# Double-Linear Approximation Algorithm to Achieve Maximum-Power-Point Tracking for Photovoltaic Arrays

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Characteristics of PV Arrays

_{ph}stands for the cell photocurrent source, D

_{j}represents the p-n junction, R

_{j}, R

_{sh}and R

_{s}are the p-n junction nonlinear impedance, intrinsic shunt resistance and intrinsic series resistance, respectively. The series resistance R

_{s}is relatively small and the shunt resistance R

_{sh}is relatively large. Therefore, the equivalent circuit can be simplified by neglecting both resistors.

_{PV}, can be described as:

_{PV}is output voltage of PV arrays, n

_{s}is the total number of cells in series, n

_{p}stands for the total number of cells in parallel, q denotes the charges of an electron (1.6 × 10

^{−19}coulomb), k is the Boltzmann constant (1.38 × 10

^{−23}J/°K), T is temperature of the PV arrays (°K), and A represents ideality factor of the p-n junction (between 1 and 5) [16]. In addition, I

_{sat}is the reversed saturation current of the PV cell, which depends on temperature of PV arrays and it can be expressed by the following equation:

_{r}is the cell reference temperature, I

_{rr}is the corresponding reversed saturation current at T

_{r}, and E

_{gap}stands for band-gap energy of the semiconductor in the PV cell. In (1), the I

_{ph}varies with irradiation S

_{i}and PV array temperature T, which can be represented as:

_{sso}is the short-circuit current while reference irradiation is 100 mW/cm

^{2}and reference temperature is set at T

_{r}, and K

_{i}is the temperature coefficient. Based on (1), output power (P

_{PV}) of PV arrays then can be determined as follows:

_{PV}varies with irradiation S

_{i}and PV-array temperature T.

_{PV}–V

_{PV}and P

_{PV}–V

_{PV}can be plotted. Simulated P

_{PV}–V

_{PV}curves under various irradiations are shown in Figure 2 for a fixed module temperature (25 °C). In the case of constant irradiation (1000 W/m

^{2}), Figure 3 shows the relationship between P

_{PV}and V

_{PV}under various module temperatures.

Model | SM55 |
---|---|

Typical peak power (P_{P}) | 55 W |

Voltage at peak power (V_{PP}) | 17.4 V |

Current at peak power (I_{PP}) | 3.15 A |

Short-circuit current (I_{SC}) | 3.45 A |

Open-circuit voltage (V_{OC}) | 21.7 V |

Temperature coefficient of open-circuit voltage | −0.077 V/°C |

Temperature coefficient of short-circuit current (K_{i}) | 1.2 mA/°C |

_{i}, will affect the generated PV power significantly. To improve system efficiency, an MPPT algorithm has to be adopted to draw maximum power from PV arrays.

## 3. The Proposed MPPT Algorithm

_{ref,MPPT}to draw maximum power from PV arrays can be obtained:

_{s}and n

_{p}are one. Then, by substituting (5) into (4), the maximum power P

_{MPPT}is expressed as:

_{MPPT}and V

_{ref,MPPT}under constant module temperature while irradiation varies from 200 to 1000 W/m

^{2}. In the case of fixed irradiation, the trajectory of P

_{MPPT}–V

_{ref,MPPT}with an increase of temperature from 25 to 65 °C is shown in Figure 5. Figure 4 and Figure 5 reveal that P

_{MPPT}is linear to V

_{ref,MPPT}approximately. In addition, based on Equation (4), the curves of V

_{ref,MPPT}–T and V

_{ref,MPPT}–S

_{i}are shown in Figure 6 and Figure 7, respectively, both of which can be approximated by straight lines. As a result, once a V

_{ref,MPPT}is obtained, the MPPT is achieved readily. A corresponding analog circuit to determine V

_{ref,MPPT}is designed and shown in Figure 8. We choose a photo-diode (PD) for irradiation sensing to realize the first linear approximation shown in Figure 7 while a negative temperature coefficient of thermal resistor (NTC) is adopted for temperature detecting to practice the second linear approximation shown in Figure 6. In the DLAA circuit, the potential E

_{i}is proportional to the magnitude of irradiation and then, the first approximation is determined by:

**Figure 4.**The relationship between P

_{MPPT}and V

_{ref,MPPT}while irradiation increases from 200 to 1000 W/m

^{2}.

**Figure 5.**The relationship between P

_{MPPT}and V

_{ref,MPPT}while module temperature increases from 25 to 65 °C.

**Figure 8.**The proposed DLAA circuit to determine a voltage command achieving MPPT with analog devices.

## 4. Implementation Example

_{f}

_{,min}are computed as:

_{o}is output voltage, P

_{o}stands for output power, and T expresses switching period. In addition, the minimum output capacitance C

_{o}

_{,min}is calculated from:

_{o}represents the peak-to-peak ripple voltage at the output. Some important parameters of the system are listed as follows:

- PV arrays: SIEMENS SM55 (four units in series),
- C
_{i}= 100 μF, C_{o}= 100 μF, - L
_{f}= 0.2 mH, - K
_{p}= 0.0295, - K
_{i}= 0.0016, - active power switch: IRF540N, and ultrafast diode: FRF1601CT.

_{ref,MPPT}corresponding to an atmospheric condition. The PV output voltage is sensed and compared with the V

_{ref,MPPT}. Through the simple PI controller an appropriate control signal is generated to regulate the PV output voltage so that the DC/DC converter draws maximum power from PV arrays. Then, the DC/DC converter injects the power into DC bus for DC-distribution applications or into utility via a grid-connection DC/AC inverter.

**Figure 9.**Illustration of an implementation example which controls PV voltage to fulfill MPPT feature with the proposed DLAA circuit.

_{M}to the energy provided by a PV simulator in the MPP. That is:

## 5. Simulated and Experimental Results

**Figure 18.**Simulated result: tracking trajectory of the converter with the POM while PV power steps up from 100 to 200 W. Note: power: 50 W/div; time: 2 s/div.

**Figure 19.**Simulated result: tracking trajectory of the converter with the DLAA while PV power steps up from 100 to 200 W. Note: power: 50 W/div; time: 1 s/div.

**Figure 20.**Simulated result: tracking trajectory of the converter with the POM while PV power steps down from 200 to 100 W. Note: power: 50 W/div; time: 2 s/div.

**Figure 21.**Simulated result: tracking trajectory of the converter with the DLAA while PV power steps down from 200 to 100 W. Note: power: 50 W/div; time: 1 s/div.

**Figure 22.**Practical measurement: tracking trajectory of the converter with the POM while PV power steps up from 100 to 200 W. Note: power: 50 W/div; time: 1 s/div.

**Figure 23.**Practical measurement: tracking trajectory of the converter with the DLAA while PV power steps up from 100 to 200 W. Note: power: 50 W/div; time: 0.5 s/div.

**Figure 24.**Practical measurement: tracking trajectory of the converter with the POM while PV power steps down from 200 to 100 W. Note: power: 50 W/div; time: 2 s/div.

**Figure 25.**Practical measurement: tracking trajectory of the converter with the DLAA while PV power steps down from 200 to 100 W. Note: power: 50 W/div; time: 0.5 s/div.

## 6. Conclusions

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**MDPI and ACS Style**

Shen, C.-L.; Tsai, C.-T.
Double-Linear Approximation Algorithm to Achieve Maximum-Power-Point Tracking for Photovoltaic Arrays. *Energies* **2012**, *5*, 1982-1997.
https://doi.org/10.3390/en5061982

**AMA Style**

Shen C-L, Tsai C-T.
Double-Linear Approximation Algorithm to Achieve Maximum-Power-Point Tracking for Photovoltaic Arrays. *Energies*. 2012; 5(6):1982-1997.
https://doi.org/10.3390/en5061982

**Chicago/Turabian Style**

Shen, Chih-Lung, and Cheng-Tao Tsai.
2012. "Double-Linear Approximation Algorithm to Achieve Maximum-Power-Point Tracking for Photovoltaic Arrays" *Energies* 5, no. 6: 1982-1997.
https://doi.org/10.3390/en5061982