## 2 Modified Topology and Control Scheme  
The conventional multilevel cascaded H-Bridge (CHB-MLI) uses an isolated DC source for every H-bridge. The number of voltage levels generated can be generalized to \((2k+1)\), where \(k\) is the number of H-bridges. An inverter with two H-bridges uses two DC sources of equal magnitude for a five-level output voltage generation. However, it requires more DC sources for each H-bridge to generate higher voltage levels, which increases the cost of the system. A capacitor can be used instead of a DC voltage source. Furthermore, the same configuration with one DC source and one capacitor at their respective H-bridges can generate five, seven, or nine voltage levels, respectively. A trade-off has to be made as increasing the voltage levels decreases the magnitude of the maximum voltage generated. The use of a capacitor raises the issue of charging and discharging so that the capacitor voltage can be balanced at the desired value. The balancing of the capacitor can be attained by cascading another control loop in addition to a grid current control loop using conventional PI-based control schemes [15, 16, 17]. However, these control loops make the overall system complex. This issue of intricacy is further augmented when additional control schemes are introduced to reduce the leakage current in grid-tied PV converters. This can be smoothly achieved with the proposed proper control scheme. The modified topology, a hybrid-MLI (HMLI), effectively solves the control challenges created by the incorporation of a capacitor in one of the H-bridges. The modified model predictive control (MPC) implemented in this work uses multivariate, multi-input, and multi-output optimisation problems. The proposed modified circuit is depicted in Figure 1. A block diagram for the overall control scheme is shown in Figure 2. The control loop takes the capacitor voltage (\(V_{\text{cap}}\)), grid voltage (\(V_{\text{g}}\)), current (\(I_{\text{g}}\)), reference current (\(I_{\text{ref}}\)), grid angle (\(\theta\)), and weighting factor. The grid angle is generated through a phase-locked loop (PLL). All these control loops use modified MPC to generate the required switching signals fed to the HMLI. The multilevel output generation property of the MLI is fully used in the proposed modified HMLI. The voltage across capacitors is maintained at a third of the input voltage (\(V_{DC}\)), which results in the maximum of different voltage levels possible obtained across two H-bridges (i.e., nine). The 16 combinations of voltage vectors are given in Table 1 to generate these nine levels. However, only nine separate vectors are required out of only 16 voltage vectors to generate the required voltage level. The modified MPC only considers these nine states, thus reducing the computation time significantly. The number of distributed states is also further minimized using optimization, which will be discussedlater in this paper. In addition, two extra switches, \(S_{5}\) and \(S_{6}\), are incorporated to minimize the leakage current problem, as shown in Figure 1.  
Figure 1: Transformer less nine-level PV-supported HMLI. Figure 2: Schematic of the overall system. The parasitic capacitance is formed between the PV module and the ground. The leakage current is generated by a change in the CMV across it. The CMV is the average value of voltage between the output and the reference point. The negative terminal of the DC voltage, (i.e., terminal \(N\)) is called the common reference point for the upper H-bridge. Similarly, for the lower H-bridge inverter, \(N^{\prime}\) is the common reference point. The parasitic capacitance is formed for the upper H-bridge. The CMV and leakage current in the two H-bridges are also the same. The CMV voltage (\(V_{\text{CM}}\)) for the upper full-bridge can be expressed in (1):  
\[V_{\text{CM}}=\frac{V_{\text{AN}}+V_{\text{BN}}}{2} \tag{1}\]  
where \(V_{\text{AN}}\) and \(V_{\text{BN}}\) are voltages between the mid-point of the upper H-bridge legs to the negative terminal of the DC link.