## 4 Operation of the Proposed MMPC
### Converter Design Analysis  
The selection of the voltages for the battery and PV modules which interface to the PM is one of the most important design parameters for the proposed MMPC as it determines the imbalance range that can be handled between the PM's batteries. During battery charging, if for a given PM the PV boost is operating at its MPP, then the minimum achievable output voltage for this PM is V\({}_{\text{pv-max}}\) which is PM's PV-module voltage at MPP.  
During battery charge/discharge (battery buck/boost), the maximum achievable output voltage of the PM is the voltage of its battery (V\({}_{\text{batt}}\)). As a result, the maximum imbalance in the SoCs of the PMs' batteries that can be handled can be derived based on Equations (4) and (6) under these conditions as:  
\[\frac{\text{SOC}_{\text{max}}}{\text{SOC}_{\text{min}}}=\frac{\text{V}_{ \text{batt}}}{\text{V}_{\text{pv-max}}} \tag{14}\]  
Figure 8: Block diagram of the proposed MMPC.  
As can be observed from Equation (14), the wider the range between battery voltage and PV-module voltage at MPP, the wider the imbalance among the PMs' batteries that can be handled.  
Determining the number of PMs is also one of the key design parameters of the proposed MMPC, as it defines the maximum utilization of the battery modules as well as the PV modules' available power. For a defined output voltage level for the proposed MMPC, increasing the number of PMs results in reducing the required terminal voltage of a single battery module which in turn reduces the number of series-connected battery cells per module and consequently better utilization of available battery capacity.  
The power extracted from a single PV-string is limited by the weakest PV-module in the string, this weakness being due to aging/degradation or shadowing. Segmenting the single PV-string into several PV-modules interfacing the different ports of the proposed converter will improve the utilization of the available power as it allows independent MPPT for each PV-module limiting the effect of weak PV-modules on the overall performance of the PV-string. This also gives flexibility in configuring PV-modules with different angles/orientations which enhances the utilization of the space available for mounting PV-modules.  
Increasing the number of PMs also decreases the voltage stress of the switches which enables the use of low voltage rating devices with smaller parasitic resistances and capacitances which may reduce the conduction and switching losses of the switches and boosts the system efficiency.  
Increasing the number of modules increases the fault tolerance of the MMPC as it decreases the impact of the faulty module on the overall system operation.  
On the other hand, increasing the number of PMs results in increased converter complexity due to the increased number of local control loops which require dedicated measurements and their own PWM. Therefore, the selection of the number of modules for a particular application needs to be the subject of a tradeoff study where the impact on system efficiency, complexity and cost is evaluated against improving the maximum utilization of battery capacity and available PV power and the fault tolerance of the system.  
Since the proposed converter is aimed to operate with very high switching frequency in order to achieve high power density, current ripple of the PV-stage inductor (L2) of the PMs that may affect the efficiency of the MPPT for the PV modules will be in very high frequency and hence a proper decoupling capacitor can be implemented at the PV-stage input.