Determination of Different Types of Controller Parameters Using Metaheuristic Optimization Algorithms for Buck Converter Systems

The steady-state operation with low error and the fast dynamic response in transient of the DC-DC converter circuits depend on the controller design. The performance of the controller used in DC-DC converters, which vary the level of DC voltage depends on the controller coefficients. Although classical methods are often used to determine these coefficients in controller design, various modern optimization methods have been recently used. In this study, the DC-DC buck converter control simulation is performed with FOPID, PID and TID controllers. Aquila Optimizer, African Vultures Optimization Algorithm and Hunger Games Search optimization algorithms are used to determine the coefficients of these controllers in the literature. However, Fitness-Distance Balance Based Runge Kutta is employed first time for PID controller in buck converter in this study. The performance indices integral absolute error, integral square error, integral time absolute error, and integral time squared error are employed to assess the outcomes. When the results obtained are examined, the FOPID controller gives the best results in the control of the buck converter. These results are obtained by using the coefficients determined by the Fitness-Distance Balance Based Runge Kutta (FDBRUN) optimization algorithm. It has better performance than the other algorithms.


I. INTRODUCTION
Power electronics converters are used in photovoltaic systems, smart grids, electric vehicles, uninterrupted power supplies, many industrial applications and products. In power electronics converter topologies, which are generally controlled by pulse width modulation semiconductor power switches are used, and they are controlled by different control strategies. Inductors, capacitors and diodes are also used in various placement depending on the topology. The converter has nonlinear behavior due to the presence of semiconductor elements such as diodes and switches. Therefore, linearization is required for modeling this type of converters.
Power electronics converters have different topologies such as DC-DC, AC-DC or DC-AC converters according to the point of use. Switched mode DC-DC converters are The associate editor coordinating the review of this manuscript and approving it for publication was Vitor Monteiro . used in many industrial and technological products which work with DC current. Various DC-DC converters such as buck, boost, buck-boost, sepic, cuk, flyback etc. are used in different applications. The buck converter is a basic nonisolated converter type used in applications where the output voltage is lower than the input voltage.
As in many power electronics converters, Proportional-Integral (PI) or Proportional-Integral-Derivative (PID) controllers are commonly used in buck converter control. In addition, fuzzy logic control [1], sliding mode control [2], hysteresis control [3], dead beat control [4] are frequently used control methods. The controlled electrical quantity varies depending on the application in which the converter is used. PID controller is used in buck converter control in various applications such as speed control in motor applications, photovoltaic (PV) panel power control solar in systems, output voltage control in switched power supplies and many applications. This controller ensures that the applied error signal reaches the zero value in the shortest time with minimum overshoot. The reference signal is obtained with the algorithm using controller, and switching signal is produced by PWM controller using reference signal. By using the output voltage reference and the measured output voltage produced in a buck converter which the output voltage is controlled, an error is obtained and given to the PID controller. The controller generates the modulation signal that reduces this error to zero, and switching signals are obtained with the help of pulse-width-modulation (PWM). In order to eliminate the effect of a disturbing input, which occurs during the operation of the converter the controller must respond in a robust, dynamic, fast, and stable manner. The dynamic response speed, steady-state response, and stability of the converter depend on the PID coefficients. The determination of these coefficients plays an essential role in the controller performance. There are various methods which are used to determine the coefficients.
The methods used for PID controller coefficient detection can be divided into 3 basic categories: rule-based, formulabased, and optimization-based [5]. In rule-based methods, a rule-based PID-like fuzzy controller was used in motor torque control by using a PD-like fuzzy controller and a parallel PI controller. The use of a minimum rule-based fuzzy controller reduced the challenges experienced in torque control [6]. The Antlion Optimization (ALO) method was used to compare the performance of the fractional-order Particle Swarm Optimization (PSO)-based PID controller with the PSO-based PID controller in the fuzzy-based fractionalorder PID controller design utilized in buck converter control. The designed controller has been proved to be more robust under various operating situations such as noise and load variations [7]. Temperature and level control were carried out with real-time experimental application using the fuzzy-rule based automatic adjustment approach in the design of the Internal Model Control (IMC) based PID controller, where only the closed loop time constant parameter needed to be adjusted [8].
The Ziegler Nichols approach, one of the formula-based methods extensively used in PID design, and the Cohen-Coon [9] method were utilized in the design of the PID controller used in the drinking water filtering system, and their performances were compared [10]. For fractional-order PID (FOPID), a novel Ziegler Nichols autotuning approach with smaller overshoot, improved durability and steady-state response, and shorter sitting time was developed [11].
Maximum power point tracking in solar systems was accomplished using a PID controller optimized with the Cuckoo Search Algorithm (CSA). The global maximum power point in the shaded condition was calculated, and the oscillation created at that point is removed [12]. The PSO technique was used to compute the PID controller coefficients of a bidirectional buck-boost converter utilized in small satellite applications. [13]. Moth-Flame Optimization (MFO) was used in wind turbine blade angle control to prevent oscillation in output voltage and power [14]. FOPID controller of a boost converter was optimized with Queen Bee assisted Genetic Algorithm (QBGA) in [15] to demonstrate the robustness of the proposed controller. In the brushless DC motor control (BLDCM), a new technique called GEO-RPFNN, which the combination of Golden Eagle Optimization (GEO) and Radial Basis Function Neural Network (RBFNN) was proposed. The parameters of the PID controller were optimized with the proposed algorithm, and better results with regarding to torque ripple, power factor and THD of stator current [16]. The coefficients of PID controller were optimized with fuzzy particle swarm optimization algorithm in the buck converter, and the result were given in the integral of time-weighted absolute error (ITAE) performance index [17]. A feedbacktype two-degree-of-freedom proportional-integral-derivative (FB2PID) controller was optimized with a bat algorithm (BA) in a parallel dc-dc converter (PDCC). The dynamic response and robustness of the PDCC was improved with the optimization [18]. A four-switch buck-boost dc-dc converter used in Proton Exchange Membrane Fuel Cell (PEMFC) is controlled with FOPID controller for voltage compensation, and the controller parameters were optimized with a stochastic inertia weight PSO algorithm. [19]. Ant colony optimization (ACO) algorithm was used to determine the optimum PID controller parameters of matrix converter [20]. A new designing method based on Strength Pareto Evolutionary Algorithm (SPEA) was proposed for fractional-order PID controller that was used for boost converter control. The better start-up response was achieved with the new method [21]. The PID parameters of zeta converter was optimized with Ant colony Optimization. The zeta converter that is a fourth order system was reduced to second order, and PID coefficients were optimized [22]. LCC resonant converter that was controlled with PID controller was presented in [23]. The PID controller was tuned with differential evolution optimization (DEO), grey wolf optimization (GWO) and grasshopper optimization (GOA) algorithms. Integral absolute error (IAE), integral square error (ISE) and integral time absolute error (ITAE) were used as performance indices. The interleaved buckboost converter was controlled with PID, and the coefficients were optimized with PSO algorithm [24]. PID, FOPID, and TID (tilt-integral-derivative) controllers were used to operate a buck type step-down converter in this study. The coefficients of these controllers have been optimized for the first time with the Fitness-Distance Balance Based Runge Kutta (FDBRUN) [25] algorithm in the literature and presented in comparison with the results of the Aquila Optimizer (AO) [26], African Vultures Optimization Algorithm (AVOA) [27] and Hunger Games Search (HGS) [28] optimization algorithms. The S-domain circuit model of the circuit is derived in order to employ these optimization approaches. The controller coefficients derived by four distinct optimization approaches are used to evaluate the converter response, and the results are presented.
In the second part of the study, the mathematical equations and modeling of the buck converter are presented. FOPID and TID controllers used in closed loop control algorithm are VOLUME 10, 2022 explained in Section 3. In Section 4, FDBRUN algorithm, which is the metaheuristic optimization algorithm proposed in the study is mentioned. The simulation study with different optimization algorithms and controllers is given in Section 5 and the obtained results are given in Section 6.

II. MODELING OF BUCK CONVERTER
Because switched-mode power supplies are more efficient, they have many applications such as television, electric vehicles, mobile devices, and uninterruptible power supply. Buck converter, which is switched-mode power supply have a variety of applications and is used to decrease the DC voltage level. The converter, which is made up of passive parts like inductors and capacitors as well as switching elements like MOSFETs and diodes is controlled by modifying the MOSFET's duty cycle. The voltage waveform produced by high-frequency switching is filtered by the LC filter at the output, a constant DC voltage is obtained.
When the PWM signal is applied to the MOSFET in the converter, the input voltage is also applied to the LC filter. The voltage created on the LC filter ends as a result of switching the MOSFET is seen in Fig. 2a. It is seen that the LC filter voltage, which is a square wave takes the value of the input voltage when the switch is turned on and becomes zero when the switch is turned off. Simultaneously, when voltage is applied, the inductance current increases, and when no voltage is applied, the current reduces. Inductance current and output voltage vary depending on the duty cycle of the switching signal as seen in Fig. 2b and Fig. 2c.
Because it is a switched-type converter, the buck converter circuit seen in Fig. 1 is classified as a nonlinear system. Different circuit equations occur and electrical quantities change depending on the conduction state of the semiconductor switch positioned at the input part of the circuit. While the switch is turned on, energy is stored in the inductance while the output is supplied. When the switch is turned off, the energy stored in the inductance is delivered into the output. The energy balance is accomplished and the desired output is obtained by applying the duty cycle computed by the control algorithm.
Firstly, the system must be linearized before designing the converter's control. The dynamic equations of the buck converter used to make the system linear are given in equations (1) and (2). In these equations, V i is the input voltage, v 0 is the output voltage, L is the inductance, C is the output capacitor, R is the output load resistance, and d is  the switching state. When writing the equations, the circuit equations that occur when the switch is turned on and off are employed. The state variables in these equations are inductance current and capacitor voltage. The transfer function of the converter can be calculated using these equations. The s-domain transfer function of the converter obtained by using equations is given in equation (3). The study examines the performances of different control structures in different optimization techniques utilizing this transfer function, which gives the fluctuation of the output voltage based on the duty cycle.
Equations (4) and (5) can be used to compute the minimal inductance and capacitor value that should be used based on the ripple amount (i L ) and switching frequency (f s ) determined while designing the buck converter. As the equations  show, increasing the switching frequency causes the inductance and capacitor values for the same current and voltage ripple to decrease.

III. CLOSED-LOOP CONTROLLED BUCK CONVERTER
Open-loop and closed-loop control are the two basic control architectures of converters. If a regulation is not required, open-loop control is applied. It can control the system if each electrical quantity is stable. However, output voltage of the converter changes when a disturbance occurs in the system. To regulate the output, closed-loop control structure is required. Fig. 4 depicts the closed-loop control block diagram of the converter. In the figure,v 0 (s),v ref (s),ê v (s) vê d(s) defines output voltage, reference value of output voltage, voltage error and duty cycle, respectively. The controller attempts to reduce the output voltage error to zero while ensuring that the desired output voltage is reached.
There are various types of regulators used in closed loop control. In this study, PID, FOPID and TID regulators were examined.

A. FRACTIONAL ORDER PROPORTIONAL INTEGRAL DERIVATIVE (FOPID) CONTROLLER
The FOPID controller transfer function is given in (6). In the equation, k p , k i , k d , λ and µ define the proportional constant, integral constant, derivative constant, integral order and fractional derivative order, respectively. In Fig. 5, the block diagram of FOPID controller is seen. The reference signal is obtained by summing the results obtained by passing the error value of the control signal through proportional, integral, and derivative blocks.
If the integral order and fractional derivative order is λ = 1 and µ = 1, λ = 0 and µ = 1, λ = 1 and µ = 0, λ = 0 and µ = 0, PID, PD, PI and P controller are obtained, respectively. A generalized FOPID control extends  PID control from a point to a plane in Fig. 6, which shows the relationship between FOPID and classical controllers. Controllers that are formed depending on the values of µ and λ parameters are seen in the figure.

B. TILT INTEGRAL DERIVATIVE (TID) CONTROLLER
The TID controller is a compensator with coefficients of k T , k i , k d and tuning parameter of n. TID controller, which has a similar structure with PID controller, has s −1/n transfer function with n parameter and k T parameter instead of proportional coefficient k p .

IV. THE PROPOSED METAHEURISTIC OPTIMIZATION ALGORITHM A. RUNGE KUTTA OPTIMIZATION ALGORITHM
The RUN algorithm was presented to the literature by [25] in 2021 as a new swarm-based optimization algorithm with stochastic components. To get the best value, the researchers employed the slope formula in the algorithm as a search logic for all solution candidates in the population. The algorithm consists of two stages: the search strategy using the Runge-Kutta theory and the improvement of the solution quality. The basic structure of the RUN algorithm will be explained in the subsections of this section.

1) INITIALIZATION POPULATION
The first part of the RUN algorithm is defined as the creation of the initial population in order to find the most suitable solution candidate. The initial population is formed between the minimum and maximum limit values of the control variables determined by the researcher. The initial population is expressed mathematically as in (7).
n represents each solution candidate in the population (n = 1, 2, 3, . . . , N ). N represents the number of randomly generated solution candidates within the limit values, in other words, the size of the population. L i and U i show the limit VOLUME 10, 2022 values of the i th variable (i = 1, 2, 3, . . . , D). rand represents a randomly generated number between [0,1].

2) UPDATING SOLUTIONS
The algorithm starts the optimization process with the candidate solutions created in the initial population. At each iteration, the candidate solutions use a search mechanism (SM) based on the Runge-Kutta technique to update their locations for the next iteration. The SM method plays an active role in providing global (discovery) and local (exploitation) search. This situation is given in detail in algorithm 1.

Algorithm 1 Exploration and Exploitation Phases in RUN Algorithm
if rand < 0.5 exploration phase The r and g used in algorithm 1 represents an integer number as 1 or −1, and a random number between [0,2], respectively. µ defines a random number and is calculated based on equation (8). The expression randn in equation (8) represents a random number with normal distribution. x s ve x ss expressions are explained mathematically as seen in equation (9). x m ve x c are formulated as in equation (10).
ϕ is expressed as a random number in the range [0,1]. x best is defined as the best solution so far, while x lbest is defined as the best solution obtained from each iteration. While SF is defined as the adaptive factor that provides the balance between exploration and exploitation, it is shown mathematically as in equation (11).
In the equation, a and b defines two constant number, i defines current iteration number, and Maxi defines maximum iteration number.

3) ENHANCED SOLUTION QUALITY
The enhanced solution quality (ESQ) method in this section is employed to prevent the solution candidates used in the optimization process from being caught in local solution points or solution traps and to increase the solution quality.
The calculation of x new2 according to this method is shown in detail in algorithm 2. In the method, a random number of w and rand are used together. The ESQ method runs while rand < 0.5, and x new2 = x new1 + r × w. x new1 − x avg + randn equation is used if w< 1, otherwise x new2 = x new1 − x avg + r × w × u × x new1 − x avg + randn is applied to enhance the solution [29].

Algorithm 2 Calculation of the x new2 via the Enhanced Solution Quality
w, x avg and x new1 are, respectively, defined as a random number, the average of three random solutions and a new solution, and given in equation (12)- (14).
The variable β expresses a random number between [0,1]. r is an integer that gets number of 1,0 or −1. c is calculated using the equation of c = 0.5 × rand. If the solution x new2 does not have a better solution than the current solution candidates, x new3 is calculated, which allows the better candidate solutions shown in algorithm 3 to be obtained.

Algorithm 3 Calculation of the x new3
if rand < w The parameter of v is calculated by the equation of v = 2 × rand.

B. THE PROPOSED METHOD: FITNESS DISTANCE BALANCE-BASED RUN ALGORITHM
Metaheuristic optimization algorithms are caught in local solution traps in the search process life cycle to find the most suitable solution. In order to eliminate this undesirable situation, [30] presented a method called fitness distance balance (FDB) to the literature in 2020. In this method, it is aimed to better select the solution candidates that guide the search process by using fitness values. Cengiz et al. used the FDB selection method to eliminate the disadvantage of RUN algorithm being caught in local solution traps. The fitness values of the solution candidates and their distance from the best solution candidate (P best ) in the population are considered in the FDB selection process. FDB selection method is applied according to the following process steps [31], [32], [33], [34], [35].
i. Optimization algorithms consist of n-unit candidate solution spaces to solve the optimization problem. The vector of solution candidates (population, P) in the solution space and the vector of fitness values (F) of these candidates are shown in equation (15).
ii. The Euclidean distance of the i-th candidate solution p i in the population to the best solution candidate p best is expressed as in equation (16).
iii. The distance of solution candidates in the population from p best is represented by the vector D p given in equation (17). (17) iv. In the calculation of the FDB value, normalization is required so that the vector of fitness values and distance values of the solution candidates do not dominate each other. Normalized fitness and distance values between [0, 1] are used to calculate the FDB values (S p ) of the solution candidates as normF and normD p , respectively.
In equation (18), w is expressed as the weight coefficient representing the effects of the fit and distance parameters on the FDB value, and the w coefficient is accepted as 0.5. v. The FDB values of the solution candidates in the solution space of the optimization problem are represented as the n-dimensional S p vector as in equation (19).  After the S P vector showing the FDB scores of the solution candidates in the solution space of the optimization problem is created, the roulette wheel method is used to select the solution candidates to guide the search process to reach the global solution point. The point where the FDB selection method is applied is given in equation (20) [36].
The flowchart of FDBRUN algorithm is shown in Fig. 8.

V. SIMULATION RESULTS
In this section, comparative simulation results of buck converter with PID, FOPID and TID control structures, optimal parameters determined by FDBRUN, AO, AVOA, and HGS optimization algorithms, are presented. Table 1 shows the circuit parameters of the buck converter. The transfer functions given in (22)- (25), as shown at the bottom of the next page, are derived depending on the parameters.

A. DESIGN OF THE DIFFERENT CONTROLLER STRUCTURES VIA METAHEURISTIC ALGORITHMS
Controllers used in power electronics converters are expected to dampen the oscillations that occur in sudden changes in the system and minimize the steady state error. In order to achieve this, the optimization of the controller coefficients has an important role. The objective function used in any optimization method is also important for better results. The objective function given in (21) was preferred in the study because it provides fast dynamic response, low overshoot, short settling time and minimum steady-state error in sudden changing system operating conditions. The function [37] was firstly used in [38] for automatic voltage regulator.
In this equation, α is defined as the weight coefficient often set to 1, E ss defines the steady-state error, the percent VOLUME 10, 2022  overshoot is expressed as M p , T s defines the time to settle in a band gap of ±2%, and T r defines the rise time. In optimization problems, it is necessary to determine the limits of the variables defined as control variables. In this study, the minimum and maximum limit values of PID, FOPID and TID controller parameters are used as in Table 2 [16]. Fig. 9 shows convergence profile of best runs of the proposed FDBRUN algorithm for FOPID controller. As seen from Fig. 9, the proposed FDBRUN algorithm converges to the lowest fitness function value.
To ensure fair performance comparison, the number of iterations as the stopping criterion and population size is set to 50 and 25 respectively for each optimization algorithm. During the determination of the controller parameters, all optimization algorithms are run 30 times. The optimal controller parameters found by the optimization algorithms are given in Table 3. When the values in Table 3 are examined in detail, it is seen that the optimized controller parameters are within the limit values in Table 2. The results obtained at the end of 30 trials are statistically evaluated in Table 4.
The FOPID controller structure utilized in the buck converter structure may be expressed as the structure with the best objective function for each optimization technique according to the PID and TID controllers when Table 4 is reviewed in detail. As can be seen from the boxplot shown in Fig. 10, the statistical performance of the FDBRUN optimization algorithm in all controllers is better than the performance of other AO, AVOA and HGS algorithms.   In the case of using FOPID as the controller in the system shown in Fig. 4, the coefficients determined by the AO, AVOA, FDBRUN and HGS algorithms and the transfer functions obtained using the system parameters seen in Table 1 are given in the equations (22)-(25), respectively.

1) TRANSIENT RESPONSE ANALYSIS OF THE BUCK CONVERTER
During the optimization process, the best parameters of PID, FOPID, and TID controllers are obtained using FDBRUN, AO, AVOA, and HGS algorithms. The time responses of the buck converter according to the obtained controller parameters are shown in Fig. 11. Furthermore, Table 5 describes the transient response criteria in detail, including overshoot, rise time, settling time, and peak time. According to these values, it is seen that the FDBRUN based FOPID controller is better than other controllers in improving the transient response of the buck converter system.

2) FREQUENCY RESPONSE AND COMPARISON ANALYSIS OF THE PERFORMANCE INDICES
The bode diagram that shows gain and phase margin of the FOPID controller which parameters are determined by optimization algorithms is given in Fig. 12. In addition, integral absolute error (IAE) [39], integral square error (ISE) [40], integral time absolute error (ITAE) [41] and integral time squared error (ITSE) [42] performance indices are used to evaluate the performance of the algorithms in more detail. The formulas of the indices are given in the equations (26)- (29).
T is the simulation time, and e(t) is the error signal, according to these equations. Performance indexes are examined in two stages. In the first stage, the response of the system under unchanged conditions is examined, while in the second stage, the system response is examined in case of changes in resistance (R), inductance (L) and capacitance (C) values.
The results of the performance indices according to the system response of the FOPID controller under continuous conditions are shown in Fig. 13 as a bar graph. The numer-VOLUME 10, 2022 ical values of the performance indices are given in Table 6. Under continuous working conditions, the FDBRUN algorithm has the best value in all performance indexes, according to Table 6. In other words, the FDBRUN algorithm for the IAE performance index has 9.8101%, 0.7911%, and 1.0284% less values than the AO, AVOA, and HGS algorithms. Fig 14 shows the results of the performance indices according to the system response of the FOPID controller depending on the change of the R-L-C parameters. According to the change of R-L-C parameters, FDBRUN algorithm is seen as the algorithm with the best value among the performance indexes in Table 6. In addition, the FDBRUN algorithm has 33.4250%, 2.0632% and 3.2324% less values than the AO, AVOA, and HGS algorithms, respectively for the IAE performance index.

VI. CONCLUSION
In this study, the performance of buck type step-down converter with PID, FOPID and TID controllers is investigated. The coefficients, which determines the performances of the controllers are determined by AO, AVOA, FDBRUN and HGS optimization methods. FDBRUN method is used for the first time to optimize the PID controller in the buck converter in the literature. When the steady state response of the buck system is examined, the FOPID control structure has the best objective function, and the best control response is obtained  with the coefficients determined by the FDBRUN optimization method. In the transient performance, the FOPID controller, the coefficients of which are determined by the FDBRUN optimization method, give the best response in the parameters of transient peak value, percent overshoot, rise time, settling time and time to peak value. When the robustness of the algorithms is examined in detail in terms of performance indices, the FDBRUN algorithm has the best value in terms of performance indices calculated depending on the controller parameters optimized by the optimization algorithms in steady state conditions. It is seen that the FDBRUN algorithm has the best value among the optimization algorithms according to the performance indices after the change of the R-L-C values in the system parameters. As a result of the coefficient optimization, which is made with the FDBRUN algorithm, the FOPID controller provides the best performance in the system response. The results reveals that the FDBRUN method achieves best performance compared to other tested methods, and is applicable for coefficient optimization in power electronics converter controller.