Advanced Limited Search Strategy for Enhancing the Performance of MPPT Algorithms

Abstract

Photovoltaic (PV) arrays are gaining popularity for electricity generation due to their simple and green energy production. However, the power transfer efficiency of PV varies depending on the load’s electrical properties, the PV panels’ temperature, and the insolation conditions. Maximum Power Point Tracking (MPPT) is a method formulated as an optimization problem that adjusts the PV output voltage to deliver maximum power to the load based on these criteria (maximum power in the P-V curve). MPPT is a convex optimization problem when the Sun’s rays completely cover the PV surface (full insolation). Several power points are formed in the Power vs. Voltage (P-V) curve, rendering MPPT as a non-convex problem during incomplete insolation (partial shadowing) on the PV surface due to barriers such as passing clouds or trees in the path of the Sun and the PV’s surface. Unfortunately, mathematical programming techniques, such as gradient ascent and momentum, are not good optimization candidate algorithms because they cannot distinguish between the local and global maximum of a function (the case of non-convex problems). On the other hand, metaheuristic algorithms have better search space exploration capability, making it easier to discern the P-V curve’s local and global power peaks. However, due to their pseudorandom search space exploration (random with some intuition), there is plenty of room for improving their performance. In this work, we elaborate on the Advanced Limited Search Strategy (ALSS), a technique we proposed in one of our previous works on MPPT. We prove its universal usefulness by applying it to other MPPT algorithms to enhance their performance. The ALSS first finds the direction where it is most probable to discover the MPP using the finite difference between two candidate duty cycles and then computes a duty cycle between two bounds designated by the previous direction. After that, the resulting duty cycle is further updated according to the metaheuristic update equation. Therefore, the single solution update is another advantage of ALSS that further improves the computational cost of the MPPT algorithms.

1. Introduction

The high demand for electricity generation due to the increasing population and technological advancements poses a heavy burden on the energy sector. This leads to a sharp decrease in the fossil fuels used for energy generation. Moreover, fossil-fuel-based energy generation produces harmful gases, which leads to global warming. The depletion of fossil fuels with environmentally unfriendly energy generation makes it necessary to shift towards renewable sources of energy that are abundantly available and do not produce hazardous gases. Among several renewable energy sources, Photovoltaic (PV) is the most popular one. Due to PV’s growing adoption in electricity generation from renewable sources, it is necessary to improve their efficiency, i.e., to deliver the maximum possible power at the output load. However, there exist several challenges in the design of PV systems.

A very challenging problem in designing PV systems is generating the maximum power at the PV output corresponding to a specific insolation condition. In order to make the photovoltaic panel capable of adjusting its voltage corresponding to the maximum power, Maximum Power Point Tracking (MPPT) is used, employing optimization algorithms capable of converging to the maximum power peak in the P-V curve. However, MPPT becomes challenging when the PV surface is partially shaded. This is known as the Partial Shading (PS) effect, which significantly reduces the power output of a PV panel due to the lesser current across the affected PV module. In order to mitigate the PS effect, bypass diodes are connected at the PV output to block the low currents through the shaded modules. However, including bypass diodes across PV modules results in multiple power peaks regarding the voltage at the PV output. This makes the Power versus Voltage (P-V) characteristic of a PV non-convex, i.e., there are several local peaks and only one global peak of power.

Several optimization algorithms have been proposed in the literature for MPPT [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30].

For MPPT, standard algorithms such as Gradient Descent (GD), hill climbing, and incremental conductance have been developed [1,2,3,4,5]. The above algorithms perform desirably in converging to the Maximum Power Point (MPP) for full insolation (no PS) conditions, but fail under PS conditions due to their inflexible search space exploration, while the P-V curve is non-convex. Artificial Intelligence (AI) algorithms have then been employed, such as Fuzzy Logic Control (FLC) and Artificial Neural Networks (ANNs) for MPPT [6,7]. The AI techniques proved to be proficient in tracking the MPP under PS (dealing with the non-convexity of P-V curve), but require many data for their training, which puts a computational burden on the microcontroller [26]. Finally, the metaheuristic algorithms for MPPT [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] have been employed to accommodate both computational efficiency and convergence to the global maximum in MPPT under PS. First, they provide a much higher probability of converging to the highest power because of a better search space exploration. Furthermore, they are less computationally expensive compared to AI techniques because they do not require training data.

Despite their advantages over AI and conventional algorithms, metaheuristic techniques need further improvements to make the MPPT controller more power-efficient and reliable. Enhancing the MPPT algorithm’s performance leads to escalated PV performance without extra hardware costs. This enhancement is profound in power and cost efficiency when seen on a large-scale PV implementation. This motivates further research to develop newer and better-performing algorithms to make PV power generation more power-efficient and cost-effective. Several recent MPPT algorithms have tried to achieve the goal mentioned above. For instance, in [8], Radial Movement Optimization (RMO) with Teaching–Learning-Based Optimization (RMOTLBO) was proposed, which combines the benefits of RMO with the learner phase of the TLBO algorithm to improve the convergence speed and power losses. In [9], the convergence speed was improved using a DeterMinistic Jaya (DM-Jaya) algorithm by removing the random coefficients in the simple Jaya algorithm [31]. The authors in [10] proposed an adaptive PSO that was made to calculate the duty cycles theoretically to solve local convergence in PSO. The authors in [12] proposed a synergism of Particle Swarm Optimization (PSO) and Differential eVolution (DE) (PSO-DV), which helped improve the local convergence issues in PSO. The Modified Butterfly Optimization Algorithm (MBOA) was proposed by the authors in [13], which avoids exploration in non-feasible regions by limiting the search area, which results in a better convergence speed and power fluctuations. The authors in [14] improved the simple Jaya algorithm using the Levy Flight (LF) steps (JayaLF) in its updating equations. The LFs take short jumps towards the optimum solution during exploration while taking long jumps occasionally, which improved the convergence speed. Moreover, in [15], another Jaya-based algorithm, namely Adaptive Jaya (AJaya), was proposed to improve power fluctuations and convergence speed by using the Time-Varying Coefficients (TVCs) in the simple Jaya updating equations. Finally, in [24], the Salp Swarm (SS) algorithm was proposed based on salps’ food foraging behavior. The algorithm was able to improve convergence speed and power losses.

This work uses the Advanced Limited Search Strategy (ALSS) proposed in [25] to significantly improve the performance of some recently proposed MPPT algorithms. In addition to contributing toward a more power-efficient and reliable MPPT controller, this work also proves the universal applicability and efficacy of the ALSS to motivate MPPT researchers to apply this technique to their algorithms for enhanced MPPT performance. This work also reveals another aspect of the ALSS that was not mentioned in [25], i.e., it reduces the number of solutions in metaheuristic algorithms to be updated in each iteration. A metaheuristic technique typically requires updating 3 to 5 solutions in each iteration. The ALSS reduces the number of updating solutions of metaheuristic algorithms to one, as simple gradient-based methods do. To summarize, the following are the contributions of this work:

• Prove the universal applicability of the ALSS technique in MPPT algorithms.
• Propose new metaheuristic MPPT algorithms by integrating the ALSS in existing MPPT algorithms.
• Reduce candidate solutions in metaheuristic algorithms.

The rest of the paper is organized as follows. Section 2 provides the background for MPPT. Section 3 explains the ALSS technique and how it can be integrated with metaheuristic algorithms. Section 4 provides experimental results justifying the improved performance of different MPPT algorithms using the ALSS, and Section 5 concludes the paper.

2. MPPT Background

2.1. Partial Shading Effect in Photovoltaics

Partial Shading (PS) occurs when natural obstacles such as trees or moving clouds block the path of sunlight to the PV surface. This significantly diminishes the current across the shaded modules. For a series-connected PV system (modules in series), the shaded module with the least current determines, due to Kirchhoff’s Current Theorem (KCT), the current flowing through the whole PV array, thus resulting in high power losses. In order to avoid this effect, bypass diodes are connected at the PV output that become short-circuited under partial shading conditions to prevent the current across the affected module from flowing into the circuit. Although bypass diodes mitigate the PS effect, they make the P-V curve non-convex, i.e., it exhibits several local power points and only one highest power, which corresponds to the maximum power (see Figure 1). In order to solve this, maximum power point tracking is used as described in the next subsection.

2.2. MPPT with Metaheuristic Algorithms

Maximum Power Point Tracking (MPPT) is a technique used to adjust the output voltage of a PV panel corresponding to the highest power in the P-V curve. Therefore, DC–DC converter circuits (e.g., boost, buck, buck–boost) are added between the PV and the output load to perform these adjustments. The duty cycle of the DC–DC converter is selected using the optimization algorithms compiled in a microcontroller (metaheuristic algorithms in this study). The metaheuristic algorithm distributes candidate duty cycles across the search space of duty cycles for exploration in each iteration. Next, the microcontroller sends the candidate duty cycles to the converter through a gate driver circuit to obtain the corresponding power values. Then, the algorithm compares the new power with the old power at each iteration and updates the duty cycle accordingly. The process continues until the maximum power is reached. Figure 2 shows the MPPT system with a PV array, power controlling unit, and output load.

Figure 1. Partial shading mitigation using the bypass diodes. The green dot corresponds to the Maximum Power Point (MPP), while the red dots correspond to the Local Maximum Power Points (LMPPs).

Figure 1. Partial shading mitigation using the bypass diodes. The green dot corresponds to the Maximum Power Point (MPP), while the red dots correspond to the Local Maximum Power Points (LMPPs).

3. Metaheuristic Algorithms with Advanced Limited Search Strategy

This section explains the integration of the ALSS technique in metaheuristic algorithms (Figure 3 shows a skeleton of how the ALSS can be integrated into metaheuristic algorithms). The ALSS is applied after the metaheuristic initialization. Thus, we first illustrate the initialization of the metaheuristic algorithms. Then, we analyze the ALSS, and finally, the metaheuristic update procedure is explained.

3.1. Initialization of Metaheuristic Algorithms

The metaheuristic initialization is shown in the box of Figure 3 under the heading “Initialization”. Metaheuristic algorithms start by spreading solutions (duty cycles) over the search space to initiate the exploration for the voltage that corresponds to the maximum power value. After comparing the power values of the candidate solutions (duty cycles), the solution (duty cycle) with the highest power value is selected as the initial best solution. The initial best solution is then used to indicate where to search next to find the global power peak in the P-V curve (There is a one-to-one relationship between duty cycles and voltages. Thus, when we say we are looking for the best duty cycle (to adjust the DC–DC boost converter), it is the one that corresponds to a voltage providing the maximum power at the output). In order to obtain the best MPPT performance with the ALSS, the number of initial duty cycles has to be greater than or equal to the number of PV panels. Moreover, the initial duty cycles must be distributed uniformly between zero and one (we assume four PV panels, and thus, we initialize four duty cycles ($D_1$, $D_2$, $D_3$, and $D_4$) in the flow chart shown in Figure 3). Moreover, the metaheuristic initialization is shown in Figure 4 under the heading “Initialization” as a running example to make the explanation more illustrative. The initial best solution is colored green.

3.2. ALSS Integration in Metaheuristic Algorithms

The Advanced Limited Search Strategy (ALSS) is depicted in the box of Figure 3 under the heading “ALSS Update”. The ALSS proposed in [25] is a technique used to confine the search space within a small region of space to help an MPPT algorithm converge faster and more efficiently to the MPP. After identifying the initial best duty cycle in the initialization phase, the ALSS estimates the direction in the search space where the possibility of finding the duty cycle corresponding to the global power peak is high. In order to do this, the ALSS calculates the finite difference derivative between the initial best duty cycle and a slight perturbation ($\delta D$ = $1\times {10}^{-3}$) to higher duty cycles. The finite difference derivative indicates the direction of the optimum duty cycle. If the derivative is positive, the optimum duty cycle lies in the higher duty cycle region; else, it lies in the lower duty cycle region. For the former case, a small constant positive perturbation c = 0.05 is made in the initial best duty cycle, while in the latter case, a small constant negative perturbation c = −0.05 is made. Suppose the duty cycle after the positive (negative) perturbation c has higher power than the initial best one. In that case, the duty cycle is updated further by a slight positive (negative) amount $c{}^{″}$ = 0.025; else, the duty cycle is updated with a value between the initial best and the duty cycle after the small perturbation c we have made to the initial best. The duty ratios corresponding to the highest and the second-highest power values among the initial best, the perturbed, and the finally updated duty cycle are then chosen for the metaheuristic update. The ALSS integration with metaheuristic is shown in Figure 4 under the heading “ALSS update” as a running example to make the explanation more illustrative.

Figure 4. Running example of the ALSS applied to a metaheuristic algorithm. Consists of three phases: the initialization, the ALSS update, and the metaheuristic update.

Figure 4. Running example of the ALSS applied to a metaheuristic algorithm. Consists of three phases: the initialization, the ALSS update, and the metaheuristic update.

3.3. Metaheuristic Algorithm Update

This section is shown in the box of Figure 3 under the heading “Metaheuristic Update”. After the ALSS update, a metaheuristic algorithm can be used to further update the candidate duty cycles until convergence. The metaheuristic update starts by updating the duty cycle that corresponds to the highest power value using the duty cycle that corresponds to the second-highest power value in the metaheuristic update formula. The power corresponding to the updated duty cycle is evaluated and compared with the previous best value. After the comparison, the duty cycle corresponding to the higher power value becomes the best, while the one with the lower power value becomes the second-best duty cycle. A counter is then incremented by one, which controls the exploration rate. For instance, after the ALSS update, the best and second-best duty cycles sometimes correspond to very close power values, whose difference can be smaller than 1%. Suppose the difference between the best and the second power is less than 1% after the ALSS update. In that case, the algorithm would converge without further metaheuristic exploration, leading to less convergence accuracy. Therefore, the algorithm does not consider the 1% change until three iterations have passed. Once the counter has reached three iterations, the algorithm starts detecting if the difference between the best and the second best power values is less than 1%. If the difference is less than 1%, the algorithm converges and triggers the boost converter switch through the best duty cycle. Otherwise, it updates the duty cycle through the metaheuristic equation. Once the former case is true (algorithm converged), it keeps on detecting if the difference between the power during convergence and the power obtained at each iteration after triggering the best duty cycle is greater than 1%. If the difference is greater than 1%, it implies an insolation change and is reinitialized. However, before reinitializing, the algorithm detects if the runtime is over, in which case, the algorithm ends. Otherwise, it is reinitialized. The metaheuristic update is shown in Figure 4 under the heading “Metaheuristic update” as a running example to make the explanation more illustrative.

4. Results and Discussion

The improvement in terms of power loss and convergence is demonstrated in this section by applying the ALSS to three very recently proposed metaheuristic algorithms. The algorithms are namely Radial Movement Optimization hybrid with the Teaching–Learning-Based Optimization (RMOTLBO), the Salp Swarm (SS) algorithm, and the Jaya algorithm with the Levy Flight (JayaLF) algorithm. Their improved versions with the ALSS are namely RMOTLBO with ALSS (RTBALSS), SS with ALSS (SSALSS), and JayaLF with ALSS (JLFALSS). Finally, all the improved algorithms are compared with their previous versions to show the significant improvement in their performances.

4.1. Experimental Setup

The results were obtained on an 11th Gen Intel(R) Core(TM) i7-1165G7 @ 2.80 GHz, with 16 Gb RAM and a 64-bit Windows operating system. We used the typhoon hardware-in-the-loop (HIL) software to obtain the results. We considered a PV array with four modules connected in series. Each PV module has a rating as follows: open-circuit voltage ($V_{ov}$) = 5.425 V, short-circuit current ($I_{sc}$) = 5.34 A, MPP voltage ($V_{MPP}$) = 4.35 V, and MPP current ($I_{MPP}$) = 5.02 A, which are the ratings of the KC85T PV module. More specifically, we considered ¼ of the voltage rating of that module to design each PV module in our system. Moreover, the DC–DC boost converter ratings are: inductance (L) = 1.15 mH, input capacitance ($C_i$) = 47 $\mathsf{\mu }$F, output capacitance ($C_o$) = 470 $\mathsf{\mu }$F, and load resistance ($R_o$) = 10 $\Omega$. The results were evaluated for both static and dynamic PS conditions.

4.2. Static PS Conditions

We used three different PS patterns in this case, namely static PS SPS1, SPS2, and SPS3, summarized in

Table 1. Percentage insolation reduction on the four PV modules taking as a reference full insolation.

Shading Patterns	Module 1	Module 2	Module 3	Module 4
SPS1	0%	1%	11%	21%
SPS2	0%	30%	45%	75%
SPS3	0%	15%	65%	90%

. The three different PS patterns correspond to different positions of power peaks on the P-V curve, which validates the performance of the proposed algorithms for different power peak positions. The comparison was made based on the following performance metrics:

• Total number of power fluctuations generated by the algorithms during MPP tracking.
• Size of the power fluctuations.
• How fast an algorithm finally converges to the MPP once it has tracked the MPP the very first time.
• The convergence speed of the algorithms.
• The robustness of an algorithm against varying shading patterns.

All these advantages led to significant improvement in power losses and how robust an algorithm was against highly fluctuating PS conditions, thereby significantly improving the overall power generation efficiency and robustness of the MPPT controller.

4.2.1. SPS1

Figure 5 shows the results for SPS1 for all algorithms. The Maximum Power Point (MPP) position is on the left region of the P-V curve.

Figure 5a compares the results between JayaLF and its improved version through the ALSS (JLFALSS). The fluctuations were measured after the initialization phase during tracking. As can be observed, JayaLF exhibits several power fluctuations, some of which are large and, thus, increase power loss. Moreover, even after tracking the MPP, there are fluctuations between the tracked and the converged instant. Conversely, in JLFALSS, first, there are no fluctuations of a large size. Second, the total number of fluctuations is much less compared to JayaLF. Third, there is no fluctuation between the power tracked and power converged instant. Finally, the convergence speed of JLFALSS is much better compared to JayaLF. The convergence time and converged power of JayaLF and JLFALSS were, respectively: 1.06 s and 38.7 W; 0.36 s and 38.852 W.

Figure 5b compares the results between RMOTLBO and its improved version through the ALSS (RTBALSS). As can be seen, RMOTLBO exhibits several large and small power fluctuations. Furthermore, even after tracking the MPP, there are fluctuations between the tracked and the converged instant. On the other side, in RTBALSS, first, there are no fluctuations of a large size. Second, the total number of fluctuations is much less compared to RMOTLBO. Third, there is no fluctuation between the power tracked and power converged instant. Finally, the convergence speed of RTBALSS is much better compared to RMOTLBO. The convergence time and converged power of RMOTLBO and RTBALSS were 0.96 s, 38.26 W and 0.46 s, 38.67 W, respectively.

Figure 5c compares the results between Salp Swarm (SS) and its improved version through the ALSS (SSALSS). The SS algorithm exhibits more power fluctuations than JayaLF and RMOTLBO in size and number. Furthermore, even after tracking the MPP, there are fluctuations between the tracked and the converged instant. However, the application of the ALSS to SS has shown significant improvements. Again, the power fluctuations in terms of size and number reduced drastically. Moreover, there is no fluctuation between the power tracked and power converged instant. Finally, the convergence speed of SSALSS is much better compared to SS. The convergence time and converged power of SS and SSALSS were 2.66 s, 38.87 W and 0.56 s, 38.866 W, respectively.

4.2.2. SPS2

Figure 6 shows the results for SPS2 for all algorithms. The MPP position resides in the middle region of the P-V curve. All the markings to illustrate the metrics related to fluctuations are given in Figure 5. These metrics will remain common for the rest of the results and, thus, are not marked on other curves.

Figure 6a compares the results between JayaLF and JLFALSS. Similar to the above results, the JayaLF algorithm exhibits several power fluctuations. However, there are no large-sized power fluctuations for this scenario. The tracked and converged instant fluctuations, similar to SPS1, are still there. On the contrary, in the case of JLFALSS, again, there is no significant size power fluctuation, and the total fluctuations during tracking are very few. Moreover, the algorithm converges once the MPP is tracked without creating fluctuations between the tracked and the converged instant. Finally, the convergence speed of JLFALSS is much better than JayaLF. The convergence time and converged power of JayaLF and JLFALSS were 1 s, 39.71 W and 0.46 s, 39.43 W, respectively.

Figure 6b compares the results between RMOTLBO and RTBALSS. Similar to the previous scenario, the RMOTLBO algorithm exhibits several power fluctuations. However, there is only one fluctuation between the tracked and the converged instants. Conversely, in the case of RTBALSS, again, the total fluctuations are very few with a tiny size. There is only one fluctuation between tracked and converged MPP instants. Finally, the convergence speed of RTBALSS is much better compared to RMOTLBO. The convergence time and converged power of RMOTLBO and RTBALSS were 0.96 s, 39.73 W and 0.43 s, 39.78 W, respectively.

Figure 6c compares the results between SS and SSALSS. The SS algorithm exhibits several power fluctuations of a small and a large size. The fluctuations between the tracked and the converged instants, similar to SPS1, are still there. Furthermore, there are several power fluctuations between the tracked and the converged instants of power. On the other hand, in SSALSS, again, there is no large-sized power fluctuation, and the total fluctuations during tracking are negligibly small. Moreover, the algorithm develops very few fluctuations between tracked and converged instants of very small sizes. Finally, the convergence speed of SSALSS is much better compared to SS. The convergence time and converged power of SS and SSALSS were 2.66 s, 39.78 W and 0.66 s, 38.82 W, respectively.

4.2.3. SPS3

Figure 7 shows the results for SPS3 for all algorithms. The MPP position is on the right region of the P-V curve.

Figure 7a compares the results between JayaLF and JLFALSS. For this case, the number of small- and large-sized fluctuations is lower compared to the previous cases, but still needs improvement. Furthermore, like in the previous scenarios, there are fluctuations between power tracked and power converged instants. At the same time, the performance of the JLFALSS remains similar. The number of power fluctuations remains very few, with a tiny size between power tracked and power converged instants. Finally, the convergence speed of JLFALSS is much faster than JayaLF. The convergence time and converged power of JayaLF and JLFALSS were 0.76 s, 74.71 W and 0.456 s, 74.74 W, respectively.

Figure 7b compares the results between RMOTLBO and RTBALSS. Similar to the previous two cases, there are several power fluctuations in RMOTLBO, and some are rather large. The fluctuations between tracked and converged instants for this case are also low. In RTBALSS, again, the number and size of power fluctuations are meager. The fluctuations between the tracked and the converged instants are also low and similar to RMOTLBO. Finally, the convergence speed of RTBALSS is much faster than RMOTLBO. The convergence time and converged power of RMOTLBO and RTBALSS are 0.96 s, 74.08 W and 0.46 s, 74.8 W, respectively.

Figure 7c compares the results between SS and SSALSS. Like the previous scenarios, the SS algorithm shows inferior performance in several large and small power fluctuations and several fluctuations between tracked and converged power instant. While similar to the previous scenarios, the SSALSS algorithm significantly outperforms SS in terms of all power fluctuation metrics. Moreover, the results above reflect another advantage of the ALSS: robustness. The convergence time and converged power of SS and SSALSS were 2.65 s, 74.72 W and 0.46 s, 74.74 W, respectively. Results for all the static conditions are summarized in

Table 2. The static PS scenario performance summary.

Algorithms	Shading Patterns
	SPS1			SPS2			SPS3
	Power (W)	True MPP (W)	Time (s)	Power (W)	True MPP (W)	Time (s)	Power (W)	True MPP (W)	Time (s)
JayaLF	38.7	38.87	1.06	39.71	39.78	1	74.71	74.86	0.76
JLFALSS	38.852		0.36	39.43		0.46	74.74		0.456
RMOTLBO	38.26		0.96	39.73		0.96	74.08		0.96
RTBALSS	38.67		0.46	39.78		0.43	74.78		0.46
SS	38.87		2.66	39.78		2.66	74.72		2.65
SSALSS	38.866		0.56	38.82		0.66	74.74		0.46

.

In summary, the application of the ALSS to the three metaheuristic algorithms provided significant enhancements in power loss and convergence time. Moreover, the results above reflect another advantage of the ALSS, which is robustness. It is clear from the results above for all three PS scenarios that the improved algorithms do not vary in their performances for any PS condition and give a similar performance in terms of power loss and convergence. This shows their robustness against varying PS conditions, unlike other algorithms, whose performance varies with different PS scenarios.

4.3. Dynamic PS conditions

The dynamic PS condition results resemble a more real-world scenario where the PS patterns on the PV surface keep on changing with time. Two types of dynamic PS conditions were used in this work for the performance evaluation of the proposed algorithms.

4.3.1. Randomly Varying Dynamic Insolation

For this case, the insolation was varied considering the varying Sun positions that will cause different shading patterns on the PV surface due to a specific obstacle.

The PS summary for the randomly varying dynamic case is given in

Table 3. Percentage insolation reduction taking as a reference the full insolation on the four PV modules for every timeslot of the “randomly-varying insolation” dynamic PS scenario.

Shading Instant	Percentage Insolation Reduction on Each Module with Respect to a Full Insolation of 1000 W/m${}^2$
	Module 1	Module 2	Module 3	Module 4
1 (0–3 s)	0%	7%	22%	37%
2 (3–6 s)	0%	37%	45%	80%
3 (6–9 s)	0%	11%	40%	50%
4 (9–12 s)	0%	19%	70%	90%

for different time instants. Figure 8 compares the results for the randomly varying dynamic insolation case.

Figure 8a compares the results between JayaLF and JLFALSS. For all four instants of the dynamic PS condition, the JayaLF algorithm exhibits the above-explained deficiencies in several small- and large-sized power fluctuations and the fluctuations between power tracked and power converged instants. Again, the JLFALSS algorithm significantly outperforms the JayaLF algorithm in terms of all the performance metrics related to power loss for all four instants. Furthermore, the convergence speed of JLFALSS is much better compared to JayaLF for all the dynamic instants.

Figure 8b compares the results between SS and SSALSS. Again, it is apparent for all four instants of dynamic shading that SSALSS significantly outperforms the SS algorithm in terms of the power loss performance metrics and convergence speed.

Figure 8c compares the results between RMOTLBO and RTBALSS. Similar to the above cases, the improved version through the ALSS significantly outperforms RMOTLBO in terms of the power loss performance metrics and the convergence time.

The summary of the power and convergence time for all the instants is given in

Table 4. Performance summary for the “randomly-varying insolation” dynamic scenario.

Algorithms	Shading Instants
	1 (0–3 s)			2 (3–6 s)			3 (6–9 s)			4 (9–12 s)
	Power (W)	True MPP (W)	Time (s)	Power (W)	True MPP (W)	Time (s)	Power (W)	True MPP (W)	Time (s)	Power (W)	True MPP (W)	Time (s)
JayaLF	70.36	70.74	0.6	39.26	39.36	0.6	62.1	62.44	0.61	37.32	37.32	1.6
JLFALSS	70.74		0.4	39.36		0.4	62.3		0.45	37.2		0.46
RMOTLBO	69.647		0.92	39.19		1.16	61.99		0.71	37.23		1.15
RTBALSS	70.62		0.46	39.36		0.46	62.44		0.47	37.32		0.46
SS	70.67		2.66	39.34		2.66	62.44		2.65	37.31		2.66
SSALSS	70.62		0.46	39.36		0.46	62.26		0.46	37.2		0.5

.

4.3.2. Increasing Dynamic Insolation

This case resembles the shading caused by moving clouds. The cloud initially covers the PV surface, decreasing the insolation value significantly and, thus, the PV output power. After some time, with cloud movement, the cloud shades gradually uncover the PV surface until the PV starts receiving full insolation (no shading).

The PS summary for the increasing dynamic insolation case is given in

Table 5. Percentage insolation reduction taking as a reference the full insolation on the four PV modules for every timeslot of the “increasing insolation” dynamic PS scenario.

Shading Instant	Percentage Insolation Reduction on Each Module with Respect to a Full Insolation of 1000 W/m${}^2$
	Module 1	Module 2	Module 3	Module 4
1 (0–3 s)	0	48%	79%	91%
2 (3–6 s)	0	33%	52%	69%
3 (6–9 s)	0	21%	32%	60%
4 (9–12 s)	0	0%	15%	35%
5 (12–15 s)	0	0%	0%	0%

for different time instants. Figure 9 compares the results for the increasing dynamic insolation case shading conditions.

Figure 9a compares the results between JayaLF and JLFALSS. For this dynamic case, JayaLF exhibits problems related to multiple large- and small-sized power fluctuations. Moreover, there are power fluctuations between the tracked and the converged instants. The JLFALSS algorithm again significantly outperforms the JayaLF algorithm in terms of all the performance metrics related to power loss for all five time instants. Finally, the convergence speed of JLFALSS is much faster than JayaLF for all the dynamic instants.

Figure 9b compares the results between SS and SSALSS. As previously, SSALSS significantly outperforms the SS algorithm in the performance metrics related to power fluctuations and convergence speed.

Figure 9c compares the results between RMOTLBO and RTBALSS. Similar to the above cases, the improved version through the ALSS significantly outperforms RMOTLBO in terms of power loss performance metrics and convergence time.

The power and convergence time summary for all instants is summarized in

Table 6. Performance summary of the “increasing-insolation” dynamic PS scenario.

Algorithms	Shading Instants
	1 (0–3 s)			2 (3–6 s)			3 (6–9 s)			4 (9–12 s)			5 (12–15 s)
	Power (W)	True MPP (W)	Time (s)	Power (W)	True MPP (W)	Time (s)	Power (W)	True MPP (W)	Time (s)	Power (W)	True MPP (W)	Time (s)	Power (W)	True MPP (W)	Time (s)
JayaLF	24.624	24.8	0.96	35.058	35.059	0.61	48.32	48.32	0.91	63.415	63.56	0.61	87.08	87.2	0.9
JLFALSS	24.704		0.4	35.04		0.65	48.32		0.46	63.54		0.46	87.02		0.4
RMOTLBO	24.69		1.1	35.05		1.15	47.855		0.98	62.2		1.17	87.01		0.81
RTBALSS	24.8		0.4	35.04		0.46	48.32		0.47	63.56		0.466	87		0.41
SS	24.8		2.46	34.79		1.47	48.32		2.66	63.47		2.67	87.08		2.75
SSALSS	24.59		0.66	35.04		0.41	48.32		0.45	63.54		0.46	86.88		0.56

. To summarize, the performance of all the improved algorithms was validated for the two dynamically varying real-world scenarios. Again, it is clear that the improved algorithms provide similar performance in power loss and convergence time for all the instants of dynamic PS. This proves the efficacy and robustness of the proposed algorithms against dynamically varying PS conditions. Furthermore, this proves how efficiently and reliably the proposed algorithms can work when implemented for real-world PV applications to serve humanity. Furthermore, the proof of the efficacy and solution reduction capability of the ALSS on metaheuristic algorithms will encourage MPPT researchers to apply the ALSS to their algorithms, which will result in further enhancements and improvements in MPPT applications.

Moreover, in addition to comparing with JayaLF, RMOTLBO, and SS, a qualitative comparison of the proposed JLFALSS, RTBALSS, and SSALSS was made with several other algorithms in the literature in addition, given in

Table 7. Qualitative comparison between the proposed and other algorithms in the literature.

Metrics	Algorithms
	JayaLF [14]	JLFALSS	RMOTLBO [8]	RTBALSS	SS [24]	SSALSS	AJaya [15]	MBOA [13]	PSODV [12]	APSO [10]
Large power fluctuations	Present	Not present	Present	Not present	Present	Not present	Present	Present	Present	Present
Convergence speed	Fast	Very fast	Moderate	Very fast	Slow	Very fast	Moderate	Fast	Very slow	Moderate
Total number of power fluctuations	Moderate	Very few	Moderate	Very few	Many	Very few	Moderate	Few	Too many	Moderate
Convergence after tracking the MPP	Moderate	Very fast	Moderate	Very fast	Slow	Very fast	Moderate	Moderate	Very slow	Moderate

, to show their comparative efficacy with other techniques. The comparison was made based on several performance metrics such as the presence of large-sized power fluctuations, convergence speed, the total number of power fluctuations, and the speed of convergence after the MPP is tracked. It is clear from Table 7 that the improved algorithms, apart from improving the performance of JayaLF, RMOTLBO, and SS, outperform other recent algorithms in the literature.

5. Conclusions

In this work, the Advanced Limited Search Strategy (ALSS) was applied to existing metaheuristic algorithms to improve their performance. We showed that the ALSS technique can be integrated directly and reduce the candidate solutions required in metaheuristic algorithms. Three recently proposed algorithms, namely JayaLF, RMOTLBO, and SS, were chosen to show their performance improvement using ALSS. The results demonstrated a remarkable improvement in power loss, convergence speed, and robustness for both static and dynamic partial shading scenarios. The average percentage improvement in convergence speed of the examined MPPT algorithms updated with ALSS over their initial versions for the static partial shading scenarios was 302.6% and for the randomly varying and increasing dynamic insolation scenarios was 326% and 290%, respectively. Moreover, in addition to improving the performance of JayaLF, RMOTLBO, and SS, the proposed method outperformed other algorithms in the literature when compared qualitatively. The improved algorithms thus contribute toward a more power-efficient and reliable MPPT controller for residential and commercial applications.