3.1.2. Simulations Results in PSIM Software

Figure 7 shows the results of the stability study of our system in the PSIM software. The PSIM software is a power electronics simulation software. We notice a similarity between the results obtained with PSIM and those obtained with MATLAB/SIMULINK. In the following, we will use the bifurcation diagrams for a more detailed study.

**Figure 5.** MATLAB simulation of evolution of the Floquet multipliers of the PV system by taking the amplitude of the carrier signal amplitude *VM* as a bifurcation parameter for different values of the current sensor bandwidth *ω<sup>c</sup>* and DC output voltage *E*. The critical values of *VM* at which period doubling bifurcation takes place are indicated. (**a**) *fc* = *fs* <sup>2</sup> , *E* = 60 V; (**b**) *fc* = *fs*, *E* = 60 V; (**c**) *fc* = <sup>2</sup> *fs*, *<sup>E</sup>* = 60 V; (**d**) *fc* = *fs* <sup>2</sup> , *E* = 48 V; (**e**) *fc* = *fs*, *E* = 48 V; (**f**) *fc* = 2 *fs*, *E* = 48 V.

**Figure 6.** Bifurcation diagrams in MATLAB software by taking the amplitude of the carrier signal amplitude *VM* as a bifurcation parameter for different values of the current sensor bandwidth *ω<sup>c</sup>* and DC output voltage *<sup>E</sup>*. (**a**) *fc* = *fs* <sup>2</sup> , *<sup>E</sup>* <sup>=</sup> 60 V. (**b**) *fc* <sup>=</sup> *fs*, *<sup>E</sup>* <sup>=</sup> 60 V. (**c**) *fc* <sup>=</sup> <sup>2</sup> *fs*, *<sup>E</sup>* <sup>=</sup> 60 V. (**d**) *fc* <sup>=</sup> *fs* 2 , *E* = 48 V. (**e**) *fc* = *fs*, *E* = 48 V. (**f**) *fc* = 2 *fs*, *E* = 48 V.

**Figure 7.** PSIM software simulation of Evolution of the Floquet multipliers of the PV system by taking the amplitude of the carrier signal amplitude *VM* as a bifurcation parameter for different values of the current sensor bandwidth *ω<sup>c</sup>* and DC output voltage *E*. The critical values of *VM* at which period doubling bifurcation takes place are indicated. (**a**) *fc* = *fs* <sup>2</sup> , *E* = 60 V. (**b**) *fc* = *fs*, *E* = 60 V. (**c**) *fc* = <sup>2</sup> *fs*, *<sup>E</sup>* = 60 V. (**d**) *fc* = *fs* <sup>2</sup> , *E* = 48 V. (**e**) *fc* = *fs*, *E* = 48 V. (**f**) *fc* = 2 *fs*, *E* = 48 V.

*3.2. Bifurcation Behavior from the Nonlinear Circuit-Level Switched Model with the Linear Model of the PV Generator from MATLAB/SIMULINK Software*

Bifurcation diagrams constitute appropriate means of recapitulating different transitions to chaos in the system in terms of different parameter values. The maximum Lyapunov exponent is complementary with bifurcation diagram, it is the tool that allows us to conclude if the system is chaotic or not and there are computed following the wellknown Wolf algorithm [37]. In this section, the numerically dynamics behavior of our system is performed in MATLAB/SIMULINK software by integrating system (3) using the most frequent fourth-order Runge–Kutta scheme [38] adapted to DC-DC converters which offers a better accuracy for solving single-step differential equations unlike the discretization method developed in the literature. First, the system has been carefully studied through simulations using the linear model of the PV generator in MATLAB software. We used a fixed time step equal to *<sup>h</sup>* = <sup>5</sup> × <sup>10</sup><sup>−</sup>8, a total number of iterations *<sup>N</sup>* = <sup>10</sup><sup>6</sup> and a transient phase-cut to *<sup>N</sup>* = <sup>8</sup> × 105 The bifurcation diagram and graphs of maximal Lyapunov exponents presented below are obtained for three values of the cut-off frequency, i.e., *fc* = *fs* <sup>2</sup> , *fc* = *fs*, *fc* = 2 *fs*with*fs* = 50 kHz , and for two values of DC output voltage (*E* = 60 *V* and *E* = 48 *V*), we use ramp amplitude *VM* as a bifurcation parameter. The local maximas of the duty cycle and the local maximas of the output power of the generator made it possible to plot bifurcation diagrams. The dynamics of the MPPT algorithm is neglected because it is usually much slower than the converter dynamics. The values of the parameters used for this study are shown in Tables 1 and 2.



**Table 2.** Parameter of the PV generator (BP 585 module).


Figure 6 shows the bifurcation diagrams when the *VM* parameter varies for different values of cut-off frequency and output voltage. For *E* = 60 V and  *fc* = *fs* <sup>2</sup> , *fc* = *fs*, *fc* = 2 *fs* , we can observe in Figure 6a–c inverse period-doubling phenomena from chaos to period-1 chaos → *period* − 8 → *period* − 4 → *period* − 2 → *period* − 1 and when the ordinate axis is a representation of the local maxima of the duty cycle. For a representation of the local maxima of the power output of pv on the ordinate axis of the same figure Figure 3 (*a*1, *b*1, *c*1) show that we start directly from chaos in period-1, we also remark that the system stabilizes at the maximum power point, which shows that our MPPT controller used is good. In the same figure, we can see that when the cut-off frequency increases, i.e., *fc* = *fs* 2 < (*fc* = *fs*) < (*fc* = 2 *fs*) the critical value *VM* represents the boundary between the periodic oscillations of period-1 and the subharmonic oscillations decrease. In other words, when the cut-off frequency increases, the system tends to lose its subharmonic behaviour. Thus, the cut-off frequency increases with the stability range, which is interesting for this study since the desired behavior for a pv energy conversion chain is the periodic behavior.

For *E* = 48 V and *fc* ∈ *fs* <sup>2</sup> , *fs*, 2 *fs* , Figure 3 (*d*1, *e*1, *f*1), we can also observe inverse period-doubling phenomena from chaos to period 1 and a faster loss of chaos. We can, therefore, conclude in this part that it is desirable to choose *E* = 48 V as the output voltage of the battery instead of *E* = 60 *V* because it is at this value and for different values of the cut-off frequency that the system tends to quickly lose subharmonic behaviors since the desire behavior for this type of application is the periodic behavior.

Figure 8 shows the Lyapunov exponent graphs complementary with the bifurcation diagrams in Figure 6. These graphs are obtained for three values of the cut-off frequency and for two values of the output voltage and in the *VM* zone between 0 ≤ *VM* ≤ 1, in this area, we observe in Figure 8 negative values of the maximum Lyapunov exponent which correspond to regular oscillations in the system and positive values corresponding to subharmonic oscillations. We can also remark that the cut-off frequency increases with stability, which justifies the bifurcation diagrams of Figure 8.

**Figure 8.** Lyapunov exponent diagrams in MATLAB/SIMULINK software by taking the amplitude of the carrier signal amplitude *VM* as a bifurcation parameter for different values of the current sensor bandwidth *<sup>ω</sup><sup>c</sup>* and DC output voltage *<sup>E</sup>*. (**a**) *fc* = *fs* <sup>2</sup> , *E* = 60 V. (**b**) *fc* = *fs*, *E* = 60 V. (**c**) *fc* = 2 *fs*, *<sup>E</sup>* = 60 V. (**d**) *fc* = *fs* <sup>2</sup> , *E* = 48 V. (**e**) *fc* = *fs*, *E* = 48 V. (**f**) *fc* = 2 *fs*, *E* = 48 V.

The bifurcation diagrams Figure 6 and Lyapunov exponent in Figure 8 show an overlap between regular and irregular behaviors in the subharmonic oscillation zone. In addition, the bifurcation diagrams in Figure 6 show border collision [39–41] respectively to the critical values *VM* = 1.42 V, *VM* = 1.31 V, *VM* = 1.25 V (see Figure 6a–c) and *VM* = 0.93 V, *VM* = 0.74 V, *VM* = 0.65 V (see Figure 6d,e,f).

**Remark 3.** *We note that for E* = 60 *V, the critical values VM, Figure 6a–c, are superior to those of E* = 48 *V Figure 6d–f; this means that the range of stability is greater when E* = 48 *V compared with E* = 60 *V. Therefore, it is desirable to choose E =* 48 *V as the battery output voltage instead of E =* 60 *V because it is at this value and for different values of the cut-off frequency that the system tends to quickly lose the subharmonic behaviours and increases the stability range.*
