Photovoltaic Systems Based on Average Current Mode Control: Dynamical Analysis and Chaos Suppression by Using a Non-Adaptive Feedback Outer Loop Controller

Abstract

This paper deals with the modeling and theoretical study of an average-current-mode-controlled photovoltaic power conversion chain. It should be noted that current mode control is a superior scheme for controlling DC–DC power electronic converters for photovoltaic applications. Bifurcation diagrams, largest Lyapunov exponents, Floquet theory, and time series are used to study the dynamics of the system. The theoretical results show the existence of subharmonic oscillations and period-1 oscillations in the system. The results of the numerical simulations showed that when the battery voltage at the output of the converter is fixed and ramp amplitude is taken as a control parameter, the photovoltaic power system exhibits the phenomenon of period doubling leading to chaotic dynamics. Furthermore, bifurcation diagrams showed that both the critical value of ramp amplitude for the occurrence of border collision bifurcation and the critical value of ramp amplitude for the occurrence of period-1 in the proposed system increased with the value of the battery terminal voltage. The numerical results are in accordance with the theoretical ones. Finally, an external control based on a non-adaptive controller having a sinusoidal function as a target is applied to the overall system for the suppression of chaotic behavior.

1. Introduction

In a difficult energy and economic context, the expectations regarding renewable energies in general and solar energy in particular are increasingly important. Reducing the costs of photovoltaic systems and increasing their efficiency are major concerns for researchers in order to make them as competitive as possible [1]. Photovoltaic (PV) systems are one of the interesting renewable energy sources. They are not only renewable but also inexhaustible and non-polluting [2,3,4,5,6,7]. The capacity to achieve maximum energy output is crucial for the optimization of the generation system [8]. Since the power output of a PV panel varies instantaneously with the variation of the ambient conditions, it is difficult for such a system to operate at its maximum power point, hence the need for a maximum power point tracking (MPPT) controller to force the (PV) system to operate at its optimum operating point. Generally speaking, a PV system consists of a PV generator, a DC/DC converter, and a control system that regulates certain parameters in order to extract the maximum energy from the PV panels and transfer it to the load for domestic and industrial purposes. The boost converter is more appropriate for these types of applications as it receives the relatively low voltage from the PV panels and raises it to a higher level appropriate for the load [9]. In the field of photovoltaic energy, most researchers have focused on the identification of PV module parameters through meta-heuristic optimization methods such as the pattern search algorithm [10], the optimization algorithm (WOA) [11,12,13,14], the flower pollination algorithm (FPA) [15], the water cycle algorithm (WCA) [16], the Jaya algorithm [17], the harmony search algorithm (HSA) [18], the multiple back learning search algorithm (MLBSA) [19], the hybrid teaching–learning artificial bee colony algorithm (TL-ABC) [20], and the multiple feedback search algorithm (MLBSA). Other researchers have focused on maximum power point tracking using new MPPT methods, namely the perturbation and observation modify (P&OM) method for partially shaded panels [21] and the technique based on particle swarms combined with neural networks [22]. However, the nonlinearity due to PV panels (inverter operation and load), makes them susceptible to generate nonlinear behaviors such as bifurcations, harmonic and subharmonic oscillations, and chaotic oscillations [23,24,25]. The issue of the nonlinear behavior of DC/DC converters for PV applications has been addressed by very few researchers in recent years. Indeed, refs [26,27,28,29] analyzed bifurcations and harmonic and subharmonic behavior for PV applications, respectively. The limitation of these papers comes down to the control level as they mostly use voltage controllers. It should be noted that current mode control is important because it is one of the most widely used techniques in industry for the control of power converters [30]. Recently, a highly efficient and fast maximum power point tracking technique for a shaded photovoltaic DC–DC power conversion system was proposed in [31]. In their work, they used an average current mode controller (ACMC) to regulate the PV array current using two feedback loops: a voltage control loop and a current control loop. The external voltage control loop regulated the PV array voltage for use by the MPPT algorithm, whereas a modulation of the reference current of the internal current control loop allowed for fast tracking of the reference current. Unfortunately, this paper did not deal with the dynamic behavior of the converter, which is a major drawback because the performance of the PV power conversion chain strongly depends on the dynamics of its converter. If the converter dynamic presents subharmonic behaviors, it could limit the performance of the photovoltaic system and quickly deteriorate it. Therefore, it remains important to perform a dynamic analysis of DC/DC converters for PV applications [32,33], with the aim avoiding energy losses for rows of system parameters that lead to subharmonic behavior in the converter. The desired behavior for these types of applications being the periodic behavior of period-1. Thus, our objectives in this paper are summarized as follows:

(i)

To propose a mathematical modeling of the DC–DC photovoltaic power conversion system studied by Hosseini et al. in [31].

(ii)

To determine by analytical and then numerical methods the dynamic behaviors of a photovoltaic system with only the inner ACMC controller.

(iii)

To propose a non-adaptive feedback outer loop controller for the suppression of chaotic behaviors in order for the photovoltaic system to exhibit optimal operation.

The rest of this paper is organized as follows: Section 2 presents the mathematical modeling of the photovoltaic DC–DC power conversion system, its stability, and its numerical simulations. The control to eliminate the subharmonic oscillations found in the photovoltaic DC–DC power conversion system is presented in Section 3. This paper ends with a conclusion.

2. Materials and Methods

2.1. Continuous Time Modelling and Theoretical Analysis of a Photovoltaic Power Conversion System Integrating ACMC Control Topology

Figure 1 shows a photovoltaic power conversion device.

Figure 1 consists of a DC/DC boost converter fed by a photovoltaic array at its input and a storage battery at its output. We choose the BP Solar SX series photovoltaic model 585 to provide cost-effective photovoltaic power for general use through direct current load operation or AC load on systems with inverters [29]. This module produces a maximum power of 85 W. The electrical characteristics of this photovoltaic module at a temperature of $25°$ and irradiation of $1000 {\mathrm{W}/\mathrm{m}}^2$ are given in the

Table 1. BP 585 module parameter [29].

Circuit Component	Values
Maximum power $P_{\max }$	$85 \mathrm{W}$
Voltage at $V_{mpp}$	$18 \mathrm{V}$
Current at $I_{mpp}$	$4.72 \mathrm{A}$
Short-circuit current $I_{sc}$	$5 \mathrm{A}$
Open-circuit voltage $V_{oc}$	$21.1 \mathrm{V}$

.

Regarding the converter, it plays a role in interfacing and adapting energy flow between the source and load [29]. The input capacitor smooths the voltage supplied by PV panel in order to avoid ripple frequencies caused by the non-linearity of the photovoltaic panel. An LC filter is also used at the output to reduce the ripples caused by the non-linearity of the converter and to provide a smooth current at the output. Another function of the DC/DC converter is to track the maximum power point (MPP) by controlling the duty cycle using the incremental conductance algorithm (InCond), which lowers or raises the duty cycle value to position the PV operating point at its MPPT at each instant in our project. The error signal is processed using two compensators that constitute the control. Each compensator has a low-frequency pole that provides a high gain to minimize the error, a first high-frequency pole necessary to cancel the zero generated by the series resistor, and a second high-frequency pole placed to attenuate the switching noise in the feedback loop [34]. The main purpose of the compensator is to adjust the frequency response of a controller so that the crossover frequency of the boost converter is optimally positioned while maintaining sufficient phase and gain margin for good dynamic response and stability [35]. These compensators are usually designed by using a transfer function given by:

$$H(s)=\frac{k_P{\omega }_z}{s}\frac{1+\frac{s}{{\omega }_z}}{1+\frac{s}{{\omega }_p}},$$

(1)

where $k_p$ is the proportional gain of the controller. ${\omega }_z$ represents the zero and ${\omega }_p$ represents the pole. The control logic is to compare the control $v_{con}$ voltage with ramp signal ($V_{ramp}$). Thus, if $v_{con}\succ V_{ramp}$, the switch $S$ is closed, and if $v_{con}\prec V_{ramp}$, the switch $S$ is open. This sequence is well illustrated in Figure 2.

In Figure 2, it is evident that the command signal u is a binary variable, taking a value of 1 when the switch is closed and 0 when it is open. It is important to note that the amplitude variation of the ramp signal can result in different converter behaviors, including harmonic, subharmonic, and periodic oscillations.

2.1.1. Mathematical Modeling

By applying Kirchhoff’s laws for both the meshes and nodes in Figure 1, we can derive the following set of equations:

$$\begin{array}{l}\frac{d{v}_{C_1}}{dt}=\frac{i_{pv}}{C_1}-\frac{i_L}{C_1},\\ \frac{d{i}_L}{dt}=\frac{v_{C_1}}{L}-\frac{r}{L}{i}_L-\frac{v_{C_2}}{L}\left(1-u\right),\\ \frac{d{v}_{C_2}}{dt}=\frac{V_{CC}-v_{C_2}}{R{C}_2}+\frac{i_L}{C_2}\left(1-u\right),\end{array}$$

(2)

where $v_{C_1}\left(v_{pv}\right) \mathrm{et} i_{pv}$, respectively, represent the voltage and current of the PV source, $i_L$ is the current of the inductor, and $u$ is the state of the switch. $C_1$ is the capacitance directly connected to the PV and its role is to smooth the voltage at the output of the PV. $L$ represents the inductance, $V_{CC}$ the output voltage of the battery. $r$ is the equivalent resistance in series with the coil and $R$ the equivalent resistance that can be felt at the output of the inverter. As demonstrated in [36], the PV can be linearized around the point of maximum $MPP\left(v_{mpp},i_{mpp}\right)$ power using a Taylor series limit expansion. Thus, stopping at the first order of the limit expansion, the PV current can be written in the following linear form:

$$i_{pv}\approx i_{mpp}-\frac{1}{R_{pv}}\left(v_{C_1}-v_{mpp}\right)$$

(3)

with

$$R_{pv}=\frac{\left(R_s+R_p\right)V_T+R_s{R}_p{I}_s{e}^{\frac{V_{mpp}+R_s{i}_{mpp}}{V_T}}}{V_T+R_p{I}_s{e}^{\frac{V_{mpp}+R_s{i}_{mpp}}{V_T}}},$$

(4)

where $I_s$ is the saturation current of the diode and $R_s$ and $R_p$ represent the series resistance and parallel resistance of the PV, respectively. The voltage and current of the PV at the maximum power point are $V_{mpp}$ and $i_{mpp}$, respectively. By applying the control law, the equations derived from this photovoltaic energy conversion system are:

$$\begin{array}{l}\frac{d{i}_e}{dt}=v_{C_1}-v_{ref},\\ \frac{d{v}_e}{dt}=i_{con}-i_L,\\ \frac{d{i}_{con}}{dt}=k_p{\omega }_p{\omega }_z{i}_e+k_p{\omega }_p\left(v_{C_1}-v_{ref}\right)-{\omega }_p{i}_{con},\\ \frac{d{v}_{con}}{dt}=k_p{\omega }_p{\omega }_z{v}_e+k_p{\omega }_p\left(i_{con}-i_L\right)-{\omega }_p{v}_{con}.\end{array}$$

(5)

By setting $\tau =\frac{1}{R{C}_2}t$, ${\alpha }_1=\frac{R{C}_2}{C_1}$, ${\alpha }_2=\frac{R{C}_2}{L}$, ${\alpha }_3=R{C}_2$, ${\alpha }_4=k_{p}R{C}_2{\omega }_p{\omega }_z$, ${\alpha }_5=k_{p}R{C}_2{\omega }_p$, ${\alpha }_6=R{C}_2{\omega }_p$, and ${\alpha }_7=\frac{R{C}_2}{R_{pv}{C}_1}$, the photovoltaic energy conversion system shown in Figure 1 can be put into the more elegant form as follows:

$$\begin{array}{l}\frac{d{v}_{C_1}}{d\tau }={\alpha }_1\left(i_{mpp}-i_L\right)-{\alpha }_7\left(v_{C_1}-v_{mpp}\right),\\ \frac{d{i}_L}{d\tau }={\alpha }_2\left(v_{C_1}-r{i}_L-v_{C_2}\left(1-u\right)\right),\\ \frac{d{v}_{C_2}}{d\tau }=V_{CC}-v_{C_2}+R{i}_L\left(1-u\right),\\ \frac{d{i}_e}{d\tau }={\alpha }_3\left(v_{C_1}-v_{ref}\right),\\ \frac{d{v}_e}{d\tau }={\alpha }_3\left(i_{con}-i_L\right),\end{array}$$

(6)

$$\begin{array}{l}\frac{d{i}_{con}}{d\tau }={\alpha }_4{i}_e+{\alpha }_5\left(v_{C_1}-v_{ref}\right)-{\alpha }_6{i}_{con},\\ \frac{d{v}_{con}}{d\tau }={\alpha }_4{v}_e+{\alpha }_5\left(i_{con}-i_L\right)-{\alpha }_6{v}_{con}.\end{array}$$

The values of the parameters of Equations (2), (5) and (6) are mentioned in

Table 2. Boost converter and ACMC controller parameters.

Circuit Component	Values	Circuit Component	Values
$C_1$	$43 \mu \mathrm{F}$	$L$	$200 \mu \mathrm{H}$
$R_1$	$100 \mathrm{m}\Omega$	$V_{CC}$	Variable
$R_2$	$200 \mathrm{m}\Omega$	${\omega }_z=1/T$	$1 \mathrm{krad}/\mathrm{s}$
$C_2$	$200\mu \mathrm{F}$	$\beta$	$275 \mathrm{ms}$
$f$	$50 \mathrm{kHz}$	$k_p$	1
${\omega }_p$	$175 \mathrm{krad}/\mathrm{s}$	$V_U$	1
$V_l$	0

.

2.1.2. Stability Study Using Floquet Theory

The periodic equation that we need to consider can be expressed in the state space representation (or matrix representation), which is suitable for solving a system of differential equations. The constants are represented as a column vector $N$, and their dynamics can be expressed as a function of a vector of variables $x$ called the state vector. Assuming that the subsystem is linear and time invariant, the evolution of each subsystem is defined by the following equation:

$$\dot{x}=\left\{\begin{array}{l}{M}_{1}x+N \mathrm{for} \mathrm{u}=1\\ M_{0}x+N \mathrm{for} \mathrm{u}=0\end{array}\right.\mathrm{with} x={\left(\begin{array}{ccccccc}{v}_{C_1}& i_L& v_{C_2}& i_e& v_e& i_{con}& v_{con}\end{array}\right)}^T$$

(7)

where $M_1, M_0$, and$N$ are matrices of appropriate dimensions. In particular, $M_1 \mathrm{and} M_0$ describe the dynamic relationship between the state vectors called dynamic matrices. The control signal $v_{con}$ represents one of the variables in (7) and is given by $v_{con}=Kx$, where $K$ is the feedback coefficient. The matrices $M_1, M_0,N \mathrm{and} K$ are given by:

$$M_1=\left(\begin{array}{ccccccc}-{\alpha }_7& -{\alpha }_1& 0& 0& 0& 0& 0\\ {\alpha }_2& -r{\alpha }_2& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0\\ {\alpha }_3& 0& 0& 0& 0& 0& 0\\ 0& -{\alpha }_3& 0& 0& 0& {\alpha }_3& 0\\ {\alpha }_5& 0& 0& {\alpha }_4& 0& -{\alpha }_6& 0\\ 0& -{\alpha }_5& 0& 0& {\alpha }_4& {\alpha }_5& -{\alpha }_6\end{array}\right),$$

(8)

$$M_0=\left(\begin{array}{ccccccc}-{\alpha }_7& -{\alpha }_1& 0& 0& 0& 0& 0\\ {\alpha }_2& -r{\alpha }_2& -{\alpha }_2& 0& 0& 0& 0\\ 0& R& -1& 0& 0& 0& 0\\ {\alpha }_3& 0& 0& 0& 0& 0& 0\\ 0& -{\alpha }_3& 0& 0& 0& {\alpha }_3& 0\\ {\alpha }_5& 0& 0& {\alpha }_4& 0& -{\alpha }_6& 0\\ 0& -{\alpha }_5& 0& 0& {\alpha }_4& {\alpha }_5& -{\alpha }_6\end{array}\right),$$

(9)

$$N\left(\begin{array}{c}{\alpha }_1{i}_{mpp}+{\alpha }_7{v}_{mpp}\\ 0\\ E\\ -{\alpha }_3{v}_{ref}\\ 0\\ -{\alpha }_5{v}_{ref}\\ 0\end{array}\right)K{\left(\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 1\end{array}\right)}^T.$$

(10)

By assuming operation in the nominal steady state of period-1 in which there is only one switching in a complete clock cycle, we can derive the following system equations:

• Before the first switching of a clock cycle, the status changes as follows [37]:

$$x\left(dT\right)=e^{M_{1}dT}x\left(0\right)+{\displaystyle {\int }_0^{dT}{e}^{M_1\left(dT-\tau \right)}}Nd\tau .$$

(11)

• After switching and until the end of the clock cycle, the status changes as follows [37]:

$$x\left(T\right)=e^{M_0\left(1-d\right)T}x\left(dT\right)+{\displaystyle {\int }_{dT}^T{e}^{M_0\left(T-\tau \right)}}Nd\tau .$$

(12)

By assuming that there is no discontinuity in the states, we can admit that the final state that precedes the switching time is equal to the initial state after the switching. Hence the condition:

$$x\left(T\right)=x\left(0\right).$$

(13)

The objective here is to express this initial condition in terms of the duty cycle. Substituting (11) into (12):

$$x\left(0\right)={\left(I-e^{\left(M_0\left(1-d\right)+M_{1}d\right)T}\right)}^{-1}\left(e^{M_0\left(1-d\right)T}{\displaystyle {\int }_0^{dT}{e}^{M_1\left(dT-\tau \right)}}Nd\tau +{\displaystyle {\int }_{dT}^T{e}^{M_0\left(T-\tau \right)}}Nd\tau \right),$$

(14)

with the identity $I$ matrix and the same dimension as $M_0 \mathrm{et} M_1$. Let us propose:

$$h\left(x,t\right)=v_{con}-V_{ramp},$$

(15)

with,

$$V_{ramp}=\frac{\left(V_M\right)}{T}t.$$

(16)

By replacing (15) with (16) gives:

$$h\left(x,t\right)=Kx-\frac{\left(V_M\right)}{T}t.$$

(17)

The converter switches from an ON to OFF configuration when the following switching condition is met:

$$h\left(x,t\right)=0.$$

(18)

By substituting (11) into (17) we get:

$$K\left(e^{M_{1}dT}x\left(0\right)+{\displaystyle {\int }_0^{dT}{e}^{M_1\left(dT-\tau \right)}}Nd\tau \right)=V_{M}d.$$

(19)

Replacing $x\left(0\right)$ in Equation (14) into Equation (19) gives:

$$K\left(e^{M_{1}dT}{\left(I-e^{\left(M_0\left(1-d\right)+M_{1}d\right)T}\right)}^{-1}\left(e^{M_0\left(1-d\right)T}{\displaystyle {\int }_0^{dT}{e}^{M_1\left(dT-\tau \right)}}Nd\tau +{\displaystyle {\int }_{dT}^T{e}^{M_0\left(T-\tau \right)}}Nd\tau \right)+{\displaystyle {\int }_0^{dT}{e}^{M_1\left(dT-\tau \right)}}Nd\tau \right)=V_{M}d.$$

(20)

It is noted that the duty cycle $d$ is the only unknown in this equation; solving this equation via the Newton Raphson method allows us to find not only the duty cycle $d$ but also the saltation matrix whose expression is [37]:

$$S=I+\frac{\left(M_0-M_1\right)x\left(dT\right)K}{K\left(M_{1}x\left(dT\right)+N\right)-\frac{\left(V_M\right)}{T}}.$$

(21)

The knowledge of saltation matrix $S$ and transition matrices before and after commutation, $M_0$ and $M_1$, respectively, allows us to determine the monodromy matrix of the period-1 orbit, which is a key element in the study of stability of this orbit. Its expression is given by the following Equation (22):

$$P=e^{M_0\left(1-d\right)T}S{e}^{M_{1}dT}.$$

(22)

These Floquet multipliers ${\eta }_i$ will consequently be the solutions of the equation $\det (P-{\eta }_{i}I)=0$ (or the eigenvalues of the monodromy matrix) [37,38]. It should be noted that the period-2 orbit will gain stability and bifurcate into period-1 if the system includes the phenomenon of inverse period splitting when a control parameter varies. Moreover, at this value of the control parameter, where period-2 is destroyed and gives way to period-1, the monodrony matrix admits an eigenvalue (Floquet multiplier) equal to −1.

Table 3. Duty cycle values and Floquet multipliers for the fixed amplitude $V_M$.

${\mathit{V}}_{\mathit{M}}$	$\mathbf{Duty} \mathbf{Cycle} \mathit{d}$	Floquet Multipliers	Special Behavior
0.4	$\begin{array}{l}0.9334\\ 0.8620\end{array}$	$\begin{array}{l}{\eta }_1=-1.019\\ {\eta }_2=0.988\\ {\eta }_3=0.966\\ {\eta }_4=0.696\\ {\eta }_5=0.359\\ {\eta }_6=0.001\\ {\eta }_7=-0.001\end{array}$
$0.404$	$0.8970$	$\begin{array}{l}{\eta }_1=-1.000\\ {\eta }_2=0.988\\ {\eta }_3=0.966\\ {\eta }_4=0.696\\ {\eta }_5=0.359\\ {\eta }_6=0.001\\ {\eta }_7=-0.001\end{array}$	Reverse Period doubling
$0.6$	$0.8974$	$\begin{array}{l}{\eta }_1=-0.409\\ {\eta }_2=0.987\\ {\eta }_3=0.971\\ {\eta }_4=0.657\\ {\eta }_5=0.356\\ {\eta }_6=-0.003\\ {\eta }_7=0.001\end{array}$
$1.0$	$0.8978$	$\begin{array}{l}{\eta }_{1,2}=0.983\pm 0.002\mathrm{i}\\ {\eta }_3= 0.519\\ {\eta }_4=0.342\\ {\eta }_5=0.250\\ {\eta }_6=0.001\\ {\eta }_7=0.005\end{array}$

shows the Floquet multipliers and duty cycle of the system studied for specific values of the ramp amplitude $V_M$ and the voltage $V_{CC}=48 \mathrm{V}$.

Indeed, for a fixed value of $V_M$, the values of the system parameters used are given in Table 1 and we solve Equation (20) using the Newton Raphson method to determine the duty cycle $d$. Once the duty cycle is known, we determine $x\left(dT\right)$, which allows us to evaluate the saltation and monodromy matrices and therefore determine the Floquet multipliers. In Table 3, for the amplitude of the ramp $V_M=0.404 \mathrm{V}$, one of the Floquet multipliers is approximately equal to −1. This indicates that at these values of gain, the orbit of period-2 stabilizes and gives way to an oscillation of period-1. The critical value allows us to ensure that the system remains stable until the maximum value of fixed work $V_M=1 \mathrm{V}$. The critical value $V_M=0.404 \mathrm{V}$ also helps us to fix the boundary between period-1 oscillations and subharmonic oscillations.

2.1.3. Numerical Method

A flowchart for the average current mode control (ACMC) used in the study is presented in Figure 3. The algorithm starts by initializing the induction current and the voltages (Figure 1) across the capacitors of the power converter (Block 2), while ensuring that the voltage reference satisfies $V_{ref,init}=0.85\times V_{MPPT,\max }$.

Blocks 3, 4, and 5 allow us to always work at the maximum power point. Indeed, with fluctuations in temperature and solar radiation, it is important to maintain the output voltage of the MPPT module in Figure 1 at the maximum power point and this voltage is represented in our equations by the voltage $V_{ref}$. Once $V_{ref}$ is determined, we use the fourth-order Runge–Kutta algorithm adapted to the power converter (block 7) and proposed in [39] to solve the system (7). This algorithm works as follows: if $u=1$ (switch closed), solve with the fourth-order Runge–Kutta algorithm the system $\dot{x}=M_{1}x+N$; however, if $u=0$ (switch open), solve with the fourth-order Runge–Kutta algorithm the system $\dot{x}=M_{0}x+N.$ On the other hand, if $u=0$ and the time is a multiple of the ramp period, close the switch and solve with the fourth-order Runge–Kutta algorithm the system $\dot{x}=M_{1}x+N.$ Furthermore, if $u=1$ and the voltage of the internal controller is lower than the value of the modulating signal, then the switch opens in this case and we solve with the fourth-order Runge–Kutta algorithm the system $\dot{x}=M_{0}x+N.$ Numerically, we obtain the duty cycle using the expression derived from Equation (18). It should be noted that all our numerical simulations are performed using MATLAB/SIMULINK. Other software such as C++, Fortran, or Python could also be used [40,41,42,43,44].

2.2. Non-adaptive Control

A control method to eliminate subharmonic oscillators in the photovoltaic power conversion chain proposed in Figure 1 is proposed in this section. Figure 4 is the block diagram of our control method that we apply here to eliminate the subharmonic and chaotic behaviors in the photovoltaic power conversion chain.

In Figure 4, the loop consisting of the internal PWM controller and the photovoltaic system is the one we studied in the first part of this paper. The obtained behaviors are chaotic behavior to period 1 behavior passing through subharmonic behaviors. Our control system consists of four blocks: a block representing the PV system [45,46,47], a block representing the internal controller that can produce the control voltage $v_{con}$, a block that provides PWM to open and close the control switch SW, and a block representing the external controller. Note that the PV system is modeled by Equation (2), the internal controller by Equation (5), and the PWM block contains a ramp (modulating signal) of fixed frequency at its comparator. This ramp will produce a pulse at the output of the comparator when it cuts off the control signal that will allow the duty cycle to be varied and consequently control the opening or closing of the power switch SW. From the above, we can say that if the control voltage is sinusoidal with a single fundamental frequency in its frequency spectrum, then the ramp is likely to intersect it uniformly and therefore the photovoltaic system will produce periodic oscillations of period-1. Thus, our objective in this part will be to bring an external control to the system in order to make the control signal $v_{con}$ tend towards a sinusoidal voltage to destroy the subharmonic behavior in the photovoltaic system. Of all the control methods in the literature, we choose the non-adaptive control method because it is easy to implement and inexpensive [48,49,50]. Applying the non-adaptive controller to the internal controller block, Equation (5) becomes:

$$\begin{array}{l}\frac{d{i}_e}{d\tau }={\alpha }_3\left(v_{C_1}-v_{ref}\right)-k_1\left(i_e-\left(a_0+a_1\sin \left(2\pi \tau \right)\right)\right),\\ \frac{d{v}_e}{d\tau }={\alpha }_3\left(i_{con}-i_L\right)-k_2\left(v_e-\left(a_0+a_1\sin \left(2\pi \tau \right)\right)\right),\\ \frac{d{i}_{con}}{d\tau }={\alpha }_4{i}_e+{\alpha }_5\left(v_{C_1}-v_{ref}\right)-{\alpha }_6{i}_{con}-k_3\left(i_{con}-\left(a_0+a_1\sin \left(2\pi \tau \right)\right)\right),\\ \frac{d{v}_{con}}{d\tau }={\alpha }_4{v}_e+{\alpha }_5\left(i_{con}-i_L\right)-{\alpha }_6{v}_{con}-k_4\left(v_{con}-\left(a_0+a_1\sin \left(2\pi \tau \right)\right)\right).\end{array}$$

(23)

where $k_i \mathrm{i}\in \left[1,4\right]$ represents the non-adaptive feedback gains and the function $f\left(t\right)=a_0+a_1\sin \left(2\pi \tau \right)$ represents the sinusoidal current ($I^{*}$) or the sinusoidal voltage ($V^{*}$) taken as the target to be achieved by $i_e$, $v_e$, $i_{con}$, and $v_{con}$. For practical and clutter reasons, it is advisable to control only one direction of the system to be monitored [48,49,50]. Thus, as it is the voltage $v_{con}$ that we are interested here, we choose to annul the gains $k_1$, $k_2$, and $k_3$ (the case of Figure 5). By setting $k_4=k$, the dynamics of the internal controller under the action of the non-adaptive external control can be put in the form:

$$\begin{array}{l}\frac{d{i}_e}{d\tau }={\alpha }_3\left(v_{C_1}-v_{ref}\right),\\ \frac{d{v}_e}{d\tau }={\alpha }_3\left(i_{con}-i_L\right),\\ \frac{d{i}_{con}}{d\tau }={\alpha }_4{i}_e+{\alpha }_5\left(v_{C_1}-v_{ref}\right)-{\alpha }_6{i}_{con},\\ \frac{d{v}_{con}}{d\tau }={\alpha }_4{v}_e+{\alpha }_5\left(i_{con}-i_L\right)-{\alpha }_6{v}_{con}-k\left(v_{con}-\left(a_0+a_1\sin \left(2\pi \tau \right)\right)\right).\end{array}$$

(24)

We deduce that the equations for the dynamics of the photovoltaic system under the influence of the external controller can be put in the form:

$$\begin{array}{l}\frac{d{v}_{C_1}}{d\tau }={\alpha }_1\left(i_{mpp}-i_L\right)-{\alpha }_7\left(v_{C_1}-v_{mpp}\right),\\ \frac{d{i}_L}{d\tau }={\alpha }_2\left(v_{C_1}-r{i}_L-v_{C_2}\left(1-u\right)\right),\\ \frac{d{v}_{C_2}}{d\tau }=V_{CC}-v_{C_2}+R{i}_L\left(1-u\right),\\ \frac{d{i}_e}{d\tau }={\alpha }_3\left(v_{C_1}-v_{ref}\right),\\ \frac{d{v}_e}{d\tau }={\alpha }_3\left(i_{con}-i_L\right),\\ \frac{d{i}_{con}}{d\tau }={\alpha }_4{i}_e+{\alpha }_5\left(v_{C_1}-v_{ref}\right)-{\alpha }_6{i}_{con},\\ \frac{d{v}_{con}}{d\tau }={\alpha }_4{v}_e+{\alpha }_5\left(i_{con}-i_L\right)-{\alpha }_6{v}_{con}-k\left(v_{con}-\left(a_0+a_1\sin \left(2\pi \tau \right)\right)\right).\end{array}$$

(25)

3. Results

3.1. Existence of Period-Doubling Bifurcation in the PV System

The representation in the complex plane of the Floquet multipliers when the amplitude $V_M$ varies in the interval $0.3 \mathrm{V}$ to $0.5 \mathrm{V}$ with a step of ${10}^{-3}$ is represented in Figure 5.

In Figure 5, for several values of the gain $V_M$, the module of the Floquet multipliers is located inside the circle of unit radius, which is synonymous with stability of the period-1 orbit. Moreover, for other values of the parameter $V_M$, the Floquet multipliers are located outside the circle of unit radius, which is synonymous with destabilization of the periodic orbit of period-1. Finally, since the circle of unit radius is in the complex plane intersects the point (−1,0) for the critical value of the gain $V_M\approx 0.404 \mathrm{V}$, we not only deduce the existence of the phenomenon of the inverse period doubling in the system (as shown in Table 3) but also the boundary between the stability and instability of the orbit of period-1.

Figure 6 shows the domains of subharmonic and stable oscillation when the ramp amplitude $V_M$ and the battery terminal voltage $V_{CC}$ are incremented in the intervals $0.3 \mathrm{V}\le V_M\le 0.5 \mathrm{V}$ and $40 \mathrm{V}\le V_{CC}\le 60 \mathrm{V}$, respectively.

Figure 6 is obtained by searching for the Floquet multipliers for each value of the ramp voltage and the battery terminal voltage. If there is a Floquet multiplier equal to −1 for a given couple $\left(V_M,V_{CC}\right)$, we retain that couple in a table. After scanning the intervals of the ramp voltage and the battery terminal voltage with reasonable steps, we obtained a set of torques $\left(V_M,V_{CC}\right)$ corresponding to the boundary between stability and instability in the photovoltaic energy conversion system. This result is especially important because it allows the engineer to determine the value of the ramp to be used in the controller for a desired battery voltage.

3.2. Dynamic Behavior of the PV System with the ACMC Inner Loop Control

The values of the fixed parameters used for this study are shown in Table 1. Figure 7 shows the bifurcation diagram and its corresponding Lyapunov exponent when $V_M$ is taken as a control parameter and $V_{CC}$ is set at 48 V. These traces were obtained by integrating via the fourth-order Runge–Kutta algorithm the differential system (6) with a fixed time step equal to $\Delta \tau ={10}^{-3}$, a total number of iterations of $N=2\times {10}^6$, and a transient phase cut off at $N^{\prime }=1.2\times {10}^6$.

The phenomenon of inverse period doubling $Chaos\to period-4\to period-2\to period-1$ is depicted in Figure 7, which extends practically in the area $0.395 \mathrm{V}\le V_M\le 1 \mathrm{V}$. Figure 7b shows negative values of the maximum Lyapunov exponent corresponding to regular oscillations in the system. This result, representing in the system and its occurrence at a value of $V_M\approx 0.404 \mathrm{V}$ (see Figure 7a), is in agreement with the result developed through the Floquet theory associated with the Filippov method in the previous section.

For $0.01 \mathrm{V}\le V_M\le 0.395 \mathrm{V}$ in Figure 7a, an infinity of local maxima of the duty cycle is found, which are bounded by the maximum value of the duty cycle $d=1$ and its minimum value $d=0.1$. This infinity of points on the bifurcation diagram is confirmed by the positive value of the Lyapunov exponent in this interval (see Figure 7b). In the range $0.01 \mathrm{V}\le V_M\le 0.395 \mathrm{V}$, the system exhibits chaotic behavior. Additionally, the diagrams of Figure 7a present border collision (BC) bifurcation, which appears in the system for $V_M=0.398 \mathrm{V}$. Therefore, for practical applications and good operation of the converter, it is important to choose the gain $V_M\ge 0.404 \mathrm{V}$ in the area where the system operates regularly with a period-1. Out of this interval, the system exhibits subharmonic behaviors that can be detrimental to photovoltaic energy conversion systems [29].

Figure 8 shows the bifurcation diagrams highlighting the value of the duty cycle in an interval of $T$ for the motion when $V_M$ is taken as the control parameter and the voltage at the terminals of the battery $V_{CC}$ is fixed at $V_{CC}=55 \mathrm{V}$ for Figure 8a and $V_{CC}=65 \mathrm{V}$ for Figure 8b. Figure 8 shows an overlap between regular and irregular behavior in the area of subharmonic oscillations. Moreover, the two diagrams in Figure 8 show border collision (BC) bifurcation at the critical values of $V_M=0.468 \mathrm{V}$ for Figure 8a and $V_M=0.574\mathrm{V}$ for Figure 8b. Therefore, we can say that the critical point of release of the border collision bifurcation grows with the value of the battery terminal voltage. At the end on these two diagrams, the boundary between the stability of period-1 and its instability (destruction of period-1) is observed for critical point values of $V_{M\_55}=0.474 \mathrm{V}$ (for Figure 8a) and $V_{M\_65}=0.578 \mathrm{V}$ (for Figure 8b). We can conclude here that the value of the critical point of the stability of period-1 grows with the increase in the battery voltage $V_{CC}$ ($V_{M\_48}<V_{M\_55}<V_{M\_65}$), which confirms the result found in [29]. Figure 9, Figure 10 and Figure 11 show the time evolutions of the boost converter integrated into the photovoltaic power conversion system. The values of the electrical components in Figure 1 are given in Table 1, and the value of the output battery voltage is set to $V_{CC}=48 \mathrm{V}$.

Figure 9, Figure 10 and Figure 11 shows a periodic orbit of period-1 for $V_M=0.6 \mathrm{V}$ (Figure 9), a periodic orbit of period-2 for $V_M=0.4 \mathrm{V}$ (Figure 10), and chaotic oscillations for $V_M=0.2 \mathrm{V}$ (Figure 11). In Figure 9, Figure 10 and Figure 11, the black curve represents the current through the coil, the red curve represents the controller voltage in ACMC mode, and the blue curve represents the ramp signal. Furthermore, we notice that decreasing the ramp amplitude $V_M$ leads to the following dynamic behaviors: for $V_M=0.6 \mathrm{V}$ (Figure 9), the system shows stable oscillations in period-1; for $V_M=0.4\mathrm{V}$ (Figure 10), we witness oscillations in period-2; and for $V_M=0.2 \mathrm{V}$ (Figure 11), we have the chaotic oscillations in the system. This sequence confirms the existence of an inverse period-doubling phenomenon in the average current mode controlled photovoltaic energy conversion system presented by the bifurcation diagram of Figure 7. At the end of these curves, we notice that when the control voltage of the switch S is higher than the ramp signal (corresponding to the switch S being closed), the current through the coil increases. On the other hand, when the inverse phenomenon occurs between the control voltage and the ramp signal (corresponding to the switch S open), the current decreases. This last remark is in good agreement with the results developed in the literature regarding the dynamics of photovoltaic energy conversion systems.

Remark: 

From Figure 9, Figure 10 and Figure 11 we notice that the induction current and the control voltage $v_{con}$ have the same dynamics. This means that for the dynamics of the induction current to be period-1, it is necessary and sufficient that the control voltage also has the dynamics of period-1. Therefore, if we can control the control voltage $v_{con}$ towards a sinusoidal function then the system will be controlled towards period-1 oscillations. This will be the main objective in the rest of this work.

3.3. Effect of Outer Loop Control on the Dynamic Behavior of the PV System

Figure 12 shows the time traces of the quantities $v_{C_1}$, $v_{C_2}$, $i_L$, ($V_{ramp}$ in red), and $v_{con}$ (in blue) when the feedback gain of the external controller is set to $k=2$ as in (25) and the rest of the parameters are the same as those used to plot Figure 11.

From Figure 12, it can be said that the value of the feedback gain for the external controller used to control the system towards periodic oscillations of period-1 is insufficient. On the other hand, for $k=10$, as shown in Figure 13, the dynamic states of the boost converter oscillate periodically of period-1.

Figure 13 demonstrates the effectiveness of a simple external controller with a non-adaptive feedback gain in achieving stable oscillations for period-1 in the photovoltaic power conversion system. However, it is important to note that increasing the feedback gain of the external controller can lead to further destabilization of the system and the emergence of intermittency phenomena, as shown in Figure 14.

Figure 15 shows bifurcation diagram of the photovoltaic system with an internal controller in the average current mode proposed in [31] under the effect of a non-adaptive feedback external controller.

Figure 15 presents the effect of external feedback gain $k$ on the stability of the photovoltaic system. When $0\le k\le 3.58$ and $27.34\le k\le 50$ (region I and III), the controlled system exhibits the subharmonic oscillations (chaotic oscillations in this case). These two regions are not desirous in engineering because the converter will not work optimally and may deteriorate very quickly. However, for $3.58<k<27.34$ (region II), the external controller is shown to be effective in suppressing chaotic behavior. In this region, the photovoltaic system now presents period-1 oscillations. Thus, in this region ($3.58<k<27.34$) we will have an optimal operation of the converter, which is interesting for engineering applications.

Figure 16 and Figure 17 show the FPGA simulation results with the DE2-115 FPGA module of the system (25). On these figures, we notice that when the signal $v_{con}$ becomes chaotic, the inductor current also becomes so. On the other hand, when the signal $v_{con}$ becomes periodic, the inductor current also becomes so. This result has already been presented by Figure 12 and Figure 13, which are the results obtained in MATLAB. We can conclude that the results obtained by FPGA implementation and those obtained in MATLAB are in good agreement.

4. Discussion

In this paper we have examined the dynamics of a photovoltaic energy conversion chain proposed by Hosseini et al. in [31]. The topology of this string consists of a photovoltaic source, a boost converter, and an output load, all controlled by a controller in average current mode. The controller is critical to the stability and accuracy of the power supply and, in practice, designers often use the pulse-width modulation (PWM) technique for regulation. Compared with other current mode control techniques used to control photovoltaic systems in the literature, the method we study here has the advantage of requiring fewer current sensors, which reduces the cost of the electrical installation. Regarding the study of the dynamics, Figure 7 and Figure 8 show the existence of the reverse period-doubling phenomenon, which is different from the period-doubling phenomenon observed in other works in the literature [29]. This difference can be explained by the fact that we use the amplitude of the ramp (modulating signal) as the control parameter, whereas other works use the gain of the corrector contained in the internal controller. The advantage of our technique is that it directly influences the variation of the duty cycle, whereas theirs indirectly influences it. Furthermore, on the curve in Figure 7a we notice that the boundary between period-1 oscillations and subharmonic oscillations is at $V_M=0.4V$. This result is significantly better than the findings of Kengne et al. in [51], where they presented a current mode controller that uses a low-pass filter after the inductor current sensor. They showed that as the filter frequency tends towards infinity (i.e., no filter), the boundary between subharmonic oscillations and period-1 oscillations converges to $V_M=0.5V$. This result shows that the average current mode control presented in this work is better for the stabilization of a photovoltaic converter string than the one developed by Kengne et al. in [51]. We can also mention that, as with most works in the literature, appropriate linearization of the PV generator model does not affect the accuracy of the model in predicting the period-doubling bifurcation of the system. Figure 7b presents the largest Lyapunov exponent, which is indeed the most reliable indicator to attest to the presence of chaos in a system. However, several works in the literature attest to the presence of chaos in their system without presenting the Lyapunov exponent [26,27,36]. This is due to the fact that many of these works are developed only on the basis of Simulink or PSIM software, which makes it difficult to plot the Lyapunov exponent. We were able to do this thanks to the numerical method using the fourth-order Runge–Kutta adapted for power converters developed in [39]. We adapted this method to a photovoltaic energy conversion string by replacing the PV with its linear model. In the numerical method developed in [39], the voltage source at the input of the converter can be replaced by the Norton generator of the PV in parallel with a filter capacitor. Regarding the control, we based ourselves on the work proposed by Hongxiang et al. in [52]. In their work, the authors developed an external controller based on a non-adaptive linear Luenberger observer but their target was the steady state of a constant value, unlike what we proposed in this work, which is a sinusoidal source. The advantage of our non-adaptive control over the one proposed in [50], is that we use less feedback gain to achieve almost the same objective. Thus, our external control loop has the advantages of reducing the cost of the electrical installation and solving the space problem. The diagram in Figure 15 is used to select the value of the non-adaptive gain for controlling chaotic and subharmonic oscillations in the system. This map will be a very important tool for the making of decisions in the field of electrical engineering.

Table 4. Comparison of our results with those in the literature.

Papers	Numerical Method	Analog and Experimental Implementations	Tools to Characterize the Chaotic Motion	Method to Suppression of Chaotic Motion
[26]	-	PSIM and experimentation	Bifurcation diagrams	-
[27]	Discret method	Experimentation	Bifurcation Diagrams	-
[36]	-	PSIM and experimentation	Time series	-
[51]	Fourth-order Runge–Kutta	PSIM	Bifurcation diagrams and their corresponding largest Lyapunov exponents	-
This work	Fourth-order Runge–Kutta	FPGA	Bifurcation diagrams and their corresponding largest Lyapunov exponents	Non-adaptive controller

clearly shows our contribution to the field of renewable energies. The stability analysis of the PV system under the influence of the external control loop can also be performed by analytical methods using Lyapunov functions. Non-adaptive control can be robust. This robustness is possible when the uncertainties and external disturbances are of low amplitude. Unfortunately, if these amplitudes become significant, the control method proposed in this document will not be able to fulfill its function. It will then be necessary to use an adaptive controller. The particularity of the adaptive controller is that its gain adjusts according to disturbances. This analytical approach and the experimental implementation will be the subject of future work. These discussions are summarized in Table 3.

5. Conclusions

This paper evaluated various dynamical behaviors of a boost converter fed by a PV generator and controlled in average current mode. The study of the stability by the Floquet method based on the monodromy matrix showed that the inverse period doubling can be detected accurately. The results of the numerical simulations in Matlab/Simulink software and based on the fourth-order Runge–Kutta algorithm showed the existence of periodic, aperiodic, and chaotic behaviors in the system. Concerning the point of appearance of the inverse period-doubling bifurcation in the system, the results of the numerical simulations in Matlab confirmed the results obtained analytically with the Floquet theory. Analysis of the bifurcation diagrams revealed that both the critical value of ramp amplitude for the occurrence of border collision bifurcation and the critical value of ramp amplitude for the occurrence of period-1 in the proposed system increase with the value of the battery terminal voltage. The positive values of the largest Lyapunov exponent for certain values of the ramp amplitude confirmed the chaotic behavior of the subharmonic oscillations obtained in these regions. This last chaotic behavior, being an undesirable behavior for PV systems, required us to apply an external controller to the system to remove it, which proves the robustness of our controller. Based on the approach used in this paper, the critical value for the control parameter for stability loss can be determined. For practical studies, this paper can then help to select values for which there is undesirable behavior. Since it has been shown in the literature that adaptive controllers are more robust than their non-adaptive counterparts [48,49,50], the application of an adaptive external controller to the PV system studied in this work, as well as the experimental implementation of the results obtained, will be the subject of future work.