Metaheuristic Optimization of Fractional Order Incremental Conductance (FO-INC) Maximum Power Point Tracking (MPPT)

School of Engineering and Applied Science, Nile University, Giza, Egypt Smart Engineering Systems Research Center (SESC), Nile University, 12588 Shaikh Zayed City, Giza, Egypt Robotics and Internet-of-ings Lab (RIOTU), Prince Sultan University, Riyadh 12435, Saudi Arabia Faculty of Computers and Articial Intelligence, Benha University, Banha, Egypt Faculty of Electronic Engineering, Menoa University, Menoa, Egypt


Introduction
Green energy sources are the primary research goal nowadays as they are viable, ecological, and cost-e ective energy sources.Solar, wind, tidal, and biomass energy have penetrated the electric power production market in recent years due to the diverse methods and their renewable nature.e bene ts of developing renewable power include reducing fossil fuel usage, mitigating the greenhouse impact, and reducing air pollution [1].In addition, control approaches and optimization have shown that the performance of photovoltaic devices depends upon climate conditions (sunlight and temperature) and load impedance [2].However, its low energy conversion e ectiveness (especially in low radiation and temperatures) is the primary disadvantage of PV systems.e MPPT needs to be operated for ideal e ciency and operation as mentioned [3].One of the pioneering challenges of the PV devices is their nonlinear current-voltage I-V relationship dynamic which generates a distinctive MPP in the power-voltage P-V relationship as noted [4].Because of the P-V relationship with climate and load circumstances, the MPPT method becomes complex.MPPT methods do not only enhance the power performance of PV and energy delivered to the load, but they also increase the operating life of the PV system [5].Previous studies have suggested several MPPT methods; most MPPT techniques demonstrate higher efficiency under stable weather [6].MPPT algorithms are usually used as electronic power conversion devices and the control signal is a duty cycle for peak load energy [7].A wide variety of methods for solving the MPPT issue have been implemented, such as the perturb and observe (P&O) method, incremental conductance (INC) algorithm, and artificial intelligence includes fuzzy logic, neural networks, and metaheuristic techniques.e P&O and INC are the most common algorithms used for PV-MPPT systems [8].e P&O technique is frugal and very easy to execute; its operation is based on the iterative measurement of the voltage and current of the PV system to obtain the duty cycle and consequently the MPP.Its main disadvantage, however, is that it provides an oscillatory power around the MPP and is also unable to manipulate PV power variations due to climatic effects or inherent disturbances of the MPPT.e INC approach is based on the behavior of PV, given that a MPP is reached by zero in a pitch of the PV curve, positive to the left and negative to the right of the PV curve.On the basis of this, the technique calculates the DC-DC converter duty cycle by relating iterative conductivity to the incremental conductivity.e primary drawback in the INC technique is that the system's reaction to the MPP may be slow under some conditions.However, the INC technique exhibits less oscillatory behavior around the MPP compared to the P&O method [9].Fractional calculus introduces the nonintegral order/ fractional of derivatives and integrals.Many of the real systems show a nonlinear and fractional order dynamical behavior, such as heat conduction in solids, electrical behavior in R-L transportation lines, mass diffusion, and electromagnetic waves [10].e nonlinear characteristics of the current-voltage of a PV cell occur because the PV cells are manufactured from semiconducting materials (crystalline silicon, c-Si).e power of PV cell depends on the inherent voltage drop across the p-n junction (energy band) which produces a photoelectric current (current source).e light and ambient temperature interaction also shows anomalous diffusion which can be described as fractional order diffusion [11].
erefore, Grunwald-Letnikov fractional approximation [12] is introduced to control the fractional order differentiation for current and voltage nonlinear dynamical behavior.To improve dynamic performance, FO-INC based on the nonlinear and fractional order changes of the PV voltage and current has been proposed to track the maximum output power [13].It is very important to select the proper converter [14] to enhance the MPPT performance.
e MPPT techniques have been compared using MATLAB and Simulink tools created by MathWorks, considering all the design and implementation specs [15].erefore, metaheuristic optimization techniques' robustness and ability to find the optimal solution in different nonlinear systems have demonstrated itself in numerous past research studies.Metaheuristic abilities are powerful techniques of resolving optimization problems for nonlinear and fractional order systems [15].In power systems, different optimization techniques have been utilized.Considering the different constraints in PV systems and difference in the nature of DC-DC converter system, the ACO algorithm has been used [16].It has been proved to be very robust, consistent, and performs better than conventional optimization techniques (e.g., PSO and GA) [16].
e experiments show computational effectiveness and time decrease in monitoring for a small PV Systems.e AntLion optimizer (ALO) is a recent metaheuristic algorithm that replicates the hunting scheme of antlions in catching ants [17].ALO also gives a good performance results in PV-MPPT systems [18]. is research aims to extract maximum power from PV systems by using FO-INC and metaheuristic optimization technique.is enhanced system efficiency in different climatic conditions using fixed and variable-step FO-INC with PSO, ACO, and ALO optimization techniques. is paper is organized as follows: Section 2 addresses the modeling of the complete PV system, and Section 3 describes the MPPT algorithm design and operation.Section 4 gives the operation of metaheuristic optimization algorithms.Sections 5 and 6 illustrate the experimental results and conclusions to show the efficiency of the proposed technique.

Photovoltaic (PV) System Modeling
and Simulation e proposed PV system is constituted by a PV module, the Buck-Boost converter as a DC-DC converter between the PV panel and the DC load, and the MPPT controller to achieve maximum power point of the PV panels.e model of the solar panels used in the proposed system will be illustrated, and the PV system is introduced [19].e inputs to MPPT are the PV voltage and current which are used to calculate and deliver the control signal (duty cycle) to the Buck-Boost converter, as shown in Figure 1.
e main function of the MPPT algorithm is to automatically track the voltage/current change of the PV panel and feed the Buck-Boost converter with the appropriate duty cycle to get the MPP under specific climatic conditions.

Modeling of PV Panel.
e nonlinear equations of the PV system which describe the relationships between the different PV model parameters are developed and solved via MATLAB and Simulink tools where the PV cell electric circuit model is shown in Figure 2. e PV output current I PV can be obtained using equation (1) where N p and N s are the number of parallel and series cells, respectively: e nonlinear equation of I-V characteristics of onediode PV model was expressed by Milici et al. [9] as follows: where V PV and I PV are the PV terminal voltage and current, respectively, R s and R sh are the series and shunt resistance, respectively, η is the ideality factor, the Boltzmann's constant is k, q is the electron charge, T k is the temperature degree in Kelvin, I G is photo-generated current, and the diode saturation current is I o .e PV panel parameters are shown in Table 1.e I-V and P-V nonlinear characteristic curves of the PV array simulated using MATLAB at different climatic conditions (temperature and irradiance) are shown in Figure 3.

DC-DC Converter.
Simulink and Simscape tools have been selected as platforms for modeling, implementation, and testing the Buck-Boost converter.
e state space modeling is primarily represented by equation (3), where A, B, C, and D are the system matrices, x is the state variable vector, x ′ is the state variable derivative vector with respect to time, u is the input, and y is the output [14]: Figure 4 shows the Buck-Boost model using Simscape which is simulated at different duty cycles and fixed load according to the state space model represented in equation ( 4), where x 1 � I L , x 2 � V C out , and d � duty cycle.e simulation results at different duty cycles are shown in Figure 5: e proposed Buck-Boost has been designed and simulated using the parameters illustrated in Table 2.

Design and Implementation of MPPT
e primary feature of the PV system is the total energy monitoring in which the power of the PV modules can be extracted in a certain climatic situation.As shown in the literature, the most commonly used MPPT algorithm is INC.
e INC algorithm is based on the reality that the PV output energy derivative for the output voltage at the MPP is zero (dP/dV � 0), positive on the left side of MPP (dP/dV > 0) and negative on the right side of MPP (dP/dV < 0) [5].

Fixed-step INC Method.
e INC algorithm is used to detect the condition of MPP via the conductance (dI/dV) behavior of the PV system.e INC-MPPT can be executed through the following sequence [20]:

Fractional Order INC Method (FO-INC).
Many computational requests for fractional order derivatives according to the definition have been suggested by Riemann-Liouville and Grunwald-Letnikov, [9].e general form of fractional order differentiator can be expressed by Kamal and Ibrahim [21]; supposed that f m (t) � t m and m � 1, 2, 3, . . ., is demonstrated at where Γ(•) represents Eular's gamma function and α is the order number of derivative, when its value is 0 < α < 1, representing physical phenomenon of fractional order [9].e FO-INC MPPT main criteria can be expressed by equations ( 7) and ( 8): e control procedure of the FO-INC algorithm can be expressed by the flowchart depicted in Figure 6. e procedure starts with measuring the PV's voltage and current to determine the MPPT action according to the following conditions: where V L is the resistive load voltage, V PV � V ref , and D is the duty cycle.
Both fixed-and variable-step FO-INC MPPT have been implemented to improve the performance of the MPP tracking of the nonlinear PV system with Buck-Boost converter and resistive load.In case of fixed step, the effective parameter of MPPT performance is alpha α.For variable step both alpha (α) and step size S are affecting the MPPT performance as shown in Figure 6.

Metaheuristic Optimization Algorithms
Genetic algorithms, Particle Swarm Optimization, and Ant Colony Optimization are among the most frequent algorithms in this field.However, these algorithms can solve many real and difficult problems.As one of the recent algorithms, the AntLion Optimizer Optimizer will be introduced along with its basic working principle, updated criteria, and pseudo algorithms.According to Pradhan et al. [20], the searching techniques of different optimizers are as follows: (1) Initialize solution randomly (2) Specify the search direction (3) Specify the update criteria (4) Specify the stopping criteria

Particle Swarm Optimization (PSO).
e inspiration of the particle swarm algorithm is to simulate the navigation and foraging of swarm of birds or school of fishes.PSO was developed by James Kennedy and Russel Eberhart in 1995 while studying the social behaviors of animals working in swarms [22].e PSO is seeking high-quality optimization by refining, iteratively, a candidate solution.e pseudo code of the PSO algorithm is illustrated in detail with the steps in Algorithm 1.In Algorithm 1, N is the number of particles, C 1 and C 2 are the acceleration coefficients, and W min and W max are the ranges of weight of particles.PSO uses fewer resources than the other optimization techniques.Usually, it does not require the problem to be differentiable as the gradient of the problem is not taken into consideration.As a result, there might be chances that PSO does not converge to optimal solution.

Ant Colony Optimization (ACO). Ant Colony Optimization (ACO) introduces an artificial algorithm motivating actual ant colonies that solve discrete optimization problem.
It was first presented by Marco Dorigo in 1992 as a major aspect of his Ph.D thesis and called it the ant system [12].While further improvements were carried out to ant colony by Gambardella Dorigo in 1997 [23].Pseudo code for ant colony optimization is implemented with the steps as Algorithm 2.

Complexity
In Algorithm 2, τ ij (t) represents the intensity of trail on connection (i, j) at time t, L(ant) is the cost function result at each ant, and λ is pheromone decay coefficient between time (t and t + 1) (i.e., 0 < λ < 1).Evaporation occurs in real trails, but it is too slow to play an important role.For continuous improvements it allows the search routine to forget errors and poor quality solution in favor of better ones.

AntLion Optimizer Optimization (ALO).
e primary motive of ALO is the running behavior of larvae of antlions.
ALO is suggested based on the Emary and Zawbaa [24] mathematical model.
e ALO algorithm simulates the interaction between the traps.e ants must move across the search area in order to model such interactions, and the antlions are permitted to chase and fit the traps.Given that ants move randomly to find food in actual life, a random walk algorithm is selected as shown in Heidari et al. [25] to model the ants' motion.
In Algorithm 3, I is a ratio, c t is the minimum of all variables at tth iteration, and d t indicates the vector including the maximum of all variables at tth iteration.I � 10 w (T/t), where t is the current iteration, T is the  8) Equation (7) = equation ( 8) 6 Complexity maximum number of iterations, and w is a constant defined based on the current iteration.X(t) is ant' movement, cumsum calculates the cumulative sum, n is the maximum number of iteration, t shows the step of random walk, and r(t) is a stochastic function.Also, a i is the minimum of random walk and b i is the maximum of random walk in ith variable.R t A is the random walk around the antlion selected by the roulette wheel at tth iteration and R t E is the random walk around the elite at tth iteration.e pseudo code of ALO for MPPT developed as mentioned in Algorithm 3.

Modeling and Simulation Results
e proposed system has been modeled and simulated using MATLAB and Simscape software environments in order to study the system behavior and MPPT performance with different metaheuristic optimization algorithms.e block diagram describing the total PV system with MPPT and optimizer is shown in Figure 7, where the MPPT algorithm is changed between conventional INC methods and FO-INC (fixed and variable step).e MPPT is optimized by one of the metaheuristic techniques (PSO, ACO, and ALO).
e operation sequence of PV with MPPT and optimization process is a closed loop as shown in Figure 8, and it starts with measuring the irradiance (G) and temperature (T) applied to the PV system to get the reference maximum power point from PV characteristics curves (P MPP ).A closed loop of PV with MPPT and Buck-Boost converter is running in Simscape environment for two seconds to measure the PV output power.
e mean squared-error (MSE) between the MPP (P MPP ) and output power of the PV system (P PV ) is the cost function of the metaheuristic optimizer calculated    8 Complexity in MATLAB environment to get the optimal MPPT parameters.e optimal parameters are applied to the chosen MPPT technique in Simscape.Performance index is calculated in MATLAB through a dynamical data exchange between MATLAB and Simscape.e proposed MPPT contribution is generated by measuring the output energy of the PV system under different solar irradiances.Simulation was conducted when solar radiation and cell temperature change with a transient method of approximately 2 sec with 0.01 sec sampling.e characteristics of the PV array will be altered when the natural radiation and cell temperature alter, which causes the I-V curves of the PV array to change.In addition, the particular irradiance ranges from 400 to 1000 W/m 2 and the cell temperature ranges from 20 °C to 40 °C which makes it more realistic as shown in Tables 3  and 4.
Figure 9 shows the I-V and P-V curves of fixed-step INC under different temperature and radiation with small step which in return gives it better results and less oscillation; however, it takes more time to get maximum power.e variable-step INC curves, shown in Figure 10,   e MPPTperformance η can be monitored as in equation ( 10) for all the abovementioned MPPT techniques.Irradiance, temperature, power, and maximum power time waveform of the system using the proposed MPPT methods have been used

Conclusion and Future Work
e output power of the PV system will be changed by irradiance and temperature according to the simulation results of the system with different climatic conditions as illustrated in Tables 3 and 4.
e proposed incremental fractional order FO-INC demonstrates better results than traditional INC under environmental changing processes and improves the efficiency of MPPT as FO-INC is able to provide a dynamical mathematical model for describing the nonlinear and fractional properties.e incremental change in the fractional order as a dynamic variable is used to adjust the MPPT service cycle.Using metaheuristic optimization enhances the performance of FO-INC and

Table 1 :Figure 3 :
Figure 3: P-V and I-V characteristic curves at different climatic conditions.

( 1 )
e voltage and current of the PV module are sensed by the MPPT controller (2) If (dI/dV < − I/V) is satisfied, the duty cycle of the converter needs to be decreased and vice versa (3) No change in the duty cycle occurs if I + V(dI/dv) � 0 is satisfied e duty cycle (PV reference voltage (V ref )) increasing or decreasing occurs with fixed step.
increase the duty cycle of the Buck-Boost converter (decrease V ref as in equation (9)) Condition 4. Calculate P o � V o × I o and P � V × I.If P > P o ⟶ terminate, otherwise update the voltage V o � V, current I o � I, and power P o � P e duty cycle of the Buck-Boost converter can be calculated based on the output of the FO-INC controller Measure the PV voltage andCurrent I & V.Calculate the fractional derivative I and Vd α I = I -αI o = ΔI d α V = (V -V o ) α = ΔV αStartCalculate the output Power of PV (P = I × V)

ComplexityResult:
MPPT parameter to achieve MPP Initialize the first population of ants and antlions randomly while the termination criteria not achieved (MPP) do For (each antlion (i)) Select an antlion using Roulette wheel algorithm [16].Simulate and calculate the MPP and the cost Function.if CostFunction(i) < Objective(MPP) then e Best Cost (i) � Cost Function (i) e elite (i) � e e Best Solution (i) else e Best Cost (i) � e Best Cost (i − 1) e elite (i) � e Best Solution (i − 1) end (a) Update c and d using equations: C t � (C t /I) and d t � (d t /I) (b) Create a random walk and normalize it using: X(t) � [0, . . ., cumsum(2r(t n ) − 1)] , where n � 1, 2, 3, . .., n and
Figure 9 shows the I-V and P-V curves of fixed-step INC under different temperature and radiation with small step which in return gives it better results and less oscillation; however, it takes more time to get maximum power.e variable-step INC curves, shown in Figure 10, give better results than the fixed INC.However, in variable-step INC, improper selection of the initial step size may require large number of steps to reach the MPP.Also, improper selection of the scaling factor may lead to oscillations.e objective of PSO, ACO, and ALO is to select the best value of α for the fixed-step FO-INC and the best values of α and S for variable step to get the maximum PV power.Fixedstep FO-INC MPPT results optimized by PSO, ACO, and ALO are shown in Figure 11.Fixed-step FO-INC-PSO gives better results than conventional INC methods, less number of MPPT steps to maximum power value, and less oscillation, yet PSO optimization needs more iterations to get the

Figure 13 :Figure 14 :
Figure 13: Optimization of variable-step FO-INC I-V and P-V curves.

Table 2 :
Buck-Boost design parameters.comparable to the standard increments; the only distinction is the calculation of the step size.Step � N * abs(dP/dV) is used in the variable step size algorithm to change the duty cycle step size, where N is the scaling factor.
MPPT parameter to achieve MPP Initialize the PSO parameters (N, C 1 , C 2 , W min , W max , V max ) while the termination criteria not achieved (MPP) do For (each Particle i) Simulate and calculate the MPP and the cost function.if CostFunction(i) < Objective(MPP) Result:

Table 4 :
Comparative results between MPPT algorithms at 1000 W/m 2 number of MPPT steps to maximum power value, and less oscillation.PSO needs more number of iterations than ACO and ALO to get the optimal MPPT parameters.ALO gives the optimum parameters for the maximum power with larger number of MPPT steps and vice-versa with ACO as shown in Figures13 and 14.

Table 5 :
MPPT algorithms efficiency.Complexity provides another dynamical variable to the MPPT control.Compared to ACO and ALO, the PSO uses less number of variables and shorter calculation time for the same number of iterations.However, sometimes it cannot achieve the optimal solution.e ALO uses larger number of variables and takes the longest calculation time, yet it gives more optimal solution compared to ACO and PSO. is work could be extended by changing the resistance using another dynamical load, e.g., DC motor, or by applying different optimization techniques on the FO-INC.