Studying the Applicability of Swarm Intelligence in Designing Optimized PI Controller for DC-DC Zeta Converter: A MATLAB-Based Approach

Studying the stability of power converters and improving the performance is a major concern for researchers in the domain of power electronics. In this context, the DC-DC Zeta converter is studied in this paper and the closed-loop operation is comprehensively investigated by employing swarm intelligence (SI) algorithms with a view to design an optimized proportionalintegral (PI) controller. ese algorithms have been increasingly used to develop and optimize power converters in recent years. e state-space averaging technique was used to design the converter’s closed-loop transfer function. Hence, the traditional and SI algorithm-based PI controllers are inspected, and comparative analysis is presented. Four objective functions termed as integral absolute error, integral time absolute error, integral square error, integral time squared error, gain values, and dierent performance parameters such as percentage of overshoot, rise time, settling time, and peak amplitude are tabulated to examine the stability of the system. Furthermore, eigenvalues have been analyzed for determining the stability of the system extensively. Finally, a detailed comparative study is shown to provide a detailed evaluation of the performances where ant colony optimization for continuous domains (ACOR)-based PI controller has shown promising results than other SI-based controllers in terms of percentage of overshoot (2.27%), rise time (1.54 μs), and settling time (0.103 μs). All the simulation results and analysis are obtained using the MATLAB-Simulink.


Introduction
In this modern era of technology, designing sophisticated power electronic devices carries immense importance in the research community. is emerging technology has made its way into various applications, including renewable energy generation, electric vehicle (EV), biomedical equipment, and tiny appliances like laptop and mobile phone chargers, LED drives, microgrid systems. [1]. In this regard, DC-DC converters are the simplest basic electronic circuits for converting DC voltage switching activity [2]. Apart from turning the source's DC output into a regulated DC voltage, the DC-DC power converter also performs handful significant control duties. For example, it can track the maximum power point (MPPT) for di erent renewable sources [3].
Buck, Boost, Buck-Boost, Cuk, SEPIC, Zeta, and other advanced DC-DC converters are all examples of DC-DC converters. Buck, Boost, and Buck-boost can perform step down, step up, and step up-down actions, respectively. However, some higher order converters like Zeta, Cuk, and SEPIC converters, allow power when the demand required a low input voltage and a high output voltage [4][5][6][7][8][9]. In correlation with Cuk and SEPIC converters, the Zeta converter is a nonlinear, noninverting fourth-order converter that can operate as buck-boost-buck with respect to input and boostbuck-boost with respect to energy output. e key advantages of the Zeta converter are reduced switching stress, exibility, and nonpulsating output current. It is dependent on the duty cycle, which can be operated in several modes such as boost and buck [10]. In addition, Zeta outperforms SEPIC in terms of maintaining a stable feedback loop for a more comprehensive input voltage range, higher load transients, reduced output voltage ripple, and simpler compensation [11]. However, an open-loop Zeta converter cannot meet expectations based on the performance characteristics of the percentage of overshoot, rise time, settling time, and peak amplitude. As a result, a controller is required to improve performance and keep the output voltage stable.
For decades, the PI controller has been one of the most basic and extensively used controllers as it uses a control loop feedback mechanism to control process variables. A proportional parameter (Kp) will lower the rising time and the steady-state error but never completely removes it. On the other hand, integral feedback (Ki) can be used to eliminate the steady-state error and lower the amount of forwarding gain required. e goal of a typical PI tuning controller technique is to get the performance index under consideration to a minimum or maximum value. However, it is challenging to identify appropriate values for Kp and Ki in a simple PI tuner by manual tuning method for proper control. In this context, different forms of smart and intelligence algorithms can be brought into action to make the whole system automated and more sophisticated. For instance, swarm intelligent (SI) algorithms have received notable attention from the power and control engineers in this regard. e potential parallelism and distributed properties of SI algorithms allow them to solve difficult nonlinear problems with enhanced capabilities in terms of self-adaptability, resilience, and searchability [12]. Among different SI-based optimization methods, particle swarm optimization (PSO), artificial bee colony (ABC), firefly algorithm (FA), shuffled frog leaping algorithm (SFLA), ant colony optimization for continuous domain (ACOR), and chimp optimization technique [13][14][15][16][17][18][19] are employed by different researches in various times in a wide range of power electronics-based application. In [20], the output regulation of the Zeta converter was addressed in the presence of model uncertainty, input voltage changes, and load variations. e linear matrix inequality tool was used to construct and solve this problem, resulting in an optimal robust state feedback controller. Finally, the performance of the proposed method was compared to the traditional PI controller. In [21], an ACOR-based PID controller was presented for the Zeta converter with the model order reduction method. e SSA technique was utilized to obtain the linear model of the Zeta converter, and the fourth-order transfer function was derived in terms of the duty ratio to the output voltage. On the other hand, the study in [22] displayed the analysis and design of a DC-DC Buck converter with a cascaded control strategy by employing PSO and gravitational search algorithm (GSA) adjusted PI and slide mode controller (SMC). In [23], an ABC-PID controller algorithm was designed and implemented in a buck converter. In terms of settling time, steady-state error, and load change scenarios, the results reveal that ABC-PID outperforms genetic algorithm (GA)-based PID in controller. However, a SEPIC converter was designed by utilizing perturbation and observation (P&O), PSO, and FA, and it was observed that FA minimizes the difficulties of determining global maximum power, reduces convergence, and improves the converter's conversion efficiency [24]. Using the whale optimization technique, a novel approach was proposed in [25] for selecting the PID controller settings of a DC-DC Buck converter. Based on simulation findings, it was discovered that the proposed method was more effective than the genetic algorithm at enhancing the transient response of the Buck converter. e innovative approach for creating fuzzy logic controllers for voltage-regulated DC-DC power converters is suggested in [26]. In order to identify the best feasible membership functions and rules for the fuzzy controller, the multi-objective PSO was utilized to find several pareto-optimal solutions in a multi-objective optimization problem. Furthermore, in [27], the performance of sine-cosine algorithm (SCA) and PSO are compared in coordinating the power of an autonomous hybrid microgrid system with adjustable loads for demand-side management. In [28], the PID controller parameters for a DC-DC Boost converter were tuned using the grey wolf optimizer (GWO) and performances were compared with GA and PSO. e study by the authors of [29] offers a PID controller design using PSO for the best selection of the controller parameters. e outcomes show that the optimization approach is capable of minimizing the aforementioned transient and steady-state faults, which reduce output voltage fluctuations despite input and load side oscillations. e performance and efficiency of the Landsman, SEPIC, and Zeta converters are evaluated in this research [30], where similar investigation was conducted using PSO. However, this study focuses on the following aspects: Proper voltage regulation of a Zeta converter is obtained in this paper utilizing a PI controller that has been optimized using several algorithms to determine optimal values for the proportional and integral gains, resulting in improved converter performance. In Section 2, the state space averaging technique is used to model the converter mathematically. e outline of the algorithms is depicted with flowcharts in Section 3, and the algorithm's objective function and SI-based PI controller design are also depicted in this section. e results and simulations, as well as the overall analysis, are presented in Section 4. e MATLAB programming platform is used to run all of the required simulations. Lastly, a comprehensive correlation analysis is provided.

State Space Average Method
State space averaging (SSA) is widely employed and a popular method for modeling power converters mathematically. All the derivatives of the circuit have been obtained with the aid of this technique [31]. Piecewise linear networks, where the topology changes at the boundaries between succeeding subintervals within a prototypical switching cycle, are frequently used to describe switched converters. e relevant state-space equations can be determined based on the state of each switching element, such as a transistor or a diode. e energy storage component is usually linked to the state variables [32].
In the Zeta converter (shown in Figure 1), there are two modes of operation. When MOSFET is on the ON-state, the diode is open. So, in this interval, -(V in + V 0 ) is found across the diode. Also, both the inductors are in the charging phase, which means the current in these inductors is increasing linearly. e capacitor C 1 will be discharged, and energy will be dissipated at the resistor; hence V 0 will increase, and it is connected in series with L 2 . e total current of the charging inductor flows through MOSFET.
By Kirchhoff's voltage law (KVL) in Figure 2, the following equations are achieved: In the next mode of operation, MOSFET becomes open, and the diode is shorted (Figure 3). Now, both inductors L 1 and L 2 will be discharged. e diode will be forward biased as the voltage polarity of the inductor is changed, and it will conduct.
At this stage, the stored voltage in L 1 and L 2 will be dissipated to capacitor C 1 and the output resistor R O . As a consequence, both inductor currents decrease linearly. By KVL, we get Current flows through capacitor C 1 with the help of KCL: e input-output voltage relationship of the Zeta converter is by applying KVL, for inductor L 1 on ON-state and OFF-state, respectively, are V L1 � V in and So, we get Now, the equation for inductor L 2 using KVL is en overall, equation will be Figure 1: Zeta converter. Figure 2: Zeta converter while Q is shorted.

Mathematical Problems in Engineering
Merging, these equations, we get e SSA equation are obtained using these formulas: where And Assuming the variables, State-space equation matrix is given as So, System matrix, control matrix, output matrix, and feedforward matrix are represented by A, B, C, and D, respectively. x, u, and y are the state vector, input vector, and output vector, accordingly. A 1 and B 1 also are mentioned for the ON-state. A 2 and B 2 are noted for the OFF-state, correspondingly.

Implementation of Swarm
Intelligence Algorithms 3.1. Outline of the Algorithms. Swarm intelligence (SI) algorithm refers to a type of optimization technique that is evolving, that is collective intellect of swarms of simple entities [33]. Developing algorithms with swarm intelligence necessitates adaptability to internal and external changes, as well as robustness in the face of individual failures. In the following section, five SI algorithms are studied and implemented for the stability operation of the Zeta converter.

Particle Swarm Optimization (PSO). PSO, proposed by
Eberhart and Kennedy in 1995, is a stochastic optimization technique based on swarm that iteratively refines solution candidates. e method is based on sharing information among particle individuals and the particles use their own positions by previous information regarding the best location in the group. It starts with a population of randomly generated solutions, particles with a velocity and a location [34]. e solution particles in this circumstance move through the search region at dynamically balanced velocities as part of a process. e local best-known position of particles and the best-known positions in the search space in iteration (t − 1) influence particle movement in iteration t toward optimum solutions are represented as pbest t−1 s and gbest t−1 s . Particle i's position and velocity in dimension s of optimization space are represented by (x t is ) and (v t is ), which are written as follows

Artificial Bee Colony (ABC).
ABC is a swarm-based meta-heuristic algorithm developed by Karaboga (2005) for numerical problem optimization. e algorithm is based on Tereshko and Loengarov's (2005) model for honey bee colony foraging behavior [35]. In the solution space, food sources distributed throughout nature are referred to as solutions. e ABC algorithm's conceptualization converts natural processes and activities into algorithmic components and functionalities, where a "food source" is translated into a "feasible solution," and a "nectar amount" is recognized as the fitness of a solution denoted by F(x i ) as provided in the following equation: Equation (17) calculates the probability of a specific food source being chosen by the ABC algorithm, while (17) generates a neighboring solution such as where x i , x n , and v i refer to the current, neighbor, and candidate solutions, respectively, "i" is random number, and N denotes the index of the food source.
where x i,j is the j th decision variable as a member of the x i solution vector; j � 1, 2, ..., D is the index, D is the total number of decision variables, and LB and UB are the decision variable's upper and lower boundary values.

Firefly Algorithm (FA).
FA was proposed by Yang created in 2008, which is a revolutionary SI algorithm that imitates the behavior of fireflies in group encounters [36]. To simplify the composition of the FA, Yang et al. included these three rules. First, there is no gender distinction among the firefly. Second, a more visible firefly will attract a less visible one. ird, the brightness of each firefly is determined by the goal function's fitness. e light intensity I(r) varies monotonically and exponentially with the distance r in its simplest form.
where, I o represents the initial light intensity and c represents the light absorption coefficient. According to the abovementioned criteria, each firefly's attraction is proportional to its light intensity; consequently, the attractiveness (beta) is stated as where "r" represents the Euclidean distance between two fireflies and "β o " represents the minimum attractiveness at r = 0. e light absorption coefficient is indicated by parameter "c," which is normally set to 1.
e Euclidean distance "r ij " between two randomly selected fireflies x i and x j in the same search space can be determined as where "D" is the dimension of the problem.
Any firefly x i moving towards a brighter firefly x j is expressed as follows: Mathematical Problems in Engineering (22) where the d th dimension values of firefly "i" and "j," respectively, are X jd and X id . Furthermore, "ε" is a random number with the range [−0.5, 0.5], and "α" is a random value with the range [0, 1]. Finally, the number of iterations is denoted by "t".

Shuffled Frog Leaping Algorithm (SFLA). Eusuff and
Lansey presented SFLA in 2003 as an evolutionary optimization strategy based on swarm intelligence that mimics the foraging behavior of frogs for searching global optimum [37]. SFLA refers to a bionic optimization algorithm inspired by frog foraging behavior that fully utilizes the individual generation approach of the memetic algorithm (MA) as well as the information exchange mechanism of the PSO. e technique for splitting the population into memeplexes is as follows: the first frog is assigned to the first memeplex M1, the second to the second memeplex M2, the third to the third memeplex Ms, the fourth to the fourth memeplex Ms, and so on.

Ant Colony Optimization for Continuous Domain (ACOR).
Ant colony optimization for continuous domain (ACOR) is a dependable population-based algorithm for solving discrete problems that mimic the foraging behavior of ants [38]. e core idea of ACOR is to switch from a discrete to a continuous probability distribution, i.e., a probability density function (PDF). In ACOR, ant samples refer to PDF. A probability density function can theoretically be any function P(x), such that In ACOR, solutions are archived in a file called solution archive. e solution archive is initialized at the start of the process by randomly generating solutions. After all solutions have been collocated, they are reserved, and the rest are discarded. e following is how the mentioned standard deviation should be determined in this direction: e positive parameter ξ is constant across all dimensions and has an effect comparable to the ACOR's pheromone evaporation rate.

Objective Functions.
e primary objective is to check the transient behavior of the converter and inspect its stability in the open-loop and closed-loop system. However, the instability can be mitigated by employing a manually tuned PI tuner. But, the process is quite time consuming and a lot of effort has to be invested. For this context, the responsibility of tuning and selecting the most optimized parameters of the controller is handed over to SI algorithms. e analysis is carried out in MATLAB and Simulink. e performance factors of the control instruments are analyzed in the form of error formulas to state the objective functions. e objective functions introduce the performance parameter, which aids in illustrating the stability and effectiveness of the converter. ree error functions are used in this paper. e formulas are as follows:  Figure 4: Flowchart of SI-based PI controller [11]. Mathematical Problems in Engineering sets of controller parameter values are particles with their values changed to minimize the objective function, which in this case is the error criteria. It is ensured that the estimated controller settings result in a reliable closed-loop system for the PI controller architecture. e shape of the closed-loop system transient response to some input signal, such as a step or ramp input, is commonly related to the performance objectives from a control standpoint, as shown in Figure 4.

Simulation Results and Stability Analysis
e relevant simulations have been carried out using MATLAB and Simulink to acquire a detailed understanding of the Zeta converter's stability. erefore, Table 1 contains the recommended parameters for converter [11].
e Zeta converter Simulink model has been built and illustrated in Figure 5 Hence, the step response of the closed-loop system has been examined, revealing a 10.5% overshoot as shown in Figure 7. e output of the conventional PI controlled Zeta converter has been shown in Table 2

4.1.
Step Response Analysis. All the parameters of ABC, ACOR, PSO, FA, and SFLA algorithms have been tabulated in Table 3, Table 4, Table 5, Table 6, and Table 7, respectively. However, the corresponding fitness functions (IAE, ITAE, ISE, and ITSE), and the relevant responses have been recorded in Table 8, Table 9, Table 10, Table 11, and Table 12, consecutively. Hence, the comparative analysis among the controllers has been presented in Figures 8-12, respectively.

Eigenvalue Analysis.
e proposed algorithms' capacity to achieve system stability is determined using eigenvalue analysis. e analysis is performed by using four specified objective functions consist of IAE, ITAE, ISE, and ITSE. e real portion of the eigenvalue offers information about the system's stability. e system is said to be stable when the real component of the eigenvalue lies on the left side of the "s" plane [39]. Table 13 demonstrates the eigenvalues for each objective function of five swarm-based algorithms. Four Eigenvalues are obtained for each objective function, including two complex values, as stated in the table. However, from Figure 13 to Figure 17, it is visually evident that their imaginary parts are placed in such a way that they appear to cancel each other out apparently. Furthermore, the Step response of the conventional PI controller.             real component of the eigenvalues from five SI-based PI controllers stays on the left side of the "s" plane, indicating that the system is stable, as shown in the tables. ere is some overlapping of points due to the extensive range of eigenvalues in the figures. As a result, each algorithm's figure is accompanied by four additional figures that represent overlapping spots in a more obvious manner.

Comparative Analysis.
Compared to four fitness functions, for ACOR-PI, FA-PI, and SFLA-PI, the ISE fitness function delivers a lower percentage of overshoot, shorter settling time, lower rise time, and lower peak amplitude. For the ABC-PI and PSO-PI controllers, the ITSE and IAE fitness functions independently yield a lower percentage of overshoot, shorter settling time, lower rising time, and lower peak amplitude when compared to other fitness functions. Furthermore, the stability of the SI-based PI controller is exhibited in figures of eigenvalues. It is evident from the eigenvalue figures that ISE objective function for ACOR-PI is more stable compared to the rest, as it is on the leftmost side of the s-plane, exhibiting the largest negative real part of eigenvalue among all five algorithms. Despite the fact that the ISE objective function for SFLA-PI illustrates remarkably similar stability and step response to ACOR-PI, the peak amplitudes for all fitness functions for each algorithm are almost identical. When comparing these five algorithms, it is apparent that the ACOR-PI controller's ISE fitness function delivers the best response. e top-performing fitness function from each algorithm is presented in Table 14, along with their associated performance parameters.
Overall comparison of percentage of overshoot, rise time, and settling time among the best performing ones are depicted in Figure 18, Figure 19, and Figure 20, respectively. Moreover, in the study [11], the Zeta converter's optimization was carried out using a PID controller that incorporated the particle swarm optimization algorithm.        However, in this case we used the PI controller which collaborated separately with five swarm intelligence algorithms to perform a comparative analysis. Since the derivative values in our work were so negligible and made no contribution to the optimization, we decided to omit the derivative controller and instead developed a PI controller. Some of the corresponding literature are studied and their outputs are shown in Table 15.

Conclusion
After observing all the simulation results, it is evident that the ACOR-PI controller outperforms the other controllers in terms of percentage of overshoot (2.27%), rise time (1.54 μs), and settling time (0.103 μs). In addition, it is worth mentioning that both the PSO-PI and ABC-PI controllers provid effective responses. Furthermore, among the four objective functions (IAE, ITAE, ISE, and ITSE), the ISE objective function offered remarkable responses for three swarmedbased PI controller (ACOR-PI, SFLA-PI, and FA-PI). For the other two, IAE for PSO-PI and ITSE for ABC-PI yielded better responses. Hence, applying eigenvalue analysis, it is observed that the real parts of eigenvalues from five swarmed-based PI controllers are negative, indicating that the system is stable for all five PI controllers. e eigenvalue figures are also depicting that the ISE objective function for ACOR-PI is providing the largest negative real value compared to all five swarmed-based PI controllers, revealing that the system is more stable for this controller. However, there is not much of a performance difference between them, especially with ACOR-PI and SFLA-PI controllers and so, it can be concluded that both controllers will be best suited for Zeta converter. In the future, we plan to use a modeling technique based on machine learning to synthesize the response of power converters which will enable us to develop an alternative design methodology for power electronic devices that relies on artificial intelligence (AI) techniques rather than conventional approaches.

D:
Duty cycle x: State vector y: Output vector u: Input vector A: System matrix B: Control matrix C: Output matrix D: Feed forward matrix x t is : Optimized position of i's particle v t is : Velocity of i's particle c 1 , c 2 : Acceleration coefficient w: Inertia weight Intensity of light c: Light absorption coefficient β: Function of attractiveness r: Euclidean distance X i , X j : Butterflies D: Dimension of the problem α, ε: Random number MA: Memetic algorithm OS: Overshoot T r : Rising time T s : Settling time.

Data Availability
e data used to support the findings of this study are available within this article.

Conflicts of Interest
e authors declare that there are no conflicts of interest.