**Electric Vehicle Efficient Power and Propulsion Systems**

Editors

**Jo ˜ao Pedro F. Trov˜ao Minh Cao Ta**

MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin

*Editors* Joao Pedro F. Trov ˜ ao˜ University of Sherbrooke Canada Polytechnic of Coimbra (IPC-ISEC) and INESC Coimbra Portugal

Minh Cao Ta Hanoi University of Science and Technology Vietnam University of Sherbrooke Canada

*Editorial Office* MDPI St. Alban-Anlage 66 4052 Basel, Switzerland

This is a reprint of articles from the Special Issue published online in the open access journal *Energies* (ISSN 1996-1073) (available at: https://www.mdpi.com/journal/energies/special issues/Electric Vehicle Efficient Power Propulsion Systems).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below:

LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. *Journal Name* **Year**, *Volume Number*, Page Range.

**ISBN 978-3-0365-5291-0 (Hbk) ISBN 978-3-0365-5292-7 (PDF)**

Cover image courtesy of Joao Pedro F. Trov ˜ ao˜

© 2022 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications.

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## **Contents**

## **About the Editors**

#### **Jo ˜ao Pedro F. Trov˜ao**

Joao Pedro F. Trov ˜ ao (Senior Member, IEEE) received M.Sc. degree and the Ph.D. degree ˜ in Electrical Engineering from the University of Coimbra, Coimbra, Portugal, in 2004 and 2013, respectively. Dr. Trovao was a Teaching Assistant, and he serves as Professor of Polytechnic of ˜ Coimbra–Coimbra Institute of Engineering (IPC–ISEC), Portugal. Since 2014, he has been a Professor with the Department of Electrical Engineering and Computer Engineering, University of Sherbrooke, Sherbrooke, QC, Canada, where he holds the Canadian Research Chair position in Efficient Electric Vehicles with Hybridized Energy Storage Systems. He is an author/coauthor of over 170 journal and conference papers. His research interests cover the areas of electric vehicles, hybridized energy storage systems, energy management and rotating electrical machines. Dr. Trovao was the General ˜ Chair of the 2018 IEEE Vehicle Power and Propulsion Conference, Chicago, US. He was a Guest Editor for the Special Issue of IET Electrical Systems in Transportation on Energy Storage and Electric Power Sub-Systems for Advanced Vehicles. He was a Guest Editor for the Special Issues of IEEE Transactions on Vehicular Technology on Electric Powertrains for Future Vehicles and on Advanced Vehicle Power Propulsion Systems. He is a founding member and the director of the electric-Transport, Energy Storage and Conversion (e-TESC) Lab of the University of Sherbrooke. He is a Senior Editor for the Automotive Electronics topic of the IEEE Vehicular Technology Magazine.

#### **Minh Cao Ta**

Minh Cao Ta (Senior Member, IEEE) received the B.S. (Hons.) degree from the Institute of Technology (now the University of West Bohemia), Pilsen, Czech Republic, in 1986, and the Ph.D. degree from Laval University, QC, Canada, in 1998, both in electrical engineering. From 1998 to 2004, he was with Kyushu University; The University of Tokyo; and NSK Steering Systems Ltd. Co., Japan. In 2009, he became an Associate Professor with the Hanoi University of Science and Technology (HUST), Vietnam, and was the Founding Director of the Center for Technology Innovation (CTI) 2009–2018, and Head of the Department of Industrial Automation, HUST, 2018–2020. He is currently with the Department of Electrical and Computer Engineering, University of Sherbrooke, Sherbrooke, QC, Canada. He is the author/coauthor of 14 Japanese/US/European patents and over 100 journal and conference papers. His main research interests include motor drives and advanced control techniques and their applications to electric vehicles and energy conversion systems. Dr. Ta was a Recipient the 2017 Nagamori Awards. He was the General Chair of the 2019 IEEE Vehicle Power and Propulsion Conference, Hanoi, Vietnam. He was a Guest Editor for several Special Issues of IEEE Transactions. He severed as Vice-President and General Secretary of the Vietnam Automation Association (VAA) 2014–2020.

## **Preface to "Electric Vehicle Efficient Power and Propulsion Systems"**

Nowadays, vehicle electrification is an important field of research since it is a large umbrella of almost all engineering activities and systems. The challenges are much more than the development of lithium-ion batteries that were proposed in the beginning of the nineties and have been the most used vehicle energy storages to this day, mostly in commercial electric vehicles. Technological advances and affordability have enabled the deployment of new energy storage systems, power electronics semiconductors, electric machines, energy management systems, and in particular, efficient systems and intelligent systems. These new-generation systems cover a broad range of implementation for which traditional approaches have practical limitations. As a result, new development in component and system level related to power efficiency and computational intelligence has been expanding at a high speed over the last two decades. Electric vehicle itself is still a current research trend. Indeed, there are hundreds of publications on this topic that may cause even a knowledgeable person in the subject to lose track of all novelties.

The main purpose of the book is therefore to present some recent and important developments related to vehicle electrification, with consciousness to the efficiency of the powertrain in EVs through state-of-the-art surveys. Future directions and research perspectives on the subject are clearly presented. This book is designed to give to the readers a better understanding of different challenges faced in electrified vehicles, from the power to the propulsion systems, to provide insights that enable the development of new approaches. The reading of this book is not intended to be linear, i.e., each chapter stands on its own. However, the chapter order follows a logical flow of ideas that may help a novice in vehicle electrification to better understand the presented concepts. Chapter 1 is an overview of this book that shows how it should be read to make important links between the contributions and positions readers to the next challenges in this important field of engineering knowledge.

The following two chapters of the book introduce recent development in the sub-field of "Energy Storage Systems". The contributions are focused on the enhancement of the battery's performance based on multi-agent consensus and on effectiveness of the estimation technique by adaptative square-root unscented Kalman filter. The fourth chapter presents the recent development on wide-bandgap (WBG) power semiconductors. Recent research on WBG semiconductors such as silicon carbide (SiC) and gallium nitride (GaN) have attracted significant attention, since the increase in converter switching frequency contributes to the decrease in global power losses. The next four chapters are related to the evolution of traction and motion control techniques and co-design technique for vehicle transmission. The last four chapters are the demonstrations of the effective usage of on-board stored energy. In this sub-field "Energy Management Systems", readers can experience methodologies such as rule-based supervisory controller, novel real-time optimization-based torque distribution strategy, energy management strategy for an industrial hybrid self-guided vehicle, considering the size of a fuel cell stack and the degradation of a battery pack. To replace fossil-fueled vehicles by EVs in logistics applications, a system to manage energy and to schedule the vehicle route has been proposed using genetic algorithm. Most of the powertrain improvement "from energy storage to wheels" presented in this book deal with trends, implementation difficulty and their limitations. In summary, the contributed book covers theoretical aspects of vehicle electrification engineering along with practical applications. This book empowers the readers to establish the appropriate research planning and methods for devoted specific application over the next generation of efficient powertrains.

The book would not be possible without the contributions of various authors. The advanced control and optimization techniques, the valuable findings and practical implementation described in the high-quality manuscripts have been very much appreciated.

**Jo˜ao Pedro F. Trov˜ao and Minh Cao Ta**

*Editors*

## *Editorial* **Electric Vehicle Efficient Power and Propulsion Systems**

**João Pedro F. Trovão 1,2,\* and Minh Cao Ta 1,3**

#### **1. Introduction**

Vehicle electrification is one of the main growing trends with an identified growth capacity of 15% until 2030. In the coming years, many internal combustion engine vehicles, hybrid vehicles and all-electric vehicles will be on the road as consumers switch to more efficient and environmentally friendly propulsion systems. To remain competitive in this electrically powered future, carmakers and researchers are investing in a wide range of propulsion technologies to increase efficiency and power capacity, developing the next generation of powertrains. The development of more efficient pure EVs, HEVs and fuel cell electric vehicles (FCEV) presents both a challenge and a definite solution to current mobility issues. A reliable EV solution should therefore harness the advantages of more efficient and powerful energy storage systems from multiple sources through their effective management and new and improved power converters, including the new generation of switching devices, and explore advanced configurations for electric motors, reducing the use of rare-earth materials.

This Special Issue entitled "Electric Vehicle Efficient Power and Propulsion Systems" was established to encourage researchers working in this field to share the latest developments on electric vehicle efficient power and propulsion systems for road, rail and air vehicles, both manned and unmanned. Therefore, this Special Issue aims to provide recent developments in any field of electric vehicle efficiency powertrains, covering a wide range of scientific topics, including, but not limited to, the following:

The Guest Editors are Prof. João Pedro F. Trovão and Minh C. Ta (University of Sherbrooke, Sherbrooke, QC, Canada). Each paper was reviewed by three independent reviewers from countries differing from each author's country of origin. In order to ensure the highest quality, only 11 submissions were accepted for inclusion in this Special Issue of *Energies*.

#### **2. Special Issue Content**

In total, 11 papers relating to electric vehicle efficiency powertrains have been accepted and published in this Special Issue. This subject encompasses a wide range of topics, as illustrated by the diversity of the accepted papers: four papers were selected on energy management algorithms and topologies, four on wheel-drives control development, transmission, and validation, two on Lithium-ion batteries and one on SiC-inverter efficiency. All four of these topics are addressed in this Special Issue.

**Citation:** Trovão, J.P.F.; Ta, M.C. Electric Vehicle Efficient Power and Propulsion Systems. *Energies* **2022**, *15*, 3863. https://doi.org/10.3390/ en15113863

Received: 16 May 2022 Accepted: 16 May 2022 Published: 24 May 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In the first sub-topic "Energy storage Systems" we received two papers. First, Barreras, J.V. et al. [1] present novel research on the distributed multi-objective control algorithm, which is based on a multi-agent consensus algorithm. The authors implemented and validated the control in simulations, considering an electro-thermal lithium-ion battery model and an electric vehicle model parameterized with experimental data. The results show that the proposed multi-functional balancing can enhance the performance of batteries with substantial cell-to-cell differences under the most demanding operating conditions. The second paper in this sub-topic focuses on state-of-charge (SOC) as a fundamental indicator of lithium-ion batteries, which has an important role in the battery's optimized operation. Ouyang, Q. et al. [2] propose an adaptive square-root unscented Kalman filter (SRUKF) to estimate the battery's SOC. The effectiveness of the estimation technique is demonstrated by their extensive experimental results, which show higher precision compared with other commonly used Kalman filter-based methods.

For the second subtopic "Power Electronics", one contribution was received focusing on wide-bandgap (WBG) power semiconductors. Recent research on WBG power semiconductor devices such as silicon carbide (SiC) and gallium nitride (GaN) have attracted significant attention, since the increase in inverter switching frequency contributes to the decrease in switching power loss. In paper [3], Ryu, J.-H. et al. investigate a method for switching the frequency determination of a SiC-inverter to improve the efficiency of the railway propulsion system. The hybrid switching method combined with the synchronous PWM and asynchronous PWM was considered in the analysis of the SiC inverter that feeds a PMSM. The efficiency curve of the propulsion system as a function of the switching frequency demonstrated the potential to serve as a guideline for switching frequency selection.

For the third subtopic "Electric machines", three papers focus on electric machines for traction/motion control, and one paper on the co-design technique of a continuously variable transmission (CVT) for EVs. Vo-Duy, T. et al. present an experimental platform for the evaluation of on-board real-time motion controllers for EVs [4], providing a good example of the hardware-in-the-loop (HiL) system, on which the most important vehicle dynamics, including longitudinal, lateral, yaw, kinematic, and position models, have been built and validated in various testing scenarios. The proposed HiL system can be employed for research on various topics of EV control. Paper [5] demonstrates an alternative point of view, using multi-agent-system theory. Nguyen, B.-M. et al. propose three different ways of modeling in-wheel-motor (IWMs) vehicles: a nonlinear model with a hierarchical structure for passivity-based motion control, a linearized model with a rank-1 interconnection matrix for stability analysis, and a time-varying state-space model for optimal control, using a linear quadratic regulator (LQR). The effectiveness of the three models and their design approaches are discussed in several examples with a Matlab/Carsim co-simulator. Additionally, paper [6] addresses the multi-motor configuration and control of EVs. De Pinto, S. et al. provide a comparison between 2-wheel-drive (2WD) and 4WD configurations for the same EV from the viewpoint of drivability and energy consumption. An optimization routine is used to calculate the energy-efficient gear state and/or torque distribution for each considered configuration. The results highlight that the single-speed 4WD layout can reduce the energy consumption during driving cycles by approximately 9% compared to the conventional 2WD layout with single-speed transmission. Finally, Wei, C. et al. consider the co-design of a continuously variable transmission (CVT) for EVs. In paper [7], a novel convex programming (CP)-based co-design method is proposed to minimize the total-cost-of-ownership (TCO). The strength of the co-design is highlighted in comparison with a sequential design, and insights into the design of a low-power EV that is energy-efficient and cost-effective for urban driving are provided. A highly integrated EM-CVT system, which is efficient, low-cost, and lightweight, can be expected for future EV applications.

The effective utilization of energy storage systems is still a critical issue in electrified vehicles. The subtopic of "Energy management systems" attracted four papers in this Special Issue [8–11]. First, Maddumage, W. et al. propose a design methodology for a rule-based supervisory controller of a pre-transmission parallel hybrid three-wheeler based on the optimal control strategy, i.e., dynamic programming (DP). The developed rule-based strategy shows performance within 10% of the DP results on WLTC and UDC-NEDC drive cycles and has the advantage of being near-optimal, easy-to-implement and computationally less demanding. Nguyễn, B.-H. et al. propose a novel real-time optimization-based torque distribution strategy for a parallel hybrid truck in [9]. The strategy aims to minimize the engine fuel consumption while ensuring battery charge-sustaining by using linear quadratic regulation in a closed-loop control scheme. By reformulating the problem, the obtained strategy does not require the information of the engine efficiency map like the previous studies in the literature. The proposed method has been evaluated via simulation in comparison with dynamic programming as a benchmark, and experimentally validated by using a power hardware-in-the-loop simulator. Additionally, Ghobadpour, A. et al. present the design of an energy management strategy (EMS) for an industrial hybrid self-guided vehicle (SGV), considering the size of a fuel cell (FC) stack and the degradation of a battery pack [10]. In the paper, a realistic energy model of the SGV was first proposed and validated, based on experiments. The performance of the developed FC/battery SGV powertrain was then validated under three EMS modes. Each mode was studied by considering four different FC sizes and three battery degradation levels. The results showed several important remarks, such that a small FC as a range extender is recommended to reduce system costs. Battery SOC must be kept at a high level during SGV operation to support the FC during SGV acceleration. Finally, EVs to replace fossil-fueled vehicles in logistics applications are studied in [11] by Wang, L. et al. The authors propose a system to manage energy and to schedule the vehicle route, which comprises two parts: (i) a case-based reasoning subsystem to forecast the energy consumption and travel time for each route section, and (ii) a genetic algorithm to optimize vehicle routing with an energy consumption situation as a new constraint. A dynamic adjustment algorithm is also adopted to achieve a rapid response to accidents in which the vehicles might be involved. Simulation study, validations on Solomon benchmarks, and result analysis have been performed to show that the proposed vehicle management system is more economical than the traditional method.

#### **3. Closing Remarks and Future Challenges**

The articles presented in this Special Issue cover important aspects of electric vehicle efficiency powertrains. With the wide range of topics covered in the selected papers, the contributions to this Special Issue can stimulate the EV community to undertake further research concerning more efficient EVs. Therefore, we believe that the presented papers will have practical importance for forthcoming developments in the automotive industries. The contributions included in this Special Issue offer new data, information, and findings to continue the R&D effort in the field, with the aim of stimulating the research community to further contribute to the development of the field.

**Author Contributions:** Conceptualization and methodology J.P.F.T. and M.C.T.; formal analysis, J.P.F.T. and M.C.T.; resources, J.P.F.T. and M.C.T.; writing—original draft preparation, J.P.F.T. and M.C.T.; writing—review and editing, J.P.F.T. and M.C.T.; visualization, J.P.F.T. and M.C.T.; supervision, J.P.F.T. and M.C.T.; project administration, J.P.F.T. and M.C.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** Each paper was reviewed by at least three reviewers; therefore, the Guest Editors would like to thank them for their work, which helped the authors to improve their manuscripts. In addition, the Guest Editors would like to thank Sherwin Chen for his help and support in the Editorial process of the Electric Vehicle Efficient Power and Propulsion Systems Special Issue of the *Energies* journal.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

## *Article* **A Consensus Algorithm for Multi-Objective Battery Balancing**

**Jorge Varela Barreras 1,2,\*, Ricardo de Castro 3, Yihao Wan <sup>4</sup> and Tomislav Dragicevic <sup>4</sup>**

**Abstract:** Batteries stacks are made of cells in certain series-parallel arrangements. Unfortunately, cell performance degrades over time in terms of capacity, internal resistance, or self-discharge rate. In addition, degradation rates are heterogeneous, leading to cell-to-cell variations. Balancing systems can be used to equalize those differences. Dissipative or non-dissipative systems, so-called passive or active balancing, can be used to equalize either voltage at end-of-charge, or state-ofcharge (SOC) at all times. While passive balancing is broadly adopted by industry, active balancing has been mostly studied in academia. Beyond that, an emerging research field is multi-functional balancing, i.e., active balancing systems that pursue additional goals on top of SOC equalization, such as equalization of temperature, power capability, degradation rates, or losses minimization. Regardless of their functionality, balancing circuits are based either on centralized or decentralized control systems. Centralized control entails difficult expandability and single point of failure issues, while decentralized control has severe controllability limitations. As a shift in this paradigm, here we present for the first time a distributed multi-objective control algorithm, based on a multi-agent consensus algorithm. We implement and validate the control in simulations, considering an electrothermal lithium-ion battery model and an electric vehicle model parameterized with experimental data. Our results show that our novel multi-functional balancing can enhance the performance of batteries with substantial cell-to-cell differences under the most demanding operating conditions, i.e., aggressive driving and DC fast charging (2C). Driving times are extended (>10%), charging times are reduced (>20%), maximum cell temperatures are decreased (>10 ◦C), temperature differences are lowered (~3 ◦C rms), and the occurrence of low voltage violations during driving is reduced (>5×), minimizing the need for power derating and enhancing the user experience. The algorithm is effective, scalable, flexible, and requires low implementation and tuning effort, resulting in an ideal candidate for industry adoption.

**Keywords:** lithium-ion battery; balancing systems; electric vehicles; consensus algorithm; distributed control; state-of-charge equalization; temperature equalization

#### **1. Introduction**

The 2015 Paris agreement has established global guidelines to tackle climate change, including electrification of transportation [1]. Supported by policies, technological improvements, and falling costs, the global fleet of light-duty passenger vehicles (LDPVs) with a higher degree of electrification, such as battery electric vehicles (BEVs) and plug-in hybrid electric vehicles (PHEVs), have expanded from 2014–2019 by an annual average of 60%, reaching about 1% of the global fleet by 2019 [2].

In 2020 the COVID-19 pandemic has brought great uncertainty for the global electric vehicle (EV) market and the auto market. Although, recent studies confirm that the future remains bright for passenger EVs in the following decades, which will gain market share supported by battery cost reductions and technological advances [3].

**Citation:** Barreras, J.V.; de Castro, R.; Wan, Y.; Dragicevic, T. A Consensus Algorithm for Multi-Objective Battery Balancing. *Energies* **2021**, *14*, 4279. https://doi.org/10.3390/en14144279

Academic Editors: João Pedro Trovao and Ta Cao Minh

Received: 17 May 2021 Accepted: 11 July 2021 Published: 15 July 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Virtually all EVs in today's market store electrical energy in batteries, an essential technology to electrify LDPVs in the next 20 years [2,3]. EVs with a higher degree of electrification, i.e., BEVs and PHEVs, are equipped with larger batteries that can be directly recharged from the grid, in addition to regenerative braking, a common feature in EVs with lower degrees of electrification. Nowadays, a BEV or PHEV battery is formed of a combination of hundreds to thousands of lithium-ion cells connected in a certain seriesparallel arrangement.

Lithium-ion battery technology is improving, showing increased energy storage capability, and their cost is falling, mainly thanks to economies of scale. However, there are still some significant technical barriers to overcome for EV adoption [2,3], particularly regarding range anxiety, charging infrastructure, battery supply chain, battery safety [4], and battery management [5].

Regarding the latter, for safety reasons, lithium-ion cell manufacturers limit their operating window by certain voltages, currents, and temperatures. Outside that so-called safety operating window (SOA), the degradation rate accelerates and there is a risk of catastrophic failure [5]. In addition, original equipment manufacturers (OEMs) typically define more restrictive operating windows to increase safety, ensure reduced degradation rates, and/or enhance the user experience. That is implemented in the form of conventional current or power derating algorithms, as reviewed by Barreras et al. [6] and Sun et al. [6]. Derating can be improved through degradation-aware algorithms, as recently shown by Sowe et al. [7] and Schimpe et al. [8,9].

Indeed, battery degradation mechanisms are complex and enhanced by certain operating and storage conditions that are fundamentally related to SOC, temperature, or current levels, as discussed by Schimpe et al. [9] orPellettier et al. [10] or For instance, extreme temperatures are a major concern. On one hand, the degradation rates are accelerated at high temperatures, due to rapid growth of the solid electrolyte interface (SEI) layer, existing even the risk of thermal runaway over certain thresholds (Figure 1). On the other hand, at low temperatures, there is accelerated aging due to lithium-plating and stripping, being more intense at higher C-rates. There is also a risk of internal short due to the formation of dendritic lithium (Figure 1). Even within a moderate temperature range (~10–50 ◦C), power and energy capability are influenced by battery temperature, since lithium-ions diffusivity in the electrodes increases with temperature. Indeed, the temperature window within which the battery lifetime (calendar and cycle life) and power and energy performance are balanced, is relatively small (~20–40 ◦C).

Regarding SOC, for example, in BEVs and PHEVs the minimum allowable SOC is typically over 5–10%, since the battery power capability, and thus the vehicle acceleration is lower at low SOC levels, and also to avoid the risk of overdischarge. Battery degradation rate is also higher at extreme SOCs, and therefore the maximum SOC window in EV batteries is set typically below 90–95%—although the user may occasionally select operating modes that allow larger SOC windows for extended range operation.

To alleviate those problems, active heating and cooling strategies are implemented in EVs, and the battery power is derated under eventual extreme SOC or temperature conditions [11]. However, the former translates into higher auxiliary loads (i.e., lower efficiency), more complex integration, extra costs, volume, weight, and reliability and maintenance issues, while the latter typically means longer charging times at low temperatures and reduced power capability at high temperatures.

To make everything more complex, the battery pack should also be managed at a cell level, due to the existence of cell-to-cell variations in SOC, temperatures, or power and energy capability (Figure 2). While cell-to-cell variations may not be substantial at the beginning-of-life (BOL), they increase cumulatively over time due to intrinsic and system-level induced issues, such as uneven power due to parallelization, non-uniform heating/cooling, heterogeneous mechanical stress, uneven presence of heat sources nearby, or battery management issues. For example, in series-connected packs, lower capacity cells charge and/or discharge faster, leading to a reduced battery operating window. In other words, less energy is charged and discharged, since the battery pack performance is limited by the cell(s) with the minimum voltage/SOC during discharge and the maximum voltage/SOC during charge.

**Figure 2.** Schematic representation of cell-to-cell differences on SOC (**left**) and temperature (**right**), showing balanced (**top**) and imbalanced modules (**bottom**) of 8 cells in series. As suggested on the bottom-left schematic, non-uniform cooling can lead to temperature imbalances, but other reasons are possible.

There are two main approaches to balance these cell-to-cell differences. The first relies on dissipative mechanisms, and it is so-called passive balancing. The idea is to dissipate the energy of the highest-charged cell in a passive element, such as a resistor, during the charging phase [12]. The main advantages of this approach are simplicity and low cost but at the expense of higher energy losses. This dissipative approach also does not fully address the balancing problem during discharge: the effective capacity and power of the battery pack are still defined by the weakest cell(s), i.e., the cell(s) with the highest/lowest SOC, lowest power capability and/or maximum/minimum temperature [13].

The second approach is non-dissipative, a so-called active balancing, which aims to continuously re-distribute the energy within the battery pack using power conversion. In the literature, there is a wide range of power conversion topologies that can be used [5,12,14,15], enabling the transfer of energy from one cell to another, a common energy buffer (e.g., a capacitance of inductance) or from the cell to the pack. This re-distribution of energy brings several advantages. Firstly, it provides higher energy efficiency than pure

dissipative approaches and can be much faster. But more importantly, it opens new possibilities from the point of view of the control algorithm beyond the conventional SOC-based, OCV-based, and terminal voltage-based balancing algorithms, which are broadly used both in passive and active balancing, as described by Fleischer et al. in [12].

As shown in pioneering works of Altaf et al. [16,17], Barreras et al. [5,18], Pinto et al. [14,19], and de Castro et al. [20], active balancing opens the way for novel balancing functions, including not only SOC and voltage balancing, but also thermal control, power capability equalization, or distributed hybridization. For instance, it is well known the connection between battery current and temperature, due to self-heating effects, as illustrated in Figure 3. This means that increase in battery temperature can be controlled, up to some extent, through current limitation [5].

**Figure 3.** Temperature and voltage in time for CC discharge at various C-rates for a pristine Kokam 53 Ah cell.

All in all, active balancing can enable improved battery performance, prolonged lifetime, and improved or additional features, as already stated by Barreras et al. in 2014 [18]. Regarding battery performance, the usable capacity of the entire pack over its lifetime can be increased, since both the charge and discharge performance of the battery pack can be dictated by the average characteristics of the cells, instead of the weakest cell, as is the case with dissipative approaches [13]. On the other hand, higher costs, complexity, and volume represent the main obstacles for the deployment of this technology in practice, especially in automotive applications [5,12,15].

From a control architecture perspective, battery balancing functions can be performed in the following ways: centralized; de-centralized, or distributed architectures. That can be related to the battery management system (BMS) topology [5], as illustrated in Figure 4.

In centralized networks, control functions are concentrated in a single unit (aka agent), which receives state information from all battery cells and centrally decides the control actions for the balancing hardware. In decentralized networks, each cell (or module) decides the control action locally, so the biggest reliability problem of centralized networks, i.e., single point of failure, is solved. However, the controllability is limited because there is no cell-to-cell communication. In distributed paradigms, the decentralization is only partial, i.e., the cells can communicate variables of interest with their neighboring cells. Therefore, the control intelligence is divided among a full network of control agents with partial information about the state of the battery.

**Figure 4.** Basic types of BMS topologies (**top**) and their corresponding control networks (**bottom**): (**a**) centralized; (**b**) decentralized (modular and master-slave topologies), and (**c**) distributed.

In general, centralized and distributed control paradigms have the potential to achieve optimal performance, since they access global state information; however, they are also more complex to solve and implement, especially for large-scale systems with dozens or hundreds of actuators. On the other hand, decentralized architectures execute the control effort only locally, thereby requiring no information exchange and lower computational power. However, since each agent uses local information of the battery pack, it is challenging to achieve optimal control performance.

From an algorithmic perspective, three classes of control methods have been proposed for advanced active battery balancing, either in centralized or de-centralized architectures:

Beyond that, a novel method for battery balancing control are consensus algorithms, a distributed architecture based on graph theory in the area of communication. Consensus algorithms only require communication links between adjacent units, i.e., previous and next cells in a series string in our case, featuring advantages of less information requirement, scalability (i.e., modularity), and robustness, while keeping the potential for optimal performance due to global information discovery [25].

So far, only a couple of papers in the literature have implemented consensus algorithms, and for a single function or objective, i.e., either cell voltage [26] or SOC equalization [27]. In contrast, in this study, we propose for the first time in the literature a multi-functional/multi-objective consensus algorithm for distributed battery balancing. Here we show how this algorithm can simultaneously perform SOC, temperature, and voltage balancing, but it could be easily extended to equalize other variables. For example, the surface concentration of electrode particles, or the rates of degradation, being able to perform derating actions, such as the ones presented in [9], at cell or module level.

The overall framework is presented in Figure 5. The concept is evaluated in simulations in the context of an e-mobility application, considering a cell-to-pack-to-cell balancing system configuration and a series-connected battery pack.

**Figure 5.** Block diagram of the distributed consensus control framework for multi-function EV battery balancing.

#### **2. System Modeling and Control Algorithm**

A single module of a battery system in an e-mobility application is modeled in this work, including 8-cells connected in series and the cell-to-pack-to-cell (CPC) balancing system. The battery power profiles are generated considering the full-pack and different driving conditions. The battery module electrical equivalent circuit model (EECM) is summarized in Figure 6.

**Figure 6.** EECM of the 8S battery module and the DC/DC converter of the CPC balancing system.

#### *2.1. Electro-Thermal Battery Model*

A battery module made of 8 large format Li-ion pouch cells (NMC cathode, 53 Ah) connected in series is simulated by means of a multi-cell model. The electro-thermal model of each cell is composed of a simple lumped EECM coupled with a simple lumped thermal equivalent circuit model (TECM), which considers heat transfer between neighboring cells. Apart from that, different sets of parameters are generated to create different scenarios for cell-to-cell variations. Experimental data from single-cell characterization tests are combined with statistical data from screening tests on more than 200 pristine cells from Barreras et al. [28] and data from the literature to parameterize the models.

#### 2.1.1. Electrical Model

Regarding the EECM of each cell *j*, the circuit implemented consists of a variable DC voltage source, OCV*<sup>j</sup>* [V], in series with a constant resistor, *Rj* [Ω] (Figure 6). The former represents the cell equilibrium voltage, usually referred to in this context as the open-circuit voltage (OCV), while the latter accounts for all the cell internal resistance effects. This is mathematically represented as:

$$w\_{\dot{j}}(t) = \text{OCV}\_{\dot{j}}(\text{SOC}\_{\dot{j}}) - R\_{\dot{j}}i\_{\dot{j}}(t), \quad j \in \{1, \dots, N\_s\} \tag{1}$$

$$\text{S\dot{O}C}\_{\dot{f}}(T) = -\frac{1}{\overline{Q}\_{\dot{f}}} i\_{\dot{f}}(t) \tag{2}$$

where *vj* [V] is the terminal voltage of the cell *j*, *Ns* [–] is the number of cells in series in each module, SOC*<sup>j</sup>* [–] is the cell SOC, *Qj* [As] is the cell actual capacity, and *ij* [A] is the cell current.

The average non-linear OCV vs. SOC characteristic is obtained from step-response tests conducted on an automated battery tester on a single cell (Figure 7). For implementation purposes, the average OCV vs. SOC relationship is linearized between 95% and 5% SOC, as illustrated in Figure 8. That is the typical SOC range of a lithium-ion battery in a full-electric vehicle. For the interest of the reader, the electrical test system setup is shown in Figure 9. This linear model is described as:

$$\text{OCV}\_{\circ}(t) = a\_{\circ} + b\_{\circ}\text{SOC}\_{\circ}(t) \tag{3}$$

where - *aj*, *bj* are constant parameters derived for the model, and equal to 3.406 V and 0.673 V, respectively.

**Figure 7.** Step-response test data at 0.5 and 25 ◦C.

**Figure 8.** OCV vs. SOC characteristic.

**Figure 9.** Test setup: (**a**) Kokam 53 Ah SLPB pouch cell; (**b**) 3-D representation of one of the cells and its holder; (**c**) from right to left, Maccor 4000 Series automated battery tester, FRA, and thermal chambers cabinets.

On the other hand, a statistical model of cell-to-cell variations is built to estimate the value of the series resistance and the cell capacity for each cell in the module at different aging states, as described in next Section 2.1.2.

#### 2.1.2. Statistical Model of Cell-to-Cell Variations

Estimation or prediction of cell-to-cell variations over the battery life is particularly challenging. Manufacturing tolerances and uneven temperature distributions are key factors behind imbalances in electrical parameters such as capacity, internal resistance, or self-discharge rate. While there is a myriad of modeling and aging studies at the singlecell level, only a very few focus on the diversion of aging of large groups of lithium-ion cells. Moreover, just two of them present experimental results for large-capacity cells, as summarized in Table 1.

**Table 1.** Overview of experimental studies on cell-to-cell variations in the literature.

<sup>1</sup> Articles sorted in descending order of cell capacity.

In this study, statistical data obtained by Barreras et al. [28] through screening tests conducted on 208 Kokam SLPB 53 Ah cells tested at Beginning-of-Life (BOL) is used to parametrize the statistical model of cell-to-cell variation at BOL. The cell capacities are derived from the discharge capacities measured during a 1C full discharge at constant current and room temperature. The internal resistances are derived from measurements of the so-called direct current resistance (DCR) around room temperature and 50% SOC. The DCR parameterization is interesting in this context because it accounts for pure ohmic plus charge-transfer polarization effects, i.e., the dominant effects in an EV battery pack during driving [44]. The two histograms of the relative frequency of discharge capacity and the DCR resistance at BOL are shown in Figure 10. These histograms are fitted to normal distributions, which are the de facto statistical models of cell-to-cell variations at BOL [28].

**Figure 10.** Histograms and normal distributions of the relative frequency of (**a**) the DCR and (**b**) the 1C discharge capacity for 208 Kokam SLPB 53 Ah lithium-ion pouch cells at BOL [5,14].

On the other hand, in the absence of more statistical data from large format cells at other aging states in the literature, the Middle-of-Life (MOL) and End-of-Life (EOL) parameter distributions are calculated through certain manipulation of the BOL distributions based on insights from the literature, following the same approach of Pinto et al. [14]. Then, as summarized in Figure 11, based on these statistical models of parameter distributions, a mixed aging scenario with substantial cell-to-cell variations is generated for our model.

**Figure 11.** Mixed aging scenario for cell-to-cell variations considered for the 8S module in this study [14].

Table 2 presents the resistance and capacity values derived for each cell of the 8S module under this scenario. The values are presented in vectors of 8 elements and normalized to the mean value of the internal resistance at the BOL (2.09 mΩ), and the nominal cell capacity (53 Ah), respectively. Further information on the methods used to determine these datasets in [14].

**Table 2.** Normalized internal resistances and capacities for the 8S module.

In addition, for all the BEV driving scenarios we assumed the next dataset for the initial SOC of each cell in the 8S module: *ϕ* = [0.925 0.935 0.932 0.930 0.931 0.922 0.930 0.938]. The complement of this dataset is assumed for all the fast charging scenarios, i.e., (1 − *ϕ*).

#### 2.1.3. Thermal Model

With respect to the thermal model, we assumed a lumped-parameter model and considered that: (1) the heat generation is evaluated based on Joule losses; (2) the cell heat capacity is constant; (3) the thermal resistance from cell to ambient is constant and formulated based on heat convection; (4) there is certain conductive heat transfer from one cell to the next/previous.

This is represented by means of a simple lumped TECM for each cell with links to neighboring cells, as illustrated in Figure 12, where *Tj* [K] is the actual temperature of cell *j*, *Tamb* [K] is the ambient temperature, *Cp* [J/K] is the cell heat capacity, *Rcond* [K/W] is the conductive thermal resistance between adjacent cells, and *Rconv* [K/W] is the convective thermal resistance between each cell and the environment. For each cell *j*, the TECM is coupled with the corresponding EECM through the heat generation term, . *Qgen*,*<sup>j</sup>* [W], which represents Joule losses and is expressed as:

$$
\dot{Q}\_{\text{geen},\dot{\jmath}} = R\_{\dot{\jmath}} i\_{\dot{\jmath}}^2(t) \tag{4}
$$

**Figure 12.** TECM of cell *j* with links to neighboring cells.

Regarding heat capacity, is assumed that *CP* = 1032 J/K. This value is derived from the cell mass (1.2 kg) and the specific heat capacity (860 J/kgK)) determined by means of flash lamp thermal characterization tests by Barreras et al. [45,46]. The tests are conducted at room conditions on a similar cell (Kokam SLPB 40 Ah pouch cell), following ASTM E1461 standard procedures [47].

From the same references [45,46], the in-plane thermal conductivity is obtained. This value is used in combination with the cell dimensions, the tab dimensions, and the thermal conductivity of the tabs, to estimate the cell-to-cell conductive thermal resistance from the next equations:

$$R\_{cond} = \mathcal{Z}R\_{cond,cell} + R\_{cond,tabs} \tag{5}$$

where *Rcond*,*cell* [K/W] is the in-plane conductive thermal resistance of a cell, and *Rcond*,*tabs* [K/W] is the conductive thermal resistance of the cell tabs that physically connect adjacent cells, and are derived as follows:

$$R\_{\text{cond,cell}} = \frac{d/2}{k\_{||}A\_{||}}\tag{6}$$

where *<sup>d</sup>* = 0.222 m is the height of the cell, *<sup>k</sup>*|| = 33 W/mK is the in-plane thermal conductivity [45,46], and *<sup>A</sup>*|| <sup>=</sup> 0.01908 m<sup>2</sup> is the averaged cross-section of the cell calculated from the datasheet.

$$R\_{cond,tabs} = \frac{L/2}{k\_{tabs}A\_{tabs}}\tag{7}$$

where *L* = 0.02 m is the distance assumed between the tabs of adjacent cells, *ktabs* = 237 W/mK is the thermal conductivity of the tabs, assuming that is equal to the tabulated value of aluminum and neglecting contact resistances between tabs, and *Atabs* = <sup>32</sup>·10−<sup>5</sup> <sup>m</sup><sup>2</sup> is the cross-section of two tabs superimposed, calculated from the datasheet.

These calculations result in *Rcond* = 6.16 K/W, *Rcond*,*cell* = 1.76 K/W, and *Rcond*,*tabs* = 2.64 K/W, suggesting that both the cell thermal resistances and the tab resistance play an important role in cell-to-cell heat transfer through the tabs of adjacent cells. Unfortunately, there is a lack of results in the literature for other cells, for example, Zhu et al. [48] just estimated a value of 1.711 K/W for large format prismatic cells. In any case, *Rcond*,*cell* is an intrinsic cell property, which suggests that *Rcond* > 3.53 K/W even assuming that *Rcond*,*tabs* could be minimized by improved thermal design of tab-interconnection.

On the other hand, *Rconv* is derived from:

$$R\_{conv} = \frac{1}{hA} \tag{8}$$

where *h* [W/m2K] is the convective heat transfer coefficient, and *A* [m2] is the interface surface area of the solid material. In this study, we estimated *Rconv* = 0.813 K/W, assuming *h* = 15 W/m2K and *A* = 0.082 m2. The former is a typical value for low forced air velocities, and the latter is the sum of the areas of the larger front and rear faces of the cell. We considered a battery module with parallel cooling/heating based on forced air. The arrangement is sketched in Figure 13. In comparison with series cooling/heating, a parallel arrangement provides a more even distribution of the temperatures [14].

**Figure 13.** 2-D schematic of the parallel air-cooling arrangement assumed for the 8S module [5].

#### *2.2. Balancing System Model*

The overall balancing system topology proposed consists of a set of bi-directional DC/DC converters connected in parallel with each cell, which can move energy from/to each cell to/from the 8S battery module. This is the so-called CPC energy transfer configuration. As shown in Figure 6, the balancing system is represented through a simplified EECM, assuming that all the conduction power losses of each DC/DC converter can be lumped together into an equivalent resistive component, assuming *RB* = 10 mΩ. On the other hand, for the calculation of the overall balancing system losses, switching, magnetic, and drivers' energy losses are assumed to be constant and equal to *PB*,*loss*,*other* = 100 mW for each converter.

#### *2.3. Battery Power Profiles Generation*

To test the algorithm under realistic demanding conditions, we have generated both fast charging and aggressive driving power profiles considering an exemplary BEV and the battery specifications.

#### 2.3.1. Aggressive Driving Power Profiles

In a BEV, the battery pack supplies power and energy to the vehicle driveline, which is composed of the electric motor(s), inverter, and mechanical transmission. This load power (*pout*xi(t) = −Lx(t)) is dependent on numerous factors, including the type of driving cycle the vehicle is operating, the energy efficiency of the driveline's components, as well as the inertial, rolling, and aerodynamic resistance forces that affect the vehicle (see [49] for details). Given that our goals consist in the design and test of the balancing control algorithm, in what follows we will assume that the load power *pout* is a known disturbance. Its value is computed considering the uCar [49] as a reference vehicle. In this paper, we use the US06 (repeated a certain number of times) as the driving cycle, since we focus on more demanding aggressive driving scenarios, but the model framework also implements other standard and non-standard driving cycles. The driving process is finished when the end-of-discharge condition is detected, i.e., when min- SOC*<sup>j</sup>* ≤ 0.05, a common limit in real BEVs.

With regard to low voltage violations, they take place when one or several cells show a voltage below the lower voltage threshold set by the application. Here we used the minimum cell voltage stated by the manufacturer to define this limit (*vj* < 2.7 V), but more restrictive windows are common for safety and reliability reasons. In real-life, the BMS would derate the BEV battery power when a low voltage violation is detected [6,11], either temporarily or permanently by forcing a low-power operating mode, sometimes called "limp home" mode. However, here we decided not to implement such a derating algorithm, to enable a fair comparison between different cases by ensuring the same power levels. Instead, we allowed BEV operation until the aforementioned lower SOC threshold and quantified the operation time with low voltage violations as a key performance indicator.

#### 2.3.2. Fast Charging Power Profiles

The DC fast charging profiles are created based on a constant power constant voltage (CP-CV) strategy, which is common in fast charging protocols. The charging process shifts to the CV stage when max- *vj* ≥ 4.2 V. The power level in the CP stage is determined by trial-and-error aiming to charge with the maximum power that does not result in cell overcurrent considering the limits defined in the manufacturer's datasheet, i.e., max- *ij* ≤ 106 A. The power level is determined considering also the maximum balancing currents, which are below 20A for all the cases evaluated. The charging process is finished when the end-of-charge condition is detected, i.e., when max- SOC*<sup>j</sup>* ≥ 0.8. This is a also common setting in fast charging protocols.

#### *2.4. Multi-Agent Consensus Algorithm*

The coordination of distributed system with multiple agents is enabled by a protocol known as consensus algorithms, which allows the agents to exchange information via the communication network. In the case of the battery system, consensus algorithms can achieve information sharing and coordination among different cells.

The coordination of distributed system with multiple agents is enabled by a protocol known as consensus algorithms, which allows the agents to exchange information via the communication network. In the case of a battery pack, each cell in a pack represents one agent. An important feature of the consensus algorithm is that information exchange occurs only between the neighboring cells and that the centralized controller does not exist. For this reason, even if some of the communication links fail, the full information exchange capability can be preserved, where the actual level of resiliency depends on the communication network topology, as discussed in [50].

When every given cell knows the average of these variables, it is possible to design local control protocols that can ensure desired behavior of both the battery pack system and each cell totally independently of the centralized pack-level controller. In this context, a single point of failure is avoided, while the performance can be designed per technical requirements without any limitation.

The consensus algorithm is applied to the controller of every given cell by locally implementing a simple difference of variables of interest (in our case cell SOCs, terminal voltages, and temperatures) between each cell and those of other adjacent cells, which can be expressed as [4] .

$$\dot{\mathbf{x}}\_{\dot{\jmath}}(t) = \sum\_{k=N\_{\dot{\jmath}}} |\mathbf{x}\_{\dot{\jmath}}(t) - \mathbf{x}\_{k}(t)|\tag{9}$$

where *xj*(*t*) and *xk*(*t*) are the variables of interest in agent *j* and *k*. *Nj* represents the whole set of neighbors of cell *j*.

From the equation above, it can be understood that the variables of interest change interactively according to the values of local measurements and the receiving measurements coming from the neighboring cells. Provided that the communication network is fully connected, all the variable values will eventually converge to an average. Therefore, based on the consensus algorithm that information is shared among units, each cell can compute an exact average value of the variables of interest, i.e., cells SOCs, terminal voltages, and temperatures, as shown in [51].

The multi-agent network of the battery pack is modeled as an undirected and connected graph. And the consensus algorithm can be expressed as follows [5,6]:

$$
\dot{x}\_{\dot{\jmath}}(t) = -Lx(t) \tag{10}
$$

where *L* is the graph Laplacian of the network, which formally describes the network's topology. The elements of the matrix are defined as

$$L\_{jk} = \begin{cases} -1, & k \in N\_j \\ d\_{j\prime} & k = j \end{cases} \tag{11}$$

where *dj* is the number of neighbors of node *j*, which means the diagonal elements *Ljj* are determined by the number of edges attached to the node.

In this application, *x*(*t*) includes the average estimated SOC, or temperature, or terminal voltage, but other variables are also possible. Each cell (or agent) *j* receives the estimated current average value from neighboring agents and updates its information based on the SOC, temperature, and terminal voltage using three consensus balancing algorithms.

The estimation of SOC is updated as

$$\overline{\text{SOC}}\_{j}(t) = \text{SOC}\_{j}(t) + \sigma\_{1} \sum\_{k=N\_{j}} \int \left[ \overline{\text{SOC}}\_{j}(t) - \overline{\text{SOC}}\_{k}(t) \right] dt \tag{12}$$

where SOC*<sup>j</sup>* denotes the agent j's average estimated SOC of all agents, SOC*<sup>j</sup>* is the actual SOC of agent *j*, and *σ*<sup>1</sup> is the SOC consensus coefficient.

Similarly, the agent j's average temperature and terminal voltage of all agents can be obtained based on the next consensus algorithms:

$$\overline{T}\_{\dot{\jmath}}(t) = T\_{\dot{\jmath}}(t) + \sigma\_2 \sum\_{k=N\_{\dot{\jmath}}} \int \left[ \overline{T}\_{\dot{\jmath}}(t) - \overline{T}\_k(t) \right] dt \tag{13}$$

where *Tj* represents the estimation of agent *j*, *Tj* is the actual temperature of agent j, and *σ*<sup>2</sup> is the temperature consensus coefficient, and

$$
\overline{\boldsymbol{\upsilon}}\_{\dot{\boldsymbol{\beta}}}(t) = \boldsymbol{\upsilon}\_{\dot{\boldsymbol{\beta}}}(t) + \boldsymbol{\sigma}\_{3} \sum\_{k=N\_{\dot{\beta}}} \int \left[ \overline{\boldsymbol{\upsilon}}\_{\dot{\boldsymbol{\beta}}}(t) - \overline{\boldsymbol{\upsilon}}\_{k}(t) \right] dt \tag{14}
$$

where *vj* represents the estimation of agent *j*, *vj* is the actual terminal voltage of agent *j*, and *σ*<sup>3</sup> is the voltage consensus coefficient.

Then, the reference balancing current for each cell *j*, i.e., *iB*,*<sup>j</sup>* [A] in Figure 6, is calculated with a correction term obtained from the cooperation between agents, incorporating the SOC balancing (Equation (12)), temperature balancing (Equation (13)), and/or terminal voltage balancing (Equation (14)), based on multi-agent. As a result, the multi-agent consensus algorithm helps each local unit to converge towards the global average of SOC, terminal voltage, and temperature.

Regarding the value of each consensus coefficient, *σ*1, *σ*2, and *σ*3, they could be formally selected taking into account the topology of the communication network defined by graph Laplacian matrix *L* (Equation (11)) to ensure fast and stable convergence. Indeed, these parameters affect the second smallest eigenvalue of the *L*, which should be maximized to ensure the fastest possible convergence, as explained in detail in [52]. However, formal stability analysis falls beyond the scope of this paper, and the interested reader is referred to the literature for more details.

In this paper, we have just followed a sub-optimal trial-and-error selection method to determine the starting values of the consensus parameters. The selection is also supported by sensitivity analysis in the more complex cases of multi-function control.

In addition, we took into account the following considerations: (1) the values of the consensus coefficients determine the convergence rate of the algorithm for each variable of interest, being faster for larger values, (2) the control actions, i.e., the balancing currents, are larger for larger values of the coefficients and larger differences in the variables of interest, and (3) the magnitude of the differences might be quite different for each variable of interest, being particularly lower for the SOCs, as they are defined as dimensionless variables with values between 0 and 1. This means that different values are expected for each consensus parameter, being particularly larger for the SOC case.

It is also worth noting that in ordinary control theory, hardware implementation of the controller is often not considered, and therefore in theory too large consensus coefficients that might lead to extremely large control signals are possible. This might result in hardware or stability issues. In practice, this situation is typically avoided by including saturation in the actuator, as discussed in [53]. Here, we included a saturation block to the controller to limit the balancing currents to ±1C (53 A), but other limits could be explored.

Finally, it is worth noting that, within this framework, static or dynamic values for the consensus coefficients can be applied. For instance, in this paper, beyond the static coefficients, we also propose a dynamic voltage consensus coefficient determined in quadratic proportion to the battery current, *iout* [A]:

$$
\sigma\_3 = k\_1 \left( 1 + k\_2 i\_{out}^2 \right) \tag{15}
$$

where *k*<sup>1</sup> and *k*<sup>2</sup> [–] are constant coefficients, and typically *k*<sup>1</sup> *k*<sup>2</sup> to avoid controller saturation. This dynamic definition of the voltage consensus coefficient virtually eliminates the widely known problem in regular balancing systems [54] of over-balancing voltages during low current periods or under-balancing voltages during high current peaks.

#### **3. Results**

Using our simulation framework, we can easily study a myriad of scenarios with respect to battery states, external conditions, or power demand. In this paper, due to space constraints, we focus only on demonstrating the effectiveness of the algorithm under more demanding scenarios. Hence, we focus on aggressive driving and fast charging conditions (see Section 2.3), in combination with substantial differences in cell-to-cell variations (see Section 2.1.2), and moderate parallel forced air cooling (see Section 2.1.3). Thus, we assumed in all the scenarios analyzed the *mixed aging* cell-to-cell variations scenario. As qualitatively shown in Figure 12, this is the case with a larger variation in cell internal resistances and capacities, since it combines cells with BOL, MOL, and EOL parameterization.

We compare side-by-side the performance of single-controllers for either SOC, voltage, and temperature balancing, and evaluate the results in comparison with the 'no control' case, which is used as a benchmark. We also evaluate multi-function dual controllers, in particular: (1) simultaneous SOC and voltage balancing, and (2) SOC and temperature balancing.

For a detailed comparison, we present plots in time of key variables, but also provide performance metrics in corresponding tables. In the case of multi-function balancing, we also conduct sensitivity analysis, aiming to provide insights on the performance of the controller depending on the tuning of the consensus gains.

The following results are displayed for the aggressive driving scenarios:

**Figure 14.** Single controllers result in time. End condition: min SOC*<sup>j</sup>* ≤ 0.5. Scenario: aggressive driving (US06), mixed agingg, parallel forced air cooling (*h* = 15 W/m2K).

**Table 3.** Key metrics of single controllers results shown in Figure 14.

**Figure 15.** Single and dual controllers results in time. End condition: min SOC*<sup>j</sup>* ≤ 0.5. Scenario: aggressive driving (US06), mixed agingg, parallel forced air cooling (*h* = 15 W/m2K).

**Table 4.** Key metrics of single and dual controllers results shown in Figure 15.

**Figure 16.** Sensitivity analysis: dual balancing (SOC and temperature): *σ*<sup>1</sup> = 2000(1 − *α*), *σ*<sup>2</sup> = 20*α*, *σ*<sup>3</sup> = 0. End condition: min SOC*<sup>j</sup>* <sup>≤</sup> 0.5. Scenario: aggressive driving (US06), mixed aging, parallel forced air cooling (*<sup>h</sup>* <sup>=</sup> 15 W/m2K).

**Table 5.** Key metrics of dual balancing (SOC and temperature) results shown in Figure 16.

**Figure 17.** Single controllers results in time. Scenario: fast charging, mixed aging, parallel forced air cooling (*h* = 15 W/m2K). All cases using the CP-CV charge profile of the 'no control' case.

**Table 6.** Key metrics of single controllers results shown in Figure 17.

**Figure 18.** Sensitivity analysis: dual balancing (SOC and voltage): *σ*<sup>1</sup> = 2000(1 − *α*), *σ*<sup>2</sup> = 0, *σ*<sup>3</sup> = 250*α*. Scenario: fast charging, mixed aging, parallel forced air cooling (*h* = 15 W/m2K). All cases using the CP-CV charge profile of the 'no control' case.

**Table 7.** Key metrics of dual balancing (SOC and voltage) results of Figure 18.

**Figure 19.** Single and dual controllers results in time. Scenario: fast charging, mixed aging, parallel forced air cooling (*h* = 15 W/m2K). Each case using its own CP-CV charge power profile optimized to reduce charging times.

#### *3.1. Aggressive Driving Results*

From Figure 14 and Table 3, we learn that near-perfect SOC balancing (ΔSOC*rms* < 0.1%) is possible with relatively low currents and can also extend driving range (+10%). In practice, it might be limited by the error in the SOC estimation, which is not considered here. However, conventional SOC balancing cannot solve the problem of low voltage violations, overheating, and thermal unbalance.

On the other hand, temperature and voltage balancing might induce larger SOC to unbalance (7.9 and 3.4%, respectively), which might shorten the driving range (−18% and −1%) but actually minimizes the occurrence of low voltage violations and thus the potential need for derating during driving. In addition, both temperature and voltage balancing can reduce the maximum temperatures (in 12.8 and 7.4 ◦C) and the cell-to-cell temperature differences. The peak balancing currents are around 20 A, i.e., 50% higher than for SOC balancing, but below 10% of the peak battery current.

The magnitude of these balancing effects is dictated by the values of the consensus gains. To illustrate that, in Figure 15 and Table 4, we analyze the impact of implementing a dynamic voltage consensus coefficient (Equation (15)). In this case, we can see that the balancing action is more pronounced during high power peaks. To keep the balancing power level within reasonable limits, we limited the balancing currents to 53A (or 1C). It is observed that the dynamic gain boosts the performance of the voltage balancing controller, with a minor increase in SOC unbalances and balancing losses. Moreover, when used in combination with SOC balancing in a dual controller fashion, we can get the advantages of both controllers while minimizing their caveats.

As previously stated, the magnitude of the balancing effects is dictated by the values of the consensus gains. To illustrate the importance of appropriate tuning for the consensus gains, in Figure 16 and Table 5 we present a sensitivity analysis for a dual balancing case. We considered SOC and temperature balancing, but similar studies can be conducted for other scenarios. To conduct the parameter sweep we re-defined the values of the SOC and temperature consensus gains as a function of a dimensionless variable *α*, which takes values from 0 to 1. This is expressed as *σ*<sup>1</sup> = 2000(1 − *α*) and *σ*<sup>2</sup> = 20*α*. In extreme cases, the dual controller behaves as a single controller, as either *σ*<sup>1</sup> or *σ*<sup>2</sup> are equal to zero. As illustrated in Figure 16, different non-linear trends are observed in the metrics, including an irregular pattern for the maximum temperatures. These results highlight the importance of developing selection methods to balance the trade-offs of each controller.

#### *3.2. Fast Charging Results*

From Figure 17 and Table 6, we learn that the algorithm is also effective during fast charging scenarios. The SOC balancing controller can equalize SOC smoothly, and the voltage balancing controller is able not only to equalize terminal voltages but also to reduce both the maximum temperatures (>8 ◦C) and the cell-to-cell temperature differences. However, it increases the SOC unbalance.

As pointed out earlier for the aggressive driving scenarios, a balance between the pros and cons of single controllers can be achieved by combining them in a multi-function controller and making an appropriate selection of the consensus gains. This possibility is investigated in the sensitivity analysis presented in Figure 18 and Table 7 for a dual controller that performs SOC and voltage balancing.

For a fair comparison, the results displayed in Figures 17 and 18, and their corresponding Tables 6 and 7, are obtained using in all cases the same fast-charging power profile. The maximum power levels are determined considering the maximum terminal voltages of the 'no control' case (Figure 17a), and the maximum balancing currents of the SOC and voltage balancing cases (Figure 17b,c).

However, as can be observed in Figure 17b,c, and thanks to the balancing actions, it would be possible to charge the battery at higher power levels in those cases without violating the upper voltage limit. That is particularly true for the voltage balancing case (Figure 17c).

This is investigated in the cases shown in Figure 19, where the battery is charged using its own CP-CV charge profiles to optimize the charging time. In comparison with the 'no control' case (Figure 19a), which requires 1811 s to bring the battery up to the end-of-charge condition, the battery with voltage balancing (Figure 19b) and dual balancing (SOC and voltage) (Figure 19c) can be charged up to the same condition in only 1400 s and 1655 s, respectively. This enables a charging time reduction of 22.7% and 8.6%, respectively.

#### **4. Discussion**

Here we discuss other qualitative aspects of this work. Firstly, we would like to acknowledge that both the EECM and the TECM proposed are relatively simple, but they are sufficient to validate the novel balancing control algorithm proposed. This is particularly true when coupled with a rational distribution of cell-to-cell parameter variations. Indeed, an equivalent modeling methodology has already been followed and validated experimentally by de Castro et al. [20].

In addition, this relatively simple electro-thermal model can be formulated with convex functions in a convex set. This means that in future work the control for the balancing system can be evaluated offline using convex optimization, following the pioneering methodology developed by Pinto et al. [14]. This enables the comparison of the sub-optimal online results obtained with the novel consensus algorithm, with the global optimal results provided by convex optimization.

Apart from that, we would like to highlight the flexibility of the algorithm, which can be easily modified to equalize other variables of interest. This can include variables related to battery internal states and degradation mechanisms, such as the surface concentration of electrode particles, or the rates of degradation derived from more or less complex degradation models. Thus, it would be possible to equalize battery lifetime at the cell level with this framework if such variables or states can be monitored, diagnosed, and/or predicted up to some extent, which is doable with the current state-of-the-art.

As a final remark, it is worth noting that the model framework is extremely fast one study case can be run in less than a second. This is very convenient when running sensitivity analysis of key variables of interest, such as the consensus gains, and opens the door to generate large amounts of synthetic data. This large amount of data could be useful, for example, in combination with big data analytics to uncover hidden patterns, correlations, and other valuable insights.

#### **5. Conclusions**

This study presents for the first time a multi-objective distributed control for balancing systems, based on a multi-agent consensus algorithm. In our view, the algorithm's outstanding effectiveness, scalability, flexibility, and ease of implementation and tuning make it an ideal candidate for industry adoption. Beyond its distributed architecture, our new consensus algorithm offers key advantages over current centralized and de-centralized algorithms in the literature, such as (1) model-based, since it does not demand prediction of future variables and does not need a model of the system; (2) machine learning, since it does not demand any data for training and validation, and has extremely low memory and computational requirements; and (3) fuzzy logic, since it offers a systematic implementation approach.

Our results show that our novel multi-functional balancing can boost the performance of batteries with substantial cell-to-cell differences under the most demanding operating conditions, i.e., aggressive driving and DC fast charging (2C). Indeed, it is well known that under high power demand battery performance might be limited by overheating and excessive unbalance in cells SOC, terminal voltages, or temperatures. However, we demonstrate that a well-tuned consensus algorithm can generate balancing current setpoints that virtually eliminate these limitations, achieving a good balance between the pros and cons of single controllers.

In comparison with no control, driving times are increased (>10%), charging times reduced (>22%), maximum cell temperatures decreased (>10 ◦C), temperature differences lowered (~3 ◦C rms), and the occurrence of low voltage violations during driving reduced (>5×). This minimizes the need for power derating and enhances the driver experience. In a broader sense, it might also have an impact on the thermal management design, by reducing the demand for intensive and uniform cooling.

The benefits of multi-functional balancing are particularly interesting in the case of (1) EVs oriented to motorsport (e.g., Formula E), (2) any EVs that contemplate aggressive driving operating modes, such as the Tesla ludicrous mode, (3) all EVs that offer supercharging, and (4) batteries with significant cell-to-cell differences. In future work, we would like to evaluate the impact of the control in battery lifetime and to investigate the applicability within battery energy storage applications.

**Author Contributions:** Conceptualization, J.V.B. and R.d.C.; Formal analysis: J.V.B., and R.d.C.; Methodology: J.V.B., R.d.C. and T.D.; Visualization, J.V.B. and R.d.C.; Writing—original draft preparation, J.V.B., R.d.C., Y.W. and T.D.; writing—review and editing, J.V.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** J.V.B. was funded by the EPSRC Faraday Institution Multi-Scale Modelling Project (EP/S003053/1, grant number FIRG003).

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

#### *Article*

## **Adaptive Square-Root Unscented Kalman Filter-Based State-of-Charge Estimation for Lithium-Ion Batteries with Model Parameter Online Identification**

#### **Quan Ouyang, Rui Ma, Zhaoxiang Wu, Guotuan Xu and Zhisheng Wang \***

College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China<EMAIL_ADDRESS>(Q.O<EMAIL_ADDRESS>(R.M<EMAIL_ADDRESS>(Z.W<EMAIL_ADDRESS>(G.X.)

**\*** Correspondence<EMAIL_ADDRESS>Tel.: +86-025-5283-2301

Received: 13 August 2020; Accepted: 17 September 2020; Published: 22 September 2020

**Abstract:** The state-of-charge (SOC) is a fundamental indicator representing the remaining capacity of lithium-ion batteries, which plays an important role in the battery's optimized operation. In this paper, the model-based SOC estimation strategy is studied for batteries. However, the battery's model parameters need to be extracted through cumbersome prior experiments. To remedy such deficiency, a recursive least squares (RLS) algorithm is utilized for model parameter online identification, and an adaptive square-root unscented Kalman filter (SRUKF) is designed to estimate the battery's SOC. As demonstrated in extensive experimental results, the designed adaptive SRUKF combined with RLS-based model identification is a promising SOC estimation approach. Compared with other commonly used Kalman filter-based methods, the proposed algorithm has higher precision in the SOC estimation.

**Keywords:** lithium-ion batteries; state-of-charge estimation; adaptive square-root unscented Kalman filter; recursive least squares

#### **1. Introduction**

Rechargeable lithium-ion batteries have been widely used in numerous applications due to their superior power performance, long life cycle, and so forth [1,2]. The state-of-charge (SOC) is an important and fundamental parameter that indicates the remaining available capacity of the battery [3]. Inaccurate SOC estimation can easily lead to the battery's overdischarge or overcharge even resulting in an explosion. It hence calls for promising strategies to accurately estimate the SOC of the battery's.

Two commonly utilized non-model-based SOC estimation approaches are the ampere-hour counting algorithm and open circuit voltage (OCV) method, respectively. Ampere-hour counting algorithm uses the time integration of the current as the indicator of the change of the SOC of the battery. Although this method is simple to implement, measurement errors accumulate since the SOC calculation process is open-loop based. Generally, it needs frequent calibrations to eliminate the battery's SOC estimation error. Derived from the identified relationship between the battery's OCV and SOC [4], the SOC can be obtained through measuring the OCV of the battery. However, the OCV can only be measured by cutting the battery off from the external circuit, which could inconvenience the battery user.

In order to get more accurate estimated SOC, model-based SOC estimation strategies attract great research attention. Currently, the commonly used battery models can be classified into the following three categories: electrochemical mechanism models [5,6], equivalent circuit models [7], and neural network models [8]. Among them, the equivalent circuit models, utilizing the circuits to

simulate the dynamic characteristics of the battery, have reasonable computational complexity and great flexibility in battery materials and sizes, which are the most suitable to be utilized for the SOC estimation. Extended Kalman filter (EKF) algorithms are the most widely studied model-based SOC estimation approaches [9,10]. Adaptive EKFs are proposed in [11] and [12] to estimate the SOC with noise covariances online update. However, the nonlinear battery model needs to be linearized in these EKF algorithms, which will reduce the SOC estimation accuracy of the battery. The unscented Kalman filter (UKF) [13] and adaptive UKF [14] are proposed as the improvements of the EKFs for the SOC estimation. The first-order linearization is not required in the UKFs. Instead, they utilize a nonlinear transformation, called unscented transform (UT), for the propagation of the battery states' mean and covariance. The square-root UKF (SRUKF) can be treated as an enhanced version of the UKF with the attached advantage that the positive semi-definiteness of the state covariance can be guaranteed [15]. In addition, it needs less computational complexity for state estimation. In [16], based on a radial basis function neural network battery model, an SRUKF is utilized for the SOC estimation. A square-root spherical UKF is designed for SOC estimation for the battery in Nanosatellite [17]. An adaptive SRUKF method is designed in [18] to estimate the battery's SOC. A proportional-integral observer is proposed for the SOC estimation of the battery in electric drive vehicles [19]. A neural network-based SOC observer is proposed in [20] with the convergence proved utilizing Lyapunov stability analysis. A disturbance observer-based strategy is designed in [21] to estimate the SOC, which can reduce the computational burden without reducing the estimation accuracy.

The above SOC estimation strategies [9–14,16–20] all run on a battery model with parameters known. However, the model parameters must be extracted by cumbersome prior experiments. It will be more convenient if the model parameters are not necessary to be identified in advance for the designed SOC estimation algorithm. Therefore, an interesting and important question can be investigated: how to design a battery's SOC estimation strategy with the model parameters treated as unknown values. In [22], a recursive least squares (RLS) algorithm is utilized to determine the model parameters of the battery, and a linear observer is adopted to estimate the battery's SOC. However, the model noise information is not considered in this method, and the SOC estimation accuracy is reduced. The battery's SOC and model parameters are simultaneously estimated by utilizing an iterated EKF in [23]. A UKF algorithm is developed to estimate the SOC with another adaptive UKF utilized for the online identification of the model parameters of the battery in [24]. Its experimental results indicate that this UKF algorithm has higher SOC estimation accuracy compared with the EKF.

In this paper, an equivalent circuit model is utilized to describe the dynamics of the battery, where the model parameters are considered to be unknown. An RLS algorithm is utilized for model parameter online identification. Next, an adaptive SRUKF based on this online identified battery model is developed to estimate the battery's SOC, where a noise statistic estimator is utilized for the noise information online update. The main novelties of this work are summarized in the following two aspects.

(1) The model parameters are treated as unknown variables, and an RLS algorithm is utilized to estimate them online. This can avoid unnecessary experiments prior to SOC estimation for parameter extraction.

(2) Extensive experiments demonstrate the effectiveness of the proposed charging strategy showing that the proposed adaptive SRUKF can provide higher SOC estimation accuracy compared with other commonly used Kalman filter-based methods.

The rest of this manuscript is arranged as follows. In Section 2, the battery's equivalent circuit model is presented. In Section 3, an RLS algorithm is utilized for the online model parameter identification, and an adaptive SRUKF is developed to estimate the SOC of the battery. Related experimental results are included in Section 4, and concluding remarks are provided in Section 5.

#### **2. Battery Model Development**

#### *2.1. Battery Equivalent Circuit Model*

A commonly used equivalent circuit model [25,26] is utilized to simulate the lithium-ion battery's dynamics, which can strike an excellent balance between accuracy and simplicity. As illustrated in Figure 1, the capacitor *Cb* denotes the battery's full charged capacity, and the resistor *R*<sup>0</sup> represents the battery's energy losses during the charging and discharging process. The RC network (*Rt*, *Ct*) characterizes the battery's transient voltage-current response. The voltage across *Cb* represents the battery's SOC quantitatively, with 0–1 V corresponding to 0–100% of the SOC [20]. The mapping from the SOC to the OCV of the battery is denoted as

$$V\_{\text{OC}} = \text{g}(\text{SOC}),\tag{1}$$

where *VOC* and *SOC* are the battery's OCV and SOC, respectively; *g*(·) is a nonlinear function. Based on the Kirchhoff's laws of current and voltage, the dynamics of the battery is obtained as

$$\begin{aligned} \dot{SOC} &= -\frac{\eta\_0}{\mathbf{C}\_b} I\_B\\ \dot{V}\_t &= -\frac{1}{R\_t \mathbf{C}\_t} V\_t + \frac{1}{\mathbf{C}\_t} I\_B\\ V\_B &= V\_{OC} - R\_0 I\_B - V\_{t\prime} \end{aligned} \tag{2}$$

where *Vt*, *VB*, and *IB* are, respectively, the voltage across the capacitor *Ct*, the terminal voltage, and the current of the battery; *η*<sup>0</sup> is the Coulomb coefficient.

**Figure 1.** Equivalent circuit model of the battery.

#### *2.2. Model Discretization*

With the current *IB* assumed to be constant during each sampling period, and the battery model (2) can be discretized as follows:

$$\begin{aligned} SOC(k) &= SOC(k-1) - \frac{T}{\mathbb{C}\_b} I\_B(k-1) \\ V\_t(k) &= \varepsilon^{-\frac{T}{R\_t \mathbb{C}\_t}} V\_t(k-1) + (1 - \varepsilon^{-\frac{T}{R\_t \mathbb{C}\_t}}) R\_t I\_B(k-1) \\ V\_B(k) &= \lg(SOC(k)) - R\_0 I\_B(k) - V\_t(k) \end{aligned} \tag{3}$$

with *T* the sampling period. To simplify the notations, with the process and measurement noise considered, the battery model (3) can be expressed in the following form

$$\begin{aligned} x(k) &= A(\theta)x(k-1) + B(\theta)u(k-1) + w(k) \\ y(k) &= h(x(k), \theta, u(k)) + v(k), \end{aligned} \tag{4}$$

with

$$A(\theta) = \begin{bmatrix} 1 & 0 \\ 0 & \theta\_2 \end{bmatrix}, \quad B(\theta) = \begin{bmatrix} -\frac{T}{C\_b} \\ \theta\_1 - \theta\_1 \theta\_2 \end{bmatrix}$$

$$h(x(k), \theta, u(k)) = g(x\_1(k)) - x\_2(k) - \theta\_3 u(k)$$

where the system output is *y*(*k*) - *VB* <sup>∈</sup> <sup>R</sup>; *<sup>u</sup>*(*k*) - *IB* <sup>∈</sup> <sup>R</sup> is the system input; the states *<sup>x</sup>*(*k*) <sup>∈</sup> <sup>R</sup><sup>2</sup> and parameters *<sup>θ</sup>* <sup>∈</sup> <sup>R</sup><sup>3</sup> are defined as *<sup>x</sup>*(*k*)=[*x*1(*k*), *<sup>x</sup>*2(*k*)]*<sup>T</sup>* - [*SOC*(*k*), *Vt*(*k*)]*<sup>T</sup>* and *θ* = [*θ*1, *θ*2, *θ*3] *<sup>T</sup>* - [*Rt*, *e* <sup>−</sup> *<sup>T</sup> RtCt* , *R*0] *<sup>T</sup>*, respectively; *w*(*k*) and *v*(*k*) are the process noise and the measurement noise of the battery model, which are usually assumed as uncorrelated zero-mean Gaussian white noises that satisfy

$$\begin{array}{l} w(k) \sim \mathcal{N}(0, Q^2) \\ v(k) \sim \mathcal{N}(0, R^2), \end{array} \tag{5}$$

with *<sup>Q</sup>*<sup>2</sup> <sup>∈</sup> <sup>R</sup>2×2, and *<sup>R</sup>*<sup>2</sup> <sup>∈</sup> <sup>R</sup> representing the covariances of the process and measurement noises. In practice, the noise statistic is correlated with the operating environment of the battery, which is generally unknown.

#### *2.3. Observability Analysis*

Referring to [27], the battery model (4) satisfies the observability condition, if the following observability matrix

$$
\theta = \begin{bmatrix} \mathcal{C}(k) \\ \mathcal{C}(k+1)A(\theta) \end{bmatrix} \tag{6}
$$

$$
\mathcal{C}(k) = \begin{bmatrix} \frac{\mathrm{d}\mathcal{Y}(\mathbf{x}\_1(k))}{\mathrm{d}\mathbf{x}\_1} & -1 \end{bmatrix}
$$

with

$$\mathcal{L}(\mathbf{v}) = \begin{bmatrix} \frac{\mathbf{d}x\_1}{\mathbf{d}x\_1} \end{bmatrix}$$

is full rank for all *x*.

As *Cb* is much larger than the value of *IB*(*k*), the SOC change is small enough in each sampling step. Hence, <sup>d</sup>*g*(*x*1(*k*)) <sup>d</sup>*x*<sup>1</sup> can be simplified to a constant *<sup>k</sup>*<sup>0</sup> > 0 [28,29]. By substituting (4) into (6), it can be derived that

$$
\theta = \begin{bmatrix} k\_0 & -1 \\ k\_0 & -\theta\_1 \end{bmatrix}. \tag{7}
$$

Since *k*<sup>0</sup> is a positive constant and *θ*<sup>2</sup> is less than 1 as *θ*<sup>2</sup> is an exponential function, *ϑ* is full rank. Hence, it can be concluded that if the model parameter vector *θ* is known, the battery system is observable, therefore the battery's SOC can be estimated.

The model parameters are assumed to be known in numerous SOC estimation algorithms in the literature such as [9–14]. However, the RC parameters *θ* in the equivalent circuit model of the battery should be identified through cumbersome and inconvenient prior experiments [30]. It motivates us to propose an SOC estimation strategy by treating the parameters *θ* in (4) as unknown, and online estimating the model parameters and the SOC of the battery simultaneously.

#### **3. Battery SOC and Model Parameter Estimation**

Based on the model developed above, an RLS algorithm is utilized for the online model parameter identification, and an adaptive SRUKF algorithm is designed to estimate the SOC of the battery.

#### *3.1. Model Parameter Identification with Rls Algorithm*

Referring to [28,29], it can be assumed that the OCV of the battery remains constant during a short sampling period that satisfies *g*(*x*1(*k*)) − *g*(*x*1(*k* − 1)) = 0. With defining *y*1(*k*) = *y*(*k*) − *y*(*k* − 1) and *u*1(*k*) = *u*(*k*) − *u*(*k* − 1), from (4), it yields

$$y\_1(k) = \mathcal{U}^T(k)\Phi + v\_1(k),\tag{8}$$

with

$$\Phi \stackrel{\triangle}{=} \begin{bmatrix} \theta\_{2\prime} - \theta\_{3\prime} \ \theta\_2 \theta\_3 - (1 - \theta\_2)\theta\_1 \end{bmatrix}^T$$

$$\mathcal{U}(k) \stackrel{\triangle}{=} \begin{bmatrix} y\_1(k-1), u\_1(k), u\_1(k-1) \end{bmatrix}^T$$

where *v*1(*k*) is the noise. Then, an RLS is proposed for the battery model parameters online identification as follows:

$$\Phi(k) = \Phi(k-1) + \Gamma(k)(y\_1(k) - lI^T(k)\Phi(k-1))\tag{9}$$

where <sup>Φ</sup><sup>ˆ</sup> (*k*)=[*φ*ˆ1(*k*), *<sup>φ</sup>*ˆ2(*k*), *<sup>φ</sup>*ˆ3(*k*)]*<sup>T</sup>* <sup>∈</sup> <sup>R</sup><sup>3</sup> is the estimation of <sup>Φ</sup>; <sup>Γ</sup>(*k*) <sup>∈</sup> <sup>R</sup><sup>3</sup> is the designed gain vector that can be updated as

$$\begin{aligned} \Gamma(k) &= \frac{P(k-1)lI(k)}{\lambda + lI^T(k)P\_\theta(k-1)lI(k)}\\ P\_\theta(k) &= \frac{P\_\theta(k-1) - \Gamma(k)lI^T(k)P\_\theta(k-1)}{\lambda} \end{aligned} \tag{10}$$

with *λ* denoting the forgetting factor that is normally chosen as a value between 0.98 and 0.995. Based on (9) and (10), the estimated battery model parameter vector ˆ *θ*(*k*) is calculated that

$$\theta(k) = [\frac{-\hat{\Phi}\_1(k)\hat{\Phi}\_2(k) - \hat{\Phi}\_3(k)}{1 - \hat{\Phi}\_1(k)}, \hat{\Phi}\_1(k), -\hat{\Phi}\_2(k)]^T. \tag{11}$$

#### *3.2. Adaptive Square-Root Ukf Based Soc Estimation*

Different from the traditional UKF, the square-root of the state covariance matrix is directly propagated in the SRUKF method, avoiding the need to calculate the square-root of the state covariance at each iteration step. The SRUKF is more advantageous with better numerical properties and guaranteed positive semi-definiteness of the state covariance matrix [31]. Due to these advantages, based on the battery model with the estimated model parameters, an adaptive SRUKF algorithm is adopted here to estimate the battery's SOC, where an adaptive noise statistic estimator is utilized to update the noise statistic information at each sampling instant to get rid of its dependence on priori noise information. The algorithmic process of the adaptive SRUKF algorithm can be divided into four steps: initialization, prediction, measurement update, and noise statistic estimation, which are provided as follows:

**Step 1: Initialization**. The initial estimated state vector is set as *x*ˆ(0) = *E*[*x*(0)] and the initial square-root of the state covariance matrix is *<sup>S</sup>*(0) = *cholupdate*{*E*[(*x*(0) − *<sup>x</sup>*ˆ(0))(*x*(0) − *<sup>x</sup>*ˆ(0))*T*]}, where *E*[·] denotes the mean and *cholupdate*{·} represents the Cholesky factorization [32]. The initial estimated covariances of the process and measurement noises are set as *Q*ˆ <sup>2</sup>(0) = *Q*<sup>2</sup> <sup>0</sup> and *<sup>R</sup>*<sup>ˆ</sup> <sup>2</sup>(*k*) = *R*2 <sup>0</sup>, respectively.

**Step 2: Prediction**. The UT technique is utilized for the propagation of the states' mean and covariance, where sigma points are chosen to approximate the mean and the covariance of the actual distribution. A set of 2*n* + 1 sigma points are selected in the following way

$$\begin{cases} \chi\_0(k-1) = \pounds(k-1),\\ \chi\_i(k-1) = \pounds(k-1) + (\sqrt{n+\lambda}\pounds(k-1)), \quad i = 1, \cdots, n, \\ \chi\_i(k-1) = \pounds(k-1) - (\sqrt{n+\lambda}\pounds\_{i-n}(k-1)), \; i = n+1, \cdots, 2n \end{cases} \tag{12}$$

where *<sup>x</sup>*ˆ(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* denotes the estimated state vector at time step *<sup>k</sup>* <sup>−</sup> 1, *<sup>n</sup>* represents the state vector's dimension , *Si*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) (<sup>1</sup> <sup>≤</sup> *<sup>i</sup>* <sup>≤</sup> *<sup>n</sup>*) is the *<sup>i</sup>*-th column of the matrix *<sup>S</sup>*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) <sup>∈</sup> <sup>R</sup>*n*×*<sup>n</sup>* with *<sup>S</sup>*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) the square-root matrix of the state covariance, *λ* is a scalar that can be calculated as

$$
\lambda = \mathfrak{a}^2 (n + \sigma) - n,\tag{13}
$$

where *α* and *σ* are the tuning parameters determining the spread of the sigma points. With utilizing ˆ *θ*(*k* − 1) to replace *θ* in (4), each point in (12) is updated through the battery model as

$$\chi\_i(k|k-1) = A(\hat{\theta}(k-1))\chi\_i(k-1) + B(\hat{\theta}(k-1))u(k-1),\tag{14}$$

for 0 ≤ *i* ≤ 2*n*. Then, the the a priori estimation of the state vector *x*ˆ(*k*|*k* − 1) is computed as follows:

$$
\hat{\mathfrak{X}}(k|k-1) = \sum\_{i=0}^{2n} \mathcal{W}\_i^{(m)} \chi\_i(k|k-1) \tag{15}
$$

with the weights

$$\begin{cases} \begin{array}{ll} \mathsf{W}\_{0}^{(m)} = \frac{\lambda}{\lambda+n}, \\ \mathsf{W}\_{i}^{(m)} = \frac{1}{2(\lambda+n)} \quad i = 1, \cdots, 2n. \end{array} \end{cases}$$

The a priori square-root of the state covariance *S*(*k*|*k* − 1) is updated as:

$$S\_1(k|k-1) = qr\{\big|\sqrt{\mathcal{W}\_1^{(c)}}(\chi\_1(k|k-1) - \hat{\mathfrak{x}}(k|k-1)) \ \cdots\}\tag{16}$$

$$\sqrt{\mathcal{W}\_{2n}^{(c)}}(\chi\_{2n}(k|k-1) - \hat{\mathfrak{x}}(k|k-1))\big{\mathcal{Q}}(k)\big{|}^{T}\}\tag{16}$$

$$S(k|k-1) = \operatorname{ch}\operatorname{dup}\operatorname{date}\{S\_1^{T}(k|k-1), \chi\_0(k|k-1) - \hat{\mathfrak{x}}(k|k-1), \mathcal{W}\_0^{(c)}\}$$

with the weights

$$\begin{cases} \begin{array}{ll} \mathcal{W}\_0^{(c)} = \frac{\lambda}{\lambda + n} + (1 - \alpha^2 + \beta) , \\ \mathcal{W}\_i^{(c)} = \frac{1}{2(\lambda + n)} & i = 1, \dots, 2n \end{array} \end{cases}$$

where *qr*{·} denotes the QR decomposition [32]; *<sup>Q</sup>*ˆ(*k*) denotes the estimation of the square-root of the process noise covariance matrix; *β* is a constant, which is usually chosen as 2 for the Gaussian distribution of the states [33]. Note that the QR decomposition of the matrix *<sup>D</sup>* = [ *W*(*c*) <sup>1</sup> (*χ*1(*k*|*k* − 1) − *<sup>x</sup>*ˆ(*k*|*<sup>k</sup>* <sup>−</sup> <sup>1</sup>)) ··· *<sup>Q</sup>*ˆ(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>)]*<sup>T</sup>* <sup>∈</sup> <sup>R</sup>3*n*×*<sup>n</sup>* is given by *<sup>D</sup>* <sup>=</sup> *<sup>Q</sup>*1*R*1, where *<sup>Q</sup>*<sup>1</sup> <sup>∈</sup> <sup>R</sup>3*n*×3*<sup>n</sup>* is an orthogonal matrix and *<sup>R</sup>*<sup>1</sup> <sup>∈</sup> <sup>R</sup>3*n*×*<sup>n</sup>* is an upper triangular matrix [32]. Here, *qr*{*D*} denotes the QR decomposition of the matrix *<sup>D</sup>* with *<sup>R</sup>*¯ <sup>1</sup> <sup>∈</sup> <sup>R</sup>*n*×*<sup>n</sup>* returned, where *<sup>R</sup>*¯ <sup>1</sup> is the upper triangular part of *<sup>R</sup>*1, i.e., *<sup>S</sup>*1(*k*|*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) = *<sup>R</sup>*¯ <sup>1</sup> in (16). The Cholesky factor of the rank 1 update *<sup>P</sup>* ± *νξξ<sup>T</sup>* is represented as *<sup>S</sup>*<sup>2</sup> = *cholupdate*{*S*1, *<sup>ξ</sup>*, ±*ν*}, where *S*<sup>1</sup> is the original Cholesky factor of *P* with *P* = *S*1*S<sup>T</sup>* <sup>1</sup> . In (16), *S*(*k*|*k* − 1) = *cholupdate*{·} denotes the Cholesky factor of *S<sup>T</sup>* <sup>1</sup> (*k*|*<sup>k</sup>* <sup>−</sup> <sup>1</sup>)*S*1(*k*|*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) + *<sup>W</sup>*(*c*) <sup>0</sup> *ξξ<sup>T</sup>* with *<sup>ξ</sup>* = *<sup>χ</sup>*0(*k*|*<sup>k</sup>* − <sup>1</sup>) − *<sup>x</sup>*ˆ(*k*|*<sup>k</sup>* − <sup>1</sup>). Note that *S<sup>T</sup>* <sup>1</sup> (*k*|*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) is used in (16) since *<sup>R</sup>*¯ <sup>1</sup> denotes the transpose of the Cholesky factor of *DD<sup>T</sup>* with *R*¯ *T* <sup>1</sup> *<sup>R</sup>*¯ <sup>1</sup> = *<sup>R</sup>*¯ *<sup>T</sup>* <sup>1</sup> *<sup>Q</sup>TQR*¯ <sup>1</sup> = *DDT*. From (16), it yields that the corresponding priori sate covariance matrix *P*(*k*|*k* − 1) is

$$\begin{split} P(k|k-1) &= S(k|k-1)S^T(k|k-1) = D^T D + \mathcal{W}\_0^{(c)} \xi\_\sharp^T \\ &= \sum\_{i=0}^{2n} \mathcal{W}\_i^{(c)} \left( \chi\_i(k|k-1) - \mathfrak{x}(k|k-1) \right) \left( \chi\_i(k|k-1) - \mathfrak{x}(k|k-1) \right)^T + \hat{\mathcal{Q}}^2(k), \end{split} \tag{17}$$

which is consistent with that of the UKF [24]. Note that *S*(*k*|*k* − 1) is utilized rather than the square-root of *P*(*k*|*k* − 1) in the designed SRUKF since it is more efficient to calculate the QR decomposition and Cholesky factorization in (16) than performing the square-root of *P*(*k*|*k* − 1) directly in the UKF [15]. The estimated output is computed by propagating the sigma points based on the model (4) as follows:

$$\mathcal{Y}(k|k-1) = \sum\_{i=0}^{2n} \mathcal{W}\_i^{(m)} \mathbf{Y}\_i(k|k-1),\tag{18}$$

*Energies* **2020**, *13*, 4968

with

$$\mathcal{Y}\_i(k|k-1) = h(\chi\_i(k|k-1), \hat{\theta}(k), u(k)), \tag{19}$$

where *<sup>y</sup>*ˆ(*k*|*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) is the estimated output through the SRUKF, and <sup>ˆ</sup> *θ*(*k*) are updated based on (9)–(11).

**Step 3: Measurement update**. Similarly, the square-root of the measurement covariance *Sy*(*k*) can be calculated by

$$S\_y(k) =
cholupdate\{ [S\_{y1}^T(k), Y\_0(k|k-1) - \mathcal{Y}(k|k-1), W\_0^{(c)}] \},\tag{20}$$

with

$$S\_{y1}(k) = \eta r \{ \lceil \sqrt{\mathbb{W}\_1^{(\varepsilon)}} (\mathbf{Y}\_1(k|k-1) - \mathfrak{H}(k|k-1)) \cdot \dots \cdot \sqrt{\mathbb{W}\_{2n}^{(\varepsilon)}} (\mathbf{Y}\_{2n}(k|k-1) - \mathfrak{H}(k|k-1)) \cdot \mathfrak{R}(k) \rceil^{\mathsf{T}} \},\tag{21}$$

with *R*ˆ(*k*) denoting the estimated square-root of the measurement noise covariance. The cross-correlation covariance *Pxy*(*k*) are obtained as

$$P\_{\rm XY}(k) = \sum\_{i=0}^{2n} W\_i^{(c)} [\chi\_i(k|k-1) - \pounds(k|k-1)] [\mathbf{Y}\_i(k|k-1) - \hat{\varrho}(k|k-1)].\tag{22}$$

Then, the estimated state vector *x*ˆ(*k*) is updated as follows:

$$\pounds(k) = \pounds(k|k-1) + L(k)(\mathcal{y}(k) - \mathcal{Y}(k|k-1)),\tag{23}$$

with the filter gain vector *L*(*k*) updated as

$$L(k) = P\_{xy}(k)(S\_y^{-1}(k))^T S\_y^{-1}(k),\tag{24}$$

where *S*−<sup>1</sup> *<sup>y</sup>* (*k*) denotes the inverse of *Sy*(*k*). Based on (20) and (24), the square-root matrix of the state covariance is updated that

$$S(k) = \text{cholupdate}\{\mathcal{S}(k|k-1), L(k)\mathcal{S}\_{\mathcal{Y}}(k), -1\}. \tag{25}$$

**Step 4: Noise statistic estimation**. A moving window of the output estimation error sequence is utilized for the noise statistic estimation. Referring to [14,24], the covariances of the process and measurement noises are updated as

$$\begin{split} \mathcal{Q}^2(k) &= L(k) \left( \sum\_{j=k-L+1}^{k} (y(j) - \mathfrak{z}(j|j-1))(y(j) - \mathfrak{z}(j|j-1))^T \right) L^T(k) \\ \mathcal{R}^2(k) &= \sum\_{j=k-L+1}^{k} (y(j) - \mathfrak{z}(j|j-1))(y(j) - \mathfrak{z}(j|j-1))^T \\ &\quad + \sum\_{i=1}^{2n} W\_i^{(c)} (\mathbf{Y}\_i(k|k-1) - y(k)) (\mathbf{Y}\_i(k|k-1) - y(k))^T, \end{split} \tag{26}$$

where *L* is the size of the selected moving window used for covariance matching.

#### **4. Experimental Results**

To validate the performance of the designed adaptive SRUKF with RLS-based model parameters online identification on battery SOC estimation, an IFP36130155-36Ah lithium-ion battery with a nominal capacity of 36 Ah [26] was chosen for the experiment. The experiment was conducted at room temperature. The mapping from the battery's SOC to its OCV was as shown in Figure 2. The ITECH dc electronics Load IT8512 was programmed to simulate the actual working condition of the battery with the recorded signals of the current and the corresponding terminal voltage shown in Figures 3 and 4, where the sampling rate is 1 Hz. In the RLS algorithm, the initial estimated model parameter vector was randomly selected as ˆ *θ*<sup>0</sup> = [0.001, 0.95, 0.02] *<sup>T</sup>*, *<sup>λ</sup>* and *<sup>P</sup><sup>θ</sup>* (0) are set as 0.99 and *diag*{0.1, 0.1, 0.1}, respectively. For the proposed adaptive SRUKF algorithm, the initial estimated state vector was randomly selected as *x*ˆ(0)=[70%, 0] *<sup>T</sup>*. The initial square-root state estimation covariance matrix and the initial estimated square-root covariances of the process and measurement noises were randomly set as *S*<sup>0</sup> = *diag*{0.01, 0.01}, *Q*<sup>0</sup> = *diag*{0.01, 0.01}, and *R*<sup>0</sup> = 0.05, respectively. The parameters *L*, *α*, and *β* were selected as 100, 0.85, and 2, respectively.

**Figure 2.** Open circuit voltage (OCV) versus state-of-charge (SOC) of the battery [26].

The experimental results in terms of the estimated model parameters *θi*(1 ≤ *i* ≤ 3) are demonstrated in Figures 5–7. The corresponding estimated *Ct* is shown in Figure 8. As a comparison, referring to [26], the parameter vector *θ* in the battery equivalent circuit model is calculated offline through utilizing the least-squares method with the results of ˆ *θ* = [0.00631, 0.9838, 0.0224] *T*.

**Figure 5.** Comparison of the estimated results of parameter *θ*1.

**Figure 6.** Comparison of the estimated results of parameter *θ*2.

**Figure 7.** Comparison of the estimated results of parameter *θ*3.

**Figure 8.** Comparison of the estimated results of *Ct*.

The actual and estimated SOCs by the adaptive SRUKF with RLS-based model parameters online identification are illustrated in Figure 9. The battery's actual SOC is gotten with using the ampere-hour counting method, where its initial SOC is obtained through measuring the initial OCV of the battery [20]. The SOC estimation error of the designed approach, calculated by subtracting the actual SOC from the estimated one, is illustrated in Figure 10. The statistic results in terms of the mean and root mean square (RMS) of the SOC estimation error are 0.13% and 0.98%, respectively, in which the mean and RMS errors are calculated as follows:

$$\begin{aligned} mean &= \frac{1}{N} \sum\_{k=1}^{N} |\mathbf{x}\_1(k) - \mathbf{\hat{x}}\_1(k)| \\ RMS &= \sqrt{\frac{1}{N} \sum\_{k=1}^{N} (\mathbf{x}\_1(k) - \mathbf{\hat{x}}\_1(k))^2} \end{aligned} \tag{27}$$

with *N* denoting the data number. To demonstrate the superior performance of the designed adaptive SRUKF with RLS-based model parameters online identification method, the estimation results of the SRUKF and the adaptive EKF with model parameters identified by RLS, and the adaptive SRUKF with off-line calculated model parameters are given as comparisons. The comparison of the measured and estimated terminal voltages is shown in Figure 11 and Table 1. It demonstrates that the methods with model parameters online identification can provide a smaller voltage estimation error compared with those with model parameter offline calculation. Their comparison results of the SOC estimation error are listed in Figure 10 and Table 2. It shows that higher estimation accuracy can be achieved by the proposed estimation method than other commonly used Kalman filter-based methods, which demonstrates its promising performance in terms of the SOC estimation.

**Table 1.** Statistics of the terminal voltage simulation errors.

**Figure 9.** Actual and estimated SOC curves.

**Figure 10.** SOC estimation error of the battery.

**Figure 11.** Comparison of the measured and simulated terminal voltages.

#### **5. Conclusions**

Accurate SOC estimation is important for the operation optimization of the battery. Based on a battery equivalent circuit model with unknown parameters, an RLS algorithm is utilized for model parameters online identification, and an adaptive SRUKF is proposed to estimate the battery's SOC, where a noise statistic estimator is utilized for the noise information online update. Extensive experimental results are provided to validate the performance of the proposed method showing that the mean and RMS of the SOC estimation error can be 0.13% and 0.98%, respectively. Higher accuracy in the SOC estimation is demonstrated compared with other commonly-used Kalman filter-based methods.

**Author Contributions:** Conceptualization, Q.O. and Z.W. (Zhisheng Wang); Methodology, Q.O.; Software, R.M.; Validation, Q.O., G.X. and R.M.; Formal Analysis, Z.W. (Zhaoxiang Wu); Investigation, R.M.; Resources, Z.W. (Zhaoxiang Wu); Data Curation, G.X.; Writing—Original Draft Preparation, Q.O.; Writing—Review & Editing, Q.O.; Visualization, R.M.; Supervision, Z.W. (Zhisheng Wang); Project Administration, Z.W. (Zhisheng Wang); Funding Acquisition, Q.O. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported in part by the National Natural Science Foundation of China (Grant No. 61903189), and in part by the Open Research Project of the State Key Laboratory of Industrial Control Technology, Zhejiang University, China (No. ICT20053).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Switching Frequency Determination of SiC-Inverter for High E**ffi**ciency Propulsion System of Railway Vehicle**

#### **Joon-Hyoung Ryu 1, June-Hee Lee <sup>1</sup> and June-Seok Lee 2,\***

Received: 17 August 2020; Accepted: 22 September 2020; Published: 24 September 2020

**Abstract:** This paper suggests the reasonable switching frequency determination method for achieving highest efficiency of the railway propulsion system consisting the silicon carbide (SiC) inverter and permanent magnet synchronous motor (PMSM). The SiC power device allows increasing the switching frequency of the inverter because it has the small switching power loss. The total efficiency is taken into account for determining the switching frequency of SiC inverter in this paper. In the efficiency analysis of SiC inverter and PMSM, the PMSM drive control is considered with the hybrid switching method combined the synchronous PWM and asynchronous PWM. The result of the analysis shows the efficiency curve of propulsion system depending on the switching frequency. The switching frequency having the minimum power loss of propulsion system is selected based on the extracted power loss curve.

**Keywords:** propulsion inverter; PMSM drive; efficiency analysis; SiC inverter

#### **1. Introduction**

Much research on the efficiency improvement of railway system has been performed for last decade [1–3]. In the railway vehicle, it has been trying to apply new technologies such as the permanent magnet synchronous motor (PMSM) and the silicon carbide (SiC) power device to the propulsion system. The PMSMs have 5–8% efficiency higher than induction motors [1,4], and SiC power devices reduce the switching loss to less than half loss of Silicon (Si) power device [5–7]. It is obvious that the propulsion system consisting the SiC inverter and PMSM, which is shown in Figure 1, guarantees the high efficiency.

**Figure 1.** Railway propulsion system consisting the silicon carbide (SiC) inverter and permanent magnet synchronous motor (PMSM).

In the railway industry, the induction motor has been selected as the traction motor for several decades [1,2,8]. The main reason is that the induction motor has the high reliability and superior maintainability beside it is inexpensive compared to PMSMs. However, since the totally enclosed structure improves the reliability and maintainability of PMSM, the number of the cases where the totally-enclosed PMSM is applied to the railway vehicle increases gradually [3,4,9–13]. In addition, the PMSMs show the higher efficiency than that of induction motor. Nowadays, the PMSMs have become attractive for railway operation company to save the operating cost. Along with the interest in PMSMs, the research on the PMSM drive control has been introduced in many paper [11–18]. The research theme in the railway vehicle can be divided into several areas: torque control, sensorless control, restarting control, and fault tolerant control. Nevertheless, these papers are based on the high efficiency operation of inverter. The existing PWM method for induction motor is applied to drive PMSM [13,14]. In the existing PWM method for induction motor, the six-step operation appears at propulsion inverter. Although the six-step operation has the large torque ripple, it is attractive PWM method to reduce the switching loss of power device.

Nowadays, the voltage class of SiC power module is announced to 3300 V [5,6,19]. Therefore, the studies applying the high voltage SiC power module into various applications are introducing in lots of papers [19–29]. The SiC power module with 1700 V class has been applied for the auxiliary converter of railway vehicle [20,25]. The switching frequency of 50 kHz is used in the resonant converter consisting 1700 V class SiC power module in [20]. In the catenary-free tram, the SiC power module with 1700 V class and hybrid SiC power with 1700 V class for the DC–DC converter used in the energy storage system are compared to Si module with 1700 V class [21]. Here, the switching frequency of 6 kHz in hybrid SiC power module has the same power loss with the Si module with the switching frequency of 2 kHz. The SiC power modules with various voltage classes have been considered in the traction inverter of railway vehicle [19,22–24,26–28]. Initially, the hybrid SiC power modules where the SiC freewheeling diode combines with silicon (Si)-IGBT were tested [19,22–24,26,27]. The mass and volume comparison results between the inverters consisting Si-IGBT modules and SiC power modules in the same switching frequency, which is respectively low switching frequency, 800 Hz, are shown in [22,24,26]. The propulsion inverter consisting of the hybrid SiC power modules can decrease the mass and volume until 60% of the conventional Si-IGBT inverter. Nonetheless, it can lead to the power loss reduction of 35% compared to the conventional Si-IGBT inverter [24]. The full SiC power modules enhance the efficiency improvement of propulsion inverter [28,29]. Since the SiC power module with 1200 V class is applied in [28], the switching frequency could be increased to 10 kHz. Although the SiC power module with 3300 V class is used in [29], the number of switching pulse increases in only the synchronous mode of PWM method. Generally, in the railway application, the SiC power module has been used to reduce the mass and volume of inverter in keeping the switching frequency as low value.

In Figure 1, the efficiency of propulsion system is determined by each efficiency of the PMSM and the SiC inverter. This paper focuses on the switching frequency because the switching frequency can influence both efficiencies of PMSM and SiC inverter. The effect of switching frequency on both the induction motor and Si-IGBT inverter is introduced in [24,29]. Ref. [29] used the triple pulse PWM method instead of the six-step operation to increase the efficiency of induction motor. Ref. [24] has proved that the six-step operation aggravates the efficiency of induction motor.

This paper proposes the reasonable switching frequency determination method for achieving highest efficiency of the railway propulsion system consisting the PMSM and SiC inverter. The efficiency analysis is implemented by considering the existing PWM strategies which are introduced in Section 2 briefly. In the efficiency analysis, the efficiencies of SiC inverter and PMSM are calculated through the co-simulation of SiC inverter (PSIM software, Powersim Inc., Rockville, MD, USA) and PMSM (J-MAG software, JSOL Corporation, Tokyo, Japan) to consider the interaction of them. Simulation results are based on the determination of the switching frequency of SiC inverter [30–32]. Based on the power loss curve of the propulsion system, the switching frequency with highest efficiency can be determined for propulsion inverter.

#### **2. PWM Strategy for SiC Inverter**

This paper focusses on how to maximize the efficiency of propulsion system. Therefore, the PWM strategy for the SiC inverter is also aimed to achieve the high efficiency of the SiC inverter. The existing PWM strategy used in the propulsion inverter guarantees maximizing the efficiency of the SiC inverter [14]. This strategy uses not only linear voltage modulation but also over-voltage modulation and six-step modulation. These modulations can be classified by magnitude of *Vo* (|*Vo*|) and dc-link voltage (*Vdc*) [15]. Figure 2 shows the output voltage (*Vo*) of existing Si-IGBT inverter depending on speed of railway vehicle. By considering that the maximum speed is 80 km/h, the six-step modulation relatively appears at low speed. Even though the six-step modulation leads to large torque ripple with vibration, it is enough to be ignored in railway vehicle. The six-step modulation appears over half of whole speed range in the existing PWM strategy. It means that the switching loss of the propulsion inverter can be minimized. The conventional PWM methods [15,16] guarantee the ideal change between modulations as |*Vo*| increases or |*Vo*| decreases. Generally, the reference voltage without any change is used to generate the desired |*Vo*| in the linear voltage modulation where the *Vo* is lower than *Vdc*/ √ 3(=0.5774 *Vdc*). However, in the over-voltage and six step modulations, the changed reference voltage is used to generate the desired |*Vo*|. This paper applies the existing PWM strategy for the SiC inverter.

**Figure 2.** The output voltage (*Vo*) of SiC inverter depending on the speed of railway vehicle.

The six-step modulation is not affected by the swishing frequency (*fsw*). The number of the switching is determined by only the fundamental frequency of *Vo* in the six-step modulation. Therefore, the inverter can show the same efficiency in six-step modulation regardless of *fsw*. In addition, the *fsw* is defined as *K*(integer) x the fundamental frequency of *Vo* in the synchronous PWM. It means that *fsw* is not changeable parameter and the efficiency is fixed as the same as that of six-step modulation if *K* is fixed. On the other hand, the variation of *fsw* influences the efficiency of inverter during the linear-voltage modulation in the asynchronous PWM. As *fsw* increases, the efficiency of inverter decreases. Consequently, it is needed to analyze the efficiency of propulsion system during the speed where the asynchronous PWM is applied.

#### **3. Permanent Magnet Synchronous Motor Control**

#### *3.1. Permanent Magnet Synchronout Motor*

This paper considers the 332-kW interior permanent magnet synchronous motor (IPMSM) as the traction motor for railway vehicle. Its specifications and cross section are shown in Table 1 and Figure 3. The B-H curves of materials (35PN230 and NdFeB) can be identified through the material datasheets. The lamination thickness of stator core and rotor core is 0.35 mm; the magnet is not laminated.

**Table 1.** PMSM specifications.

**Figure 3.** The cross section of interior permanent magnet synchronous motor (IPMSM) used in this paper.

#### *3.2. Control Method*

Maximum torque per ampere (MTPA) operation is required to control the IPMSM. Lots of papers have suggested control methods for the MTPA operation [33–35]. This paper uses the current table-based control method [35]. The control block diagram is shown in Figure 4.

**Figure 4.** Control block diagram for driving the IPMSM.

The d-axis current (*Id,ref*) and q-axis current (*Iq,ref*) are determined based on the MTPA tables according to the reference flux (λ*ref*) and reference torque (*Tref*), respectively. To eliminate the flux error, the compensation of the d-axis current (*Id,comp*) is used. To deal with the whole speed range with six-step modulation, the additional considerations (flux-weakening method and maximum torque per flux) should be reflected into the control method. However, this paper applies the simple control method of Figure 4 for the MTPA operation because the effect on the variation of *fsw* exists in the low speed range where the MTPA operation is needed.

#### **4. Loss Analysis of Propulsion System**

The loss analysis of the propulsion system can be divided into two parts: SiC inverter and IPMSM. Two parts affect each other as shown in Figure 5. The non-sinusoidal electromotive force of IPMSM affects the current control of IPMSM; therefore, it leads the low harmonics frequency components in the inverter currents. Finally, the low harmonics frequency components can increase the loss of IPMSM. In addition, the current ripple generated by *fsw* affects the power loss of IPMSM. Therefore, this paper conducts the loss analysis through the co-simulation between the SiC inverter (Power SIM software) and the IPMSM (J-MAG software) to consider the interaction of them. It is obvious that the simulation results are reliable in the comparison and analysis [30–32].

The loss analysis of propulsion system is implemented in the valid speed range (below 1200 r/m) where the effect of *fsw* variation exists.

**Figure 5.** Interaction between SiC inverter and IPMSM.

#### *4.1. SiC Inverter*

The SiC inverter consists of six SiC MOSFETs. The 3300 V/800 A 2in1 SiC module where two SiC MOSFETs are connected to configure one leg, which is MSM800FS33ALT manufactured by HITACHI, is selected in this paper. The gate on and off resistances which decide *Eon* (Turn-on Loss per Pulse), *Eo*ff (Turn-off Loss per Pulse), and *Err* (Reverse Recovery Loss per Pulse) are 1 Ω and 1.5 Ω, respectively. The additional data for power loss analysis of SiC MOSFET is shown in datasheet of MSM800FS33ALT. It is assumed that the SiC MOSFETs are closely connected to the DC-link capacitor to mitigate the effect of stray inductance enough [36,37].

The operation conditions of the propulsion inverter are as follows: *Vdc* is 1500 V, the control period is 1/(2x*fsw*), and *Tref* is 1522 Nm.

Figure 6 shows the three-phase current waveforms depending on *fsw* at 400 r/m. The |*Vo*| at 400 r/m is 188 V. The currents do not seem the completed sinusoidal waveform. There are several frequency components in currents: fundamental frequency component, low harmonics frequency components, and *fsw* components. Nevertheless, it is confident that as *fsw* increases, the current ripple caused by *fsw* is reduced.

**Figure 6.** Three-phase current waveforms depending on *fsw* at 400 r/m: (**a**) 750 Hz; (**b**) 1500 Hz; (**c**) 2250 Hz; (**d**) 3000 Hz.

Figure 7 shows the efficiency of one SiC MOSFET depending on the *fsw* and speed (r/m). Since the SiC MOSFET consists of MOSFET and Diode in one device, the loss analysis is implemented in two parts separately. As *fsw* increases, only the switching losses of MOSFET and Diode increase as shown in Figure 7a. One the other hand, the speed variation does not give enough impact on the total loss as shown in Figure 7b. The speed variation of IPMSM leads to the change of |*Vo*|. As the speed increase, |*Vo*| increases. The much current flows through MOSFET in the high |*Vo*| compared to Diode. It is identified that conduction loss of Diode is reduced as the speed increases.

Figure 8 shows the total loss of SiC inverter depending on *fsw*. The inverter loss at 3000 Hz is more than twice inverter loss at 750 Hz. In terms of SiC inverter, the low *fsw* is required to reduce the power loss.

#### *Energies* **2020**, *13*, 5035

**Figure 7.** Loss analysis of SiC MOSFET depending on *fsw*: (**a**) 400 r/m; (**b**) 750 Hz.

**Figure 8.** Total loss of SiC inverter depending on *fsw*.

#### *4.2. IPMSM*

The finite elements method (FEM) model of IPMSM, which is Figure 9, is developed to analyze the power loss of IPMSM. The currents with ripple components of Figure 6 are injected to IPMSM through the co-simulation to reveal the effect of current ripple components on the power loss. The conditions for loss analysis are the same as those of loss analysis of SiC inverter. In the loss analysis, the electrical resistivity of NdFeB magnet is 1.80 <sup>×</sup> 10−<sup>5</sup> <sup>Ω</sup>·m, which is used to analyze the eddy current loss of magnet. The phase resistance (*Rcoil*) of stator coil for the copper loss analysis is 0.021882 Ω. The iron loss of stator and rotor cores is estimated by applying the loss density curve of Figure 10.

**Figure 9.** The finite elements method (FEM) model of 322 kW IPMSM considered in this paper.

To verify the estimated power loss of IPMSM from co-simulation, it is compared to the result of the power loss calculation equation. The loss components are divided into the coil copper loss (*Pcopper*), the core iron loss (*Piron*), and the magnet eddy current loss (*Pm,eddy*). In this paper, *Pcopper* and *Piron* are calculated and compared to estimated power loss from J-MAG software. Although *Pm,eddy* is not considered in the verification, the several researches have proved its accuracy [38–40].

To calculate *Pcopper*, the skin effect is ignored because the stator coil consists of many thin coils with the small dimension [41]. Therefore, *Pcopper* can be calculated as

$$P\_{\text{copper}} = 3I\_{\text{phase}}^2 R\_{\text{coil}} \tag{1}$$

The loss curve for *Piron* is often supplied by the core material manufactures [42]. The *Piron* is represented as

$$P\_{Ivm} = P\_{Loss}(f\_B) \tag{2}$$

where *fB* is a frequency component of the magnet flux density and *PLoss* is loss density curve of Figure 10.

**Figure 10.** The loss density of stator and rotor core material (0.35 mm) depending on the magnetic flux density.

Figure 11 shows accuracy of the estimated power losses of IPMSM from J-MAG software and calculated power losses from (1) and (2). These values are expressed as per unit. Two copper losses are similar, and its error ratio is under 2%. In addition, the error ratio is about 5% in the iron loss comparison results.

**Figure 11.** Comparison of estimated power losses and calculated power losses.

Figures 12 and 13 show injected currents and the power loss analysis result of IPMSM at 1200 r/m. The magnitude of current ripple is largest at *fsw* of 750 Hz and is smallest at *fsw* of 3000 Hz. The difference between current ripples at *fsw* of 2250 Hz and *fsw* of 3000 Hz is relatively small. The *fsw* over 2000 Hz has less impact on the current ripple as shown in Figure 12. The IPMSM power loss can be divided into four losses as shown in Figure 13. The stator winding has the copper loss. The steel, which is the material of the stator and rotor, results in the stator and rotor iron losses containing the hysteresis

and eddy current losses. Only the eddy current loss is considered in the magnet. High *fsw* reduces the magnitude of current ripple as shown in Figure 12, and it mainly influences on the three losses such as stator iron loss, rotor iron loss, and magnet eddy current loss. Three losses decrease as the magnitude of the ripple current decreases. However, the effect of loss reduction is not large enough since *fsw* is about 2000 Hz. It can be estimated from the similar ripple currents of 2250 Hz and 3000 Hz. On the other hand, the copper loss does not have the enough impact by the variation of *fsw*.

**Figure 12.** The currents injected to IPMSM at 1200 r/m: (**a**) injected three-phase current at 750 Hz; (**b**) a-phase current depending on switching frequency.

**Figure 13.** The power loss analysis result of IPMSM at 1200 r/m depending on *fsw*.

Figure 14 shows the hysteresis losses of stator and rotor in the frequency domain depending on *fsw* at 1200 r/m. It is identified that the low harmonics frequency components are almost same regardless of *fsw* and *fsw* components have the different value each other. *fsw* of 750 Hz leads to the current ripple of 1500 Hz and its value is larger than that of 3000 Hz. Consequently, the high *fsw* reduces the power loss of IPMSM and it is opposite to the condition for the low power loss of SiC inverter.

(**a**)

**Figure 14.** The hysteresis losses of IPMSM in frequency domain at 1200 r/m: (**a**) *fsw* is 750 Hz; (**b**) *fsw* is 3000 Hz.

#### *4.3. fsw Determination for Propulstion System*

The power loss of propulsion system depending on the speed (r/m) of IPMSM can be represented by considering the results of Figures 8 and 13. Figure 14 show the total power loss of the propulsion system. As mentioned in Sections 4.1 and 4.2, the power loss of SiC inverter increases and IPMSM decreases as *fsw* increases. Figure 15 shows the individual power losses and the total loss. The total power loss curve of propulsion system has the minimum loss point between 750 Hz and 3000 Hz. This point is 2250 Hz and guarantees maximizing the efficiency of the propulsion system.

**Figure 15.** The power loss analysis result of the propulsion system as 1200 r/m.

Figure 16 shows the total power loss of propulsion system depending on speed (r/m). The *fsw* for the minimum loss point at 400 r/m, 800 r/m, and 1200 r/m is 750 Hz, 1500 Hz, and 2250 Hz, respectively. The power loss of IPMSM at the low speed is small relatively compared to the power loss of SiC inverter. It means the power loss of SiC inverter is dominant at low speed. Therefore, as the speed increases, *fsw* for the minimum loss point increases. In addition, it can be estimated that the switching frequency upper than 3000 Hz aggravates the power efficiency of propulsions system under 1200 r/m.

**Figure 16.** The total power loss of the propulsion system depending on speed (r/m).

#### **5. Conclusions**

This paper determines the switching frequency (*fsw*) guaranteeing the minimum power loss of the propulsion system consisting SiC inverter and 332 kW IPMSM. The co-simulation was implemented and the effect of the low harmonic frequency and *fsw* components was reflected in the power loss analysis. As *fsw* increases, the power loss of SiC inverter increases because the number of switching increases, but the power loss of IPMSM decreases because the magnitude of current ripple caused by *fsw* is reduced. In addition, it was identified that as the motor speed increases, *fsw* for maximum efficiency operation of propulsion inverter, increases.

The implemented process suggests that both inverter and IPMSM should be considered in determining *fsw* of inverter to maximize the efficiency of the propulsion system. If the switching device or IPMSM specification is changed, the results of Figure 16 are not valid. However, it is important and meaningful that *fsw* can be determined through the process implemented in this paper.

**Author Contributions:** Conceptualization, project administration, writing—original draft preparation, J.-H.R.; data curation and software, J.-H.L.; supervision and writing—review and editing, J.-S.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Railroad Technology Research Program through the Ministry of Land, Infrastructure and Transport of Korean Government under Grant 20RTRP-B146008-03.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Experimental Platform for Evaluation of On-Board Real-Time Motion Controllers for Electric Vehicles**

#### **Thanh Vo-Duy <sup>1</sup>, Minh C. Ta 1,2,\*, Bảo-Huy Nguyễn 1,2 and João Pedro F. Trovão 2,3,4**

Received: 31 October 2020; Accepted: 3 December 2020; Published: 6 December 2020