*3.3. Bifurcation Behavior from the Nonlinear Circuit-Level Switched Model with the Nonlinear Model of the PV Generator from PSIM Software*

The circuit diagram for the current-mode-controlled boost converter is constructed using the PSIM simulation software. The selection of component parameters is consistent with the numerical simulation in Tables 1 and 2. In this part, nonlinear phenomena such as bifurcations are exhibited, using the nonlinear and linear models of the PV generator, the *VM* bifurcation parameter, the different values of the output voltage (*E* = 60 V and *E* = 48 V) and the different values of the cut-off frequency *fc* = *fs* <sup>2</sup> , *fc* = *fs*, *fc* = 2 *fs* with*fs* = 50 kHz being equal to those used in numerical simulations in MATLAB. Comparing the simulation bifurcation of the nonlinear model of the pv generator in Figure 9 ( PSIM simulation) with that in Figure 6 (MATLAB simulation), it can be seen that the results are in almost perfect agreement.

Note here that the results presented in these two figures are identical, for example, for E = 60 and  *fc* = *fs* 2 , the critical value obtained is the same in both cases and is equal to *VM* = 1.42 V. We want to show through these two graphs that linearizing appropriately the PV generator model does not affect the accuracy of the model in predicting the perioddoubling bifurcation of the system. The aim is to study the linear model of PV generator, which is going to help us to study the stability of our system, as it is very difficult to study stability with the nonlinear model of PV.

**Figure 9.** Bifurcation diagrams obtained 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 linearized model of the PV generator close to the MPP was used.

**Remark 4.** *Figure 9 shows the bifurcation with the nonlinear model of the PV generator substituted by its linearized model constituting a linear resistance V*mpp/*I*mpp *and a current source* 2*I*mpp*. It can be observed that the matching between these bifurcation diagrams and those obtained in Figure 6 is excellent. It is widely believed among the power electronics community that with values of steadystate duty cycles of less than* 0.5*, no external ramp is needed to achieve a stable system. However, from the previous bifurcation diagrams, it is clear that the system is still prone to period-doubling bifurcation even for D* < 0.5 *if the cut-off frequency of the current sensor is relatively low. Therefore, a ramp slope compensation is also necessary for duty cycle values less than* 0.5*.*

#### *3.4. Stability Boundaries in the Parameter Space*

Now that the nonlinear behaviour of the PV system under peak current-mode control is understood and the mechanisms of losing stability are known, the stability boundary for useful practical design will be determined.

The focus of this paper is on the period-doubling bifurcation boundary. The results helps in setting the design parameters as they show the stable region in the parametric space.

One of the ways to locate the subharmonic oscillation boundary is by using the expression of the characteristic equation det(**P** − *λ***I**) = 0, imposing the period-doubling condition in the eigenvalue *λ* and solving the resulting equation together with the steadystate condition. Therefore, at the boundary of this bifurcation, the following conditions hold

$$\det(\mathbf{P} + \mathbf{I}) = 0,\tag{24a}$$

$$\mathbf{x}(DT) - \mathbf{x}((D+1)T) = \mathbf{0},\tag{24b}$$

where **<sup>0</sup>** <sup>∈</sup> <sup>R</sup><sup>4</sup> is a null vector. Note that the integral state variable can be determined by (24b).

Figure 10 presents the stability study in two dimensions of the photovoltaic system obtained in MATLAB/SIMULINK. It clearly shows the regions of stability (period-1 oscillations) and instability (subharmonic oscillations) of the system in the E-plane for three values of the cut-off frequency: blue *fc* = *fs*/2, red *fc* = *fs* and yellow *fc* = 2 *fs*. The other system parameters are taken as in Tables 1 and 2. We see on this map that when 20 V ≤ *E* ≤ 30 V, the system does not show period-doubling phenomenon except for *fc* = *fs*/2. In this case, this map is not sufficient to conclude on the system stability. Moreover, for 30 V ≤ *E* ≤ 70 V, the stability domain of the photovoltaic system increases with the filter cut-off frequency. Finally, for 70 V ≤ *E* ≤ 90 V, the filter cut-off frequency has no influence on the stability of the PV system. These results are more complete than the results obtained in the previous sections, so this map will be a crucial tool in the decision-making process for engineers in industries.

**Figure 10.** Stability boundaries in the plane *E* − *VM* for different values of cut-off frequency in MATLAB/SIMULINK software. *fs* is fixed at *fs* = 50 kHz. Where: A(60v,1.42v), B(60v,1.42v), C(60v,1.25v), D(48v,0.93v), E(48v,0.74v), and F(48v,0.65v) are the period splitting type bifurcation occur in the system. A and D are for fc = fs/2 ; B and E are for fc = fs ; and C and E are for fc = 2fs

Figure 11a shows the critical curves in the plane *D* − *VM* for different values of *fc*. The conventional boundary with *ω<sup>c</sup>* → ∞ is also shown. Figure 11b shows the critical curves in the plane *E* − *VM* for different values of *fc*. The critical points for *E* = 48 and *E* = 60 are indicated in the figure. The critical values of the ramp amplitude for the different values of *E* and *fc* remarkably coincide with the critical values obtained by the bifurcation diagrams in the previous section. For instance, for point A, one has *E* = 60 V and *fc* = *fs*/2 and the critical point of *VM* is 1.42 V agreeing with the bifurcation diagram of Figure 6a. Above the curves, the system is stable and below it, it exhibits period-doubling bifurcation. The critical curve in the plane *<sup>D</sup>* <sup>−</sup> *VM* passes very close to the point (0, <sup>1</sup> <sup>2</sup> ) as long as *fc* is relatively small. This curve becomes concave in this plane when *VM* is increased. By increasing the ramp amplitude *VM*, the region of stability is widened for *D* > <sup>1</sup> <sup>2</sup> and is reduced for *D* < <sup>1</sup> <sup>2</sup> . Furthermore, the stable region becomes wider when increasing the cut-off frequency. These last results obtained with PSIM software are in perfect agreement with those of Figure 10 obtained with MATLAB/SIMULINK software.

**Figure 11.** Stability boundaries for different values of cut-off frequency in PSIM software: (**a**) in the plane *D* − *VM*; (**b**) in the plane *E* − *VM*. Where: A(60v,1.42v), B(60v,1.42v), C(60v,1.25v), D(48v,0.93v), E(48v,0.74v), and F(48v,0.65v) are the period splitting type bifurcation occur in the system. A and D are for fc=fs/2 ; B and E are for fc = fs ; and C and E are for fc = 2fs.

#### **4. Conclusions**

This paper focuses on the nonlinear behaviour and stability of the current-mode-controlled boost converter with the battery load. First, numerical analysis of its state equations, bifurcation diagrams and Lyapunov exponent was conducted in MATLAB/SIMULINK Software using a linear model of PV. Secondly, analogy simulations using PSIM were performed using a nonlinear models of the PV generator. The stability of a boost converter supplied by a PV panel was studied. To make an analytical study possible, the nonlinear PV generator has been linearizing around its MPPT. The simulation results considered three values of the cut-off frequency and two values of the output voltage. They showed through the bifurcation diagrams and the Lyapunov exponent that the system presents nonlinear phenomena such as chaos and periodic motion, which are influenced by system parameters and topological structure. The numerical results obtained in MATLAB/SIMULINK software are remarkable in their agreement with the analogy simulations in PSIM. We can also mention that linearizing appropriately the PV generator model does not affect the accuracy of the model in predicting the period-doubling bifurcation of the system. In general, we have seen that the stability increases with the frequency of the cut which is interesting about this study since the type of behaviour desired for a PV is the period-1.

**Author Contributions:** Conceptualization, E.R.M.K., A.S.T.K.; Data curation, M.S.S., T.T.T., A.R.M., M.T., Z.I.K.; Formal analysis, E.R.M.K., A.S.T.K., M.S.S., T.T.T., A.T.A., A.R.M., M.T., Z.I.K.; Investigation, T.T.T., A.T.A., A.R.M., M.T. Methodology, E.R.M.K., A.S.T.K., M.S.S., T.T.T., A.T.A., A.R.M., M.T., Z.I.K.; Resources, E.R.M.K., A.S.T.K., M.S.S., T.T.T., A.T.A., A.R.M., M.T., Z.I.K.; Software, A.S.T.K., M.S.S., T.T.T., A.R.M., M.T., Z.I.K.; Supervision, E.R.M.K.; Validation, M.S.S., A.T.A., A.R.M.; Visualization, A.T.A. and Z.I.K.; Writing – original draft, E.R.M.K., A.S.T.K.; Writing – review & editing, E.R.M.K., A.S.T.K., M.S.S., T.T.T., A.T.A., A.R.M., M.T., Z.I.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Prince Sultan University, Riyadh, Saudi Arabia.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank Prince Sultan University, Riyadh, Saudi Arabia for funding the Article Processing Charges (APCs) of this publication. Special acknowledgments are given to Automated Systems & Soft Computing Lab (ASSCL), Prince Sultan University, Riyadh, Saudi Arabia.

**Conflicts of Interest:** The authors declare no conflict of interest.
