## 2 Differential Boost AC Module Under Three-Loop Control  
The system studied in this paper is a grid-tied differential boost inverter supplied by a PV generator for AC module applications. The circuit schematic diagram of its power stage and the block diagram of its controller are shown in Figure 1. The system is under a differential CMC strategy to decide the state of its four switches. Specifically, its strategy is based on three control loops: a first loop which oversees regulating the DC PV voltage to the value provided by the MPPT controller, a second loop synchronizes the grid current to the grid voltage and a third differential peak CMC loop for limiting the peak value of the difference between its inductor currents. Specifically, a comparator is used to compare the signal \(R_{s}(i_{1}-i_{2})+v_{r}\) with the signal \(R_{s}i_{\text{ref}}\), where \(v_{r}\) is an artificial ramp signal with slope \(m_{r}=V_{M}/T\), \(V_{M}\) being its amplitude and \(T\) its period. The output of the comparator is connected to the reset pin of a flip-flop and a periodic clock signal is connected to its set input. The Q and \(\overline{\text{Q}}\) outputs of the flip-flop are the driving pulses for the switches \(\text{Q}_{1}\text{-Q}_{4}\) as shown in Figure 1. A type-III controller was demonstrated to be effective for controlling the grid current in the system [10]. Its transfer function in the Laplace \(s\)--domain can be expressed as follows [28]:  
\[H_{c}(s)=\kappa_{\text{pc}}\frac{\omega_{z}}{s}\frac{(s/\omega_{z}+1)^{2}}{(s/ \omega_{p}+1)^{2}}. \tag{1}\]  
where \(\kappa_{\text{pc}}\) is a suitable proportional gain of the controller, \(\omega_{z}\) is its double zero and \(\omega_{p}\) is its double pole. The placement of the poles and zeros can be performed according to the rules explained in [28]. The signal \(R_{s}i_{\text{ref}}\) for the current difference \(R_{s}(i_{1}-i_{2})\) is the output of the type-III controller which is responsible for making the grid current \(i_{g}\) to track its reference \(R_{sg}i_{\text{gref}}\). For active power control, this reference signal is made proportional to the grid voltage \(v_{g}=V_{g}\sqrt{2}\sin(\omega_{g}t)\), \(\omega_{g}=2\pi f_{g}\). This is achieved by means of a Phase-Locked Loop (PLL) which generates a sinusoidal signal \(\sin(\omega_{g}t)\) in phase with the grid voltage. The signal obtained by multiplying the output of the PLL and the output of the PV voltage controller gives the time varying reference \(R_{sg}i_{\text{gref}}\) that can be expressed as follows  
\[R_{sg}i_{\text{gref}}=R_{sg}I_{\text{gref}}\sin(\omega_{g}t) \tag{2}\]  
The PV voltage \(v_{\text{dc}}\) is regulated by a simple PI controller that aims to make it equal on average to the reference voltage \(V_{\text{mpp}}\). This reference voltage is decided by an MPPT controller. To reduce the harmonic content in the feedback loop, a low-pass filter is also placed after the PV voltage sensor. Therefore, the controller processes the error voltage \(v_{f}-V_{\text{mpp}}\), where \(v_{f}\) is the filtered PV voltage. The cut-off frequency \(\omega_{c}\) of the filter is  
consists of a string of four series-attached modules. The parameter values of one module are specified in [30] and outlined in Table 3.  
The focus here is on fast-scale system stability as affected by time varying sinusoidal voltage reference under different weather conditions. For that, only steady-state operation of the closed-loop system will be shown. A fixed value of temperature (\(\Theta=25\)\({}^{\circ}\)C) was used while changing the irradiance. The output of the PV-fed differential boost AC module is tied to a 230 V AC, 50 Hz grid voltage performing active power control as described previously. The MPPT is performed using a conventional Perturb and Observe (P&O) algorithm providing the voltage reference \(V_{\text{mpp}}\) for the average value of the DC PV voltage \(v_{\text{dc}}\).