Modified Levy Flight Optimization for a Maximum Power Point Tracking Algorithm under Partial Shading

This paper presents a novel modified Levy flight optimization for a photovoltaic PV solar energy system. Conventionally, the Perturb and Observe (P&O) algorithm has been widely deployed in most applications due to its simplicity and ease of implementation. However, P&O suffers from steady-state oscillation and stability, besides its failure in tracking the optimum power under partial shading conditions and fast irradiance changes. Therefore, a modified Levy flight optimization is proposed by incorporating a global search of beta parameters, which can significantly improve the tracking capability in local and global searches compared to the conventional methods. The proposed modified Levy flight optimization is verified with simulations and experiments under uniform, non-uniform, and dynamic conditions. All results prove the advantages of the proposed modified Levy flight optimization in extracting the optimal power with a fast response and high efficiency from the PV arrays.


Introduction
Greater awareness of clean energy has undoubtedly forced many governments and private sectors to adopt renewable energy resources [1,2]. Typically, the energy demand in a household is around 7% of the total energy usage daily [3,4]. Mainly, the energy supply is driven by fossil fuels. For many decades, fossil fuel has been used in modern civilization as the main source of electricity. The burning of fossil fuels causes global warming. The carbon emitted from fossil fuel has become an environmental pollutant. Thus, the need for low carbon-based sources of electricity (green energy) such as wind, solar, biomass, etc., is essential in the coming future [5,6]. Among these sources, solar photovoltaic (PV) energy is seen as the best potential candidate due to its availability and cleanliness [7][8][9][10]. Thus, due to this interest, focusing on solar PV energy is prioritized. Operating a PV system to the utmost efficiency is essential for more commercial and economic reasons [2]. However, due to the non-linear characteristics of PV cells and rapid in the search of MPPT, and the formula of Levy flight with modification of the β parameter to the global best particles is included. This has significantly improved the performance of the conventional Levy flight with the global search capability in the algorithm. The response time and efficiency of MPPT in tracking local and global optimums are significantly improved. Section 2 explains the PV configuration, modeling, and converter system. Section 3 presents the overview of Levy flight optimization. Section 4 describes MPPT with the modified Levy Flight Optimization. Sections 5 and 6 present the results and discussion from the simulation and experiment, respectively. Lastly, the conclusion is given in Section 7.

PV System Configuration
Generally, a standalone PV system is configured which consists of a PV array as the input source, dc-dc boost converter, and a resistive load as shown in Figure 1 [3,46,47].
In this system, a MOSFET is used as a switching device. The gate signal is controlled by a PWM control technique. The variation of the duty cycle is controlled by the MPPT algorithm to track the maximum power point of the PV arrays. This paper presents a modified LFO algorithm that incorporates some modifications to the conventional Levy flight search. The proposed method is inspired by PSO behavior in the search of MPPT, and the formula of Levy flight with modification of the β parameter to the global best particles is included. This has significantly improved the performance of the conventional Levy flight with the global search capability in the algorithm. The response time and efficiency of MPPT in tracking local and global optimums are significantly improved. Section 2 explains the PV configuration, modeling, and converter system. Section 3 presents the overview of Levy flight optimization. Section 4 describes MPPT with the modified Levy Flight Optimization. Section 5 and 6 present the results and discussion from the simulation and experiment, respectively. Lastly, the conclusion is given in Section 7.

PV System Configuration
Generally, a standalone PV system is configured which consists of a PV array as the input source, dc-dc boost converter, and a resistive load as shown in Figure 1 [3,46,47]. In this system, a MOSFET is used as a switching device. The gate signal is controlled by a PWM control technique. The variation of the duty cycle is controlled by the MPPT algorithm to track the maximum power point of the PV arrays.

Modeling of the PV Module
A single-diode model of a PV cell is depicted in Figure 2 [13]. The accuracy of the single-diode model is high and its complexity is low [48][49][50][51][52] compared to two-diode modeling [52][53][54][55]. A single-diode model is widely used by researchers [13,56] and is a reliable model for MPPT algorithm analysis [48,[57][58][59][60]. The physics of the p-n junction of a PV cell are governed by five main parameters which are photon current Iph, output current I, terminal voltage V, shunt resistor Rsh, and series resistance Rs. The shunt resistor, Rsh, shows the leakage current of the p-n junction, while series resistance, Rs, represents the combination of the contact resistance, metal grid, and p and n layers [59].

Modeling of the PV Module
A single-diode model of a PV cell is depicted in Figure 2 [13]. The accuracy of the single-diode model is high and its complexity is low [48][49][50][51][52] compared to two-diode modeling [52][53][54][55]. A single-diode model is widely used by researchers [13,56] and is a reliable model for MPPT algorithm analysis [48,[57][58][59][60]. The physics of the p-n junction of a PV cell are governed by five main parameters which are photon current I ph , output current I, terminal voltage V, shunt resistor R sh , and series resistance R s . The shunt resistor, R sh , shows the leakage current of the p-n junction, while series resistance, R s , represents the combination of the contact resistance, metal grid, and p and n layers [59]. This paper presents a modified LFO algorithm that incorporates some modifications to the conventional Levy flight search. The proposed method is inspired by PSO behavior in the search of MPPT, and the formula of Levy flight with modification of the β parameter to the global best particles is included. This has significantly improved the performance of the conventional Levy flight with the global search capability in the algorithm. The response time and efficiency of MPPT in tracking local and global optimums are significantly improved. Section 2 explains the PV configuration, modeling, and converter system. Section 3 presents the overview of Levy flight optimization. Section 4 describes MPPT with the modified Levy Flight Optimization. Section 5 and 6 present the results and discussion from the simulation and experiment, respectively. Lastly, the conclusion is given in Section 7.

PV System Configuration
Generally, a standalone PV system is configured which consists of a PV array as the input source, dc-dc boost converter, and a resistive load as shown in Figure 1 [3,46,47]. In this system, a MOSFET is used as a switching device. The gate signal is controlled by a PWM control technique. The variation of the duty cycle is controlled by the MPPT algorithm to track the maximum power point of the PV arrays.

Modeling of the PV Module
A single-diode model of a PV cell is depicted in Figure 2 [13]. The accuracy of the single-diode model is high and its complexity is low [48][49][50][51][52] compared to two-diode modeling [52][53][54][55]. A single-diode model is widely used by researchers [13,56] and is a reliable model for MPPT algorithm analysis [48,[57][58][59][60]. The physics of the p-n junction of a PV cell are governed by five main parameters which are photon current Iph, output current I, terminal voltage V, shunt resistor Rsh, and series resistance Rs. The shunt resistor, Rsh, shows the leakage current of the p-n junction, while series resistance, Rs, represents the combination of the contact resistance, metal grid, and p and n layers [59].  The characteristics of the single-diode model of the PV cell are illustrated in Equation (1) [61,62]. It is noted that the characteristics of the PV cell involve nonlinearity [59].
I represents the output current of the PV cell, I ph indicates the photon current, I o represents the reverse saturation current of the cell, q indicates the electronics charge, V is the output voltage of the cell, n is the ideality factor, K is Boltzmann's constant, and T is the absolute operating temperature. N s is the total number of cells in series. I sh indicates the current flow through the shunt resistor.
S represents the actual irradiance (W/m 2 ), S ref is the irradiance under the reference condition (W/m 2 ), I ph,ref is the light current under the reference condition (A), µ Isc is the manufacturer-supplied temperature coefficient of the short-circuit current (A/K), T C is the actual cell temperature (K), and T C,ref is the actual reference temperature (K).
The reserve saturation current is given by Equation (3) [64].
E go is the band gap energy, and n is the ideality factor.

Dc-dc Boost Converter Model
The selection of the converter for the MPPT controller depends on the option of using it either with or without a battery [3]. The presence of the battery requires the design of the converter to meet its voltage criteria at a constant load. For a system without a battery, the output voltage depends on the load chosen (load matching), and the power transfer follows the maximum power transfer theory [3]. Maximum power transfer can only occur when the load resistance matches the internal resistance of the PV panel as shown in Figure 3 [3]. The characteristics of the single-diode model of the PV cell are illustrated in Equation (1) [61,62]. It is noted that the characteristics of the PV cell involve nonlinearity [59].
I represents the output current of the PV cell, Iph indicates the photon current, Io represents the reverse saturation current of the cell, q indicates the electronics charge, V is the output voltage of the cell, n is the ideality factor, K is Boltzmann's constant, and T is the absolute operating temperature. Ns is the total number of cells in series. Ish indicates the current flow through the shunt resistor. The photon current is denoted by Equation (2) Ego is the band gap energy, and n is the ideality factor.

Dc-dc Boost Converter Model
The selection of the converter for the MPPT controller depends on the option of using it either with or without a battery [3]. The presence of the battery requires the design of the converter to meet its voltage criteria at a constant load. For a system without a battery, the output voltage depends on the load chosen (load matching), and the power transfer follows the maximum power transfer theory [3]. Maximum power transfer can only occur when the load resistance matches the internal resistance of the PV panel as shown in Fig  The complexity to design the converter comes in due to the non-linearity of the PV module characteristics [13]. A few important steps in designing the converter are considered. Firstly, the range of the output resistance, Rload, needs to be determined. The range of Rload is calculated based on Equation (4) [13]. The complexity to design the converter comes in due to the non-linearity of the PV module characteristics [13]. A few important steps in designing the converter are considered. Firstly, the range of the output resistance, R load , needs to be determined. The range of R load is calculated based on Equation (4) [13].
where R m is the value of the resistance at the maximum power point and D is the duty cycle. The range of the optimum R load is determined as shown in Equation (5) [13].
where R m(max) denotes the resistance at the maximum power point at maximum irradiance, D min is the minimum duty cycle, R load(opt) is the optimum load resistance, R m(min) is resistance at the maximum power point at minimum irradiance, and D max is the maximum duty cycle. The determination of the range of the load resistance is crucial to ensure the maximum power transfer is achievable. The output voltage of the boost converter is determined by Equation (6) [13].
where V om is the output voltage and V mG is the maximum voltage at maximum irradiance. The value of the inductor, L, is given in Equation (7) [13].
where ∆i L represents inductor ripple current and f is the switching frequency. The minimum input capacitance, C in, is calculated as shown in Equation (8) [13].
where γ Vm is the voltage factor at the maximum power point. The minimum output capacitance, C out, is determined by Equation (9) [13].
Here, γ V o represents the ripple factor of the output voltage.

Overview of Levy Flight
This algorithm is inspired by the brood parasitism strategy of cuckoo birds [40,65]. Typically, this obligate parasite lays its eggs without taking care of its young but letting the host take care of them. Cuckoo birds have an interesting behavior that can imitate the host color and shape of the eggs. Another incredibly interesting action to take note of is that the cuckoo egg tends to hatch earlier than the host bird. As early as its hatches, the cuckoo wipes off the host eggs. However, the probability of being noticed by the host bird is there as once caught by the host bird, the egg is destroyed. This will force the cuckoo bird to migrate and build another new nest [40]. The crucial part of a cuckoo bird's brood parasitism strategy is hunting for the host nest. The hunting for the nest is randomly executed, and trajectories of the direction are modeled using mathematical equations. Typically, a random search is modeled with Levy flight [40,65]. In a cuckoo search, Levy flight reflects the hunting effort for the host nest.
The random distribution of Levy flight is expressed in Equation (10). The distribution pattern is based on the following power law.
where ι is the length of the flight, λ represents the variance, and q is infinite variance at 1 < λ < 3. The distribution pattern of the Levy flight is shown in Figure 4 [40]. It is observed that the pattern of the Levy distribution is inconsistent as the step varies in the form of a large step, small step, and long-distance jump [40,65]. Thus, if this pattern is incorporated in metaheuristic algorithms, especially in nonlinear situations, it will significantly improve the search efficiency [40]. where ι is the length of the flight, λ represents the variance, and q is infinite variance . The distribution pattern of the Levy flight is shown in Figure 4 [40]. It is observed that the pattern of the Levy distribution is inconsistent as the step varies in the form of a large step, small step, and long-distance jump [40,65]. Thus, if this pattern is incorporated in metaheuristic algorithms, especially in nonlinear situations, it will significantly improve the search efficiency [40]. In the cuckoo search algorithm, the algorithm starts with the brood parasitism mechanism [40,66]. Firstly, the algorithm is executed based on the hunting process for the host nest. Then, the best host nest is found, and lastly, there is the possibility that the host will discover the cuckoo's eggs with given probability, Pa which is 0 < Pa < 1. Once the cuckoo's eggs are found by the host, the host either leaves the nest or wipes off the eggs.

Variables and Equations of the Proposed MPPT
There are few modifications in incorporating the modified Levy flight algorithm in MPPT. Initially, the assigned particles are needed to determine the search space. The particles represent the duty cycle, D, i.e., Di (i = 1, 2, … n). Here, n denotes the total number of particles. Next, the step size, α needs to be determined. Then, the fitness function, f', is the power of the PV.
At the start, the initialization is first performed, where all particles are executed in the search space, and the initial value of the fitness function is the power at all respective particles. The maximum power is considered initially, and the best duty cycle at the respective power is considered the best particle sample. Soon after, the next stage is executed following the equation for Levy flight as shown in Equation (11) [38,41]; the new particles at the respective power are considered.

( )
A simplified modified Levy distribution is given by Equation (12).
, and P is the Levy multiplying coefficient. The coefficients u and υ are obtained from normal distribution curves as in Equation (13)  In the cuckoo search algorithm, the algorithm starts with the brood parasitism mechanism [40,66]. Firstly, the algorithm is executed based on the hunting process for the host nest. Then, the best host nest is found, and lastly, there is the possibility that the host will discover the cuckoo's eggs with given probability, P a which is 0 < P a < 1. Once the cuckoo's eggs are found by the host, the host either leaves the nest or wipes off the eggs.

Variables and Equations of the Proposed MPPT
There are few modifications in incorporating the modified Levy flight algorithm in MPPT. Initially, the assigned particles are needed to determine the search space. The particles represent the duty cycle, D, i.e., D i (i = 1, 2, . . . n). Here, n denotes the total number of particles. Next, the step size, α needs to be determined. Then, the fitness function, f', is the power of the PV.
At the start, the initialization is first performed, where all particles are executed in the search space, and the initial value of the fitness function is the power at all respective particles. The maximum power is considered initially, and the best duty cycle at the respective power is considered the best particle sample. Soon after, the next stage is executed following the equation for Levy flight as shown in Equation (11) [38,41]; the new particles at the respective power are considered A simplified modified Levy distribution is given by Equation (12).
, and P is the Levy multiplying coefficient. The coefficients u and υ are obtained from normal distribution curves as in Equation (13) [38,41].
Variables σ u and σ υ are expressed in Equation (14) with symbol Γ to denote the integral gamma function.
The power of the PV module is calculated and compared for the respective particle. The power in the respective duty cycle is compared, and the new best particle is selected. The best particle is once again compared with the global best particle at the respective power. If the best particle is not the current best particle, the new global best particle is assigned. Consequently, the iterative process is executed for all particles, and the current global best particle is evaluated over f'. This iterative process is executed until all particles achieve the maximum power point (MPP). Figure 5 shows the mechanism of the movement of the particles for the proposed modified Levy flight algorithm. The particles are considered to be distributed initially along the entire P-V curve. Four initial duty cycles are chosen [38,40]. The particles are distributed at locations D 1 , D 2 , D 3, and D 4 . The power at each respective particle, D, is read. The highest value of power at the corresponding particle, D pm, is considered as the best particle value. Other particles are forced to move towards this particle, D pm, as the best particle is the mark point for the movement of all other particles. The movement of the particles is guided by Levy flight as in Equation (11). The step sizes are varied towards the best particles, either in small or large jumps depending upon the boundaries of the search area. The step size becomes small as the particle moves nearer to MPP. A randomized step size in Levy flight ensures that the search would cover all the points along the P-V curve. Once all the particles reach the best particle, another evaluation is executed by comparing it with the global best particle. The comparison between the best particle and global best particle is obtained as shown in Equation (12). The new global best particle is assigned if the current best particle exceeds the global best particle. The position and velocity as shown in Equations (11) and (12) are updated. The convergence stops once all particles reach their MPP. All particle step sizes reach zero once MPP is achieved. Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 27

Simulation Analysis
The effectiveness of the proposed algorithm is verified with simulation in the MATLAB/Simulink environment. The circuit model is shown in Figure 6. This system consists of a PV panel, boost converter, MPPT controller, and load [67]. The boost converter is chosen in this application due to its popularity among researchers as it has a low voltage ripple [11]. The parameters of the boost converter are listed in Table 1. The converter is connected to a resistive load of 50 Ω. A PV module, PMS50W, with specifications as listed in Table 2 is used.

Simulation Analysis
The effectiveness of the proposed algorithm is verified with simulation in the MAT-LAB/Simulink environment. The circuit model is shown in Figure 6. This system consists of a PV panel, boost converter, MPPT controller, and load [67]. The boost converter is chosen in this application due to its popularity among researchers as it has a low voltage ripple [11]. The parameters of the boost converter are listed in Table 1. The converter is connected to a resistive load of 50 Ω. A PV module, PMS50W, with specifications as listed in Table 2 is used.  Initial particles D are set to 0.1, 0.2, 0.5, and 0.7. This is highly important as the initial execution process is to scan the full range before converging on the best solution. The duty cycle limit is also set from dmin to dmax. The setting of limitations on the duty cycle is included in the algorithm. Here, dmin is set to 0.1 and dmax is set to 0.75. The stopping criterion in this algorithm is set by assigning maximum iteration in the proposed algorithm. Here, the maximum iteration is set to 10. The other parameters are also set as follows: the P coefficient is set to 0.8, β1 is set to 1.5, β2 is set to 2.0, and υ is set to 1. In this simulation, the sampling time is 0.1 s with a sampling frequency of 10 Hz. Table 3 summarizes the parameter setting of the proposed algorithm.
Firstly, the proposed algorithm is evaluated under uniform irradiance (STC). The tested conditions are investigated under three different irradiances which are high (1000 W/m 2 ), medium (600 W/m 2 ), and low (200 W/m 2 ) as shown in Figure 7. Based on Figure 7, it is noted that the MPP of the PV module varies with irradiance. Higher irradiance contributes to a higher value of maximum power at a constant temperature. It is also observed that there is only one peak power point for each irradiance. This peak is known as the local maximum power point (LMPP).  Initial particles D are set to 0.1, 0.2, 0.5, and 0.7. This is highly important as the initial execution process is to scan the full range before converging on the best solution. The duty cycle limit is also set from d min to d max . The setting of limitations on the duty cycle is included in the algorithm. Here, d min is set to 0.1 and d max is set to 0.75. The stopping criterion in this algorithm is set by assigning maximum iteration in the proposed algorithm. Here, the maximum iteration is set to 10. The other parameters are also set as follows: the P coefficient is set to 0.8, β 1 is set to 1.5, β 2 is set to 2.0, and υ is set to 1. In this simulation, the sampling time is 0.1 s with a sampling frequency of 10 Hz. Table 3 summarizes the parameter setting of the proposed algorithm.  Figure 7. Based on Figure 7, it is noted that the MPP of the PV module varies with irradiance. Higher irradiance contributes to a higher value of maximum power at a constant temperature. It is also observed that there is only one peak power point for each irradiance. This peak is known as the local maximum power point (LMPP).   Next, the proposed algorithm is tested under the partial shading condition (PSC) as shown in Figure 8. The characteristics of the I-V and P-V curves under PSCs are shown for three different conditions. Based on Figure 8, it can be noted that multiple peaks occur along the curve. This phenomenon happens due to non-uniform irradiances that penetrated into the PV module as shown in Figure 9. Multiple peaks here include LMPP and GMPP. The irradiance patterns of the PSCs are summarized in Table 4. Next, the proposed algorithm is tested under the partial shading condition (PSC) as shown in Figure 8. The characteristics of the I-V and P-V curves under PSCs are shown for three different conditions. Based on Figure 8, it can be noted that multiple peaks occur along the curve. This phenomenon happens due to non-uniform irradiances that penetrated into the PV module as shown in Figure 9. Multiple peaks here include LMPP and GMPP. The irradiance patterns of the PSCs are summarized in Table 4.   Firstly, the proposed algorithm is tested under uniform irradiance. Figure 10 shows simulation results of the proposed modified LFO at 1000 W/m 2 irradiance. As can be seen, the proposed modified LFO has successfully tracked the maximum power at 50 W under 1000 W/m 2 STC. It is also observed from the simulation results that the modified LFO is stable and no oscillation at steady-state occurs. Figure 11 shows the simulation results of the proposed modified LFO at 600 W/m 2 STC. Similarly, the proposed algorithm is able to track optimum power at medium irradiance which is 30 W. The proposed algorithm of modified LFO is stable with almost zero steady-state oscillation. Figure 12 shows the simulation results of the proposed modified LFO at low irradiance, 200 W/m 2 . The proposed modified LFO successfully tracks the optimum power at 10 W. Based on the simulation results, the proposed modified LFO is stable with zero steady-state oscillation.   Firstly, the proposed algorithm is tested under uniform irradiance. Figure 10 shows simulation results of the proposed modified LFO at 1000 W/m 2 irradiance. As can be seen, the proposed modified LFO has successfully tracked the maximum power at 50 W under 1000 W/m 2 STC. It is also observed from the simulation results that the modified LFO is stable and no oscillation at steady-state occurs. Figure 11 shows the simulation results of the proposed modified LFO at 600 W/m 2 STC. Similarly, the proposed algorithm is able to track optimum power at medium irradiance which is 30 W. The proposed algorithm of modified LFO is stable with almost zero steady-state oscillation. Figure 12 shows the simulation results of the proposed modified LFO at low irradiance, 200 W/m 2 . The proposed modified LFO successfully tracks the optimum power at 10 W. Based on the simulation results, the proposed modified LFO is stable with zero steady-state oscillation.  Firstly, the proposed algorithm is tested under uniform irradiance. Figure 10 shows simulation results of the proposed modified LFO at 1000 W/m 2 irradiance. As can be seen, the proposed modified LFO has successfully tracked the maximum power at 50 W under 1000 W/m 2 STC. It is also observed from the simulation results that the modified LFO is stable and no oscillation at steady-state occurs. Figure 11 shows the simulation results of the proposed modified LFO at 600 W/m 2 STC. Similarly, the proposed algorithm is able to track optimum power at medium irradiance which is 30 W. The proposed algorithm of modified LFO is stable with almost zero steady-state oscillation.  Further testing is conducted under PSCs as described in Table 4. The shadow patterns are created as in Table 4. Non-uniform irradiances are penetrated into PV modules as in Table 4 to generate a pattern for PSCs. Based on Figure 8b, the global maximum power point is 17.5 W for PSC 1. Here, the proposed modified LFO is tested. Figure 13 shows the simulation results of the proposed modified LFO algorithm under PSC 1. The results show that the proposed algorithm can track GMPP at 17.5 W. The algorithm is stable, and no steady-state oscillation is observed. Further testing is conducted under PSCs as described in Table 4. The shadow patterns are created as in Table 4. Non-uniform irradiances are penetrated into PV modules as in Table 4 to generate a pattern for PSCs. Based on Figure 8b, the global maximum power point is 17.5 W for PSC 1. Here, the proposed modified LFO is tested. Figure 13 shows the simulation results of the proposed modified LFO algorithm under PSC 1. The results show that the proposed algorithm can track GMPP at 17.5 W. The algorithm is stable, and no steady-state oscillation is observed. Further testing is conducted under PSCs as described in Table 4. The shadow patterns are created as in Table 4. Non-uniform irradiances are penetrated into PV modules as in Table 4 to generate a pattern for PSCs. Based on Figure 8b, the global maximum power point is 17.5 W for PSC 1. Here, the proposed modified LFO is tested. Figure 13 shows the simulation results of the proposed modified LFO algorithm under PSC 1. The results show that the proposed algorithm can track GMPP at 17.5 W. The algorithm is stable, and no steady-state oscillation is observed.  Table 4 is created. The proposed modified LFO is tested under PSC 2. Based on Figure 14, the simulation results indicate that the proposed modified LFO can distinguish GMPP at 11.1 W. The algorithm is stable, and no steady-state oscillations occur. PSC 3 is created by non-uniform irradiances on the PV panels as shown in Table 4. The modified LFO is tested. Again, as can be seen, the modified LFO has successfully  Table 4 is created. The proposed modified LFO is tested under PSC 2. Based on Figure 14, the simulation results indicate that the proposed modified LFO can distinguish GMPP at 11.1 W. The algorithm is stable, and no steady-state oscillations occur.  Table 4 is created. The proposed modified LFO is tested under PSC 2. Based on Figure 14, the simulation results indicate that the proposed modified LFO can distinguish GMPP at 11.1 W. The algorithm is stable, and no steady-state oscillations occur. PSC 3 is created by non-uniform irradiances on the PV panels as shown in Table 4. The modified LFO is tested. Again, as can be seen, the modified LFO has successfully   Table 4. The modified LFO is tested. Again, as can be seen, the modified LFO has successfully tracked GMPP at 16 W as shown in Figure 15. Notably, the proposed modified LFO is stable, and no steady-state oscillation is observed. tracked GMPP at 16 W as shown in Figure 15. Notably, the proposed modified LFO is stable, and no steady-state oscillation is observed. Next, the proposed modified LFO is tested under dynamic conditions. The dynamic condition is set from STC at time 0 to 0.1 s and then it changes from STC to PSC 1 at time 0.1 s. Based on the results in Figure 16, it is noted that the proposed algorithm is capable of detecting MPP at STC, which is 50 W, and then GMPP at PSC 1, which is 17.5 W, with a 3.0 s response time. Based on these results, the proposed algorithm is stable, and no steady-state oscillation is observed. Next, the proposed modified LFO is tested under dynamic conditions. The dynamic condition is set from STC at time 0 to 0.1 s and then it changes from STC to PSC 1 at time 0.1 s. Based on the results in Figure 16, it is noted that the proposed algorithm is capable of detecting MPP at STC, which is 50 W, and then GMPP at PSC 1, which is 17.5 W, with a 3.0 s response time. Based on these results, the proposed algorithm is stable, and no steady-state oscillation is observed. tracked GMPP at 16 W as shown in Figure 15. Notably, the proposed modified LFO is stable, and no steady-state oscillation is observed. Next, the proposed modified LFO is tested under dynamic conditions. The dynamic condition is set from STC at time 0 to 0.1 s and then it changes from STC to PSC 1 at time 0.1 s. Based on the results in Figure 16, it is noted that the proposed algorithm is capable of detecting MPP at STC, which is 50 W, and then GMPP at PSC 1, which is 17.5 W, with a 3.0 s response time. Based on these results, the proposed algorithm is stable, and no steady-state oscillation is observed.

Experimental Results
The effectiveness of the proposed modified LFO is further tested with a prototype. The experimental prototype has a similar rated power as that in the simulation for fair comparison. The boost converter with the same component values as in the simulation is used here. The boost converter is used as the main component in searching for MPPT and also for supplying the optimum power to the load. The PV array output is generated by a PV simulator. Two sensors are used: LA-55p and LV-25. The Hall Effect current sensor, LA-55p, is used to measure the PV input current, while the voltage sensor, LV-25, is used to measure the PV input voltage. These measured signals are fed to the Analog-to-Digital converter (ADC) of DSP TMS320F28335. The signals are used as the input for MPPT algorithm execution. The Pulse Width Modulation (PWM) is the output signal produced by the MPPT algorithm. Typically, the PWM generated by DSP is in the range of 0-5 V; thus, a gate driver, HCPL 3120, is chosen. The driver HCPL 3120 boosts up the PWM signal to ±15 V, sufficient to trigger the power switch in the boost converter. The resistive load is connected at the output. The waveforms are displayed and captured with a digital oscilloscope. The schemes of the experimental rig and experimental setup are shown in Figure  17a,b, respectively.

Experimental Results
The effectiveness of the proposed modified LFO is further tested with a prototype. The experimental prototype has a similar rated power as that in the simulation for fair comparison. The boost converter with the same component values as in the simulation is used here. The boost converter is used as the main component in searching for MPPT and also for supplying the optimum power to the load. The PV array output is generated by a PV simulator. Two sensors are used: LA-55p and LV-25. The Hall Effect current sensor, LA-55p, is used to measure the PV input current, while the voltage sensor, LV-25, is used to measure the PV input voltage. These measured signals are fed to the Analog-to-Digital converter (ADC) of DSP TMS320F28335. The signals are used as the input for MPPT algorithm execution. The Pulse Width Modulation (PWM) is the output signal produced by the MPPT algorithm. Typically, the PWM generated by DSP is in the range of 0-5 V; thus, a gate driver, HCPL 3120, is chosen. The driver HCPL 3120 boosts up the PWM signal to ±15 V, sufficient to trigger the power switch in the boost converter. The resistive load is connected at the output. The waveforms are displayed and captured with a digital oscilloscope. The schemes of the experimental rig and experimental setup are shown in Figure 17a,b, respectively.

Experimental Results
The effectiveness of the proposed modified LFO is further tested with a prototype. The experimental prototype has a similar rated power as that in the simulation for fair comparison. The boost converter with the same component values as in the simulation is used here. The boost converter is used as the main component in searching for MPPT and also for supplying the optimum power to the load. The PV array output is generated by a PV simulator. Two sensors are used: LA-55p and LV-25. The Hall Effect current sensor, LA-55p, is used to measure the PV input current, while the voltage sensor, LV-25, is used to measure the PV input voltage. These measured signals are fed to the Analog-to-Digital converter (ADC) of DSP TMS320F28335. The signals are used as the input for MPPT algorithm execution. The Pulse Width Modulation (PWM) is the output signal produced by the MPPT algorithm. Typically, the PWM generated by DSP is in the range of 0-5 V; thus, a gate driver, HCPL 3120, is chosen. The driver HCPL 3120 boosts up the PWM signal to ±15 V, sufficient to trigger the power switch in the boost converter. The resistive load is connected at the output. The waveforms are displayed and captured with a digital oscilloscope. The schemes of the experimental rig and experimental setup are shown in Figure  17a,b, respectively. Firstly, the proposed modified LFO algorithm is experimentally tested under uniform irradiance. The numerical data of the I-V and P-V curves for high (1000 W/m 2 ), medium (600 W/m 2 ), and low (200 W/m 2 ) irradiances at STC are loaded into the PV simulator. Once the respective numerical data are loaded, the patterns of the I-V and P-V curves are created as shown in Figure 18.  Figure 19 shows experimental results for the proposed modified LFO algorithm tested at 1000 W/m 2 irradiance. Based on the results, the proposed algorithm is able to track LMPP, which is 50 W. From the results, the proposed modified LFO has excellent searching capability since it uses a random search. Once the optimum is found, the speed slows and reaches zero speed when all the particles have found the optimum point. Firstly, the proposed modified LFO algorithm is experimentally tested under uniform irradiance. The numerical data of the I-V and P-V curves for high (1000 W/m 2 ), medium (600 W/m 2 ), and low (200 W/m 2 ) irradiances at STC are loaded into the PV simulator. Once the respective numerical data are loaded, the patterns of the I-V and P-V curves are created as shown in Figure 18. Firstly, the proposed modified LFO algorithm is experimentally tested under uniform irradiance. The numerical data of the I-V and P-V curves for high (1000 W/m 2 ), medium (600 W/m 2 ), and low (200 W/m 2 ) irradiances at STC are loaded into the PV simulator. Once the respective numerical data are loaded, the patterns of the I-V and P-V curves are created as shown in Figure 18.  Figure 19 shows experimental results for the proposed modified LFO algorithm tested at 1000 W/m 2 irradiance. Based on the results, the proposed algorithm is able to track LMPP, which is 50 W. From the results, the proposed modified LFO has excellent searching capability since it uses a random search. Once the optimum is found, the speed slows and reaches zero speed when all the particles have found the optimum point.  Figure 19 shows experimental results for the proposed modified LFO algorithm tested at 1000 W/m 2 irradiance. Based on the results, the proposed algorithm is able to track LMPP, which is 50 W. From the results, the proposed modified LFO has excellent searching capability since it uses a random search. Once the optimum is found, the speed slows and reaches zero speed when all the particles have found the optimum point.     Further testing is conducted under partial shading conditions. The same patterns as those used in the simulation are tested here. All the numerical data of the shading patterns given in Table 4 are loaded into the PV simulator. Three patterns are shown in Figure 22, i.e., PSC 1, PSC 2, and PSC 3.   Further testing is conducted under partial shading conditions. The same patterns as those used in the simulation are tested here. All the numerical data of the shading patterns given in Table 4 are loaded into the PV simulator. Three patterns are shown in Figure 22, i.e., PSC 1, PSC 2, and PSC 3. Further testing is conducted under partial shading conditions. The same patterns as those used in the simulation are tested here. All the numerical data of the shading patterns given in Table 4 are loaded into the PV simulator. Three patterns are shown in Figure 22, i.e., PSC 1, PSC 2, and PSC 3.  Figure 23 shows the experimental results for the proposed modified LFO tested under PSC 1. Based on the experimental results, the proposed algorithm is able to track GMPP at 17 W. It clearly shows that the proposed modified LFO has zero steady-state oscillation and it is stable. The proposed modified LFO is more stable due to its random searching behavior, where it has ability to control the speed, i.e., either longer or shorter jumps. Once the optimum point is reached, all the points slow down and have a shorter jump. Thus, this characteristic will facilitate the proposed modified LFO to perform better in the searching effort under PSC.  Figure 24 shows the experimental results of the proposed modified LFO under PSC 2. As can be seen, the proposed algorithm can achieve GMPP at 11 W. Here, the proposed  Figure 23 shows the experimental results for the proposed modified LFO tested under PSC 1. Based on the experimental results, the proposed algorithm is able to track GMPP at 17 W. It clearly shows that the proposed modified LFO has zero steady-state oscillation and it is stable. The proposed modified LFO is more stable due to its random searching behavior, where it has ability to control the speed, i.e., either longer or shorter jumps. Once the optimum point is reached, all the points slow down and have a shorter jump. Thus, this characteristic will facilitate the proposed modified LFO to perform better in the searching effort under PSC.  Figure 23 shows the experimental results for the proposed modified LFO tested under PSC 1. Based on the experimental results, the proposed algorithm is able to track GMPP at 17 W. It clearly shows that the proposed modified LFO has zero steady-state oscillation and it is stable. The proposed modified LFO is more stable due to its random searching behavior, where it has ability to control the speed, i.e., either longer or shorter jumps. Once the optimum point is reached, all the points slow down and have a shorter jump. Thus, this characteristic will facilitate the proposed modified LFO to perform better in the searching effort under PSC.  Figure 24 shows the experimental results of the proposed modified LFO under PSC 2. As can be seen, the proposed algorithm can achieve GMPP at 11 W. Here, the proposed  Figure 24 shows the experimental results of the proposed modified LFO under PSC 2. As can be seen, the proposed algorithm can achieve GMPP at 11 W. Here, the proposed modified LFO has 99% efficiency. Further, the proposed modified LFO is stable and has negligible steady-state oscillation.     Next, the prototype is experimentally tested under dynamic conditions. The same patterns as those in the simulation are tested here, i.e., STC to PSC 1. All the numerical data of the patterns are listed in Table 4 and are loaded into the PV simulator. The dynamic patterns are shown in Figure 26.  Figure 27 shows the experimental results for the proposed modified LFO tested under dynamic conditions, i.e., STC to PSC 1. As can be observed, the proposed modified LFO is capable of detecting the LMPP under STC at irradiance 1000 W/m 2 with optimum power at 49.9 W, and then the operating condition is changed to PSC 1. Immediately after the change, the proposed modified LFO manages to detect the new GMPP at 17.4 W under PSC 1 with 3.0 s response time. The result in Figure 27 also indicates that the proposed modified LFO is stable, and negligible steady-state oscillation is observed. Table 5 summarizes the performance of the proposed modified LFO. It has excellent performance under uniform and non-uniform conditions. The proposed modified LFO has successfully tracked the LMPP under STC and GMPP under PSCs. The proposed modified LFO is also stable and has negligible steady-state oscillations. Next, the prototype is experimentally tested under dynamic conditions. The same patterns as those in the simulation are tested here, i.e., STC to PSC 1. All the numerical data of the patterns are listed in Table 4 and are loaded into the PV simulator. The dynamic patterns are shown in Figure 26. Next, the prototype is experimentally tested under dynamic conditions. The same patterns as those in the simulation are tested here, i.e., STC to PSC 1. All the numerical data of the patterns are listed in Table 4 and are loaded into the PV simulator. The dynamic patterns are shown in Figure 26.  Figure 27 shows the experimental results for the proposed modified LFO tested under dynamic conditions, i.e., STC to PSC 1. As can be observed, the proposed modified LFO is capable of detecting the LMPP under STC at irradiance 1000 W/m 2 with optimum power at 49.9 W, and then the operating condition is changed to PSC 1. Immediately after the change, the proposed modified LFO manages to detect the new GMPP at 17.4 W under PSC 1 with 3.0 s response time. The result in Figure 27 also indicates that the proposed modified LFO is stable, and negligible steady-state oscillation is observed. Table 5 summarizes the performance of the proposed modified LFO. It has excellent performance under uniform and non-uniform conditions. The proposed modified LFO has successfully tracked the LMPP under STC and GMPP under PSCs. The proposed modified LFO is also stable and has negligible steady-state oscillations.  Figure 27 shows the experimental results for the proposed modified LFO tested under dynamic conditions, i.e., STC to PSC 1. As can be observed, the proposed modified LFO is capable of detecting the LMPP under STC at irradiance 1000 W/m 2 with optimum power at 49.9 W, and then the operating condition is changed to PSC 1. Immediately after the change, the proposed modified LFO manages to detect the new GMPP at 17.4 W under PSC 1 with 3.0 s response time. The result in Figure 27 also indicates that the proposed modified LFO is stable, and negligible steady-state oscillation is observed. Table 5 summarizes the performance of the proposed modified LFO. It has excellent performance under uniform and non-uniform conditions. The proposed modified LFO has successfully tracked the LMPP under STC and GMPP under PSCs. The proposed modified LFO is also stable and has negligible steady-state oscillations.  The proposed algorithm is then compared with the two most famous methods which are P&O and PSO. P&O is chosen since this method is widely used either in practice or research, while PSO is chosen due to the fact that it is the most famous and simple searchbased method. Table 6 shows a comparison between P&O, PSO, and the proposed algorithm. The proposed algorithm improves the efficiency of the system compared to P&O and PSO. The efficiency of all the tested conditions is higher than 98%. The operation of the proposed algorithm under partial shading is also guaranteed.   The proposed algorithm is then compared with the two most famous methods which are P&O and PSO. P&O is chosen since this method is widely used either in practice or research, while PSO is chosen due to the fact that it is the most famous and simple search-based method. Table 6 shows a comparison between P&O, PSO, and the proposed algorithm. The proposed algorithm improves the efficiency of the system compared to P&O and PSO. The efficiency of all the tested conditions is higher than 98%. The operation of the proposed algorithm under partial shading is also guaranteed.

Conclusions
The modified LFO is proposed in this paper based on a random search of Levy flight for a PV energy system. The new variable β is introduced to solve the global best in obtaining optimum power. The proposed algorithm has excellent performance in tracking the global maximum point under PSCs. The proposed algorithm solves three major problems, mainly intermittent tracking under non-uniform irradiances, steady-state oscillation, and stability. Simulation studies are carried out under STC, PSC, and dynamic conditions. The proposed algorithm shows excellent performance in tracking MPPT. In addition, the performance of the proposed algorithm is verified with a prototype. Both simulation and experimental results show that the proposed algorithm has zero steadystate oscillation and it is stable. The efficiency of the proposed algorithm is higher than 98%. The simulation and experimental results show that the modified LFO has successfully tracked the global maximum point for all operating conditions in which the maximum power is always extracted and fast response time is achieved.