# Model Predictive Control Using Multi Virtual Voltage Vectors for a Two-Level Inverter in Electric Propulsion ship

Chan Roh, Euicheol Shin, Hyeongyo Chae

► To cite this version:

Chan Roh, Euicheol Shin, Hyeongyo Chae. Model Predictive Control Using Multi Virtual Voltage Vectors for a Two-Level Inverter in Electric Propulsion ship. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0103 . hal-05088998

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Distributed under a Creative Commons Attribution 4.0 International License
Main Title:

Model Predictive Control Using Multi Virtual Voltage Vectors for a Two-Level Inverter in Electric Propulsion ship

Chan Roh, Euicheol Shin, Hyeongyo Chae
KOREA MARITIME & OCEAN UNIVERSITY

727 Taejong-ro, Yeongdo-Gu
Busan 49112, South Korea

Tel.: +82 / (51) - 410.4285.

E-Mail: rohchan@kmou.ac.kr, shineuicheol@g.kmou.ac.kr, chae@g.kmou.ac.kr

URL: https://www.kmou.ac.kr/english/main.do

Acknowledgements

This work was supported by the Technology Innovation Program (RS-2024-00437087) funded by the Ministry of Trade, Industry and Energy (MOTIE, Korea).

Keywords

«Electric propulsion ship», «Two-level voltage source inverter», «Virtual voltage vector», «Model predictive control», «Output performance improvement».

Abstract

In response to environmental concerns, electric propulsion ships are emerging as a key technology in the maritime industry. The inverter, which controls the propulsion motor, is a critical component in electric propulsion ships. The 2-level voltage source inverter is widely implemented in small and medium-sized ships, but its output performance remains constrained due to limited number of voltage vectors. The proposed approach enables 2-level voltage source inverters to achieve multilevel inverter-comparable performance by adding virtual voltage vectors. While it incurs increased computational burden, the proposed method achieves performance enhancement without computational burden through the application of pre-voltage selection. Experiments have shown improvements in output current THD and current error. The results demonstrate potential advantages for electric propulsion systems.

Introduction

Climate change has emerged as one of
humanity's greatest challenges [1, 2], prompting
various industries dependent on fossil fuels to

reassess and transform their environmental
impact. Maritime transport, which accounts for
over 80% of international trade, faces increasing
pressure to address its environmental footprint. In
response, the shipping industry has intensified its
efforts to develop environmentally sustainable
vessels to minimize ecological damage [3].
Electric propulsion systems have emerged as a
promising solution for next-generation ships,
offering reductions in atmospheric pollutants [4,
5]. The inverter serves as a crucial component in
electric propulsion vessels, controlling the ship's
propulsion motor. The system's overall efficiency
and reliability heavily depend on the inverter's
performance and effectiveness [6].

For small to medium-sized electric propulsion
vessels, 2-level Voltage Source Inverters (VSI)
are commonly deployed due to their
straightforward design, control simplicity, and
cost-effectiveness. However, these systems suffer
from high switching losses and frequency
limitations, resulting in reduced system efficiency
and power generation constraints [7, 8]. While
multi-level VSI systems offer solutions to these
challenges, they necessitate higher costs and more
complex control methods [9, 10, 11]. Model
Predictive Control (MPC), an advanced inverter
control technique, anticipates future system
outputs using model-based predictions [12, 13].
MPC provides benefits such as computational
efficiency and performance optimization [14, 15].
However, enhancing MPC performance requires
increased switching frequency. For 2-level VSI,
frequency enhancement faces limitations due to
restricted switching states [16, 17], while multi-
level inverters struggle with frequency increases
due to extended computation times resulting from
numerous voltage vectors [18].

This research introduces an enhanced model
predictive current control (MPCC) methodology
incorporating virtual voltage vectors and a pre-
selection vector approach to improve 2-level VSI output performance. This technique creates virtual voltage vectors alongside the traditional 8 voltage vectors of 2-level inverters, enabling output voltage generation comparable to multi-level VSI systems. While this approach demands increased computational resources due to additional voltage vectors, the computational burden is mitigated through vector pre-selection. This streamlines the optimization process and implements minimal cost switching states to address existing limitations. The proposed methodology selects potential voltage vectors based on reference voltage polarity and sector analysis. This approach simplifies optimization calculations, reduces system behavior prediction requirements, and enables efficient implementation of minimum cost switching states. Consequently, the computational demands decrease, allowing for more straightforward MPC implementation. Simulation studies and experimental validations have demonstrated the effectiveness of this approach under conditions identical to conventional model predictive current control. Experimental results confirmed a 50% improvement in both output current THD and current error metrics, significantly enhancing electric propulsion vessel performance and efficiency. These advancements are anticipated to facilitate the widespread adoption of electric propulsion vessels while contributing to marine ecosystem preservation.

## Conventional Finite-Control-Set Model Predictive Control For Two-Level Voltage Source Inverter

<figure><img src="image_2.png" /><figcaption>Fig. 1: Two-level three phase VSI circuit topology</figcaption></figure>

The VSI converts a fixed DC voltage into three-phase AC voltage. The inverter uses two switches connected in series and operating complementarily. In a three-phase inverter, each

phase (a, b, c) has independent switches to output each phase voltage as shown in Fig. 1.

<table><caption>Table I: The switching status (Sx) of each phase and output voltage</caption><thead><tr><td>Status</td><td>Sx(x=a,b,c)</td><td>Vxp(x=a,b,c)</td></tr></thead><tbody><tr><td>1</td><td>On</td><td>Vdc</td></tr><tr><td>0</td><td>Off</td><td>0</td></tr></tbody></table>

The pole voltages of the 2-level VSI according to switching states in Table I.

<figure><img src="image_3.png" /><figcaption>Fig. 2: Voltage vectors generated by two-level VSI</figcaption></figure>

The 2-level inverter has 8 different switching states depending on the combinations of switching states for each phase. However, since $V_0$ and $V_7$ generate the same zero voltage vector, the actual number of voltage vectors is 7 (Fig. 2).

To obtain the continuous-time state-space equations for the R-L load of each phase, the differential equation for the load current is given by:

$$v = R \cdot i + L \cdot \frac{di}{dt} \quad (1)$$

By projecting the equation onto the stationary $\alpha-\beta$ frame using Clarke transformation (abc to $\alpha\beta$ transformation), we obtain:

$$V_{\alpha} = \frac{2}{3} \cdot (V_{a} - 0.5V_{b} - 0.5V_{c}) \quad (2)$$

$$V_{\beta} = \frac{2}{3} \cdot (0.5\sqrt{3}V_{b} - 0.5\sqrt{3}V_{c}) \quad (3)$$

The continuous-time state-space equations for the load current after applying Clarke transformation are:
$$ \begin{bmatrix} i_{\alpha} \\ i_{\beta} \end{bmatrix} = \begin{bmatrix} -R/L & 0 \\ 0 & -R/L \end{bmatrix} \begin{bmatrix} i_{\alpha} \\ i_{\beta} \end{bmatrix} + \begin{bmatrix} 1/L & 0 \\ 0 & 1/L \end{bmatrix} \begin{bmatrix} v_{\alpha} \\ v_{\beta} \end{bmatrix} \quad (4) $$

By using the Euler-forward equation to express this as a discrete-time system, the discrete-time equation for the future load current $i$ is obtained. Here, $T_s$ is the sampling time and $k$ is the current sampling instant:

$$ \begin{bmatrix} i_{\alpha}(k+1) \\ i_{\beta}(k+1) \end{bmatrix} = \begin{bmatrix} 1-T_s \cdot R/L & 0 \\ 0 & 1-T_s \cdot R/L \end{bmatrix} \begin{bmatrix} i_{\alpha}(k) \\ i_{\beta}(k) \end{bmatrix} + \begin{bmatrix} T_s/L & 0 \\ 0 & T_s/L \end{bmatrix} \begin{bmatrix} v_{\alpha}(k) \\ v_{\beta}(k) \end{bmatrix} \quad (5) $$

Equation (5) is used to predict the load current for each switching possibility. Conventional Finite Control Set Model Predictive Control (FCS-MPC) computes the error between the reference current ($i_{ref.\alpha,\beta}(k+1)$) and the predicted current ($i_{\alpha,\beta}(k+1)$), then selects the switching state that minimizes the cost function ($g$) for application to the inverter.

The cost function can be defined with absolute value terms as follows:

$$ g = |i_{ref.\alpha}(k+1) - i_{\alpha}(k+1)| + |i_{ref.\beta}(k+1) - i_{\beta}(k+1)| \quad (6) $$

## Proposed Model Predictive Control Based On Virtual Voltage Vectors

<figure><img src="image_4.png" /><figcaption>Fig. 3: Proposed Model Predictive Control of block diagram</figcaption></figure>

Adding virtual voltage vectors to the existing 2-level inverter voltage vectors and MPC can achieve the effects of a multi-level inverter. This implies improved performance of the output current. Conventional MPC considers all possible voltage states that the inverter can create at every step. This means that as the number of voltage vectors increases, the computation also increases. In this paper, we reduce the computation by using pre-selecting vectors based on the polarity and

sector of the output voltage. The block diagram of the proposed method is shown in Fig. 3.

<figure><img src="image_5.png" /><figcaption>Fig. 4: Vector diagram of virtual voltage vectors based model predictive control with output voltage of 3-level</figcaption></figure>

The voltage vector diagram of a 2-level VSI including virtual voltage vectors is shown in Fig. 4. 12 virtual voltage vectors ($V_{mx}$) are added to the existing 7 voltage vectors ($V_n$). As a result, a total of 19 voltage vectors need to be considered in the cost function.

Fig. 4 shows an example of the output voltage level set to 3-level. The number of virtual voltage vectors ($N_{virtual-vectors}$) depends on the output voltage level (m) and can be calculated using Equation (7).

$$ N_{virtual-vectors} = 2 * \sum_{n=0}^{m-2} (m+n) + (2m-1) \quad (7) $$

For the proposed method, the number of voltage vectors to be considered depends on the output level setting, as shown below:

<table><caption>Table II: The number of voltage vectors based on the output level of the proposed method</caption><thead><tr><th>m-level</th><th>3-level</th><th>4-level</th><th>5-level</th></tr></thead><tbody><tr><td>Real Vector</td><td>7</td><td>7</td><td>7</td></tr><tr><td>Multi-Vector</td><td>12</td><td>30</td><td>54</td></tr><tr><td>Total</td><td>19</td><td>37</td><td>61</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>Fig. 5: Pre-selected voltage vectors shown in the voltage vector diagram considering virtual voltage vectors : (a) 3-level, (b) 4-level, (c) 5-level</figcaption></figure>

The number of voltage vectors to be calculated increases with the output voltage level (Fig. 5). To solve this issue, the proposed method reduces computation by considering only a limited number of voltage vectors based on the sector and polarity.

To determine the sector and polarity, the reference voltage for the next step (k+1)th must be calculated based on the reference current and actual current. Additionally, to compensate for time delay, the reference current for the next step can be shifted as follows [13]. The equations for determining the sector and polarity using the predicted reference voltage are:

$$ 
\begin{aligned}
\begin{bmatrix} v_{\text{ref.}\alpha}(k+1) \\ v_{\text{ref.}\beta}(k+1) \end{bmatrix} &= (R - T_{\text{sp}}^{-1} L_{\text{load}}) \begin{bmatrix} i_{\alpha}(k+1) \\ i_{\beta}(k+1) \end{bmatrix} + L_{\text{load}}^{-1} T_{\text{sp}} \begin{bmatrix} i_{\text{ref.}\alpha}(k+2) \\ i_{\text{ref.}\beta}(k+2) \end{bmatrix} \quad (8)
\end{aligned}
$$

The equation for identifying the sector where the voltage vector is located is as follows.

$$ 
\begin{aligned}
|v_{\text{ref.}\alpha\beta}(k+1)| &= \sqrt{(v_{\text{ref.}\alpha}6(k+1))^2 + (v_{\text{ref.}\beta}(k+1))^2} \quad (9) \\
\theta &= \tan^{-1}\left(\frac{v_{\alpha}}{v_{\beta}}\right) \quad (10)
\end{aligned}
$$

<figure><img src="image_7.png" /><figcaption>Fig. 6: Pre-selection of voltage vectors based on the reference voltage location in the proposed method</figcaption></figure>

For example, when the reference voltage is located in Sector 1 as shown in Fig. 6, the polarity of the reference voltage across each sector are (+, -, -). The angle of the reference voltage vector can be represented by Equation (10). Based on this, the proposed method pre-selects nearby voltage vectors, specifically $V_2, V_0, V_{m2}$, and $V_{m7}$, to make a list of candidate voltage vectors.

<figure><img src="image_8.png" /><figcaption>Fig. 7: Flowchart of the proposed method</figcaption></figure>

The pre-selected voltage vector candidates determine the optimized switching state based on the reference voltage and the cost function in equation (11). The flowchart of the proposed method is shown in Fig. 6.

$$ g = |mV_{\text{ref.}\alpha}(k+1) - V_{\alpha}(k+1)| + |mV_{\text{ref.}\beta}(k+1) - V_{\beta}(k+1)| \quad (11) $$
Simulation And Experiment Results

To analyze the performance of the proposed method, experiments were conducted with an input DC voltage(Vin=100V), an RL load (Rload = 1.2 Ω, Lload = 5.3 mH). The reference current was set to 15A with a modulation index of 0.7. Table III summarizes the values of the parameters used for the experiments. The proposed method employed a 3-level with multi-vector, considering 19 voltage vectors. The performance of the proposed method was compared with conventional model predictive control under both steady-state and transient conditions.

<table><caption>Table III: Parameter for Experiment</caption><thead><tr><td>Parameters</td><td>Values</td></tr></thead><tbody><tr><td>Vin (DC voltage)</td><td>100 V</td></tr><tr><td>R<sub>load</sub> (Load Resistance)</td><td>1.233 Ω</td></tr><tr><td>L<sub>load</sub> (Load inductance)</td><td>9.873 mH</td></tr><tr><td>C<sub>dc</sub> (DC capacitance)</td><td>4400 μF</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>Fig. 8: Photograph and configuration diagram of the prototype setup to verify the performance of the proposed methods</figcaption></figure>

The proposed method using pre-selection with multi-vector control was implemented by configuring a prototype setup as shown in Fig. 8. The experimental setup consisted of a 3-phase VSI with an IGBT module and an RL load. A digital signal processor (DSP, TMS320F28335) was used to execute the switching algorithm of the VSI.

<figure><img src="image_10.png" /><figcaption>Fig. 9: Experimental results of the three-phase output currents (i<sub>a</sub>, i<sub>b</sub>, i<sub>c</sub>), the line to line voltage(v<sub>ab</sub>), and the a-phase voltage(V<sub>an</sub>) and the frequency spectrum of the a-phase current during steady-state: (a) the conventional method and (b) the proposed method.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(b)</figcaption></figure>

Fig. 10: Experimental results of the three-phase output currents ($i_a$, $i_b$, $i_c$), the line to line voltage($v_{ab}$), and the a-phase voltage($v_{an}$) and the frequency spectrum of the a-phase current during transient-state with step and frequency changes: (a) the conventional method and (b) the proposed method.

The proposed method demonstrates better performance compared to the conventional MPC in Fig. 9 and Fig. 10. Such performance improvements usually lead to an increase in computation time. However, the application of the pre-selection vector method enables the processing time to be almost the same with the conventional MPC as shown in Table 2. This means that the computation time did not increase.

<table><caption>Table IV: Power losses under symmetric and asymmetric operation for AC/DC converter in rated operation</caption><thead><tr><th></th><th>Execution Time [µs]</th><th>reduction rate of DSP time [-]</th></tr></thead><tbody><tr><td>Conventional MPC</td><td>25.23</td><td>-</td></tr><tr><td>Proposed MPC(3-level)</td><td>19.34</td><td>0.76</td></tr><tr><td>Proposed MPC(4-level)</td><td>24.87</td><td>0.98</td></tr><tr><td>Proposed MPC(5-level)</td><td>29.41</td><td>1.16</td></tr></tbody></table>

As shown in Table IV, the proposed method considers more vectors as the output levels increase, which leads to an increase in DSP processing time. However, it can be seen that the pre-selection vector method does not significantly increase the processing time.

<figure><img src="image_13.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(b)</figcaption></figure>

Fig. 11: Performance comparison of the conventional model predictive control and the proposed method according to the sampling frequency change: (a) current error, (b) output current THD
<figure><img src="image_15.png" /><figcaption>Fig. 12: Performance comparison of the conventional model predictive control and the proposed method according to the modulation index change: (a) current error, (b) output current THD</figcaption></figure>

The output performance of the proposed MPC was compared the conventional MPC under different sampling frequencies and modulation conditions. The proposed method showed lower current error in all regions and conditions compared to the conventional method. Additionally, the THD of the output current was improved with the proposed method, as shown in Fig. 11 and 12.

## Conclusion

This research introduces an enhanced control strategy utilizing virtual voltage vectors, designed to address the inherent constraints of traditional 2-level VSI systems. While conventional 2-level VSI technology has been restricted to eight voltage vectors, limiting its operational capabilities, our innovative approach implements virtual voltage vectors to achieve performance comparable to multilevel inverter systems. The methodology's scalability to m-level configurations offers enhanced operational flexibility.

Through the implementation of pre-voltage selection mechanisms, we demonstrated improved output characteristics without compromising computational efficiency.

Empirical testing under diverse load scenarios confirmed the method's efficacy, yielding significant improvements including a 50% decrease in both output current THD and current error measurements. These results indicate substantial potential for advancing electric propulsion ship.

Ongoing research directions include field implementation on maritime vessels to assess sustained performance metrics and durability characteristics. Additional focus areas encompass the development of adaptive control paradigms for diverse operational scenarios. Our research trajectory also includes investigation into comprehensive energy efficiency optimization through energy storage system integration. These advancements are anticipated to facilitate broader adoption of electric propulsion technology in maritime applications while supporting environmental preservation objectives.

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