# Maximum Power Point Tracking of Photovoltaic Generation System Using Improved Quantum-Behavior Particle Swarm Optimization

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The PV Circuit and the Effects of Environmental Conditions

#### 2.1. The Equivalent Circuit of PV Cells

_{s}, a parallel diode D

_{j}, a series resistance R

_{s}, a PV output current I

_{pv}, and a PV output voltage V

_{pv}. The output current can be represented by Equation (1), where k is the Boltzmann’s constant, n is the ideality factor, T is the temperature in Kelvin, q is the electron charge, and I

_{d}is the saturation current. Thus, the output power of the PV cells can be obtained as Equation (2). Figure 2 shows the Power–Voltage (P–V) curve of PV cells, which is a nonlinear curve due to the MPP.

#### 2.2. The Effects of Irradiance, Temperature, and PSC

## 3. IQPSO Algorithm

#### 3.1. QPSO Algorithm

_{1}is a random variable in the interval [0, 1]. In Equation (6), ${M}^{k}$ denotes the mean best position of the particle swarm at the present iteration, and u is a random variable within [0, 1]. The value of the random variable u determines the sign in the Equation (3). If u is greater than 0.5, a negative sign is adopted; otherwise, a positive sign is used in Equation (3). In Equation (7), N denotes the number of particles. The QPSO algorithm fundamentally operates based on the quantum motion of the delta potential well. The Monte Carlo approach is integrated to guarantee a balanced exploration–exploitation trade-off during the search process. The particles have a certain probability of being dispersed throughout the search area, increasing the likelihood of finding the globally optimal solution across the entire domain.

#### 3.2. IQPSO Algorithm

^{k}of the particle swarm. To expedite the convergence speed of the QPSO algorithm, the contraction–expansion coefficient β is typically designed to decrease with the increase in generations, as depicted in Equation (4). However, during the initial phases of the QPSO algorithm, β may easily lead particles to exceed the search space and become confined to the search boundary. In the later stages of the QPSO algorithm, if β becomes too small when nearing the steady state, it may impede particle mobility, causing them to linger near the optimal point and failing to attain the true optimal value. Consequently, this may result in premature convergence issues, leading to the failure to attain the globally optimal solution. To circumvent these premature convergence problems, this study exploits the characteristic that the reciprocal of the natural exponent with a small power has a larger value in the early stages of the algorithm and gradually decreases to a positive number that is significant as it approaches the steady state. Equation (4) is refined into to Equation (8), enabling the contraction–expansion coefficient β to decrease in a natural exponential fashion as the number of iterations increases. Consequently, the proposed improved QPSO algorithm can expedite the convergence speed in the early stages and enhance tracking accuracy in the steady state.

^{k}represents, solely, the mean’s best position of the particle swarm, as delineated in Equation (7). However, this calculation of M

^{k}does not consider the relationship between the fitness value and the movement of individual particles towards the best position of the particle swarm. Furthermore, if the particle positions are unevenly distributed, the mean best value of the particle swarm may be distant from the globally optimal solution, thereby resulting in slower convergence or even convergence only to a locally optimal solution. Consequently, this paper comprehensively considers the fitness value, individual particle position, and the best position of the particle swarm to modify M

^{k}, as depicted in Equation (11), aiming to enhance the success rate of tracking within the proposed QPSO algorithm.

^{k}exhibits a superior tracking rate compared to the existing M

^{k}, regardless of the number of particles utilized in the algorithm.

## 4. MPPT Circuit

_{1}into buck mode and the power switch M

_{2}into boost mode, respectively. Upon receiving the signals Vpv and Ipv from the feedback circuits, the microprocessor TMS320F28069 calculates the duty ratio using the IQPSO program stored within the chip. Based on the calculated duty ratio, the corresponding pulse width modulation (PWM) signal drives the insulated gate bipolar transistor (IGBT) power switch in either buck mode or boost mode, maximizing the output power of the PGS.

_{now}represents the current particle position of MPPT, while ΔD represents the change step of the duty ratio. Additionally, the symbol D_M

_{1}denotes the duty ratio of the power switch M

_{1}, and D_M

_{2}denotes the duty ratio of the power switch M

_{2}. Throughout the MPPT mode switching process, D_M

_{1}is set to 1, D_M

_{2}is set to 0, and a delay of 50 ms is implemented to stabilize the input voltage before transitioning modes to reduce surges.

## 5. IQPSO MPPT

## 6. Experimental Results

^{2}every 10 seconds. Specifically, the irradiance decreases from 1000 W/m

^{2}to 600 W/m

^{2}and then increases back to 1000 W/m

^{2}cyclically. The experiment lasts a total of ninety seconds. Experimental data obtained from the 62100H-600S programmable DC power supply is utilized to construct the power curve, as depicted in Figure 23. Throughout the irradiance decrease from 1000 W/m

^{2}to 600 W/m

^{2}, the temperature remains fixed at 25 °C. The global maximum PV output power ${P}_{PV,GMAX}$ corresponds to 1999.92 W, 1800 W, 1600 W, 1400 W, and 1200 W, respectively.

^{2}, the maximum power output reaches 2 kW. Figure 26 illustrates the tracking responses of the PV arrays under single-peak conditions using various MPPT algorithms. These experiments were conducted under similar ambient conditions (irradiance and temperature), with no shading on the PV arrays. Figure 26a depicts the MPPT response using IQPSO, characterized by the fastest response and highest accuracy. Figure 26b illustrates the MPPT response using QPSO, which exhibits a slower but accurate response. Figure 26c shows the MPPT response using FA, featuring a fast response speed but lower accuracy. Figure 26d presents the MPPT response using PSO, which exhibits the slowest response and less accuracy. Table 8 provides the experimental data presented in Figure 26. In terms of tracking accuracy, IQPSO performs the best while FA performs the worst. Regarding tracking time, IQPSO is the fastest, and PSO is the slowest.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 20.**MPPT responses of the single-peak P–V curve: (

**a**) IQPSO; (

**b**) QPSO; (

**c**) FA; (

**d**) PSO (Ppv: 500 W/div, Vpv: 200 V/div, Ipv: 5 A/div, time: 2 s/div).

**Figure 22.**MPPT responses of the multi-peak P–V curve: (

**a**) IQPSO; (

**b**) QPSO; (

**c**) FA; (

**d**) PSO (Ppv: 500 W/div, Vpv: 200 V/div, Ipv: 5 A/div, time: 2 s/div).

**Figure 24.**MPPT responses under varying irradiance conditions: (

**a**) IQPSO; (

**b**) QPSO; (

**c**) FA; (

**d**) PSO (Ppv: 500 W/div, Vpv: 200 V/div, Ipv: 5 A/div, time: 10 s/div).

**Figure 26.**MPPT responses of the PV arrays under single-peak conditions: (

**a**) IQPSO; (

**b**) QPSO; (

**c**) FA; (

**d**) PSO (Ppv: 500 W/div, Vpv: 200 V/div, Ipv: 5 A/div, time: 2 s/div).

**Figure 27.**MPPT responses of the PV arrays under multi-peak conditions: (

**a**) IQPSO; (

**b**) QPSO; (

**c**) FA; (

**d**) PSO (Ppv: 500 W/div, Vpv: 200 V/div, Ipv: 5 A/div, time: 2 s/div).

System Power | 2000 W |

Input voltage V_{pv} | 10–560 V |

Output voltage V_{o} | 100–400 V |

Inductor L_{M} | 1.2 mH |

Input capacitor C_{in} | 560 μF |

Output capacitor C_{out} | 560 μF |

Method | PSO | FA | QPSO | IQPSO | ||||
---|---|---|---|---|---|---|---|---|

Parameter | W | 0.3 | A | 0.02 | β_{max} | 1.0 | γ | 2.0 |

C_{1} | 0.5 | Β | 0.50 | β_{min} | 0.1 | |||

C_{2} | 0.5 | Γ | 0.50 |

Maximum Power (W) | Tracking Power (W) | Tracking Accuracy | Tracking Time (s) | |
---|---|---|---|---|

IQPSO | 1999.92 | 1980.59 | 99.03 % | 1.32 |

QPSO | 1999.92 | 1969.62 | 98.48 % | 3.93 |

FA | 1999.92 | 1926.04 | 96.31 % | 2.81 |

PSO | 1999.92 | 1954.81 | 97.74 % | 4.51 |

Tracking Power (W) | Tracking Accuracy | Tracking Time (s) | |
---|---|---|---|

IQPSO w/o Equation (8) CHG | 1970.12 | 98.51 % | 2.91 |

IQPSO w/o Equation (9) CHG | 1977.32 | 98.87 % | 2.32 |

IQPSO w/o Equation (10) CHG | 1972.32 | 98.62 % | 2.51 |

IQPSO w/o Equation (11) CHG | 1976.12 | 98.81 % | 2.38 |

Maximum Power (W) | Tracking Power (W) | Tracking Accuracy | Tracking Time (s) | |
---|---|---|---|---|

IQPSO | 1260.02 | 1244.99 | 98.81 % | 1.93 |

QPSO | 1260.02 | 1239.55 | 98.38 % | 6.78 |

FA | 1260.02 | 1215.66 | 96.48 % | 2.61 |

PSO | 1260.02 | 1205.78 | 95.70 % | 5.41 |

Tracking Power (W) | Tracking Accuracy | Tracking Time (s) | |
---|---|---|---|

IQPSO w/o Equation (8) CHG | 1240.09 | 98.42 % | 4.51 |

IQPSO w/o Equation (9) CHG | 1243.99 | 98.73 % | 2.42 |

IQPSO w/o Equation (10) CHG | 1241.48 | 98.53 % | 3.83 |

IQPSO w/o Equation (11) CHG | 1243.75 | 98.71 % | 3.25 |

Average Tracking Accuracy | Average Tracking Time (s) | |
---|---|---|

IQPSO | 99.84 % | 2.04 |

QPSO | 98.73 % | 3.60 |

FA | 98.55 % | 2.37 |

PSO | 98.51 % | 3.29 |

Irradiance (W/m ^{2}) | Temperature (°C) | Maximum Power (W) | Tracking Power (W) | Tracking Accuracy | Tracking Time (s) | |
---|---|---|---|---|---|---|

IQPSO | 825 | 35.0 | 1314 | 1307 | 99.47 % | 1.35 |

QPSO | 791 | 35.8 | 1317 | 1304 | 99.01 % | 4.42 |

FA | 780 | 38.1 | 1339 | 1308 | 98.35 % | 2.35 |

PSO | 804 | 34.5 | 1272 | 1255 | 98.66 % | 5.43 |

Irradiance (W/m ^{2}) | Temperature (°C) | Maximum Power (W) | Tracking Power (W) | Tracking Accuracy | Tracking Time (s) | |
---|---|---|---|---|---|---|

IQPSO | 986 | 39.6 | 1065 | 1064 | 99.91 % | 1.73 |

QPSO | 1010 | 37.8 | 1005 | 997 | 99.20 % | 3.86 |

FA | 975 | 38.1 | 967 | 948 | 98.03 % | 1.93 |

PSO | 977 | 38.8 | 968 | 941 | 97.21 % | 4.63 |

IQPSO | QPSO | FA | PSO | |
---|---|---|---|---|

Day1 | 6703.66 | 6388.45 | 6422.04 | 6000.40 |

Day2 | 10,521.22 | 10,408.85 | 10,319.25 | 10,164.69 |

Day3 | 9382.16 | 9284.92 | 9211.54 | 9072.42 |

Total | 26,607.04 | 26,082.22 | 25,952.83 | 25,237.51 |

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**MDPI and ACS Style**

Yu, G.-R.; Chang, Y.-D.; Lee, W.-S.
Maximum Power Point Tracking of Photovoltaic Generation System Using Improved Quantum-Behavior Particle Swarm Optimization. *Biomimetics* **2024**, *9*, 223.
https://doi.org/10.3390/biomimetics9040223

**AMA Style**

Yu G-R, Chang Y-D, Lee W-S.
Maximum Power Point Tracking of Photovoltaic Generation System Using Improved Quantum-Behavior Particle Swarm Optimization. *Biomimetics*. 2024; 9(4):223.
https://doi.org/10.3390/biomimetics9040223

**Chicago/Turabian Style**

Yu, Gwo-Ruey, Yong-Dong Chang, and Weng-Sheng Lee.
2024. "Maximum Power Point Tracking of Photovoltaic Generation System Using Improved Quantum-Behavior Particle Swarm Optimization" *Biomimetics* 9, no. 4: 223.
https://doi.org/10.3390/biomimetics9040223