## 1 Introduction
The study in [28] proposed adaptive neuro-fuzzy inference system (ANFIS) reduces fuel consumption in parallel HEV. Other control techniques, such as conventional proportional integral (PI), fuzzy PI, and rule-based were compared with the proposed controller, and the simulation results proved the proposed strategy's superiority. Most of the literature dedicated to investigating the theoretical implementation of the nonlinear control methods for hybrid vehicle applications. The neural network could be computationally intensive. The poor prediction in MPC makes the system performance worse and requires a precise system model. However, the adaptive controllers are excellent algorithms for nonlinear systems but can be sensitive to parameter tuning. According to the literature, optimization-based EMS approaches, such as GA, PSO, DP, and many more, were combined with control theory to realize effective control in HEVs. Over the decades, the PID control algorithm has been widely adopted for industrial process control owing to its robustness and practicality for commercial meechatronics products. In contrast, most of the advanced controllers discussed earlier are often difficult to implement due to their complexity and lack of intuition for practical implementation, making PIDs suitable and alternative control candidates. The implemented vehicle controller in the hardware system in this research is a classical PID controller. However, linear controllers may not be suitable control candidates for EVs due to the system's complexity and intricate nonlinearities, which may compromise the system's stability. To mitigate this effect, an optimization method is combined with control theory for robust performance in order to achieve optimal fuel consumption, reduce CO\({}_{2}\) emissions, and extend the range of the battery pack in HEVs. GA is characterized by its global solid search [29], more straightforward implementation, robustness and ability to handle complex and multi-optimization problems effectively. Similarly, PSO has a fast convergence time [29], and requires less parameter tuning than GAs. Due to these advantages, these two optimization methods are commonly adopted to realize optimized system efficiency. The combined effect of these optimization techniques and control theory has yielded satisfactory performance in hybrid vehicles. Therefore, optimization methods have been used to search for an optimal performance in the field of vehicle engineering [30]. Nevertheless, the PID control algorithm has been optimized with PSO (PSO-PID), GA (GA-PID), grey wolf optimization (GWO-PID) and rule-based technique such as fuzzy logic (Fuzzy-PID) to realize robust performance. The study in [31] proposed a multi-operating point Fuzzy PID control strategy (MOPFPCS) and adaptive PSO-based fuzzy PID for the optimal fuel consumption in HEV. The two proposed methods have reduced fuel consumption by 18.3% and 15%, respectively. In recent years, nonlinear versions of PID such as fractional-order controllers (FOCs) in [32], sigmoid PID (SPID) in [33], neuroendocrine PID (NPID) in [34], nonlinear PID, adaptive PID, and many more have received extensive attention due to their robustness and ability to capture the nonlinear dynamic of complex system performance compared to integer-order controllers. Furthermore, the FOPID offers robust performance due to its ability to handle nonlinear systems compared to the classical PID integer order controller and can be more flexible and less complex compared to sigmoid and neuroendocrine PIDs. However, compared to integer order strategy, the tuning process in FOPID can be challenging due to higher number of tuning parameters. Apart from the more tuning parameters in SPID, its increased complexity due to the nonlinear function can be challenging. Moreover, higher number of tuning parameters makes the design and optimization of NPID often difficult. The controllers mentioned above could be computationally intensive and, although robust, might not be realistic with our applications.