# Maximum Power Point Tracking Constraint Conditions and Two Control Methods for Isolated Photovoltaic Systems

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{sc}, V

_{oc}, I

_{m}and V

_{m}) provided by manufacturers and, based on the derivation of the circuit model, to simplify the modeling process, which is called engineering modeling. Under standard test conditions (STC; solar irradiance S is 1000 W/m

^{2}, and PV cell temperature T is 25 °C), the PV cell engineering model is obtained using Equations (1)–(3), where I, V, I

_{sc}, V

_{oc}, I

_{m}and V

_{m}represent the output current, voltage of the PV cell, short-circuit current, open-circuit voltage, MPP current and voltage of the PV cell at STC, respectively [4].

- (1)
- The mathematical models of isolated PV systems are established, and the mathematical relationships between the output power of the PV systems and the weather conditions are found.
- (2)
- The MCCs of isolated PV systems are found based on the engineering model and the MPP linear model. The relationships between MCCs and the weather conditions, circuit parameters and system structure are obtained.
- (3)
- The practicality of the MPPT control algorithm can be enhanced. The problem of MPPT failure can be avoided by fully considering the MCCs in the design and improvement of the MPPT algorithm. Therefore, two MPPT methods, which are applicable to different PV system structures, are proposed to improve the stability, applicability and rapidity of MPPT control.

## 2. Materials and Methods

#### 2.1. Integrative Model of Isolated PV Systems

_{o}and V

_{o}denote the output current and output voltage of the isolated DC/DC converter, respectively. R

_{i}and R

_{L}denote the equivalent resistances after the PV cell and after the isolated DC/DC converter, respectively.

- (1)
- All circuit components are ideal;
- (2)
- The isolated DC/DC converter operates in the continuous-current mode (CCM).

_{o}denotes the output power of the PV system.

_{1}/N

_{2}.

_{r}and I

_{r}denote the RMS values of the output AC voltage and AC current for the inverter, respectively.

_{omax}appears in the slope of the curve at 0. Therefore, in order to find the MCCs of PV systems with different structures, their mathematical models are analyzed by substituting each of them into Equation (29).

_{3}is represented by Equation (34).

_{3}is only related to the parameters of the PV cell itself (S and T). The simulation experiments revealed that P

_{omax}is only affected by S and T and is independent of R

_{L}and n. Therefore, only the values of C

_{3}and P

_{omax}under different weather conditions are required to derive the relationship between D

_{max}and R

_{L}, n. This leads to the MPPT control of isolated PV systems to improve the efficiency. The C

_{3}-S, C

_{3}-T, P

_{omax}-S and P

_{omax}-T curves under different weather conditions were plotted using MATLAB, and by applying the curve-fitting method, Equations (35) and (36) can be obtained.

_{3}and P

_{omax}can be easily derived from the weather conditions. Meanwhile, in order to find the MCCs and improve the MPPT methodology of isolated PV systems, D

_{max}can also be derived by combining the circuit parameters R

_{L}and n.

_{iMPP}, V

_{MPP}and I

_{MPP}represent the values of R

_{i}, V and I at the MPP in Figure 2, respectively.

_{sM}is involved in the design of MPPT as the output signal of the model.

#### 2.2. MCCs Based on the Engineering Model

_{max}represents D at the MPP.

_{L}range in which the PV-Forward system can successfully track the MPP.

_{L1}, while their maximum D is D

_{U1}, and the minimum D of the half-bridge, full-bridge and push–pull converters is D

_{L2}, while their maximum D is D

_{U2}. At this point, the duty cycle ranges of the forward and flyback converters can be expressed by Equation (49), and the half-bridge, full-bridge and push–pull converter duty cycle ranges can be expressed by Equation (50).

_{L}range in which the PV-Forward system can successfully track the MPP in practical applications.

#### 2.3. MCCs Based on the MPP Linear Model

#### 2.3.1. Expression of MCCs

_{sM}and R

_{sM}are quantities that vary with the weather conditions (S and T).

_{i}of the PV-Forward system can be expressed by Equation (55), and R

_{i}will vary with the different output devices and the transformations of isolated DC/DC converters.

_{i}of the PV-Forward-INV system can be expressed by Equation (56).

_{i}, R

_{L}and n) and the control signals (D and M). On the basis of these expressions, the MCCs can be found.

_{L}range in which the PV-Forward system can successfully track the MPP.

_{L}or n value always exists in the PV-Flyback system to match the conditions for the use of the MPP linear model. Also, Table 3 shows that under ideal conditions, a V

_{Dbus}or n value always exists in the PV-Flyback-Dbus system to match the use of the linear model. In contrast, for other PV systems, some constraints always exist. In addition, the use of inverters in isolated PV systems also affects the ranges of R

_{L}and n. For the PV-Forward-INV, PV-Half-bridge-INV and PV-Full-bridge-INV systems, the presence of inverters narrows the ranges of R

_{L}and n. Obviously, the expressions shown in Table 3 are the theoretical expressions of the MCCs, which can be used as the basis for designing the MPPT control process and proposing the MPPT control strategy under ideal conditions.

_{L}range in which MPPT control can be successfully realized in practical applications for the PV-Forward system.

_{L}and V

_{Dbus}when compared with those in Table 3. Unlike the ideal case, the PV-Flyback, PV-Flyback-Dbus and PV-Flyback-INV systems have certain constraints in practical applications. Obviously, the expressions in Table 4 provide a theoretical basis for isolated PV systems on the basis of the MPP linear model in practical applications.

#### 2.3.2. Range of MCCs

_{sM}and R

_{sM}have been derived for changing weather conditions. Therefore, the extreme values of MCCs for practical applications are shown in Table 5. It can be seen that the maximum range of R

_{L}(or V

_{Dbus}) is necessary for each PV system to be modeled with the MPP linear cell. By contrast, the minimum range of R

_{L}(or V

_{Dbus}) is a sufficient condition for every PV system to use the MPP linear model. Similarly, the maximum and minimum ranges of the variable ratio n can be derived analogously.

#### 2.4. Two New MPPT Methods Based on MPP Linear Modeling

#### 2.4.1. MPPT Method for PV Systems with Resistive Output (RMPPT)

_{max}to the weather conditions (S and T) and the circuit parameters (R

_{L}and n) when the PV-Flyback system operates at the MPP.

_{L}and n are measured or known. Equation (62) is the theoretical basis of RMPPT, which can be described as follows: by measuring or knowing S and T as well as R

_{L}and n, the duty cycle D

_{max}at the MPP for the isolated PV system can be calculated, and the microcontroller or chip can realize MPPT control by controlling D = D

_{max}.

_{max}value of the PV system when it is located at the MPP attachment can be simply calculated by using a microcontroller or chip to measure or know the weather parameters (S and T) and the circuit parameters (n, V

_{o}and I

_{o}), calculating the load resistor R

_{L}and then substituting these parameters into Equation (62). When the input is a PV array, the cost of the sensor can be reduced by sharing the irradiance sensor if S is uniform in a certain area. Also, the cost of voltage sampling and current sampling can be reduced if R

_{L}is essentially the same for each PV system. It can be seen that the implementation of RMPPT requires only a simple process with low computational complexity, which can greatly reduce the hardware cost and program design of an isolated PV system.

#### 2.4.2. MPPT Method with Output as DC Bus (BMPPT)

_{max}to the weather conditions (S and T) and the circuit parameters (V

_{Dbus}and n) when the PV-Flyback-Dbus system operates at MPP.

_{Dbus}and n can be measured or known, BMPPT can be used. Equation (63) is the theoretical basis of BMPPT, which can be described as follows: From the measured or known S and T, as well as V

_{Dbus}and n, the duty cycle at the MPP D

_{max}of the isolated PV system can be calculated. Then, the microcontroller or chip makes the duty cycle of the PWM wave equal to D

_{max}, thereby achieving MPPT control. In contrast to RMPPT, BMPPT need not collect the output current. Eliminating the current-sampling device from the hardware design reduces the design difficulty and cost of the PV system and also reduces the current-sampling program designed for the software. When the output is a DC bus, BMPPT has an obvious advantage.

_{max}for a PV system located at the MPP attachment can be simply calculated by using a microcontroller or chip to measure or know the weather conditions (S and T) and the circuit parameters (n and V

_{Dbus}) and then substituting these parameters into Equation (63). Similarly, in the case of multiple PV arrays at the input, the cost of the sensors can be reduced by sharing irradiance sensors if S is uniform in a certain area. At the same time, multiple PV cells simplify the design of voltage-sampling circuits and reduce hardware and software costs by sharing a common set of DC buses.

## 3. Results

#### 3.1. Simulation Verification of MCCs Based on the Engineering Model

_{L}= 5 Ω or V

_{Dbus}= 500 V for three cases, respectively. The experimental results are shown in Figure 8. The four factory parameter settings of this PV cell model are the same as in the first PV cell (1Soltech 1STH-215-P) of the PV array module in MATLAB/Simulink, which are I

_{sc}= 7.84 A, V

_{oc}= 36.3 V, I

_{m}= 7.35 A and V

_{m}= 29 V, respectively.

_{L1}, D

_{U1}, D

_{L2}and D

_{U2}are taken as 0.2, 0.8, 0.1 and 0.45, respectively, that R

_{L}= 5 Ω or V

_{Dbus}= 500 V for the study of the range of n, and that n = 0.1 for the study of the range of R

_{L}or V

_{Dbus}. The calculated maximum and minimum values of the circuit parameter range for a PV system with a forward converter and a full-bridge converter capable of successful MPPT are shown in Table 6, where R

_{Lmax}and R

_{Lmin}denote the maximum and minimum values of R

_{L}, respectively, n

_{max}and n

_{min}denote the maximum and minimum values of n, respectively, and V

_{Dmax}and V

_{Dmin}denote the maximum and minimum values of V

_{Dbus,}respectively. These data are compared with Figure 8 to analyze the reasonableness and accuracy of the MCCs.

_{L}= R

_{Lmin}or V

_{Dbus}= V

_{Dmin}is satisfied, the MPP is reached exactly at D = D

_{L1}. When R

_{L}= R

_{Lmax}or V

_{Dbus}= V

_{Dmax}is satisfied, the MPP is reached exactly at D = D

_{U1}. When R

_{L}or V

_{Dbus}is certain and n = n

_{min}is satisfied, the MPP is reached exactly at D = D

_{L1}. When n = n

_{max}, the MPP is reached exactly at D = D

_{U1}.

_{L2}when n is certain and R

_{L}= R

_{Lmin}or V

_{Dbus}= V

_{Dmin}is satisfied. When R

_{L}= R

_{Lmax}or V

_{Dbus}= V

_{Dmax}is satisfied, the MPP is reached exactly at D = D

_{U2}. When R

_{L}or V

_{Dbus}is certain and n = n

_{min}is satisfied, the MPP is reached exactly at D = D

_{L2}. When n = n

_{max}is satisfied, the MPP is reached exactly at D = D

_{U2}.

#### 3.2. Simulation Verification of MCCs Based on MPP Linear Model

#### 3.2.1. Accuracy Verification of MCCs

_{L1}, D

_{U1}, D

_{L2}and D

_{U2}are taken as 0.2, 0.8, 0.1 and 0.45, respectively, and R

_{L}is equal to 0.5 Ω. The simulation results are shown in Figure 9. Figure 9 compares the curves of D

_{max}variation with n for the PV-Forward, PV-Flyback, PV-Half-bridge and PV-Full-bridge systems under four weather conditions. Meanwhile, the MCCs in Table 4 are calculated, and the results are shown in Table 8. They can verify the accuracy of the simulation results in Figure 9 and Table 4.

_{max}remains at 0.2 when $n<{D}_{\mathrm{L}1}\sqrt{{R}_{\mathrm{sM}}}/\sqrt{{R}_{\mathrm{L}}}$ and 0.8 when $n>{D}_{\mathrm{U}1}\sqrt{{R}_{\mathrm{sM}}}/\sqrt{{R}_{\mathrm{L}}}$, which implies that the MPP does not exist outside the range of n, and the MPP linear model cannot be used. In Figure 9b, it can be seen that, for the PV-Flyback system, D

_{max}stays at 0.2 when $n<{D}_{\mathrm{L}1}\sqrt{{R}_{\mathrm{sM}}}/[\sqrt{{R}_{\mathrm{L}}}(1-{D}_{\mathrm{L}1})]$, while when $n>{D}_{\mathrm{U}1}\sqrt{{R}_{\mathrm{sM}}}/[\sqrt{{R}_{\mathrm{L}}}(1-{D}_{\mathrm{U}1})]$, Dmax stays at 0.8, which means that the MPP does not exist outside the range of n, and the MPP linear model cannot be used. In Figure 9c, it can be seen that, for the PV-Half-bridge system, D

_{min}stays at 0.1 when $n<{D}_{\mathrm{L}2}\sqrt{{R}_{\mathrm{sM}}}/\sqrt{{R}_{\mathrm{L}}}$, while D

_{max}stays at 0.45 when $n>{D}_{\mathrm{U}2}\sqrt{{R}_{\mathrm{sM}}}/\sqrt{{R}_{\mathrm{L}}}$, which implies that the MPP does not exist outside of the range of n, and the MPP linear model cannot be used. In Figure 9d, it can be seen that the PV-Full-bridge system maintains D

_{min}at 0.1 when $n<2{D}_{\mathrm{L}2}\sqrt{{R}_{\mathrm{sM}}}/\sqrt{{R}_{\mathrm{L}}}$, while D

_{max}remains at 0.45 under the condition of $n>2{D}_{\mathrm{U}2}\sqrt{{R}_{\mathrm{sM}}}/\sqrt{{R}_{\mathrm{L}}}$, which implies that the MPP does not exist outside the range of n, and the MPP linear model cannot be used.

_{max}of the PV system varies with n when n is within the MCCs. In this case, the MPP always exists, and the MPP linear model can be used for these four PV systems. The simulation results shown in Figure 9 are consistent with the corresponding data in Table 8, whereas the D

_{max}-n curves of PV systems under different weather conditions differ significantly. Therefore, it can be concluded that the practical expressions of MCCs for various isolated PV systems in Table 4 are accurate for the PV-Forward, PV-Flyback, PV-Half-bridge and PV-Full-bridge systems.

#### 3.2.2. Comparison of MCCs

_{L1}, D

_{L2}, D

_{U1}, D

_{U2}, n, R

_{L}and V

_{Dbus}are the same as in Section 3.2.1. In this case, Table 9 shows the calculated values according to Table 5.

_{LminFD}and R

_{LmaxHB}denote the maximum and minimum values of R

_{L}for the PV-Forward system and PV-Half-bridge system, respectively. Other circuit parameter boundaries are also presented in Figure 10.

_{L}< 4R

_{sM}, only the PV system using the half-bridge converter can successfully realize MPPT control. When the weather parameters and load resistance are certain, the maximum value of load resistance for the PV system using the full-bridge converter is about two times that for the PV system using the half-bridge converter. Meanwhile, only the PV system using the half-bridge converter can successfully realize MPPT control when n < 0.28. When an inverter is connected to the PV system, no matter what kind of converter is used as the MPPT control circuit, the range of circuit parameters is reduced to a certain extent. When the flyback converter is used, the load resistance, transformer ratio or bus voltage range is much larger than that of other isolated PV systems. Since both R

_{sM}and V

_{sM}are functions of S and T, the load, transformer ratio or bus voltage range changes with S and T. In addition, Figure 10 not only shows the range of variation in R

_{L}, n and V

_{Dbus}but also verifies the accuracy of the boundary values given in Table 9. The MPP linear model can be used only if the MPP is always present in the isolated PV system within this range.

#### 3.3. Simulation Analysis of RMPPT

_{L}are equal to 2 and 1.7 Ω, respectively. In addition, the capacitors, inductors and transformers in the circuit are ideal components, the switching components are MOSFETs, and the PWM wave frequency is 15 kHz.

_{max}and D

_{max1}denote D values at the MPP when the RMPPT and P&O methods are used, respectively. P

_{omax}and P

_{omax1}denote the maximum output power values of the PV cell when the RMPPT and P&O methods are used, respectively. P

_{omax2}denotes the maximum output power of the PV system. The parameter settings are n = 1/10 and R

_{L}= 500 Ω. The P&O method step size is set to 0.005.

_{max}and P

_{omax}calculated by RMPPT are basically equal to D

_{max1}and P

_{max1}, respectively. This proves the practicality of RMPPT. In addition, it can be seen from P

_{omax1}and P

_{omax2}that there is a difference between them due to the loss of the circuit components, the average value of which is the circuit loss, which is calculated to be about 2.41W.

- (1)
- Simulation experiment of irradiance change

^{2}and T = 25 °C; at 0.3~0.7 s, S = 1200 W/m

^{2}and T = 25 °C; and at 0.7~1 s, S = 400 W/m

^{2}and T = 25 °C. Figure 11 shows the simulation results.

_{max}, which is the reason why the output power of the P&O method oscillates at the MPP, while the RMPPT stabilizes at the MPP. It can also be seen in Figure 11 that, after the sudden change in S, D is actively adjusted to the new D

_{max}, and the P

_{omax}of the PV cell is also stabilized to the new P

_{omax}after a rapid adjustment, which also proves the correctness of the conclusion in Section 2.1.

- (2)
- Simulation experiment of R
_{L}change

_{L}. It can also be seen in Figure 12 that D

_{max}is actively adjusted to the new D

_{max}after a sudden change in R

_{L}, but P

_{omax}remains at the same value after a short transient adjustment.

#### 3.4. Simulation Analysis of BMPPT

_{Dbus}= 25 V, the capacitors, inductors and transformers in the circuit are ideal components, the switching components are MOSFETs, and the PWM wave frequency is 15 kHz. The simulation experiment results under varying temperature and DC bus voltage conditions are shown in Figure 13.

_{Dbus}. It can also be seen in Figure 13b,c that, after a sudden change in T, D

_{max}is actively adjusted to the new D

_{max}, and P

_{omax}is also stabilized to the new P

_{omax}after a rapid stepwise adjustment.

## 4. Discussion

## 5. Conclusions

- (1)
- The overall mathematical models of twenty isolated PV systems are established. And the relationships between the output power of isolated PV systems, the parameters of the PV cell and circuit parameters are found.
- (2)
- The MCCs are found for isolated PV systems with different topologies and outputs on the basis of the PV cell engineering model and MPP linear model, respectively. They are a good guide for the circuit design, theoretical derivation and product selection of PV systems.
- (3)
- Based on the MPP linear model and MCC, two MPPT methods (RMPPT and BMPPT) applicable to different output conditions are proposed. The experimental results verify the speed and accuracy of the two proposed MPPT methods. The MPPT time is improved from 0.23 s to 0.03 s. These two methods have the advantages of a simple program, small computational volume and low hardware and software costs.

- (1)
- The theoretical derivation in this paper makes some idealized assumptions. However, there may be more complicated situations in the practical circuit, and determining how to establish the MCCs and MPPT methods for more complicated situations is an important research direction.
- (2)
- The two MPPT methods proposed put forward higher requirements on the speed, accuracy and economy of the irradiance and temperature sensors. If irradiance and temperature sensors with lower costs, higher accuracy and faster speed can be developed, the MPPT control method proposed in this paper can be more widely used.
- (3)
- The MCCs proposed in this paper are based on the premise that the irradiance of all PV cells is uniform, but due to the environmental changes that may occur in the case of the partial shading of PV cells, it is also an important direction to consider the MCCs and the MPPT method in the case of non-uniform irradiance.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MPP | Maximum power point | STC | Standard test conditions |

PV | Photovoltaic | PWM | Pulse-width modulation |

MCC | MPPT constraint conditions | DC | Direct current |

MPPT | Maximum power point tracking | AC | Alternating current |

VWP | Variable-weather parameter |

## Nomenclature

I | Output current of PV cell (A) | n | Transformer ratio of isolated DC/DC converter |

V | Output voltage of PV cell (V) | M | SPWM wave modulation ratio |

S | Solar irradiance (W/m^{2}) | V_{r} | Output voltage of inverter (V) |

T | Cell temperature (°C) | I_{r} | Output current of inverter (A) |

I_{o} | Output current of isolated DC/DC converter (A) | R_{sM} | Internal resistance of linear cell model (Ω) |

V_{o} | Output voltage of isolated DC/DC converter (V) | V_{sM} | Open-circuit voltage of MPP linear model (V) |

D | Duty cycle of the PWM signal of converter | D_{max} | D at the MPP |

I_{sc} | Short-circuit current of PV cell under STC (A) | P_{omax} | Output power at MPP (W) |

V_{oc} | Open-circuit voltage of PV cell under STC (V) | R_{iMPP} | Value of R_{i} at MPP (Ω) |

I_{m} | MPP current of PV cell under STC (A) | V_{MPP} | Value of V at MPP (Ω) |

V_{m} | MPP voltage of PV cell under STC (V) | I_{MPP} | Value of I at MPP (Ω) |

R_{i} | Input resistance of isolated DC/DC converter (Ω) | V_{Dbus} | Voltage of DC bus (V) |

R_{L} | Load or equivalent load resistance of PV system (Ω) | V_{Abus} | Voltage of AC bus (V) |

D_{L1} | Minimum D for forward and flyback converters | D_{L2} | Minimum D for half-bridge, full-bridge, push–pull converter |

D_{U1} | Maximum D for forward and flyback converters | D_{U2} | Maximum D for half-bridge, full-bridge, push–pull converter |

P_{o} | Output power of PV system (W) |

## References

- Zhao, D.; Qian, M.; Ma, J.; Yamashita, K. Photovoltaic generator model for power system dynamic studies. Sol. Energy
**2020**, 210, 101–114. [Google Scholar] [CrossRef] - Koohi-Kamali, S.; Rahim, N.A.; Mokhlis, H.; Tyagi, V.V. Photovoltaic electricity generator dynamic modeling methods for smart grid applications: A review. Renew. Sustain. Energy Rev.
**2016**, 57, 131–172. [Google Scholar] [CrossRef] - Moreira, H.S.; Silva, J.L.D.S.; Reis, M.V.G.D.; Mesquita, D.B.; Paula, B.H.K.; Villalva, M.G. Experimental comparative study of photovoltaic models for uniform and partially shading conditions. Renew. Energy
**2021**, 164, 58–73. [Google Scholar] [CrossRef] - Li, Q.; Zhao, S.; Wang, M.; Zou, Z.; Wang, B.; Chen, Q. An improved perturbation and observation maximum power point tracking algorithm based on a PV module four-parameter model for higher efficiency. Appl. Energy
**2017**, 195, 523–537. [Google Scholar] [CrossRef] - Yin, O.W.; Babu, B.C. Simple and easy approach for mathematical analysis of photovoltaic (PV) module under normal and partial shading conditions. Optik
**2018**, 169, 48–61. [Google Scholar] [CrossRef] - Zsiborács, H.; Baranyai, N.H.; Vincze, A.; Pintér, G. An economic analysis of the shading effects of transmission lines on photovoltaic power plant investment decisions: A case study. Sensors
**2021**, 21, 4973. [Google Scholar] [CrossRef] - Hsieh, Y.; Yu, L.; Chang, T.; Liu, W.; Wu, T.; Moo, C. Parameter identification of one-diode dynamic equivalent circuit model for photovoltaic panel. IEEE J. Photovolt.
**2020**, 10, 219–225. [Google Scholar] [CrossRef] - Si, X.; Chen, Q.; Gao, Z. A piecewise linearization based equivalent model of photovoltaic array. Power Syst. Technol.
**2014**, 38, 947–951. [Google Scholar] - Wang, Y.J.; Hsu, P.C. Modelling of solar cells and modules using piecewise linear parallel branches. IET Renew. Power Gener.
**2011**, 5, 215–222. [Google Scholar] [CrossRef] - Bharadwaj, P.; John, V. Linearised model for PV panel power output variation with changes in ambient conditions. Sādhanā
**2017**, 42, 2183–2187. [Google Scholar] [CrossRef] - Golshani, A.; Bathaee, S.M.T.; Moghaddas-Tafreshi, S.M. Small signal stability analysis of photovoltaic array based on averaged switch modeling technique. J. Renew. Sustain. Energy
**2012**, 4, 043117. [Google Scholar] [CrossRef] - Li, S. Linear equivalent models at the maximum power point based on variable weather parameters for photovoltaic cell. Appl. Energy
**2016**, 182, 94–104. [Google Scholar] [CrossRef] - Amir, A.; Amir, A.; Che, H.S.; Khateb, A.E.; Rahim, N.A. Comparative analysis of high voltage gain DC-DC converter topologies for photovoltaic systems. Renew. Energy
**2019**, 136, 1147–1163. [Google Scholar] [CrossRef] - Başoğlu, M.E.; Çakır, B. Comparisons of MPPT performances of isolated and non-isolated DC-DC converters by using a new approach. Renew. Sustain. Energy Rev.
**2016**, 60, 1100–1113. [Google Scholar] [CrossRef] - Brito, M.A.G.; Galotto, L.; Sampaio, L.P.; Melo, G.A.; Canesin, C.A. Evaluation of the main MPPT techniques for photovoltaic applications. IEEE Trans. Ind. Electron.
**2013**, 60, 1156–1167. [Google Scholar] [CrossRef] - Seyedmahmoudian, M.; Horan, B.; Soon, T.K.; Rahmani, R.; Oo, A.M.T.; Mekhilef, S.; Stojcevski, A. State of the art artificial intelligence-based MPPT techniques for mitigating partial shading effects on PV systems—A review. Renew. Sustain. Energy Rev.
**2016**, 64, 435–455. [Google Scholar] [CrossRef] - Hassan, T.U.; Abbassi, R.; Jerbi, H. A novel algorithm for MPPT of an isolated PV system using push pull converter with fuzzy logic controller. Energies
**2020**, 13, 4007. [Google Scholar] [CrossRef] - Islam, H.; Mekhilef, S.; Shah, N.B.M.; Soon, T.K.; Seyedmahmousian, M.; Horan, B.; Stojcevski, A. Performance evaluation of maximum power point tracking approaches and photovoltaic systems. Energies
**2018**, 11, 365. [Google Scholar] [CrossRef] - Bollipo, R.B.; Mikkili, S.; Bonthagorla, P.K. Hybrid, optimization, intelligent and classical PV MPPT techniques: Review. CSEE J. Power Energy Syst.
**2021**, 7, 25. [Google Scholar] - Mohapatra, A.; Nayak, B.; Das, P.; Mohanty, K.B. A review on MPPT techniques of PV system under partial shading condition. Renew. Sustain. Energy Rev.
**2017**, 80, 854–867. [Google Scholar] [CrossRef] - Li, S.; Ping, A.; Liu, Y.; Ma, X.; Li, C. A variable-weather-parameter MPPT method based on a defined characteristic resistance of photovoltaic cell. Sol. Energy
**2020**, 199, 673–684. [Google Scholar] [CrossRef] - Li, S. Circuit parameter range of photovoltaic system to correctly use the mpp linear model of photovoltaic cell. Energies
**2021**, 14, 3997. [Google Scholar] [CrossRef] - Chen, J.; Kang, Y. Power Electronics: Power Electronic Conversion and Control Technology, 3rd ed.; Higher Education Press: Beijing, China, 2011; pp. 81–93. [Google Scholar]
- Qiu, G.; Luo, X. Circuits, 5th ed.; Higher Education Press: Beijing, China, 2013. [Google Scholar]

**Figure 5.**Isolated PV system based on MPP linear model. (* indicates the eponymous end of the induced electromotive force of the winding).

**Figure 8.**P

_{o}-D curves of the different outputs and n. (

**a**) P

_{o}-D curves of PV-Flyback system for different R

_{L}; (

**b**) P

_{o}-D curves of PV-Flyback system for different n; (

**c**) P

_{o}-D curves of PV-Full-bridge system for different R

_{L}; (

**d**) P

_{o}-D curves of PV-Full-bridge system for different n; (

**e**) P

_{o}-D curves of PV-Flyback-Dbus for different V

_{Dbus}; (

**f**) P

_{o}-D curves of PV-Flyback-Dbus for different n; (

**g**) P

_{o}-D curves of PV-Full-bridge-Dbus for different V

_{Dbus}; (

**h**) P

_{o}-D curves of PV-Full-bridge-Dbus for different n.

**Figure 9.**P

_{o}-D curves of different PV systems. (

**a**) D

_{max}-n curves of PV-Forward system; (

**b**) D

_{max}-n curves of PV-Flyback system; (

**c**) D

_{max}-n curves of PV-Half-bridge system; (

**d**) D

_{max}-n curves of PV-Full-bridge system.

**Figure 10.**Comparison of curves of MCCs. (

**a**) PV-Half-bridge compared with PV-Full-bridge system; (

**b**) PV-Forward compared with PV-Forward-INV system; (

**c**) PV-Full-bridge compared with PV-Full-bridge-INV system; (

**d**) PV-Forward-Dbus compared with PV-Half-bridge-Dbus system.

**Figure 11.**Simulation experiment of irradiance change. (

**a**) S curve variation with t; (

**b**) comparison of P

_{omax}-t curves of RMPPT and P&O methods; (

**c**) comparison of D-t curves of RMPPT and P&O methods.

**Figure 12.**Simulation experiment of R

_{L}change. (

**a**) R

_{L}curve variation with t; (

**b**) comparison of P

_{omax}-t curves of RMPPT and P&O methods; (

**c**) comparison of D-t curves of RMPPT and P&O methods.

**Figure 13.**Simulation experiment of T and V

_{Dbus}changes. (

**a**) T curve variation with t; (

**b**) comparison of P

_{omax}-t curves of T change; (

**c**) comparison of D-t curves of T change; (

**d**) V

_{Dbus}curve variation with t; (

**e**) comparison of P

_{omax}-t curves of V

_{Dbus}change; (

**f**) comparison of D-t curves of V

_{Dbus}change.

PV System | Range of the Output | Range of n |
---|---|---|

PV-Forward | $0<{R}_{\mathrm{L}}<\frac{{C}_{3}^{2}}{{n}^{2}{P}_{\mathrm{omax}}}$ | $0<n<\frac{{C}_{3}}{\sqrt{{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Flyback | $0<{R}_{\mathrm{L}}$ | $0<n$ |

PV-Half-bridge | $0<{R}_{\mathrm{L}}<\frac{{C}_{3}^{2}}{4{n}^{2}{P}_{\mathrm{omax}}}$ | $0<n<\frac{{C}_{3}}{2\sqrt{{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Full-bridge | $0<{R}_{\mathrm{L}}\le \frac{{C}_{3}^{2}}{{n}^{2}{P}_{\mathrm{omax}}}$ | $0<n\le \frac{{C}_{3}}{\sqrt{{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Forward-Dbus | $0<{V}_{\mathrm{Dbus}}<\frac{{C}_{3}}{n}$ | $0<n<\frac{{C}_{3}}{{V}_{\mathrm{Dbus}}}$ |

PV-Flyback-Dbus | $0<{V}_{\mathrm{Dbus}}$ | $0<n$ |

PV-Half-bridge-Dbus | $0<{V}_{\mathrm{Dbus}}<\frac{{C}_{3}}{2n}$ | $0<n<\frac{{C}_{3}}{2{V}_{\mathrm{Dbus}}}$ |

PV-Full-bridge-Dbus | $0<{V}_{\mathrm{Dbus}}\le \frac{{C}_{3}}{n}$ | $0<n\le \frac{{C}_{3}}{{V}_{\mathrm{Dbus}}}$ |

PV-Forward-INV | $0<{R}_{\mathrm{L}}<\frac{{M}^{2}{C}_{3}^{2}}{2{n}^{2}{P}_{\mathrm{omax}}}$ | $0<n<\frac{M{C}_{3}}{\sqrt{2{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Flyback-INV | $0<{R}_{\mathrm{L}}$ | $0<n$ |

PV-Half-bridge-INV | $0<{R}_{\mathrm{L}}<\frac{{M}^{2}{C}_{3}^{2}}{8{n}^{2}{P}_{\mathrm{omax}}}$ | $0<n<\frac{M{C}_{3}}{2\sqrt{2{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Full-bridge-INV | $0<{R}_{\mathrm{L}}\le \frac{{M}^{2}{C}_{3}^{2}}{2{n}^{2}{P}_{\mathrm{omax}}}$ | $0<n\le \frac{M{C}_{3}}{\sqrt{2{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Forward-INV-Abus | $0<{V}_{\mathrm{Abus}}<\frac{{C}_{3}M}{\sqrt{2}n}$ | $0<n<\frac{{C}_{3}M}{\sqrt{2}{V}_{\mathrm{Abus}}}$ |

PV-Flyback-INV-Abus | $0<{V}_{\mathrm{Abus}}$ | $0<n$ |

PV-Half-bridge-INV-Abus | $0<{V}_{\mathrm{Abus}}<\frac{{C}_{3}M}{2\sqrt{2}n}$ | $0<n<\frac{{C}_{3}M}{2\sqrt{2}{V}_{\mathrm{Abus}}}$ |

PV-Full-bridge-INV-Abus | $0<{V}_{\mathrm{Abus}}\le \frac{{C}_{3}M}{\sqrt{2}n}$ | $0<n\le \frac{{C}_{3}M}{\sqrt{2}{V}_{\mathrm{Abus}}}$ |

PV System | Range of the Output | Range of n |
---|---|---|

PV-Forward | $\frac{{D}_{\mathrm{L}1}^{2}{C}_{3}^{2}}{{n}^{2}{P}_{\mathrm{omax}}}\le {R}_{L}\le \frac{{D}_{\mathrm{U}1}^{2}{C}_{3}^{2}}{{n}^{2}{P}_{\mathrm{omax}}}$ | $\frac{{D}_{\mathrm{L}1}{C}_{3}}{\sqrt{{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}\le n\le \frac{{D}_{\mathrm{U}1}{C}_{3}}{\sqrt{{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Flyback | $\frac{{D}_{\mathrm{L}1}^{2}{C}_{3}^{2}}{{n}^{2}{(1-{D}_{\mathrm{L}1})}^{2}{P}_{\mathrm{omax}}}\le {R}_{L}\le \frac{{D}_{\mathrm{U}1}^{2}{C}_{3}^{2}}{{n}^{2}{(1-{D}_{\mathrm{U}1})}^{2}{P}_{\mathrm{omax}}}$ | $\frac{{D}_{\mathrm{L}1}{C}_{3}}{(1-{D}_{\mathrm{L}1})\sqrt{{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}\le n\le \frac{{D}_{\mathrm{U}1}{C}_{3}}{(1-{D}_{\mathrm{U}1})\sqrt{{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Half-bridge | $\frac{{D}_{\mathrm{L}2}^{2}{C}_{3}^{2}}{{n}^{2}{P}_{\mathrm{omax}}}\le {R}_{L}\le \frac{{D}_{\mathrm{U}2}^{2}{C}_{3}^{2}}{{n}^{2}{\mathrm{P}}_{\mathrm{omax}}}$ | $\frac{{D}_{\mathrm{L}2}{C}_{3}}{\sqrt{{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}\le n\le \frac{{D}_{\mathrm{U}2}{C}_{3}}{\sqrt{{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Full-bridge | $\frac{4{D}_{\mathrm{L}2}^{2}{C}_{3}^{2}}{{n}^{2}{P}_{\mathrm{omax}}}\le {R}_{L}\le \frac{4{D}_{\mathrm{U}2}^{2}{C}_{3}^{2}}{{n}^{2}{P}_{\mathrm{omax}}}$ | $\frac{2{D}_{\mathrm{L}2}{C}_{3}}{\sqrt{{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}\le n\le \frac{2{D}_{\mathrm{U}2}{C}_{3}}{\sqrt{{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Forward-Dbus | $\frac{{C}_{3}{D}_{\mathrm{L}1}}{n}\le {V}_{\mathrm{Dbus}}\le \frac{{C}_{3}{D}_{\mathrm{U}1}}{n}$ | $\frac{{C}_{3}{D}_{\mathrm{L}1}}{{V}_{\mathrm{Dbus}}}\le n\le \frac{{C}_{3}{D}_{\mathrm{U}1}}{{V}_{\mathrm{Dbus}}}$ |

PV-Flyback-Dbus | $\frac{{C}_{3}{D}_{\mathrm{L}1}}{n(1-{D}_{\mathrm{L}1})}<{V}_{\mathrm{Dbus}}\le \frac{{C}_{3}{D}_{\mathrm{U}1}}{n(1-{D}_{\mathrm{U}1})}$ | $\frac{{C}_{3}{D}_{\mathrm{L}1}}{{V}_{\mathrm{Dbus}}(1-{D}_{\mathrm{L}1})}<n\le \frac{{C}_{3}{D}_{\mathrm{U}1}}{{V}_{\mathrm{Dbus}}(1-{D}_{\mathrm{U}1})}$ |

PV-Half-bridge-Dbus | $\frac{{C}_{3}{D}_{\mathrm{L}2}}{n}\le {V}_{\mathrm{Dbus}}\le \frac{{C}_{3}{D}_{\mathrm{U}2}}{n}$ | $\frac{{C}_{3}{D}_{\mathrm{L}2}}{{V}_{\mathrm{Dbus}}}\le n\le \frac{{C}_{3}{D}_{\mathrm{U}2}}{{V}_{\mathrm{Dbus}}}$ |

PV-Full-bridge-Dbus | $\frac{2{C}_{3}{D}_{\mathrm{L}2}}{n}\le {V}_{\mathrm{Dbus}}\le \frac{2{C}_{3}{D}_{\mathrm{U}2}}{n}$ | $\frac{2{C}_{3}{D}_{\mathrm{L}2}}{{V}_{\mathrm{Dbus}}}\le n\le \frac{2{C}_{3}{D}_{\mathrm{U}2}}{{V}_{\mathrm{Dbus}}}$ |

PV-Forward-INV | $\frac{{M}^{2}{C}_{3}^{2}{D}_{\mathrm{L}1}^{2}}{2{n}^{2}{P}_{\mathrm{omax}}}\le {R}_{\mathrm{L}}\le \frac{{M}^{2}{C}_{3}^{2}{D}_{\mathrm{U}1}^{2}}{2{n}^{2}{P}_{\mathrm{omax}}}$ | $\frac{M{D}_{\mathrm{L}1}{C}_{3}}{\sqrt{2{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}\le n\le \frac{M{D}_{\mathrm{U}1}{C}_{3}}{\sqrt{2{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Flyback-INV | $\frac{{M}^{2}{C}_{3}^{2}{D}_{\mathrm{L}1}^{2}}{2{n}^{2}{P}_{\mathrm{omax}}{(1-{D}_{\mathrm{L}1})}^{2}}\le {R}_{\mathrm{L}}\le \frac{{M}^{2}{C}_{3}^{2}{D}_{\mathrm{U}1}^{2}}{2{n}^{2}{P}_{\mathrm{omax}}{(1-{D}_{\mathrm{U}1})}^{2}}$ | $\frac{M{D}_{\mathrm{L}1}{C}_{3}}{(1-{D}_{\mathrm{L}1})\sqrt{2{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}\le n\le \frac{M{D}_{\mathrm{U}1}{C}_{3}}{(1-{D}_{\mathrm{U}1})\sqrt{2{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Half-bridge-INV | $\frac{{M}^{2}{C}_{3}^{2}{D}_{\mathrm{L}2}^{2}}{2{n}^{2}{P}_{\mathrm{omax}}}\le {R}_{\mathrm{L}}\le \frac{{M}^{2}{C}_{3}^{2}{D}_{\mathrm{U}2}^{2}}{2{n}^{2}{P}_{\mathrm{omax}}}$ | $\frac{M{D}_{\mathrm{L}2}{C}_{3}}{\sqrt{2{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}\le n\le \frac{M{D}_{\mathrm{U}2}{C}_{3}}{\sqrt{2{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Full-bridge-INV | $\frac{2{M}^{2}{C}_{3}^{2}{D}_{\mathrm{L}2}^{2}}{{n}^{2}{P}_{\mathrm{omax}}}\le {R}_{\mathrm{L}}\le \frac{2{M}^{2}{C}_{3}^{2}{D}_{\mathrm{U}2}^{2}}{{n}^{2}{P}_{\mathrm{omax}}}$ | $\frac{\sqrt{2}M{D}_{\mathrm{L}2}{C}_{3}}{\sqrt{{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}\le n\le \frac{\sqrt{2}M{D}_{\mathrm{U}2}{C}_{3}}{\sqrt{{P}_{\mathrm{omax}}{R}_{\mathrm{L}}}}$ |

PV-Forward-INV-Abus | $\frac{{C}_{3}M{D}_{\mathrm{L}1}}{\sqrt{2}n}\le {V}_{\mathrm{Abus}}\le \frac{{C}_{3}M{D}_{\mathrm{U}1}}{\sqrt{2}n}$ | $\frac{{C}_{3}M{D}_{\mathrm{L}1}}{\sqrt{2}{V}_{\mathrm{Abus}}}\le n\le \frac{{C}_{3}M{D}_{\mathrm{U}1}}{\sqrt{2}{V}_{\mathrm{Abus}}}$ |

PV-Flyback-INV-Abus | $\frac{{C}_{3}M{D}_{\mathrm{L}1}}{\sqrt{2}n(1-{D}_{\mathrm{L}1})}\le {V}_{\mathrm{Abus}}\le \frac{{C}_{3}M{D}_{\mathrm{U}1}}{\sqrt{2}n(1-{D}_{\mathrm{U}1})}$ | $\frac{{C}_{3}M{D}_{\mathrm{L}1}}{\sqrt{2}{V}_{\mathrm{Abus}}(1-{D}_{L1})}\le n\le \frac{{C}_{3}M{D}_{\mathrm{U}1}}{\sqrt{2}{V}_{\mathrm{Abus}}(1-{D}_{\mathrm{U}1})}$ |

PV-Half-bridge-INV-Abus | $\frac{{C}_{3}M{D}_{\mathrm{L}2}}{\sqrt{2}n}\le {V}_{\mathrm{Abus}}\le \frac{{C}_{3}M{D}_{\mathrm{U}2}}{\sqrt{2}n}$ | $\frac{{C}_{3}M{D}_{\mathrm{L}2}}{\sqrt{2}{V}_{\mathrm{Abus}}}\le n\le \frac{{C}_{3}M{D}_{\mathrm{U}2}}{\sqrt{2}{V}_{\mathrm{Abus}}}$ |

PV-Full-bridge-INV-Abus | $\frac{\sqrt{2}{C}_{3}M{D}_{\mathrm{L}2}}{n}\le {V}_{\mathrm{Abus}}\le \frac{\sqrt{2}{C}_{3}M{D}_{\mathrm{U}2}}{n}$ | $\frac{\sqrt{2}{C}_{3}M{D}_{\mathrm{L}2}}{{V}_{\mathrm{Abus}}}\le n\le \frac{\sqrt{2}{C}_{3}M{D}_{\mathrm{U}2}}{{V}_{\mathrm{Abus}}}$ |

PV System | Range of the Output | Range of n |
---|---|---|

PV-Forward | $0<{R}_{\mathrm{L}}<\frac{{R}_{\mathrm{sM}}}{{n}^{2}}$ | $0<n<\sqrt{\frac{{R}_{\mathrm{sM}}}{{R}_{\mathrm{L}}}}$ |

PV-Flyback | $0<{R}_{\mathrm{L}}$ | $0<n$ |

PV-Half-bridge | $0<{R}_{\mathrm{L}}<\frac{{R}_{\mathrm{sM}}}{4{n}^{2}}$ | $0<n<\frac{1}{2}\sqrt{\frac{{R}_{\mathrm{sM}}}{{R}_{\mathrm{L}}}}$ |

PV-Full-bridge | $0<{R}_{\mathrm{L}}\le \frac{{R}_{\mathrm{sM}}}{{n}^{2}}$ | $0<n\le \sqrt{\frac{{R}_{\mathrm{sM}}}{{R}_{\mathrm{L}}}}$ |

PV-Forward-Dbus | $0<{V}_{\mathrm{Dbus}}<\frac{{V}_{\mathrm{sM}}}{2n}$ | $0<n<\frac{{V}_{\mathrm{sM}}}{2{V}_{\mathrm{Dbus}}}$ |

PV-Flyback-Dbus | $0<{V}_{\mathrm{Dbus}}$ | $0<n$ |

PV-Half-bridge-Dbus | $0<{V}_{\mathrm{Dbus}}\le \frac{{V}_{\mathrm{sM}}}{4n}$ | $0<n<\frac{{V}_{\mathrm{sM}}}{4{V}_{\mathrm{Dbus}}}$ |

PV-Full-bridge-Dbus | $0<{V}_{\mathrm{Dbus}}\le \frac{{V}_{\mathrm{sM}}}{2n}$ | $0<n\le \frac{{V}_{\mathrm{sM}}}{2{V}_{\mathrm{Dbus}}}$ |

PV-Forward-INV | $0<{R}_{\mathrm{L}}<\frac{{M}^{2}{R}_{\mathrm{sM}}}{2{n}^{2}}$ | $0<n<M\sqrt{\frac{{R}_{\mathrm{sM}}}{2{R}_{\mathrm{L}}}}$ |

PV-Flyback-INV | $0<{R}_{\mathrm{L}}$ | $0<n$ |

PV-Half-bridge-INV | $0<{R}_{\mathrm{L}}<\frac{{M}^{2}{R}_{\mathrm{sM}}}{8{n}^{2}}$ | $0<n<\frac{M}{2}\sqrt{\frac{{R}_{\mathrm{sM}}}{2{R}_{\mathrm{L}}}}$ |

PV-Full-bridge-INV | $0<{R}_{\mathrm{L}}\le \frac{{M}^{2}{R}_{\mathrm{sM}}}{2{n}^{2}}$ | $0<n\le M\sqrt{\frac{{R}_{\mathrm{sM}}}{2{R}_{\mathrm{L}}}}$ |

PV System | Range of the Output | Range of n |
---|---|---|

PV-Forward | $\frac{{D}_{\mathrm{L}1}^{2}{R}_{\mathrm{sM}}}{{n}^{2}}\le {R}_{L}\le \frac{{D}_{\mathrm{U}1}^{2}{R}_{\mathrm{sM}}}{{n}^{2}}$ | ${D}_{\mathrm{L}1}\sqrt{\frac{{R}_{\mathrm{sM}}}{{R}_{\mathrm{L}}}}\le n\le {D}_{\mathrm{U}1}\sqrt{\frac{{R}_{\mathrm{sM}}}{{R}_{\mathrm{L}}}}$ |

PV-Flyback | $\frac{{D}_{\mathrm{L}1}^{2}{R}_{\mathrm{sM}}}{{n}^{2}{(1-{D}_{\mathrm{L}1})}^{2}}\le {R}_{L}\le \frac{{D}_{\mathrm{U}1}^{2}{R}_{sM}}{{n}^{2}{(1-{D}_{\mathrm{U}1})}^{2}}$ | $\frac{{D}_{\mathrm{L}1}}{(1-{D}_{\mathrm{L}1})}\sqrt{\frac{{R}_{\mathrm{sM}}}{{R}_{\mathrm{L}}}}\le n\le \frac{{D}_{\mathrm{U}1}}{(1-{D}_{\mathrm{U}1})}\sqrt{\frac{{R}_{\mathrm{sM}}}{{R}_{\mathrm{L}}}}$ |

PV-Half-bridge | $\frac{{D}_{\mathrm{L}2}^{2}{R}_{\mathrm{sM}}}{{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}2}^{2}{R}_{\mathrm{sM}}}{{n}^{2}}$ | ${D}_{\mathrm{L}2}\sqrt{\frac{{R}_{\mathrm{sM}}}{{R}_{\mathrm{L}}}}\le n\le {D}_{\mathrm{U}2}\sqrt{\frac{{R}_{\mathrm{sM}}}{{R}_{\mathrm{L}}}}$ |

PV-Full-bridge | $\frac{4{D}_{\mathrm{L}2}^{2}{R}_{\mathrm{sM}}}{{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{4{D}_{\mathrm{U}2}^{2}{R}_{\mathrm{sM}}}{{n}^{2}}$ | $2{D}_{\mathrm{L}2}\sqrt{\frac{{R}_{\mathrm{sM}}}{{R}_{\mathrm{L}}}}\le n\le 2{D}_{\mathrm{U}2}\sqrt{\frac{{R}_{\mathrm{sM}}}{{R}_{\mathrm{L}}}}$ |

PV-Forward-Dbus | $\frac{{D}_{\mathrm{L}1}{V}_{\mathrm{sM}}}{2n}\le {V}_{\mathrm{Dbus}}\le \frac{{D}_{\mathrm{U}1}{V}_{\mathrm{sM}}}{2n}$ | $\frac{{D}_{\mathrm{L}1}{V}_{\mathrm{sM}}}{2{V}_{\mathrm{Dbus}}}\le n\le \frac{{D}_{\mathrm{U}1}{V}_{\mathrm{sM}}}{2{V}_{\mathrm{Dbus}}}$ |

PV-Flyback-Dbus | $\frac{{D}_{\mathrm{L}1}{V}_{\mathrm{sM}}}{2n(1-{D}_{\mathrm{L}1})}<{V}_{\mathrm{Dbus}}\le \frac{{D}_{\mathrm{U}1}{V}_{\mathrm{sM}}}{2n(1-{D}_{\mathrm{U}1})}$ | $\frac{{D}_{\mathrm{L}1}{V}_{\mathrm{sM}}}{2{V}_{\mathrm{Dbus}}(1-{D}_{\mathrm{L}1})}<n\le \frac{{D}_{\mathrm{U}1}{V}_{\mathrm{sM}}}{2{V}_{\mathrm{Dbus}}(1-{D}_{\mathrm{U}1})}$ |

PV-Half-bridge-Dbus | $\frac{{D}_{\mathrm{L}2}{V}_{\mathrm{sM}}}{2n}\le {V}_{\mathrm{Dbus}}\le \frac{{D}_{\mathrm{U}2}{V}_{\mathrm{sM}}}{2n}$ | $\frac{{D}_{\mathrm{L}2}{V}_{\mathrm{sM}}}{2{V}_{\mathrm{Dbus}}}\le n\le \frac{{D}_{\mathrm{U}2}{V}_{\mathrm{sM}}}{2{V}_{\mathrm{Dbus}}}$ |

PV-Full-bridge-Dbus | $\frac{{D}_{\mathrm{L}2}{V}_{\mathrm{sM}}}{n}\le {V}_{\mathrm{Dbus}}\le \frac{{D}_{\mathrm{U}2}{V}_{\mathrm{sM}}}{n}$ | $\frac{{D}_{\mathrm{L}2}{V}_{\mathrm{sM}}}{{V}_{\mathrm{Dbus}}}\le n\le \frac{{D}_{\mathrm{U}2}{V}_{\mathrm{sM}}}{{V}_{\mathrm{Dbus}}}$ |

PV-Forward-INV | $\frac{{D}_{\mathrm{L}1}^{2}{M}^{2}{R}_{\mathrm{sM}}}{2{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}1}^{2}{M}^{2}{R}_{\mathrm{sM}}}{2{n}^{2}}$ | $M{D}_{\mathrm{L}1}\sqrt{\frac{{R}_{\mathrm{sM}}}{2{R}_{\mathrm{L}}}}\le n\le M{D}_{\mathrm{U}1}\sqrt{\frac{{R}_{\mathrm{sM}}}{2{R}_{\mathrm{L}}}}$ |

PV-Flyback-INV | $\frac{{D}_{\mathrm{L}1}^{2}{M}^{2}{\mathrm{R}}_{\mathrm{sM}}}{2{n}^{2}{(1-{D}_{\mathrm{L}1})}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}1}^{2}{M}^{2}{R}_{\mathrm{sM}}}{2{n}^{2}{(1-{D}_{\mathrm{U}1})}^{2}}$ | $\frac{M{D}_{\mathrm{L}1}}{(1-{D}_{\mathrm{L}1})}\sqrt{\frac{{R}_{\mathrm{sM}}}{2{R}_{\mathrm{L}}}}\le n\le \frac{M{D}_{\mathrm{U}1}}{(1-{D}_{\mathrm{U}1})}\sqrt{\frac{{R}_{\mathrm{sM}}}{2{R}_{\mathrm{L}}}}$ |

PV-Half-bridge-INV | $\frac{{D}_{\mathrm{L}2}^{2}{M}^{2}{R}_{\mathrm{sM}}}{2{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}2}^{2}{M}^{2}{R}_{\mathrm{sM}}}{2{n}^{2}}$ | $M{D}_{\mathrm{L}2}\sqrt{\frac{{R}_{\mathrm{sM}}}{2{R}_{\mathrm{L}}}}\le n\le M{D}_{\mathrm{U}2}\sqrt{\frac{{R}_{\mathrm{sM}}}{2{R}_{\mathrm{L}}}}$ |

PV-Full-bridge-INV | $\frac{2{D}_{\mathrm{L}2}^{2}{M}^{2}{R}_{\mathrm{sM}}}{{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{2{D}_{\mathrm{U}2}^{2}{M}^{2}{R}_{\mathrm{sM}}}{{n}^{2}}$ | $M{D}_{\mathrm{L}2}\sqrt{\frac{2{R}_{\mathrm{sM}}}{{R}_{\mathrm{L}}}}\le n\le M{D}_{\mathrm{U}2}\sqrt{\frac{2{R}_{\mathrm{sM}}}{{R}_{\mathrm{L}}}}$ |

PV System | Maximum Range | Minimum Range |
---|---|---|

PV-Forward | $\frac{{D}_{\mathrm{L}1}^{2}{R}_{\mathrm{sMmin}}}{{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}1}^{2}{R}_{\mathrm{sMmax}}}{{n}^{2}}$ | $\frac{{D}_{\mathrm{L}1}^{2}{R}_{\mathrm{sMmax}}}{{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}1}^{2}{R}_{\mathrm{sMmin}}}{{n}^{2}}$ |

PV-Flyback | $\frac{{D}_{\mathrm{L}1}^{2}{R}_{\mathrm{sMmin}}}{{n}^{2}{(1-{D}_{\mathrm{L}1})}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}1}^{2}{R}_{\mathrm{sMmax}}}{{n}^{2}{(1-{D}_{\mathrm{U}1})}^{2}}$ | $\frac{{D}_{\mathrm{L}1}^{2}{R}_{\mathrm{sMmax}}}{{n}^{2}{(1-{D}_{\mathrm{L}1})}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}1}^{2}{R}_{\mathrm{sMmin}}}{{n}^{2}{(1-{D}_{\mathrm{U}1})}^{2}}$ |

PV-Half-bridge | $\frac{{D}_{\mathrm{L}2}^{2}{R}_{\mathrm{sMmin}}}{{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}2}^{2}{R}_{\mathrm{sMmax}}}{{n}^{2}}$ | $\frac{{D}_{\mathrm{L}2}^{2}{R}_{\mathrm{sMmax}}}{{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}2}^{2}{R}_{\mathrm{sMmin}}}{{n}^{2}}$ |

PV-Full-bridge | $\frac{4{D}_{\mathrm{L}2}^{2}{R}_{\mathrm{sMmin}}}{{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{4{D}_{\mathrm{U}2}^{2}{R}_{\mathrm{sMmax}}}{{n}^{2}}$ | $\frac{4{D}_{\mathrm{L}2}^{2}{R}_{\mathrm{sMmax}}}{{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{4{D}_{\mathrm{U}2}^{2}{R}_{\mathrm{sMmin}}}{{n}^{2}}$ |

PV-Forward-Dbus | $\frac{{D}_{\mathrm{L}1}{V}_{\mathrm{sMmin}}}{2n}\le {V}_{\mathrm{Dbus}}\le \frac{{D}_{\mathrm{U}1}{V}_{\mathrm{sMmax}}}{2n}$ | $\frac{{D}_{\mathrm{L}1}{V}_{\mathrm{sMmax}}}{2n}\le {V}_{\mathrm{Dbus}}\le \frac{{D}_{\mathrm{U}1}{V}_{\mathrm{sMmin}}}{2n}$ |

PV-Flyback-Dbus | $\frac{{D}_{\mathrm{L}1}{V}_{\mathrm{sMmin}}}{2n(1-{D}_{\mathrm{L}1})}<{V}_{\mathrm{Dbus}}\le \frac{{D}_{\mathrm{U}1}{\mathrm{V}}_{\mathrm{sMmax}}}{2n(1-{D}_{\mathrm{U}1})}$ | $\frac{{D}_{\mathrm{L}1}{V}_{\mathrm{sMmax}}}{2n(1-{D}_{\mathrm{L}1})}<{V}_{\mathrm{Dbus}}\le \frac{{D}_{\mathrm{U}1}{V}_{\mathrm{sMmin}}}{2n(1-{D}_{\mathrm{U}1})}$ |

PV-Half-bridge-Dbus | $\frac{{D}_{\mathrm{L}2}{V}_{\mathrm{sMmin}}}{2n}\le {V}_{\mathrm{Dbus}}\le \frac{{D}_{\mathrm{U}2}{V}_{\mathrm{sMmax}}}{2n}$ | $\frac{{D}_{\mathrm{L}2}{V}_{\mathrm{sMmax}}}{2n}\le {V}_{\mathrm{Dbus}}\le \frac{{D}_{\mathrm{U}2}{V}_{\mathrm{sMmin}}}{2n}$ |

PV-Full-bridge-Dbus | $\frac{{D}_{\mathrm{L}2}{V}_{\mathrm{sMmin}}}{n}\le {V}_{\mathrm{Dbus}}\le \frac{{D}_{\mathrm{U}2}{V}_{\mathrm{sMmax}}}{n}$ | $\frac{{D}_{\mathrm{L}2}{V}_{\mathrm{sMmax}}}{n}\le {V}_{\mathrm{Dbus}}\le \frac{{D}_{\mathrm{U}2}{V}_{\mathrm{sMmin}}}{n}$ |

PV-Forward-INV | $\frac{{D}_{\mathrm{L}1}^{2}{M}^{2}{R}_{\mathrm{sMmin}}}{2{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}1}^{2}{M}^{2}{R}_{\mathrm{sMmax}}}{2{n}^{2}}$ | $\frac{{D}_{\mathrm{L}1}^{2}{M}^{2}{R}_{\mathrm{sMmax}}}{2{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}1}^{2}{M}^{2}{R}_{\mathrm{sMmin}}}{2{n}^{2}}$ |

PV-Flyback-INV | $\frac{{D}_{\mathrm{L}1}^{2}{M}^{2}{R}_{\mathrm{sMmin}}}{2{n}^{2}{(1-{D}_{\mathrm{L}1})}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}1}^{2}{M}^{2}{R}_{\mathrm{sMmax}}}{2{n}^{2}{(1-{D}_{\mathrm{U}1})}^{2}}$ | $\frac{{D}_{\mathrm{L}1}^{2}{M}^{2}{R}_{\mathrm{sMmax}}}{2{n}^{2}{(1-{D}_{\mathrm{L}1})}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}1}^{2}{M}^{2}{R}_{\mathrm{sMmin}}}{2{n}^{2}{(1-{D}_{\mathrm{U}1})}^{2}}$ |

PV-Half-bridge-INV | $\frac{{D}_{\mathrm{L}2}^{2}{M}^{2}{R}_{\mathrm{sMmin}}}{2{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}2}^{2}{M}^{2}{R}_{\mathrm{sMmax}}}{2{n}^{2}}$ | $\frac{{D}_{\mathrm{L}2}^{2}{M}^{2}{R}_{\mathrm{sMmax}}}{2{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{{D}_{\mathrm{U}2}^{2}{M}^{2}{R}_{\mathrm{sMmin}}}{2{n}^{2}}$ |

PV-Full-bridge-INV | $\frac{2{D}_{\mathrm{L}2}^{2}{M}^{2}{R}_{\mathrm{sMmin}}}{{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{2{D}_{\mathrm{U}2}^{2}{M}^{2}{R}_{\mathrm{sMmax}}}{{n}^{2}}$ | $\frac{2{D}_{\mathrm{L}2}^{2}{M}^{2}{R}_{\mathrm{sMmax}}}{{n}^{2}}\le {R}_{\mathrm{L}}\le \frac{2{D}_{\mathrm{U}2}^{2}{M}^{2}{R}_{\mathrm{sMmin}}}{{n}^{2}}$ |

PV System | R_{Lmin} or V_{Dmin} | R_{Lmax} or V_{Dmax} | n_{min} | n_{max} |
---|---|---|---|---|

PV-Flyback | 25.79 Ω | 6601 Ω | 0.2271 | 3.634 |

PV-Full-bridge | 16.5 Ω | 334.2 Ω | 0.1817 | 0.8175 |

PV-Flyback-Dbus | 74.25 V | 1188 V | 0.01485 | 0.2376 |

PV-Full-bridge-Dbus | 59.4 V | 267.3 V | 0.0594 | 0.2673 |

Weather Conditions | (a) | (b) | (c) | (d) |
---|---|---|---|---|

S (W/m^{2}) | 1300 | 850 | 550 | 350 |

T (℃) | 40 | 25 | 20 | 15 |

Weather Conditions | (a) | (b) | (c) | (d) |
---|---|---|---|---|

PV-Forward | 0.497 | 0.615 | 0.760 | 0.957 |

1.989 | 2.462 | 3.038 | 3.829 | |

PV-Flyback | 0.622 | 0.769 | 0.949 | 1.197 |

9.947 | 12.31 | 15.19 | 19.15 | |

PV-Half-bridge | 0.249 | 0.308 | 0.380 | 0.479 |

1.119 | 1.385 | 1.709 | 2.154 | |

PV-Full-bridge | 0.497 | 0.615 | 0.760 | 0.957 |

2.238 | 2.770 | 3.418 | 4.308 |

PV System | Calculated MCC Values | |
---|---|---|

PV-Forward | $0.28\sqrt{{R}_{\mathrm{sM}}}\le n\le 1.13\sqrt{{R}_{\mathrm{sM}}}$ | $4{R}_{\mathrm{sM}}\le {R}_{\mathrm{L}}\le 64{R}_{\mathrm{sM}}$ |

PV-Flyback | $0.35\sqrt{{R}_{\mathrm{sM}}}\le n\le 5.66\sqrt{{R}_{\mathrm{sM}}}$ | $6.25{R}_{\mathrm{sM}}\le {R}_{\mathrm{L}}\le 1600{R}_{\mathrm{sM}}$ |

PV-Half-bridge | $0.14\sqrt{{R}_{\mathrm{sM}}}\le n\le 0.636\sqrt{{R}_{\mathrm{sM}}}$ | ${R}_{\mathrm{sM}}\le {R}_{\mathrm{L}}<20.25{R}_{\mathrm{sM}}$ |

PV-Full-bridge | $0.28\sqrt{{R}_{\mathrm{sM}}}\le n\le 1.272\sqrt{{R}_{\mathrm{sM}}}$ | $4{R}_{\mathrm{sM}}\le {R}_{\mathrm{L}}<81{R}_{\mathrm{sM}}$ |

PV-Forward-Dbus | $0.01{V}_{\mathrm{sM}}\le n\le 0.04{V}_{\mathrm{sM}}$ | ${V}_{\mathrm{sM}}\le {V}_{\mathrm{Dbus}}\le 4{V}_{\mathrm{sM}}$ |

PV-Flyback-Dbus | $0.013{V}_{\mathrm{sM}}\le n\le 0.2{V}_{\mathrm{sM}}$ | $1.25{V}_{\mathrm{sM}}\le {V}_{\mathrm{Dbus}}\le 20{V}_{\mathrm{sM}}$ |

PV-Half-bridge-Dbus | $0.005{V}_{\mathrm{sM}}\le n\le 0.02{V}_{\mathrm{sM}}$ | $0.5{V}_{\mathrm{sM}}\le {V}_{\mathrm{Dbus}}<2.25{V}_{\mathrm{sM}}$ |

PV-Full-bridge-Dbus | $0.01{V}_{\mathrm{sM}}\le n\le 0.045{V}_{\mathrm{sM}}$ | ${V}_{\mathrm{sM}}\le {V}_{\mathrm{Dbus}}\le 4{.5\mathrm{V}}_{\mathrm{sM}}$ |

PV-Forward-INV | $0.16\sqrt{{R}_{\mathrm{sM}}}\le n\le 0.64\sqrt{{R}_{\mathrm{sM}}}$ | $1.28{R}_{\mathrm{sM}}\le {R}_{\mathrm{L}}\le 20.48{R}_{\mathrm{sM}}$ |

PV-Flyback-INV | $0.2\sqrt{{R}_{\mathrm{sM}}}\le n\le 3.2\sqrt{{R}_{\mathrm{sM}}}$ | $2{R}_{\mathrm{sM}}\le {R}_{\mathrm{L}}\le 512{R}_{\mathrm{sM}}$ |

PV-Half-bridge-INV | $0.08\sqrt{{R}_{\mathrm{sM}}}\le n\le 0.36\sqrt{{R}_{\mathrm{sM}}}$ | $0.32{R}_{\mathrm{sM}}\le {R}_{\mathrm{L}}<6.48{R}_{\mathrm{sM}}$ |

PV-Full-bridge-INV | $0.16\sqrt{{R}_{\mathrm{sM}}}\le n\le 0.72\sqrt{{R}_{\mathrm{sM}}}$ | $1.28{R}_{\mathrm{sM}}\le {R}_{\mathrm{L}}<25.92{R}_{\mathrm{sM}}$ |

(S,T)/(W/m^{2}, °C) | D_{max} | D_{max1} | P_{omax} | P_{omax1} | P_{omax2} |
---|---|---|---|---|---|

(750, 15) | 0.4865 | 0.4821 | 152.19 | 152.13 | 149.63 |

(1000, 15) | 0.5175 | 0.5204 | 214.7 | 214.89 | 212.5 |

(1250, 15) | 0.54 | 0.5373 | 281.77 | 281.72 | 279.92 |

(750, 25) | 0.4929 | 0.5007 | 151.29 | 151.22 | 148.79 |

(1000, 25) | 0.524 | 0.5221 | 213.4 | 213.69 | 211.2 |

(1250, 25) | 0.547 | 0.5455 | 280.87 | 280.83 | 277.93 |

(750, 35) | 0.4994 | 0.5013 | 150.39 | 150.51 | 148.43 |

(1000, 35) | 0.5308 | 0.5269 | 212.9 | 212.69 | 210.09 |

(1250, 35) | 0.5539 | 0.5520 | 279.97 | 280.14 | 277.64 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fan, J.; Li, S.; Liu, S.; Deng, X.; Zhu, X.
Maximum Power Point Tracking Constraint Conditions and Two Control Methods for Isolated Photovoltaic Systems. *Processes* **2023**, *11*, 3245.
https://doi.org/10.3390/pr11113245

**AMA Style**

Fan J, Li S, Liu S, Deng X, Zhu X.
Maximum Power Point Tracking Constraint Conditions and Two Control Methods for Isolated Photovoltaic Systems. *Processes*. 2023; 11(11):3245.
https://doi.org/10.3390/pr11113245

**Chicago/Turabian Style**

Fan, Jingxun, Shaowu Li, Sanjun Liu, Xiaoqing Deng, and Xianping Zhu.
2023. "Maximum Power Point Tracking Constraint Conditions and Two Control Methods for Isolated Photovoltaic Systems" *Processes* 11, no. 11: 3245.
https://doi.org/10.3390/pr11113245