# Minimization of the circulating currents of an interleaved converter for particle accelerator using inversion-based control

Gabriel Godart, Fulvio Boattini, Philippe Delarue, Alain Bouscayrol

► To cite this version:

Gabriel Godart, Fulvio Boattini, Philippe Delarue, Alain Bouscayrol. Minimization of the circulating currents of an interleaved converter for particle accelerator using inversion-based control. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0260. hal-05085407

HAL Id: hal-05085407

https://utc.hal.science/hal-05085407v1

Submitted on 26 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Minimization of the circulating currents of an interleaved converter for particle accelerator using inversion-based control

Gabriel GODART¹,² Fulvio BOATTINI¹, Philippe DELARUE², Alain BOUSCAYROL²

¹ CERN, 1211 Genève 23, Suisse

² Univ. Lille, Arts et Metiers Institute of Technology, Centrale Lille, Junia, ULR 2697 - L2EP, F-59000 Lille, France

Corresponding Author: gabriel.godart@cern.ch

## Keywords

«DC-DC power converter», «converter control», «particle accelerator», «Energetic macroscopic representation».

## Abstract

An interleaved DC/DC converter is used to supply a magnet for particle accelerator. A very high accuracy of 50 parts per million is required to control the particle beam. Any disturbance can reduce this accuracy. An inversion-based control is derived from the Energetic Macroscopic Representation of this power converter. This control is composed of several control loops and compensations. The control leads to the right accuracy despite some ripple due to current measurement.

## I. Introduction

The CERN, European Organization for Nuclear Research, is the largest nuclear research centre in Europe. Its main objective is to uncover the nature of the universe and to understand how it works [1]. The Proton Synchrotron Booster (PSB) is the first circular accelerator of the Large Hadron Collider (LHC) injectors chain. It receives the particle beams from the Linac4 (linear accelerator), accelerates them and sends them to the Proton Synchrotron (PS) to be further accelerated on their way to the LHC.

Particles are very sensitive to the smallest perturbation in the magnetic field during the acceleration phase. It is therefore extremely important to assure a very high accuracy in the control of the current in the electromagnets. Typically, an accuracy of 10-50 ppm (parts per million) is needed, which demands the utmost attention to reduce all sources of ripple and disturbances, particularly in the DC/DC converter.

The POPS-B magnetic field / current control is ensured by a RST polynomial regulator [3].

The power converter utilizes interleaving to reduce current ripple on the load, primarily by interleaving the six parallel legs of the DC/DC converter (Figure 1). This technique compensates low frequency harmonics associated with the 333 Hz switching frequency. As a result of this technique, the theoretical first harmonic in the output voltage of the DC/DC converter, would be 2000Hz.

The transient behavior of the output currents in the DC/DC converter legs, combined with unavoidable asymmetries in the legs' impedance, disrupts the steady-state compensation achieved by interleaving. As a result, some original switching frequency components, as well as their harmonics, persist in the output voltage. Although these components are minimal, they can interfere with the control of the DC/DC converter output currents, potentially causing the power converter to shut down.

Parallel interleaved NPCs are used in many domains such as solar energy [4] and active power filtering [5].

The typical approach to current sharing control, is to use average current sharing techniques [6][7]. This control is typically decoupled from the standard reference, it is the approach initially followed in POPS-B. However, the system exhibited some shutdowns due to failing to control the current sharing. To address this challenge, a novel control strategy is required to improve the regulation of current imbalance in the DC/DC converter legs, thereby improving robustness and reducing the risk of power converter trips.

Energetic Macroscopic Representation (EMR) is a graphical formalism that allows to organize the control of complex energy conversion systems [8]. It decomposes the system in elementary sections which are inverted by a dedicated control operation including explicit disturbance compensations. If this method leads to an
important number of control operations, it enables a high accuracy. CERN has already applied the EMR and inversion-based methodology for the SIRIUS power converter [9] [10].

The objective of the paper is to propose a new control structure of the POPS-B DC/DC power converter that minimizes the circulating currents in the output three phase inductors. The control is derived from an inversion of the EMR of the system.

In section II, the model of the POPS-B converter is organized using EMR. In section III, the inversion-based control is derived from the EMR. In section IV, the control performances are studied.

## II. Interleaved converter for particle acceleration

POPS-B is composed of a DC bus, an interleaved 6-leg NPC converter, 2 three-phase coupled inductive filters, a capacitor filter to supply an electromagnet (Figure 1).

<figure><img src="image_2.png" /><figcaption>Figure 1: POPS-B power converter</figcaption></figure>

### II.1. Modelling of the converter

The DC bus is modelled using a constant voltage source:

$$u_{dc} = 5000 \text{ V} \qquad (1)$$

A voltage divider is used to create the 3 voltage levels of the NPC.

$$u_h = u_l = \frac{u_{dc}}{2} \qquad (2)$$

The interleaved DC/DC converter is modelled in such a way to that the interleaved currents are decoupled from the current in the magnet.

In order to simplify the NPC model, an equivalent scheme is considered [11] [12] (Figure 2). The $f_{xy}$ functions are defined from the switching function $s_{xy}$ (Table I) where $x$ is the number of the NPC leg. The combination of the equivalent power switches is a commutation cell:

$$f_{x1} + f_{x2} + f_{x3} = 1 \qquad (3)$$

<figure><img src="image_3.png" /><figcaption>Figure 2: Equivalent electrical circuit</figcaption></figure>

<table><caption>Table 1: Voltage $v_x$ vs. $s_{xy}$ and $f_{xz}$ functions</caption><thead><tr><th>v<sub>x</sub></th><th>S<sub>x1</sub></th><th>S<sub>x2</sub></th><th>S<sub>x3</sub></th><th>S<sub>x4</sub></th><th>f<sub>x1</sub></th><th>f<sub>x2</sub></th><th>f<sub>x3</sub></th></tr></thead><tbody><tr><td>u<sub>dc</sub>/2</td><td>1</td><td>1</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td></tr><tr><td>0</td><td>0</td><td>1</td><td>1</td><td>0</td><td>0</td><td>1</td><td>0</td></tr><tr><td>-u<sub>dc</sub>/2</td><td>0</td><td>0</td><td>1</td><td>1</td><td>0</td><td>0</td><td>1</td></tr></tbody></table>

To model the interleaved inductances, a neutral point N is considered (Figure 3). The modulated voltages $v_x$ are a combination of the $f_{xy}$ functions and the bridge voltage, $u_h$ and $u_l$:

<figure><img src="image_4.png" /><figcaption>Figure 3: Interleaved inductances</figcaption></figure>

$$v_1 = f_{11}u_h - f_{13}u_l = m_1 \frac{u_{dc}}{2} \qquad (4)$$

$$v_2 = f_{21}u_h - f_{23}u_l = m_2 \frac{u_{dc}}{2} \qquad (5)$$

$$v_3 = f_{31}u_h - f_{33}u_l = m_3 \frac{u_{dc}}{2} \qquad (6)$$

$$v_4 = f_{41}u_h - f_{43}u_l = m_4 \frac{u_{dc}}{2} \qquad (7)$$

$$v_5 = f_{51}u_h - f_{53}u_l = m_5 \frac{u_{dc}}{2} \qquad (8)$$

$$v_6 = f_{61}u_h - f_{63}u_l = m_6 \frac{u_{dc}}{2} \qquad (9)$$

A global modulation index $m_x$ can also be defined.
The inductive filters are modelled by combining the self-inductance L and the mutual inductance M. The inductance current $i_x$ is thus dependant on the modulated voltage $v_x$ and the output voltages, $v_A$ and $v_B$:

$$v_1 - v_A = L \frac{di_1}{dt} + M \frac{di_2}{dt} + N \frac{di_3}{dt} + ri_1 \quad (10)$$

$$v_2 - v_A = M \frac{di_1}{dt} + L \frac{di_2}{dt} + M \frac{di_3}{dt} + ri_2 \quad (11)$$

$$v_3 - v_A = M \frac{di_1}{dt} + M \frac{di_2}{dt} + L \frac{di_3}{dt} + ri_3 \quad (12)$$

$$v_4 - v_B = L \frac{di_4}{dt} + M \frac{di_5}{dt} + N \frac{di_6}{dt} + ri_4 \quad (13)$$

$$v_5 - v_B = M \frac{di_4}{dt} + L \frac{di_5}{dt} + M \frac{di_6}{dt} + ri_5 \quad (14)$$

$$v_6 - v_B = M \frac{di_4}{dt} + M \frac{di_5}{dt} + L \frac{di_6}{dt} + ri_6 \quad (15)$$

The total voltage $u_f$ can thus be obtained.

$$v_A - v_B = u_f \quad (16)$$

Moreover, the filter current $i_f$ is a combination of the currents in the inductances:

$$i_f = i_1 + i_2 + i_3 = -i_4 - i_5 - i_6 \quad (17)$$

Adding (10), (11), (12) and replacing the currents by $i_f$ using (17), a new differential equation is obtained:

$$v_1 + v_2 + v_3 - 3v_A = (L + 2M) \frac{di_f}{dt} + ri_f \quad (18)$$

A second one is obtained in the same way from (13), (14), (15) and (17):

$$v_4 + v_5 + v_6 - 3v_B = -(L + 2M) \frac{di_f}{dt} - ri_f \quad (19)$$

New equivalent voltages are defined:

$$V^+ = (v_1 + v_2 + v_3)/3 \quad (20)$$

$$V^- = (v_4 + v_5 + v_6) \quad (21)$$

$$u = V^+ - V^- \quad (22)$$

$$v_1 + v_2 + v_3 - v_4 - v_5 - v_6 - 3u_f = 2(L + 2M) \frac{di_f}{dt} + 2ri_f \quad (23)$$

That can be rewritten using (20), (21), (22)

$$u - u_f = \frac{2}{3}(L + 2M) \frac{di_f}{dt} + \frac{2}{3}ri_f \quad (24)$$

The circulating currents in the interleaved inductance are defined:

$$j_{12} = i_1 - i_2 \quad (25)$$

$$j_{13} = i_1 - i_3 \quad (26)$$

$$j_{45} = i_4 - i_5 \quad (27)$$

$$j_{46} = i_4 - i_6 \quad (28)$$

These are due to the interference between the modulated voltages:

$$u_{12} = v_1 - v_2 = (L - M) \frac{dj_{12}}{dt} + rj_{12} \quad (29)$$

$$u_{13} = v_1 - v_3 = (L - M) \frac{dj_{13}}{dt} + rj_{13} \quad (30)$$

$$u_{45} = v_4 - v_5 = (L - M) \frac{dj_{45}}{dt} + rj_{45} \quad (31)$$

$$u_{46} = v_4 - v_6 = (L - M) \frac{dj_{46}}{dt} + rj_{46} \quad (32)$$

In this new model, the initial inductance currents are replaced by 5 state variables, the circulating currents, $j_{12}$, $j_{13}$, $j_{45}$, $j_{46}$, and the filter current $i_f$ (eq. 24 and 29 to 32). This equivalent model is of interest for the control to impose the right filter current while reducing the circulating current.

The filter capacitor is modelled by an ideal capacitors $C_{f1}$.

$$i_f - i_{int} = C_{f1} \frac{du_f}{dt} \quad (33)$$

The magnet is modelled as a pure inductance $L_m$ supplied by the filter voltage $u_f$ and generating the magnet current $i_m$.

$$u_f = L_m \frac{di_m}{dt} + R_m i_m \quad (34)$$

## II.2. EMR of the converter

The EMR formalism allows to organize the model according to the principle of interaction and causality. By using EMR it is possible to deduce a systemic control organization based on the principle of inversion [5].

The DC bus is a source element (green oval, Figure 4). The voltage divider is a coupling element (overlapped orange square).

The NPC converter is a conversion element (orange square) with the modulation vector $\underline{m}=[m_1, m_2, m_3, m_4, m_5, m_6]^t$ as tuning vector to impose the modulated voltage vector $\underline{v}=[v_1, v_2, v_3, v_4, v_5, v_6]^t$. This vector leads to the inductance current vector $\underline{i}=[i_1, i_2, i_3, i_4, i_5, i_6]^t$ as a reaction. The decomposition of these currents in the filter and circulation currents is another coupling elements (overlapped orange square). Equations (29) to (32) are an accumulation element (crossed orange rectangle) of dimension 4 with the voltage vector $\underline{u}=[u_{12}, u_{13}, u_{45}, u_{56}]^t$ as input and the
circulating current vector j=[j12, j13, j45, j56]t as output (state variables). Equation (24) is depicted by another accumulation element with u and u_f as inputs and i_f as output (state variable).

The filter capacitor is another accumulation elements with u_f as output.

The magnet current is a last accumulation element with the magnet current i_m as output.

The EMR of POPS-B (Figure 4) highlights the filter current to control the magnet current. Thanks to the proposed decomposition a decoupling is obtained between this filter current and the circulating currents. A decoupled control will thus be possible.

<figure><img src="image_5.png" /><figcaption>Figure 4: EMR of the POPS-B power converter</figcaption></figure>

## II.3. Validation of the converter model

A simulation of the power converter was performed in MATLAB-Simulink© using an EMR library. To validate the model, a step of 0.4 was applied to the modulation vectors shown in Figure 5.a. The simulated current error between the present POPS-B controller and the new controller is approximately 0.1 A for a 13.5 kA output (Figure 5.b). The difference in circulating current is slightly larger, with an error of 1 A for a 100 A output (Figure 5.c).

# III. Inversion-based control of the converter

## III.1. Maximal Control structure

The analysis of the EMR allows the deduction of a control scheme using the inversion principle. A tuning path is first defined from the tuning vector m to the current magnet i_m (Figure 6).

<figure><img src="image_6.png" /><figcaption>Figure 6: Tuning path of POPS-B</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5: Comparison of the EMR-Based simulation and the CERN results a) current profile, b) error in the magnet current, c) error in the circulating current j12.</figcaption></figure>

The EMR elements are inverted along this tuning path. Accumulation elements are inverted using a closed-loop control. Other elements are directly inverted.

The inversion of the magnet (34) cannot be performed directly without the use of derivative relations. It is forbidden by the EMR formalism because of its non-causal aspect. Thus, a controller (closed-loop) needs to be used to invert the relation between i_m and u_f-ref:

$$u_{f\_ref} = C_1(i_{m\_ref} - i_{m\_meas}) \quad (35)$$

With C_l(-) a controller, which can be P or PI or other type. The inversion of (33) requires another control loop of u_f and a compensation of i_m, to define the reference i_f-ref:

$$i_{f\_ref} = C_2(u_{f\_ref} - u_{f\_meas}) + i_{m\_meas} \quad (36)$$

With C_2(-) another controller. The inversion of (24) also requires a control loop of i_f and a compensation of u_f, to define the reference u_ref:

$$u_{ref} = C_3(i_{f\_ref} - i_{f\_meas}) + u_{f\_meas} \quad (37)$$

With C_3(-) another controller. The inversions of (29) to (32) requires a last control loop of i_jxy to define the reference u_{xy-ref}:
<figure><img src="image_8.png" /><figcaption>Figure 7: Inversion-based control of POPS-B</figcaption></figure>

$$u_{12-ref} = C_{12}(j_{12-ref} - j_{12-meas}) \quad (38)$$

$$u_{13-ref} = C_{13}(j_{13-ref} - j_{13-meas}) \quad (39)$$

$$u_{45-ref} = C_{45}(j_{45-ref} - j_{45-meas}) \quad (40)$$

$$u_{46-ref} = C_{46}(j_{46-ref} - j_{46-meas}) \quad (41)$$

The inversion of (22) is a coupling inversion. Since two outputs must be deduced from a single input, a distribution input named $k_{dv}$ must be deduced for the coupling. This input defines the sharing between the modules.

$$V_{ref}^{+} = k_{dv} u_{ref} \quad (42)$$

$$V_{ref}^{-} = (k_{dv} - 1) u_{ref} \quad (43)$$

This distribution input is defined by a strategy. From the references of $V^+$, $V$ and the of the modulated voltages, the leg voltage references are computed:

$$v_{1ref} = V_{ref}^{+} + (u_{12ref} + u_{13ref})/3 \quad (44)$$

$$v_{2ref} = V_{ref}^{+} + u_{13ref}/3 - 2u_{12ref}/3 \quad (45)$$

$$v_{3ref} = V_{ref}^{+} + u_{12ref}/3 - 2u_{13ref}/3 \quad (46)$$

$$v_{4ref} = V_{ref}^{-} + (u_{45ref} + u_{46ref})/3 \quad (47)$$

$$v_{5ref} = V_{ref}^{-} + u_{46ref}/3 - 2u_{45ref}/3 \quad (48)$$

$$v_{6ref} = V_{ref}^{-} + u_{45ref}/3 - 2u_{46ref}/3 \quad (49)$$

The last inversions of (4) to (9) are direct and lead to the reference of the modulation functions from the modulated voltage references:

$$m_{1\_ref} = \frac{2v_{1\_ref}}{u_{dc\_meas}} \quad (50)$$

$$m_{2\_ref} = \frac{2v_{2\_ref}}{u_{dc\_meas}} \quad (51)$$

$$m_{3\_ref} = \frac{2v_{3\_ref}}{u_{dc\_meas}} \quad (52)$$

$$m_{4\_ref} = \frac{2v_{4\_ref}}{u_{dc\_meas}} \quad (53)$$

$$m_{5\_ref} = \frac{2v_{5\_ref}}{u_{dc\_meas}} \quad (54)$$

$$m_{6\_ref} = \frac{2v_{6\_ref}}{u_{dc\_meas}} \quad (55)$$

To go from the modulated vectors $\underline{m}$ to $f_{xy}$ function, each of the modulated voltages are compared to 2 triangular signals: one going from 0 to 1 and the other going from -1 to 0. The triangular signals are shifted by 60° for each modulated vector to ensure an output of 2000Hz. Thus, when $m_{x\_ref}$ is above the positive triangle $f_{x1}$ is equal to 1, when $m_{x\_ref}$ is between the positive triangle and the negative one, it is equal to 0. When $m_{x\_ref}$ is under the negative triangular signal, $f_{x1}$ is equal to -1.

With those equations it is possible to build the inversion-based control structure (Figure 7) with
blue parallelograms as control operations (crossed pictograms indicates a control loop).

### III.2. Strategy for reduction of the circulating currents

A strategy block has to provide the distribution input and the references for the circulating currents.

In this study, the voltage divider is considered as perfect with a strict balance of the voltage. That can be expressed by $V^+ = -V^-$ and

$$k_{dv} = 0.5 \qquad (56)$$

If there is some imbalance, it can be solved by another value of the distribution input $k_{dv}$.

To reduce the circulating currents in the interleaved inductances, these currents must be imposed to zeros.

$$j_{12\_ref} = 0 \qquad (57)$$

$$j_{13\_ref} = 0 \qquad (58)$$

$$j_{45\_ref} = 0 \qquad (59)$$

$$j_{46\_ref} = 0 \qquad (60)$$

As they are now independent of the filter current $i_f$, it is possible to impose these zero values by their controller, (38) to (41).

### III.3. Simulation of the inversion-based control

The inversion-based control is simulated using MATLAB-Simulink©. The ripple of the measurement of the magnet current are also introduced. The reference current is well tracked (Figure 8.a) and the current error is lower than the requested accuracy of 50 ppm, if outside the transient and if the ripple is neglected (Figure 8.b). All the circulating currents are cancelled despite some transients (Figure 8.c).

## IV. Performances of the inversion-based control

### IV.1. Comparison with the initial control

In the initial control, the circulating currents are compensated using the measurement of 6 currents

in the interleaved inductances. The voltages references are compensated by the following equations:

$$\begin{bmatrix} m_{1\_ref} \\ m_{2\_ref} \\ m_{3\_ref} \end{bmatrix} = \left(\frac{2}{u_{dc\_meas}}\right) \left(\frac{u_{ref} - \frac{333.3}{2+3}}{2} \begin{bmatrix} 2L - 2M & M - L & M - L \\ M - L & 2L - 2M & M - L \\ M - L & M - L & 2L - 2M \end{bmatrix} \begin{bmatrix} i_1 \\ i_2 \\ i_3 \end{bmatrix} \right) \qquad (61)$$

$$\begin{bmatrix} m_{4\_ref} \\ m_{5\_ref} \\ m_{6\_ref} \end{bmatrix} = \left(\frac{2}{u_{dc\_meas}}\right) \left(\frac{u_{ref} - \frac{333.3}{2+3}}{2} \begin{bmatrix} 2L - 2M & M - L & M - L \\ M - L & 2L - 2M & M - L \\ M - L & M - L & 2L - 2M \end{bmatrix} \begin{bmatrix} i_4 \\ i_5 \\ i_6 \end{bmatrix} \right) \qquad (62)$$

For the magnet current profile (Figure 9.a), the current error is still under the accuracy limit (Figure 9.b) despite some more important transients. It can be observed that the circulating currents are not really cancelled during this operation (Figure 9.c). If the global accuracy is not impacted for the magnet current, this effect leads to more energy consumption due to the losses of the circulating currents. As POPS-B is in operation 24h a day and 7 days a week, the new inversion-based control can significantly reduce the energy consumption.

## IV.2. Robustness of the inversion-based control

In order to test the robustness of the inversion-based control, the parameters of the coupled inductances are changed: variation of +20% of $L$, +20% of $M$ and +25% of $r$. Both controls still use the initial parameters to tune the controllers. The error of the magnet current in the initial control (Figure 10.a) is higher than the one of the inversion-based control (Figure 10.c). The circulating currents of the initial control (Figure 10.b) are also higher than the ones of the inversion-based control (Figure 10.d). The inversion-based control is thus more robust to parameter variation than the initial control.

A second test is focused on the ripple due to the measurements. A new value is considered at 750 Hz. The error in the current magnet and the circulating currents are plotted (Figure 11). The results of the initial control (Figure 11.a and Figure 11.b) are more affected than the results of the inversion-based control (Figure 11.c and Figure 11.d).
<figure><img src="image_9.png" /><figcaption>Figure 8: Simulation of the inversion-based control a) magnet current, b) error in current $i_m$, c) the 4 circulating currents j.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 9: Simulation of the initial control a) magnet current, b) error in current $i_m$, c) the 4 circulating currents</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 10: Simulation of the robustness test a) current error for the initial control, b) circulating current for the initial control, c) error for the inversion-based control, d) circulating currents for the inversion-based control</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 11: Simulation of the ripple test for 750 Hz a) current error for the initial control, b) circulating current for the initial control, c) error for the inversion-based control, d) circulating currents for the inversion-based control</figcaption></figure>
Conclusion

The EMR of the POPS-B DC/DC power
converter has been developed by decomposing
every energy storage element, including the
inductive filter and its circulating currents. This
decomposition enables accurate control of the
magnet current while imposing zero circulating
current.

An inversion-based control is obtained from the
inversion of the EMR. According to simulation,
this control achieves a magnet current regulation
accuracy of 50 ppm. The cancellation of the
circulating currents enables such high accuracy.
Simulation further indicates that the approach
provides increased robustness to other types of
disturbances, albeit at the cost of additional
computational power due to the inclusion of five
more controllers.

The next steps include testing the strategy
presented in this paper against the typical
approach of average current-sharing control,
utilizing a rapid control prototyping platform
coupled with a 1/100 down-scaled model of the
POPS-B power converter.

References

[1]. https://home.cern/about/who-we-are/our-mission,
Website visited in 2024.

[2]. R. Chen et al., "Coupled Inductor Design for Interleaved Three-level Active Neutral Point Clamped Inverters Considering EMI Noise Reduction," 2019 IEEE Applied Power Electronics Conference and Exposition (APEC), Anaheim, CA, USA, 2019, pp. 257-264, doi: 10.1109/APEC.2019.8721817.

[3]. S. Rocca, F. Boattini and L. M. de Paco Soto, "Development of a control strategy for the 18 MW power converter of the CERN PS Booster accelerator," 2018 IEEE 19th Workshop on Control and Modeling for Power Electronics (COMPEL), Padua, Italy, 2018, pp. 1-8, doi: 10.1109/COMPEL.2018.8460164.

[4]. Y. Yamamoto and T. Mannen, "Three-Phase Unbalance Compensator Using Neutral Point Clamped Multilevel Interleaved Inverter," 2018 IEEE 4th Southern Power Electronics Conference (SPEC), Singapore, 2018, pp. 1-7, doi: 10.1109/SPEC.2018.8636092.

[5]. L. Wu, H. Lin, L. Zhao and G. Chen, "Analysis of DC-link Current of Parallel Three-phase Inverters under Unbalanced Output Currents," IECON 2019 - 45th Annual Conference of the IEEE Industrial Electronics Society, Lisbon, Portugal, 2019, pp. 3381-3386, doi: 10.1109/IECON.2019.8927043.

[6]. D. Jovanović, H. Pezeshki, M. A. H. Broadmeadow, G. R. Walker and G. F. Ledwich, "Current-balancing technique for paralleled interleaved inverters with magnetically coupled inductors," 2017 IEEE 12th International Conference on Power Electronics and Drive Systems (PEDS), Honolulu, HI, USA, 2017, pp. 789-794, doi: 10.1109/PEDS.2017.8289224.

[7]. G. J. Capella, J. Pou, S. Ceballos, J. Zaragoza and V. G. Agelidis, "Current-Balancing Technique for Interleaved Voltage Source Inverters With Magnetically Coupled Legs Connected in Parallel," in IEEE Transactions on Industrial Electronics, vol. 62, no. 3, pp. 1335-1344, March 2015, doi: 10.1109/TIE.2014.2345345.

[8]. A. Bouscayrol, B. Lemaire-Semail, "Energetic Macroscopic Representation and Inversion-Based Control ", Encyclopedia of electrical and electronic power engineering, Vol. 3, pp 365-375, Elsevier, DOI : 10.1016/B978-0-12-821204-2.00117-3, ISBN : 978-0-12-823211-8, 2023.

[9]. L. Horrein and J.-M. Cravero, "Hybrid capacitor discharge/switch-mode converter for pulsed applications: Topology and control design," 2017 19th European Conference on Power Electronics and Applications (EPE'17 ECCE Europe), Warsaw, Poland, 2017, pp. P.1-P.10, doi: 10.23919/EPE17ECCEEurope.2017.8099101.

[10]. L. Horrein, J. M. Cravero, P. Delarue, A. Bouscayrol, D. Aguglia, C. Ortega-Perez, “Dead-Time influence on fast switching pulsed power converters design – A high current application for accelerator’s magnets”, EPE’20 ECCE Europe, Lyon (France), September 2020.

[11]. T. Peng et al., "A Uniform Modeling Method Based on Open-Circuit Faults Analysis for NPC-Three-Level Converter," in *IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 66, no. 3, pp. 457-461, March 2019, doi: 10.1109/TCSII.2018.2856862.

W. Li, J. Han, L. Zhou, C. Deng and M. Yan, "A Novel SVPWM Method for NPC three-level interleaved H-bridge inverter," 2020 IEEE Energy Conversion Congress and Exposition (ECCE), Detroit, MI, USA, 2020, pp. 2606-2611, doi: 10.1109/ECCE44975.2020.9236112