# Methodology for pre-sizing an MMC by optimization for an aeronautical application

Stephane Ngo Bui Hung, Cyrille Gautier, Pniak Lucas, Loïc Quéval, Bertrand Revol, Eric Labouré

► To cite this version:

Stephane Ngo Bui Hung, Cyrille Gautier, Pniak Lucas, Loïc Quéval, Bertrand Revol, et al. Methodology for pre-sizing an MMC by optimization for an aeronautical application. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0127. hal-05022968

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<figure><img src="image_1.png" /></figure>
# Methodology for pre-sizing an MMC by optimization for an aeronautical application

Stéphane NGO BUI HUNG¹⁻², Cyrille GAUTIER¹, Lucas PNIAK¹, Loïc QUÉVAL²,
Bertrand REVOL¹, Éric LABOURÉ²

(1): Pôles Systèmes Électriques et Électroniques - Safran Tech, Châteaufort, France Rue des jeunes Bois, 78117 Châteaufort

(2): Université Paris-Saclay, CentraleSupélec, CNRS, Laboratoire de Génie Électrique et Électronique de Paris, 91192, Gif-sur-Yvette, France.

Sorbonne Université, CNRS, Laboratoire de Génie Électrique et Électronique de Paris, 75252,
Paris, France 11 Rue Joliot Curie, 91192 Gif-sur-Yvette

E-Mail: stephane.ngobuihung@centralesupelec.fr

## Keywords

«Modular Multilevel Converters (MMC)»,
«Aerospace», «DC-AC converter», «Active
rectifier», «Multi-objective optimization».
$S_1$

## Abstract

This article presents the pre-sizing of $L_{arm}$, $C_{eq}$ components of a 150 kW MMC converter in steady state used as an active rectifier, associated with a variable frequency PMSM generator, and designed to generate an 800 VDC aircraft electrical network. It details the simulation and modelling used and the optimisation process implemented.

## Introduction

Due to the increasing number of on-board electrical systems in the design of modern aircraft, the growing demand of electrical power for aircraft electrical networks, exceeding 500 kW on programs such as the B787 and A350, presents a significant challenge. Currently, voltages are limited to 230 $V_{ac}$ or 270 $V_{dc}$ in some military aircrafts. To reduce cable weight, this voltage could be increased to 540 $V_{dc}$, or even 800 $V_{dc}$ in the future, particularly in electrical networks dedicated to propulsion systems.

In this study, we examine a specific case where the aircraft's on-board network generator is a permanent magnet synchronous machine (PMSM) driven by the turbofan, as shown in Fig. 1.

<figure><img src="image_2.png" /><figcaption>Fig. 1: On-board network electrical generation diagram</figcaption></figure>

The project specification requires the turbofan to drive a variable-speed electrical machine to generate three-phase power. This power must then be transferred to the DC bus via a controlled AC/DC converter. The converter must be capable of transferring a nominal 150 kW of three-phase power at variable voltage and frequency while maintaining a steady 800V on the DC bus. The two-level active rectifier is commonly used for AC/DC conversion due to its simplicity and technological maturity. In this application, it is essential to use transistors rated as voltages exceeding the DC bus voltage. To ensure an acceptable level of reliability, the power components selected must have a voltage rating much higher than 800V such as 1200V or 1700V. However, as the blocking voltage increases, both switching and conduction losses also increase for a given voltage and switching current [2][3]. Therefore, to maintain good efficiency, it is necessary to limit the switching frequency [1]. Additionally, the system is designed for use in an aircraft, where low atmospheric pressure and high voltage conditions can lead to partial discharge. This phenomenon is particularly prominent when a 2-level active rectifier is used in combination with a low-inductive machine and cables.
<figure><img src="image_3.png" /><figcaption>Fig. 2: Voltage at the end of an aircraft cable</figcaption></figure>

As shown in Fig. 2, experimental results with a 2-level inverter and an aircraft cable demonstrate significant overshoot, approaching 100%, due to high ripple.

So, in order to reduce stress on the electrical machine, such as overvoltage's caused by high dv/dt, a sine or anti-dv/dt filter is required [4]. These filters are typically large and heavy. An alternative solution is to use multi-level topologies that is investigated, which reduce the amplitude of voltage steps and thereby lower filtering requirements. Additionally, some of these topologies allow for the use of semiconductors with lower voltage ratings, which in turn reduces switching losses [2]. Multi-level topologies that comply with aeronautical constraints and enhance power density have been explored in the literature, such as the parallel multi-cell inverter [3]. Additionally, the increase in frequency in these topologies, combined with the use of low-loss semiconductors enables an improvement in power density [5]. The active rectifier used in this study must comply with aeronautical constraints, including voltage ripple on the DC bus, harmonic distortion, overvoltage protection, electromagnetic compatibility (EMC), temperature limits, etc.

Consequently, a multilevel topology is investigated, and in particular the modular multilevel converter (MMC), in order to determine whether there are any advantages to its use in the future aeronautical power system. This structure has the advantage of achieving the desired voltage levels by duplicating identical modules, distinguishing it from other multi-level topologies. It is already an established industrial solution in electrical networks. However, unlike its application in terrestrial networks with fixed frequency and

voltage, the challenge here is to ensure that this converter operates effectively over variable AC frequency and voltage ranges. Additionally, it involves integrating low-voltage GaN semiconductors (with a voltage rating below 200V) and PCB technology, aiming to achieve improvements in power density [5].

This article focuses on the pre-sizing of a converter designed to support an optimization algorithm for steady-state operation. First, an average mathematical model is developed for use in the optimization algorithm to compute the optimum inductance-capacitance Pareto front for various operating points. Next, both switching and average simulation models are presented for time-domain simulations. Finally, after selecting the values of inductances and capacitors through optimization, the results are validated by comparing them with time-domain simulations.

## MMC analytical model

The analytical model is developed based on the assumptions of the average model, with the presumption that the oscillations at 3ω and 6ω are adequately filtered [10]. Consequently, these terms are excluded from the results.

## Description of the converter

Fig. 4 illustrates the electromechanical conversion chain. A half-bridge (HB) MMC connects the PMSM to the 800 V DC bus. The converter must transmit a maximum power of 150 kW, over a frequency range [500 Hz; 1500 Hz], with the machine operating at a line-to-neutral voltage $e_{ac}$ in the range [153 Vpk; 460 Vpk]. The machine is controlled to get maximum active power transfer and $V_{AN} \le 400 V_{pk}$.

Fig. 3 shows the electrical diagram of an HB module and the current pathing through the module denoted as $i_M$. The state of the switch $\{0,1\}$ is called $s$:

- $s = 0$, the capacitor is not inserted.

- $s = 1$, the capacitor is inserted.
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3: Status of an HB module. In red, the current path $i_M$</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4: (a) Schematic of the conversion chain (b) Average model of half bridges in one arm.</figcaption></figure>

## Notation definition

Let $x_{(u,l),p}^{(k)}$ be a variable. The indexes $u$ and $l$ stand for "upper" and "lower". The index $p$ refers to the phase (a, b or c) and the index $k$ refers to the $k^{th}$ module. It is assumed that the converter has $N_m$ modules per arm.

## Equivalent module model

The voltage across the $k^{th}$ module and the current through the capacitor are:

$$v_{M,(u,l),p}^{(k)} = s_{(u,l),p}^{(k)} \times v_{C,(u,l),p}^{(k)} \quad (1)$$

$$i_{C,(u,l),p}^{(k)} = s_{(u,l),p}^{(k)} \times i_{(u,l),p} \quad (2)$$

The dynamic behavior of the capacitors are:

$$C \frac{dv_{C,(u,l),p}^{(k)}}{dt} = s_{(u,l),p}^{(k)} \times i_{(u,l),p} \quad (3)$$

The voltage of the capacitors on each arm is assumed to be balanced and is equal to:

$$v_{C,(u,l),p}^{(k)} = \frac{v_{C_{eq},(u,l),p}}{N_m} \quad (4)$$

Summing the voltages from the $k$ modules, as described by equation (1) and incorporating the relation from equation (4), results in:

$$\begin{aligned} v_{M,(u,l),p} &= \frac{v_{C_{eq},(u,l),p}}{N_m} \sum_{k}^{N_m} s_{(u,l),p}^{(k)} && (5) \\ &= m_{(u,l),p} \times v_{C_{eq},(u,l),p} \end{aligned}$$

Thus, the modulation function $m$ is defined by:

$$m_{(u,l),p} = \frac{1}{N_m} \sum_{k}^{N_m} s_{(u,l),p}^{(k)} \in [0; 1] \quad (6)$$

Summing the $k$ modules equations (3) and incorporating the relation from equation (4) yields:

$$C_{eq} \frac{dv_{C_{eq},(u,l),p}}{dt} = m_{(u,l),p} \times i_{(u,l),p} \quad (7)$$

With $C_{eq} = \frac{C}{N_m}$.

From equations (5) and (7) the equivalent diagram enclosed in the red box in Fig. 4 can be deduced.

## Methodology for obtaining the mathematical model in dq0 space

This section outlines the methodology for deriving the state representation in the Park coordinate system.

### Mathematical model in abc

Assuming that the voltages across the module capacitors are balanced, the modules can be represented by the average model of Fig. 4. The generator is modelled using the Behn Eschenburg model, as illustrated in the blue box. The mathematical model developed is based on research from thesis [8][9]. By applying Kirchoffs' laws to the upper and lower arms of each phase, along with the equations from the average model, we obtain a set of 18 equations. For a machine with smooth and
balanced poles, it follows that $V_{NM} = 0$ at low frequencies.

## Mathematical model in $\Delta\Sigma_{abc}$

According to the article [10], the $\Delta\Sigma$ representation, which utilizes the difference or sum of quantities, simplifies the steady-state analysis. In this representation, the variables in $\Delta$ encompass the fundamental frequency, while the variables $\Sigma$ include the harmonics at $-2\omega$ and the DC component. The method for deriving the variables in the $\Delta\Sigma$ representation is detailed in (8) according to [8].

$$ 
\begin{aligned}
x_{ul} &= [v_{Ml} \ v_{Mu} \ i_l \ i_u \ v_{cl} \ v_{cu}]^T \\
x_{\Delta\Sigma} &= [v_{M\Delta} \ v_{M\Sigma} \ i_{\Delta} \ i_{\Sigma} \ v_{c\Delta} \ v_{c\Sigma}]^T \\
m_{ul} &= [m_l \ m_u]^T \\
m_{\Delta\Sigma} &= [m_{\Delta} \ m_{\Sigma}]^T
\end{aligned}
$$

$$ x_{\Delta\Sigma}(t) = \begin{pmatrix} \frac{1}{2} & -\frac{1}{2} & 0 & 0 & 0 & 0 \\ \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2} & -\frac{1}{2} \\ 0 & 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \end{pmatrix} x_{ul}(t) \quad (8) $$
$$ m_{\Delta\Sigma}(t) = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} m_{ul}(t) $$

$$ 
\left\{ 
\begin{aligned}
L_{\Delta} &= \frac{1}{2}(L_{arm} + 2L_{ac}) \\
R_{\Delta} &= \frac{1}{2}(R_{arm} + 2R_{ac}) \\
L_{\Sigma} &= L_{arm} \\
R_{\Sigma} &= R_{arm}
\end{aligned}
\right. $$

where $L_{arm}$ et $R_{arm}$ represent the arm inductor and equivalent losses in the arm, $L_{ac}$ and $R_{ac}$ represent the inductor and losses in the AC line. Summing and differencing upper and lower equations for each phase (p) and using transformation $\Delta\Sigma$ yields:

$$ A = \begin{pmatrix} -\frac{R_{\Delta}}{L_{\Delta}} & \omega & 0 & 0 & 0 & \frac{2m_{\Delta\Sigma} + m_{\Sigma\Delta}}{4L_{\Delta}} & -\frac{m_{\Sigma\Delta}}{4L_{\Delta}} & 0 & \frac{m_{\Delta\Delta}}{4L_{\Delta}} & -\frac{m_{\Delta\Delta}}{4L_{\Delta}} & \frac{m_{\Delta\Delta}}{2L_{\Delta}} \\ -\omega & -\frac{R_{\Delta}}{L_{\Delta}} & 0 & 0 & 0 & -\frac{m_{\Sigma\Delta}}{4L_{\Delta}} & \frac{2m_{\Sigma\Delta} - m_{\Sigma\Delta}}{4L_{\Delta}} & 0 & -\frac{m_{\Sigma\Delta}}{4L_{\Delta}} & -\frac{m_{\Sigma\Delta}}{4L_{\Delta}} & \frac{m_{\Sigma\Delta}}{2L_{\Delta}} \\ 0 & 0 & -\frac{R_{\Delta}}{L_{\Delta}} & 0 & 0 & 0 & 0 & \frac{m_{\Sigma\Delta}}{2L_{\Delta}} & 0 & 0 & \frac{m_{\Sigma\Delta}}{2L_{\Delta}} \\ 0 & 0 & 0 & -\frac{R_{\Sigma}}{L_{\Sigma}} & -2\omega & 0 & -\frac{m_{\Delta\Delta}}{4L_{\Sigma}} & \frac{m_{\Delta\Delta}}{4L_{\Sigma}} & 0 & -\frac{m_{\Sigma\Delta}}{2L_{\Sigma}} & 0 & -\frac{m_{\Sigma\Delta}}{2L_{\Sigma}} \\ 0 & 0 & 0 & 2\omega & -\frac{R_{\Sigma}}{L_{\Sigma}} & 0 & \frac{m_{\Delta\Delta}}{4L_{\Sigma}} & \frac{m_{\Delta\Delta}}{4L_{\Sigma}} & 0 & 0 & -\frac{m_{\Sigma\Delta}}{2L_{\Sigma}} & -\frac{m_{\Sigma\Delta}}{2L_{\Sigma}} \\ 0 & 0 & 0 & 0 & 0 & -\frac{R_{\Sigma}}{L_{\Sigma}} & -\frac{m_{\Delta\Delta}}{4L_{\Sigma}} & -\frac{m_{\Delta\Delta}}{4L_{\Sigma}} & -\frac{m_{\Delta\Delta}}{2L_{\Sigma}} & -\frac{m_{\Delta\Delta}}{4L_{\Sigma}} & -\frac{m_{\Delta\Delta}}{2L_{\Sigma}} & \frac{m_{\Sigma\Delta}}{2L_{\Sigma}} \\ -\frac{2m_{\Sigma\Delta} + m_{\Delta\Delta}}{8C_{eq}} & \frac{m_{\Sigma\Delta}}{8C_{eq}} & 0 & \frac{m_{\Delta\Delta}}{4C_{eq}} & -\frac{m_{\Delta\Delta}}{4C_{eq}} & \frac{m_{\Delta\Delta}}{2C_{eq}} & 0 & \omega & 0 & 0 & 0 & 0 \\ \frac{m_{\Sigma\Delta}}{8C_{eq}} & \frac{2m_{\Sigma\Delta} - m_{\Delta\Delta}}{8C_{eq}} & 0 & -\frac{m_{\Delta\Delta}}{4C_{eq}} & -\frac{m_{\Delta\Delta}}{4C_{eq}} & \frac{m_{\Delta\Delta}}{2C_{eq}} & -\omega & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -\frac{m_{\Sigma\Delta}}{4C_{eq}} & 0 & 0 & \frac{m_{\Delta\Delta}}{2C_{eq}} & 0 & 0 & 0 & 0 & 0 \\ -\frac{m_{\Delta\Delta}}{8C_{eq}} & \frac{m_{\Delta\Delta}}{8C_{eq}} & 0 & \frac{m_{\Sigma\Delta}}{2C_{eq}} & 0 & \frac{m_{\Sigma\Delta}}{2C_{eq}} & 0 & 0 & 0 & 0 & -2\omega & 0 \\ \frac{m_{\Delta\Delta}}{8C_{eq}} & \frac{m_{\Delta\Delta}}{8C_{eq}} & 0 & 0 & \frac{m_{\Sigma\Delta}}{2C_{eq}} & \frac{m_{\Sigma\Delta}}{2C_{eq}} & 0 & 0 & 0 & 2\omega & 0 & 0 \\ -\frac{m_{\Delta\Delta}}{8C_{eq}} & -\frac{m_{\Delta\Delta}}{8C_{eq}} & -\frac{m_{\Delta\Delta}}{4C_{eq}} & -\frac{m_{\Delta\Delta}}{4C_{eq}} & \frac{m_{\Sigma\Delta}}{4C_{eq}} & \frac{m_{\Sigma\Delta}}{4C_{eq}} & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}; B = \begin{pmatrix} \frac{e_{\Delta}}{L_{\Delta}} & \frac{e_{\Delta}}{L_{\Delta}} & \frac{e_{\Delta}}{L_{\Delta}} & \frac{e_{\Delta}}{L_{\Delta}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} $$

$$ v_{M,\Delta,p} = \frac{1}{2} (m_{\Delta,p} v_{c\eq\Sigma,p} + m_{\Sigma,p} v_{c\eq\Delta,p}) \quad (9) $$

$$ v_{M,\Sigma,p} = \frac{1}{2} (m_{\Delta,p} v_{c\eq\Delta,p} + m_{\Sigma,p} v_{c\eq\Sigma,p}) \quad (10) $$

$$ \frac{di_{\Delta,p}}{dt} = \frac{v_{M,\Delta,p} - R_{\Delta}i_{\Delta,p} - e_p}{L_{\Delta}} \quad (11) $$

$$ \frac{di_{\Sigma,p}}{dt} = \frac{v_{dc} - 2v_{M,\Sigma,p} - 2R_{\Sigma}i_{\Sigma,p}}{2L_{\Sigma}} \quad (12) $$

$$ \frac{dv_{c\eq\Delta,p}}{dt} = \frac{2m_{\Delta,p}i_{\Sigma,p} - m_{\Sigma,p}i_{\Delta,p}}{4C_{eq}} \quad (13) $$

$$ \frac{dv_{c\eq\Sigma,p}}{dt} = \frac{2m_{\Sigma,p}i_{\Sigma,p} - m_{\Delta,p}i_{\Delta,p}}{4C_{eq}} \quad (14) $$

where $v_{c\eq\Delta,p}$ is the equivalent capacitor voltage, $v_{m,\Delta\Sigma}$ is the modulated voltage, $m_{\Delta\Sigma}$ are the modulation functions and $i_{\Delta\Sigma}$ are the currents in the representation in the $\Delta\Sigma$ representation.

## Mathematical model in $\Delta\Sigma_{dqo}$

Using the previous model, we perform the following Park transform:

$$ X_{dqo}(t) = P(\theta)X_{abc}(t) \quad (15) $$

With:

$$ P(\theta) = \begin{pmatrix} \cos(\theta) & \cos\left(\theta - \frac{2\pi}{3}\right) & \cos\left(\theta - \frac{4\pi}{3}\right) \\ -\sin(\theta) & -\sin\left(\theta - \frac{2\pi}{3}\right) & -\sin\left(\theta - \frac{4\pi}{3}\right) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{pmatrix} \quad (16) $$

The following vector is defined:

$$ X_{dqo} \triangleq \begin{bmatrix} i_{\Delta dqo} & v_{c\eq\Delta dqo} & v_{c\eq\Sigma dqo} \end{bmatrix}^T \quad (17) $$

And:

$$ \frac{dX_{dqo}(t)}{dt} = AX_{dqo}(t) + B \quad (18) $$

By applying the transform $P(\theta)$ to $\Delta$ equations and $P(-2\theta)$ to $\Sigma$ equations, and assuming that the oscillations at $3\omega$ and $6\omega$ are well filtered, we obtain the matrices below.
Pre-sizing of MMC passive
components by optimization

This section aims to determine the values of the
arm inductors and module capacitors to meet
the operating points while minimizing their
sizes, thereby maximizing the converter's power
density through optimization. We utilize the
previously developed mathematical model and
analyze the steady-state conditions, defined by
equation (19):

$$
\frac{dX_{dq0}(t)}{dt} = 0 \qquad (19)
$$

<table><caption>Table I : System Parameters</caption>
  <thead>
    <tr>
      <th></th>
      <th>Variable</th>
      <th>Value</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Number of module</td>
      <td>N<sub>m</sub></td>
      <td>6</td>
    </tr>
    <tr>
      <td>Frequency</td>
      <td>f</td>
      <td>[500 - 1500]<br>Hz</td>
    </tr>
    <tr>
      <td>DC grid voltage</td>
      <td>V<sub>dc</sub></td>
      <td>800 V</td>
    </tr>
    <tr>
      <td>AC grid voltage</td>
      <td>e<sub>ac</sub></td>
      <td>[153 - 460] V</td>
    </tr>
    <tr>
      <td>AC Power machine</td>
      <td>P<sub>ac</sub></td>
      <td>[80 - 150] kW</td>
    </tr>
    <tr>
      <td>AC Inductance</td>
      <td>L<sub>ac</sub></td>
      <td>100 &mu;H</td>
    </tr>
    <tr>
      <td>AC Resistance</td>
      <td>R<sub>ac</sub></td>
      <td>10 m&Omega;</td>
    </tr>
    <tr>
      <td>Arm Resistance</td>
      <td>R<sub>arm</sub></td>
      <td>10 m&Omega;</td>
    </tr>
    <tr>
      <td>Modulation limit</td>
      <td>m<sub>max</sub></td>
      <td>1</td>
    </tr>
    <tr>
      <td>Eqv. capacitor<br>voltage ripple</td>
      <td>&Delta;v<sub>c<sub>eq,pp</sub></sub></td>
      <td>200 V</td>
    </tr>
    <tr>
      <td>Maximum<br>distortion harmonic<br>fundamental 2</td>
      <td>DHF<sub>2</sub></td>
      <td>5 %</td>
    </tr>
  </tbody>
</table>

Note that the $DHF_2$ of the current is calculated
as follows:

$$
DHF_2 = 100 \times \frac{I_{h2}}{I_f} \qquad (20)
$$

With $I_f$ and $I_{h2}$ the amplitude of the
fundamental and second harmonic components
of the current in one phase of the MMC. The
number of modules is set at six to accommodate
the use of 200V GaN transistors for the Half
Bridge.

The problem is addressed using the multi-
objective particle swarm optimization
(MOPSO) algorithm. This method is inspired
by the social behavior of swarms [11]. In each
iteration, the particles in the swarm move to
new positions and are evaluated based on the
defined objectives.

The optimization problem is defined as follows:

• Objectives

o Minimize $L_{arm}$

o Minimize $C_{eq}$

Note that the mass and volume of these
components are directly related to their
values. An optimization problem focusing
on minimizing mass will be addressed in
future works.

Parameters

• Parameters

o AC network ($e_{ac}$, $L_{ac}$, $R_{ac}$, $f$)

o DC network ($V_{dc}$)

• Variables

o $C_{eq}$

o $L_{arm}$

o $m_{\Delta\Sigma dq}$

• Constraints

o $P_{ac} < P_i$

o $\Delta v_{c_{eq,pp}} < 200$ V

o $DHF_{2\%} < 5\%$

o $0 \le m_{ul}(t) \le 1$

o $|i_{\Delta d}| \le 10$ A (depend of the operating point)

<figure><img src="image_7.png" /><figcaption>Fig. 5 : Diagram of the optimization principles</figcaption></figure>

Optimization result

A minimum arm inductance of 20 µH is assumed for the MMC to mitigate current ripples associated with the switching frequency on the DC bus. Additionally, values below this threshold values lead the optimization algorithm to converge to pairs of values where the system operates at an unstable equilibrium point.Optimization is conducted at the following operating point:

- $P_i = -100$ kW (active rectifier operation)

- $f = 500$ Hz

- $e_{ac} = 153$ V<sub>pk</sub>
<figure><img src="image_8.png" /><figcaption>Fig. 6: Optimization result. Red point is studied</figcaption></figure>

**Fig. 6** displays the optimization result as a Pareto front for (L,C). It is important to note that the MOPSO algorithm relies on the random selection of particles, which evolve based on their performances relative to the objectives. Consequently, the optimal outcome is not guaranteed in every iteration [10].

According to [12] [13], the equation for $C_{eq}$ and $L_{arm}$ are:

$$C_{eq} = \frac{P_{ac}}{3mV_{c_{eq}}\Delta V_{c_{eq},pp}\omega_{0}\cos(\varphi)} \left(1 - \left(\frac{m\cos(\varphi)}{2}\right)^2\right)^{\frac{3}{2}} \quad (21)$$

$$L_{arm} = \frac{1}{8\omega_0^2 C_{eq} V_{c_{eq}}} \left( \frac{P_{ac}}{3I_{2f}} + V_{dc} \right) \quad (22)$$

Where $\varphi$ is the phase shift between $v_{A'N}$ and $i_{\Delta a}$. These formulas are valid when $m_{\Sigma d,q} = 0$ (no control on current harmonic 2).

For the red point, without controlling the second current harmonic, **Table II** provides a comparison between theoretical calculations and optimization results.

<table><caption>**Table II:** Theory VS optimization $L_{arm}$, $C_{eq}$</caption><thead><tr><th></th><th>Variable</th><th>Value</th></tr></thead><tbody><tr><td>Optimization Capacitor</td><td>$C_{eq, opt}$</td><td>350 μF</td></tr><tr><td>Theoretical Capacitor</td><td>$C_{eq, theo}$</td><td>360 μF</td></tr><tr><td>Optimization Inductance</td><td>$L_{arm, opt}$</td><td>49 μH</td></tr><tr><td>Theoretical Inductance</td><td>$L_{arm, theo}$</td><td>53 μH</td></tr><tr><td>DC grid voltage</td><td>$V_{dc}$</td><td>800 V</td></tr><tr><td>Modulation</td><td>$m$</td><td>0,2</td></tr><tr><td>Current harmonic 2</td><td>$I_{h2}$</td><td>86,2 A</td></tr><tr><td>Mean eqv. capacitor voltage</td><td>$V_{c_{eq}}$</td><td>786 V</td></tr><tr><td>Eqv. capacitor voltage ripple</td><td>$\Delta v_{c_{eq}, pp}$</td><td>240 V</td></tr></tbody></table>

These results demonstrate that theoretical formula align well with the simulation results obtained in open loop using the average model. However, they do not account for the second harmonic injection carried by $m_{\Sigma dq}$ which can impact both $I_{2f}$ and $v_{c_{eq}, pp}$.

In the following section, we selected the red optimization point for validation through simulation. This point serves as an example to demonstrate and verify the effectiveness of our approach.

## Simulation model

Simulation models were established to compare the results from mathematical model with those from the detailed-switching (SW) and average (AVE) time models.

The sinusoidal waveform phase references are determined by the machine's output voltage $e$:

$$e_{abc}(t) \triangleq E \begin{pmatrix} \sin(\omega t) \\ \sin\left(\omega t - \frac{2\pi}{3}\right) \\ \sin\left(\omega t - \frac{4\pi}{3}\right) \end{pmatrix} \quad (23)$$

The simulation model operates in an open-loop configuration and under steady state conditions. The system inputs are the modulation indices, which are defined as follows:

$$\begin{align}
m_{\Delta abc}(t) &\triangleq M_{\Delta} \begin{pmatrix} \sin(\omega t + \varphi_{m\Delta}) \\ \sin\left(\omega t - \frac{2\pi}{3} + \varphi_{m\Delta}\right) \\ \sin\left(\omega t - \frac{4\pi}{3} + \varphi_{m\Delta}\right) \end{pmatrix} \nonumber \\
m_{\Sigma abc}(t) &\triangleq M_{0\Sigma} \begin{pmatrix} \sin(-2\omega t + \varphi_{m\Sigma}) \\ \sin\left(-2\omega t - \frac{2\pi}{3} + \varphi_{m\Sigma}\right) \\ \sin\left(-2\omega t - \frac{4\pi}{3} + \varphi_{m\Sigma}\right) \end{pmatrix} \tag{24}
\end{align}$$

with $M_{0\Sigma} = 1$ to center the modulation in abc frame.

## SW model

The HB MMC block is employed in each arm, with the assumption that the switches are ideal. To maintain capacitors balanced, a balancing block is incorporated according to the algorithm outlined in [9]. The PWM technique used for the converter is a Phase-Shifted PWM (PSPWM) [6].

<figure><img src="image_9.png" /><figcaption>Fig. 7: Schematic diagram of the PSPWM applied to the MMC</figcaption></figure>
Fig. 7 displays the schematic diagram of the SW model. The carrier frequency is set at 50 kHz. In this diagram, $m_{ul}(t)$ represents the modulation signal for the upper and lower arms, $N_{ul}$ denotes the number of modules switched on per arm, $v_{c,ul}(t)$ is the voltage across the capacitors in each module, $i_{ul}(t)$ is the current flowing through the arms and $g_{ul1...N}$ are the switch control commands for each module.

<figure><img src="image_10.png" /><figcaption>Fig. 8: Simulink® MMC model</figcaption></figure>

For SW simulation, carrier frequency for the PWM is set to $f_c = 50$ kHz. The sampling frequency is set to $100 \times N_m \times f_{sw}$ and the frequency execution of the balancing algorithm is set to $N_m \times f_c$.

## AVE Model

To achieve reasonable simulation times, the average model is employed. The average model defined in Fig. 4, replaces the module block in the SW model. The simulation time ratio is 1:20 resulting in a simulation time of approximately 1 minute per operating point.

Upon completion of this modeling phase, we obtain a model that enables rapid determination of operating points as a function of the modulation indices.

For the AVE simulation, the sampling frequency is set to 1000 times the EMF frequency of the machine ($1000 \times f_c$).

## Validation

In this section, we compare the results of the three models for the red optimization point highlighted in Fig. 6.

<table><caption>Table III: values for the optimized red point</caption><thead><tr><td>Parameter</td><td>Variable</td><td>Value</td></tr></thead><tbody><tr><td>Modulation Δ</td><td>$m_{\Delta dq0}$</td><td>[-0.306; -0.271; 0]</td></tr><tr><td>Modulation Σ</td><td>$m_{\Sigma dq0}$</td><td>[-0.036; 0.0175; 1]</td></tr><tr><td>Capacitance</td><td>$C_{eq}$</td><td>350 μF</td></tr><tr><td>Arm inductance</td><td>$L_{arm}$</td><td>49,1 μH</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>Fig. 9: Arm current and capacitor voltage – phase a. Red: Math, Green: AVE, Blue: SW</figcaption></figure>

According to Fig. 9, the mathematical model provides estimates that are closed to those of both the AVE and SW models for the current in the arm and the capacitor voltages. However, the comparison reveals a discrepancy in the DHF. This discrepancy may indicate that the assumption regarding the filtering of the 3rd and 6th harmonics, as illustrated in Fig. 10, for the circulating current in the axes $d$, is not fully valid for the given arm inductance value. The error in the mean value between the “math” and “ave” model is due to the invalidity of the hypothesis about harmonics.

<figure><img src="image_12.png" /><figcaption>Fig. 10: Current $i_{\Sigma d}$. Existence of 6th harmonics</figcaption></figure>

The following table shows the difference between the mathematical model and the average model for DC currents, $I_1$ and $I_2$.
<table><caption>**Table IV :** Error between mathematic model and average model simulation on current</caption><thead><tr><th>L<sub>arm</sub> [µH]</th><th>I<sub>dc</sub> %</th><th>I<sub>f</sub> %</th><th>I<sub>h2</sub> %</th></tr></thead><tbody><tr><td>30</td><td>0,29</td><td>0,16</td><td>179</td></tr><tr><td>40</td><td>0,33</td><td>0,101</td><td>63</td></tr><tr><td>50</td><td>0,31</td><td>0,04</td><td>17</td></tr><tr><td>60</td><td>0,13</td><td>0,01</td><td>7</td></tr><tr><td>70</td><td>0,47</td><td>0,05</td><td>13</td></tr></tbody></table>

<table><caption>**Table V :** $HDF_2$ of average model and mathematic model</caption><thead><tr><th>L<sub>arm</sub> [µH]</th><th>HDF<sub>2AVE</sub> %</th><th>HDF<sub>2MATH</sub> %</th></tr></thead><tbody><tr><td>30</td><td>1,48</td><td>1,12</td></tr><tr><td>40</td><td>0,7</td><td>0,65</td></tr><tr><td>50</td><td>0,96</td><td>0,93</td></tr><tr><td>60</td><td>0,8</td><td>0,95</td></tr><tr><td>70</td><td>0,87</td><td>0,76</td></tr></tbody></table>

The following table shows the difference between the mathematical model and the average model for DC currents, $I_1$ and $I_2$.

**Table IV** shows that the mathematical model struggles to estimate the 2nd harmonic accurately and will cause an error in the $HDF_2$ displays in **Table V**. However, it estimates the amplitude of the mean DC current and the fundamental with good accuracy.

<figure><img src="image_13.png" /><figcaption>Fig. 11 : DC Current ($f_{sw}$ = 50 kHz)</figcaption></figure>

Fig. 11, reveals a harmonic at $2\omega$ on the DC side in the SW simulation. Additionally, **Fig. 12** demonstrates that the amplitude of the current ripple is associated with the switching frequency.

<figure><img src="image_14.png" /><figcaption>Fig. 12 : DC current vs frequency carrier</figcaption></figure>

As the switching frequency increases, the current ripples on the DC bus decrease. The ripple frequency is given by $N_m \times f_{dec}$. Therefore, selecting an appropriate number of modules and switching frequency will help in choosing the technologies best suited to the application.

The results show that the mathematical model is effective for making initial selections of values for arm inductance and module capacitance. Additionally, mathematical model in the optimization algorithm facilitated the rapid generation of a Pareto front, typically within about a minute, compared to optimization coupled with simulation. Specifically, an AVE simulation takes approximately one minute, while a SW simulation at a frequency of 50 kHz requires around twenty minutes.

## Conclusion

This paper has presented a method for pre-sizing the arm inductors and capacitors of HB modules for a MMC through optimization. The process used to develop the mathematical model has been detailed, and the MOPSO optimization algorithm was employed to determine the optimum values for minimizing $L_{arm}$ and $C_{eq}$ while adhering to constraints. This approach allows for rapid computation of optimization results, typically within a few minutes.

The optimization results were compared with time simulation results from both, the AVE and SW model, showing that the mathematical model's results are consistent with the simulation outcomes. However, the assumptions made in the mathematical model do not fully account for harmonics in the DC
current, leading to some deviation in the
amplitude of the 2nd harmonic.

Future works will focus on refining the
optimization problem to better address the
objectives of minimizing mass and increasing
power density.

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