1. Introduction
The development of renewable energy resources have been drastically increases due to the effect of greenhouse gases, depletion of fossil fuels and high demand of electricity [
1]. Solar photovoltaic (PV) generation has excellent potential due to the absence of fuel cost and limited maintenance compared to other energy resources [
2]. Based on the World Energy Outlook 2018 reports, the PV power generation will become increases significantly by 2040 and exhibits a higher global generation capacity than all other forms of energy [
3]. The PV generated power greatly depends on the environmental conditions like irradiation and temperature as well as the partial shading (PS) phenomena. These factors drastically reduces the PV generated power [
4,
5]. Researchers focused to develop maximum power point tracking (MPPT) techniques [
6] to reduce the influence of these factors and to improve the PV system efficiency. The non-linear characteristics of solar PV, and partial shading condition causes multiple peaks on the P-V characteristics. However, to reduce these multi-peaks into a global peak is a challenging task [
7].
With this inspiration, researchers and practitioners developed numerous MPPT techniques which can be categorized into two types based on their implementation. Those are traditional MPPT techniques and soft computing techniques. The traditional MPPT methods like perturbation and observation (P&O) [
8], hill climbing (HC) [
9] and incremental conductance (IncCond) [
10] are widely used due to simplicity in design, easy implementation, replicates less number of sensors. Despite of these features, P&O method suffers from a high rate of perturbations and generates more oscillations around maximum power point (MPP). The IncCond method exhibits weak convergence, and the HC method is suitable only for low power applications. Moreover, these three methods are failed to achieve global peak under rapid irradiation changes [
1].
The evolutionary techniques were widely preferred and developed as they overcome the drawbacks of traditional methods like fixed step size. In addition, few other advantages like, capability of handling non-linear and multi-modal objective functions, extensive exploration in search space [
11] increases the development of evolutionary techniques. Thereby, numerous optimization algorithms have been developed by various researchers. Notable algorithms proposed in the recent years are, moth-flame optimization algorithm (MFO) [
12], ant colony optimizer [
13], bat algorithm [
14], flower pollination algorithm (FPA) [
15,
16], non-linear backstepping control technique [
17], mine blast optimization (MBO) and teaching learning-based optimization (TLBO) algorithms [
18,
19], golden section search based algorithm [
20], bypass diode scanning algorithm [
21] and wind-driven optimization (WDO) algorithm [
22]. The modified versions of stochastic algorithms are introduced such as modified cat swarm optimization (MCSO) [
7], improved particle swarm optimization (IPSO) [
23], modified PSO (MPSO) method is presented in [
24], and a new version of PSO named leader PSO (LPSO) is proposed in [
25]. Additionally, other MPPT methods are proposed to alleviate a random number in conventional cuckoo search (CS) algorithm [
26] whereas Distributed MPPT technique is proposed in [
27].
Another new trend named hybrid techniques were introduced to improve the performance of classical algorithms. In this, the properties of two or more meta-heuristic optimization algorithms are combined or hybridized between the conventional methods with the meta-heuristic optimization techniques. A simplified accelerated particle swarm optimization (SAPSO) was proposed in [
28] by combining a variant of the particle swarm optimization (PSO) algorithm and the classical HC algorithm. Hybrid gaussian process regression-jaya (GPR-Jaya) algorithm in [
29] improved the performance of jaya algorithm by introducing the GPR model. Other hybrid grey wolf optimization with fuzzy logic control (GWO-FLC) method is proposed to solve the problem of Local maximum power point (LMPP) and oscillations around Maximum power point (MPP). Another advanced searching technique named PSO-based MPPT algorithm TSPSOEM is proposed [
30] by utilizing properties of particle swarm optimization (PSO) and shuffled frog leaping algorithm (SFLA). The other hybrid method proposed by incorporating properties of FLC and three-point weight method in [
31].
Another novel method using artificial vision to track MPP is proposed by authors in [
32]. In this method, a webcam is used to recognize the shadow irradiation progressively and give the reference voltage that supplies the maximum power irrespective of the number of peaks on the P-V curve. This method requires high efficient webcams, which increases the cost.
From the presented literature, it is understood that, there exist some limitations in the methods, that are listed as follows:
Conventional MPPT techniques may settle at any one of LMPP. Adaptive step-size methods take a longer time to reach MPP. In addition, these methods require complex calculations to estimate step size and exhibits slow convergence. Further, these methods are more efficient, only in-case of uniform irradiation conditions [
33].
For the evolutionary algorithm (EA) based MPPT techniques, the commonly noticeable drawbacks are the trade-off between exploration-exploitation which is very less, which results in fluctuations during the process of optimization. These methods may fall in LMPP during wider (strong) shading conditions. In addition, these methods fail to reach new GMPP once they change their position since the search particles will be busy around previous MPP and lack of consistency [
34]. Therefore, due to these limitations, the efficiency of the system will get reduce. Additionally, that open-up a room to improve the performance of MPPT techniques further.
The various research gaps observed from the literature review are summarized in
Table 1.
Based on the previous limitations and gap analysis, it is noted that none of the researchers have made an attempt to involve the chaotic features into MPPT, which extensively increases the performance of non-linear objectives. The chaos theory to improve meta-heuristic optimization algorithms has become the dominant topic in the field of optimization algorithms. The dynamical and randomization properties of chaos maps help the optimization algorithms to balance between the intensification (exploitation) and diversification (exploration) phases [
35,
36].
With this motivation, authors introduced new MPPT technique titled novel fractional chaotic flower pollination algorithm (FC-FPA) with a combination of fractional order chaos maps.
The dynamic properties of fractional order chaos maps enhances the perofrmance of meta-heuristic algorithms. This has not been implemented by any of the researcher so far. Therefore, in this article, authors proposed a new and unique technique of introducing chaotic variants to achieve the GMPP irrespective of environmental conditions, type of PV module, uniform and dynamic change in irradiation conditions. In the proposed FC-FPA technique, three fractional chaos maps are co-operated with the FPA algorithm for the initialization phase and for tuning its parameters. The considered fractional chaos maps are fractional logistic map, fractional sine map, and fractional tent chaos map.
The main contributions of the paper are:
In this article authors introduced a unique novel method of chaotic variants (based on fractional-order chaos maps) to track maximum power point.
The proposed method is tested with two different types of models under two configurations like Multi-crystalline type (S36) with four series combination (4S) and Mono-crystalline type (SM55) PV model with four-series-two-parallel (4S2P) configuration.
The effectiveness of the proposed method is validated using 3 different shade pattern with above mentioned configurations.
The robustness of the algorithm variants are evaluated in tracking the GMPP during a sudden change in the irradiation conditions.
The proposed variants are compared with the basic version of the FPA algorithm over all the stages of analysis.
Extensive statistical analysis has been performed to demonstrate the superiority of the proposed variants and recommended the best chaos map that helps the FPA in achieving fast-tracking for GMPP with most consistent and accurate behavior.
The recommended MPPT algorithm variant is compared with the traditional perturb and observe MPPT (P&O) over the nonuniform distribution of radiation and step change in its levels.
The organization of the manuscript is as follows;
Section 2 discusses the photovoltaic modeling, DC-DC boost converter, and the effect of the PSC over the considered PV system.
Section 3 contains the proposed optimization algorithm variants, and implementation steps for the application of MPPT.
Section 4 discusses the results and analysis.
Section 5 exhibits the comparison between the recommended MPPT variant and P&O technique. Finally,
Section 6 summarizes the main outcomes and the conclusions.
4. Simulation and Results
To highlight the superiority of the proposed chaotic variants, the simulation results are compared with FPA. The simulations are carried-out for uniform and partial shade conditions using 4S and 4S2P configurations. A set of 5 shade patterns were tested and analyzed its performance. Several performance parameters like tracking speed, convergence time and efficiency are considered for evaluation. In addition, statistical analysis is performed to test the superiority and reliability of the introduced variants. Each algorithm is simulated for 10 independent runs to perform statistical analysis. The formulae used for performing the statistical analysis are presented below:
where,
are the estimated for each run
, the average power over number of runs and measured global PV power.
K is the total number of the runs.
The schematic diagram of the setup shown in
Figure 5. The considered PV configurations are connected to the DC-DC boost converter with input inductance of 30 mH, output capacitor value 100
F, connected to resistive load of 100
and operating at switching frequency 10 kHz. The experimentation’s were performed using a laptop of Core i7-6500U CPU with
GHz of speed and 4 GB of RAM and “MATLAB 2018” environment.
The experimentation’s carried-out under uniform and partial shading conditions are presented in the
Section 4.1. In this subsection authors presented results obtained for both 4S and 4S2P configurations. To verify the performance of the system even under dynamic change in irradiation using different patterns are performed and are discussed in the
Section 4.2.
4.1. Simulation Validation of Proposed Chaotic Variants in Comparison with FPA under Partial Shading Conditions
The performance of proposed system is analyzed under various parameters, the
Section 4.1.1 deals the analysis based on tacking speed, computational time and accuracy. While the
Section 4.1.2 is investigated based on the performed statistical analysis.
4.1.1. Tracking Speed, Time and Accuracy Factors
After performing the rigorous simulations over 4S configuration, the obtained mean convergence curves for the PV power, voltage, current and duty cycle are presented in
Table 4. These plotted curves were obtained by performing number of independent runs.
For the 1st string of 4S connected S36 modules: three shade patterns are considered with the 1st configuration. The patterns 1, 2, and 3 indicate uniform, medium and heavy shade conditions respectively.
Pattern 1: a single global peak (GMPP) exists in the characteristics of the PV string at
W, it can be notice from
Figure 3c of pattern 1. After performing the simulations with introduced variants the obtained convergence curves for Pattern 1 are presented in 1st row of
Table 4. From the listed figures, it can be observe that, FPA tracks power of 145.336 W in 3.999 s including high oscillations around MPP and also there exist wide range of switching particles. FC-FPA variants (fractional logistic, fractional sine and fractional tent maps) able to track 148.513 W, 148.255 W and 148.514 W in a duration of 2.432 s, 1.228 s and 0.992 s respectively. From the figures it can be visualized that, the FC-FPA variants exhibits less number of oscillations around MPP and converges to maximum power than FPA in a less time period. The difference in power levels represents the poor exploitation capability of FPA. FPA-fractional tent map shows better performance, it converges in 0.992 s which is saving 70% of tracking time and shows more stability than FPA.
Pattern 2: Based on the receiving irradiation, pattern 2, generates two peaks over PV characteristics, as shown in
Figure 3c. The two peaks occurred at power value of 69.742 W (GMPP), 43.978 W (LMPP) and are highlighting with points P2 and P3 respectively. After performing the simulations, the obtained convergence curves for Pattern 2 are presented in 2nd row of
Table 4. FPA offers power value equaled to 69.623 W at 2.333 s with high amount of oscillations. FC-FPA variants (fractional logistic, fractional sine and fractional tent maps) tracks 69.71 W, 69.643 W, and 69.71 W in 1.433 s, 1.688 s and 1.353 s respectively. In this condition, it can be observe that, there exist negligible power difference between FPA and FC-FPA variants, however the FC-FPA variants converges in less time and than FPA. Fractional tent map converges in a very less time 1.353 s with minimal oscillations around MPP.
Pattern 3 is considered as the strong shade condition as it receives nonhomogeneous irradiation levels. Sequentially four peaks are produced in the P-V characteristic, as shown in
Figure 3. The GMPP located at 57.616 W and other LMPPs are located at 51.005 W, 46.415 W and 30.301 W, respectively. In spite of the rigorous test of pattern 3, the obtained plots are presented in last row of
Table 4. From the plotted curves, it is observed that FC-FPA variants prove their robustness as they produce higher values of power in shorter tracking time. FPA tracks mean power 56.121 W at 3.33 s with high oscillation around MPP. The FC-FPA variants with fractional maps track 57.389 W, 57.424 W, and 57.467 W in 1.330 s, 1.681 s and 1.083 s, respectively. Therefore cooperating the fractional chaos maps with the basic version of FPA enhances in providing more power even under high shade conditions with reduced tracking time nearly for 50% from consumed by standard FPA and with zero fluctuation around MPP.
For the 2nd configuration of 4S2P connected SM55 module: Similar to the previous case, the mean convergence curves for the PV power, voltage, current and duty cycle for the Pattern 4 and pattern 5 are presented in
Table 5. Pattern 4 and pattern 5 are derived for the configuration of 4S2P to test proposed method under high rated power capacity.
Pattern 4: Due to the presence of different shades over Pattern 4, there exist three peaks and are shown in
Figure 4c. Three power peaks are produced with magnitude of 212.718 W, 178.648 W, and 143.336 W. The GMPP is located at left side of PV curve with power values of 212.718 W. The obtained convergence curves for Pattern 4 are presented in 1st row of
Table 5. From the presented figures it can be noticed that, FC-FPA variants shown their success in tracking the power which is closer to the GMPP in shorter period of time and achieves high stability compared with FPA. FC-FPA with fractional logistic map settles at values of 212.350 W, in 0.820 s. In the case of fractional sine map, the tracked power is 212.265 W at 1.358 s. The extracted power in case of fractional tent map is 212.452 W at 1.202 s. Meanwhile, FPA generates 205.254 W at 2.718 s with the presence of oscillations around MPP. Based on the obtained results, by using FC-FPA variants, able to track 3.4% power higher and also reduces 50% of tracking time than FPA. The attained 3.4% higher power reflects great significant in achieving more power in high rated PV systems which generates more income in less period of time.
Pattern 5: The pattern 5 generates, four peaks in P-V curve can observe the same from
Figure 4c. The power values of four peaks are 94.065 W, 197.172 W, 213.468 W and 143.867 W. The third peak
is the GMPP while the others are considered as LMPP. The last row of
Table 5 shows the convergence curves plotted for the successful execution of pattern 5 using FPA and proposed FC-FPA variants. The three FC-FPA variants enhance their superiority as they track 213.156 W, 213.134 W and 213.089 W in 1.111 s, 1.562 s and 1.084 s, respectively. However, FPA generates power of 203.492 W in 1.931 s. The power generated by FPA is very less than FC-FPA variants and consumes more time. There exit high amount of oscillations due to wide range of switching particles.
From the presented discussions it can be observed that, the FC-FPA variants exhibits extensive features like, they are not characterized by high initial oscillations and also confirms the tracking maximum power in a shorter period of time irrespective of shade conditions. The oscillations and switching of particles before converging shows high impact on switching devices. The wide range of switching oscillations generates thermal stress over switching devices. which results, failure of switching devices in regular intervals. Introducing the properties of chaotic variants to FPA, it reduces the oscillations of initial switching of particles. Thereby, it reduces thermal stress over the switch, which results improving the life time of switching devices. In addition, FC-FPA variants, shows high stability in searching for the global duty cycle. From the considered 5 patterns, it confirms that, fractional logistic and fractional tent maps show superior performance than other methods. Consequently, utilizing the FC-FPA variants based on the tracker system helps in saving wasted power and cost.
4.1.2. Statistical Analysis
The statistical analysis is the one, which indicates the performance of any system qualitatively. Therefore, authors considered it as important factor to evaluate the performance proposed variants compare with FPA. By performing the calculations using Equations (
13)–(
16), the obtained values are presented in
Table 6 for the 5 shade patterns. The discussion on statistical analysis carried-out using 4S and 4S2P configurations are presented below:
By the end of this subsection, it confirms that integrating the fractional chaotic maps with the FPA enhances the reliability of the basic technique under all shade conditions. FC-FPA variants exhibits higher power than FPA with shorter tracking time by 50% of that FPA, especially with the fractional logistic and fractional tent maps. Further, introduced variants helps to get rid of higher switching stress over switching devices. Further, the chaotic variants significantly improved system performance and helps to maintain its randomness for any sort of irradiation conditions.
4.2. Validation of Proposed Method under Dynamic Change in Irradiation’s
It is a fact that, environmental conditions will not be uniform for long time and it will change often. Therefore, it is mandatory to validate the proposed method, under dynamic change in irradiation’s. In order to test the dynamic ability of introduced variants under shade conditions, various simulations have been performed and are discussed as follows:
4.2.1. 4S Configuration Designed with S36 Modules
In order to test dynamic ability of the proposed method, simulations have been programmed in such a way that, the irradiation distribution changes from pattern 1 to pattern 2 and finally to pattern 3. First, the distribution on the PV array change from pattern 1 to pattern 2 at 2.7 s, while the second step change in the irradiation level occurred at 5.4 s to pattern 3. The step change in irradiation and power convergence curves including duty cycles changes can be observed from the figures presented in
Figure 6.
From the figures, it is seen that the FPA method takes longer time to reach MPP and including with high oscillation around MPP. The same behavior is continued during change in patterns. However, the introduced chaotic variants achieved MPP in shorter period of time more importantly with negligible oscillations. In addition, the features of chaotic variants helps the proposed technique to identify the global MPP in a less time. This confirms the strong exploitation ability of the fractional chaotic variants especially in cases of fractional logistics and fractional tent. Moreover, that validates the suitability of proposed variants for the application of MPPT even under high dynamic change in irradiation conditions.
4.2.2. 4S2P Configuration Design with SM55 Modules
A similar dynamic test is performed with 4S2P configuration to show the robustness, reliability and efficiency of the proposed variants. In this configuration, the system is operated with pattern 4 until 4 s, and then shifted to pattern 5. The obtained convergence curves, which reflects the behavior of proposed method including with FPA is presented in
Figure 7. The presented results authenticates the converging the system into a global power. But the, FPA exhibits high amount of oscillations, converges to GMPP for pattern 4 at 2.294 s, after shifted to pattern 5, it failed to reach global and produces continuous oscillations. After occurrence of dynamic change, FPA is able to tack only 178.032 W. FC-FPA variants shows superior performance tracks global power without any oscillations around MPP. After performing dynamic change, FC-FPA variants is able to track 213.286 W, which is much higher than FPA. However, FC-FPA variants, able to track global power for the both pattern 4 and pattern 5. MPPT with fractional chaotic tent map shows faster convergence than other methods then fractional chaotic logistics map. The efficiency of the FC-FPA MPPT methods achieves 99.8% in shorter time meanwhile that of FPA is not exceeded by 98%. With the performed analysis, once again it proves the potential in exploration and exploitation behavior of proposed variants. From the presented results it is noteworthy to mention that, the proposed fractional chaotic variants performs superior performance in terms of GMPP and faster convergence and computational time.
5. Comparison with Traditional Perturb and Observe Technique (P&O)
In this section, the recommended fractional chaotic tent map combined with FPA based MPPT technique compared with the well known traditional Perturb and Observe algorithm (P&O). The considered flowchart of P&O is illustrate at
Figure 8.
The comparison is carried out on two aspects, the former one exhibits the results of the algorithm over pattern 5 of 4S2P array reconfiguration. The latter one, however, shows the response of the algorithms during the dynamic change for the irradiance levels from pattern 1 to 2 then to 3 at time samples 2.7 s and 5.4 s from the simulation time as described in
Section 4.2.1. The settings of P&O method are the initial duty cycle value (
d =
) and the step change in the duty cycle (
d =
). The duty cycle selected such that to simplify the task on P&O especially in the dynamical change in the irradinace levels. It is noticed that the global peak exists at duty cycle near to
from the previous subsections. Therefore, P&O starts the perturbation from this value of duty cycle (
d =
).
The response of the implemented P&O and F-tent map over pattern 5 of shading is illustrated in
Figure 9. By inspecting the figure it can be seen that, the P&O response reaches the higher value of the power compared to that obtaining by FC-FPA mppt in case of F-tent after longer time of fluctuation.
For the dynamical changes in the irradiance levels from patterns 1 to 2 and finally to 3, the dynamical performances by the two considered techniques are shown in
Figure 10. The figure indicates the efficiency of the proposed mppt technique with F-tent in tracking the rapid changes in the irradiation levels over the considered modules of the PV array. P&O shows high fluctuation and trapping in local peak. Accordingly, the authors endorse the FC-FPA mppt in case of using the F-tent to overcome the limitations and drawbacks of P&O with rapid change in the shading situations.