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Article

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

by
Jingxun Fan
1,2,
Shaowu Li
1,*,
Sanjun Liu
1,*,
Xiaoqing Deng
1,3 and
Xianping Zhu
1,4
1
College of Intelligent Systems Science and Engineering, Hubei Minzu University, Enshi 445000, China
2
Jiangmen Pengjiang Power Supply Bureau of Guangdong Power Grid Co., Ltd., Jiangmen 529000, China
3
Hubei Chuangsinuo Electrical Technology Corp., Enshi 445000, China
4
Guilin Power Supply Bureau of Guangxi Power Grid Co., Ltd., Guilin 541000, China
*
Authors to whom correspondence should be addressed.
Processes 2023, 11(11), 3245; https://doi.org/10.3390/pr11113245
Submission received: 30 September 2023 / Revised: 27 October 2023 / Accepted: 8 November 2023 / Published: 17 November 2023
(This article belongs to the Special Issue Process Design and Modeling of Low-Carbon Energy Systems)

## Abstract

:
A maximum power point (MPP) always exists in photovoltaic (PV) cells, but a mismatch between PV system circuit parameters, weather conditions and system structure leads to the possibility that the MPP may not be tracked successfully. In addition, the introduction of an isolation transformer into a basic PV system allows for moderate values of the converter duty cycle and electrical isolation. However, there is no comprehensive research on MPPT (maximum power point tracking) constraint conditions for different isolated PV systems, which seriously hinders the application of isolated PV systems and the development of a related linear control theory. Therefore, in this paper, the overall mathematical models of different isolated PV systems are first established based on the PV cell engineering model and the MPP linear model, and then, two sets of constraint conditions are found for the successful realization of MPPT. These MPPT constraint conditions (MCCs) describe in detail the direct mathematical relationships between PV cell parameters, weather conditions and circuit parameters. Finally, based on the MPP linear model and MCCs, two new MPPT methods are designed for isolated PV systems. Considering the MCCs proposed in this paper, a suitable range of load and transformer ratios can be estimated from the measured data of irradiance and temperature in a certain area, and the range of MPPs existing in PV systems with different structures can be estimated, which is a good guide for circuit design, theoretical derivation and product selection for PV systems. Meanwhile, comparative experiments confirm the rapidity and accuracy of the two proposed MPPT methods, with the MPPT time improving from 0.23 s to 0.03 s, and they have the advantages of a simple program, small computational volume and low hardware cost.

## 1. Introduction

To carry out a theoretical analysis and practical verification of a PV system, an accurate model of the PV cell should be established first. Nowadays, a large number of studies on PV systems and PV cells are carried out, and they have led to a lot of breakthroughs and innovations in mathematical and circuit model optimization, as well as MPPT and parameter extraction methods for PV cells. However, the model used cannot be completely compatible with the required accuracy, the complexity of the calculations and the environmental conditions [1]. There are nine commonly used circuit models and mathematical models of PV cells categorized in Ref. [2], which can accurately reflect the output characteristics of PV cells but are not convenient for engineering applications, so simplified engineering models of PV cells have been widely investigated [3]. Many scholars have investigated how to model PV cells using four important parameters (Isc, Voc, Im and Vm) 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/m2, and PV cell temperature T is 25 °C), the PV cell engineering model is obtained using Equations (1)–(3), where I, V, Isc, Voc, Im and Vm 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].
$I = I sc [ 1 − C 1 ( e V C 2 V oc − 1 ) ]$
$C 1 = ( 1 − I m I sc ) e − V m C 2 V oc$
$C 2 = V m V oc − 1 ln ( 1 − I m I sc )$
However, when there are obstacles such as tall buildings and trees, the illumination of PV modules is no longer uniform, resulting in partial shadow problems in which the power curve has multiple peaks. So, it is necessary to establish a PV model under partial shadow conditions and to simulate and analyze its output characteristics [5,6]. This is a steady-state model of PV cells, but MPPT is a dynamic optimization process, so the dynamic characteristics of PV cells have also been studied in a number of ways [7]. All of the above models are nonlinear models of PV cells, which require complex iterations and calculations to extract parameters and conduct studies, so scholars have proposed some linearized models, such as segmented linear models, which replace the nonlinear PV relationship with multi-segmented linear equations through segmented linearization [8]. In [9], the authors proposed a new segmented linear shunt branch model that approximates the nonlinear I-V curve of a PV cell via an equivalent circuit. The segmented linearized model simplifies the workload in the nonlinear PV cell model and obtains comparable accuracy under certain conditions, but the number of segments must be increased in the segmented linear model if higher accuracy is required, which undoubtedly increases the computational complexity. The authors of [10] derived a linearized model that relates changes in the inputs to the system, such as irradiance and temperature, to its outputs, such as the array current and power. The authors of [11] derived a set of nonlinear state-space equations based on the average switching technique, which was implemented using MATLAB2016b. The authors of [12] linearized the voltage–current characteristics of PV cells at the MPP in order to completely remove the obstacle of nonlinear PV cells to the overall linearization of the PV system by proposing two equivalent linear models, the Thevenin equivalent model and the Norton equivalent model, as shown in Figure 1. In contrast, the MPP linear model can better overcome these problems in the segmented linear model. On this basis, it is feasible and reasonable to linearize the PV system as a whole, and the PV system can be conveniently studied using the traditional linear theory or law.
The DC/DC converters in PV systems are categorized into non-isolated and isolated DC/DC converters. Non-isolated DC/DC converters, such as buck, boost, buck-boost and Sepic converters, are widely used as the MPPT control circuits of PV systems. Isolated DC/DC converters usually include forward, flyback, push–pull, half-bridge and full-bridge converters. The introduction of an isolation transformer into a basic non-isolated DC/DC converter can realize electrical isolation between the converter’s input power supply and load. Meanwhile, it can improve the safety and reliability of converter operation and electromagnetic compatibility. In addition, it can make the duty cycle of the DC/DC converter change near a moderate value. Usually, in this case, a high boosted voltage can be achieved by using a high-transformation-ratio transformer and a voltage multiplier [13]. The analysis shown in Reference [14] verifies that isolation not only ensures safety but also increases the MPPT capability. Meanwhile, it shows that isolated converters have the highest MPPT capability without considering the hardware implementation.
At present, MPPT methods can be classified into five categories: (1) classical methods, such as perturbation observation, constant voltage and conductance increment methods [15]; (2) intelligent methods, such as artificial neural networks (ANNs), fuzzy logic controllers (FLCs) and sliding-mode control (SMC) [16,17]; (3) optimization methods, such as cuckoo search (CS), the particle swarm algorithm (PSO), the gray wolf algorithm (GWO), the ant colony algorithm (ACO) and the artificial bee colony algorithm (ABC) [18,19]; (4) hybrid methods, such as fuzzy particle swarm optimization (FPSO) and the adaptive neuro-fuzzy inference system (ANFIS) [20]; (5) other methods, such as the variable-weather parameter (VWP) method [21]. Under specific environmental conditions and requirements, good performance can be obtained with all five of the above-mentioned MPPT methods. However, the nonlinear model of the PV cell is one of the fundamental reasons why the linear control theory cannot be widely applied in the MPPT control of PV systems at present. And since the MPP must always exist in the process of use, it is easy to cause errors if its constraints are not analyzed. In order to solve this problem, some expressions have been proposed in Reference [22] to ensure the existence of the MPP in PV systems with buck, boost, buck/boost and other non-isolated DC/DC converters.
Therefore, the research objective of this paper is as follows: to find the relationship between the circuit parameters and the control signals of an isolated PV system by directly utilizing the weather conditions so as to find the range of circuit parameters for which it is capable of successful MPPT control and, accordingly, to propose two new MPPT methods.
The innovations and contributions of this work are as follows:
(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.
The section arrangement of this paper is as follows: Two MPPT constraint conditions and two new MPPT methods are presented in Section 2. Some simulation experiments are presented in Section 3. Finally, a discussion and some conclusions are given in Section 4 and Section 5, respectively.

## 2. Materials and Methods

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

The structure of the isolated PV system is shown in Figure 2. I and V denote the output current and voltage of the PV cell, respectively. Io and Vo denote the output current and output voltage of the isolated DC/DC converter, respectively. Ri and RL denote the equivalent resistances after the PV cell and after the isolated DC/DC converter, respectively.
The basic circuits of isolated DC/DC converters include the forward converter, flyback converter, half-bridge converter, full-bridge converter and push–pull converter. They are associated with the PV cell to produce the PV-Forward system, PV-Flyback system, PV-Half-bridge system, PV-Full-bridge system and PV-push–pull system, respectively. The isolated DC/DC converter is generally connected to a resistor, DC bus, inverter or AC bus (shown in Figure 3). The different system structures also lead to differences in the mathematical model and MPPT method.
In order to derive a theoretical mathematical model, two assumptions need to be made for isolated PV systems:
(1)
All circuit components are ideal;
(2)
The isolated DC/DC converter operates in the continuous-current mode (CCM).
Firstly, according to Figure 2, it can be obtained by the power balance relationship:
$V I = V o I o = P o$
$R i = V I$
$R L = V o I o$
Po denotes the output power of the PV system.
The input-and-output-voltage relationships of forward, flyback, half-bridge, full-bridge and push–pull converters can be expressed by Equations (7)–(11), respectively [23]. D denotes the duty cycle of the PWM wave for the isolated DC/DC converter, and the isolation transformer ratio n is equal to N1/N2.
$V o = D V n$
$V o = D V n ( 1 − D )$
$V o = D V n$
$V o = 2 D V n$
$V o = D V n$
It can be seen that Equations (7), (9) and (11) are the same, which means that the input–output-voltage relationships are the same for forward, half-bridge and push–pull converters.
According to Figure 2, Equation (12) is satisfied.
$P o = V o 2 R L$
The mathematical model of the PV-Forward system can be obtained by combining Equations (1), (4), (7) and (12).
Since the forward, half-bridge and push–pull converters have the same input–output-voltage relationships, the mathematical models of the PV-Forward, PV-Half-bridge and PV-Push–pull systems are also the same, all of which are expressed in Equation (13) and will not be repeated below.
Similarly, the mathematical models of the PV-Flyback and PV-Full-bridge systems can also be obtained.
For the DC bus, Equation (16) is satisfied.
$V o = V Dbus$
The mathematical model of the PV-Forward-Dbus system can be obtained by combining Equations (1), (4), (7) and (16).
Similarly, the mathematical models of the PV-Flyback-Dbus and PV-Full-bridge-Dbus systems can also be obtained.
The mathematical models of the inverter (SPWM control) and AC load can be represented by Equations (20) and (21), respectively. M denotes the SPWM wave modulation ratio. Vr and Ir denote the RMS values of the output AC voltage and AC current for the inverter, respectively.
$V r = M V o 2$
$R L = V r I r$
The mathematical model of the PV-Forward-INV system can be obtained by combining Equations (1), (4), (7), (20) and (21).
Similarly, the mathematical models of the PV-Flyback-INV and PV-Full-bridge-INV systems can also be obtained.
For the AC bus, Equation (25) is satisfied.
$V r = V Abus$
The mathematical model of the PV-Forward-INV-Abus system can be obtained by combining Equations (1), (10), (13) and (25).
Similarly, the mathematical models of the PV-Flyback-INV-Abus and PV-Full-bridge-INV-Abus systems can also be obtained.
Equations (13)–(15), (17)–(19), (22)–(24) and (26)–(28) are the theoretical basis for the MCCs of PV systems with these five isolated DC/DC converters connected to the load, DC bus, inverter and AC bus, respectively.
It can be concluded that Pomax 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).
$d P o d D = 0$
For the PV-Forward, PV-Flyback, PV-Full-bridge and PV-Forward-Dbus systems, substituting Equations (13)–(15) and (17) into Equation (29), respectively, give Equations (30)–(33), where the parameter C3 is represented by Equation (34).
$D max = n P omax R L C 3$
$D max = P omax R L C 3 / n + P omax R L = 1 − C 3 / n C 3 / n + P omax R L$
$D max = n P omax R L 2 C 3$
$V = C 3$
$C 3 = C 2 V oc [ lambertw ( e × 1 + C 1 C 1 ) − 1 ]$
According to Equation (34), it can be concluded that the value of C3 is only related to the parameters of the PV cell itself (S and T). The simulation experiments revealed that Pomax is only affected by S and T and is independent of RL and n. Therefore, only the values of C3 and Pomax under different weather conditions are required to derive the relationship between Dmax and RL, n. This leads to the MPPT control of isolated PV systems to improve the efficiency. The C3-S, C3-T, Pomax-S and Pomax-T curves under different weather conditions were plotted using MATLAB, and by applying the curve-fitting method, Equations (35) and (36) can be obtained.
$C 3 = 0.0057 × S − 0.086 × T + 26.15$
According to Equations (35) and (36), C3 and Pomax can be easily derived from the weather conditions. Meanwhile, in order to find the MCCs and improve the MPPT methodology of isolated PV systems, Dmax can also be derived by combining the circuit parameters RL and n.
Figure 4 shows the equivalent model of the isolated PV system at the MPP [12], where RiMPP, VMPP and IMPP represent the values of Ri, V and I at the MPP in Figure 2, respectively.
At the MPP, Equations (37) and (38) can be given by the circuit theorem [24].
$R iMPP = V MPP I MPP$
$P omax = V MPP × I MPP$
Equations (33), (37) and (38) are combined to obtain Equation (39).
$R iMPP = C 3 2 P omax$
According to the maximum power transfer theorem [24], the isolated PV system can operate at the MPP when Equation (40) is satisfied.
$R iMPP = R sM$
Meanwhile, according to the circuit theorem [24], Equation (41) is satisfied.
$V sM = 2 C 3$
Using Equations (35), (36), (39) and (41), Equations (42) and (43) can be obtained.
$R sM ( S , T ) = [ C 3 ( S , T ) ] 2 P omax ( S , T )$
$V sM ( S , T ) = 2 C 3 ( S , T )$
According to Equations (42) and (43), the MPP linear model of the PV cell can be built using MATLAB/Simulink. When the weather conditions change, RsM is involved in the design of MPPT as the output signal of the model.

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

The relationship between circuit parameters, weather conditions and control parameters has been derived in Section 2.1 when the output of the isolated DC/DC converter is a load resistor. This section continues to derive the MCCs for isolated PV systems with different topologies and outputs on the basis of the engineering cell model.
The circuit topologies of forward and flyback converters determine their D to satisfy Equation (44), those of half-bridge and push–pull converters determine their D to satisfy Equation (45), and that of the full-bridge converter determines its D to satisfy Equation (46) [23]. These three formulas are also the basis of the analysis of MCCs carried out in a later section. Dmax represents D at the MPP.
$0 < D max < 1$
$0 < D max < 0 . 5$
$0 < D max ≤ 0.5$
Substituting Equation (30) into Equation (44), it can be seen that Equation (47) is satisfied. This is the RL range in which the PV-Forward system can successfully track the MPP.
$0 < R L < C 3 2 n 2 P omax$
If the transformer ratio n is the object of study, Equation (47) can be replaced by Equation (48).
$0 < n < C 3 P omax R L$
Similarly, the MCCs in the ideal case using the different PV systems are displayed in Table 1. These expressions are the prerequisites of successful MPPT control for isolated PV systems in the ideal case.
From the practical application point of view, the isolated PV system is a non-ideal circuit, and the expressions in Table 1 need to be improved. The duty cycle of the isolated DC/DC converter cannot be too small or too large due to the losses of the switching devices and the isolation transformer itself, the limitations on the switching device’s opening and closing times and the through-current withstand voltage, the transmission delay of the controller and the PWM sampling delay. Therefore, in order to find the MCCs in practical applications, it is assumed that the minimum D of the forward and flyback converters is DL1, while their maximum D is DU1, and the minimum D of the half-bridge, full-bridge and push–pull converters is DL2, while their maximum D is DU2. 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).
$D L 1 ≤ D max ≤ D U 1$
$D L 2 ≤ D max ≤ D U 2$
Substituting Equation (30) into Equation (49), it can be seen that Equation (51) can be obtained. This is the RL range in which the PV-Forward system can successfully track the MPP in practical applications.
$D L 1 2 C 3 2 n 2 P omax ≤ R L ≤ D U 1 2 C 3 2 n 2 P omax$
If the transformer ratio n is the object of study, Equation (51) can be replaced by Equation (52).
$D L 1 C 3 P omax R L ≤ n ≤ D U 1 C 3 P omax R L$
Similarly, the MCCs of various isolated PV systems can be derived when the D limitation in a practical situation is considered, as shown in Table 2. These expressions are the prerequisites of successful MPPT control for isolated PV systems in practical applications.

#### 2.3.1. Expression of MCCs

The analysis in Section 2.2 has produced the ranges of circuit parameters for twenty isolated PV systems capable of MPPT control based on the engineering model. This section continues with an in-depth study of these circuit parameter ranges based on the MPP linear model. After the engineering model has been linearized by using the methodology in Section 2.1, the isolated PV system structure can be replaced by the system shown in Figure 5. The flyback converter is selected as an example, where VsM and RsM are quantities that vary with the weather conditions (S and T).
In order to find the MCCs in the ideal case, according to the maximum power transfer theorem, it can be seen that Equations (53) and (54) are satisfied when the PV system is operating at the MPP.
$R i = R sM$
$V sM = 2 V$
The Ri of the PV-Forward system can be expressed by Equation (55), and Ri will vary with the different output devices and the transformations of isolated DC/DC converters.
$R i = n 2 R L D 2$
The Ri of the PV-Forward-INV system can be expressed by Equation (56).
$R i = 2 n 2 R L M 2 D 2$
Equation (56) reveals the mathematical relationship between the circuit parameters (Ri, RL and n) and the control signals (D and M). On the basis of these expressions, the MCCs can be found.
When the output of the PV cell is connected to a resistor, Equation (55) is substituted into Equation (53), and then Equation (57) can be obtained.
$D max = n R L R sM$
Substituting Equation (56) into Equation (53), it can be seen that Equation (58) is satisfied. This is the RL range in which the PV-Forward system can successfully track the MPP.
$0 < R L < R sM n 2$
If the transformer ratio n is the object of study, Equation (58) can be replaced by Equation (59).
$0 < n < R sM R L$
Similarly, the MCCs of the different PV systems in the ideal case are displayed in Table 3. These expressions are the prerequisites of successful MPPT control for isolated PV systems in the ideal case.
Table 3 shows that under ideal conditions, an RL 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 VDbus 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 RL 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 RL 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.
From the practical application point of view, the isolated PV system is a non-ideal circuit, and the expressions in Table 3 need to be improved. The duty cycle of the isolated DC/DC converter cannot be too small or too large. Therefore, in order to find the range of circuit parameters in practical applications, the duty cycle ranges of the forward, flyback, half-bridge, full-bridge and push–pull converters are expressed by Equations (49) and (50).
Substituting Equation (57) into Equation (49), it can be seen that Equation (60) is satisfied. This is the RL range in which MPPT control can be successfully realized in practical applications for the PV-Forward system.
$D L 1 2 R sM n 2 ≤ R L ≤ D U 1 2 R sM n 2$
If the transformer ratio n is the object of study, Equation (60) can be replaced by Equation (61).
$D L 1 R sM R L ≤ n ≤ D U 1 R sM R L$
Similarly, the ranges of circuit parameters in which various isolated PV systems are capable of successfully realizing MPPT are shown in Table 4, when considering the limited range of D in practical situations. These expressions are the prerequisites of successful MPPT control for isolated PV systems in practical situations.
Table 4 shows significantly smaller ranges for RL and VDbus 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

The ranges of VsM and RsM 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 RL (or VDbus) is necessary for each PV system to be modeled with the MPP linear cell. By contrast, the minimum range of RL (or VDbus) 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.
In practical applications, Table 3, Table 4 and Table 5 are a good guide for the circuit design, theoretical derivation and product selection of isolated PV systems. On the one hand, it is complicated to adjust the output under changing weather conditions. In order to realize MPPT control, they can be used to select the types of isolated DC/DC converters and circuit components. On the other hand, they can also be used as a basis for the study of MPPT methods. Meanwhile, they can be used to estimate the MPPT effect based on the recorded historical data of S and T in the application area. In addition, the results shown in Table 5 can provide a theoretical basis when the overall linearized model of the isolated PV system is investigated.

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

Two new MPPT methods based on the MPP linear model are proposed. Here, the PV-Flyback and PV-Flyback-Dbus systems are used as examples.

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

Substituting Equations (4)–(6), (8) and (42) into (53), Equation (62) is satisfied. It relates Dmax to the weather conditions (S and T) and the circuit parameters (RL and n) when the PV-Flyback system operates at the MPP.
$D max = n R L R sM ( S , T ) + n R L$
According to Equation (62), it can be seen that RMPPT can be used when RL 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 RL and n, the duty cycle Dmax at the MPP for the isolated PV system can be calculated, and the microcontroller or chip can realize MPPT control by controlling D = Dmax.
The structure of the isolated PV system using RMPPT is shown in Figure 6. As can be seen in Figure 6, the Dmax 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, Vo and Io), calculating the load resistor RL 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 RL 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)

Substituting Equations (8), (16), (43) into (54), Equation (63) can be obtained. It relates Dmax to the weather conditions (S and T) and the circuit parameters (VDbus and n) when the PV-Flyback-Dbus system operates at MPP.
$D max = 2 n V Dbus V sM ( S , T ) + 2 n V Dbus$
According to Equation (63), it can be seen that when VDbus 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 VDbus and n, the duty cycle at the MPP Dmax of the isolated PV system can be calculated. Then, the microcontroller or chip makes the duty cycle of the PWM wave equal to Dmax, 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.
The structure of the isolated PV system using BMPPT is shown in Figure 7. As can be seen in Figure 7, the value of Dmax 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 VDbus) 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

In Table 2, it can be seen that the MCCs for PV systems with forward, half-bridge, full-bridge and push–pull converters are similar, as are the MCCs for PV systems with and without inverters. In this section, only the PV-Flyback, PV-Full-bridge, PV-Flyback-Dbus and PV-Full-bridge-Dbus systems are verified, and other PV systems with different structures can be verified analogously. In order to verify the accuracy of Table 2, some simulation experiments were carried out for PV systems with a flyback converter and full-bridge converter at STC with n = 1/10 or RL = 5 Ω or VDbus = 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 Isc = 7.84 A, Voc = 36.3 V, Im = 7.35 A and Vm = 29 V, respectively.
Assume that DL1, DU1, DL2 and DU2 are taken as 0.2, 0.8, 0.1 and 0.45, respectively, that RL = 5 Ω or VDbus = 500 V for the study of the range of n, and that n = 0.1 for the study of the range of RL or VDbus. 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 RLmax and RLmin denote the maximum and minimum values of RL, respectively, nmax and nmin denote the maximum and minimum values of n, respectively, and VDmax and VDmin denote the maximum and minimum values of VDbus, respectively. These data are compared with Figure 8 to analyze the reasonableness and accuracy of the MCCs.
According to Figure 8a,b,e,f, for the PV-Flyback and PV-Flyback-Dbus systems, when n is certain and RL = RLmin or VDbus = VDmin is satisfied, the MPP is reached exactly at D = DL1. When RL = RLmax or VDbus = VDmax is satisfied, the MPP is reached exactly at D = DU1. When RL or VDbus is certain and n = nmin is satisfied, the MPP is reached exactly at D = DL1. When n = nmax, the MPP is reached exactly at D = DU1.
In Figure 8c,d,g,h, it can be seen that, for the PV-Full-bridge and PV-Full-bridge-Dbus systems, the MPP is reached exactly at D = DL2 when n is certain and RL = RLmin or VDbus = VDmin is satisfied. When RL = RLmax or VDbus = VDmax is satisfied, the MPP is reached exactly at D = DU2. When RL or VDbus is certain and n = nmin is satisfied, the MPP is reached exactly at D = DL2. When n = nmax is satisfied, the MPP is reached exactly at D = DU2.
In Table 6, it can be seen that the range of MCCs for the PV-Flyback and PV-Flyback-Dbus systems is much larger than that of the PV-Full-bridge and PV-Full-bridge-Dbus systems. However, in the small-load and low-variable-ratio segments, the range of MCCs for the PV-Full-bridge and PV-Full-bridge-Dbus systems is slightly larger than that for the PV-Flyback and PV-Flyback-Dbus PV systems.
In conclusion, according to Figure 8, the MCCs shown in Table 2 are accurate in practical applications when the duty cycle constraints of isolated DC/DC converters are considered.
Obviously, the MCCs of PV systems are influenced by the changing irradiance and temperature. Therefore, in the research and application of PV systems, we can judge the effect of MPPT control and estimate the range of its circuit parameters according to local historical meteorological data.

#### 3.2.1. Accuracy Verification of MCCs

Table 7 shows the four weather conditions of the PV system, and simulation experiments were conducted for the MCCs. Meanwhile, the results in Table 3, Table 4 and Table 5 and other weather conditions can be verified analogously.
When the output of the PV system is resistive, DL1, DU1, DL2 and DU2 are taken as 0.2, 0.8, 0.1 and 0.45, respectively, and RL is equal to 0.5 Ω. The simulation results are shown in Figure 9. Figure 9 compares the curves of Dmax 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.
In Figure 9a, it can be seen that, for the PV-Forward system, Dmax remains at 0.2 when $n < D L 1 R sM / R L$ and 0.8 when $n > D U 1 R sM / R L$, 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, Dmax stays at 0.2 when $n < D L 1 R sM / [ R L ( 1 − D L 1 ) ]$, while when $n > D U 1 R sM / [ R L ( 1 − D U 1 ) ]$, 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, Dmin stays at 0.1 when $n < D L 2 R sM / R L$, while Dmax stays at 0.45 when $n > D U 2 R sM / R L$, 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 Dmin at 0.1 when $n < 2 D L 2 R sM / R L$, while Dmax remains at 0.45 under the condition of $n > 2 D U 2 R sM / R L$, which implies that the MPP does not exist outside the range of n, and the MPP linear model cannot be used.
Comparing Figure 9, the Dmax 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 Dmax-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

The fifteen PV systems studied in this section can be applied under a wide range of practical requirements. However, the choice of the right PV system is complex. Therefore, it is essential to compare their MCCs. Here, it is assumed that the values of DL1, DL2, DU1, DU2, n, RL and VDbus are the same as in Section 3.2.1. In this case, Table 9 shows the calculated values according to Table 5.
Some simulations based on Table 9 were performed to further analyze the MCCs. The simulation results are shown in Figure 10. In Figure 10, RLminFD and RLmaxHB denote the maximum and minimum values of RL for the PV-Forward system and PV-Half-bridge system, respectively. Other circuit parameter boundaries are also presented in Figure 10.
In conclusion, both the different choices of isolated DC/DC converters and changing weather parameters may lead to changes in the MCCs.

#### 3.3. Simulation Analysis of RMPPT

In order to verify the practicality of RMPPT and test its MPPT capability, the PV-Flyback system model was built by using Simulink. In this case, the MPP linear model in Section 2.1 is used. Meanwhile, n and RL 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.
Simulation experiments on the practicality of RMPPT were conducted, and the results are shown in Table 10. Dmax and Dmax1 denote D values at the MPP when the RMPPT and P&O methods are used, respectively. Pomax and Pomax1 denote the maximum output power values of the PV cell when the RMPPT and P&O methods are used, respectively. Pomax2 denotes the maximum output power of the PV system. The parameter settings are n = 1/10 and RL = 500 Ω. The P&O method step size is set to 0.005.
In Table 10, it can be seen that the values of Dmax and Pomax calculated by RMPPT are basically equal to Dmax1 and Pmax1, respectively. This proves the practicality of RMPPT. In addition, it can be seen from Pomax1 and Pomax2 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.
Two sets of simulation experiments were performed for RMPPT using Simulink. And the MPPT methods were judged on the basis of stability and speed.
(1)
In order to simulate a sudden weather change situation, it is assumed that at 0~0.3 s, S = 800 W/m2 and T = 25 °C; at 0.3~0.7 s, S = 1200 W/m2 and T = 25 °C; and at 0.7~1 s, S = 400 W/m2 and T = 25 °C. Figure 11 shows the simulation results.
In Figure 11b, it can be seen that the tracking time and numerical stability of the MPP are much better than in the traditional P&O method when RMPPT is used in the isolated PV system with sudden changes in weather conditions (S). In Figure 11c, it can be seen that the P&O method itself has a step-length limitation, which causes D to oscillate around Dmax, 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 Dmax, and the Pomax of the PV cell is also stabilized to the new Pomax after a rapid adjustment, which also proves the correctness of the conclusion in Section 2.1.
(2)
Simulation experiment of RL change
Figure 12 shows the simulation results. In Figure 12b, it can be seen that the tracking time and numerical stability of the MPP are much better than those of the P&O method when RMPPT is used with sudden changes in RL. It can also be seen in Figure 12 that Dmax is actively adjusted to the new Dmax after a sudden change in RL, but Pomax remains at the same value after a short transient adjustment.
Therefore, it can be concluded that RMPPT outperforms the conventional P&O method in terms of MPPT rapidity and stability, regardless of changing weather conditions or circuit parameters.
Although only the MPPT method for the PV-Flyback system based on Equation (62) is proposed and verified in this section, the remaining MPPT methods for different isolated PV systems can be proposed analogously, which makes it easy for researchers and users of PV systems to select the corresponding MPPT methods.

#### 3.4. Simulation Analysis of BMPPT

In order to verify the practicality of BMPPT and test its MPPT capability, the PV-Flyback-Dbus system model shown in Figure 8 was built by using Simulink. The parameter settings are n = 2 and VDbus = 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.
In Figure 13b,e, it can be seen that the tracking time and numerical stability of the MPP are much better than in the P&O method when BMPPT is used in isolated PV systems with sudden changes in T or VDbus. It can also be seen in Figure 13b,c that, after a sudden change in T, Dmax is actively adjusted to the new Dmax, and Pomax is also stabilized to the new Pomax after a rapid stepwise adjustment.
Therefore, it can be concluded that BMPPT is far superior to the traditional P&O method in terms of rapidity and stability under changing weather conditions or circuit parameters.
Although only the MPPT method for the PV-Flyback-Dbus system based on the theory of Equation (63) is proposed and validated in this section, the remaining MPPT methods for different isolated PV systems can also be proposed analogously.
In this section, two MPPT methods (RMPPT and BMPPT) are verified when a load resistance and DC bus are selected as the output of the PV system, respectively. The conventional P&O method is compared with two MPPT methods implemented in Matlab/Simulink under varying weather conditions (irradiance and temperature) and circuit parameters (DC bus voltage and load resistance). The experimental results verify the high speed and accuracy of the two proposed MPPT methods and show the advantages of a simple program, small computational volume and low cost of hardware and software. They also verify the correctness and practicability of the MPP linear model established in Section 2.1.

## 4. Discussion

Table 1, Table 2, Table 3, Table 4 and Table 5 show the constraint conditions that enable the successful realization of MPPT control for isolated PV systems on the basis of the PV cell engineering model and MPP linear model. However, in practical applications, these constraint conditions usually play an important role in the hardware design, theoretical study and product installation of the PV system. On the one hand, since the boundaries of these constraints always change with the weather parameters, it is difficult to adjust the operating system in real time based on whether the load (or bus voltage) varies within the MCC range. For hardware designers, the MCCs can be utilized to select system configurations and circuit components. For the theoretical researcher, the MCCs can be used as a basis for ensuring the usability of the proposed control method. For the system installer, the MCCs can be used to estimate the MPPT effect based on solar irradiance and temperature recordings in the installation area. On the other hand, in practical applications, the maximum selected value of the load (or bus voltage) can be reflected by the MCCs. In other words, for a PV system, if the selected value of the load (or bus voltage) is not within the corresponding interval, the MPP cannot be successfully tracked, regardless of the used MPPT method, in which case, of course, the MPP linear model cannot be used. In addition, the MCCs can provide a theoretical basis when the MPP linear model is used to study the overall linearized model of the PV system.
However, in practice, the MPPT control of PV systems is usually affected by some other factors, such as the installed PV power, non-ideal DC/DC converter, non-ideal inverter and transmission efficiency. Therefore, the conclusions of this paper will be influenced by these factors to some extent. However, these factors are negligible. The reasons are as follows. On the one hand, the use of ideal isolated DC/DC converters and inverters can greatly simplify the theoretical study, just like in other studies. On the other hand, the aim of this work is to reveal the governing relationships between PV cell parameters and the load resistance or bus voltage when the MPP of the PV system is always present. Obviously, obtaining these relationships is very beneficial for the study of MPPT control methods using both PV cell models. Finally, the two constraint conditions in this paper represent the key results on the basis of which other factors can be easily considered and involved in practical applications.

## 5. Conclusions

For isolated PV systems, this paper solves the problem of when to apply the MPP linear model of the PV cell and proposes two faster and more accurate MPPT methods on the basis of MCCs, which are important for studying the overall linearization of isolated PV systems. In practical applications, the MCCs are a good guide for the circuit design, theoretical derivation and product selection of isolated PV systems. Theoretical researchers, hardware circuit designers and PV equipment installers can select the suitable isolated PV system according to different load and DC bus range requirements and make a preliminary estimation of MPPT effects. The main work in this paper is summarized as follows:
(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.
Although this thesis finds some direct mathematical relationships between weather parameters (irradiance and temperature), circuit parameters (load resistance, transformer ratio and bus voltage) and control signals (PWM wave duty cycle) for isolated PV systems and proposes two MPPT methods applicable to different topologies and load types, there is still a lot of follow-up work to be carried out.
(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

Conceptualization, J.F. and S.L. (Shaowu Li); methodology, J.F.; software, J.F.; validation, J.F., S.L. (Shaowu Li), S.L. (Sanjun Liu) and X.Z.; formal analysis, J.F.; investigation, J.F., S.L. (Shaowu Li), S.L. (Sanjun Liu), X.D. and X.Z.; resources, S.L. (Shaowu Li) and S.L. (Sanjun Liu); data curation, J.F.; writing—original draft preparation, J.F. and S.L. (Shaowu Li); writing—review and editing, J.F., S.L. (Shaowu Li) and S.L. (Sanjun Liu); visualization, J.F. and S.L. (Shaowu Li); supervision, S.L. (Shaowu Li), S.L. (Sanjun Liu), X.D. and X.Z.; project administration, S.L. (Shaowu Li) and S.L. (Sanjun Liu); funding acquisition, S.L. (Shaowu Li), S.L. (Sanjun Liu) and X.D. All authors have read and agreed to the published version of the manuscript.

## Funding

This research was funded by the National Natural Science Foundation of China (Nos. 61961016 and 61963014).

## Data Availability Statement

All relevant data are within the paper.

## Acknowledgments

The authors would like to sincerely thank the editor and anonymous reviewers for their valuable comments and suggestions to improve the quality of the article.

## Conflicts of Interest

Author J.F. was employed by the company Jiangmen Pengjiang Power Supply Bureau of Guangdong Power Grid Co., Ltd. Author X.D. was employed by the company Hubei Chuangsinuo Electrical Technology Corp. Author X.Z. was employed by the company Guilin Power Supply Bureau of Guangxi Power Grid Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict 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/m2) Vr Output voltage of inverter (V) T Cell temperature (°C) Ir Output current of inverter (A) Io Output current of isolated DC/DC converter (A) RsM Internal resistance of linear cell model (Ω) Vo Output voltage of isolated DC/DC converter (V) VsM Open-circuit voltage of MPP linear model (V) D Duty cycle of the PWM signal of converter Dmax D at the MPP Isc Short-circuit current of PV cell under STC (A) Pomax Output power at MPP (W) Voc Open-circuit voltage of PV cell under STC (V) RiMPP Value of Ri at MPP (Ω) Im MPP current of PV cell under STC (A) VMPP Value of V at MPP (Ω) Vm MPP voltage of PV cell under STC (V) IMPP Value of I at MPP (Ω) Ri Input resistance of isolated DC/DC converter (Ω) VDbus Voltage of DC bus (V) RL Load or equivalent load resistance of PV system (Ω) VAbus Voltage of AC bus (V) DL1 Minimum D for forward and flyback converters DL2 Minimum D for half-bridge, full-bridge, push–pull converter DU1 Maximum D for forward and flyback converters DU2 Maximum D for half-bridge, full-bridge, push–pull converter Po Output power of PV system (W)

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Figure 1. Relationship between single-diode model and MPP linear model.
Figure 1. Relationship between single-diode model and MPP linear model.
Figure 2. Isolated PV system structure.
Figure 2. Isolated PV system structure.
Figure 3. Four types of output. (a) Load; (b) DC bus; (c) inverter; (d) AC bus.
Figure 3. Four types of output. (a) Load; (b) DC bus; (c) inverter; (d) AC bus.
Figure 4. Isolated PV system with MPP linear model.
Figure 4. Isolated PV system with MPP linear model.
Figure 5. Isolated PV system based on MPP linear model. (* indicates the eponymous end of the induced electromotive force of the winding).
Figure 5. Isolated PV system based on MPP linear model. (* indicates the eponymous end of the induced electromotive force of the winding).
Figure 6. Isolated PV system structure of RMPPT.
Figure 6. Isolated PV system structure of RMPPT.
Figure 7. Isolated PV system structure of BMPPT.
Figure 7. Isolated PV system structure of BMPPT.
Figure 8. Po-D curves of the different outputs and n. (a) Po-D curves of PV-Flyback system for different RL; (b) Po-D curves of PV-Flyback system for different n; (c) Po-D curves of PV-Full-bridge system for different RL; (d) Po-D curves of PV-Full-bridge system for different n; (e) Po-D curves of PV-Flyback-Dbus for different VDbus; (f) Po-D curves of PV-Flyback-Dbus for different n; (g) Po-D curves of PV-Full-bridge-Dbus for different VDbus; (h) Po-D curves of PV-Full-bridge-Dbus for different n.
Figure 8. Po-D curves of the different outputs and n. (a) Po-D curves of PV-Flyback system for different RL; (b) Po-D curves of PV-Flyback system for different n; (c) Po-D curves of PV-Full-bridge system for different RL; (d) Po-D curves of PV-Full-bridge system for different n; (e) Po-D curves of PV-Flyback-Dbus for different VDbus; (f) Po-D curves of PV-Flyback-Dbus for different n; (g) Po-D curves of PV-Full-bridge-Dbus for different VDbus; (h) Po-D curves of PV-Full-bridge-Dbus for different n.
Figure 9. Po-D curves of different PV systems. (a) Dmax-n curves of PV-Forward system; (b) Dmax-n curves of PV-Flyback system; (c) Dmax-n curves of PV-Half-bridge system; (d) Dmax-n curves of PV-Full-bridge system.
Figure 9. Po-D curves of different PV systems. (a) Dmax-n curves of PV-Forward system; (b) Dmax-n curves of PV-Flyback system; (c) Dmax-n curves of PV-Half-bridge system; (d) Dmax-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 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 Pomax-t curves of RMPPT and P&O methods; (c) comparison of D-t curves of RMPPT and P&O methods.
Figure 11. Simulation experiment of irradiance change. (a) S curve variation with t; (b) comparison of Pomax-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 RL change. (a) RL curve variation with t; (b) comparison of Pomax-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 RL change. (a) RL curve variation with t; (b) comparison of Pomax-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 VDbus changes. (a) T curve variation with t; (b) comparison of Pomax-t curves of T change; (c) comparison of D-t curves of T change; (d) VDbus curve variation with t; (e) comparison of Pomax-t curves of VDbus change; (f) comparison of D-t curves of VDbus change.
Figure 13. Simulation experiment of T and VDbus changes. (a) T curve variation with t; (b) comparison of Pomax-t curves of T change; (c) comparison of D-t curves of T change; (d) VDbus curve variation with t; (e) comparison of Pomax-t curves of VDbus change; (f) comparison of D-t curves of VDbus change.
Table 1. Theoretical expressions of MCCs.
Table 1. Theoretical expressions of MCCs.
PV SystemRange of the OutputRange of n
PV-Forward$0 < R L < C 3 2 n 2 P omax$$0 < n < C 3 P omax R L$
PV-Flyback$0 < R L$$0 < n$
PV-Half-bridge$0 < R L < C 3 2 4 n 2 P omax$$0 < n < C 3 2 P omax R L$
PV-Full-bridge$0 < R L ≤ C 3 2 n 2 P omax$$0 < n ≤ C 3 P omax R L$
PV-Forward-Dbus$0 < V Dbus < C 3 n$$0 < n < C 3 V Dbus$
PV-Flyback-Dbus$0 < V Dbus$$0 < n$
PV-Half-bridge-Dbus$0 < V Dbus < C 3 2 n$$0 < n < C 3 2 V Dbus$
PV-Full-bridge-Dbus$0 < V Dbus ≤ C 3 n$$0 < n ≤ C 3 V Dbus$
PV-Forward-INV$0 < R L < M 2 C 3 2 2 n 2 P omax$$0 < n < M C 3 2 P omax R L$
PV-Flyback-INV$0 < R L$$0 < n$
PV-Half-bridge-INV$0 < R L < M 2 C 3 2 8 n 2 P omax$$0 < n < M C 3 2 2 P omax R L$
PV-Full-bridge-INV$0 < R L ≤ M 2 C 3 2 2 n 2 P omax$$0 < n ≤ M C 3 2 P omax R L$
PV-Forward-INV-Abus$0 < V Abus < C 3 M 2 n$$0 < n < C 3 M 2 V Abus$
PV-Flyback-INV-Abus$0 < V Abus$$0 < n$
PV-Half-bridge-INV-Abus$0 < V Abus < C 3 M 2 2 n$$0 < n < C 3 M 2 2 V Abus$
PV-Full-bridge-INV-Abus$0 < V Abus ≤ C 3 M 2 n$$0 < n ≤ C 3 M 2 V Abus$
Table 2. Practical expressions of MCCs.
Table 2. Practical expressions of MCCs.
PV SystemRange of the OutputRange of n
PV-Forward$D L 1 2 C 3 2 n 2 P omax ≤ R L ≤ D U 1 2 C 3 2 n 2 P omax$$D L 1 C 3 P omax R L ≤ n ≤ D U 1 C 3 P omax R L$
PV-Flyback$D L 1 2 C 3 2 n 2 ( 1 − D L 1 ) 2 P omax ≤ R L ≤ D U 1 2 C 3 2 n 2 ( 1 − D U 1 ) 2 P omax$$D L 1 C 3 ( 1 − D L 1 ) P omax R L ≤ n ≤ D U 1 C 3 ( 1 − D U 1 ) P omax R L$
PV-Half-bridge$D L 2 2 C 3 2 n 2 P omax ≤ R L ≤ D U 2 2 C 3 2 n 2 P omax$$D L 2 C 3 P omax R L ≤ n ≤ D U 2 C 3 P omax R L$
PV-Full-bridge$4 D L 2 2 C 3 2 n 2 P omax ≤ R L ≤ 4 D U 2 2 C 3 2 n 2 P omax$$2 D L 2 C 3 P omax R L ≤ n ≤ 2 D U 2 C 3 P omax R L$
PV-Forward-Dbus$C 3 D L 1 n ≤ V Dbus ≤ C 3 D U 1 n$$C 3 D L 1 V Dbus ≤ n ≤ C 3 D U 1 V Dbus$
PV-Flyback-Dbus$C 3 D L 1 n ( 1 − D L 1 ) < V Dbus ≤ C 3 D U 1 n ( 1 − D U 1 )$$C 3 D L 1 V Dbus ( 1 − D L 1 ) < n ≤ C 3 D U 1 V Dbus ( 1 − D U 1 )$
PV-Half-bridge-Dbus$C 3 D L 2 n ≤ V Dbus ≤ C 3 D U 2 n$$C 3 D L 2 V Dbus ≤ n ≤ C 3 D U 2 V Dbus$
PV-Full-bridge-Dbus$2 C 3 D L 2 n ≤ V Dbus ≤ 2 C 3 D U 2 n$$2 C 3 D L 2 V Dbus ≤ n ≤ 2 C 3 D U 2 V Dbus$
PV-Forward-INV$M 2 C 3 2 D L 1 2 2 n 2 P omax ≤ R L ≤ M 2 C 3 2 D U 1 2 2 n 2 P omax$$M D L 1 C 3 2 P omax R L ≤ n ≤ M D U 1 C 3 2 P omax R L$
PV-Flyback-INV$M 2 C 3 2 D L 1 2 2 n 2 P omax ( 1 − D L 1 ) 2 ≤ R L ≤ M 2 C 3 2 D U 1 2 2 n 2 P omax ( 1 − D U 1 ) 2$$M D L 1 C 3 ( 1 − D L 1 ) 2 P omax R L ≤ n ≤ M D U 1 C 3 ( 1 − D U 1 ) 2 P omax R L$
PV-Half-bridge-INV$M 2 C 3 2 D L 2 2 2 n 2 P omax ≤ R L ≤ M 2 C 3 2 D U 2 2 2 n 2 P omax$$M D L 2 C 3 2 P omax R L ≤ n ≤ M D U 2 C 3 2 P omax R L$
PV-Full-bridge-INV$2 M 2 C 3 2 D L 2 2 n 2 P omax ≤ R L ≤ 2 M 2 C 3 2 D U 2 2 n 2 P omax$$2 M D L 2 C 3 P omax R L ≤ n ≤ 2 M D U 2 C 3 P omax R L$
PV-Forward-INV-Abus$C 3 M D L 1 2 n ≤ V Abus ≤ C 3 M D U 1 2 n$$C 3 M D L 1 2 V Abus ≤ n ≤ C 3 M D U 1 2 V Abus$
PV-Flyback-INV-Abus$C 3 M D L 1 2 n ( 1 − D L 1 ) ≤ V Abus ≤ C 3 M D U 1 2 n ( 1 − D U 1 )$$C 3 M D L 1 2 V Abus ( 1 − D L 1 ) ≤ n ≤ C 3 M D U 1 2 V Abus ( 1 − D U 1 )$
PV-Half-bridge-INV-Abus$C 3 M D L 2 2 n ≤ V Abus ≤ C 3 M D U 2 2 n$$C 3 M D L 2 2 V Abus ≤ n ≤ C 3 M D U 2 2 V Abus$
PV-Full-bridge-INV-Abus$2 C 3 M D L 2 n ≤ V Abus ≤ 2 C 3 M D U 2 n$$2 C 3 M D L 2 V Abus ≤ n ≤ 2 C 3 M D U 2 V Abus$
Table 3. Theoretical expressions of MCCs.
Table 3. Theoretical expressions of MCCs.
PV SystemRange of the OutputRange of n
PV-Forward$0 < R L < R sM n 2$$0 < n < R sM R L$
PV-Flyback$0 < R L$$0 < n$
PV-Half-bridge$0 < R L < R sM 4 n 2$$0 < n < 1 2 R sM R L$
PV-Full-bridge$0 < R L ≤ R sM n 2$$0 < n ≤ R sM R L$
PV-Forward-Dbus$0 < V Dbus < V sM 2 n$$0 < n < V sM 2 V Dbus$
PV-Flyback-Dbus$0 < V Dbus$$0 < n$
PV-Half-bridge-Dbus$0 < V Dbus ≤ V sM 4 n$$0 < n < V sM 4 V Dbus$
PV-Full-bridge-Dbus$0 < V Dbus ≤ V sM 2 n$$0 < n ≤ V sM 2 V Dbus$
PV-Forward-INV$0 < R L < M 2 R sM 2 n 2$$0 < n < M R sM 2 R L$
PV-Flyback-INV$0 < R L$$0 < n$
PV-Half-bridge-INV$0 < R L < M 2 R sM 8 n 2$$0 < n < M 2 R sM 2 R L$
PV-Full-bridge-INV$0 < R L ≤ M 2 R sM 2 n 2$$0 < n ≤ M R sM 2 R L$
Table 4. Practical expressions of MCCs.
Table 4. Practical expressions of MCCs.
PV SystemRange of the OutputRange of n
PV-Forward$D L 1 2 R sM n 2 ≤ R L ≤ D U 1 2 R sM n 2$$D L 1 R sM R L ≤ n ≤ D U 1 R sM R L$
PV-Flyback$D L 1 2 R sM n 2 ( 1 − D L 1 ) 2 ≤ R L ≤ D U 1 2 R s M n 2 ( 1 − D U 1 ) 2$$D L 1 ( 1 − D L 1 ) R sM R L ≤ n ≤ D U 1 ( 1 − D U 1 ) R sM R L$
PV-Half-bridge$D L 2 2 R sM n 2 ≤ R L ≤ D U 2 2 R sM n 2$$D L 2 R sM R L ≤ n ≤ D U 2 R sM R L$
PV-Full-bridge$4 D L 2 2 R sM n 2 ≤ R L ≤ 4 D U 2 2 R sM n 2$$2 D L 2 R sM R L ≤ n ≤ 2 D U 2 R sM R L$
PV-Forward-Dbus$D L 1 V sM 2 n ≤ V Dbus ≤ D U 1 V sM 2 n$$D L 1 V sM 2 V Dbus ≤ n ≤ D U 1 V sM 2 V Dbus$
PV-Flyback-Dbus$D L 1 V sM 2 n ( 1 − D L 1 ) < V Dbus ≤ D U 1 V sM 2 n ( 1 − D U 1 )$$D L 1 V sM 2 V Dbus ( 1 − D L 1 ) < n ≤ D U 1 V sM 2 V Dbus ( 1 − D U 1 )$
PV-Half-bridge-Dbus$D L 2 V sM 2 n ≤ V Dbus ≤ D U 2 V sM 2 n$$D L 2 V sM 2 V Dbus ≤ n ≤ D U 2 V sM 2 V Dbus$
PV-Full-bridge-Dbus$D L 2 V sM n ≤ V Dbus ≤ D U 2 V sM n$$D L 2 V sM V Dbus ≤ n ≤ D U 2 V sM V Dbus$
PV-Forward-INV$D L 1 2 M 2 R sM 2 n 2 ≤ R L ≤ D U 1 2 M 2 R sM 2 n 2$$M D L 1 R sM 2 R L ≤ n ≤ M D U 1 R sM 2 R L$
PV-Flyback-INV$D L 1 2 M 2 R sM 2 n 2 ( 1 − D L 1 ) 2 ≤ R L ≤ D U 1 2 M 2 R sM 2 n 2 ( 1 − D U 1 ) 2$$M D L 1 ( 1 − D L 1 ) R sM 2 R L ≤ n ≤ M D U 1 ( 1 − D U 1 ) R sM 2 R L$
PV-Half-bridge-INV$D L 2 2 M 2 R sM 2 n 2 ≤ R L ≤ D U 2 2 M 2 R sM 2 n 2$$M D L 2 R sM 2 R L ≤ n ≤ M D U 2 R sM 2 R L$
PV-Full-bridge-INV$2 D L 2 2 M 2 R sM n 2 ≤ R L ≤ 2 D U 2 2 M 2 R sM n 2$$M D L 2 2 R sM R L ≤ n ≤ M D U 2 2 R sM R L$
Table 5. Ranges of MCCs.
Table 5. Ranges of MCCs.
PV SystemMaximum RangeMinimum Range
PV-Forward$D L 1 2 R sMmin n 2 ≤ R L ≤ D U 1 2 R sMmax n 2$$D L 1 2 R sMmax n 2 ≤ R L ≤ D U 1 2 R sMmin n 2$
PV-Flyback$D L 1 2 R sMmin n 2 ( 1 − D L 1 ) 2 ≤ R L ≤ D U 1 2 R sMmax n 2 ( 1 − D U 1 ) 2$$D L 1 2 R sMmax n 2 ( 1 − D L 1 ) 2 ≤ R L ≤ D U 1 2 R sMmin n 2 ( 1 − D U 1 ) 2$
PV-Half-bridge$D L 2 2 R sMmin n 2 ≤ R L ≤ D U 2 2 R sMmax n 2$$D L 2 2 R sMmax n 2 ≤ R L ≤ D U 2 2 R sMmin n 2$
PV-Full-bridge$4 D L 2 2 R sMmin n 2 ≤ R L ≤ 4 D U 2 2 R sMmax n 2$$4 D L 2 2 R sMmax n 2 ≤ R L ≤ 4 D U 2 2 R sMmin n 2$
PV-Forward-Dbus$D L 1 V sMmin 2 n ≤ V Dbus ≤ D U 1 V sMmax 2 n$$D L 1 V sMmax 2 n ≤ V Dbus ≤ D U 1 V sMmin 2 n$
PV-Flyback-Dbus$D L 1 V sMmin 2 n ( 1 − D L 1 ) < V Dbus ≤ D U 1 V sMmax 2 n ( 1 − D U 1 )$$D L 1 V sMmax 2 n ( 1 − D L 1 ) < V Dbus ≤ D U 1 V sMmin 2 n ( 1 − D U 1 )$
PV-Half-bridge-Dbus$D L 2 V sMmin 2 n ≤ V Dbus ≤ D U 2 V sMmax 2 n$$D L 2 V sMmax 2 n ≤ V Dbus ≤ D U 2 V sMmin 2 n$
PV-Full-bridge-Dbus$D L 2 V sMmin n ≤ V Dbus ≤ D U 2 V sMmax n$$D L 2 V sMmax n ≤ V Dbus ≤ D U 2 V sMmin n$
PV-Forward-INV$D L 1 2 M 2 R sMmin 2 n 2 ≤ R L ≤ D U 1 2 M 2 R sMmax 2 n 2$$D L 1 2 M 2 R sMmax 2 n 2 ≤ R L ≤ D U 1 2 M 2 R sMmin 2 n 2$
PV-Flyback-INV$D L 1 2 M 2 R sMmin 2 n 2 ( 1 − D L 1 ) 2 ≤ R L ≤ D U 1 2 M 2 R sMmax 2 n 2 ( 1 − D U 1 ) 2$$D L 1 2 M 2 R sMmax 2 n 2 ( 1 − D L 1 ) 2 ≤ R L ≤ D U 1 2 M 2 R sMmin 2 n 2 ( 1 − D U 1 ) 2$
PV-Half-bridge-INV$D L 2 2 M 2 R sMmin 2 n 2 ≤ R L ≤ D U 2 2 M 2 R sMmax 2 n 2$$D L 2 2 M 2 R sMmax 2 n 2 ≤ R L ≤ D U 2 2 M 2 R sMmin 2 n 2$
PV-Full-bridge-INV$2 D L 2 2 M 2 R sMmin n 2 ≤ R L ≤ 2 D U 2 2 M 2 R sMmax n 2$$2 D L 2 2 M 2 R sMmax n 2 ≤ R L ≤ 2 D U 2 2 M 2 R sMmin n 2$
Table 6. The extreme values of MCCs.
Table 6. The extreme values of MCCs.
PV SystemRLmin or VDminRLmax or VDmaxnminnmax
PV-Flyback25.79 Ω6601 Ω0.22713.634
PV-Full-bridge16.5 Ω334.2 Ω0.18170.8175
PV-Flyback-Dbus74.25 V1188 V0.014850.2376
PV-Full-bridge-Dbus59.4 V267.3 V0.05940.2673
Table 7. Simulated weather parameters of PV system.
Table 7. Simulated weather parameters of PV system.
Weather Conditions(a)(b)(c)(d)
S (W/m2)1300850550350
T (℃)40252015
Table 8. Calculated values of MCCs.
Table 8. Calculated values of MCCs.
Weather Conditions(a)(b)(c)(d)
PV-Forward0.4970.6150.7600.957
1.9892.4623.0383.829
PV-Flyback0.6220.7690.9491.197
9.94712.3115.1919.15
PV-Half-bridge0.2490.3080.3800.479
1.1191.3851.7092.154
PV-Full-bridge0.4970.6150.7600.957
2.2382.7703.4184.308
Table 9. Calculated values of MCCs.
Table 9. Calculated values of MCCs.
PV SystemCalculated MCC Values
PV-Forward$0.28 R sM ≤ n ≤ 1.13 R sM$$4 R sM ≤ R L ≤ 64 R sM$
PV-Flyback$0.35 R sM ≤ n ≤ 5.66 R sM$$6.25 R sM ≤ R L ≤ 1600 R sM$
PV-Half-bridge$0.14 R sM ≤ n ≤ 0.636 R sM$$R sM ≤ R L < 20 . 25 R sM$
PV-Full-bridge$0.28 R sM ≤ n ≤ 1.272 R sM$$4 R sM ≤ R L < 81 R sM$
PV-Forward-Dbus$0.01 V sM ≤ n ≤ 0.04 V sM$$V sM ≤ V Dbus ≤ 4 V sM$
PV-Flyback-Dbus$0.013 V sM ≤ n ≤ 0.2 V sM$$1.25 V sM ≤ V Dbus ≤ 20 V sM$
PV-Half-bridge-Dbus$0.005 V sM ≤ n ≤ 0.02 V sM$$0.5 V sM ≤ V Dbus < 2 . 25 V sM$
PV-Full-bridge-Dbus$0.01 V sM ≤ n ≤ 0.045 V sM$$V sM ≤ V Dbus ≤ 4 . 5 V sM$
PV-Forward-INV$0.16 R sM ≤ n ≤ 0.64 R sM$$1 . 28 R sM ≤ R L ≤ 20 . 48 R sM$
PV-Flyback-INV$0.2 R sM ≤ n ≤ 3.2 R sM$$2 R sM ≤ R L ≤ 512 R sM$
PV-Half-bridge-INV$0.08 R sM ≤ n ≤ 0.36 R sM$$0.32 R sM ≤ R L < 6 . 48 R sM$
PV-Full-bridge-INV$0.16 R sM ≤ n ≤ 0.72 R sM$$1 . 28 R sM ≤ R L < 25 . 92 R sM$
Table 10. Experimental results for practicability of RMPPT.
Table 10. Experimental results for practicability of RMPPT.
(S,T)/(W/m2, °C)DmaxDmax1PomaxPomax1Pomax2
(750, 15)0.48650.4821152.19152.13149.63
(1000, 15)0.51750.5204214.7214.89212.5
(1250, 15)0.540.5373281.77281.72279.92
(750, 25)0.49290.5007151.29151.22148.79
(1000, 25)0.5240.5221213.4213.69211.2
(1250, 25)0.5470.5455280.87280.83277.93
(750, 35)0.49940.5013150.39150.51148.43
(1000, 35)0.53080.5269212.9212.69210.09
(1250, 35)0.55390.5520279.97280.14277.64
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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

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