Differential flat & PSO based photovoltaic maximum power point tracking control under partial shading condition

The use of maximum power point tracking (MPPT) technology has significantly increased the conversion efficiency of PV modules. However, the presence of partial shading in PV arrays can lead to multi-peaked output curves, which traditional MPPT methods struggle to track due to falling into local maximum power points. The paper proposes a MPPT control algorithm based on the combination of differential flat control (DFBC) and adaptive particle swarm optimization (APSO) algorithm. The PSO output value is used as the feed-forward feedback input of differential flat, and a second-order controller is used to track the reference flat trajectory, achieving global MPPT through differential flat control. The algorithm can overcome the system oscillation caused by the randomness of the PSO algorithm with the initialized particle position and the existence of control lag misjudgment. Simulation and experimental results show that the algorithm not only solves the problem that the traditional MPPT algorithm cannot find the global maximum power point, but also solves the problems that the traditional particle swarm algorithm has large randomness, slow convergence speed, and easy to produce large oscillations. The algorithm has greatly improved the tracking accuracy, tracking speed and response speed, realizing fast and accurate response to external changes, reducing energy loss, and improving the dynamic tracking performance of the system.


Introduction
With the increasing depletion of traditional fossil energy sources, clean and environmentally friendly solar energy has become an important part of the energy field.Photovoltaic cells are capable of converting solar energy into electrical energy, and when irradiance and temperature change, the output voltage and output current of Photovoltaic cells will change, exhibiting nonlinear output characteristics 1 In order to improve the conversion efficiency of the PV module, the PV module is always operated at the maximum power point by the corresponding control strategy, and this process is called maximum power point tracking (MPPT).MPPT technology has become an important part of PV systems and is very important to improve the conversion efficiency of PV modules.The commonly used MPPT methods are constant voltage (CVT) method, 2 perturbation observation (P&O) 3,4 method, and conductance increment method (INC). 5,6hese traditional methods cannot take into account the requirements of tracking time, tracking accuracy, and response speed due to their own limitations, and they are easy to fall into the local maximum power point in the shaded case and cannot track the global maximum power point, which causes great power loss.Therefore, the study of maximum power point tracking in the shaded case becomes especially important. 7any new approaches have been proposed to solve the multi-peak problem for this problem.Literature 8 uses a control strategy combining differential flat \right)control and perturbation observation method, but it requires initialization of the reference voltage and cannot accurately track the PV maximum power point under sudden changes in irradiance, so there are some limitations.Literature 9 proposed a combination of a sliding mode variable structure control and a perturbation observation method, but both methods have the potential to miss the maximum power point.Current research on maximum power tracking in shaded situations has focused on population intelligence optimization algorithms. 10,11Particle swarm algorithm, as a global algorithm with simple principle and strong global search capability, is widely used on MPPT. 12,13The literatures 14,15 used particle swarm algorithm by introducing adaptive parameter algorithm and random particles, but there are problems such as large oscillation amplitude, slow convergence speed, and dependence on initial conditions. 16,17Literature 18 proposed a hybrid algorithm based on the perturbation observation method and PSO, while the tracking time can be reduced by more than 35% compared to the original PSO, but the system oscillates more and the tracking accuracy is not high.
Therefore, this paper proposes a differential flatness control & APSO based MPPT control algorithm.Differential flatness theory is a novel control method for nonlinear systems. 19In this paper, the differential flattening theory is applied to the MPPT of PV system, and the APSO output value is used as the feedforward input of differential flattening, and the reference flat trajectory is tracked using a second-order controller, and finally the global MPPT is achieved by differential flattening control.
The rest of this paper is organized as follows: The second section introduces the model and characteristics of the photovoltaic system.The third section briefly describes the maximum power control algorithm of the photovoltaic system.The MPPT algorithm design for DFBC and APSO is provided in Section 4, followed by Section 5 Analysis of Experimental Results, finally, concluding remarks appear in Section 6.

Photovoltaic cell mathematical model
The equivalent circuit model of the solar cell is shown in Figure 1.Where the current source represents the photoreceptor current, which depends on the semiconductor material properties of the cell, the current source is connected in parallel with the diode and the equivalent parallel resistance R sh , and the equivalent series resistance R s on the series, I and U are the output current and output voltage of the solar cell, which can be seen from the equivalent circuit model of the solar cell. 20 Where: I ph is the current generated by the photovoltaic cell due to the photoelectric effect, I d is the internal dark current of the photovoltaic cell.The expression for the I-U output characteristics of a photovoltaic cell is Where: U is the PV cell output voltage; I is the PV cell output current; I sh is the current flowing through; I o is the reverse saturation current.q is the electronic charge; K is the Boltzmann constant; T is the absolute temperature value of solar cell operation; A is the diode ideal factor; R s is a series resistor; R sh is a parallel resistor.

PV array output characteristics in complex shading environments
A photovoltaic array is a physical combination of solar PV modules connected in series and parallel several times.Let the PV array of M 3 N consists of N p parallel branches, each parallel branch consists of N s PV modules in series, according to equation (2), the total output current I can be obtained at this time.
In the case of partial shading, the shaded PV module cannot generate electricity and becomes a load, which can lead to a hot spot effect, resulting in power loss and permanent damage to the PV module.To solve this problem, a bypass diode is usually connected in parallel at both ends of the PV module in practical engineering.When the PV array is under normal lighting conditions, each PV cell works normally and the bypass diode is in reverse cutoff at this time.When there is a part of the module is shaded, the bypass diode is in the conduction state at this time, short-circuiting the shaded module, preventing the current breakdown and playing a protective role, which also leads to the multi-peak characteristics of the PV P-U curve.When the PV array consists of multiple PV modules connected in series, the analysis process is the same as that for a single PV module.
In this paper, the TCT structured PV array is built according to the PV cell parameters in Table 1.Firstly, 10 photovoltaic cells are connected in parallel to achieve the required output voltage, and then these strings are connected in series to ensure that the output voltage reaches the voltage range required for normal operating conditions, finally forming a photovoltaic array, the principle of photovoltaic array composition is shown in Figure 2.
To simulate a shaded PV array, different light intensities were applied to rows 1-3 of the TCT structure to achieve the effect of a partially shaded PV array.Experiments were conducted according to the light intensity in Table 2 to study the output characteristics of the PV array under complex shading environment, and the output characteristic curves were obtained as shown in Figure 3.
As can be seen from Figure 3, the output power of the PV array has a nonlinear relationship with the output voltage and shows a single-peak characteristic in the unshaded environment, and with the change of light intensity, the PV array shows a multi-peak characteristic in the shaded environment.Although the output characteristics curve of photovoltaic cells show non-linear, but when the external environment remains unchanged, at a certain point there is bound to be an extreme value, the point is the maximum power point, and the voltage corresponding to it is the maximum power point voltage.

Single-phase PV array MPPT control system structure
From the above P-U characteristic curve of PV cell, we can see that adjusting the output voltage of PV cell, its power output will change accordingly.The maximum output power is obtained by controlling the output voltage of the PV cell so that it operates at that pole point, which is the principle of PV maximum power tracking. 21he structure of the resulting single-phase PV MPPT control system is shown in Figure 4 and consists of four main parts: PV array modules consisting of PV cells connected in series and parallel to provide energy to the entire system.The controller module, based on the P-U output characteristics of the PV array, performs the MPPT control algorithm operation, and the algorithm output is used to control the duty cycle of the IGBT switches.Boost circuit, the control signal output by the controller according to the MPPT algorithm is loaded to the IGBT switch of the Boost circuit, which regulates the output voltage of the PV array by controlling the duty cycle of the IGBT, thus controlling the output power of the PV array for the purpose of maximum power tracking.A load module that converts the electrical energy generated by the PV array into other forms of energy for utilization.

PSO algorithm
In the case of PV arrays with shading, traditional MPPT algorithms such as constant voltage (CVT) method, perturbation observation (P&O) method, and conductivity increment method are prone to fall into the local maximum power point due to their own limitations, and cannot do global optimization search.To solve this problem, a particle swarm algorithm (PSO) algorithm can be introduced to perform the optimization search.The PSO algorithm is a meta-heuristic method that enables global optimization of multipole functions.Each particle in the particle swarm algorithm has three parameters, namely the current position, velocity, and direction of the particle.When alliterative updating the position of a particle, the position of each particle is determined by the current historical optimal value P best of that particle and the optimal value G best of the particle for the whole population.The updated equation for the Where, k -the number of iterations.v -inertia weights.c 1 and c 2 -learning factors to adjust the weight of individual and group experiences, respectively.r 1 and r 2 -a random number between (0,1).The position of the particle in the particle swarm algorithm is used as the output voltage value of the PV array in the maximum power seeking of the PV array.

APSO algorithm
The ordinary particle swarm algorithm is more sensitive to the initial conditions because of the fixed parameters.The learning factors c 1 and c 2 can affect the speed of particles moving to the individual optimal position and the global optimal position, and the inertia weights v reflect the degree of the algorithm's inheritance of particle speed, which has a large impact on the algorithm's search ability.In order to improve the efficiency of the whole search, this paper proposes the APSO algorithm, which dynamically changes the parameters with the number of iterations on the basis of the ordinary particle swarm algorithm.
The PSO algorithm have good performance in global optimization seeking, but some application studies in recent years have shown that particle swarm algorithms in the traditional sense substantially randomize the initialization of particle positions leading to system oscillations, while the actual algorithms have problems such as control lag misclassification.For this reason, some scholars at home and abroad began to try to combine the PSO algorithm with the traditional MPPT algorithm to solve the contradiction between tracking speed and tracking accuracy.

Differential flatness control
The differential flatness control theory (DFBC) was proposed by Martin in the form of differential algebra, then studied in depth by Fliess and others, and applied to control theory. 22The system with differential flatness based control does not need to transform the nonlinear system into a linear system, and has better stability, tracking, and robustness. 23In recent years, it has been gradually introduced into the control field of power generation, electric motors, power electronics, etc. 24 The basic meaning is that in a nonlinear system, if it is possible to find a set of output vectors and that set of output vectors and its finite order differential algebra can represent all the input vectors and state vectors of the system without any integration.Then this nonlinear system is considered to be flat and can be called differential flat system.For a better understanding, assuming that the nonlinear system state variable is x and the input variable is u, if you can find a system output quantity y, satisfying: where i and j are positive integers, that is, the control variables and state variables of the system can be represented by the output variables of the system and their finite-order differentiation, and no integration is required in the differential expression.Such a nonlinear system is called a differential flat system.For differential flat systems, differential flat control theory can be used for control.The control principle is shown in Figure 5, and the whole control consists of two stages: feed-forward control and error feedback.When the feed-forward control trajectory is generated, the desired output is selected, the feed-forward control reference trajectory is generated in the state space, and finally the expected feed-forward reference trajectory is compensated by the error feedback formed by the flat output and the expected output.

Proof that single-phase PV MPPT are DFBC systems
According to the single-phase PV array MPPT control system structure (Figure 4), its system mathematical model can be obtained.
Where, U pv is the PV array output voltage; i pv is the PV array output current; U load is the load voltage; C pv is the input capacitance.
Select state variables: From equations ( 10) and ( 11), the equation of state of the system is obtained as: The charge of the input capacitor C pv and its first and second order derivatives are chosen as the flat output quantities.
The charge Q of the capacitor C pv is used as the flat output and the duty cycle D is used as the flat input, as shown in equation ( 14).
where _ Q is the first-order derivative of the charge Q, € Q is the second order derivative of the charge Q. U load is the load resistance voltage of the PV system.I(x) indicates that the output voltage is the output current of the x PV array.
From the above derivation, the system shown in Figure 3 can be expressed in the form of equation ( 9), so the system is a differential flat system, and the duty cycle D of the boost circuit is chosen as the flat input quantity and the charge C pv of the capacitor as the flat output quantity.

Design of differential flat controller
The relationship between the duty cycle and the output and load voltages of the PV array can be obtained from the operating principle of the Boost circuit: It can be seen that the MPPT problem of PV system is the relationship between duty cycle and output voltage of PV array and load voltage, and the control algorithm changes the output voltage of PV array by changing the duty cycle to achieve the purpose of MPPT.The state variables and input variables of the differential flat control system can be uniquely determined by a set of flat outputs and their nth-order derivatives.Therefore, applying the differential flat control theory to the MPPT of PV system can realize the direct control of the PV array output voltage, which not only shortens the seeking time but also improves the robustness of the whole system.
Assuming that the output voltage of the desired PV system is the reference voltage U ref , its corresponding duty cycle D ref can be obtained according to equation (15) above to generate the desired output.Since the value of the input capacitor C pv is fixed, a one-to-one mapping can be established between the array output voltage U pv and the flat output quantity Q, which is defined by the capacitor charge quantity as follows: Therefore the setting of the flat output capacitance trajectory can be converted into the setting of the reference voltage trajectory, and finally the output voltage is tracked to the reference voltage.
To eliminate the steady-state error of the system, a second-order controller is used as an error feedback to correct for Figure 6 shows a block diagram of the reference voltage control to control the output voltage of the array.
In summary, it can be seen that direct tracking of the output voltage of the PV array can be achieved using differential flattening control.

Design of MPPT for DFBC and APSO
In this paper, a combination of DFBC and APSO algorithm is used to implement MPPT control.The voltage of the PV array is selected as the location of the particle in the PSO algorithm to provide a reference voltage for the differential flat control of the PV array.The total output power of the PV array is used as an adaptation function with a particle population of 3 and a maximum number of 100 iterations.To cope with light intensity changes, the restart condition . 0:03 is set.The value tends to 0 when the system is stable, while the output power changes abruptly when the light amplitude changes.The overall algorithm flow is shown in Figure 7.
Compared to the traditional PSO algorithm that changes the output voltage of the PV array by changing the duty cycle to achieve the MPPT purpose.A combination of DFBC and APSO algorithm is used to directly control the output voltage of the PV array for MPPT, which can shorten the seeking time and improve the robustness of the whole system.It ensures that all possible peak points are searched, and possible peak points are not lost, and at the same time, it can ensure that the algorithm will not fall into local extreme points, and finally obtain the global maximum power point.

Analysis of experimental results
To verify the effectiveness of this control strategy, the simulation model is built in MATLAB2018a/ SIMULINK as shown in Figure 4.The parameters of the simulated PV cells are shown in Table 1 and constitute the PV array as in Figure 2, and the light intensity is shown in Table 2. Different light intensities are provided according to the parameters in Table 2 to simulate the comparison of this control algorithm with the PSO algorithm under unshaded and shaded conditions.Among them, the important parameter settings of the traditional particle swarm algorithm are shown in Table 3.

Simulation without shade
The light intensity is provided for the battery cell according to Table 2 without shading.Simulations were performed with PSO, APSO, and DFBC and APSO control, respectively, and the P-t curves of the three tracking algorithms are shown in Figure 8, and the performance comparisons are shown in Table 4.
From the simulation results, it can be seen that the DFBC and APSO tracking time is significantly improved for tracking to the same maximum power point

Simulation with shade
Set the light intensity of the battery cell to provide light intensity according to Table 2 without shade mode before 2.5 s and according to Table 2 with shade mode after 2.5 s.Simulations were performed with PSO, APSO, and DFBC and APSO in the presence of shading, respectively.The P-t curves of the three algorithms are shown in Figure 9, and the performance comparisons are shown in Table 5.
From the simulation results, it can be seen that the DFBC and APSO control algorithm used in this paper overcomes the problems of system oscillation caused by the PSO algorithm in the traditional sense of substantial random initialization of particle positions, and the existence of control lag misjudgment in the actual algorithm.Increasing the system response speed and the ability to quickly track to the maximum power in single-peak as well as multi-peak situations, increasing its convergence around the optimum.

Experimental verification
In order to prove the practical feasibility of the control strategy, the experimental platform of this paper is built by using Boost circuit, load, PV panel, and the main control chip is TMS320F28335.The actual PV board parameters are shown in Table 6, the physical object is shown in Figure 10.
The experiments were conducted by placing the 2 3 2 TCT structured PV arrays under outdoor sunlight and partially shading two of the series panels.By adjusting the value of the sliding resistor, the voltage and current values are collected by the A/D module, and the data is imported into Origin for data fitting curve, which roughly depicts the characteristic curve of the PV panel output, Figure 11 shows the PV panel output P-U fitting curve.
The output power of the PV panels is low because the light intensity and temperature in the test environment do not meet the requirements of the standard case.From Figure 11, it can be seen that the output characteristic curve of the volt cell shows a single peak when it is not shaded and two peaks when it is shaded, which is consistent with the output characteristic in the case of partial shadow shading.Therefore, experiments are conducted to verify the performance of the algorithm in the case of local shading.The experimental results are shown in Figures 12-14     the oscilloscope.The horizontal axis is time, and each grid is 0.5 s. the vertical axis is voltage, and each grid is 10 V.The collected data is imported into Origin through the A/D module for data fitting, and Figure 15 shows the output power curve by data fitting.The results show that under partial shade conditions, the solar panels are re-stabilized to operate at the maximum power point under shade conditions with the control system.This control strategy can significantly improve the tracking accuracy and speed, which verifies the correctness of this control strategy in the maximum power point tracking control.

Conclusions
In this paper, DFBC and APSO is proposed for PV maximum power tracking control to perform MPPT by directly controlling the output voltage of PV arrays, which can both shorten the seeking time and improve the robustness of the whole system.It overcomes the problems that the traditional algorithm cannot track the global maximum point under shadow conditions, and has long tracking time, poor tracking accuracy and slow response time.Compared with the traditional particle swarm algorithm, the convergence speed of particle swarm is accelerated and the global search capability is improved.The MPPT method proposed in this paper has significant improvement in tracking time, tracking accuracy, and response speed compared with the traditional PSO algorithm, and finally achieves the purpose of improving the conversion efficiency of PV modules.Simulation and experimental results show that this control algorithm has excellent tracking effect even under     the condition of drastic irradiance change, which can significantly improve the conversion efficiency of PV modules and increase the utilization rate of solar

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Figure 1 .
Figure 1.Equivalent circuit model of solar cell.

Figure 4 .
Figure 4. Sketch of MPPT control system structure of PV array.

Figure 5 .
Figure 5. Block diagram of the DFBC system.
by observing the output voltage waveform of the PV panel measured by

Figure
Figure Photovoltaic cell output fitting curve.

Table 4 .
Comparison of performance without shade.

Table 5 .
Comparison of performance with shading.

Table 6 .
Parameters of practical PV panel.Power output curve without shade.