Enhanced Output Performance of Two-Level Voltage Source Inverters Using Simplified Model Predictive Control with Multi-Virtual-Voltage Vectors

Abstract

Interest in electric propulsion ships has garnered attention to reduce ship exhaust emissions. This has sparked extensive research on inverters. While two-level voltage source inverters are commonly utilized in small- and medium-sized ships owing to their simple structure and cost-effectiveness, they have limitations, such as high switching losses and reduced output performance. To address these issues, a model predictive control technique based on virtual voltage vectors is proposed in this study. Conventional two-level voltage source inverters are restricted to using only eight voltage vectors, which limits their output performance. By incorporating virtual voltage vectors, similar performance to multilevel converters can be achieved. The proposed technique involves a pre-voltage selection method that enhances output performance without increasing computational load. Through simulation and experiments, improved output current THD and current error were observed under various load conditions. This showcases the potential for enhancing the efficiency and performance of electric propulsion ships.

1. Introduction

Since the 1980s, global warming has emerged as a pressing global concern, driven by the heightened awareness of climate change. The widespread utilization of fossil fuels has resulted in the rapid escalation of carbon dioxide levels. Additionally, the proliferation of artificial compounds, such as chlorofluorocarbons, hydrofluorocarbons, and perfluorocarbons, has exacerbated this issue [1,2]. According to the United Nations Conference on Trade and Development, approximately 80% of global merchandise trade is transported by ships [3]. To reduce ship emissions, the International Maritime Organization’s Marine Environment Protection has set ambitious targets to reduce the carbon intensity of international shipping by at least 50% by 2050, compared with levels recorded in 2008 [4,5]. Ships contribute significantly to air pollution, emitting substantial amounts of particulate matter, CO₂, and sulfur oxides. Projections indicate that CO₂ emissions from ships could surge by 250% by 2050, with the shipping industry accounting for over 15% of global NOx emissions [6].

In light of mounting societal pressure for carbon neutrality, the development of ecologically friendly ships has gained momentum within the environmental and energy sectors. The development of fuel cells, hydrogen- and ammonia-powered ships, and electric propulsion ships is underway [7]. Electric propulsion ships, in particular, have emerged as a pivotal technology in the future maritime industry owing to their significant reduction in air pollutants [8,9]. Notably, electric propulsion ships eschew traditional engine facilities, relying instead on inverters to drive the propulsion motor, as shown in Figure 1. The efficiency and performance of the inverter influence the overall system’s efficiency and stability [10,11].

Inverters designed for electric propulsion ships employ various high-capacity voltage source converters to accommodate the substantial power requirements of ship motors [10]. Two-level voltage source inverters (VSIs) are commonly utilized in small- and medium-sized electric propulsion ships owing to their straightforward design, easy-to-implement control methods, and cost-effectiveness. However, these two-level VSIs may suffer from decreased system efficiency owing to high switching losses and limited high-frequency switching capabilities [12]. These limitations pose challenges in generating high-capacity power and addressing issues with high total harmonic distortion (THD) [13]. To overcome these limitations, current research is focused on minimizing switching losses of power semiconductors and enhancing input/output efficiency in two-level VSIs [14,15].

Multilevel VSIs, on the other hand, offer the ability to extend output voltage to three levels or more. By reducing voltage stress on switching devices, multilevel VSIs effectively lower switching losses and provide additional benefits, such as a harmonic reduction in output voltage and a decreased voltage fluctuation rate. Furthermore, multilevel inverters with higher output voltage levels exhibit improved THD characteristics [16,17]. However, multilevel inverters are uneconomical owing to additional switching devices and require advanced control techniques owing to increased control targets [18,19].

Model predictive control (MPC) has been extensively investigated to enhance output quality and ensure stable operation by predicting future outputs based on system models [20,21]. Unlike pulse-width modulation (PWM) control methods, MPC does not require proportional-integral controllers, thereby eliminating the need for gain value calculations. By considering the physical state of the inverter through the inherent discrete characteristics of power converters, MPC offers optimal performance [22,23]. However, increasing the switching frequency is crucial for MPC performance. This poses a challenge for multilevel inverters, as longer calculation times are required to consider numerous voltage vectors [24]. To address this issue, finite state model predictive control (FS-MPC) has been developed [24]. This approach simplifies the problem by optimizing a finite number of switching states and predicting the system’s next action, resulting in a rapid dynamic response [25]. Model predictive current control (MPCC) utilizes a cost function to pre-select vectors and employs one vector during each sampling period to minimize current error [23,26]. However, two-level VSIs face limitations in output performance owing to their limited number of switching states [27,28]. Recent studies have highlighted performance limitations, such as increased power loss from high-frequency switching [27] and torque ripple issues [28]. While optimization algorithms and hybrid control techniques have been proposed to address these challenges, they introduce complexity due to more complex mathematical models and computational processes and encounter difficulties in real-time applications. Therefore, research on control methods to enhance output performance applicable to two-level VSIs is necessary [29].

This study proposes a novel MPCC method that leverages virtual voltage vectors to enhance the output performance of two-level VSIs. This innovative method utilizes virtual voltage vectors instead of the eight voltage vectors found in conventional two-level inverters, resulting in an output voltage similar to that of multilevel VSIs. The proposed method allows for the expansion of virtual voltage vectors to N-levels, enabling the selection of a multilevel output based on specific performance requirements. Although this method is computationally burdensome compared with conventional two-level inverter models owing to increased voltage vectors, the calculation complexity can be mitigated through a pre-vector selection. The proposed method streamlines the optimization process by pre-selecting candidate voltage vectors based on the polarity and sector of the reference voltage. By applying the minimum cost switching state, the system operation prediction is significantly reduced.

The effectiveness of the proposed method was validated through simulations and experiments conducted under identical conditions as conventional MPCC. These were conducted under various conditions, including different sampling frequencies, modulation indexes, and switching frequencies. This approach provides a more comprehensive performance comparison. The experimental results verified that the proposed method achieves an approximately 50% reduction in output current THD and current error compared to conventional approaches while also improving computational efficiency by reducing DSP execution time by 24% under three-level conditions.

The remainder of this paper is organized as follows. Section 2 outlines conventional two-level VSIs, selects a load model, evaluates the cost function, predicts future states using existing FCS-MPC, and selects the switching state minimizing the cost function. Section 3 delves into the proposed simple MPC based on virtual voltage vectors. Section 4 validates the performance of the proposed method through simulation and experiments. Finally, Section 5 offers a summary of this study and draws conclusions based on the findings.

2. Conventional Finite Control Set Model Predictive Control for Two-Level Voltage Source Inverter

2.1. Two-Level Voltage Source Inverter Layout

The VSI is a device utilized to convert fixed DC voltage into three-phase AC voltage with variable magnitude and frequency. A two-level VSI three-phase power converter is the simplest type of VSI as it only has two voltage levels. The basic circuit composing the inverter is composed of two series-connected switching elements, S and S′. The proposed inverter topology is shown in Figure 2, with the three-phase inverter circuit comprising three poles: a, b, and c. Each pole switches independently to generate one phase’s voltage output. In the inverter, three pairs of complementarily controlled switches are utilized in each inverter phase or leg: (Sa, Sa′), (Sb, Sb′), and (Sc, Sc′). The variation in the pole voltage of the two-level VSI with switching states is presented in

Table 1. Switching states of each phase (Sx) and corresponding output voltages.

Status	Sx (x = a, b, c)	Vxp (x = a, b, c)
1	On	Vdc
0	Off	0

. A capacitor is typically connected in parallel between the DC power source and inverter input in the DC link section to maintain a constant DC voltage.

Depending on the combination of switching states for each phase, the two-level inverter has eight different switching states, resulting in six active voltage vectors (V₁, V₂, V₃, V₄, V₅, and V₆) and two zero voltage vectors (V₀ and V₇). The voltage vector diagram and switching states of the two-level VSI are shown in Figure 3. Because V₀ and V₇ generate the same zero voltage vector (V₀ = V₇), only seven distinct voltage vectors exist. In other words, while a three-phase two-level VSI has eight different switching combinations, it can only deliver seven distinct voltage vectors. The switching states and output voltages corresponding to the voltage vectors of the two-level VSI are listed in

Table 2. Switching states and load phase voltages based on voltage vectors.

	Vector States	Switching State (a, b, c)	Line Voltage			Phase Voltage
			${\mathit{v}}_{\mathbf{a}\mathbf{b}}$	${\mathit{v}}_{\mathbf{b}\mathbf{c}}$	${\mathit{v}}_{\mathbf{c}\mathbf{a}}$	${\mathit{v}}_{\mathbf{a}\mathbf{n}}$	${\mathit{v}}_{\mathbf{b}\mathbf{n}}$	${\mathit{v}}_{\mathbf{c}\mathbf{n}}$
Zero vector	0	000	0	0	0	0	0	0
	7	111	0	0	0	0	0	0
Active vector	1	100	Vdc	0	−Vdc	${{\displaystyle \frac{2}{3}}V}_{\mathrm{d}\mathrm{c}}$	${-{\displaystyle \frac{1}{3}}V}_{\mathrm{d}\mathrm{c}}$	${-{\displaystyle \frac{1}{3}}V}_{\mathrm{d}\mathrm{c}}$
	2	110	0	Vdc	−Vdc	${{\displaystyle \frac{1}{3}}V}_{\mathrm{d}\mathrm{c}}$	${{\displaystyle \frac{1}{3}}V}_{\mathrm{d}\mathrm{c}}$	${-{\displaystyle \frac{2}{3}}V}_{\mathrm{d}\mathrm{c}}$
	3	010	−Vdc	Vdc	0	${-{\displaystyle \frac{1}{3}}V}_{\mathrm{d}\mathrm{c}}$	${{\displaystyle \frac{2}{3}}V}_{\mathrm{d}\mathrm{c}}$	${-{\displaystyle \frac{1}{3}}V}_{\mathrm{d}\mathrm{c}}$
	4	011	−Vdc	0	Vdc	${-{\displaystyle \frac{2}{3}}V}_{\mathrm{d}\mathrm{c}}$	${{\displaystyle \frac{1}{3}}V}_{\mathrm{d}\mathrm{c}}$	${{\displaystyle \frac{1}{3}}V}_{\mathrm{d}\mathrm{c}}$
	5	001	0	−Vdc	Vdc	${-{\displaystyle \frac{1}{3}}V}_{\mathrm{d}\mathrm{c}}$	${-{\displaystyle \frac{1}{3}}V}_{\mathrm{d}\mathrm{c}}$	${{\displaystyle \frac{2}{3}}V}_{\mathrm{d}\mathrm{c}}$
	6	101	Vdc	−Vdc	0	${{\displaystyle \frac{1}{3}}V}_{\mathrm{d}\mathrm{c}}$	${-{\displaystyle \frac{2}{3}}V}_{\mathrm{d}\mathrm{c}}$	${{\displaystyle \frac{1}{3}}V}_{\mathrm{d}\mathrm{c}}$

.

2.2. Load Model

To derive the continuous-time state–space equation for the R-L load of each phase, the differential equation of the load current is expressed as follows:

$$\mathit{v}=R·\mathit{i}+L·d\mathit{i}/dt$$

(1)

These equations can be transformed into the stationary coordinate α-β frame using the Clarke transformation (abc to αβ transformation). The Clarke transformation of the output load voltage is defined as follows:

$$V_{\mathsf{\alpha }}=2/3·\left(V_{\mathrm{a}}-0.5V_{\mathrm{b}}-0.5V_{\mathrm{c}}\right)$$

(2)

$$V_{\mathsf{\beta }}=2/3·\left(0.5\sqrt{3}{V}_{\mathrm{b}}-0.5\sqrt{3}{V}_{\mathrm{c}}\right)$$

(3)

By applying the Clarke transformation to the load current, the equation can be expressed in the stationary coordinate α-β frame. The continuous-time state–space equation of the load then becomes the following:

$$\left[\begin{array}{c}{i}_{\mathsf{\alpha }}\\ i_{\mathsf{\beta }}\end{array}\right]=\left[\begin{array}{cc}-R/L& 0\\ 0& -R/L\end{array}\right]\left[\begin{array}{c}{i}_{\mathsf{\alpha }}\\ i_{\mathsf{\beta }}\end{array}\right]+\left[\begin{array}{cc}1/L& 0\\ 0& 1/L\end{array}\right]\left[\begin{array}{c}{v}_{\mathsf{\alpha }}\\ v_{\mathsf{\beta }}\end{array}\right]$$

(4)

To obtain a discrete-time system representation, we utilize the Euler-forward equation (Equation (5)) to derive a discrete-time equation for the future load current where T_s represents the sampling time and k represents the current sampling point.

$$\mathit{i}\approx i\left(k+1\right)-i\left(k\right)/T_{\mathrm{s}}$$

(5)

$$\left[\begin{array}{c}{i}_{\mathsf{\alpha }}(k+1)\\ i_{\mathsf{\beta }}(k+1)\end{array}\right]=\left[\begin{array}{cc}1-T_{\mathrm{s}}·R/L& 0\\ 0& 1-T_{\mathrm{s}}·R/L\end{array}\right]\left[\begin{array}{c}{i}_{\mathsf{\alpha }}\left(k\right)\\ i_{\mathsf{\beta }}\left(k\right)\end{array}\right]+\left[\begin{array}{cc}{T}_{\mathrm{s}}/L& 0\\ 0& T_{\mathrm{s}}/L\end{array}\right]\left[\begin{array}{c}{v}_{\mathsf{\alpha }}\left(k\right)\\ v_{\mathsf{\beta }}\left(k\right)\end{array}\right]$$

(6)

The load current for each switching possibility is predicted using Equation (6). The future value of the load current is determined by evaluating the cost function g for each of the seven possible voltage vectors generated using the two-level VSI, as shown in Table 2. The voltage vector that minimizes the cost function is then selected and applied during the next sampling point.

2.3. Conventional Finite Control Set Model Predictive Control (FCS-MPC)

Conventional FCS-MPC does not utilize a PWM block. Instead, it employs a discrete model to predict future system states and determines the optimal switching state by minimizing a cost function. FCS-MPC is commonly utilized as a current controller for two-level VSIs. The block diagram of a VSI converter operating under conventional finite control set model prediction current control (FCS-MPCC) is shown in Figure 4. i_ref. denotes the reference current for predictive current control, i (k) denotes the measured value at time (k)^th, and i (k + 1) denotes the predicted value of possible switching states at time (k + 1)^th. The discrepancy between the reference and predicted values is calculated to minimize the cost function, leading to the selection of the switching state that minimizes the cost function. The switching signal S corresponding to the selected state is applied to the converter.

Generally, the control algorithm can be summarized as follows [7]:

(1)

The load current is measured.

(2)

The load current is predicted at the next sampling point for all possible switching states.

(3)

The cost for each prediction is evaluated.

(4)

The optimal switching state that minimizes the cost function is selected.

(5)

The new switching state is applied.

The optimal voltage vector is selected from a pool of the seven voltage vectors, and the corresponding switching state required to deliver this voltage vector is determined. To identify the optimal voltage vector for application in the k + 1 cycle, the cost function must be minimized in each cycle.

The cost function can be defined as an absolute value term based on the reference and predicted current values, as shown in Equation (7).

$$g=\left|i_{\mathrm{r}\mathrm{e}\mathrm{f}.\mathsf{\alpha }}\left(k+1\right)-i_{\mathsf{\alpha }}(k+1)\right|+\left|i_{\mathrm{r}\mathrm{e}\mathrm{f}.\mathsf{\beta }}\left(k+1\right)-i_{\mathsf{\beta }}(k+1)\right|$$

(7)

The reference current i_ref (k + 1) at (k + 1)^th, required in Equation (7), can be predicted by utilizing the reference currents at the current time (k)^th, (k − 1)^th, and (k − 2)^th.

$${\mathit{i}}_{\mathrm{r}\mathrm{e}\mathrm{f}}\left(k+1\right)=3{\mathit{i}}_{\mathrm{r}\mathrm{e}\mathrm{f}}\left(k\right)-3{\mathit{i}}_{\mathrm{r}\mathrm{e}\mathrm{f}}\left(k-1\right)-3{\mathit{i}}_{\mathrm{r}\mathrm{e}\mathrm{f}}\left(k-2\right)$$

(8)

The prediction model is utilized to measure the current i (k) to predict the reference current i_ref (k + 1) at (k + 1)^th. However, when the optimal vector selected operates at (k + 1)^th, the measured current is updated to i (k + 1), resulting in a one-step delay in the control system. This method utilizes extrapolation, which analyzes a few known discrete data points to predict the trend of periodic functions such as sine waves and estimate future values. Delay compensation becomes essential to eliminating this delay phenomenon. Delay compensation involves predicting i (k + 1) at (k + 1)^th and then using this predicted value to predict i (k + 2) [30]. This enables time compensation that is inevitable in digital signal processing (DSP).

3. Proposed Model Predictive Control Based on Virtual Voltage Vectors

Virtual Voltage Vector-Based FCS-MPC Concept

Conventional MPC systems have several inherent disadvantages, such as unfixed switching frequency, large tracking current error, and significant harmonic current. To address these challenges, this study proposes a simplified MPC approach that leverages virtual voltage vectors. This method offers enhanced control performance compared with conventional single vector-based MPC techniques. By employing virtual multi-voltage vectors, the proposed method not only boosts performance but also reduces computational burden through pre-vector selection. The block diagram of the proposed method is shown in Figure 5.

The proposed method can reduce THD by generating output voltage similar to that of multilevel converters. Unlike conventional two-level inverters that rely on eight fixed voltage vectors, this method leverages virtual voltage vectors that can be expanded to N-levels based on specific output requirements. In other words, the number of virtual voltage vectors can be increased when the performance of the output current needs to be further enhanced. Furthermore, the method streamlines the selection process of candidate voltage vectors by considering the polarity and sector of the output voltage. This approach minimizes computational load, even when utilizing virtual vectors. In the case of virtual vectors based on three levels, the computational load is actually reduced.

The voltage vector diagram of a two-level inverter, including the virtual voltage vectors of the proposed method, is shown in Figure 6. In conventional two-level inverters, voltage vectors are categorized as ACTIVE V_n (n = 1, 2, 3, 4, 5, and 6) and zero voltage vectors V_n (n = 0, 7) based on the switching state. The number of virtual voltage vectors varies with the output voltage level. Virtual vectors can be represented as V_mx (m represents a multi-vector and x denotes the number), with the voltage vector diagram when the output voltage level is set to three shown in Figure 6. In this case, 12 virtual voltage vectors are introduced, indicating that 19 voltage vectors must be considered in the cost function, along with 7 voltage vectors of the two-level inverter excluding the existing duplicate vectors. The number of voltage vectors (${\mathit{N}}_{\mathrm{v}\mathrm{i}\mathrm{r}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}-\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}$) to be considered based on the output voltage level (m) can be calculated as follows:

$${\mathit{N}}_{\mathrm{v}\mathrm{i}\mathrm{r}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}-\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}=2\ast {\sum }_{n=0}^{m-2}\left(m+n\right)+\left(2m-1\right)$$

(9)

The number of voltage vectors to be considered is determined by the output level configuration, as presented in

Table 3. Number of voltage vectors based on the output levels of the proposed method.

m-Level	3-Level	4-Level	5-Level
Real vector	7	7	7
Multi-vector	12	30	54
Total	19	37	61

.

As shown in Figure 7, the number of voltage vectors to be considered increases significantly with the output voltage level. While this enhances output current performance, it also results in a higher computational load. Therefore, the proposed method reduces the number of voltage vectors to be considered by utilizing the polarity and sector of the reference voltage. By selectively choosing voltage vectors based on polarity and sector, the computational load is minimized without compromising performance. To determine the sector and polarity, the predicted reference voltage must be calculated. This can be achieved by calculating the reference and actual currents of the next step (k + 1)^th. Additionally, the reference current for the next step can be adjusted using the time delay compensation method to compensate for delays as follows [31,32]:

$$\left[\begin{array}{c}{v}_{\mathrm{r}\mathrm{e}\mathrm{f}.\mathsf{\alpha }}(k+1)\\ v_{\mathrm{r}\mathrm{e}\mathrm{f}.\mathsf{\beta }}(k+1)\end{array}\right]=\left(R-T_{\mathrm{s}\mathrm{p}}^{-1}{L}_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\right)\left[\begin{array}{c}{i}_{\mathsf{\alpha }}(k+1)\\ i_{\mathsf{\beta }}(k+1)\end{array}\right]+L_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}^{-1}{T}_{\mathrm{s}\mathrm{p}}\left[\begin{array}{c}{i}_{\mathrm{r}\mathrm{e}\mathrm{f}.\mathsf{\alpha }}(k+2)\\ i_{\mathrm{r}\mathrm{e}\mathrm{f}.\mathsf{\beta }}(k+2)\end{array}\right]$$

(10)

The sector and polarity information can be obtained from the predicted reference voltage shown in Equation (10). By calculating the magnitude and angle $\theta$ of the reference voltage vector in the α-β plane using Equations (11) and (12), the sector in which the reference voltage vector is located can be identified.

$$\left|v_{\mathrm{r}\mathrm{e}\mathrm{f}.\mathsf{\alpha }\mathsf{\beta }}(k+1)\right|=\sqrt{{\left(v_{\mathrm{r}\mathrm{e}\mathrm{f}.\mathsf{\alpha }}(k+1)\right)}^2+{\left(v_{\mathrm{r}\mathrm{e}\mathrm{f}.\mathsf{\beta }}(k+1)\right)}^2}$$

(11)

$$\theta ={\tan }^{-1}\left({\displaystyle \frac{v_{\mathsf{\alpha }}}{v_{\mathsf{\beta }}}}\right)$$

(12)

For example, when the reference voltage is located in Sector 1, as shown in Figure 8, the polarity of each reference voltage phase is (+, −, −), and the angle of the reference voltage vector can be expressed using Equation (12). This allows for the pre-selection of nearby voltage vectors V₂, V₀, V_m2, and V_m7 as potential candidates for the voltage vector.

The proposed method involves selecting the optimal switching state from the pre-screened voltage vector candidates based on the reference voltage shown in Equation (10) and the cost function. The flowchart of the proposed method is shown in Figure 9. The cost function of the proposed method can be calculated as follows:

$$g=\left|V_{\mathrm{r}\mathrm{e}\mathrm{f}.\mathsf{\alpha }}\left(k+1\right)-V_{\mathsf{\alpha }}(k+1)\right|+\left|V_{\mathrm{r}\mathrm{e}\mathrm{f}.\mathsf{\beta }}\left(k+1\right)-V_{\mathsf{\beta }}(k+1)\right|$$

(13)

4. Simulations and Experimental Results

Performance analysis was conducted through simulations using an input DC voltage (Vdc = 100 V) and an RL load (R = 1.2 Ω, L = 5.3 mH). The reference current magnitude was set at 15 A, based on a modulation index of 0.7. The values of variables used for the simulation are listed in

Table 4. Simulation parameters.

Parameters	Values
Vin (DC voltage)	100 V
Rload (load resistance)	1.233 Ω
Lload (load inductance)	9.873 mH
Cdc (DC capacitance)	4400 µF

.

Unlike conventional MPC for two-level inverters, which typically considers six active vectors and two zero vectors, this novel approach incorporated virtual multi-voltage vectors. This strategy achieved a multilevel effect, thereby enhancing output current performance. Moreover, despite the utilization of multi-vectors, the computational load remained unchanged owing to a pre-selection approach that focused only on voltage vectors near the reference voltage. For performance comparison, three-level conditions were employed as multi-vectors, considering 19 voltage vectors. Comparisons with conventional MPC were conducted under both steady-state and transient conditions.

The output performance of the conventional and proposed MPC methods under steady-state conditions was evaluated. The comparison included three-phase output currents, an a-phase reference current, line-to-line voltage, and load phase voltage. The proposed method demonstrated superior output current performance, which is attributed to its utilization of virtual multi-vectors. This enhancement was particularly evident in the load phase voltage, which closely resembled a sine wave. The fast Fourier transform (FFT) waveform of the a-phase output current further validated the enhanced performance of the proposed method, as evidenced by a lower THD value. Notably, despite the application of multi-vectors, the computational load remained unchanged owing to the pre-selection method (Figure 10).

The output performance of both methods was compared under transient conditions with varying output current magnitudes. The comparison included three-phase output currents, an a-phase reference current, line-to-line voltage, and load phase voltage. The proposed method achieved a fast response similar to Conv-MPC during transient states, even when considering multi-vectors. Furthermore, the proposed method exhibited superior output current performance under changing low load conditions. Notably, the computational load was not impacted by considering multi-vectors (Figure 11).

Furthermore, the performance of both methods was evaluated under transient conditions with changing output frequencies. The comparison included three-phase output currents, a-phase reference current, line-to-line voltage, and load phase voltage under these conditions. The proposed method maintained a fast response similar to that of the conventional MPC during transient states, despite considering multi-vectors. Additionally, the proposed method demonstrated superior output current performance under low-frequency conditions. Once again, the consideration of multi-vectors did not impact the computational load (Figure 12).

The output performance of the proposed method under the load profile conditions of an electric propulsion ship was presented. The proposed method accurately controlled three-phase output currents under small electric propulsion ship load profile conditions. Furthermore, it achieved a fast response under varying ship speed conditions by utilizing the pre-vector selection method (Figure 13).

The comparison of output performance between Conv-MPC and Prop-MPC under varying sampling frequency conditions is shown in Figure 14. In particular, a comparison of the current error performance of both approaches is shown in Figure 14a. The current error can be calculated as follows:

$$error\left(i_x\right)={\sum }_{x=a,b,c}{\displaystyle \frac{1}{N}}{\sum }_{k=1}^N\left|i_x^{\ast }-i_x\right|$$

(14)

With N set to 10,000, the error represents the average of these 10,000 calculated values. The proposed method, which considered multi-vectors, demonstrated lower current errors across all sampling frequency ranges compared with the conventional MPC. Additionally, a comparison of the output current THD of both methods is shown in Figure 14b. The output current THD can be calculated as follows:

$${\mathrm{\%}\mathrm{T}\mathrm{H}\mathrm{D}}_i={\displaystyle \raisebox{1ex}{$\sum _{x=a,b,c}\sqrt{i_{x2}^2+i_{x3}^2+\cdots +i_{xn}^2}$}\!\left/ \!\raisebox{-1ex}{$\sum _{x=a,b,c}{i}_{x1}$}\right.}$$

(15)

where i_x1⋯i_xn denote the fundamental and nth-harmonic components of the output currents in phase x, respectively. The simulation considered up to the 8335th-harmonic components. Similarly to current error performance, the proposed method exhibited superior output current THD compared with the conventional MPC.

The output performance comparison of both methods under varying modulation index conditions is shown in Figure 15. This comparison allowed for performance evaluation under changing load conditions. The proposed method outperformed the conventional method in terms of current error performance and output THD across all modulation index conditions. This enhancement in output current performance can be attributed to the consideration of multi-vectors in the proposed method.

The output performance of both methods was compared under varying switching frequency conditions, as shown in Figure 16. While both methods demonstrated similar performance at the same switching frequency, the conventional method encountered limitations in increasing the switching frequency owing to the requirement of a high sampling frequency. However, the proposed method showed a slightly lower THD at the same switching frequency owing to more accurate reference voltage tracking.

A comparison of the total losses and efficiency of both methods under varying switching frequency conditions is shown in Figure 17. Notably, Prop-MPC showcased marginally lower losses and higher efficiency compared with Conv-MPC at the same switching frequency. This is because Conv-MPC increased switching in areas of increasing current to elevate the switching frequency.

Figure 18 illustrates the changes in current error and current total harmonic distortion (THDi) as the number of virtual voltage vectors increases in the proposed method. As the proposed method’s virtual vectors increase from 3-level to 5-level voltage, there is a significant reduction in both current error (approximately 80%) and THDi (approximately 85%) compared to the conventional method. This indicates that increasing the number of virtual vectors allows for a more precise fulfillment of the load’s requirements.

The implementation of the proposed method utilizing multi-vectors with pre-selection was performed with a prototype setup, as shown in Figure 19. The experimental setup comprised a three-phase VSI with IGBT modules and an RL load. A digital signal processor (DSP, TMS320F28335) was employed to execute the switching algorithm for the VSI.

The experimental results, as shown in Figure 20 and Figure 21, illustrate the superior performance of the proposed method compared with the conventional MPC. This performance superiority was consistent with the findings from simulations. This superior performance was evident in both steady-state (Figure 20) and transient conditions (Figure 21). The enhanced output performance of the proposed method can be attributed to the utilization of multi-vectors. Despite this advancement, the execution time of the proposed method remained on par with that of conventional MPC, as indicated in

Table 5. Comparison of the DSP execution time between the conventional FCS-MPC method and proposed method.

	Conventional MPC	Proposed MPC (3-Level)	Proposed MPC (4-Level)	Proposed MPC (5-Level)
Execution time [µs]	25.23	19.34	24.87	29.41
Reduction rate of DSP time [-]	-	0.76	0.98	1.16

. This can be attributed to the application of the pre-vector selection method. Furthermore, as the output level increased, the number of vectors considered increased, resulting in an increased DSP execution time. However, the pre-vector selection method effectively mitigated a significant increase in execution time.

Model parameter uncertainty is a critical factor in model predictive control (MPC) systems, particularly when applied to load models. The impact of load resistance and inductance uncertainties on both the proposed and conventional methods is shown in Figure 22. The comparison of load current performance reveals that uncertainty in load resistance has a minimal effect on the performance of both methods. However, when the load inductance is underestimated, it results in larger current errors and total harmonic distortion (THD) compared to overestimation. This analysis emphasizes the importance of the accurate modeling of inductive components, such as inductance, in maintaining the accuracy and stability of the control system. This underscores that mismatches in inductance values can adversely affect current regulation and overall system performance.

5. Conclusions

This study proposed a novel MPC method based on virtual voltage vectors to address the limitations of two-level VSIs commonly utilized in small- and medium-sized electric propulsion ships. Conventional two-level VSIs were limited in output performance by their utilization of only eight voltage vectors. However, the proposed method leveraged virtual voltage vectors to achieve performance similar to those of multilevel converters. This approach offered scalability up to m-levels, ensuring flexibility.

Notably, the implementation of a pre-voltage selection technique enhanced output performance without increasing computational load. Through simulations and experiments conducted under various load conditions, the efficacy of the proposed method was validated, showcasing a reduction of approximately 50% in output current THD and current error. This represents a significant advancement in enhancing the efficiency and performance of electric propulsion ships.

Future research should focus on the practical application of the proposed technique on real ships to assess long-term performance and reliability. Furthermore, the development of optimal control strategies tailored to diverse operating conditions is imperative. Additionally, efforts are underway to explore avenues for maximizing the overall energy efficiency of electric propulsion ships through the integration of energy storage systems. This approach is poised to expedite the commercialization of electric propulsion ships and contribute significantly to marine environmental protection.