COMPLEXITY Complexity 1099-0526 1076-2787 Hindawi 10.1155/2019/7687891 7687891 Research Article Metaheuristic Optimization of Fractional Order Incremental Conductance (FO-INC) Maximum Power Point Tracking (MPPT) Ammar Hossam Hassan 1 2 https://orcid.org/0000-0002-7869-6373 Azar Ahmad Taher 3 4 https://orcid.org/0000-0002-3639-3665 Shalaby Raafat 1 5 Mahmoud M. I. 5 Zeng Jianwu 1 School of Engineering and Applied Science Nile University Giza Egypt nu.edu.eg 2 Smart Engineering Systems Research Center (SESC) Nile University 12588 Shaikh Zayed City Giza Egypt nu.edu.eg 3 Robotics and Internet-of-Things Lab (RIOTU) Prince Sultan University Riyadh 12435 Saudi Arabia psu.edu.sa 4 Faculty of Computers and Artificial Intelligence Benha University Banha Egypt bu.edu.eg 5 Faculty of Electronic Engineering Menofia University Menofia Egypt menofia.edu.eg 2019 28112019 2019 25 07 2019 07 10 2019 18 10 2019 28112019 2019 Copyright © 2019 Hossam Hassan Ammar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper seeks to improve the photovoltaic (PV) system efficiency using metaheuristic, optimized fractional order incremental conductance (FO-INC) control. The proposed FO-INC controls the output voltage of the PV arrays to obtain maximum power point tracking (MPPT). Due to its simplicity and efficiency, the incremental conductance MPPT (INC-MPPT) is one of the most popular algorithms used in the PV scheme. However, owing to the nonlinearity and fractional order (FO) nature of both PV and DC-DC converters, the conventional INC algorithm provides a trade-off between monitoring velocity and tracking precision. Fractional calculus is used to provide an enhanced dynamical model of the PV system to describe nonlinear characteristics. Moreover, three metaheuristic optimization techniques are applied; Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), and AntLion Optimizer (ALO) are used for tuning the FO parameters of the proposed INC-MPPT. A MATLAB-Simulink-based model of the PV and optimization have been developed and simulated for different INC-MPPT techniques. Different techniques aim to control the boost DC-DC converter towards the MPP. The proposed optimization algorithms are, also, developed and implemented in MATLAB to tune the target parameters. Four performance indices are also introduced in this research to show the reliability of the comparative analysis of the proposed FO-INC with metaheuristic optimization and the conventional INC-MPPT algorithms when applied to a dynamical PV system under rapidly changing weather conditions. The simulation results show the effective performance of the proposed metaheuristic optimized FO-INC as a MPPT control for different climatic conditions with disturbance rejection and robustness analysis.

Prince Sultan University
1. Introduction

2. Photovoltaic (PV) System Modeling and Simulation

The 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. The model of the solar panels used in the proposed system will be illustrated, and the PV system is introduced . The 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. The 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.

Complete PV system model using MATLAB and Simscape.

2.1. Modeling of PV Panel

The 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. The PV output current IPV can be obtained using equation (1) where Np and Ns are the number of parallel and series cells, respectively:(1)IPV=Np×IGIDIsh.

PV cell equivalent electric circuit model.

The nonlinear equation of I-V characteristics of one-diode PV model was expressed by Milici et al.  as follows:(2)IPV=Np×IGNs×Ioeq/ηkTk×VPV/Ns+IPV/NpRs1NpRsh×VPVNs+IPVRsNp,where VPV and IPV are the PV terminal voltage and current, respectively, Rs and Rsh are the series and shunt resistance, respectively, η is the ideality factor, the Boltzmann’s constant is k, q is the electron charge, Tk is the temperature degree in Kelvin, IG is photo-generated current, and the diode saturation current is Io. The PV panel parameters are shown in Table 1.

PV panel parameters.

Parameter value Value
Max power Pmax 735.72 W
Open circuit voltage Voc 65.9 V
Short circuit current Ish 15.21 A
Temperature coefficient of VOC –1.23 × 10e−1 V/C
Temperature coefficient of Ish 3.18 × 10e−3 A/C

The 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.

P-V and I-V characteristic curves at different climatic conditions.

2.2. DC-DC Converter

Simulink and Simscape tools have been selected as platforms for modeling, implementation, and testing the Buck–Boost converter. The 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:(3)x=Ax+Bu,y=Cx+Du.

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 x1 = IL, x2 = VCout, and d = duty cycle. The simulation results at different duty cycles are shown in Figure 5:(4)x1x2=01dL1dC1RCx1x2+dL0u1,y1y2=1001ILVCout+00u1.

Buck–Boost Simscape model.

Buck–Boost output voltage at different duty cycles.

The proposed Buck–Boost has been designed and simulated using the parameters illustrated in Table 2.

Buck–Boost design parameters.

Parameter value Value
Filter inductance 1 mH
Output filter capacitance 4700 μF
Input filter capacitance 47 μF
Switching frequency 25000 Hz
3. Design and Implementation of MPPT

The 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. The 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) .

3.1. Fixed‐step INC Method

The INC algorithm is used to detect the condition of MPP via the conductance (dI/dV) behavior of the PV system. The INC-MPPT can be executed through the following sequence :

The voltage and current of the PV module are sensed by the MPPT controller

If (dI/dV< −I/V) is satisfied, the duty cycle of the converter needs to be decreased and vice versa

No change in the duty cycle occurs if I+VdI/dv=0 is satisfied

The duty cycle (PV reference voltage (Vref)) increasing or decreasing occurs with fixed step.

3.2. Variable‐Step INC Method

The INC variable step size algorithm proposed by Motahhir et al.  can improve the MPPT controller’s tracking effectiveness. The algorithm sequences are mostly comparable to the standard increments; the only distinction is the calculation of the step size. Step=NabsdP/dV is used in the variable step size algorithm to change the duty cycle step size, where N is the scaling factor.

3.3. 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, . The general form of fractional order differentiator can be expressed by Kamal and Ibrahim ; supposed that fmt=tm and m = 1, 2, 3, … , is demonstrated at(5)DαttmΓm+1Γm+1αtmα,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 . The FO-INC MPPT main criteria can be expressed by equations (7) and (8):(6)dαIVdVα=limΔV0IVαVΔVΔVα,(7)dαIdVαIαI0VV0α,(8)dαdVαI0V0=1VΓ2Γ2αI01αI0Γ0ΓαV01α.

The control procedure of the FO-INC algorithm can be expressed by the flowchart depicted in Figure 6. The procedure starts with measuring the PV’s voltage and current to determine the MPPT action according to the following conditions:

Condition 1. If ΔVα0&dαI/dVα=dα/dVαI0/V0 or ΔVα=0&ΔI=0, keep the current duty cycle, fix the duty cycle

Condition 2. If ΔVα0&dαI/dVα>dα/dVαI0/V0 or ΔVα=0&ΔI>0, decrease the duty cycle of the Buck–Boost converter (increase Vref as in equation (9))

Condition 3. If ΔVα0&dαI/dVα<dα/dVαI0/V0 or ΔVα=0&ΔI<0, increase the duty cycle of the Buck–Boost converter (decrease Vref as in equation (9))

Condition 4. Calculate Po=Vo×Io and P=V×I. If P>Poterminate, otherwise update the voltage Vo=V, current Io=I, and power Po=P

Fractional order INC MPPT flowchart.

The duty cycle of the Buck–Boost converter can be calculated based on the output of the FO-INC controller Vref as in(9)VL=D1D×VPV then,D=VLV0±ΔVα+VL,where VL is the resistive load voltage, VPV=Vref, 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.

4. 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. , the searching techniques of different optimizers are as follows:

Initialize solution randomly

Specify the search direction

Specify the update criteria

Specify the stopping criteria

4.1. Particle Swarm Optimization (PSO)

The 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 . The PSO is seeking high-quality optimization by refining, iteratively, a candidate solution. The 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, C1 and C2 are the acceleration coefficients, and Wmin and Wmax 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.

<bold>Algorithm 1: </bold>PSO-MPPT.

Result: MPPT parameter to achieve MPP

Initialize the PSO parameters (N, C1, C2, Wmin, Wmax, Vmax)

while the termination criteria not achieved (MPP) do

For (each Particle i)

Simulate and calculate the MPP and the cost function. if CostFunctioni<ObjectiveMPP

then

The Best Cost (i) = Cost Function (i)

The Best Solution (i) = The particle (i)

else

The Best Cost (i) = The Best Cost (i − 1)

The Best Solution (i) = The Best Solution (i − 1)

end

(a) Update the velocity (V): Vit+1=Vit+C1r1PiXit+C2r2PgXit

where Vit is inertia, C1r1PiXit is cognitive component and C2r2PgXit is social component

(b) Update the position of particles: Xit+1=Xit+1+Vit

end

4.2. 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 . While further improvements were carried out to ant colony by Gambardella Dorigo in 1997 . Pseudo code for ant colony optimization is implemented with the steps as Algorithm 2.

<bold>Algorithm 2: </bold>ACO-MPPT.

Result: MPPT parameter to achieve MPP

Initialize the ACO parameters

while the termination criteria not achieved (MPP) do

For (each ant i)

Simulate and calculate the MPP and the cost Function.

if CostFunctioni<ObjectiveMPP then

The Best Cost (i) = Cost Function (i)

The Best Solution (i) = The particle (i)

else

The Best Cost (i) = The Best Cost (i − 1)

The Best Solution (i) = The Best Solution (i − 1)

end

(a) Update pheromone for each ant: τijt+1, Δτijt=ant=1antsΔτijantt

(i) Calculate the solution Δτijantt=1/Lbest if antt travels on edgei,j0otherwise

(b) Apply evaporation and globally update the ants position, according to the optimum solutions calculated earlier

end

In Algorithm 2, τijt 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.

4.3. AntLion Optimizer Optimization (ALO)

The primary motive of ALO is the running behavior of larvae of antlions. ALO is suggested based on the Emary and Zawbaa  mathematical model. The ALO algorithm simulates the interaction between the traps. The 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.  to model the ants’ motion.

In Algorithm 3, I is a ratio, ct is the minimum of all variables at tth iteration, and dt indicates the vector including the maximum of all variables at tth iteration. I=10wT/t, where t is the current iteration, T is the 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, ai is the minimum of random walk and bi is the maximum of random walk in ith variable. RAt is the random walk around the antlion selected by the roulette wheel at tth iteration and REt is the random walk around the elite at tth iteration. The pseudo code of ALO for MPPT developed as mentioned in Algorithm 3.

<bold>Algorithm 3: </bold>ALO-MPPT.

Result: 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 .

Simulate and calculate the MPP and the cost Function. if CostFunctioni<ObjectiveMPP

then

The Best Cost (i) = Cost Function (i)

The elite (i) = The The Best Solution (i)

else

The Best Cost (i) = The Best Cost (i − 1)

The elite (i) = The Best Solution (i − 1)

end

(a) Update c and d using equations: Ct=Ct/I and dt=dt/I

(b) Create a random walk and normalize it using: Xt=0,,cumsum2rtn1, where n = 1, 2, 3, …, n and Xit=Xitai×dicit/ditai+ci

(c) Update the position of antlions using antit=RAt+REt/2

(d) Calculate the fitness of all ants according to the optimum solutions calculated earlier

end

5. Modeling and Simulation Results

The 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. The 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). The MPPT is optimized by one of the metaheuristic techniques (PSO, ACO, and ALO).

The proposed PV system with metaheuristic optimization.

The 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 (PMPP). 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. The mean squared-error (MSE) between the MPP (PMPP) and output power of the PV system (PPV) is the cost function of the metaheuristic optimizer calculated in MATLAB environment to get the optimal MPPT parameters. The 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.

The proposed system implementation flowchart.

The 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. The 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/m2 and the cell temperature ranges from 20°C to 40°C which makes it more realistic as shown in Tables 3 and 4.

Comparative results between MPPT algorithms at 800 W/m2.

Climatic condition
MPPT 800 W/m2 and 25°C 800 W/m2 and 35°C
Max power (Watt) MPP steps Oscillation avg. (Watt) No. of iterations Max power (Watt) MPP steps Oscillation avg. (Watt) No. of iterations
Fixed-step INC 321.84 137 5.87 319.73 131 4.92
Variable-step INC 324.63 125 4.89 319.90 127 4.17
FO-INC fixed step + PSO 420.28 121 4.28 100 400.01 127 4.34 100
FO-INC fixed step + ACO 470.62 115 4.067 100 462.24 121 3.089 100
FO-INC fixed step + ALO 487.22 118 3.067 100 474.24 125 3.089 100
FO-INC variable step + PSO 490.28 128 3.28 100 480.01 132 3.34 100
FO-INC variable step + ACO 510.62 115 3.067 100 490.24 121 3.089 100
FO-INC variable step + ALO 515.52 120 3.269 100 510.26 127 3.802 100

Comparative results between MPPT algorithms at 1000 W/m2

Climatic condition
MPPT 1000 W/m2 and 25°C 1000 W/m2 and 35°C
Max power (Watt) MPP steps Oscillation avg. (Watt) No. of iterations Max power (Watt) MPP steps Oscillation avg. (Watt) No. of iterations
Fixed-step INC 428.84 145 4.87 419.73 153 5.92
Variable-step INC 420.73 155 5.87 419.90 139 5.17
FO-INC fixed step + PSO 436.88 120 4.88 100 410.51 121 4.84 100
FO-INC fixed step + ACO 650.82 101 2.067 100 632.24 112 2.089 100
FO-INC fixed step + ALO 667.32 118 3.87 100 664.24 120 3.889 100
FO-INC variable step + PSO 590.28 86 2.28 100 580.23 92 2.34 100
FO-INC variable step + ACO 720.62 65 2.067 100 705.24 71 2.079 100
FO-INC variable step + ALO 725.32 80 2.369 100 710.63 96 2.802 100

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. The 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.

Fixed-step INC I-V and P-V curves.

Variable-step INC I-V and P-V curves.

The 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. Fixed-step 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 optimal MPPT parameter. Fixed-step FO-INC-ACO gives less number of MPPT steps than PSO and ALO, however, not the optimum parameter that can be obtained by ALO with fewer number of MPPT steps. The output power and voltage of the PV system generated using the optimization of fixed-step FO-INC-ACO gives less oscillation yet not as much maximum power as given by ALO as shown in Figure 12. Variable-step FO-INC gives better results than conventional INC and fixed-step FO-INC methods, less 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 Figures 13 and 14.

Optimization of fixed‐step FO-INC I-V and P-V curves.

Optimization of fixed FO-INC output response.

Optimization of variable-step FO-INC I-V and P-V curves.

Optimization of variable FO-INC output response.

The MPPT performance 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 to get η. The irradiance and temperature variation mean that output power follows the highest maximum power profile very closely, and the higher performance is obtained in case of variable-step FO-INC with ALO (98.1) as noted from Table 5:(10)η=PPVt dtPMPP.

MPPT algorithms efficiency.

MPPT method Efficiency (η)
Fixed-step INC 75.9
Variable-step INC 82.1
Fixed-step FO-INC + PSO 90.2
Fixed-step FO-INC + ACO 92.3
Fixed-step FO-INC + ALO 93.2
Variable-step FO-INC + PSO 94.7
Variable-step FO-INC + ACO 97.5
Variable-step FO-INC + ALO 98.1
6. Conclusion and Future Work

The 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. The 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. The 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 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. The ALO uses larger number of variables and takes the longest calculation time, yet it gives more optimal solution compared to ACO and PSO. This 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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank Prince Sultan University, Riyadh, Saudi Arabia for supporting and funding this work. Special acknowledgment to Robotics and Internet-of-Things Lab (RIOTU) at Prince Sultan University, Riyadh, SA.

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