# Global Maximum Power Point Tracking of a Photovoltaic Module Array Based on Modified Cat Swarm Optimization

^{*}

## Abstract

**:**

## 1. Introduction

## 2. PVMA Output Characteristics under Different Shading Conditions

## 3. Conventional Cat Swarm Optimization

- Cat quantity: This ensures the number of cats needed for the algorithm, which affects the converging capability and speed of CSO. In this study, we used 6 cats.
- Initial location (duty cycle of converter): The initial location of each cat should be set right from the start; in this study, they were set randomly.
- Speed (tracking pace): After setting the initial location, the initial speed can be calculated accordingly.
- Cat flag: This parameter refers to the Boolean value indicated with only “yes” or “no”. “Yes” means that the cat is in seeking mode and “no” means that the cat is in tracking mode.
- Maximum number of iterations: The condition for iteration termination was set here to ensure that the CSO operation would cease within a certain time frame.

_{best}) of the cat with the optimal fitness value was saved. In CSO, the cats are assigned to different modes according to the set mixture ratio (MR). The MR represents the proportion of cats distributed to tracking mode, and has a value between 0 and 1. In the program, the MR was often set with smaller numbers. For example, if 10 cats were utilized in CSO with an MR of 0.2, 2 cats would be assigned to tracking mode and 8 cats would be assigned to seeking mode. The flag for cats assigned to seeking mode would be set as “yes”, and the main task of seeking mode is to update the cat location. The flowchart of CSO in Figure 2 describes seeking and tracking modes on the left and right sides, respectively.

#### 3.1. Seeking Mode

#### 3.2. Tracking Mode

- Update tracking speed v
_{i}according to Equation (1):

- 2.
- Update cat location x
_{i}according to Equation (2):$${x}_{i}(t+1)={x}_{i}(t)+{v}_{i}(t+1)$$

## 4. Proposed MCSO

#### 4.1. MCSO with Fixed Initial Tracking Voltage

^{2}), there was no extensive difference in MPP voltage. To improve the tracking speed of the conventional CSO, the initial tracking voltage V

_{begin}was first fixed at 0.8 times the MPP voltage V

_{mp}(i.e., V

_{begin}= 0.8 V

_{mp}). Therefore, each tracking would commence from V

_{begin}. Despite the change in solar irradiance, the initial tracking voltage was approximately around the GMPP; therefore, this method can swiftly and steadily track the GMPP.

#### 4.2. MCSO with Fixed Initial Tracking Voltage Combined with Tracking Pace Adjusted with Slope of P-V Curve

- (1)
- When slope m is less than zero, it indicates that the system has tracked to the right side of the MPP, and the tracking direction is heading toward the MPP at the left.
- (2)
- When slope m is greater than zero, it indicates that the system has tracked to the left side of the MPP, and the tracking direction is heading toward the MPP at the right.
- (3)
- When slope m is zero, it indicates that the system has tracked the MPP. The slope (m) and power variation (dP) are defined by Equations (3) and (4), respectively:

_{k}and V

_{k}are the power and voltage values of the current operating point, respectively, and P

_{k−1}and V

_{k−1}are the power and voltage values of the previous operating point.

#### 4.3. MCSO with Fixed Initial Tracking Voltage Combined with Tracking Pace Adjusted by Inertia Weight

_{mp}, the learning rate was combined to adjust the tracking pace. The equation for updating the MCSO speed can be expressed as Equation (5):

#### 4.4. MCSO with Fixed Initial Tracking Voltage Combined with Tracking Pace Adjusted by the Slope of P-V Curve and Inertia Weight

_{mp}, it is combined with the slope of the P-V curve and the inertia weight at the same time to adjust the tracking pace. This tracking method can help to determine the status of the current work point more precisely; at the same time, the tracking pace can be increased by combining multiple parameters, which reduces the oscillation amplitude and enhances the tracking efficiency.

## 5. Design of MPPT Controlling Converter

## 6. Simulated Results

- (1)
- Test of Case 1

^{2}and zero shading, with the MPP at 239.1 and 121.1 W, respectively. Figure 7 shows the simulation results of the conventional CSO and four MCSO methods proposed in this paper. From the figure, it can be seen that under a solar irradiance change from 1000 W/m

^{2}to 500 W/m

^{2}and back to 1000 W/m

^{2}for all methods, only the conventional CSO could not track the GMPP when the solar irradiance suddenly changed to 500 W/m

^{2}, and the maximum iteration number (t) was set as 30 times, while all four proposed MCSO methods could track the GMPP. Although the conventional CSO could track the GMPP at a solar irradiance of 1000 W/m

^{2}, it could clearly be seen that the oscillation amplitude was greater compared to the four modified methods. Among them, the MCSO with the initial tracking voltage fixed at 0.8 times the MPP voltage (0.8 V

_{mp}) under STC combined with a tracking pace adjusted with the slope of the P-V curve and inertia weight provided the optimal tracking speed response and steady performance.

- (2)
- Test of Case 2

^{2}with a single module under 50% shading, where the true MPP was at 201.7 and 102.1 W, respectively, and the GMPP was to the right. Figure 9 shows the simulation results for MPPT with the conventional CSO and four proposed MCSO methods. From the figure, it can be seen that under a solar irradiance change from 1000 W/m

^{2}to 500 W/m

^{2}and back to 1000 W/m

^{2}, all five tracking methods could track the GMPP successfully. Although all five tracking methods were stuck with an LMPP of 187.4 W, the MCSO with an initial tracking voltage fixed at 0.8 V

_{mp}combined with pace adjusted with the slope of the P-V curve and inertia weight could escape from the LMPP swiftly. The other methods could also escape but required a longer duration, and the oscillation amplitude was greater. In view of this, the MSCO proposed in this paper with the initial tracking voltage fixed at 0.8 V

_{mp}combined with pace adjusted with the slope of the P-V curve and inertia weight provided an optimal tracking speed response, and the oscillation amplitude after tracking the GMPP was lower compared to the other methods; hence, the steady performance was also better.

- (3)
- Test of Case 3

^{2}with two modules under 80% and 100% shading, where three peaks appeared with true MPP at 159.4 and 80.7 W, respectively, and the GMPP was at the far right. Figure 11 shows the simulation results for MPPT with the conventional CSO and four proposed MCSO methods. From the figure, it can be seen that under a solar irradiance change from 1000 to 500 W/m

^{2}and back to 1000 W/m

^{2}, although all five tracking methods could track the GMPP successfully, each was stuck with an LMPP at 121.7 and 154.8 W. Even though all five tracking methods could track to the GMPP eventually, it could be clearly observed that the conventional CSO required a longer duration, and the oscillation amplitude during the tracking process was greater. In view of this, the proposed MSCO with an initial tracking voltage fixed at 0.8 V

_{mp}combined with pace adjusted with the slope of the P-V curve and inertia weight not only provided the fastest tracking speed response, but the oscillation amplitude near the GMPP was lower compared to the other methods.

- (4)
- Test of Case 4

^{2}with two modules under 100% and 50% shading, where three peaks appeared with true MPP at 170.2 and 84.4 W, respectively, and the GMPP was in the middle. Figure 13 shows the simulation results for MPPT with the conventional CSO and four proposed MCSO methods. From the figure, it can be seen that under a solar irradiance change from 1000 to 500 W/m

^{2}and back to 1000 W/m

^{2}, each method was stuck with an LMPP at 121.7 W. The four modified methods could escape from the stuck point swiftly and track the GMPP successfully. The conventional CSO under a solar irradiance of 1000 W/m

^{2}could not track the GMPP successfully within the set of 30 iterations. Although the four proposed tracking methods could track the GMPP successfully, the MCSO with the initial tracking voltage fixed at 0.8 V

_{mp}combined with pace adjusted with the slope of the P-V curve and inertia weight provided the fastest tracking speed response, and the oscillation amplitude near the MPP was minor. Therefore, the power generation efficiency of the PVMA could be enhanced.

- (5)
- Test of Case 5

^{2}with nine modules under 30%, 60%, and 90% shading, where four peaks appeared with true MPP at 85.8 and 42.9 W, respectively, and the GMPP was at the second peak from the left. Figure 15 shows the simulation results for MPPT with the conventional CSO and four proposed MCSO methods. From the figure, it can be seen that under a solar irradiance change from 1000 to 500 W/m

^{2}and back to 1000 W/m

^{2}, only the conventional CSO under a solar irradiance of 1000 W/m

^{2}could not track the GMPP. Although all four modified tracking methods proposed in this paper were stuck with an LMPP at 51.9 W, they could escape from the LMPP swiftly and track the GMPP successfully. Among them, the MCSO with the initial tracking voltage fixed at 0.8 V

_{mp}combined with pace adjusted with the slope of the P-V curve and inertia weight could escape from the LMPP (51.9 W) the fastest. Therefore, this modified method provided optimal tracking speed response and steady performance.

- (6)
- Test of Case 6

^{2}with nine modules under 30%, 50%, and 80% shading, where four peaks appeared with true MPP at 98.2 and 48.7 W, respectively, and the GMPP was at the third peak from the left. Figure 17 shows the simulation results for MPPT with the conventional CSO and four proposed MCSO methods. From the figure, it can be seen that under a solar irradiance change from 1000 to 500 W/m

^{2}and back to 1000 W/m

^{2}, only the conventional CSO under a solar irradiance of 1000 W/m

^{2}could not track the GMPP. Although all four modified tracking methods proposed in this paper were stuck with an LMPP at 51.9 and 85.8 W, they could escape swiftly and track the GMPP successfully under the two solar irradiance conditions. Among them, the MCSO with the initial tracking voltage fixed at 0.8 V

_{mp}combined with pace adjusted with the slope of the P-V curve and inertia weight provided optimal tracking speed response and steady performance.

^{2}to 500 W/m

^{2}, the maximum power point voltage did not change significantly because the shading conditions did not change. The previous tracking did not capture the global maximum power point, but it was close. As a result, tracking the global power point under 500 W/m

^{2}solar exposure took only a few iterations.

## 7. Conclusions

_{mp}, MCSO with a fixed initial traction voltage at 0.8 V

_{mp}combined with pace adjusted with the slope of P-V curve, MCSO with a fixed initial traction voltage at 0.8 V

_{mp}combined with pace adjusted with the inertia weight, and MCSO with the initial tracking voltage fixed at 0.8 V

_{mp}combined with pace adjusted with the slope of P-V curve and inertia weight. The simulation results show that these modified methods provided a better tracking speed response and steady performance compared to the conventional CSO. Among the four modified methods, MCSO with the initial tracking voltage fixed at 0.8 V

_{mp}combined with pace adjusted with the slope of P-V curve and inertia weight provided a better tracking speed response and steady performance compared to the conventional CSO and the other three modified methods. Through the simulation results, it was also proven that, under a sudden change in solar irradiance on a specific day, all four MCSO methods proposed in this paper could track the GMPP swiftly. Among them, MCSO with the initial tracking voltage fixed at 0.8 V

_{mp}combined with pace adjusted with the slope of P-V curve and the inertia weight provided the optimal tracking speed response. After tracking the MPP, it could also reduce the back-and-forth tracking of the oscillation amplitude. Therefore, the power loss was decreased and the power generation efficiency of the PVMA was enhanced.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**I-V and P-V characteristic curves derived from module simulation at 25 °C and different irradiance levels.

**Figure 3.**P-V characteristic curve upon sudden change in solar irradiance with partial module shading.

**Figure 4.**Interval division relations for parameters automatically adjusted according to slope of P-V characteristic curve under MCSO.

**Figure 7.**Simulation results for case 1: MPPT with conventional CSO and four modified methods under solar irradiance change from 1000 to 500 W/m

^{2}and back to 1000 W/m

^{2}.

**Figure 9.**Simulation results for case 2: MPPT with conventional CSO and four modified methods under solar irradiance change from 1000 to 500 W/m

^{2}and back to 1000 W/m

^{2}.

**Figure 11.**Simulation results for case 3: MPPT with conventional CSO and four modified methods under solar irradiance change from 1000 to 500 W/m

^{2}and back to 1000 W/m

^{2}.

**Figure 13.**Simulation results for case 4: MPPT with conventional CSO and four modified methods under solar irradiance change from 1000 to 500 W/m

^{2}and back to 1000 W/m

^{2}.

**Figure 15.**Simulation results for case 5: MPPT with conventional CSO and four modified methods under solar irradiance change from 1000 to 500 W/m

^{2}and back to 1000 W/m

^{2}.

**Figure 17.**Simulation results for case 6: MPPT with conventional CSO and four modified methods under solar irradiance change from 1000 to 500 W/m

^{2}and back to 1000 W/m

^{2}.

**Table 1.**Specifications of electrical performance parameters for SWM20W photovoltaic modules made by MPPTSun [22].

Parameter | Value |
---|---|

Maximum output power (P_{max}) | 20 W |

Current of maximum output power (I_{mpp}) | 1.1 A |

Voltage of maximum output power (V_{mpp}) | 18.18 V |

Short-circuit current (I_{sc}) | 1.15 A |

Open-circuit voltage (V_{oc}) | 22.32 V |

Overall dimensions of single module | 395 × 345 × 17 mm |

**Table 2.**Adjustment relations between slope intervals of P-V characteristic curve and speed coefficient, c.

Slope Interval of P-V Characteristic Curve | Speed Coefficient (c) in Equation (1) |
---|---|

Interval m_{1}: $m>2$ | c = 1.4 |

Interval m_{2}: $2>m\ge 1.3$ | c = 1.3 |

Interval m_{3}: $1.3>m\ge 0.8$ | c = 1.2 |

Interval m_{4}: $0.8>m\ge 0$ | c = 1.1 |

Interval m_{5}: $0>m\ge -1$ | c = 1.0 |

Interval m_{6}: $m\le -1$ | c = 1.2 |

**Table 3.**Adjustment relations between distance change |${x}_{best}(t)-{x}_{i}(t)$| and inertia weight, w.

$$\mathit{a}=\left|{\mathit{x}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}(\mathit{t})-{\mathit{x}}_{\mathit{i}}(\mathit{t})\right|$$
| The Inertia Weight (w) in Equation (5) |
---|---|

$$1\ge a\ge 0.6$$
| w = 0.4 |

$$0.6>a\ge 0.5$$
| w = 0.35 |

$$0.5>a\ge 0.4$$
| w = 0.3 |

$$0.4>a\ge 0.3$$
| w = 0.25 |

$$0.3>a\ge 0.2$$
| w = 0.2 |

$$0.2>a\ge 0.1$$
| w = 0.15 |

$$0.1>a\ge 0$$
| w = 0.1 |

**Table 4.**Component specifications for boost converter [23].

Component | Specifications |
---|---|

Input capacitor, C_{in} | $220\mathsf{\mu}\mathrm{F}$, withstand voltage: 400 V |

Filter capacitance, C_{out} | $470\mathsf{\mu}\mathrm{F}$, withstand voltage: 450 V |

Energy storage inductance, L_{m} | 1.67 mH, withstand voltage: 450 V |

Fast diode, D (IQVD60E60A1) | Withstand voltage: 600 V, withstand current: 60 A |

Switch, S (IREP460B) | Withstand voltage: 500 V, withstand current: 20 A |

**Table 5.**Peak numbers on P-V characteristic curve of 4-series, 3-parallel PVMA under 6 shading conditions.

Scenario | Peak Numbers on P-V Characteristic Curve | 4-Series, 3-Parallel Shading % |
---|---|---|

1 | Single-peak | (0% shading + 0% shading + 0% shading + 0% shading)// (0% shading + 0% shading + 0% shading + 0% shading)// (0% shading + 0% shading + 0% shading + 0% shading) |

2 | Double-peak (MPP at right) | (50% shading + 0% shading + 0% shading + 0% shading)// (0% shading + 0% shading + 0% shading + 0% shading)// (0% shading + 0% shading + 0% shading + 0% shading) |

3 | Triple-peak (MPP at right) | (0% shading + 80% shading + 0% shading + 100% shading)// (0% shading + 0% shading + 0% shading + 0% shading)// (0% shading + 0% shading + 0% shading + 0% shading) |

4 | Triple-peak (MPP at middle) | (0% shading + 100% shading + 0% shading + 50% shading)// (0% shading + 0% shading + 0% shading + 0% shading)// (0% shading + 0% shading + 0% shading + 0% shading) |

5 | Quadruple-peak (MPP at the second peak) | (0% shading + 30% shading + 60% shading + 90% shading)// (0% shading + 30% shading + 60% shading + 90% shading)// (0% shading + 30% shading + 60% shading + 90% shading) |

6 | Quadruple-peak (MPP at the third peak) | (30% shading + 50% shading + 80% shading + 0% shading)// (30% shading + 50% shading + 80% shading + 0% shading)// (30% shading + 50% shading + 80% shading + 0% shading) |

Parameter | Conventional | Modified |
---|---|---|

Maximum iteration number (t) | 30 | |

Random number (r) | [0~1] | |

Inertia weight (w) | Zero usage w | [0.1~0.4] |

Speed coefficient (c) | 1.4 | [1.0~1.4] |

SRD | 0.2% |

Case | Number of Peaks on P-V Curve | Tracking Speed | ||||
---|---|---|---|---|---|---|

Conventional CSO Algorithm | CSO with Fixed Initial Tracking Voltage | CSO with Fixed Initial Tracking Voltage Combined with Tracking Pace Adjusted with Slope of P-V Curve | CSO with Fixed Initial Tracking Voltage Combined with Tracking Pace Adjusted by Inertia Weight | CSO with Fixed Initial Tracking Voltage Combined with Tacking Pace Adjusted by Slope of P-V Curve and Inertia Weight | ||

1 | Single peak | 0.077 s | 0.05 s | 0.048 s | 0.042 s | 0.038 s |

2 | Double peak (MPP at right) | 0.061 s | 0.051 s | 0.049 s | 0.003 s | 0.028 s |

3 | Triple peak (MPP at right) | 0.082 s | 0.059 s | 0.056 s | 0.04 s | 0.039 s |

4 | Triple peak (MPP at middle) | Fail | 0.058 s | 0.055 s | 0.043 s | 0.038 s |

5 | Quadruple peak (MPP at second peak) | Fail | 0.037 s | 0.028 s | 0.023 s | 0.02 s |

6 | Quadruple peak (MPP at third peak) | Fail | 0.061 s | 0.057 s | 0.056 s | 0.054 s |

Case | Number of Peaks on P-V Curve | Maximum Oscillation Amplitude | ||||
---|---|---|---|---|---|---|

Conventional CSO Algorithm | CSO with Fixed Initial Tracking Voltage | CSO with Fixed Initial Tracking Voltage Combined with Tracking Pace Adjusted with Slope of P-V Curve | CSO with Fixed Initial Tracking Voltage Combined with Tracking Pace Adjusted by Inertia Weight | CSO with Fixed Initial Tracking Voltage Combined with Tacking Pace Adjusted by Slope of P-V Curve and Inertia Weight | ||

1 | Single peak | 30 W | 21 W | 11 W | 12 W | 12 W |

2 | Double peak (MPP at right) | 18 W | 17 W | 13 W | 14 W | 13 W |

3 | Triple peak (MPP at right) | 30 W | 10 W | 9 W | 12 W | 11 W |

4 | Triple peak (MPP at middle) | 17 W | 10 W | 8 W | 10 W | 5 W |

5 | Quadruple peak (MPP at second peak) | 18 W | 8 W | 10 W | 10 W | 7 W |

6 | Quadruple peak (MPP at third peak) | 21 W | 10 W | 12 W | 10 W | 6 W |

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## Share and Cite

**MDPI and ACS Style**

Chao, K.-H.; Nguyen, T.B.-N.
Global Maximum Power Point Tracking of a Photovoltaic Module Array Based on Modified Cat Swarm Optimization. *Appl. Sci.* **2024**, *14*, 2853.
https://doi.org/10.3390/app14072853

**AMA Style**

Chao K-H, Nguyen TB-N.
Global Maximum Power Point Tracking of a Photovoltaic Module Array Based on Modified Cat Swarm Optimization. *Applied Sciences*. 2024; 14(7):2853.
https://doi.org/10.3390/app14072853

**Chicago/Turabian Style**

Chao, Kuei-Hsiang, and Thi Bao-Ngoc Nguyen.
2024. "Global Maximum Power Point Tracking of a Photovoltaic Module Array Based on Modified Cat Swarm Optimization" *Applied Sciences* 14, no. 7: 2853.
https://doi.org/10.3390/app14072853