An Adaptive Maximum Power Point Tracker for Photovoltaic Arrays Using an Improved Soft Computing Algorithm

Abstract

This paper presents an improved version of the firefly algorithm (FA) by which a maximum power point (MPP) tracker was developed to track down the global maximum power point (GMPP) of a partially shaded photovoltaic module array (PVMA). As the first step, our team developed a high-voltage step-up converter where a coupled inductor was used to store the energy so that the duty cycle can be reduced so as to raise the voltage gain. The single-peaked P-V output characteristic curve of a PV array turns out to contain multiple peaks when the array is partially shaded. As a consequence, conventional MPP trackers occasionally track down a local maximum power point (LMPP), instead of the desired GMPP, and the output power of the array falls accordingly. Therefore, an improved version of the FA is proposed as a way to ensure that the GMPP can be tracked down in a more efficient way. Using the Matlab software, the MPP tracking performance of the proposed tracker was finally simulated in five scenarios. As it turned out, the proposed converter provided a high voltage gain at a relatively low duty cycle, and the improved version of the FA outperformed the original in terms of tracking time.

1. Introduction

Solar power has been acknowledged as one of the most promising renewable energy sources, particularly in places with long hours of sunshine [1,2]. A maximum power point tracker is employed to increase the power generation efficiency of a solar energy system. The perturbation and observation (P&O) [3,4,5,6,7,8] algorithm and the incremental conductance (INC) algorithm [9,10,11] are two of the commonest algorithms to track the MPP on a P-V curve of a PV array. However, these two algorithms occasionally do not track down the GMPP of a shaded PV array as expected. Instead, they get trapped in an LMPP.

Over the years, there has been a multitude of publications on tracking the MPP of a shaded PV array in the literature. Here are a number of frequently used algorithms on this issue: fuzzy logic control algorithms [12,13,14], neural networks (NN) [15,16,17], the grey wolf optimization (GWO) algorithm [18,19,20], the differential evolution (DE) algorithm [21,22,23], the artificial bee colony (ABC) algorithm [24,25,26], the firefly algorithm (FA) [27,28,29,30], the variable step size perturbation and observation (P&O) [31] and the shuffled frog-leaping (SFL) algorithm [32]. In the fuzzy control algorithms [12,13,14], input data are converted into fuzzy data, a membership function is used to map the fuzzy data into member attributes. Then, the attributes are expressed as a sequence of if–then rules for subsequent fuzzy operations. Finally, the fuzzy data are converted back to the non-fuzzy data as output. Consequently, fuzzy algorithms require a large amount of time to perform complex operations, that is, a high time complexity. NNs such as those in [15,16,17] mimic the way that biological neurons signal to one another and have been used to simplify information processing in complicated nonlinear models and particularly well applied to artificial intelligence (AI) and deep learning recently. However, a pyrometer and an exact fill factor [33] are required when an NN is applied to deal with the MPP tracking issue. As a consequence, the hardware cost is raised, and the MPP tracking accuracy remains an issue as well. In [18,19,20], the GWO algorithm mimics the social hierarchy and the hunting mechanism of grey wolves in nature. In a pack of grey wolves, there are three hierarchies responsible for seeking, encircling and finally attacking prey, respectively, in order to optimize the hunting performance. Although the GWO algorithm is simple and requires a smaller number of parameters to run global optimization, its major disadvantages include poor tracking accuracy, slow convergence and particularly the likelihood of getting stuck in a local solution. The DE algorithm in [21,22,23] is essentially a heuristic model and works in a similar way to the genetic algorithm (GA). It performs real number coding on a specific population. The global optimum is sought by evaluating the variance and using the one-to-one competitive strategy for survival. However, a major disadvantage is that it requires more operations and consequently takes a longer period of time to perform a global search when there is a genetic mutation in an individual. However, inappropriate parameter setting frequently leads to over-speedy convergence. As a result, the DE algorithm gets trapped in a local maximum and fails to reach the global maximum. The ABC algorithm [24,25,26] is a global optimization algorithm that mimics the intelligent foraging behavior of honeybees in nature. Employed bees search for food sources. As it turns out, the ABC algorithm provides high stability and requires a smaller number of parameters, while suffering the disadvantage of over-speedy convergence as the GA does. Proposed by Xin-She Yang in 2008, the FA [27,28,29,30] mimics the flashing behavior of fireflies. In nature, fireflies are attracted to each other by blinking signals. The more brightly a firefly glows, the more fireflies it attracts. Particularly, fireflies are randomly attracted to different light sources with the same intensity. Consequently, the brightest firefly is the optimal solution. The FA requires a smaller number of parameters and provides a wide search scope, fast convergence and high stability, while being very likely to get trapped in a local optimum solution, particularly in a multi-dimensional problem. The variable step size P&O method [31] is considerably sensitive to the initial parameter values, so the performance of its maximum power tracking controller is highly dependent on the selection of initial parameters. Therefore, the parameters must be selected and controlled carefully for this tracking technology, or there would be a significant impact on the tracking performance of its algorithm. Moreover, it is necessary to pay attention to the selection of the sample distribution and the adjustment of the parameters in order to improve the performance of the algorithm and increase the convergence speed. As for the shuffled frog-leaping (SFL) algorithm [32], a large amount of storage space and computing resources are usually required, and the increase in the number of groups will slow down the execution of the algorithm. Even more, insufficient computer memory capacity might be a result. Moreover, more parameters need adjustment while different parameters might need to be used in different application scenarios. Therefore, it becomes more difficult to adjust the parameters of the algorithm.

In light of this, this paper presents an improved version of the FA to locate the GMPP of a P-V curve with multiple peaks. The improved version is simple and straightforward; the step size in particular is made adaptive to the slope of a P-V characteristic curve. In this manner, it takes the improved version a shorter period of time to get away from a LMPP than the original, and the GMPP can be tracked down more efficiently.

Due to advances in power electronics, converters have been widely applied to MPP trackers so as to raise the power generation efficiency of a PV array. However, a major disadvantage of conventional DC-DC converters is the low voltage ratio due to the typical duty cycle [34,35,36]. For this sake, a great amount of effort has been put over the years into developing converters with a high voltage ratio. For example, the voltage ratio was raised as intended using coupled inductors with a high turn ratio [37,38,39,40]. Nonetheless, this move increased the output ripple voltage of a converter. Based on the same configuration as in [37,38,39,40], a converter is presented in this work as a solution to the above-referred problem. Inductors were replaced with coupled inductors therein so as to raise the voltage ratio, and clamp capacitors were also used to suppress the output voltage ripples in an effort to improve the performance of the converter. The presented converter was then simulated using PSIM software [41] for performance validation.

The organization of this paper is arranged as follows. The working principle of the adopted high boost ratio converter and its component design are described herein in Section 2. Next, the P-V and I-V output characteristics of a PV module under shading are described in Section 3. Subsequently, the working principle of the conventional FA is described, followed by proposing an improved FA, in Section 4. Finally, some simulation tests are performed in Section 5 to verify the effectiveness of the proposed improved FA for MPPT.

2. High Voltage Boost Converter

The adopted high-voltage boost converter is illustrated in Figure 1 [42]. The rated input and output voltages of the converter are 80 and 400 V, respectively.

For simplicity, the voltage drops across each forward-biased diode; the main switch $S_1$ when remaining on and the inductor $L_{k1}$ are all neglected. The duty cycle D is defined as

$$D\triangleq \frac{t_{on}}{T}$$

(1)

where $t_{on}$ represents the period of time that $S_1$ remains on, and T is the operation period of the converter.

(1)

Mode 1 ($t_0~t_1$)

The circuit conduction situation in this mode is indicated in Figure 2. $S_1$ remains on, and the input voltage $V_{in}$ is directly applied to the primary side $N_1$ of the coupled inductor. Accordingly, the current through the secondary side $N_2$ charges the capacitor $C_1$, and the diode $D_2$ remains forward-biased until $S_1$ is turned off at the end of this mode. The voltage drops across components and the currents therethrough are governed by

$$\mathsf{\Delta }{i}_{Lm1(closed)}=\frac{V_{in}}{L_{m1}}DT$$

(2)

$$v_{N2}=v_{C1}=V_{in}\frac{N_2}{N_1}$$

(3)

(2)

Mode 2 ($t_1~t_2$)

The circuit conduction situation in this mode is indicated in Figure 3. $S_1$ is switched off, and $D_1$ is forward-biased. The energy stored in $N_1$ is released to $C_1$, and the polarity of $N_2$ is reversed. As a consequence, the diode $D_3$ is forward-biased, and the capacitor $C_2$ is charged until the energy stored in $N_1$ is completely released. The governing equation is expressed as

$$v_{N1}=v_{Lm1}=v_{N2}\frac{N_1}{N_2}=v_{C2}\frac{N_1}{N_2}$$

(4)

(3)

Mode 3 ($t_2~t_3$)

The circuit conduction situation in this mode is indicated in Figure 4. $S_1$ remains off. The energy stored in $C_1$, $C_2$ and $L_{m1}$ is released to the load by way of the diode $D_4$, and an operation cycle is completed accordingly.

The inductor volt-second balance theory states that, in the steady state, the net change in the current flowing through an inductor is zero over a cycle. Therefore, applying the principle to Equations (2) and (4) gives

$$V_{in}DT=\frac{N_1}{N_2}{v}_{C2}\left(1-D\right)T$$

(5)

Rearranging Equation (5) gives

$$v_{C2}=\frac{N_2}{N_1}{V}_{in}\frac{D}{1-D}=\frac{ND}{1-D}{V}_{in}$$

(6)

The output voltage is given by

$$V_o=V_{in}+v_{C1}+v_{C2}$$

(7)

Substituting Equations (3) and (6) into Equation (7) gives the voltage ratio

$$\frac{V_o}{V_{in}}=\frac{N_1+N_2}{N_1}+\frac{N_2}{N_1}\frac{D}{1-D}=\frac{1+N-D}{1-D}$$

(8)

Assuming that all the components are lossless, the input power is equal to the output power, i.e., $P_{in}=P_{out}$, and

$$V_{in}{I}_{Lm1}=\frac{V_o{}^2}{R}$$

(9)

Substituting Equations (8) and (9) into Equation (7) gives

$$I_{Lm1}=\frac{V_o{}^2}{V_{in}R}=\frac{V_o{}^2}{V_{in}{}^2}\frac{V_{in}}{R}={(\frac{1+N-D}{1-D})}^2\frac{V_{in}}{R}$$

(10)

where $(\frac{1+N-D}{1-D})$ is the voltage ratio of the converter.

Derived from Equation (2), the increment of $i_{Lm1}$, over the period of time DT that $S_1$ is on, is given by

$$\mathsf{\Delta }{i}_{Lm1(closed)}=\frac{V_{in}}{L_{m1}}DT$$

(11)

Accordingly, the maximum and the minimum magnetizing current of the inductor, $I_{Lm1(\max )}$ and $I_{Lm1(\min )}$, are respectively given by

$$I_{Lm1\left(\max \right)}=I_{Lm1}+\frac{\mathsf{\Delta }{i}_{Lm1}}{2}=V_{in}\left[{(\frac{1+N-D}{1-D})}^2\frac{1}{R}+\frac{D}{2L_{m1}f}\right]$$

(12)

$$I_{Lm1\left(\min \right)}=I_{Lm1}-\frac{\mathsf{\Delta }{i}_{Lm1}}{2}=V_{in}\left[{(\frac{1+N-D}{1-D})}^2\frac{1}{R}-\frac{D}{2L_{m1}f}\right]$$

(13)

The inductor current is continuous on the condition that $I_{Lm1(\min )}$ ≥ 0, that is,

$$L_{m1\left(\min \right)}\ge \frac{D}{2f}\frac{{\left(1-D\right)}^{2}R}{{\left(1+N-D\right)}^2}$$

(14)

The rated output power and the output voltage are specified as 300 W and 400 V, respectively, in

Table 1. Specifications of the presented converter.

Input voltage ($V_{in}$)	80 V
Output voltage ($V_O$)	400 V
Output power ($P_O$)	300 W
Switching frequency of power switch (f)	25 kHz
Turn ratio of inductor ($N=N_2/N_1$)	2

, and the load $R_O$ is evaluated as 533.33 Ω accordingly. It is requested that the inductor $L_{m1}$ works anytime in the continuous conduction mode (CCM), that is, $I_{Lm1\left(\min \right)}\ge 0$ anytime, regardless of any change in the solar irradiance. As illustrated in Figure 5, the peak of the curve y = $\frac{D{(1-D)}^2}{{\left(1+2-D\right)}^2}$ occurs at D = 0.394. Substituting D = 0.394, $R_O$ = 533.33 Ω, f = 25 kHz and the turn ratio N = 2 into Equation (14) gives $L_{m1}$ = 227 μH. However, $L_{m1}$ is multiplied by a factor of 1.25, that is, $L_{m1}$ is raised from 227 to 284 μH, to ensure that the converter works all the time in the CCM as expected. Since $V_{c1}$ is evaluated as 160 V in Equation (3), a 2 μF/250 V film capacitor is used as $C_1$. The maximum duty cycle of $S_1$ is specified as 0.75. Substituting D = 0.75 into Equation (6) gives $V_{c2}$ = 480 V, and a 5 μF/630 V film capacitor is used as $C_2$ accordingly.

3. Shading Dependence of Output Characteristics of a PV Array

A single-peaked P-V characteristic curve of a PV array happens to contain multiple peaks if the array is partially shaded in different series and parallel combinations. In this work, SW20W PV modules, manufactured by MPPTSUN Co. Ltd., China, were used to build a PV array under test. Specifications of an SWM20W PV module are listed in

Table 2. Specifications of an SWM20W PV module.

Parameter	Specification
Maximum output power ($P_{\max }$)	20 W
Current of MPP ($I_{mpp}$)	1.10 A
Voltage of MPP ($V_{mpp}$)	18.18 V
Short current ($I_{sc}$)	1.15 A
Open voltage ($V_{oc}$)	22.32 V
Size of module	395 × 345 × 17 mm

. Using the Matlab software, a family of I-V and P-V characteristic curves of a 4 series-3 parallel (4S-3P) PV array, as shown in Figure 6, were simulated under Standard Test Conditions (STCs), that is, with the solar irradiance = 1 kW/m², T = 25 °C and air mass (AM) = 1.5. As illustrated in Figure 7, the improved version of the firefly algorithm was applied to track down the GMPP of a P-V array by controlling the gate of the main switch in the converter.

4. Firefly Algorithms

As referenced previously, the FA is actually a heuristic model proposed by Xin-She Yang in 2008. Fireflies communicate to each other by blinking signals. In nature, fireflies emitting flashes of light weakly are attracted to those emitting strongly, while moving randomly between two light sources with the same intensity. The more a firefly glows, the better a position it has. Accordingly, the firefly which glows most is the optimal solution. Compared to other algorithms, the FA requires a smaller number of parameters, has a wide search scope and provides high stability. However, with a poor starting point for a search or a poor choice of a step size, the FA would fail to track down the global optimum, especially in a multi-dimensional problem. Instead, it could get trapped in a local optimum.

4.1. The Conventional FA

The FA works under the following assumptions.

(1)

Fireflies are attracted to each other despite their gender.

(2)

Level of attraction is simply a function of the light intensity and the distance between fireflies, and it decreases as the distance increases. Fireflies which glow with lower brightness are attracted to those with higher brightness—which move randomly.

Here are the steps to find the optimum solution.

Step 1:

Initialize the number of fireflies $m$, the maximum number of iterations $I{t}_{\max }$ and the step size $a$.

Step 2:

The relative brightness between a firefly and another is given by

$$\beta ={\beta }_0{e}^{-\alpha d_{ji}^2}$$

(15)

where ${\beta }_0$ represents the brightness emitted by the firefly, and $\alpha$ the absorption coefficient of a medium. ${\beta }_0$ decays as the distance increases, and $\alpha$ is usually considered a constant.

Step 3:

In an n-dimensional space, the distance between fireflies i and j is expressed as

$$d_{ji}=||x_j-x_i||={\left[{\displaystyle \sum _{k=1}^n{\left(x_{j,n}-x_{i,n}\right)}^2}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$

(16)

where $x_{j,n}$ represents the n coordinate of firefly j. In this work, the FA is applied to track the MPP of a PV array, and the n-dimensional space is therefore simplified to a plane. Hence, Equation (16) becomes

$$d_{ji}={\left[{\left(x_j-x_i\right)}^2+{\left(y_j-y_i\right)}^2\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$

(17)

Step 4:

Level of attraction is given by

$$\gamma ={\gamma }_0{e}^{-\alpha d_{{}^{ji}}^2}$$

(18)

where ${\gamma }_0$ represents the maximum of $\gamma$.

Step 5:

Firefly i is attracted to firefly j, and the position of firefly i is updated using

$$x_i^{it+1}=x_i^{it}+{\gamma }_0{e}^{-\alpha d_{ji}^2}\left(x_j^{it}-x_i^{it}\right)+a{r}_i^{it}$$

(19)

where r is a random number between 0 and 1.

Step 6:

Go back to Step 1 until the maximum number of iterations is reached, and then output $x_i$.

4.2. The Improved FA

Firstly, the upper and the lower bounds of $a$ are specified, and $a$ is updated using

$$a=(1-\frac{it}{I{t}_{max}})a_{max}+\frac{it}{I{t}_{max}}\times a_{min}$$

(20)

where $a_{max}$ and $a_{min}$ represent the upper and the lower bound of $a$, respectively, and it represents the number of iterations. Subsequently, $a$ is slightly tuned according to the slope s of a P-V curve, as illustrated in Figure 8.

Table 3. Correlation between s and the slightly tuned $a$.

$s\stackrel{\mathsf{\Delta }}{=}\frac{P_{(it+1)}-P_{(it)}}{V_{(it+1)}-V_{(it)}}$	Fine-Tuning$a$of Equation (20) $$\begin{gathered} \left(\begin{array}{l}\mathsf{\Delta }P=P_{(it+1)}-P_{(it)}\\ \\ \mathsf{\Delta }P>0\end{array}\right) \end{gathered}$$
$s>2$	$a$ of Equation (20) + s $\times$0.01
$2\ge s>1.5$	$a$ of Equation (20) + s $\times$0.01
$1.5\ge s>1$	$a$ of Equation (20) − s $\times$0.01
$1\ge s>0.5$	$a$ of Equation (20) − s $\times$0.03
$0.5\ge s>0$	$a$ of Equation (20) − s $\times$0.04
$s=0$	$a$
$0>s\ge -0.5$	$a$ of Equation (20) + s $\times$0.04
$-0.5>s\ge -1$	$a$ of Equation (20) + s $\times$0.03
$-1>s\ge -1.5$	$a$ of Equation (20) + s $\times$0.01
$-1.5>s\ge -2$	$a$ of Equation (20) − s $\times$0.01
$s<-2$	$a$ of Equation (20) − s $\times$0.01

lists the correlation between s and the slightly tuned $a$, and it must be stressed that $a$ is tuned on the condition that ∆P > 0. In this fashion, the GMPP can finally be tracked down. A flowchart combining the conventional FA and the improved FA in Figure 9 is used to illustrate the working steps of the conventional FA and the improved FA in this study.

5. Simulation Results

The proposed high-voltage step-up converter was simulated using the PSIM software, and MPPT was then simulated using the Matlab software.

5.1. Simulation Results of the Proposed Converter

Figure 10 gives waveforms of the simulated input voltage $V_{in}$, output voltage $V_O$ and the duty cycle D signal at a full load of 300 W. An input voltage of 80 V was raised to an output voltage of 400 V when D was 0.567.

5.2. Simulation Results of the Improved FA

Table 4. Parameter settings in the original version of the FA.

Parameter Name	Value
Number of fireflies ($m$)	4
Maximum number of iterations ($I{t}_{max}$)	30
Step size ($a$)	2

and

Table 5. Parameter settings in the improved version of the FA.

Parameter Name	Value
Number of fireflies ($m$)	4
Maximum number of iterations ($I{t}_{max}$)	30
Upper bound of $a$ step size ($a_{\max }$)	2.5
Lower bound of $a$ step size ($a_{\min }$)	1.5

list the parameter settings used in the conventional and the improved version of the firefly algorithm.

Table 6. Shading description of Scenarios 1–5.

Scenario	Module Connection and Shading Condition	Number of P-V Curve Peaks
1	(0% shading + 0% shading + 0% shading +0% shading)// (0% shading + 0% shading + 0% shading + 0% shading)// (0% shading + 0% shading + 0% shading +0% shading)	Single
2	(0% shading + 0% shading + 0% shading + 50% shading)// (0% shading + 0% shading + 0% shading + 0% shading)// (0% shading + 0% shading + 0% shading + 0% shading)	Double (MPP was on the rightmost peak)
3	(0% shading + 0% shading + 50% shading + 70%shading)// (0% shading + 0% shading + 0% shading + 0% shading)// (0% shading + 0% shading + 0% shading + 0% shading)	Triple (MPP was on the rightmost peak)
4	(0% shading + 30% shading + 50% shading + 70% shading)// (0% shading + 0% shading + 0% shading + 0% shading)// (0% shading + 0% shading + 0% shading + 0% shading)	Triple (MPP was on the rightmost peak)
5	(0% shading + 70% shading + 80% shading + 90% shading)// (0% shading + 70% shading + 80% shading + 90% shading)// (0% shading + 70% shading + 80% shading + 90% shading)	Quadruple (MPP was on the leftmost peak)

Note: “+” represents series and “//” represents parallel.

lists five scenarios, representing five different shading conditions, in which shading effects on P-V characteristic curves of a 4 series-3 parallel (4S-3P) configuration were demonstrated. As will be seen below, multiple peaks occurred in the P-V curves in Scenarios 2–5 where the GMPP lay near the left or right endpoint of the P-V curves. The GMPP tracking performance was simulated in each scenario and then compared to highlight the advantage of this work over others.

(1)

Scenario 1

In this scenario, the PV array was unshaded and provided a maximum output power of 239.1 W, as illustrated in Figure 11. The tracking performance is compared in Figure 12. It can be easily seen that this work outperformed the original FA and the traditional P&O method.

(2)

Scenario 2

In this scenario, 50% of a module was shaded. As illustrated in Figure 13, there were two peaks on the P-V curve, and the maximum power fell from 239.1 to 201.7 W. Similar to Scenario 1, the GMPP lay near the right endpoint of the curve. As can be seen in Figure 14, the GMPP can be tracked down well using any of the three approaches, and it is noted that this work outperformed others again.

(3)

Scenario 3

In this scenario, 50% of a PV module and 70% of another were shaded, and three peaks occurred on the P-V curve, as illustrated in Figure 15. The maximum power was further reduced to 184.7 W, and there was one more peak on the P-V curve than in Scenario 2. The GMPP lay near the right endpoint of the curve again. The tracking performance is compared in Figure 16. As it appears, this work outperformed others again. After getting trapped in a LMPP, it took the improved version of the FA a shorter period of time to get away from the trap than the original version. Here, it must be stressed that the conventional P&O method failed to track down the GMPP even after 30 iterations.

(4)

Scenario 4

In this scenario, 70% of a module was shaded in addition to the two shaded modules in Scenario 3. As illustrated in Figure 17, the maximum output power was further reduced to 184.7 W, and there was one more peak on the P-V curve than in Scenario 3. The GMPP lay near the right endpoint of the curve again. As can be seen in Figure 18, both the improved and the original version of the FA succeeded in tracking down the GMPP as expected, while the P&O method failed again. As before, this work outperformed the original version of the FA.

(5)

Scenario 5

As listed in Table 6, nine modules were shaded in this scenario, that is, 70% of the first third, 80% of the second third and 90% of the last third modules were shaded. As illustrated in Figure 19, there were 4 peaks on the P-V curve, and the global maximum output power was as low as 55.8 W. Unlike in Scenarios 1–4, the GMPP instead lay near the left endpoint of the P-V curve this time. In Figure 20, it is observed that the P&O method got trapped in a LMPP more easily than others, even though all the three algorithms finally succeeded in tracking down the GMPP. Above all, this work outperformed others again.

5.3. Discussion of the Simulation Results

For readers’ convenience, a tracking performance comparison in Scenarios 1–5 is listed in

Table 7. A comparison of the average number of iterations and tracking time required to track down the GMPP in 10 trials.

Scenario	No. of Peaks on Respective P-V Curve	Average Number of Iterations/Average Tracking Time
		Conventional P&O	Conventional FA	Improved FA
1	Single	10.4/5.2 s	5.6/2.8 s	3.4/0.7 s
2	Double (MPP was on the right peak)	19.5/7.6 s	8.4/3.4 s	5.7/0.9 s
3	Triple (MPP was on the rightmost peak)	not available	14.1/4.0 s	12.7/1.3 s
4	Quadruple (MPP was on the rightmost peak)	not available	14.3/4.2 s	12.8/1.6 s
5	Quadruple (MPP was on the leftmost peak)	22.8/5.8 s	17.2/4.7 s	14.1/2.0 s

. The tracking performance was measured as the average number of iterations and the average tracking time required to track down the GMPP in 10 trials. Obviously, this work outperformed its counterparts, which highlights the robustness of this work against various shading conditions.

At present, four test cases have been selected to conduct tracking performance tests for the number of peaks and the different locations of the peaks in different P-V characteristic curves generated under different shading conditions; they are compared with the conventional perturbation and observation (P&O) [3], the conventional grey wolf optimization (GWO) [18] and the conventional artificial bee colony (ABC) [24] algorithms, respectively, for tracking performance, the test results of which are summarized in

Table 8. Comparison of simulation test results for four scenarios with different MPPT methods.

Scenario	No. Peak(s) of the P-V Curve	Method Proposed in [3]	Method Proposed in [18]	Method Proposed in [24]	Proposed in this Study
		Average Tracking Time	Average Tracking Time	Average Tracking Time	Average Tracking Time
1	Single-peak	10.4 s	8.6 s	8.58 s	3.4 s
2	Double-peak	19.5 s	12.5 s	20.01 s	5.7 s
3	Triple-peak	None *	15.4 s	31.3 s	12.7 s
4	Quadruple-peak	None *	20.8 s	24.21 s	12.8 s

Note: The symbol “None *” indicates that this reference does not provide the test results of this scenario.

.

Take Scenario 2 from the simulation results of the maximum power tracking conducted by the conventional P&O algorithm, the conventional FA and the improved FA in Figure 21 as an example; if the area of the region (such as the interval with black dotted lines) from the start point of tracking to the global maximum power point tracked by all methods acts as the optimal output energy of the PVMA in the tracking process, and the area of the region between the tracking curve for various methods to the tracked global maximum power point acts as its actual output energy, followed by calculating its tracking efficiency from the ratio of both areas, then the tracking efficiencies of five scenarios using different methods can be listed in

Table 9. A tracking efficiency comparison of the five scenarios.

Scenario	No. of Peaks on Respective P-V Curve	Tracking Efficiency (%)
		Conventional P&O	Conventional FA	Improved FA
1	Single	78.83%	98.32%	99.29%
2	Double (MPP was on the right peak)	79.25%	98.73%	99.09%
3	Triple (MPP was on the rightmost peak)	67.88%	93.23%	98.01%
4	Quadruple (MPP was on the rightmost peak)	42.48%	82.96%	96.74%
5	Quadruple (MPP was on the leftmost peak)	77.99%	76.1%	89.76%

for comparison. It can be seen from Table 9 that, in different scenarios, the tracking efficiencies of the improved firefly optimization algorithm are the most optimal compared to the other methods, and all of them are above 89.76%. Whereas, in Scenarios 3 and 4, the conventional P&O achieves tracking efficiencies of as low as 67.88–42.48% within 30 iterations because the global maximum power point cannot be tracked.

Table 6 shows five test cases of the PVMA in 4 series-3 parallel form used under different shading conditions herein, in which all P-V characteristic curves of the PVMA are obtained from the simulation under standard test conditions (STC). Although the changes in both temperature and sunshine would affect its maximum power output, the impact on the MPPT tracking process is not significant because of its little variation in the shape of the P-V characteristic curve.

In this paper, the maximum power tracking test has been conducted under five different shading conditions for the PVMA, and each of the shading conditions is equivalent to the change in the temperature and sunshine parameter so that it is identical to the robustness test of maximum power tracking (MPPT) under the consideration of parameter uncertainty in [43,44]. Moreover, from the test results in Table 7, Table 8 and Table 9, the proposed improved FA MPPT method has better performance in tracking speed and efficiency than other methods for the location of the global maximum power point (GMPP) in different cases, so that it is evident that the proposed MPPT method is indeed robust.

6. Conclusions

This work presents an improved version of the FA which happened to outperform the original version in terms of the GMPP tracking performance. Moreover, a high-voltage step-up converter was designed to raise an input voltage of 80 V to an output voltage of 400 V. This was done simply because the converter worked at a relatively low duty cycle, and the conversion efficiency was raised accordingly. Unlike in the original version of the FA, the step size was made adaptive to the slope of a P-V curve and the number of iterations. As a consequence, once having gotten trapped in a LMPP, the presented MPP tracker can easily escape from the trap and then succeed in tracking down the GMPP expeditiously as intended. The performance of the proposed MPP tracker was tested in five scenarios, and it has, accordingly, been validated as robust against various shading conditions. Subsequently, the TMS320F2809 digital signal processor (DSP) produced by Texas Instruments will be used as the control core for implementation in the future to ensure the effectiveness and robustness of the proposed method.