# Improved Energy Management Strategy for Minimizing Capacitors Storage in Power Converters for Particle Accelerators

Dimitris Xystras, Ivan Josifovic, Olivier Michels, Gilles Le Godec

► To cite this version:

Dimitris Xystras, Ivan Josifovic, Olivier Michels, Gilles Le Godec. Improved Energy Management Strategy for Minimizing Capacitors Storage in Power Converters for Particle Accelerators. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0255. hal-05087215

HAL Id: hal-05087215

https://utc.hal.science/hal-05087215v1

Submitted on 27 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
# Improved Energy Management Strategy for Minimizing Capacitors Storage in Power Converters for Particle Accelerators

Dimitris Xystras, Ivan Josifovic, Olivier Michels, Gilles Le Godec

European Organization for Nuclear Research (CERN)
Esplanade des Particules 1, 1211 Geneva, Switzerland

E-mail: d.xystras@cern.ch

URL: https://home.cern/

## Keywords

«Energy management», «Capacitors», «Energy storage», «Control strategy», «Power cycling», «Switched-mode power supply», «Cascaded H-Bridge», «Particle accelerators».

## Abstract

This study presents a new energy management strategy for electromagnetic loads supplied by power electronic converters with integrated energy recovery capability. It aims to minimize the energy storage requirements of the power converters, thereby reducing their cost and volume. The study covers a theoretical analysis, verified by simulation and experimental results.

## Introduction

Power conversion with energy recuperation has been used to minimize energy consumption in a wide range of power cycling applications, including railway transportation [1-2], electric vehicles [3-4] and particle accelerators [5-6].

In particle accelerators, power converters supply electromagnets (or simply “magnets”), which define the trajectory of particle beams. Operating often in power cycling mode with a trapezoidal current waveform (Fig. 1), the reactive energy from the magnets is recovered into the energy storage of power converters during ramp-down and subsequently reused during the ramp-up. This energy exchange between magnets and energy banks, minimizes the power intake from the grid and it is executed according to a certain energy management strategy.

In the medium power range (up to 500 kW) and low-voltage applications (up to 1000 V), energy recovery is typically performed with energy banks composed of a number of series and parallel electrolytic capacitors.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Magnet's current and voltage waveforms during a power cycle.</figcaption></figure>

Past research on magnet energy recovery using electrolytic capacitors for energy storage has highlighted two different approaches [5-6].

The first approach involves controlled limitation of input power, while the energy storage provides the remaining energy required by the magnet over a full power cycle, including the flat-top phase. Although its implementation is simple, it is suitable only for short power cycling applications (e.g., flat-top duration of 0.1 s) because, in the case of longer power cycles (e.g., flat-top duration of 10 s), the energy stored by the electrolytic capacitor bank would be completely discharged during the magnet's flat-top phase.

The second approach is the Energy Balance (EB) strategy, where energy is exchanged between the magnet and the energy storage only during the current ramping phases. Although this strategy requires knowledge of the magnet and energy storage parameters ($L_{mag}, C_{st}$), it is an effective solution for energy recovery, as the energy storage remains charged during the long flat-top phases (Fig. 2).

This work introduces the Improved Energy Balance (IEB) strategy, which, compared to the standard EB strategy, allows for significant reduction in the required energy storage capacitors, minimizing cost and volume while maintaining the same energy consumption from the electrical grid.
<figure><img src="image_3.png" /><figcaption>Fig. 2: Current and power waveforms of the magnet, along with the voltage waveform of the energy storage, in the EB strategy.</figcaption></figure>

## POLARIS Power Converter

POLARIS (Fig. 3) is a modular power converter with built-in energy storage that can be configured with up to four power bricks in series or in parallel to serve the needs of high-power magnets. POLARIS will be employed after a renovation program for particle accelerators at CERN to supply 400 magnet circuits, storing a total energy of 20 Megajoules. It consists of a three-phase power transformer for voltage adaptation, followed by a diode rectifier, a two-quadrant (2Q) converter for managing the energy exchange between the magnet and the energy storage capacitors ($C_{st}$), and an H-bridge functioning as the magnet power supply.

The energy storage voltage in the 2Q converter of POLARIS is controlled using cascaded voltage and current loops [7] (Fig. 3), while the energy storage voltage reference ($v_{st,ref}$) is generated by the EB or IEB strategies.

<figure><img src="image_4.png" /><figcaption>Fig. 3: POLARIS power converter topology with cascaded voltage and current control loops.</figcaption></figure>

## Energy Balance Strategy

The voltage reference ($v_{st,ref}$) of the EB strategy is derived from the law of energy conservation between the magnet ($L_{mag}$) and the energy storage capacitors ($C_{st}$) during the ramp-up or ramp-down phases of the magnet current. This derivation results in the block diagram of the EB strategy, as illustrated in Fig. 4 [5-6].

In the EB strategy (Fig. 5), the reactive energy ($E_{mag}$) is exchanged between the magnet and the energy storage during both ramp-up and ramp-down phases, while the resistive loss energy ($E_{loss}$) is constantly supplied by the grid:

$$v_{st,ref}(t) = \sqrt{V_{st,max}^2 - \frac{L_{mag}}{C_{st}} \cdot i_{mag}^2(t)} \quad (1)$$

## Improved Energy Balance Strategy

The Improved Energy Balance (IEB) strategy is based on the principle that part of the energy stored in the magnet can be used to supply the resistive losses of the magnet during ramp-down ($E_{loss}^{down}$), rather than sourcing it from the grid as in the EB strategy (Fig. 5). This means that the amount of energy to be recovered by the electrolytic capacitors ($E_{st}$) is reduced by the energy dissipated in the resistance of the magnet during ramp-down:

$$E_{st} = E_{mag} - E_{loss}^{down} \quad (2)$$

<figure><img src="image_5.png" /><figcaption>Fig. 4: Block diagram of the EB strategy.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5: Power flow from the grid and energy storage to the magnet in the EB strategy.</figcaption></figure>
This energy gain can be exploited either by reducing the number of energy storage capacitors for the same voltage swing or by reducing the voltage swing for a given number of capacitors.

On the other hand, in addition to the resistive losses of the magnet during ramp-up, the grid must supply the same amount of energy dissipated as losses during ramp-down ($E_{loss}^{down}$). Effectively, the IEB strategy redistributes the energy provided by the grid during the current ramping phases, ensuring that the total grid energy contribution over one power cycle remains the same as in the EB strategy.

The additional energy supplied by the grid during ramp-up ($E_{loss}^{down}$) can be provided using different power waveforms, resulting in different variants of the IEB strategy. In this paper, two variations of the IEB strategy are described:

* IEB with Grid Overloading (IEB-GO) with the power flow waveforms illustrated in Fig. 6. The IEB-GO strategy offers simple implementation but features grid power overloading during ramp-up.

* IEB with Peak power Shaving (IEB-PS) with the power flow waveforms illustrated in Fig. 7. The IEB-PS strategy offers a more complex implementation but eliminates grid overloading.

## IEB-GO strategy

The power waveform of the IEB-GO strategy that supplies the additional energy from the grid during ramp-up ($p_{up}^{GO}$) is the same as the power waveform of the resistive loss of the magnet during ramp-down ($p_{loss}^{down}$):

$$p_{up}^{GO}(t) = p_{loss}^{down}(t) = i_{mag}^{2}(t) \cdot R_{mag} \quad (3)$$

The instantaneous additional energy provided by the grid during ramp-up ($e_{up}^{GO}$) is derived by integrating (3), where the absolute value of the magnet current slew rate is defined by SR:

<figure><img src="image_7.png" /><figcaption>Fig. 6: Power flow from the grid and energy storage to the magnet in the IEB-GO strategy.</figcaption></figure>

$$e_{up}^{GO}(t) = e_{loss}^{down}(t) = \int p_{up}^{GO}(t) \cdot dt \quad (4)$$

$$e_{up}^{GO}(t) = e_{loss}^{down}(t) = \frac{1}{3} \cdot \frac{R_{mag}}{SR} \cdot i_{mag}^{3}(t) \quad (5)$$

The energy storage, being fully charged at $E_{st,max}$ at the beginning of ramp-up, will provide the reactive energy of the magnet ($e_{mag}$) reduced by the additional energy provided by the grid ($e_{up}^{GO}$):

$$E_{st,max} - e_{st}(t) = e_{mag}(t) - e_{up}^{GO}(t) \quad (6)$$

The instantaneous energy stored in the magnet and the capacitor storage is defined by the generic equations below. The energy of the capacitor storage reaches its maximum value at the maximum energy storage voltage ($V_{st,max}$):

$$e_{mag}(t) = \frac{1}{2} \cdot L_{mag} \cdot i_{mag}^{2}(t) \quad (7)$$

$$e_{st}(t) = \frac{1}{2} \cdot C_{st} \cdot v_{st}^{2}(t) \quad (8)$$

$$E_{st,max} = \frac{1}{2} \cdot C_{st} \cdot V_{st,max}^{2} \quad (9)$$

The voltage reference of the energy storage for the IEB-GO strategy is derived by replacing (5), (7), (8), and (9) in (6) and is illustrated in the block diagram of Fig. 8:

$$v_{st,ref}^{GO}(t) = \sqrt{V_{st,max}^{2} - \frac{L_{mag}}{C_{st}} \cdot i_{mag}^{2}(t) + \frac{2}{3} \cdot \frac{R_{mag}}{C_{st}} \cdot \frac{i_{mag}^{3}(t)}{SR}} \quad (10)$$

## IEB-PS strategy

The power waveform of the IEB-PS strategy that supplies the additional energy during ramp-up ($p_{up}^{PS}$) can be selected to eliminate the grid overloading at the end of ramp-up, which is created by the IEB-GO strategy (Fig. 6):

$$p_{up}^{PS}(t) = [2 \cdot I_{top} \cdot i_{mag}(t) - 2 \cdot i_{mag}^{2}(t)] \cdot R_{mag} \quad (11)$$

In this case, the power waveform of the additional energy provided by the grid, is not the same as the power waveform of the resistive loss of the magnet during ramp-down.

<figure><img src="image_8.png" /><figcaption>Fig. 7: Power flow from the grid and energy storage to the magnet in the IEB-PS strategy.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 8: Block diagram of the IEB-GO strategy.</figcaption></figure>

Therefore, the voltage reference of the energy storage in IEB-PS strategy will be used only during ramp-up and the voltage reference of the IEB-GO strategy will be used during ramp-down. The instantaneous additional energy provided by the grid during ramp-up ($e_{up}^{PS}$) can be derived by integrating (11) as a function of the $e_{up}^{GO}$:

$$e_{up}^{PS}(t) = \int p_{up}^{PS}(t) \cdot dt \qquad (12)$$

$$e_{up}^{PS}(t) = e_{up}^{GO}(t) - \frac{R_{mag}}{SR} \cdot i_{mag}^{3}(t) + \frac{I_{top} \cdot R_{mag}}{SR} \cdot i_{mag}^{2}(t) \qquad (13)$$

Similarly to (6), the capacitor storage in the IEB-PS strategy will provide the reactive energy of the magnet ($e_{mag}$) reduced by the additional energy provided by the grid ($e_{up}^{PS}$):

$$E_{st,max} - e_{st}(t) = e_{mag}(t) - e_{up}^{PS}(t) \qquad (14)$$

The voltage reference of the energy storage for the IEB-PS strategy is derived by replacing (7), (8), (9) and (13) in (14) and the block diagram is illustrated in Fig. 9. The transition from the IEB-PS voltage reference to the IEB-GO voltage reference should occur only after the ramp-up phase is completed:

$$v_{st,ref}^{PS}(t) = \sqrt{v_{st,ref}^{GO} \cdot (t) - \frac{2 \cdot R_{mag} \cdot i_{mag}^{3}(t)}{C_{st} \cdot SR} + \frac{2 \cdot I_{top} \cdot R_{mag} \cdot i_{mag}^{2}(t)}{C_{st} \cdot SR}} \qquad (15)$$

## Comparison of the EB and IEB strategies

The following sections provide a comparison of the EB, IEB-GO and IEB-PS strategies, including the required capacitors of the energy storage, the peak current of the energy storage and the grid power overloading during ramp-up.

### Energy Storage Requirements

The energy storage of the EB strategy is always designed for the total energy stored in the magnet ($E_{mag}$), regardless of the current slew rate (SR):

$$E_{st}^{EB} = E_{mag} \qquad (16)$$

<figure><img src="image_10.png" /><figcaption>Fig. 9: Block diagram of the IEB-PS strategy.</figcaption></figure>

On the other hand, for the IEB strategies (IEB-GO and IEB-PS) the required energy storage depends on the current slew rate during ramp-down. The higher the slew rate, the less energy will be dissipated in the resistance of the magnet during ramp-down, and more energy will be recovered in the energy storage. The total energy stored in the magnet and the total energy dissipated as losses during ramp-down is derived by (5) and (7) for the maximum value of the magnet current ($I_{top}$):

$$E_{mag} = \frac{1}{2} \cdot L_{mag} \cdot I_{top}^{2} \qquad (17)$$

$$E_{loss}^{down} = \frac{1}{3} \cdot \frac{R_{mag}}{SR} \cdot I_{top}^{3} \qquad (18)$$

The minimum slew rate (SR$_{min}$) is derived for the case where the entire energy stored in the magnet is dissipated as resistive losses during ramp-down. This practically means that the energy storage will remain unused, and the whole energy of the magnet (both active and reactive) will be provided by the grid. The actual slew rate (SR) of the magnet current should always be larger than SR$_{min}$ by a factor $m > 1$:

$$E_{mag} = E_{loss}^{down} \qquad (19)$$

$$SR_{min} = \frac{2}{3} \cdot \frac{R_{mag}}{L_{mag}} \cdot I_{top} \qquad (20)$$

$$SR = m \cdot SR_{min} \qquad (21)$$

The required energy storage of the IEB-GO and IEB-PS strategies is derived below as a function of $m$ by replacing (17), (18) and (21) in (2):

$$E_{st}^{IEB} = E_{st}^{IEB-GO} = E_{st}^{IEB-PS} = E_{mag} \cdot \left(1 - \frac{1}{m}\right) \qquad (22)$$

Compared to the EB strategy, the energy storage saving offered by the IEB-GO and IEB-PS strategies is given by (23). With a typical range of $m = 2-3$, the energy storage requirement can be reduced by 30% to 50%, as illustrated in Fig. 10:

$$E_{st,saving} = \frac{E_{st}^{IEB} - E_{st}^{EB}}{E_{st}^{EB}} \cdot 100 = \frac{1}{m} \cdot 100 \qquad (23)$$
## Peak Current of the Energy Storage

The current of the energy storage is given by the following generic equation:

$$i_{st}(t) = C \cdot \frac{dv_{st}}{dt} \qquad (24)$$

The peak current in the energy storage of the EB, IEB-GO and IEB-PS strategies is given for the maximum magnet current ($I_{top}$) by substituting (1), (10) and (15) in (24). It derives that the peak current in the energy storage of the IEB-PS strategy is the same as that in the EB strategy:

$$I_{st,peak}^{EB} = I_{st,peak}^{IEB-PS} = \frac{L_{mag} \cdot I_{top} \cdot SR}{U_{st,top}} \qquad (25)$$

$$I_{st,peak}^{IEB-GO} = \frac{L_{mag} \cdot I_{mag,top} \cdot SR - I_{top}^2 \cdot R_{mag}}{U_{st,top}} \qquad (26)$$

The peak current in energy storage as a function of $m$ is derived by replacing (21) in (25) and (26):

$$I_{st,peak}^{EB} = I_{st,peak}^{IEB-PS} = \frac{\frac{2}{3} \cdot m \cdot I_{top}^2 \cdot R_{mag}}{V_{st,top}} \qquad (27)$$

$$I_{st,peak}^{IEB-GO} = \frac{\left(\frac{2}{3} \cdot m - 1\right) \cdot I_{top}^2 \cdot R_{mag}}{V_{st,top}} \qquad (28)$$

Assuming that the capacitance of the energy storage is such that the voltage swing is the same for all the EB and IEB strategies, the reduction in the peak current of the energy storage offered by the IEB-GO strategy is given as follows:

$$I_{st,pk,reduct} = \frac{I_{st,peak}^{EB} - I_{st,peak}^{IEB-GO}}{I_{st,peak}^{EB}} \cdot 100 = \frac{3}{2 \cdot m} \cdot 100 \qquad (29)$$

Compared to the EB and IEB-PS strategies, the IEB-GO strategy results in a lower peak current in the energy storage, as illustrated in Fig. 11.

<figure><img src="image_11.png" /><figcaption>Fig. 10: Energy storage savings with IEB-GO or IEB-PS strategies compared to the EB strategy as a function of the SR value.</figcaption></figure>

With a typical range of the factor $m = 2-3$, the peak current in the energy storage can be reduced by 50% to 75%, reducing the thermal stress of the capacitors in the energy storage.

## Grid Overloading

The total power provided by the grid during ramp-up is determined by the sum of the resistive losses of the magnet ($p_{loss}^{up}$) and the additional energy provided by the grid ($p_{up}$):

$$p_{up}(t) = i_{mag}^2(t) \cdot R_{mag} \qquad (30)$$

$$p_{grid,up}(t) = p_{loss}^{up}(t) + p_{up}(t) \qquad (31)$$

For the EB strategy, the additional energy provided by the grid is zero, while for the IEB-GO and IEB-PS strategies, by substituting (3) and (11) in (31), it derives:

$$p_{grid,up}^{EB}(t) = i_{mag}^2(t) \cdot R_{mag} \qquad (32)$$

$$p_{grid,up}^{IEB-PS}(t) = [2 \cdot I_{top} \cdot i_{mag}(t) - i_{mag}^2(t)] \cdot R_{mag} \qquad (33)$$

$$p_{grid,up}^{IEB-GO}(t) = 2 \cdot i_{mag}^2(t) \cdot R_{mag} \qquad (34)$$

The total grid power receives its maximum value at the end of ramp-up for the maximum value of the magnet current ($I_{top}$):

$$p_{grid,up}^{EB} = I_{top}^2 \cdot R_{mag} \qquad (35)$$

$$p_{grid,up}^{IEB-PS} = I_{top}^2 \cdot R_{mag} \qquad (36)$$

$$p_{grid,up}^{IEB-GO} = 2 \cdot I_{top}^2 \cdot R_{mag} \qquad (37)$$

In the case of the EB and IEB-PS strategies the grid peak power during ramp-up equals the flat-top value. However, for the IEB-GO strategy, the grid peak power is twice as high as the flat-top value, which may result in oversized upstream electrical grid infrastructure to handle the higher grid currents.

<figure><img src="image_12.png" /><figcaption>Fig. 11: Energy storage peak current reduction with IEB-GO strategy compared to the EB and IEB-PS strategies as a function of the SR value.</figcaption></figure>
## Summary of the Comparison

The previously developed EB strategy has straightforward implementation and does not impose an overload on the electrical network, but it requires a large energy storage capacity. To the contrary the IEB-GO and IEB-PS strategies, as described above, can significantly reduce the required number of parallel capacitors in the energy storage. Compared to the IEB-GO strategy, the IEB-PS strategy is the preferred choice as it eliminates grid overloading, provided that the current stress on the energy storage capacitors is not a limiting factor. However, in cases where the maximum permissible current in the energy storage capacitors would be exceeded using the IEB-PS strategy, the IEB-GO strategy may serve as an alternative solution. This, however, would come at the cost of increased electrical grid overloading and the need for a more extensive electrical grid infrastructure.

## Simulation Results

The energy managements strategies are validated through simulations performed on POLARIS power converter using PSIM®. The power converter is configured with four power bricks in parallel, as illustrated in Fig. 12. The complete list of power converter and magnet parameters is presented in Table I.

Equations (2) and (16) are used to define the maximum energy ($E_{st}$) for which the energy storage shall be designed. The required number of capacitors of the energy storage is:

<figure><img src="image_13.png" /><figcaption>Fig. 13: Transition from the EB strategy (a) to the IEB-GO strategy (b) and the IEB-PS strategy (c).</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12: POLARIS power converter with 2Q stage in four power bricks parallel configuration.</figcaption></figure>

$$N_{caps} = \left[ \frac{E_{st}}{N_{bricks} \cdot E_{c}} \right] \cdot N_{bricks} \quad (38)$$

The simulation is performed for the trapezoidal waveform of the magnet current illustrated in Fig. 1, and the SR value set to $2 \cdot SR_{min}$ ($m = 2$). The energy storage is composed of $N_{caps} = 152$ capacitors and it is designed for the maximum energy of the magnet ($E_{mag}$).

Fig. 13 shows that a transition from the EB (Fig. 13a) to the IEB-GO strategy (Fig. 13b) leads to smaller voltage swing in the capacitors of the energy storage, because of their reduced contribution to the magnet's energy. The remaining part of the energy is supplied by the grid during ramp-up, creating a grid peak power twice as high as the flat-top value.

The transition from the IEB-GO (Fig. 13b) to the IEB-PS (Fig. 13c) strategy requires the same energy contribution from the energy storage to the magnet, creating the same voltage swing in the energy storage capacitors.
<table><caption>Table I: Simulation parameters of POLARIS.</caption><thead><tr><td>Parameter</td><td>Symbol</td><td>Value</td></tr></thead><tbody><tr><td>Magnet resistance</td><td>R<sub>mag</sub></td><td>55 mΩ</td></tr><tr><td>Magnet inductance</td><td>L<sub>mag</sub></td><td>176 mH</td></tr><tr><td>Max. magnet current</td><td>I<sub>top</sub></td><td>1400 A</td></tr><tr><td>Max. magnet energy</td><td>E<sub>mag</sub></td><td>172.5 kJ</td></tr><tr><td>Capacitance of one capacitor</td><td>C</td><td>25 mF</td></tr><tr><td>Max. capacitor energy</td><td>E<sub>c</sub></td><td>1.15 kJ</td></tr><tr><td>Max. energy storage voltage</td><td>V<sub>st,max</sub></td><td>330 V</td></tr><tr><td>Min. energy storage voltage</td><td>V<sub>st,min</sub></td><td>130 V</td></tr><tr><td>Max. energy storage current</td><td>I<sub>st,max</sub></td><td>400 A</td></tr><tr><td>Minimum SR</td><td>SR<sub>min</sub></td><td>292 A/s</td></tr><tr><td>Flat-top duration</td><td>T<sub>top</sub></td><td>4.8 s</td></tr><tr><td>Power cycle duration</td><td>T<sub>cycle</sub></td><td>14.4 s</td></tr></tbody></table>

Compared to the IEB-GO, the IEB-PS strategy results in higher current in the energy storage, but it eliminates the grid overlading during ramp-up.

## Experimental Validation

The energy balance strategies are validated through experimental results conducted in POLARIS converter prototype illustrated in Fig. 14. The prototype is configured with two power bricks in parallel forming output current and voltage ratings of 800 A and 360 V. The complete list of power converter and magnet parameters is presented in Table II. The input and the intermediate stages of POLARIS are located in the power bricks, while the output power is delivered through the main brick.

The experimental results illustrated in Fig. 15 and Fig. 16, present the magnet current ($i_{mag}$), the energy storage voltage ($v_{st}$) and the current of the DC reactor ($i_{dc}$) in both bricks during an online transition through the three energy balance strategies with constant number of capacitors in the energy storage.

<table><caption>Table II: Experimental parameters of POLARIS.</caption><thead><tr><td>Parameter</td><td>Symbol</td><td>Value</td></tr></thead><tbody><tr><td>Magnet resistance</td><td>R<sub>mag</sub></td><td>61.3 mΩ</td></tr><tr><td>Magnet inductance</td><td>L<sub>mag</sub></td><td>67.4 mH</td></tr><tr><td>Max. magnet current</td><td>I<sub>top</sub></td><td>300 A</td></tr><tr><td>Max. magnet energy</td><td>E<sub>mag</sub></td><td>3 kJ</td></tr><tr><td>Energy storage capacitance</td><td>C<sub>st</sub></td><td>104 mF</td></tr><tr><td>Max. energy storage voltage</td><td>V<sub>st,max</sub></td><td>330 V</td></tr><tr><td>Min. energy storage voltage</td><td>V<sub>st,min</sub></td><td>130 V</td></tr><tr><td>Max. energy storage current</td><td>I<sub>st,max</sub></td><td>400 A</td></tr><tr><td>Minimum SR</td><td>SR<sub>min</sub></td><td>600 A/s</td></tr><tr><td>Flat-top duration</td><td>T<sub>top</sub></td><td>2 s</td></tr><tr><td>Power cycle duration</td><td>T<sub>cycle</sub></td><td>4.8 s</td></tr></tbody></table>

<figure><img src="image_15.png" /><figcaption>Fig. 14: POLARIS power converter prototype used for experimental validation.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 15: Experimental results for an online transition from the EB strategy (left) to the IEB-GO strategy (right).</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 16: Experimental results for an online transition from the IEB-GO strategy (left) to the IEB-PS strategy (right).</figcaption></figure>
After the transition from the EB to the IEB-GO strategy the voltage swing in the energy storage is reduced, while the current of the DC reactor, image of the grid power, emerges a peak value two times higher than the flat-top value. The transition from the IEB-GO to the IEB-PS strategy results in the same voltage swing in the energy storage, while the peak value of the DC reactor current is eliminated.

## Conclusion

The new Improved Energy Balance strategies, IEB-GO and IEB-PS, utilize the magnet's stored energy to cover the resistive losses during ramp-down, allowing for a significant reduction in the number of capacitors of the energy storage. Compared to the EB strategy, the reduction of the required number of capacitors for POLARIS, a converter for particle accelerators, ranges from 30% to 50%.

The IEB-GO strategy is more suitable for scenarios where the peak current of the energy storage needs to remain low, although this comes at the cost of electrical grid overloading and the potential need for larger grid AC infrastructure.

When the peak current of the energy storage is not a concern, the IEB-PS strategy is the preferred solution, as it eliminates the grid peak power caused by the IEB-GO strategy and minimizes the cost of the electrical grid infrastructure.

## References

[1] M. Khodaparastan, A. A. Mohamed and W. Brandauer, "Recuperation of Regenerative Braking Energy in Electric Rail Transit Systems," in IEEE Transactions on Intelligent Transportation Systems, vol. 20, no. 8, pp. 2831-2847, Aug. 2019.

[2] I. Župan, A. Lasić, D. Krušelj, V. Šunde and Ž. Ban, "Power converter circuits for recuperation of the regenerative braking energy in rail vehicles," 42nd International Convention on Information and Communication Technology, Electronics and Microelectronics (MIPRO), Opatija, Croatia, 2019, pp. 174-179.

[3] P. Spichartz and C. Sourkounis, "Measurement of braking energy recuperation in electric vehicles," 15th European Conference on Power Electronics and Applications (EPE), Lille, France, 2013, pp. 1-9.

[4] G. Stana and V. Brazis, "Analyses of Trolleybus Recuperation Energy Utilisation Losses Considering Different Efficiency Ratios of Traction Inverter and DC/DC Converter," 20th International Scientific Conference on Electric Power Engineering (EPE), Kouty nad Desnou, Czech Republic, 2019, pp. 1-6.

[5] S. Rossini, K. Papastergiou, G. Le Godec, R. G. Retegui and S. Maestri, "Power converter topologies with energy recovery and grid power limitation for inductive load applications," 17th European Conference on Power Electronics and Applications (EPE'15 ECCE-Europe), Geneva, Switzerland, 2015, pp. 1-9.

[6] S. Maestri, R. G. Retegui, G. Uicich, M. Benedetti, K. Papastergiou and G. LeGodec, "Control strategies for 2-quadrant converter used in grid power flow control," 17th European Conference on Power Electronics and Applications (EPE'15 ECCE-Europe), Geneva, Switzerland, 2015, pp. 1-10.

[7] G. Fernandez, "Cascaded voltage control," Accessed: Mar 23, 2021. [Online]. Available: https://imperix.com/doc/implementation/cascaded-voltage-control.