curves for the three controllers optimized using EDO in the third normal case study. The API controller exhibits the smoothest power curve with minimal ripples, despite variations in wind speed. In contrast, the PD² and PID controllers display noticeable oscillations in their power outputs.

The API controller reaches the rated power level the fastest, settling at 5.33 s and maintaining it consistently. The PD² controller settles at 8.73 s, while the PID controller reaches rated power slightly later, at 8.8 s. In terms of undershoot performance, the PD² controller outperforms the PID controller. The PD² controller experiences a maximum undershoot of −0.3 p.u. at 7.69 s, whereas the PID controller exhibits a deeper undershoot of −0.46 p.u. at 7.77 s. Overall, this comparison confirms that the API controller delivers the smoothest and most stable power response. Furthermore, the PD² controller demonstrates better performance than the PID controller. The API controller’s advantages stem from its time-varying gains K_P and K_i, which allow for quick responses to wind profile changes. Figures 34 and 35 illustrate the time variation of gains K_P and K_i, respectively. Table 16 presents a summary of the optimization performance comparison for Case Study 2, highlighting the superiority of API controller over PID and PD² controllers.

8.4. Fourth Case: Normal Operation, Without Fault, and Fourth Wind Speed Profile

Figure 36 illustrates the wind speed profile over time for the fourth normal case. This wind speed data was applied to the model presented in Figure 3. Table 17 displays the parameter values for the three controllers, optimized using EDO.

Figure 37 depicts the pitch angle curves of the three controllers for the fourth normal case study. The API controller exhibits a smoother pitch angle response compared to the PID and PD² controllers throughout the wind variations.

Figure 38 presents the extracted active power curves of the three controllers optimized using EDO for the fourth normal case study. The API controller demonstrates superior performance, characterized by minimal oscillations and the fastest settling time reaching rated power at 5.4 s. This is significantly quicker than the PD² and PID controllers, which settle at 8.83 s and 9.1 s, respectively. While both the PD² and PID controllers exhibit oscillatory behavior in their power curves, the PD² controller shows a lower maximum undershoot. Specifically, the PD² controller reaches an undershoot peak of −1.0 p.u. at 7.9 s, compared to the PID controller’s more pronounced undershoot of −1.26 p.u. at 8.04 s. This comparison underscores the superior stability and responsiveness of the API controller in terms of active power regulation. It also indicates that the PD² controller outperforms the PID controller. The API controller’s advantages stem from its time-varying gains K_P and K_i, enabling quick responses to wind profile variations. Figures 39 and 40 illustrate the time variation of gains K_P and K_i, respectively.

Table 18 presents a summary of the optimization performance comparison for Case Study 4, highlighting the superiority of API controller over PID and PD² controllers.