*2.2. Mathematical Modeling*

Considering the system shown in Figure 2, applying Kirchhoff's laws, we obtain the following system of equations:

$$\frac{dv\_{pv}}{dt} \quad = \begin{array}{c} i\_{pv} \\ \hline \mathcal{C}\_1 - \frac{i\_L}{\mathcal{C}\_1} \end{array} \tag{3}$$

$$\frac{di\_L}{dt} \quad = \frac{v\_{pv}}{L} - \frac{r}{L}i\_L - \frac{v\_2}{L}(1-u)\_\prime \tag{4}$$

$$\frac{dv\_2}{dt} = \frac{E - v\_2}{RC\_2} + \frac{i\_L}{C\_2}(1 - u),\tag{5}$$

where *vpv* and *ipv* represent the voltage and current of the PV source, respectively. These two variables are related with a highly nonlinear and implicit Equation (3). *iL* is the current of the inductor, and *u* is the state of the switch. *C*<sup>1</sup> 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, *VCC* the output voltage of the battery. *r* is the equivalent resistance in series with the coil and the equivalent resistance that can be felt at the output of the converter is *R*.

**Figure 3.** Flowchart of the proposed technique.

A PV generator represents a fundamental power source of a PV system. The output current/voltage characteristics depend on the solar irradiance and temperature. The PV generator has nonlinear electrical model with a single maximum power point (MPP). The performance of a PV generator is evaluated under standard test conditions, where the irradiance is normalized at 1 kW/m<sup>2</sup> and the temperature is defined at 25 ◦C. A PV generator is characterized by its current–voltage characteristic (I–V) which can be subdivided into three operating zones, a linear zone with a practically constant current, a concave zone

with almost constant voltage and an MPP which is the desired point for operation. Since this is the optimal point, the nonlinear characteristic (I–V) is linearized close to the MPP using Taylor's series expansion and ignoring high-order terms. The (I–V) equation of the PV model can be approximated by the following linear Norton equivalent model [11]:

$$
\hat{i}\_{pv} \approx 2\dot{i}\_{mpp} - \frac{I\_{\rm mpp}}{V\_{\rm mpp}} \vartheta\_{pv} + \mathcal{0}(\vartheta\_{pv}^2). \tag{6}
$$

Although numerical simulations could be performed using the nonlinear PV model, the linear model is useful to perform steady-state analysis, for controller design, stability analysis and for prediction of bifurcations. The state-space model of the power stage (3)–(5) together with (2) describing the current sensor and the switching logic determining the value of the binary control signal *u* appearing in (4) and (5) represent the closed-loop model of the PV system. While the nonlinear model of the PV generator can only be used for performing numerical simulations, the switched model with linearized PV generator model can be used for mathematically predicting the onset of period-doubling bifurcation in the system.

#### *2.3. Steady-State Analysis*

Obtaining the steady-state duty cycle *D* requires performing steady-state analysis. Under MPPT conditions and in steady-state operation, the following equalities hold

$$V\_2(1 - D) = V\_{\rm mpp} - rI\_{\rm mpp} \qquad V\_2 = E + RI\_{\rm mpp}(1 - D). \tag{7}$$

Solving both equations for the steady-state duty cycle *D*, one obtains

$$D = 1 - \frac{1}{2RI\_{\rm mpp}} \left( \sqrt{4RI\_{\rm mpp}(V\_{\rm mpp} - rI\_{\rm mpp}) + E^2} - E \right). \tag{8}$$

Note that when the parasitic resistance *r* and *R* are negligible, one has which is the well-known expression of the duty cycle of an ideal boost converter with input *V*mpp and output *E*. Since, under MPPT control and steady-state operation, the PV current will be equal to *I*mpp and since the average capacitor voltage at the input port of the converter is zero, the inductor current average value in steady state will be also equal to *I*mpp. For this reason, the reference current under steady-state operation must be given by the following expression

$$I\_{\rm ref} = I\_{\rm mpp} + (\frac{m\_a}{R\_\text{s}} + \frac{m\_1}{2})DT - \frac{1}{R\_\text{s}} \frac{m\_1(1 - e^{-DT\omega\_\text{c}})}{\omega\_\text{c}} \tag{9}$$

where

$$m\_1 = \frac{V\_{\rm mpp} - rI\_{\rm mpp}}{L}, \quad m\_a = \frac{V\_M}{T}. \tag{10}$$

Note that if *ma* = 0 (no ramp compensation) and *ω<sup>c</sup>* → ∞ (ideal current sensor), the previous expression becomes

$$i\_{\rm ref} = I\_{\rm mpp} + \frac{m\_1}{2}DT = I\_{\rm mpp} + \Delta i\_{L\prime} \tag{11}$$

where Δ*iL* = *m*1*DT*/2 is the ripple amplitude of the inductor current. Since *i*ref is the peak value under the previous conditions, this guarantees that the average value of the inductor current will coincide with *I*mpp in steady state and will also ensure the PV current in the sense of the average steady-state input capacitor current is zero.

Figure 4 shows the evolution of the reference current *iref* in terms of the ramp amplitude *VM* and for two values of the output voltage E.

**Figure 4.** The evolution of the current reference *iref* in terms of the ramp amplitude *VM* according to (9). (**a**) *E* = 60 V. (**b**) *E* = 48 V.

We notice from the above tables, that, whatever the value of *ωc*, the reference current increases with the amplitude of the ramp VM. Moreover, as *ω<sup>c</sup>* increases, the curves *iref* in terms of *VM* tend to be confused.

It is good to note that to maintain the same average inductor current and to make it equal to the MPP current, the current reference in the peak of the current loop control has been adapted according to (9).

#### *2.4. Floquet Theory*
