# Design of PIDD(r) Controller for Robust Performance of Process Plants
## 1 Introduction
### Power System Applications  
#### 1.1.1 AVR Systems  
Ensuring the reliability of power systems is crucial, and the automatic voltage regulator system plays a critical role in regulating the voltage of synchronous generators. However, this system is prone to insufficient oscillating transient response, maximum overshoot, additional settling time, and steady-state errors. Despite extensive research to develop a control strategy for the AVR system, finding the best approach requires significant time and effort. The authors in [1] proposed a PIDD2 controller for the synchronous generator to enhance the system's robustness. The controller has been practically implemented and demonstrated better performance than its counterparts. An iterative procedure obtains the controller parameters through a constrained optimisation problem. A PIDD2 controller tuned with an improved Runge Kutta optimiser has been developed to enhance the AVR systems' transient and robustness performance [9]. The controller is designed to strengthen transient and robustness performance, and simulation results showed better transient response characteristics than other controllers. Another PIDD2 controller tuned with the coronavirus herd immunity optimiser algorithm is presented in [10] to achieve optimal system control. The practical implementation of this controller exhibited superior performance in ITAE and ISAE compared to PIDD2 controllers tuned with other algorithms.  
In control systems, optimal control is crucial to ensure efficient performance. Using a PIDD2 controller, tuned with the Archimedes optimisation algorithm [11], has proven effective in achieving optimal control, as demonstrated in [12]. Additionally, an equilibrium optimiser has successfully optimised the design of a PIDD2 controller for controlling an AVR system, as presented in [13]. Incorporating hybrid optimisation, considering the excitation voltage limitations of an AVR system [14], has led to the development of an optimal PIDD2 controller from the conventional PID. Such controllers have been proposed in [12; 13; 14] and exhibited superior transient responses in comparison to alternative controllers. Further research in this field has resulted in the implementation of a hybrid algorithm consisting of simulated annealing and manta ray foraging to optimise the PIDD2 controller, which has been shown to outperform other controllers in terms of transient response characteristics [15]. These findings highlight the significance of optimal control design in achieving desirable system performance.  
In this study, a PIDD2 controller for the AVR system is developed using whale optimisation [16]. The performance of the proposed controller design is evaluated based on transient response characteristics and compared to other controller designs and tuning algorithms, demonstrating significant improvement. Previous works, such as [7; 17], have utilised particle swarm and arithmetic optimisation algorithms to tune PIDD2 controllers for AVR systems, resulting in improved performance in all step response characteristics compared to their counterparts. Additionally, a PID controller based on a linear quadratic method is introduced for the system, which, combined with a PIDD2 controller, demonstrated outstanding performance compared to a standard PID controller [3]. Although the PIkND2N2 controller tuned using a cyoote optimisation algorithm proposed in [18] outperformed its competitors, it had a more complex control structure due to the need to tune around seven parameters, compared to only four for PIDD2 and five for FOPID. Recently,a fractional-order PIDD\({}^{2}\) controller with optimal algorithm-tuned control parameters is proposed for the system, providing superior performance compared to all conventional controllers [19].