# An Efficient Variable Step Solar Maximum Power Point Tracking Algorithm

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

- A first-order linear Kalman filtering algorithm is employed to pre-process the sensor detection data, which improves the stability and accuracy of the subsequent MPPT control algorithm.
- The calculation method of the step adjustment factor of the traditional variable step conductance increment is improved to increase the efficiency of the photovoltaic system.
- The MQTT protocol is applied to transmit information when the photovoltaic system is in operation, and the corresponding client is designed to remotely monitor the working status of the photovoltaic cells.

## 2. Photovoltaic Mathematical Model and Output Characteristics Analysis

#### 2.1. Photovoltaic Cell Mathematical Modeling and Analysis

_{ph}is a photo-generated current whose magnitude varies with temperature and light intensity, I

_{0}is the reverse saturation current of the diode, I

_{d}is the current flowing through the diode, I

_{sh}is the current flowing through the shunt resistor R

_{sh}, q is the electronic charge, I and U are the output current and output voltage of the solar cells, respectively. A is the ideal factor of the diode, T is the absolute temperature value when the photovoltaic cells is operating, and K is Boltzmann’s constant, which is defined in (3).

^{2}, T = 25 °C), it requires further simplification. The mathematical model required for its simulation model can be (5)~(10), and the meanings of the symbols are shown in Table 1.

#### 2.2. Photovoltaic Cell Output Characteristics Analysis

_{ref}= 25 °C, light intensity S

_{ref}= 1000 W/m

^{2}, short-circuit current I

_{sc}= 1.2 A, open-circuit voltage U

_{oc}= 21.6 V, maximum power current I

_{m}= 1.1 A, and maximum power voltage U

_{m}= 18 V. According to this parameter, to build the MATLAB simulation model, the P-U and I-U output characteristic curves of the single solar panel used are shown in Figure 3. As shown by the P-U characteristic curve in Figure 3, when the light intensity and temperature of the environment in which the photovoltaic cells are located are fixed values, its output power is a curve that varies with the output voltage, and its change law is first increasing and then decreasing, and there is a unique maximum point, which is the maximum power point [23]. When the surrounding environmental conditions changes, the value of the maximum power point also changes; in order to make the photovoltaic cells always work in the best condition, it is necessary to implement the strictest power point tracking control algorithm [24].

^{2}to 3000 W/m

^{2}in steps of 500 W/m

^{2}to obtain the curve of the effects of light intensity on the output characteristics of photovoltaic cells. Then, the light intensity was set to 2000 W/m

^{2}and the ambient temperature increased from 25 °C to 35 °C in steps of 5 °C to obtain the curve of the effect of ambient temperature on the output characteristics of photovoltaic cells [25,26].

_{oc}with the increase in temperature, and the temperature has an obvious effect on the open circuit voltage. While the output characteristic curve in the linear region of constant current source is not much changed by temperature, the short-circuit current I

_{sc}is only slightly increased with the increase in temperature.

_{oc}.

## 3. MPPT Control Principle Analysis

_{0}is expressed as (11):

_{i}and R

_{i}be constants and the value of R

_{0}be a variable, which the formula (11) calculated for (12) and (13) by obtaining the first-order and second-order derivative of R

_{0}:

_{i}= R

_{0}and d

^{2}(P)/d(${R}_{0}^{2}$) < 0, at which time the load can obtain max power transmission. Buck converter is a dc-dc with lower voltage, while boost is a dc-dc converter with higher voltage. In this paper, the output voltage of solar panels is approximately 15 V, so the buck converter is used to charge lead acid batteries. In order to achieve maximum power matching, the photovoltaic cells have been output at maximum power, where the photovoltaic array and the load can be connected to a buck circuit to change the equivalent resistance of the external load in real time to make sure the internal resistance of the solar panel is matched to maximum power transfer [34]. The maximum power transfer structure of the photovoltaic cells is shown in Figure 8.

## 4. Traditional Conductance Increment Method Analysis

## 5. Design and Validation of an Improved Three-Stage Variable Step Incremental Conductivity Method

#### 5.1. Principle of Kalman Filter Algorithm

#### 5.2. Processing of Photovoltaic Cell Output Information by Kalman Filtering Algorithm

^{2}and 38 °C, the photovoltaic cell output voltage and current are detected in real time by INA226 power detection IC, and the collected information is transmitted into the Kalman filter for data pre-processing. In the process of Kalman filter algorithm parameter rectification, the process covariance matrix Q is 0.05 and the measurement noise covariance matrix R is 0.5. The photovoltaic cell output voltage after Kalman filter real-time processing effect is shown in Figure 12a, the error curve is shown in Figure 12b. The photovoltaic cell output current after Kalman filtering real-time processing effect is shown in Figure 12c, the error curve is shown in Figure 12d. After calculating, the average error between the sensor value and the real value after Kalman filtering is 0.23, and the average error between the real-time sensor detection value and the real value is 0.55. Using Kalman filtering algorithm, the sensor measurement error is reduced to 32%, which effectively improves the control accuracy of the system.

#### 5.3. Improved MPPT Control Algorithm by Kalman Filter

^{2}and temperature of 25 °C.

_{max}and queueing lower limit N

_{min}for S(k), if S(k) ≥ N

_{max}, the step size is chosen as ΔU

_{max}; if N

_{min}< S(k) < N

_{max}, the step size is chosen as ΔU; if S(k) ≤ N

_{min}, the variable step adjustment is used, and the step size is chosen as S(k) × ΔU. This method can automatically adjust the variable step size according to the change in light intensity region, which can still maintain good dynamic tracking speed and steady-state accuracy when the light intensity is changing rapidly. However, the determination of the step adjustment coefficient S(k) is too complicated, and the upper and lower limits of S(k) will change under different light intensities, which may result in the local optimal point serving as the operating point of tracking rather than the global power quality improvement. To solve this problem, combined with the characteristics of S(k) curve in Figure 13, this study suggests a new step update rule, and Figure 14 illustrates the control flow of its enhanced integer variables step conductance increment approach. In order to locate the maximum power point under the current conditions, it is first necessary in practical applications to gather the filtered PV cell output voltage and current using a power sensor, store the sensor data at the current moment into an array, and compare it with the state value at the previous moment. The purpose of tracking the maximum power point is then accomplished by adjusting perturbation step S(k) by computing the change in voltage and power.

_{f}= 550 mV, I

_{o}= 3 A) are selected as current continuity diodes in the Buck converter to protect the components from damage by induced voltage.

## 6. Hardware Circuit Design for MPPT Control

#### 6.1. Hardware Circuit Overall Structure

#### 6.2. Analysis of Test Results

^{2}and the ambient temperature to 25 °C. The light intensity was set to fluctuate at intervals, as illustrated in Figure 19 for various MPPT control algorithms, to test the way in which photovoltaic cells produced their output when the light intensity changed quickly.

^{2}and 1000 W/m

^{2}, and the average output power of different MPPT control algorithms when the light intensity reached the steady state is shown in Table 5.

^{2}and 1000 W/m

^{2}. The tracking accuracy of the perturbation observation method is the worst because the process of its maximum power point tracking has been in a fluctuating state. The analysis of the simulation results shows that both MPPT algorithms can achieve monitoring the highest power point, but the tracking speed and tracking accuracy of their algorithms are very different. Among them, the quickest dynamic reaction is provided by the constant voltage approach, but its tracking accuracy is only 96.5%, and as can be seen from Figure 19a, after tracking the maximum power point, its power generated is not constant. The perturbation observation method produces misjudgment when the light intensity changes drastically, which seriously affects the dynamic tracking accuracy, and its simulation tracking accuracy was only 86.7%. For the fixed step incremental conductance method, the perturbation step is proportional to the dynamic response speed and inversely proportional to the steady-state tracking accuracy, and the value of the fixed step has a very strong influence on the tracking effect and is easily caught in the local optimum. In this paper, the newly developed step size incremental conductance approach has very quick dynamic response times in addition to excellent tracking accuracy; meanwhile, it can solve the problems of local optimum under partial shading conditions.

## 7. Conclusions

- (1)
- By examining the effect of environmental elements on the output properties of photovoltaic cells, it can be determined that the temperature has little effect and that the short-circuit current and open-circuit voltage of photovoltaic cells both increase with an increase in light intensity.
- (2)
- The measurement error of the sensor can be decreased to 32% by utilizing the Kalman filtering technique to pre-process the output voltage and current of the photovoltaic cells. This significantly increases the steady-state accuracy of the MPPT control.
- (3)
- When Kalman filter parameters were adjusted, it was discovered that the output of the filter is strongly correlated with both the process noise covariance matrix (Q) and the measurement noise covariance matrix (R). When the value of Q increases, the dynamic response of the system becomes faster, but the convergence at stabilization becomes worse. When the value of R increases, the dynamic response of the system slows down, though the convergence stability of the system improves. In the process of parameter adjustment, the values of Q and R cannot be zero at the same time, and the output effect of Kalman filter is only related to the ratio of Q and R.
- (4)
- In calculating the disturbance step based on (24), there is no cyclic judgment condition, which largely improves the calculation speed of MPPT control. After physical testing, the photovoltaic MPPT control system designed in this paper has a tracking accuracy of 99.6% and low fabrication cost and fastest dynamic response time of 0.01 s in comparison to traditional MPPT control algorithms, which can meet the scenario of small power photovoltaic power generation applications.
- (5)
- In practical applications, the solar power system designed in this paper can be applied to street lights, unmanned boats and other scenarios that require power supply in the natural environment. The core of the MPPT control algorithm lies in the calculation of the perturbation step. In this paper, a three-stage perturbation step calculation method is proposed and verified, and the parameters of the perturbation step can be adjusted according to needs to meet particular design applications.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Effect of temperature on the output characteristics of the photovoltaic cells. (

**a**) I-U. (

**b**) P-U.

**Figure 5.**Effect of light intensity on the output characteristics of the photovoltaic cells. (

**a**) I-U. (

**b**) P-U.

**Figure 12.**(

**a**) Results of Kalman filtering algorithm for output voltage. (

**b**) Error between the voltage filtered value and the real value. (

**c**) Results of Kalman filtering algorithm for output current. (

**d**) Error between the current filtered value and the real value.

**Figure 16.**(

**a**) Outdoor practical working test chart. (

**b**) Manufactured PCB for the DC–DC converter optimally designed for the photovoltaic application. (

**c**) Main control board details.

**Figure 19.**(

**a**) Constant voltage method. (

**b**) Perturbation observation method. (

**c**) Constant step conductance increment method. (

**d**) Three-stage variable step conductance increment method based on Kalman filter.

Symbol | Quantity | Units |
---|---|---|

t_{ref} | Ambient temperature under STC | °C |

S_{ref} | Light intensity under STC | W/m^{2} |

I_{sc} | Short Circuit Current | A |

U_{oc} | Open Circuit Voltage | V |

a | 0.00255 | °C |

b | 0.5 | \ |

c | 0.00288 | °C |

Symbol | Quantity |
---|---|

A | State shift matrix |

B | Input control matrix |

H | Observational model matrix |

P_{k} | Error covariance matrix |

Q | Process noise covariance matrix |

R | Measurement noise covariance matrix |

I | Unit matrix |

K | Kalman gain |

- | represents the priori value |

^ | represents the estimated value |

Parameters | Values |
---|---|

Maximum Power | 19.8 W |

Open circuit voltage (Voc) | 21.6 V |

Short-circuit current (Isc) | 1.2 A |

Voltage at maximum power point (Vmp) | 18 V |

Current at maximum power point (Imp) | 1.1 A |

Filter Capacitor (C) | 47 uF |

Load resistance (R) | 1 K |

Power Inductor (L) | 6.8 uH |

Response Time | |||
---|---|---|---|

MPPT Algorithms | Start-up time | Light reduction | Light increase |

Constant voltage method | 0.1 | 0.005 | 0.004 |

Perturbation observation method | 1.02 | 0.25 | 0.19 |

Constant step conductance increment method | 1.2 | 0.28 | 0.35 |

Improved MPPT algorithms | 0.12 | 0.01 | 0.06 |

MPPT Algorithms | 800 W/m^{2} | 1000 W/m^{2} | Tracking Accuracy |
---|---|---|---|

Constant voltage method | 15.76 | 19.69 | 96.5% |

Perturbation observation method | 15.35 | 19.08 | 86.7% |

Constant step conductance increment method | 15.59 | 19.77 | 97.5% |

Improved MPPT algorithms | 15.85 | 19.78 | 99.6% |

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## Share and Cite

**MDPI and ACS Style**

Meng, Y.; Chen, Z.; Cheng, H.; Wang, E.; Tan, B. An Efficient Variable Step Solar Maximum Power Point Tracking Algorithm. *Energies* **2023**, *16*, 1299.
https://doi.org/10.3390/en16031299

**AMA Style**

Meng Y, Chen Z, Cheng H, Wang E, Tan B. An Efficient Variable Step Solar Maximum Power Point Tracking Algorithm. *Energies*. 2023; 16(3):1299.
https://doi.org/10.3390/en16031299

**Chicago/Turabian Style**

Meng, Yang, Zunliang Chen, Hui Cheng, Enpu Wang, and Baohua Tan. 2023. "An Efficient Variable Step Solar Maximum Power Point Tracking Algorithm" *Energies* 16, no. 3: 1299.
https://doi.org/10.3390/en16031299