## 2 Conventional Finite Control Set-Model Predictive Control for Two-Level Voltage Source Inverter
### Conventional Finite Control Set-Model Predictive Control (FCS-MPC)  
Finite control set-model predictive control (FCS-MPC) is a widely used control method for power electronic systems. It can be used to optimize the inverter switching states in real-time to control the load current. In this method, a predictive model is employed to forecast the load current during the subsequent sampling period, (k + 1). The three-phase two-level voltage source inverters comprise seven distinct future current vectors, \(\mathbf{i}(k+1)\). The cost function is used to determine the vector used to minimize the difference from the reference value, \(\mathbf{i}_{ref}(k+1)\). This control operates by applying a switching state that generates the required voltage vector to the inverter, which then tracks the corresponding load current during the next sampling period. Figure 3 depicts a block diagram of the conventional finite control set-model predictive control. The complete control process is given as follows:  
1. Current state measurement: The current load currents, \(\mathbf{i}_{ao}(k)\), \(\mathbf{i}_{bo}(k)\), and \(\mathbf{i}_{co}(k)\), are measured using sensors. 2. Future state prediction: The load currents, \(\mathbf{i}_{a}(k+1)\) and \(\mathbf{i}_{\beta}(k+1)\), are predicted for all the possible switching states using the predictive model. 3. Cost function evaluation: The \(\mathbf{i}_{a}(k+1)\) and \(\mathbf{i}_{\beta}(k+1)\) values that minimize the difference from the reference currents, \(\mathbf{i}_{a,ref}(k+1)\) and \(\mathbf{i}_{\beta,ref}(k+1)\), are determined using the following cost function: \[J=\sqrt{\left(\mathbf{i}_{a}(k+1)T_{s}-\mathbf{i}_{a.ref}(k+1)T_{s}\right)^{2}+\left( \mathbf{i}_{\beta}(k+1)T_{s}-\mathbf{i}_{a.ref}(k+1)T_{s}\right)^{2}}\] (7)
4. Switching state application: The optimal voltage vector required to implement the current that minimizes the cost function, \(J\), is determined, and the required switching state is applied. In summary, the FCS-MPC determines the most appropriate voltage vector among seven options by performing cost function calculations in each cycle. Figure 3: Block diagram of conventional model predictive control (MPC). This presents relatively fast response times, high control performance, and high inverter efficiency. Additionally, it presents a simple, intuitive structure that can be easily implemented. However, this study faces several limitations. Firstly, the irregular switching frequency increases the THD, thereby deteriorating the voltage and current waveform quality and reducing the system performance. Secondly, the limited voltage vectors (seven in total) make it difficult to fine-tune the output voltage, thereby increasing the current error and reducing the control accuracy. Lastly, the high computational load complicates real-time implementation, requiring high-performance hardware such as digital signal processors (DSPs) or FPGAs. [PERSON] and [PERSON] [35] proposed a method using multi-virtual-voltage vectors (MVVVs) and pre-selection. They added MVVVs to the seven voltage vectors of a two-level inverter to achieve a performance similar to that of multi-level inverters. The increased number of voltage vectors enabled more precise switching control, thereby improving the output performance. Their results demonstrated a reduction of approximately 50% in the output current THD. They applied virtual voltage vector pre-selection to mitigate the increased computational load due to the additional voltage vectors, thereby reducing the calculations required and enhancing the feasibility of real-time implementation. However, the increased number of vectors causes more frequent switching, thereby causing additional switching losses. This can reduce the inverter efficiency and increase the heating of the semiconductor device. For small- to medium-sized vessels (approximately 100 kW-400 kW), the inverter system losses and heat generation directly affect both the safety and cooling system dimensions, making them critical control factors.