Optimization and Control of a BLDC Motor for Improving Electric Vehicle Efficiency Using the Giant Armadillo Optimization Algorithm

1.1 Background and motivation

The rapid electrification of transportation change has pushed the requirement for effective and steady motor drive systems for the new electric vehicles (EVs). EVs should be able to ensure good dynamic performance with the lowest power losses possible [1, 2]. Today, the brushless direct current (BLDC) motor has been used in EVs due to its high efficiency, small size, high torque-to-weight ratio, and low maintenance. However, BLDC drives are highly nonlinear and thus very sensitive to noise and parameter changes, besides the strong coupling between torque and speed, as well as the sensitivity to parameter variations and external load fluctuations [3, 4]. Hence, the problem of finding a control system that is effective and capable of optimal speed and torque responses even under these uncertainties remains a major challenge in electric vehicle control systems. Therefore, due to their simple structure and easy implementation, traditional proportional–integrative–derivative (PID) controllers are generally employed in the speed control of BLDC [5-7]. Accordingly, a classical PID controller with fixed gains cannot satisfactorily deal with the nonlinearities of the system, changes in parameters, and dynamic loads that are typical for the operations of ATVs. In order to increase the rigidity and extend the capabilities, a fractional-order PID (FOPID) controller was proposed by several authors [8, 9]. These settings provide the system more freedom, which improves the frequency response, makes it less likely to be disturbed, and makes it more accurate in the steady state. The five parameters of the FOPID controller also affect how well the control system works. This system may be very sensitive to these values, and the search area is nonlinear and multimodal, which means that small changes in the parameters can lead to significant variations in the system's performance and stability.

As a result, smart optimization techniques are used for defining the best FOPID parameters in BLDC motor control [10].

1.2 Key challenges

Recently, a large number of metaheuristic algorithms mimicking nature have been suggested for solving motor control issues. Particle swarm optimization (PSO), Genetic Algorithm (GA), Grey Wolf Optimizer (GWO), and Whale Optimization Algorithm (WOA) are the most recognized examples of such algorithms [11-15]. While these algorithms have been successful in achieving their goals, they are still prone to the problem of premature convergence, low exploitation capability in subsequent iterations, and stagnation at local optima when dealing with complex dynamic models. Therefore, it is necessary to develop more robust and adaptive optimization methods that not only can maintain a balance between exploration and exploitation over the course of the search but also can carry out calculations with less computational effort. Because of these limits, this study offers a fast-speed FOPI controller optimized by the new Giant Armadillo Optimization (GAO) algorithm. GAO is a nature-inspired metaheuristic that models the feeding and defensive activities of giant armadillos and, by incorporating adaptive learning and cooperative features, depicts the global search in the initial stages and refined local search near the solution. This algorithm was proposed in the study [14]. This inherent equilibrium enables GAO to efficiently traverse high-dimensional search spaces, thereby making it a very appropriate tool for the nonlinear and coupled dynamics of BLDC control systems. Here, the GAO method is used to perform automatic tuning of the five FOPID variables as well as the three PID parameters to minimize a composite performance index consisting of the integral time absolute error (ITAE) and integral time square error (ITSE), thus ensuring an optimal trade-off between transient response and steady-state precision.

1.3 Previous work

In the past three years, several algorithms have been proposed to tune the parameters of the FOPID and PID controllers to enhance the performance of the BLDC control system. A hardware trans-conductance amplifier (OTA-C)-based fractional-order PID (FOPID) controller for BLDC motor speed regulation in a previous study [16]. A hardware-efficient analog structure capable of continuous-time control with adjustable fractional dynamics while still keeping high accuracy and low power consumption was the focus of the work. The paper has shown that the OTA-C implementation provides faster speed tracking, lower overshoot, and better dynamic stability as compared to conventional PID and digital FOPID control schemes. Nevertheless, the method is constrained by the limited performance of analog OTA components that are affected by temperature, are difficult to tune, and whose parameters drift with time.

To enhance steady-state accuracy and transient performance amidst fluctuating load conditions, an Equilibrium Optimizer (EO)-based FOPID controller was developed for the speed regulation of a BLDC motor [17]. Their goal was to use EO's balancing mechanism, which shows the dynamic balance between exploration and exploitation, to find the best values for the five FOPID parameters that would lead to faster convergence and fewer errors. The speed response was found to be excellent with very low settling time when comparing GA- and PSO-based techniques to the method proposed by the authors. However, the limitations of the method include its reliance on simulation environments with no hardware experiments and no robustness tests against parameter uncertainties and sensor noise. In addition, EO’s computational cost and the possibility of premature convergence in a high-dimensional search space were not fully considered. In addition, a dual fuzzy logic–based PID controller optimized by the Harmony Search Algorithm (HSA) for BLDC speed control was proposed in a previous study [18]. The primary goal was to improve the system's adaptability and its ability to handle nonlinearity by employing two fuzzy inference subsystems: one for on-the-fly gain adjustment of the PID parameters and another for dynamic error correction, while HSA was used to identify the best scaling factors for reducing steady-state error and overshoot. The simulation results indicated that the proposed controller achieved better dynamic stability, faster response, and higher robustness as compared to classical PID and single-fuzzy schemes.

However, the torque ripple in BLDC motor drives can be reduced through the use of a modified Luo converter with an FOPID controller [19]. Its main goal was to achieve the highest possible converter voltage gain and dynamic performance while also lowering torque pulsations and acoustic noise. The reconfigured topology exhibited higher efficiency and more stable torque than typical DC-DC converters with PID control. Nevertheless, the limitations of the approach arise from its reliance on a fixed controller structure without any optimization or self-tuning features; thus, its adaptability to different operating conditions is limited. The analysis only slightly considered the tolerances of the converter components and the switching losses, which may affect the results' accuracy in the real world. The lack of a complete sensitivity study or robustness examination against load disturbances makes this system less suitable for deployment in the real world.

An enhanced adaptive cascade FOPID controller to handle the speed of a BLDC motor in an EV [20]. The goal was to combine adaptive learning and fractional-order control to realize stable operation under parameter uncertainties, nonlinearity, and external disturbances. The technique changed the control gains on its own from the real-time feedback; thus, it was able to achieve superior tracking accuracy and disturbance rejection as compared to classical FOPID and adaptive PID controllers. However, the approach still has a few drawbacks, such as the reinforcement learning requiring a lot of computational power, the tuning complexity being increased, and there being no experimental validation except for the simulation. Besides, the scalability of the framework to multi-motor or distributed EV systems has not been discussed.

An adaptive Bacterial Foraging Optimisation (BFO)-based PID controller for a dual-motor electric vehicle powered by BLDC machines [21]. The simulation results showed that the adaptive BFO was able to improve the convergence speed and reduce the overshoot as compared to the GA- and PSO-tuned controllers. Nevertheless, the method’s heavy dependence on the computationally intensive adaptation and bacterial chemotaxis mechanisms makes it less feasible in real-time for high-frequency control loops. The authors also did not consider nonlinearities such as inverter switching delays and mechanical backlash, which may have an effect on the performance of practical systems.

Besides, a hierarchical FOPID controller whose parameters were tuned by the Pelican Optimization Algorithm (POA) to control the speed of a DC motor in the study [22]. Their main intention was to utilize the hierarchical FOPID structure to achieve a very good dynamic response and robustness, while the POA algorithm was finding the controller parameters that would lead to the minimum integral of the error indices. Simulation results revealed that their method reached faster convergence and a lower steady-state error as compared to the PSO- and GWO-based designs. Nevertheless, the model only considered a single-motor DC scenario and did not consider nonlinear effects typical of BLDC or multi-machine systems.

Also, the classical PID controller optimized by the WOA algorithm to increase the efficiency of the BLDC motor, thereby enhancing the transient response and robustness against load changes [23]. A comparative study of GA, PSO, and GWO-derived systems suggested that the application resulted in a major increase in tracking accuracy and energy efficiency. However, the method has some drawbacks, such as the need for proper WOA parameter tuning and the risk of becoming stuck at local minima. Furthermore, the controller was only evaluated through simulation studies without any noise consideration in hardware or sensors.

The tuning of the PID gains using the teaching-learning-based optimization (TLBO) algorithm was proposed in order to enhance the response of the controller and provide smooth speed BLDC control [24]. The achieved results are compared with the classical PID controllers under different operation conditions. In addition, a new intelligent optimization algorithm called adaptive input-output feedback linearization was applied on the BLDC motor drive to enhance the motor’s performance under nonlinear decoupling conditions [25]. The new method effectively improved the transient response and tracking error reduction largely in comparison with the conventional nonlinear controllers. The method is characterized by high algorithmic complexity and significant computational demands, which raises concerns about the real-time applicability of embedded EV systems. Additionally, the controller design relies on precise system modeling, making it sensitive to measurement noise and unmodeled dynamics.

A hybrid PSO-TLBO algorithm to tune the gains of the PID method [26]. The comparison tests showed that TLBO had a faster convergence and better stability margins than PSO, thus the transient response was improved and the torque ripple was reduced. However, the method is constrained by its reliance on fixed optimization parameters, which means that it cannot be very flexible when the solar conditions change quickly.

However, a comparative study for optimizing the PID controllers under different metaheuristic techniques such as PSO, GA, and DE algorithms was conducted in reference [27]. It was found that PSO had a superior convergence rate and robustness while DE resulted in smoother torque and lower oscillations. Nevertheless, the study was confined to static load conditions, and the authors did not consider the adaptability under dynamic torque or speed disturbances. Also, the omission of stochastic uncertainty modeling and the absence of hybrid or adaptive mechanisms limit the possibility of generalizing the results to real-world EV applications. Besides, a new optimization called a driving-training-based optimization algorithm (DTOA) was used to tune the parameters of the BLDC controller. The DTOA-PID control method showed quicker convergence and lower overshoot than exciting methods [28]. However, the optimization framework's computational complexity and parameter sensitivity might restrict real-time adaptability in embedded environments. Besides, the algorithm depended on empirically created learning rules, thus being a problem-specific method and less scalable for different motor ratings or inverter topologies.

The developed PID controller based on a heuristic Adaptive Lyrebird Optimization Algorithm (HALOA) was applied to improve the performance of the BLDC motor drive [29]. Nevertheless, the issue of the algorithm's convergence reliability in high-dimensional search spaces and dynamic disturbances was hardly addressed. However, the integration of hybrid learning or online adaptive methods could enhance the practicality and scalability of the electric vehicle and industrial motor-drive systems in the real world.

The speed overshoot, settling time, and steady-state error of the BLDC are improved by using different optimization methods for controlling the BLDC motor while maintaining a smooth torque response and drive efficiency in industrial applications [30]. The results indicated that metaheuristic-based tuning brought about a substantial dynamic change to the system response in comparison to the traditional Ziegler–Nichols methods. Also, an adaptive Tabu Search Algorithm (TSA) to control the speed of a BLDC motor by tuning the parameters [31]. Accordingly, the dynamic response can be made faster, the overshoot reduced, and the steady-state accuracy enhanced by simply adapting the memory and neighborhood search strategies to the TSA. The adaptive method was able to get better convergence than GA and PSO by changing the control parameters online. The results support the enhancement of speed tracking and torque smoothness during load variation.

Furthermore, a BES-based PI controller for a BLDC motor speed control system using a Bald Eagle Search Algorithm (BES) [32]. The main idea was to use BES's inherent balance between exploration and exploitation to find the best proportional and integral gains, thus minimizing the transient error and overshoot. The results of the simulation indicated that the BES-PI controller has improved rise time, stability, and robustness compared to PSO and slide mode methods. Besides, the control strategy only concentrated on single-objective optimization and did not consider multi-criteria trade-offs like the energy efficiency and torque ripple.

The paper [33] presented a nature-inspired optimization framework to enhance the performance of a solar-battery-supercapacitor hybrid energy storage system for a BLDC motor electric vehicle. The main goal was to manage the flow of energy and control the dynamics of the system using PID controllers that were optimized for each energy source. This led to better power management and a longer driving range.

An adaptive PID-fuzzy controller to control the speed of a BLDC motor while simultaneously reducing harmonics [34]. The foremost purpose was to integrate ordinary PID control with fuzzy-logic adaptability in order to automatically modify control gains and at the same time suppress current harmonics and torque ripple. The experimental outcomes showed that the dynamic response, total harmonic distortion (THD), and stability were substantially better as compared to the fixed-gain PID and PI-fuzzy controllers. The redesign of the control system by the addition of fuzzy-metaheuristic tuning and real-time embedded testing could be a way to further improve the control accuracy and energy saving in EV applications.

The novelty of this work lies in developing a GAO-based fractional-order PI tuning framework for EV motor speed control, where the proportional gain, integral gain, and fractional integral order are simultaneously optimized. In contrast to classical PI/PID controllers and conventional optimizer-based tuning methods, the proposed GAO-FOPI structure provides an additional fractional degree of freedom, enabling a better trade-off between overshoot suppression, rise time, settling time, and steady-state accuracy.

1.4 Contributions

This study proposes a new BLDC speed scheme using the GAO based FOPI controller. By optimizing the FOPI gains, the suggested GAO-FOPI aims to improve the performance of the BLDC motor under different speed profiles. The GAO-FOPI controller effectively limits the overshoot of the motor’s speed, rise time, and steady-state error for optimal operation and enhanced tracking speed time, improving the performance and reducing the trainset issues. The presented approach ensures the GAO rapid convergence by effectively tuning the FOPI gains with various iterations and population sizes.

The key contributions of this article are as follows:

1. Proposing a new method for controlling motor speed using the GAO algorithm with a Fractional-Order PI (FOPI) controller. Moreover, the proposed method was compared with classical methods such as PI and PSO-PI methods, thus improving both transient and steady-state performances.
2. The proposed controller is tested in three case studies representing different scenarios: step changes, sudden dropping in speed, and random speed variations. Performance is measured via the main indices such as the rise time, magnitude of overshoot, and steady-state speed error.
3. The achieved results confirm that the GAO-FOPI controller can limit the maximum overshoot to only 2.13%, reduce the rise time to 1.02 sec, and limit the steady-state error to 0.088 rpm. These results represent improvements of 68%, 40%, and 58% in overshoot, response speed, and tracking accuracy, respectively, compared to the classical PI controller.
4. The GAO-FOPI controller maintains a high level of performance without the need to increase the complexity of the control structure; hence, it is compatible with practical implementation and advancement of electric drive systems.

The rest of the article is as follows: the methods and materials are discussed in Section 2. Section 3 introduces the GAO algorithm. Section 4 presents the results and discussion. Section 5 summarizes the conclusion of the work.

3.1 Giant Armadillo Optimization algorithm overview

The GAO algorithm is one of the newer bio-inspired metaheuristics that rely on nature to design their strategies, in this case, the armadillo's search for food, digging, and staying safe from danger. GAO was proposed in reference [15] to tackle the most difficult nonlinear and multimodal optimization problems, which classical and swarm-based algorithms can hardly solve because they usually remain prematurely stuck in a local optimum. The "armadillo" in the algorithm represents a solution vector, so the whole population of potential solutions is interpreted as a group of armadillos searching for the best one. In GAO, an armadillo changes its position either by its foraging movement (looking for food here and there) or its burrowing activity around a promising nest (good solution) found previously, and so, there is intensive exploitation around a region. The algorithm is equipped with position-updating strategies that take into account and appropriately distribute the needed amount of global exploration and local exploitation during the search [15]. Figure 5 shows the Giant armadillo.

Figure 5. Giant armadillo taken and edited based on reference [15]

At the beginning of the optimization, the cells of the search space are explored via large random movements of the armadillos, enabling them to cover vast areas, thus increasing population diversity and impeding premature convergence. Subsequently, the armadillo's activities become more oriented towards the burrow, where the solution refinement is achieved in a much more localized manner. The idea of defensive escape of the armadillo is metaphorically incorporated as an algorithmic strategy to intentionally cause disruption in the direction and/or the magnitude of the position update, providing an opportunity for the algorithm to come out of its local optimum.

The way a giant armadillo hunts consists of two main activities: first, it locates and approaches termite mounds, and second, it digs in the termite mounds to eat the termites. Mathematical description of the giant armadillo's natural hunting behaviors is used in the development of the novel GAO method, which is elaborated below.

3.2 Mathematical analysis

3.2.1 Exploration phase

Giant armadillos form the population of the algorithm that can be mathematically represented by a matrix, as in Eq. (20). The initial locations of giant armadillos in the solution space are randomly generated at the start of the algorithm run as per Eq. (21) [15].

$X=\left[\begin{array}{c}X_1 \\ \vdots \\ X_i \\ \vdots \\ X_N\end{array}\right]=\left[\begin{array}{ccccc}x_{11} & \ldots & x_{1 d} & \ldots & x_{1 m} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ x_{i 1} & \ldots & x_{i d} & \ldots & x_{i m} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ X_N & \ldots & x_{N d} & \ldots & x_{N m}\end{array}\right]$          (20)

$x_{i d}=l b_d+r \times\left(u b_d-l b_d\right)$          (21)

where, $X$ is the optimizer matrix, $X_i$ represents the i^th candidate solution, $x_{i d}$ is the decision variable in dimension search space d^th, $N$ denotes the number of agents, $m$ is the variables of decision, $u b_d$ and $l b_d$ are the upper and lower limits of the search space, respectively, and $r$ is a random number [0,1].

The position of each giant armadillo in the problem-solving space reflects that it is a candidate solution for the problem, and thus, a value for the objective function can be evaluated corresponding to each giant armadillo. Thus, the objective function was written as follows:

$F=\left[\begin{array}{c}F_1 \\ \vdots \\ F_i \\ \vdots \\ F_N\end{array}\right]=\left[\begin{array}{c}F\left(X_1\right) \\ \vdots \\ F\left(X_i\right) \\ \vdots \\ F\left(X_N\right)\end{array}\right]$          (22)

where, $F$ is the objective function vector, $F_i$ represents the evaluated objective function.

The first phase of the GAO algorithm updates the positions of the population members in the problem-solving space by imitating the hunting behavior of giant armadillos when they attack termite mounds. The inspiration comes from a giant armadillo's movement to the termite mounds, which is mathematically modeled to lead the search process. Just considering the modeling of the movement of giant armadillos to termite mounds, a different location is determined for every individual of the population according to Eq. (23). Next, this new position becomes the current position of the joint if it increases the objective function value according to Eq. (24).

$x_{i, j}^{p 1}=x_{i, j}+r_{i, j} \times\left(s t m_{i, j}-I_{i, j} \times x_{i, j}\right)$          (23)

$X_i=\left\{\begin{array}{c}X_i^{p 1}, \quad F_i^{p 1} \leq F_i \\ X_i,  else,\end{array}\right.$          (24)

where, $s t m_{i, j}$ represents the i^th giant armadillo termite mound, $X_{i, j}^{p 1}$ is the j^th dimension of the new agent, $X_i^{p 1}$ denotes the new search agent, $F_i^{p 1}$ denotes the objective value, $r_{i, j}$ is the random number and $I_{i, j}$ is the random coefficient which equals 1 or 2 [15].

3.2.2 Exploitation phase

In the GAO framework, the digging ability of the giant armadillo in the termite mounds is mathematically described to change the location of each individual in the population, as shown in Eq. (25). This act of digging is like an intensive local search procedure that allows the armadillo to carefully exploit the areas near the good solutions in the solution space. Once the new location is calculated, it is evaluated by the objective function. In case the new location results in a better solution, it is used to replace the previous location of the individual in the population, as in Eq. (26) [15].

$x_{i, j}^{p 2}=x_{i, j}+\left(1-2 r_{i, j}\right) \times \frac{u b_j-l b_j}{t}$          (25)

$X_i=\left\{\begin{array}{c}X_i^{p 2}, \quad F_i^{p 2} \leq F_i \\ X_i, else,\end{array}\right.$          (26)

where, $x_{i, j}^{p 2}$ is the j^th dimension of the new agent for d the igging phase, $X_i^{p 2}$ denotes the new search agent for digging phase, $F_i^{p 2}$ denotes the objective value, $r_{i, j}$ is the random number and $t$ is the iteration counter.

The flowchart of the AGO is shown in Figure 6. This figure displays the step-by-step execution of the GAO algorithm for the optimal PI controller tuning based on the ITAE performance index. The two-phase search mechanism of GAO, consisting of exploration (Phase 1) and exploitation (Phase 2), allows attaining a proper trade-off between global search diversification and local search intensification. The candidate solutions are progressively refined according to Eqs. (23)-(26). Thus, the algorithm not only prevents stagnation at local optima but also continuously enhances the tracking performance. However, choosing ITAE as the fitness function is a clear signal of the focus on the long-term error minimization, which eventually leads to faster settling time and better steady-state accuracy in the BLDC motor speed control.

3.3 Comparison the Giant Armadillo Optimization with other classical algorithms

The GAO algorithm, which stands for Group Armadillo Optimization, is one of the most recent metaheuristic optimizers. The foraging and defensive behavior of armadillos inspired the GAO algorithm. If we compare GAO with traditional algorithms such as GWO, GA, WOA and PSO, it can be seen that the former is capable of providing a better, more adaptive balance between exploration and exploitation. As a result, this approach not only improves the convergence speed but also decreases the possibility of premature convergence. There is a previous study highlighting the capability of GAO to efficiently solve nonlinear optimization problems, including parameter tuning and energy system optimization. Nonetheless, its use in fractional-order controller tuning for EV motor speed control is still a virgin area, which marks one of the main contributions of this paper.

Table 1 presents a basic comparative analysis of the most commonly used algorithms in relation to the GAO. As presented in Table 1, the classical algorithms have either strong exploitation or strong exploration, but usually, they struggle to strike a good balance between both activities. Techniques like GWO and WOA enhance this balance but sometimes experience slow convergence issues at the fine-tuning phases. On the other hand, GAO has a quite capable exploration and exploitation adaptive mechanism, which leads to faster convergence and a lower chance of premature stagnation. For this reason, GAO is very suitable for the tuning of fractional-order controllers in nonlinear EV motor control systems, wherein both global searching ability and local adjustments matter.

Table 1. Comparative analysis of GAO with other algorithms

For comparison, the PSO is widely used, but the risk of stagnation remains; WOA is a good balance, but fine-tuning takes time; GWO is effective but susceptible to parameter changes; and GA is a robust global search, but it demands a lot of computation. Unlike the used GAO, it's a great choice for nonlinear control optimization.