# Multi-Objective PSO for the Sizing of a 12-Pulse Diode Rectifier: A Comparative Analysis at Room and Cryogenic Temperatures

Yasmine Baazizi, Ana Luiza Haas Bezerra, Lauro Ferreira, Simon Meunier, Tanguy Phulpin, Loïc Quéval

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Yasmine Baazizi, Ana Luiza Haas Bezerra, Lauro Ferreira, Simon Meunier, Tanguy Phulpin, et al. Multi-Objective PSO for the Sizing of a 12-Pulse Diode Rectifier: A Comparative Analysis at Room and Cryogenic Temperatures. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0206 . hal-05100940

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# Multi-Objective PSO for the Sizing of a 12-Pulse Diode Rectifier:
A Comparative Analysis at Room and Cryogenic Temperatures

Yasmine BAAZIZI<fnref n="1" />,<fnref n="2" />, Ana Luiza Haas Bezerra<fnref n="1" />,<fnref n="2" />, Lauro FERREIRA<fnref n="1" />,<fnref n="2" />, Simon MEUNIER<fnref n="1" />,<fnref n="2" />, Tanguy PHULPIN<fnref n="1" />,<fnref n="2" />, Loïc QUÉVAL<fnref n="1" />,<fnref n="2" />

E-Mail: yasmine.baazizi@centralesupelec.fr

URL: www.geeps.centralesupelec.fr

## Keywords

« Power electronics », « Cryogenics », « 12-pulse diode rectifier », « Optimization », « PSO ».

## Abstract

This paper compares the optimal sizing of a 12-pulse diode rectifier operating at room temperature (293 K) and at cryogenic temperature (77 K). First, a diode database is constructed from diode measurements at both temperatures. Then, using the PSO (Particle Swarm Optimization) algorithm, the converters are sized to maximise efficiency and minimise volume. The results show that the converter designed to operate at 77 K achieves efficiency levels comparable to those at 293 K (without cooling penalty). However, the cryogenic design offers significant volume advantages over the room temperature design, making it particularly beneficial for applications where volume and weight are limited.

## Introduction

Superconductivity has huge potential for a wide range of applications, including wind power generation [1], electric aircraft [2] and maglev transport [3]. In superconducting conversion chain, generators, cables, and motors operate at cryogenic temperatures, while power electronic converters operate at room temperature. The transition from cryogenic to room temperature requires specific interface components that lead to increased complexity, weight, and volume [4]. It could therefore be advantageous to operate power electronics directly at cryogenic temperatures. In addition, cryogenic operation

could reduce the power electronics conduction and switching losses [5].

Laboratory tests have shown that commercially available industrial silicon diodes can withstand temperatures as low as 77 K may perform differently than at room temperature [6]. This complicates the design of power converters. Optimization techniques can overcome this challenge and assist in the sizing of converters.

In this study, we perform the optimal sizing of a 12-pulse diode rectifier first at room temperature (293 K) and then at cryogenic temperature (77 K). To do so, we built a diode database by measuring the characteristics of a large number of diodes at both 293 K and 77 K. Using the PSO (Particle Swarm Optimization) algorithm coupled to the rectifier electrical and thermal models, we obtained the optimal sizing. Then we compare their key performance indicators. We chose to study a 12-pulse diode rectifier because it is common in many high-power applications, particularly due to its robustness and efficiency. To our knowledge, no study has performed the sizing of a converter both at room temperature and cryogenic temperature. First, we describe the 12-pulse diode rectifier. Then, we present the experimental method used to characterize the power diodes at room and cryogenic temperatures, which allowed us to compile the diode database. After that, we model the rectifier and describe the optimization problem. Finally, we discuss the results of the optimization.

## 12 pulse diode rectifier

Assume a 12-pulse diode rectifier connected on the AC side to a transformer, and feeding a resistive load on the DC side (Figure 1). The transformer has a, to achieve the 30° phase shift.

<footnote n="1">¹ Université Paris-Saclay, CentraleSupélec, CNRS, GeePs, 91192, Gif-sur-Yvette, France.</footnote>

<footnote n="2">² Sorbonne Université, CNRS, GeePs, 75252, Paris, France.</footnote>
This configuration offers several additional
advantages over the 6-pulse setup, including
reduced diode currents and lower harmonic
distortion [9].

<figure><img src="image_2.png" /><figcaption>Fig. 1: 12-pulse diode rectifier.</figcaption></figure>

Diode characterization

<figure><img src="image_3.png" /><figcaption>Fig. 2: Forward voltage test.
DUT: device under test.</figcaption></figure>

To build the diode database, we performed
electrical measurements on dozens of different
diodes to obtain their I-V characteristic, at both
room and cryogenic temperatures. This is done by
conducting a 'direct voltage test', as shown in
Fig. 2.

Fig. 3 shows the corresponding experimental
setup for diode testing in an open bath of liquid
nitrogen (LN2) at 77 K. For this test, the diode is
gradually cooled down. The process includes a
20-minute pre-cooling phase with nitrogen gas,
followed by 10 minutes in the liquid nitrogen
bath, after which the direct voltage test is
conducted. Additional details are provided in
previous work [6].

<figure><img src="image_4.png" /><figcaption>Fig. 3: Diode testing in an open bath of LN2.</figcaption></figure>

Fig. 4 compares the I-V characteristics of a few
diodes from our dataset, illustrating that a diode
can behave differently when cooled down at
77 K. For example, the DD89NK diode shows an
increase in threshold voltage at 77 K compared to
293 K, while its internal resistance remains nearly
unchanged. The RBQ30NS100A diode also
exhibits a higher threshold voltage at 77 K but a
lower internal resistance at this temperature. The
VS-E5PX3006LHN3 and the STTH15RQ06DY
diodes exhibit an unusual behavior (see its
characteristics at 77 K for a voltage of ~1.2 V/
2V). This behavior might be due to charge carrier
freezing [10]. We are currently further
investigating this phenomenon.

System modelling

Electrical model

The electrical model of the 12-pulse diode
rectifier is implemented in MATLAB Simulink
using the Simscape toolbox (Fig. 1). We choose a
nominal AC voltage of 50 V RMS, a nominal
frequency of 50 Hz, and a nominal power of
10 kW. The transformer is assumed ideal with a
turns ratio of 1:1. The diode is modelled using its
I-V characteristic tabulated in the ‘Diode’ block
[11], using the experimental data from our
dataset. The operating temperature is set using the
thermal port, with a junction temperature of
293 K for room temperature and 77 K for
cryogenic conditions. With this model, we
consider only conduction losses for diodes while
neglecting switching losses. The outputs of the
electrical model are the AC and DC active power,
the AC and DC current and the DC voltage
<figure><img src="image_5.png" /><figcaption>a) DD89NK</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>b) RBQ30NS100A</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>c) VS-E5PX3006LHN3</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>d) STTH15RQ06DY</figcaption></figure>

<caption>Fig. 4: Measured current-voltage characteristic of several diodes.</caption>

## Thermal model

Assuming that all the diodes in the rectifier are identical, the equivalent thermal circuit is shown in Fig. 5. $P_D$ is the total diode losses [W], J is the junction node, C is the case node, H is the heatsink node and A is the ambient node. $R_{JC}$ is the equivalent junction-case conduction thermal resistance [K/W], $R_{CH}$ is the equivalent case-heatsink conduction thermal resistance [K/W], $R_{HA}$ is the heatsink-ambient convection thermal resistance [K/W]. The input of the thermal model is the total diode losses $P_D$ and the outputs are the node temperatures.

The temperature difference between the junction and the ambient nodes is:

$$T_J - T_A = (R_{JC} + R_{CH} + R_{HA}) \cdot P_D \quad (2)$$

We assume that the conduction thermal resistance between the junction and the case, as well as the one between the case and the heatsink, remain constant regardless of temperature.

In this study, the length of the heatsink, L, is optimised to adjust the heat dissipation of the system and ensure that the junction temperature of the diodes remains below the specified maximum limit. The thermal resistance between the heatsink and the environment, $R_{HA}$, depends directly on the size of the heatsink, and therefore on its length.

## Definition of the optimization problem

### Optimization variables

Table 1 presents the optimization variables. For this study, our database includes the I-V characteristics of 55 different diodes, at both room and cryogenic temperatures. And we consider 80 references of heatsinks available on the market.
<figure><img src="image_9.png" /><figcaption>Fig. 5: Equivalent thermal circuit.</figcaption></figure>

<table><caption>Table 1: Optimization variables</caption><thead><tr><th>Symbol</th><th>Name</th><th>Type</th></tr></thead><tbody><tr><td><i>d</i></td><td>Diode reference</td><td>Discrete</td></tr><tr><td><i>hs</i></td><td>Heatsink reference</td><td>Discrete</td></tr><tr><td><i>L</i></td><td>Heatsink length</td><td>Continuous</td></tr></tbody></table>

## Objective and constraints

The objective functions of the optimization are to maximize the rectifier efficiency $\eta$ (without cooling penalty) and to minimize its volume Vol. We set a first constraint on the junction temperature $T_j$ that must remain below the maximum junction temperature $T_{j,max}$ specified in the diode datasheet. The second constraint is that the surface area $S$ of the heatsink must be large enough to accommodate the 12 diodes. The bi-objective optimization problem can be then formulated as,

$$ \begin{array}{ll} \min_{x} & f_1, f_2 \\ \text{subject to} & (c_1 < 0 \text{ and } c_2 < 0) \end{array} \qquad (3) $$

with,

$$ \begin{array}{ll} f_1 & = -\eta \qquad (4) \\ f_2 & = Vol \qquad (5) \\ c_1 & = T_j(d) - T_{j,max}(d) \qquad (6) \\ c_2 & = S(hs) - 12S(d) \qquad (7) \end{array} $$

## Optimization algorithm

For the optimization, we couple the electrical and thermal models with a Multi-Objective Particle Swarm Optimization (MOPSO) algorithm [12]. It is a widely used algorithm that can be applied to both continuous and discrete optimization problems [7], [8]. The weighting factor is set at 0.8. The social and cognitive learning factors are both set at 1. The size of the swarm is set at 60

particles and 30 iterations. Leaders are selected from an elitist archive.

## Results

The Pareto optimal points are shown in Fig. 6 at room temperature (293 K) and at cryogenic temperature (77 K), and summarized in Table 2. Fig. 7 shows the value of the constraints along the Pareto front. All the constraints are negative, meaning that the constraints are satisfied. On the one hand, the converter designed to operate at 77 K achieves an efficiency similar to that obtained at 293 K, 98.5% < $\eta$ < 99.5%. This shows that, for this case study, operation at cryogenic temperature has no significant impact on efficiency. This seems to contradict previous studies, which have shown better efficiency at lower temperatures [13]. This difference may be explained by the fact that the model does not take into account all the real factors affecting efficiency, such as interconnection losses or other parasitic effects.

On the other hand, the converter designed to operate at 77 K has a significantly reduced volume (reduction of a factor of five compared with its room-temperature equivalent). This is because the operation at 77 K necessitates only a small heat sink to maintain, the junction temperature $T_j$ below $T_{jmax}$ and avoid overheating. This is particularly interesting for applications where volume and weight are major constraints.(e.g., aviation and naval applications). The selection of these diode references for the rectifier at 77 K can be explained by the fact that, although their threshold voltage is higher at 77 K, it remains relatively low (0.6 V), and their internal resistance is much lower at 77 K than at 293 K (behavior similar to RBQ30NS100A in Fig 4 b).

<figure><img src="image_10.png" /><figcaption>Fig. 6: Pareto optimal points at 293 K (in red) and 77 K (in blue).</figcaption></figure>
<table><caption>Table 2: Optimization results</caption><thead><tr><th></th><th>d</th><th>hs</th><th>L [m]</th><th>η</th><th>Vol [L]</th></tr></thead><tbody><tr><td colspan="6" style="text-align: center;">293 K</td></tr><tr><td>①</td><td>MBR4035</td><td>SK 436</td><td>0.16</td><td>99.11%</td><td>0.51</td></tr><tr><td>②</td><td>30FQ030</td><td>SK 436</td><td>0.14</td><td>98.96%</td><td>0.43</td></tr><tr><td>③</td><td>45L120</td><td>SK 436</td><td>0.13</td><td>98.47%</td><td>0.42</td></tr><tr><td colspan="6" style="text-align: center;">77 K</td></tr><tr><td>①</td><td>MBR4035</td><td>SK 595</td><td>0.025</td><td>98.98%</td><td>0.18</td></tr><tr><td>②</td><td>BYS350</td><td>SK 709</td><td>0.028</td><td>98.91%</td><td>0.15</td></tr><tr><td>③</td><td>30FQ030</td><td>SK 410</td><td>0.035</td><td>98.77%</td><td>0.131</td></tr><tr><td>④</td><td>FERD20H60CTS</td><td>KGR 150</td><td>0.025</td><td>98.59</td><td>0.095</td></tr></tbody></table>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 7: Constraints along the Pareto front at 293 K (in red) and 77 K (in blue).</figcaption></figure>

## Conclusion

In this study, we compared the optimal sizing of a 12-pulse diode rectifier operating at room temperature (293 K) to the one of a rectifier operating at cryogenic temperature (77 K). The optimization aimed to maximize the converter efficiency while minimizing its volume, using the following design variables: diode reference, heatsink reference and heatsink length.

Using the PSO (Particle Swarm Optimization) algorithm coupled to the rectifier electrical and thermal models, we obtained the optimal sizing. The results showed that the converter designed for cryogenic operation achieved an efficiency similar to that obtained at room temperature, but with only 1/5th of its volume. This could be particularly interesting for applications where volume and weight are major constraints.

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