**1. Introduction**

In the difficult energy and economic context, expectations in terms of renewable energies in general and solar Photovoltaic (PV) energy in particular is increasing [1–3]. Reducing the costs of PV systems, improving their performance and increasing their efficiency are major concerns for researchers, in order to make them as competitive as possible [4]. PV generators an interesting renewable energy source because it is not only renewable but also inexhaustible and non-polluting. The ability to achieve the maximum energy output is crucial for the optimization of generation system [5]. The output power of a PV generator varies with weather conditions. A Maximum Power Point Tracking (MPPT) controller is needed to force the PV system to operate at its optimal operating point [6–10].

**Citation:** Kengne, E.R.M.; Kammogne, A.S.T.; Siewe, M.S.; Tamo, T.T.; Azar, A.T.; Mahlous, A.R.; Tounsi, M.; Khan, Z.I. Bifurcation Analysis of a Photovoltaic Power Source Interfacing a Current-Mode-Controlled Boost Converter with Limited Current Sensor Bandwidth for Maximum Power Point Tracking. *Sustainability* **2023**, *15*, 6097. https://doi.org/ 10.3390/su15076097

Academic Editors: Prince Winston David and Praveen Kumar B

Received: 1 March 2023 Revised: 25 March 2023 Accepted: 28 March 2023 Published: 31 March 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In general, a PV system consists of a PV generator, a DC-DC converter and a control system that regulates certain electrical variables in order to extract the maximum energy from the PV generator and transfer it to the load. The boost converter is the most used in small PV systems as it converts the relatively low voltage of the PV panels and raises it to a higher level, appropriate for the load [11]. To achieve MPPT, direct duty cycle control [12], voltage mode control [13] and two-loop current-mode control [14–17] have been used. In direct duty cycle control, the MPPT algorithm directly dictates the desired duty cycle for MPPT (Figure 1a). The advantage of this approach is the simplicity of the scheme. However, the performances of this strategy are very poor and severe oscillatory behavior may be produced after any step change due to the MPPT P&O algorithm. Furthermore, none of the previous structures provide an over-current protection, making impossible the paralleling of converters in a PV system. To overcome this problem, voltage regulation can be used as shown in (Figure 1b). The oscillatory behavior due to MPPT step changes may be damped. However, the settling time could still be large. Under current-mode control, the PV voltage regulation is conventionally carried out by means of cascaded feedback loops, where the inner loop controls the inductor current and the outer loop regulates the PV voltage (Figure 1c). With this control scheme, two current sensors are used—one for the PV current and one for the inductor current [15] or the input capacitor current [13]. In [18], the inductor current used for the current loop control is also used for estimating the PV power and performing MPPT (Figure 1d).

**Figure 1.** Different MPPT control strategies. (**a**) direct duty cycle control, (**b**) single-loop voltage mode control, (**c**) two-loop current mode control with voltage loop closed and (**d**) current mode control with voltage loop open with a single current sensor for both current and MPPT controls.

In single-loop voltage mode control and two-loop current-mode control, voltage regulation has been always used to attain predetermined closed-loop performance in terms of the system settling time due to changes in the weather conditions and/or MPPT parameter step changes [14,19,20]. In both control schemes, PI controllers have been used with the aim to make steady state error zero. This dynamic controller may slow down the system response. In reality, the main objective in a PV system is to track the maximum power and not to regulate the voltage. Thus, the two-loop current-mode control strategy is not necessary. In this paper, instead of using single loop voltage control and two-loop involving both voltage and current control, the single-loop current control is used. It offers many advantages in a PV system such as fast response among others [21]. Furthermore, peak current-mode control is used without the need for an integrator which may slow down the system response. However, a noticeable ripple (up to 30%) is present in the inductor current of a switching DC-DC converter. Since we only need the DC component of the current to perform MPPT, a low-pass filter *Hi*(*s*) is necessary for the current loop. The technique prevents from sensing the PV generator current. A shunt sensor is placed at the input of the power converter and used for both current-mode control and PV power estimation. It is worth noting that this low-pass filter is naturally existing in some current sensors with limited cut-off frequency. These results are simple but efficient and feature a fast-tracking capability. The proposed technique will be validated with numerical simulations, showing that the transient duration under irradiance variations or step changes due to the P&O MPPT controller is greatly reduced as compared to other existing techniques, thereby realizing the fast current response and MPPT under current mode control.

Figure 2 shows a boost converter under a single-loop current-mode control for MPPT with two sensors and with a single sensor with pre-filtering. The scheme of Figure 2b depicts the proposed solution for the control of the DC-DC converter which is based on pure peak current-mode control without an outer voltage loop. This control method provides an efficient cycle by cycle over current protection. The concept can be extended to any topology of DC-DC converter and another advantage of peak current-mode control in PV systems is that the transfer function to be compensated is non-minimum phase for all the converter topologies.

**Figure 2.** Boost converter fed by a PV generator with MPPT and current mode controller. (**a**) the MPPT control is performed by using the PV current. (**b**) The MPPT control is performed by using the filtered inductor current.

It is well known that DC-DC converters under current model control may exhibit a rich variety of nonlinear phenomena [22]. In particular, when filtering is added to the current loop, the conventional results are widely known in the power electronics community [23] to become inaccurate in predicting the onset of period-doubling bifurcation [11]. Thus, it is necessary to use an appropriate model that allows to predict mathematically the onset of this bifurcation. Such a model would also offer useful physical insights into the

behavior of the system without the need for excessive numerical simulations. In this paper, we propose to use the single-loop control scheme for DC-DC converters when used in a PV system. We suggest to use the inductor current both for controlling the converter to a reference provided by a P&O MPPT algorithm as well as to estimate the average value of the PV source power which is used by the same algorithm. It is shown that period-doubling bifurcation arises from the instability of the inner current loop of the DC-DC converter and is not significantly related to the nonlinearity of the PV generator. Therefore, linearizing appropriately the PV generator model does not affect the accuracy of the model in predicting the period-doubling bifurcation of the system. This is especially beneficial for modeling PV systems and analytically predicting their period-doubling bifurcation behavior. Circuit-level simulations from a switched model verify the theoretical findings. As an example, we present a study of a PV-fed boost converter used in microinverter applications.

The scope of this work falls into the DC-DC converter technology. We recall that the current-mode control is a predominant strategy in controlling DC-DC switched-mode power electronic converters for different applications. This is due to many advantages such as fast system response, better system performances and inherent over current protection, the easy parallel operation but it is rarely used in PV applications. A PV system under study consists of a PV generator interlinked to a DC-DC boost converter which is subject to various nonlinear phenomena under the current-mode control. Facing the challenges previously established, the main contribution of this paper are as follows:


The rest of the paper is organized as follows: Section 2 presents materials and methods. Section 3 presents Results and Discussions and finally, concluding remarks are presented in Section 4.
