Battery System Modeling 0323904726, 9780323904728

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BATTERY SYSTEM MODELING

BATTERY SYSTEM MODELING SHUNLI WANG Southwest University of Science and Technology

YONGCUN FAN Southwest University of Science and Technology

DANIEL-IOAN STROE Aalborg University

CARLOS FERNANDEZ Robert Gordon University

CHUNMEI YU Southwest University of Science and Technology

WEN CAO Southwest University of Science and Technology

ZONGHAI CHEN University of Science and Technology of China

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2021 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-323-90472-8 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

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C H A P T E R

1 Lithium-ion battery characteristics and applications 1.1 Introduction to lithium-ion battery technology The lithium-ion battery industry is a great direction of global high-tech development. The lithium-ion battery has advantages of high specific energy, high specific power, high conversion rate, long cycle life, and no pollution. It is widely used in electric vehicles and various energy storage devices due to its good electrochemical stability, high energy density, long battery life, and no need for maintenance. At present, the application range of lithium-ion batteries is more extensive. Its application mainly includes five fields: transportation, electric energy storage, mobile communication, new energy storage, and aerospace/military. Its application in electric vehicles could not only replace oil with electricity and reduce greenhouse gas emissions, but also store excess electricity from the grid. As lithium-ion batteries gradually enter the market, the use and consumption of lithium resources in the world are increasing substantially, so the derived industrial chain has great development potential and broad prospects.

1.1.1 Development history Along with the rapid development of new energy vehicles, the power battery industry chain has gradually shifted from policy-driven to market-driven conditions. As can be known from the perspective of market demand for power batteries, the installed capacity might no longer explode but stabilize. Under such circumstances, the power battery industry has overall excess capacity and insufficient structural performance. However, the rising industry-leading enterprises are accelerating the expansion of power battery capacity, which makes the market competition intense for industry differentiation. Due to lithium energy density and other requirements, new material and technological breakthroughs are imminent. From the perspective of power battery preparation, there is a huge development space of the current power battery equipment, which is expected to improve the localization

Battery System Modeling https://doi.org/10.1016/B978-0-323-90472-8.00003-2

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Copyright # 2021 Elsevier Inc. All rights reserved.

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1. Lithium-ion battery characteristics and applications

rate of equipment. The current power battery equipment development space is huge, and the equipment localization rate is expected to be improved. When the lithium-ion battery was invented, it was found that many compound atoms in laminates interacted with each other in strong covalent bonds. The layered or columnar chemical substance can be provided with intermolecular forces, such as clay, silicate, phosphate, and so on. Lithium-ion batteries can be formed by the reversible reaction against lithium metal. In the early 1970s, the layered structure was proposed as the most representative cathode, and lithium metal was used as the anode of the battery system. Its reliability was also confirmed when conducting the system level application. And then, Exxon took a closer look at battery systems with hopes of commercialization. The system soon revealed many fatal flaws. First of all, the active metal lithium can easily lead to the decomposition of organic electrolytes, leading to internal pressure on the battery. Due to the uneven potential distribution of the lithium electrode surface, lithium metal is deposited in the cathode, resulting in lithium dendrites. When the crystal deviates from equilibrium conditions, it is easy to grow like a branch and form dendritic crystals. It causes a reversible capacity loss of the embedded lithium. The dendrites can penetrate the diaphragm and connect the negative electrode. This can cause a short circuit of the battery, instantly absorbing a large amount of heat and causing an explosion, which may lead to serious safety hazards. These factors can lead to the deterioration of the recycling and safety performance of lithium metal batteries, so the system was not commercialized. In 1980, Armand first put forward the idea of a rocking chair battery [1]. In the chargedischarge process, the lithium-ion is in the motion of a pole-negative and pole-positive state, in which both ends of the rocking chair are battery poles. Lithium compounds with low embedded lithium potential are used instead of lithium metal as the anode, and lithium compounds with high embedded lithium potential are used as the cathode. In the same year, Professor Goodenough of the University of Texas proposed a series of lithium transition metal oxides as battery anode materials such as Co, Ni, or Mn. In 1987, the researcher Auburn successfully assembled a concentration difference and proved the feasibility of the rocking chair battery idea. Due to the high embedding potential for the negative electrode material of 0.7–2.0 V vs the Li/lithium-ion with low lithium capacity, it did not have high specific capacity advantages for the high-voltage lithium-ion secondary battery. In 1987, Sony used lithium embedded coke LixC6 instead of lithium metal as the anode through a battery system, using reversible embedded carbon materials for the lithium negative. It was used to prolong the service lifespan and at the same time maintain high voltage stability. The lithium-ion secondary battery cycle life is low with a poor safety performance. Pure lithium-ion batteries started in 1989 from the invention by Nagura in Japan with petroleum coke as the positive electrode and lithium-ion cobalt as the negative electrode. In the same year, the company officially launched the first-generation market structure of the commercial batteries as coke, which adopted the battery concept for the first time. Since then, this graphite positive battery has been commercialized with the continuous deepening of the material and systematic research. Due to the advent of the rapid battery development era, it has taken the largest share in the small secondary battery market of cameras, mobile phones, notebook computers, power tools, and so on. In recent years, the lithium-ion battery has also achieved rapid development in electric vehicles.

1.1 Introduction to lithium-ion battery technology

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In lithium-ion battery development history, three development characteristics are found in the world industry. First, the green environmental protection battery grows rapidly. Second, battery transition is a sustainable development strategy. Third, photovoltaic cells are becoming ever smaller and thinner. In the commercialization process of batteries, the proportion of lithium-ion batteries is the highest, especially polymer batteries, which can realize thin rechargeable batteries. It grows rapidly because of its small size, high energy, light weight, rechargeability, and pollution-free advantages. In recent years, the development of electronic information has brought many market opportunities. Because of its unique safety advantage, the lithium-ion battery has replaced the traditional battery gradually into the mainstream The polymer type of this is called the 21st-century battery. Its development prospects are very promising. The energy structures with fossil fuels, oil, and natural gas as the main energy sources have caused increasingly serious environmental pollution, so the resulting global warming and ecological environment deterioration have attracted more attention. Therefore, the development of renewable energy has become one of the most decisive influences in the future technological field and the future economic world. The lithium-ion battery is used as a new secondary clean energy and renewable energy. It has the advantages of high working voltage, light mass, and high energy density. It has been used widely, which shows a strong trend of development.

1.1.2 Energy storage technologies Lithium-ion batteries are mainly used in a variety of portable electronic products, the application range of which continues to expand along with its application progress, material performance, and design technology. According to the cathode material, it can be classified into different types such as lithium-iron-phosphate, lithium-cobalt acid, lithium-manganese acid, lithium-nickel acid, and ternary materials. All have advantages and disadvantages, and they also have suitable application scenarios correspondingly. This chapter introduces different types of lithium-ion batteries and summarizes their advantages and disadvantages. Their working characteristics are then analyzed. (1) Applications in electronic products Due to high volume-specific energy, lithium-ion batteries can be made smaller and lighter, and they have been widely used in portable electronics. With the popularity of mobile phones, digital cameras, cameras, laptops, and handheld game consoles, the battery market has maintained rapid growth. With the high current charge-discharge performance improvement, lithium-ion batteries have also expanded to the field of wireless phones and power tools. (2) Applications in electric bicycles Public transportation has been recognized as a main method of urban transportation development in the future. However, transportation can form a broad network, and it is hard to meet different points of service. Besides, electric bicycles (e-bike) have more advantages and practicability by analyzing the objective factors of national conditions. Electric bicycles no longer produce pollution in the application process. The industrial development of electric vehicles is in line with national conditions. Especially in recent years, the global development boom has been rising again to solve energy and pollution

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1. Lithium-ion battery characteristics and applications

problems. Therefore, the lithium-ion battery plays an active role in solving social problems with a shortage of oil resources and the aggravation of environmental pollution. (3) Applications in electric vehicles Promoting the development of electric vehicles can reduce greenhouse gas emissions, in line with the scientific outlook on development, which is a strategic opportunity for the automobile industry. Therefore, the key to electric vehicles should be the research focus. Promoting the industrialization of electric vehicles is a strategic choice of national conditions, and an important way to ensure energy security. At the Beijing Olympic Games, 50 electric buses claimed a record of zero breakdowns and zero failures, showing the charming style of the green Olympics. At the World Expo in Shanghai, more than 1000 new energy vehicles were used for the first time, including fuel cell vehicles, hybrid electric vehicles, ultracapacitance vehicles, and pure electric vehicles. It is estimated that this expo can save 10,000 tons of traditional fuels, eliminating 118 tons of harmful gas emissions and 28,400 tons of greenhouse gas emissions. Also, electric vehicle charging stations and other related facilities have been built with practical applications, according to which the development of the electric vehicle industry is increasingly mature. (4) Applications in aerospace Lithium-ion batteries are used on rovers and are planned for future missions. NASA considers lithium-ion batteries for its space missions, in addition to other space agencies. Currently, lithium-ion batteries in aviation are primarily used for launch, flight correction support, and ground operations. Battery efficiency and night operation support have also improved. (5) Application in energy storage devices A storage plant stores the electricity generated by a power plant from a low peak period when the electricity price is low. When selling the stored electricity during peak hours, the electricity price of the storage electric field has certain advantages over the gradient electricity price peak electricity of the power plant. Peak-valley regulation is a difficult problem. More power plants are usually needed to ensure peak demand, but it increases the cost of investment energy by keeping power plants running when demand is low. When selling electricity during peak hours, the electricity price of the energy storage field has certain advantages over the peak electricity price of the power plant. Therefore, some companies put forward the idea of investing in the construction of energy storage power stations, reducing the purchase of large and medium-sized energy storage equipment. The charged electricity in the peak period of low power consumption and in time-sharing forms a win-win situation. As a kind of green power supply object, the lithium-ion battery is recognized as an ideal choice for high-power batteries because of its good cycling performance, high energy density, and high charge retention.

1.2 Battery working mechanism The lithium-ion battery system is complex, integrating chemical, electrical, and mechanical characteristics. Consequently, the requirements of various characteristics must be considered for the battery system design. The safety and life attenuation characteristics cannot be

1.2 Battery working mechanism

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measured directly, as they are contained in the battery chemical characteristics, which are also not easy to predict in a short time. Therefore, when designing a battery system, it is necessary to adopt battery technology, group technology, and management system technology. This also takes into account battery safety, reliability, and durability. Lithium-ion batteries usually come in cylindrical and rectangular shapes.

1.2.1 Characteristic analysis The internal cylindrical battery uses a spiral-wound structure, which is made of a very fine and highly permeable thin septum isolation material. It is spaced between the positive and negative electrodes. The main materials are polyethylene, polypropylene, and a composite material [2–4]. The rectangular lithium-ion battery is formed by laminate sheets, placing a separator on the positive electrode, then placing the negative electrode, and successively stacking. The positive pole consists of a lithium-ion collector consisting of lithium-containing materials such as lithium-cobalt, lithium-manganate, Ni-cobalt-manganese oxide, and current collector materials consisting of the aluminum film. The negative electrode is composed of a lithium-ion collector composed of layered carbon material and a current collector composed of a thin copper septum. The battery is filled with organic electrolyte solution and equipped with a safety valve together with positive temperature coefficient components. This has the advantages of small thermal resistance, high heat-to-transfer efficiency, noncombustion, safety, and reliability. It can effectively prevent the battery from being hurt when it is in an abnormal state or when the output is a short circuit. The lithium-ion battery is affected by abnormal factors such as short circuits, overheating, and overcharging. High-pressure gas is likely to be generated inside the battery, which causes a deformation of the battery shell and even the risk of an explosion. As for safety application, the battery must be equipped with safety valves, which is used to avoid its abnormal discharge as well as the explosion [5]. When the pressure in the battery container rises to an abnormal state, the safety valve can quickly open and expel the gas, providing protection in case of an abnormal situation [6]. Because the positive temperature coefficient element is in a low resistance state at a normal temperature, the current can be adjusted to prevent the battery from overheating when it is abnormal. When the battery overheats due to an unusually large current caused by a short circuit or overcharging, the positive temperature coefficient element is converted into an extremely high resistance state to reduce the current in the loop [7]. Therefore, it is usually used to prevent the overcurrent of the battery and overheat caused by it to protect the battery. A single lithium-ion battery generally has a voltage of 2.8–4.20 V with a capacity of 1.5 Ah while a large-capacity one generally has a single capacity of 2–200 Ah. For new energy vehicles, several hundred volts of voltage should be introduced to meet the current range requirements [8]. However, a single battery cannot provide high voltage and energy. Therefore, the single battery cells are often processed in series or parallel to form battery packs to meet the voltage and power requirement from the power supply system.

1.2.2 Components and working principle The lithium-ion battery is mainly composed of four parts: the positive electrode material, the negative electrode material, the diaphragm, and the electrolyte. Anode materials provide

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1. Lithium-ion battery characteristics and applications

lithium ions for batteries. Common materials include lithium manganate, lithium cobalt oxide, and lithium nickel cobalt manganate [9]. The cathode material is mainly graphite. The main function of lithium-ion batteries can be introduced to store lithium ions, which realizes embedding and disembedding in the charge-discharge process [10]. The diaphragm is a special composite mode that prevents electrons from moving freely between positive and negative electrodes in a battery, but lithium ions can move freely in the electrolyte. The electrolyte is generally composed of lithium salts together with organic solvents, which conduct lots of lithium ions. Electrons cannot exist independently on a carrier, and the diaphragm is essentially an insulator that cannot contain free electrons. Consequently, it cannot conduct electricity. In a battery, elements are ions that can easily pass through the membrane while electrons escape from elements to a new carrier no matter whether its positive or negative [11]. When in contact with the membrane, it cannot absorb free electrons from the electrode, thus preventing electrons from passing through. The common materials of the diaphragm are single-layer polypropylene, polyethylene, and composite three-layer polypropylene-polyethylene-polypropylene septum. The electrolyte realizes the conduction of the lithium ions between the positive and negative electrodes of the battery. Currently, LiPF6 is the most widely used electrolyte. It is an objective representation of its internal structure. There are lithium ions, metal ions, oxygen ions, and carbon layers inside the battery [12]. The lithium-ion batteries are mainly composed of compounds. Internal reactions take place through the movement of ions in the battery. And then, the diaphragm of the battery acts as a barrier to keep the two poles of the battery apart as shown in Fig. 1.1. The lithium-ion battery is an indispensable portable energy storage element. Its performance is characterized by many external parameters such as voltage, current, and internal

anode

diaphragm

cathode

Charge

Discharge Current Collector

Lithium-ion

Current Collector

Metal ion

FIG. 1.1 Schematic diagram of the lithium-ion battery structure.

Oxygen ion

1.2 Battery working mechanism

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resistance. The reason it is used more than other batteries is that it has many advantages compared with other batteries. First, it has a high heat of combustion, which is the amount of heat given off per unit. Second, it is more environmentally friendly and can meet the requirements of green social development [13]. Third, it has a long cycle life. Under normal conditions, it can be charged and discharged hundreds of times. Consequently, it can be used for a long time [14]. Finally, it has no memory effect. The battery working process causes the battery capacity to be lost, resulting in less and less capacity; this is the memory effect that does not exist on lithium-ion batteries [15]. It has many other advantages for its widely used conditions, such as good safety performance, low self-discharge, fast-charging, and wide operating temperature range. The internal chemical reaction of the lithium-ion battery is a redox reaction, which is also the working principle of the battery in the application process. It converts electrical energy into heat energy through a chemical reaction. According to the chemical reaction equation, the battery charge-discharge process is the embedding and disembedding process of the lithium ions. When a battery is charged, the positively charged lithium atom undergoes an oxidation reaction, losing electrons and becoming lithium ions. Numerous lithium ions are produced by the oxidation reaction of the positive electrode. These lithium ions start from the positive electrode and pass through the electrolyte solution to the carbon layer of the negative electrode. The battery capacity is related to the number of lithium ions that are produced in the positive electrode reaction. It is related to the number of lithium ions that are exchanged with a negative electrode through the electrolyte [16]. In the discharge process, an oxidation reaction occurs in the negative electrode. In this process, lithium ions embedded in the negative carbon layer come out and move back to the positive electrode. The more lithium ions returning to the positive electrode, the higher the discharge capacity [17]. Similarly, when charging, the lithium ions are generated in the positive electrode of the battery, which moves to the negative electrode through the electrolyte. The lithium ions in the negative electrode can be embedded in the pores of the carbon layer [18]. When more lithium ions are embedded, the charging capacity becomes higher. The internal chemical reaction process of the lithium-ion battery is described as shown in Fig. 1.2. Electrolytes are dissolved organic solutions to lithium salts [19]. In general, the electrochemical reaction process of the lithium-ion battery is the exchange of the lithium ions by the back and forth transformation from positive to negative poles. The positive and negative electrode reaction, according to which the total reaction equations are described, is as follows. The positive electrode reaction, negative electrode reaction, and the total battery response can be described as shown in Eq. (1.1): 8 +  > < P : LiM x Oy ¼ Lið1xÞ Mx Oy + xLi + xe N : nC + xLi + + xe ¼ Lix Cn (1.1) > : T : LiM x Oy + nC ¼ Lið1xÞ Mx Oy + Lix Cn In the above three equations, M can be Co, Mn, Fe, and Ni, respectively, representing lithium-cobalt-oxide, lithium-manganese oxide, lithium-iron-phosphate, and lithium-nickeloxide batteries. Its operating principle is different from the oxidation-reduction process of ordinary batteries, but the embedding stripping process of the lithium ions can be reversibly

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1. Lithium-ion battery characteristics and applications

FIG. 1.2 Lithium-ion battery working principle.

embedded or extricated from the main material. In two stages of charge-discharge, they are embedded and deembedded from the positive to negative electrodes. In the charging process, it is first deembedded from the positive pole through the electrolyte to the negative pole, which is then embedded into the negative pole. Its negative electrode realizes a lithium-rich state at this time point. Rich lithium is a positive electrode material made by doping a small amount of lithium with positive electrode active substances such as LiMn2O4. It can make the volume changeless in the charge-discharge process of cell contraction, which improves the structural stability and cycling performance of the material. The charge-discharge processes are opposite to each other. The cathode material of the lithium-ion battery is composed of a lithium embedded compound. If there is an external electric field, ions in the cathode material can be released and embedded from the lattice under the action of the electric field. The battery reaction takes place against its positive pole and negative electrode at the same time, according to which the general equation can be described as shown in Eq. (1.2). 8 +  > < P : LiCoO2 ! xLi + Li1x CoO2 + xe +  (1.2) N : xe + xLi + 6C ! Lix C6 > : T : LiCoO2 + 6C , Li1x CoO2 + Lix C6 It has three important parts called the diaphragm, the positive electrode, and the negative electrode. Its work mainly relies on the movement back and forth between negative ions, which is caused by the lithium-ion concentration difference between ions on both ends. During the charging process, the lithium ions are disembedded from the positive electrode and embedded into the negative electrode through the corresponding electrolyte. After a series of chemical reactions, the positive electrode is in a state of less lithium and the negative electrode is in a state of more lithium [20]. Meanwhile, it compensates for the charge from the external circuit of the negative electrode. In the discharge process, the lithium ion is dislodged from the negative electrode and inserted into the positive electrode again by the action of the electrolyte.

1.2 Battery working mechanism

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1.2.3 Lithium-ion battery construction The composition of lithium-ion batteries is some compounds. The reaction is completed by the movement of ions by using the diaphragm as a barrier. The diaphragm separates the two electrodes of the battery effectively as shown in Fig. 1.3. The segmented charging of the lithium-ion battery can ensure that it can be filled quickly but not charged and play a certain repair role of the battery that cannot be discharged over a long period. At present, there are two main charging modes for it: constant-current and constant-voltage charging strategies [21]. Whether it is a constant-current or constant-voltage condition, the charging mode can be mainly divided into five stages: trickle charging, lowvoltage precharging, constant-current charging, constant-voltage charging, and termination of charging.

1.2.4 Charge-discharge strategies The charging method is a constant-current and voltage limit, most of which are controlled by an IC chip. Typical charging methods are introduced. First, the voltage to be recharged is tested. If the voltage falls below 3.00 V, it needs to be recharged. At this time, the charging current is generally one-tenth of the set current. Until the voltage rises steadily to the terminal voltage, it enters the standard charging process, which is described as follows. First, constantcurrent charging is conducted at the set current. When the battery voltage rises to 4.20 V, it is changed by using the constant-voltage charging treatment. The charging voltage continues to be 4.20 V for charging. After charging for a period, the charging current gradually drops. When it drops to one-tenth of the set current, the charging process is finished.

FIG. 1.3 Lithium-ion battery packing structure.

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1. Lithium-ion battery characteristics and applications

The first stage is the trickling charge, which is mainly used for precharging recovery charging or fully discharged battery cells. C is a representation of the nominal capacity of the battery of current. For example, if the battery currently has a capacity of 1000 mAh, then 1 C is the charging current of 1000 mA. Trickle charging is used when the battery voltage is lower than 3.00 V that is set as the limited value [22]. The trickle charging current is one-tenth of the current in constant-current charging mode, namely 0.1C. If the constant charging current value is 1.00 A, the trickle charging current currently is set as 100 mA for the terminal value. The second stage is the constant-current charging. When the voltage value of the battery rises above the trickle charging threshold, the charging current at this time is increased to the constant-current charging. In general, the current value of constant-current charging should be varying from 0.2C to 1.0C. The voltage of lithium-ion batteries gradually rises with the constant-current charging process. Generally, the voltage value set by a single battery is 3.00 V to 4.20 V. The whole structure of the lithium-ion battery packing system is described in Fig. 1.4. The third stage is the constant-voltage charging. When the voltage value of the lithium-ion battery rises to 4.20 V, the constant-current charging stage ends and the constant-voltage charging stage begins. At this time, the change of current value is determined according to the saturation degree of the cell. With the charging process, the charging current gradually decreases from the maximum value. When it decreases to 0.05C, the charging is considered to terminate [23]. The fourth stage is the charging termination, in which there are two typical methods of charging termination. The minimum charging current is used to distinguish, or the timer is used, or both are combined. The minimum current method is used to monitor the current value in the constant-voltage charging stage and the charging current value is terminated when it decreases to 0.05C or ranges from 0.02C to 0.07C. The second method can be adopted as the timing treatment approach. The time of the constant-voltage charging stage is the initial time point and the charging process is terminated after 2 h of continuous charging. The above four-stage charging method takes approximately 2.5–3 h to complete the charging of a fully discharged battery. After charging, if it is detected that the battery voltage is lower than 3.89 V, it is recharged. In the charge-discharge process, there is a certain pressure in the battery of empirical data onto 0.30–0.60 mPa. Under the same pressure, the larger the stressed area, the more serious the deformation of the battery wall of the shells. In the first charge-discharge process of the liquid lithium-ion battery, the electrode material and electrolyte react to the solid-liquid interface, forming a passivation layer covering the surface of the The Li-ion battery pack

Sampling resistance

Temperat ure sensor

Battery Cores

Battery cell

Structural base

The shell and cover

Cover plate

Current collector

FIG. 1.4 Lithium-ion battery packing system.

Detection module

Heating component

Heating sheet

Socket

Heat relay

Combined shell cover

1.3 Lithium-ion battery chemistries

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electrode material [24]. The formed passivation film can effectively prevent the passage of solvent molecules. Meanwhile, the lithium ions can be freely inserted and extracted through the passivation layer. This passivation film has the characteristics of a solid electrolyte, so it is named as Solid Electrolyte Interface (SEI) film.

1.3 Lithium-ion battery chemistries According to the classification of positive electrode materials, lithium-ion batteries can be classified into lithium-iron-phosphate, lithium-cobalt, lithium-manganate, lithium-nickelic, and ternary materials.

1.3.1 Lithium-ion battery family The phase-changing material could change its physical state in a certain temperature range [25]. It is used as the thermal conductivity medium, which is attached to the surface of the single cell; the heat dissipation effect has been greatly improved. Besides, there are also plans to combine the heat conduction materials with water cooling, so that the water cooling system heats the conduction materials absorbed by the heat transfer of the outside of the system. As for the lithium-ion battery system to prevent the thermal runaway problem, the ideal situation is to be able to directly detect the temperature, voltage, and current parameters of each cell. Therefore, even if there is no new type of sensor with high quality, low price, and good performance, the early warning and prevention of thermal runaway conditions may be successful. The number of battery cells in the system is small, and it is an important competitiveness of square batteries. Compared with soft-pack and square lithium-ion batteries, cylindrical 18,650 batteries were the earliest available commercially. They are the most automated and cheapest power battery type. With years of support from Tesla, it is a three-way race against soft packs and square batteries. The cylindrical battery family also has one more star after Tesla announced that Model 3 uses the 21,700 cylindrical lithium-ion battery. The following is a brief description of several technical points related to the cylindrical battery of those not specifically stated in the process, and the cylindrical batteries are specifically the 18,650 battery type. The cylindrical battery is the most studied and discussed type. It is mainly composed of the positive electrode, negative electrode, diaphragm, safety valve, overcurrent protection device, insulating parts, and shell [26]. There is more steel shell in the early stage and the aluminum shell is the main one at present. Each manufacturer cell overcurrent protection device design is not the same. The price is completely different according to different security requirements. General safety devices mainly have positive temperature coefficient resistance and fuse two categories. The excessive current, resistance heating, and temperature accumulation promote the rise of the positive temperature coefficient resistance value. When the temperature exceeds a threshold, the positive coefficient resistance value suddenly increases, which can separate the fault cell from the overall circuit effectively to avoid its further thermal runaway conditions. In principle, the fusing device is a fuse. In the case of excessive current, the fuse fuses, and the circuit is disconnected [27]. The difference between the two types of protection is that the former can be restored. The latter protection can only be

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1. Lithium-ion battery characteristics and applications

used once. Consequently, once the failure occurs, the system must replace the failed cell to work normally. Cylindrical 18,650 cells are the highest of the three main types of cells because of their structural characteristics, the standardization of their models, and the level of automation in the production of cylindrical cells [28]. This makes it possible to achieve a high degree of consistency and the yield increases accordingly. Data show that Samsung, Panasonic, and other major foreign manufacturers can reach 98%. Its consistency is good. It has good mechanical properties. Compared with square and soft-pack batteries, the closed cylinder can obtain the highest bending strength under the approximate size. The technology is mature and the cost is low. Consequently, the space for cost optimization is almost exhausted. Single battery cell energy is small and easy to be controlled in case of an accident. In electric vehicles, the number of cylindrical cells in the battery system is very large, which increases the complexity of the battery system. Regardless of the organization or management system compared with the other two types of batteries, the cost of cylindrical cells at the system level is relatively high. Under uneven temperature environmental conditions, the characteristic difference of internal-connected battery cells increases as well. Consequently, it is considered to be a helpless choice when Tesla chose the 18650 type batteries as the main power system components. Ten years ago, only cylindrical batteries were qualified for mass production [29]. The battery safety and thermal management requirements were the driving force for the development of the powerful electronic control system. The space for energy density to rise is very small. Its superpower has achieved a single capacity of 4050 mAh with the cell energy of 306 Wh/kg. Since then, no higher record has been seen. The open-circuit voltage is typically 3.60 V while the open-circuit power of NiMH and NiCad batteries is only 1.20 V. High energy storage density is the core value of lithium-ion batteries. Under the same output power conditions, it is not only half the weight of a nickel-metal hydride battery, but also 20% smaller in volume. It typically has a long cycle life of 1000 cycles, compared with 500 cycles for Ni-MH hydride and nickel-cadmium batteries. It can be charged quickly, taking only 1–2 h to reach the optimal state. Meanwhile, its self-discharging current rate is low, far lower than the 20% per month of a nickel-cadmium battery, and the 30% per month of a nickel-metal hydride battery. Its operating temperature range is wide, ranging from 20°C to 60°C. As it has no memory effect, it can fully realize the chargedischarge treatment for reducing the battery capacity of lightweight green environmental protection and other advantages. Lithium-ion batteries have a high internal impedance. Because its electrolyte is an organic solvent, its electrical conductivity is much lower than that of nickel-cadmium battery electrolyte solution and metal hydride nickel battery, so its internal impedance is 11 times higher than that of nickel-metal-hydride and nickel-cadmium batteries. The operating voltage varies greatly. When the battery is discharged to 80% of the rated capacity, the voltage change of the NiCad battery is very small, only 20%, while its voltage change is 40%, which is a serious defect of its power supply. Due to its high discharge voltage, it is easy to detect battery residual power and its electrode material cost is relatively high.

1.3.2 Battery with different materials A lithium-iron-phosphate battery refers to a battery using lithium iron phosphate as a positive electrode material, which has the following advantages and characteristics. The

1.3 Lithium-ion battery chemistries

13

requirements for battery assembly are also stricter and need to be completed under low-humidity conditions. As the battery structure is more complex, a special protection circuit is required. The battery uses organic electrolytes, so it has a certain safety hazard. The PdO bond between the lithium-iron-phosphate crystals is stable and difficult to decompose. Even in the case of high temperatures or overcharge, it will not collapse to generate heat or form strong oxidizing substances. Therefore, it has good safety. Even if the battery is damaged, it does not burn or explode, and its safety is the best. It is reported that some samples were burned without exploding during the needle-punching or short circuit test of the battery. In the overcharge experiment, this phenomenon still occurs when it is charged with a high voltage many times higher than its discharge voltage [30, 31]. However, its overcharge protection has been greatly improved when compared to other batteries. The lithium-iron-phosphate batteries have a long cycle life, with a standard charge with a 5 h rate of up to 2000 times. Lead-acid batteries have a maximum life of 1 -1.5 years, while lithium iron phosphate batteries with the same weight have a theoretical life of 7 -8 years when they are used under the same conditions. Considered comprehensively, its performance-price ratio is four times that of a lead-acid battery in theory. High current discharge can reach 2C for rapid charge-discharge conditions. Under a special charger, the battery can be fully charged within 40 min after 1.5C charging. The starting current can reach 2C, which is twice the battery nominal capacity, while the lead-acid battery has no such performance. According to tests, after 500 cycles of charge-discharge, the discharge capacity of this battery type can still reach more than 95%. The peak value of the lithium-iron-phosphate battery can reach 350–500°C while the peak value of lithium-manganate and lithium-cobalt batteries is only about 200°C. The lithiumiron-phosphate battery has a wide working temperature range from 20°C to +75°C that has high-temperature resistance, which greatly expands the use of the lithium-ironphosphate battery. When the external temperature is 65°C, the internal temperature can reach 95°C. When the battery is discharged, it can reach 160°C. The structure of the battery is safe and intact. It has a larger capacity than ordinary batteries such as lead-acid, ranging from 5 Ah to 1000 Ah for a single battery cell. The standard discharging current rate is 2–5C and the continuous high-current discharging treatment can reach 10C, in which the instantaneous pulse discharge for 10 s can reach 20C. If the battery has been running at full capacity and not discharged fully, the capacity quickly falls below the rated capacity value. This phenomenon is called the memory effect. The battery is generally considered environmentally friendly and does not contain any heavy metals or rare metals. Ni-MH batteries need rare metals. In compliance with the European RoHS regulation, lithium iron phosphate batteries can be used as SGS certified nontoxic products. The reason why it is favored by the industry is mainly due to environmental protection considerations. Some experts said that the environmental pollution caused by lead-acid batteries mainly occurred in their irregular production and the recycling processes of enterprises. Therefore, although belonging to the new energy industry, it cannot avoid the problem of heavy metal pollution. During the processing of metal materials, elements such as lead, arsenic, cadmium, mercury, and chromium may be released into dust. The battery itself is a chemical substance, so it is likely to produce two kinds of pollution. The first type is the pollution from the process site in production engineering. The second is battery pollution after the scrap.

14

1. Lithium-ion battery characteristics and applications

There is a large amount of lead in lead-acid batteries. If not disposed of properly after being discarded, this battery still causes secondary pollution to the environment. The lithium-ironphosphate materials have no pollution of production and application. The most promising anode materials for power lithium-ion batteries are mainly lithium-manganate, shortened as LiMn2O4. The lithium-iron-phosphate is shortened as LiFePO4. The nickel-cobalt-lithium-manganate ternary materials are shortened as LiNiCoMnO2. Due to the shortage of cobalt resources, the high cost of nickel and cobalt, and large price fluctuations, it is generally believed that nickel-cobalt lithium-manganate ternary materials will have difficulty in becoming mainstream power lithium-ion batteries for electric vehicles. It can be mixed with spinel lithium-manganate within a certain range. The lithium-iron-phosphate is generally selected as the positive electrode material. Market analysts from governments, scientific research institutions, enterprises, and even securities companies are optimistic about this material for the development direction of power lithium-ion batteries. Compaction density refers to the mass per unit volume measured after the powder in the container is not compacted under specified conditions. If the synthesized LiFeP4 powders have irregular morphology, they cannot be densely packed, resulting in the low tap density of the product. Spherical materials have excellent fluidity and distensibility. Their particle surfaces are easy to coat into complete, uniform, and firm modified layers. It is conducive to surface modification to improve its comprehensive properties. Therefore, one of the important ideas to improve the tap density of the LiFeP4 material is to prepare spherical particles. It equals areal density and material thickness, which is closely related to sheet specific capacity, efficiency, internal resistance, and battery cycle performance. In general, the battery capacity can be higher with higher compaction density. The synthesis of lithium-iron-phosphate is a complex reaction process, including a solid phosphate, iron oxide, lithium salt, carbon precursor, and reducing gas phase. In this complicated reaction process, it is difficult to ensure the consistency of the reaction. As the battery positive material, it is widely used in new energy vehicles and other fields. The problem of poor high-temperature cycling and storage performance of lithiummanganate can be solved. It has great application potential for the power battery due to its advantages of low cost and high rate performance. Because the battery performance is especially suitable for power applications, the word power is added to the name. It is also called a lithium-iron power battery, shortened to LiFe. In the metal trading market, cobalt, shortened to Co, is the most expensive and has little storage. Nickel is shortened to Ni and manganese is shortened to Mn while iron, shortened as Fe, is the cheapest type. The price of cathode materials is also in line with the price of these metals. Therefore, the lithium-ion battery made of LiFePO4 anode material is the cheapest, and does not pollute the environment. The requirements of rechargeable battery materials are large capacity, high output voltage, cyclic charge-discharge performance, stable output voltage, large current charge-discharge capability, and electrochemical stability. There should be no combustion or explosion caused by an improper operation such as overcharge, overdischarge, or short circuit. The other characteristics are the wide working temperature range, no toxicity or less toxicity, and no pollution to the environment. The lithium-iron-phosphate battery using LiFePO4 as the anode has good performance requirements, especially in large discharging current rate discharging

1.3 Lithium-ion battery chemistries

15

with 5–10C, stable discharging voltage, safety with no combustion, no explosion, number of life cycles, and no pollution to the environment. It is currently the best large current output power battery. When charged, the lithium ions in the positive electrode migrate to the negative electrode through the polymer separator. In the discharge process, the lithium ions in the negative electrode migrate to the positive electrode through the separator. It is named after the lithium ions migrating back and forth in the charge-discharge process. The nominal voltage of the LiFePO4 battery is 3.20 V, and the termination charging voltage is 3.60 V. Its termination discharging voltage is 2 V. Due to the different quality and manufacturing processes of the positive electrode, negative electrode, and electrolyte materials adopted by various manufacturers, their performances are also different. As for the same model as a standard battery packaged in the same package, the battery capacity is different from 10% to 20%. The capacity of the lithium-iron-phosphate power batteries varies greatly. It can be classified into three categories: small ones ranging from a few tenths of milliampere hours, medium ones ranging from tens of milliampere hours, and large ones ranging from several hundred milliampere hours. Similar parameters of different types of batteries also have some differences. The lithium-ion battery pack is used under the following ambient temperature conditions. The charging environment temperature varies from 10°C to 55°C. The discharge environment temperature varies from 20°C to 60°C. The core performance in the battery module is consistent. Each cell in the battery module should be a product made by the same manufacturer with the same structure and chemical composition, and it should meet the following requirements. The difference between the maximum and minimum shelved open-circuit voltages of each cell in the battery module should not be greater than 0.05 V. The difference between the maximum and minimum value of shelved internal resistance should be considered in the battery module shell. The absolute value of deviation should not exceed 0.5 mW when it is below 10 mW. The deviation should not exceed 5% of the averaged values when it is above 10 mW. The difference between the maximum and minimum capacity values of each cell in the battery module should not exceed 1% of the averaged value. The capacity of the battery pack should not be less than 95% of the rated value and its cycle life should be 800–2000 times. The battery pack should be discharged after being charged according to regulations. Its appearance should be free from obvious deformation, rust, smoke, or explosion. The battery pack is equipped with a dedicated battery management system. The error between various parameter values displayed by the battery management system and the parameter values of the battery pack shall meet the requirements. The battery voltage is tested according to regulations, in which the charging voltage display accuracy should be better than 1%. The battery current is tested according to regulations, and the display accuracy of the charge-discharge current should be better than 2.5%. The battery capacity should be tested according to regulations, and the display accuracy of the battery should be better than 5%. The temperature display error should be less than 3°C. The temperature compensation function is included, in which the battery charging shall have a temperature compensation function. When the battery is overcharged, the charging circuit should be cut off and the alarm should be given. It shall not produce liquid, smoke, fire, or explosion. After it is discharged to the termination voltage, the discharge circuit should be cut off and an alarm should be given. It should work well without producing liquid, smoke, fire, or explosion. In case of a

16

1. Lithium-ion battery characteristics and applications

short circuit for the packing battery output end, the circuit should be cut off immediately and the alarm should be given. After troubleshooting, the work should be resumed manually or automatically. After instantaneous charging, the battery voltage shall not be less than the nominal voltage and a test should be carried out. After troubleshooting, it should be able to resume work automatically. As for instantaneous charging, the voltage of the battery pack shall not be less than the nominal voltage. When the battery discharging current reaches the overload protection current value, the circuit should be cut off and the alarm should be given. The battery pack shall not produce liquid, smoke, fire, or explosion. When the temperature reaches the protection point range, the battery pack shall cut off the circuit and give an alarm. Except for the high-temperature protection against the battery management system components inside the battery pack, it shall resume operation automatically when the temperature reaches the recovery point range. The battery pack should be free from liquid leakage, smoke, fire, or explosion, as shown in Table 1.1. The classification of safety performance can be described as follows. It should be tested according to regulations and shall not catch fire or explode. Its module should be tested according to regulations, in which it shall not catch fire or explode. The battery core should be tested according to regulations and shall not produce liquid, smoke, fire, or explosion. Its module is tested according to regulations. No part of the explosive battery penetrated the screen and no part of the battery protruded from the screen. Its module should be tested according to regulations and shall not catch fire or explode. After it is tested according to regulations, its appearance should be free from obvious deformation, and its capacity shall not be less than 90% of the rated value. It should be tested according to regulations, in which its appearance should be free from obvious damage and leakage of liquid. It should be tested according to regulations, and its appearance should be free from obvious damage, leakage of liquid, smoke, or explosion, and shall work well. As for cycling, the battery pack should be tested according to the provisions. It shall have no crack at the appearance, no mass loss, and the capacity should be not less than 70% of the initial state. As for the flame-retardant performance, the test should be conducted according to the provisions for the battery pack with the plastic casing and protective cover. For the metal casing battery packs, the insulation resistance of the positive and negative electrode interfaces of the battery pack shall not be less than 2 mW to the metal shell of the battery pack. The lithium-ion battery has a stable structure, high capacity ratio, and outstanding comprehensive performance. However, its safety is poor and the cost is very high. It is mainly used for small and medium-sized batteries, which are widely used in notebook computers, TABLE 1.1 Battery temperature protection for different conditions. Projects

Protection point

Recovery point

Charging protection for high ambient temperature

60°C  2°C

35°C  2°C

No discharge protection for high-temperature environment

70°C  2°C

55°C  2°C

Protection against discharge for low ambient temperature

30°C  2°C

15°C  2°C

Charging protection for low ambient temperature

15°C  2°C

10°C  2°C

1.3 Lithium-ion battery chemistries

17

mobile phones, and other small electronic equipment with a nominal voltage of 3.70 V. The ternary Li-polymer battery uses lithium-nickel-cobalt-manganese or nickel-cobalt-lithium aluminate as its cathode material. There are many kinds of cathode materials, mainly including lithium-cobaltate, lithium-manganate, lithium-nickelate, ternary materials, and lithiumiron-phosphate. The lithium-iron-phosphate battery as the anode material has a long charge-discharge cycle life, but its disadvantages are that there are large gaps between energy density, high-low temperature performance, and charge-discharge current rate characteristics, so the production cost is high. The ternary positive electrode material with balanced capacity and safety has better cycle performance than normal lithium-cobaltate. Consequently, its nominal voltage is only 3.50–3.60 V due to technical reasons for the early stage, which is limited to the application range. However, the nominal voltage of the battery has reached 3.70 V with continuous improvement and structure improvement; it is easier to reach or even exceed the application voltage of the lithium-ion battery failure level. The majority of small high-rate power batteries use ternary anode materials, which makes the ternary anode materials gradually become mainstream. Lithium-titanate material is used as the negative electrode in the secondary battery, which can form 2.40 V or 1.90 V with positive electrode materials such as lithium-manganate, ternary material, or lithium-iron-phosphate. Besides, it can also be used as a positive electrode to form a 1.50 V lithium secondary battery of a metal or alloy negative electrode. Thanks to the characteristics of high safety, high stability, long cycle life, and environmental protection, it is likely to be a widely used cathode material for a new generation of lithium-ion batteries within a few years. Especially in new energy vehicles and electric motorcycles, its application requires high safety, high stability, and long periods. The operating voltage of the battery is 2.40 V with the highest voltage of 3.00 V. The charging current is greater than 2C when the current is twice the battery capacity value. Lithium-titanate is used as the negative electrode material in batteries. Due to its characteristics, the electrolyte material is easy to interact with each other, and the gas is separated from the charge-discharge cycle reaction process. Therefore, ordinary batteries can easily generate gas expansion, which leads to cell bulging and greatly reduces the electrical performance. Consequently, it greatly reduces its theoretical cycling life. The experimental test results show that ordinary batteries suffer from flatulence after 1500–2000 cycles, which leads to failure conditions. This is also an important factor that restricts its large-scale application. The performance improvement on the lithium-titanate battery is a comprehensive embodiment of the performance improvement on the individual materials and the organic integration of key materials. To meet the rapid charging and long service lifespan requirements, other key raw materials are also targeted besides the negative electrode, including the positive electrode, separator, and electrolyte. Meanwhile, a gas-free battery product is finally formed into a special engineering process experience. Its batch application in the electric bus is realized for the first time. The test data shows that the cycle life of the lithium-titanate battery can exceed 25,000 times and the remaining capacity can exceed 80% under the 6C charge, 6C discharge, and 100% depth of discharge working conditions. Meanwhile, the gas expansion phenomenon generated by the battery core is not obvious and does not affect its life. However, the application of rapid charging for a pure

18

1. Lithium-ion battery characteristics and applications

electric bus also shows that the batteries are grouped. Consequently, the electricity storage performance is quite excellent, which can ensure the daily commercial operation of pure electric buses. Replacing fuel vehicles with new energy vehicles is the best choice to solve urban environmental pollution. For this, lithium-ion batteries have attracted extensive attention from researchers. To meet the requirements of new energy vehicles for onboard ionic power batteries, the development of negative electrode materials with high safety and good performance is necessary. The long service lifespan is a difficult point, and is the research direction of lithium-titanate batteries. Commercial lithium-ion batteries mainly use carbon as a negative electrode, but batteries with carbon as a negative electrode still have some disadvantages in application: (1) It is easy to separate the lithium dendrites of overcharge, causing a short circuit and affecting the safety performance of the lithium-ion battery. (2) The solid-electrolyte-interphase septum can be formed easily, which leads to low initial charge-discharge efficiency and large irreversible capacity. (3) The platform voltage of the carbon material is relatively low, close to metal lithium, which can easily cause the decomposition of the electrolyte, thus bringing about potential safety hazards. Compared with carbon cathode materials, lithium-titanate has a high lithium-ion diffusion coefficient with 2  108 cm2/s and a high charge-discharge current rate. Its potential is higher than that of pure metal lithium, so lithium dendrites are not easy to generate, thus providing a basis to ensure the safety of lithium-ion batteries. There are also some disadvantages. Compared with other types, the energy density is lower. The problem of flatulence has hindered its application. There are still differences in battery consistency, which gradually increases along with the increase in cyclic charge-discharge times. In the rapid pursuit of high energy density, lithium-titanate batteries with high safety performance were once neglected due to their low energy density. It is not easy to produce lithium crystal branches with strong metal activity in the battery charge-discharge reaction. It hardly produces solid-electrolyte-interphase septum in the graphite negative electrode. This solid-electrolyte-interphase septum has extremely poor thermal stability. It is easy to decompose into high temperatures and releases a large amount of heat, which leads to thermal runaway, fire, explosion, and other risks. Its materials that hardly form the solid-electrolyteinterphase septum can perfectly avoid this problem. Through battery testing, it still will not smoke, catch fire, or explode under harsh conditions such as puncture, extrusion, or short circuit. Under the condition of 20  5°C, the single battery is kept for 1 h and then burned in a blazing fire. The battery starts to react after 50 min, which is much more stable than other lithium-ion batteries. The ternary lithium is sought after under the influence of many factors such as mileage, high energy density, and safety performance batteries. As the lithium-titanate battery has high safety performance, its weak energy density is neglected repeatedly. Passenger car mileage anxiety does not apply to buses. Different batteries are used in different fields, which is the nature of the market and technology. As a fixed station and fixed mileage operating vehicle, the bus does not need long-endurance mileage. Its 6-min fast-charging advantage enables the bus to be quickly charged in a short period without affecting its operational efficiency. The Ministry of Industry and Information Technology would speed up the implementation of mandatory national standards for the safety of new energy vehicles, which strengthens the

19

1.3 Lithium-ion battery chemistries

safety supervision of the new energy vehicle industry. The development of power batteries determines the future of pure electric vehicles. The frequent fire accidents of new energy vehicles show that it is indeed biased to regard the energy density of batteries as the only indicator. The research and development of products should be based on the rational judgment of the market, the innovative development of materials technology, and the safety of lives and property. Consequently, it is usually divided into two categories. Lithium metal batteries generally use manganese dioxide as the positive electrode material, lithium metal or its alloy metal as the negative electrode material, and nonaqueous electrolyte solution. Lithium-ion batteries generally use lithium alloy metal oxide as the positive electrode material, graphite as the negative electrode material, and a nonaqueous electrolyte. Although lithium metal batteries have a high energy density, they can reach 3860 Wh/kg theoretically. However, they cannot be used as a power battery of repeated use due to the unstable nature and inability to charge. Lithium-ion batteries have been developed as the main power batteries due to their ability to recharge repeatedly.

1.3.3 Solid-state lithium-ion battery Lithium-ion batteries have not only been widely used in the consumer, communications, and computer fields, but they also have broad development prospects in the fields of new energy vehicles and smart grids. However, because the organic electrolyte of traditional batteries has potential safety hazards such as flammability, explosion, and poor thermal stability, traditional lithium-ion batteries have been slightly weak in energy storage. The emergence of solid-state batteries is considered to be able to solve this problem. The solid-state lithium-ion battery is a secondary battery of cathode materials, anode materials, and electrolytes, which all use solid materials. It is a battery energy storage technology that has been developed since the 1950s. The structure of solid-state batteries is simpler than traditional batteries. A structural comparison diagram of solid-state lithium-ion batteries and traditional batteries is described in Fig. 1.5. The solid-state electrolyte can not only conduct lithium ions but also act as a diaphragm to avoid leakage problems. Meanwhile, solid-state electrolytes are not volatile and flammable, which improves the safety of the battery. In the construction process, the solid-state lithiumion battery does not need to use a liquid electrolyte and diaphragm, which greatly simplifies the assembly steps. A comparison of the characteristics of traditional and solid-state lithiumion batteries is described in Table 1.2.

eLoad

Anode

ePower

+

(A)

Solid electrolyte

e-

-

Load

e-

Anode

Power Diaphragm Liquid eletrolyte + Cathode

Cathode

(B)

FIG. 1.5 Structure diagram of traditional and solid-state lithium-ion batteries. (A) Solid-state lithium-ion battery. (B) Traditional lithium-ion battery.

20

1. Lithium-ion battery characteristics and applications

TABLE 1.2 Property comparison of conventional and solid lithium-ion batteries. Category

Solid-state lithium-ion battery

Traditional lithium-ion battery

Electrolyte

Inorganic materials

Organic electrolyte

Advantages

High security, cycle life, energy density, long-term storage

Widely applied in the fields of electronics and communications

Disadvantages

High cost, immature technology

Flammable and explosive

As they are driven by social development needs and potential market demands, new chemical energy storage systems are constantly emerging based on new concepts, new materials, and new technologies. Chemical energy storage technology is developing in the direction of safety, reliability, long cycle life, large scale, low cost, and pollution free. At the same time, the development of high-energy-density lithium-ion batteries has become an important subject of scientific research and industrial production. As solid-state lithium-ion batteries are filled with high energy density and safety, they have attracted interest from many companies and scientific research institutions around the world. At the 2014 international meeting on lithium-ion batteries, Toyota pointed out that if various technical obstacles are overcome, solid-state batteries could be commercialized in 2030. The urgency of developing solid-state batteries has been fully recognized. A simplified development direction of lithium-ion batteries is described in Fig. 1.6. There are currently three main development directions for solid-state lithium-ion batteries: (1) Thin-film solid-state lithium-ion battery: Thin-film lithium-ion batteries are prepared onto thin films in the order of anode, electrolyte, and cathode, which are finally packaged to form a whole battery. The Oak Ridge National Laboratory has been vigorously focusing on it. The thickness of a battery cell is only 0.17 mm. When the depth of discharge is 100% and the 1C charge is repeated, it has a life span of more than 10,000 times. It has realized 70C high rate discharges and a working temperature in the range of 40°C to 85°C. It embodies many advantages of solid-state thin-film lithium-ion batteries.

Future power&energy storage direction Existing Li-ion battery

High energy density High security

Improved Li-ion battery

Larger capacity

Polymer solid-state Li-ion battery Inorganic solidstate Li-ion battery

Solid-state Li-ion Battery Higher security Thin-film solidstate Li-ion battery

2010

2020

FIG. 1.6 Simplified development direction of lithium-ion batteries.

2030

21

1.3 Lithium-ion battery chemistries

(2) Large-capacity polymer solid-state lithium-ion battery: This is a solid-state lithium-ion battery that uses a polymer electrolyte as the preparation material. Polymer materials have many advantages such as light mass, good electrochemical stability, and high lithium-ion migration number. This kind of solid electrolyte material has aroused great interest from industry researchers since its appearance in the early 1970s. In the past 30 years, research in this field has received widespread attention. (3) Large-capacity inorganic solid-state lithium-ion battery: This type of solid-state lithium-ion battery is made of inorganic solid materials and has a very high degree of safety. Companies are increasing investment in this area. A highly competitive inorganic all-solid-state battery will be used in power or energy storage batteries in the future. Its safety and long cycle life are ensured by increasing its energy density as well as reducing costs. It lays a solid foundation for its large-scale practical application basis.

1.3.4 Comparative battery types analysis Due to the combination of different elements, the composition of the positive electrode material has great performance differences in various aspects, resulting in increased disputes over the route of the positive electrode material in the industry. The most frequently mentioned power batteries are mainly the lithium-iron-phosphate, lithiummanganate, lithium-cobaltate, and ternary nickel-cobalt-manganese lithium ternary types. A comparison of the advantages and disadvantages of these battery types is described in Table 1.3. Theoretically, the required battery should have the advantages of high energy density, high volume density, good safety, high-temperature resistance, low-temperature resistance, long cycle life, and high-power charge-discharge. All advantages should be integrated at a low cost. However, there is currently no such battery, so there is a trade-off between the advantages and disadvantages of different types of batteries. Moreover, different power systems have different requirements for batteries, which are based on long-term requirements for control systems.

TABLE 1.3

Advantages and disadvantages of different lithium-ion battery types.

Type

Advantages

Disadvantages

Lithium-iron-phosphate

Long service life span and low cost

Low energy and tap density

Ternary lithium

High energy density

Poor safety, high-temperature resistance

Lithium-manganate

High density and low cost

Poor high-temperature resistance

Lithium-cobaltate

High energy density

Poor safety

22

1. Lithium-ion battery characteristics and applications

1.4 Lithium-ion battery characteristics After analyzing the structural principle of the lithium-ion battery, the parameters and common concepts are introduced one by one. Parameters include terminal voltage, electromotive force, capacity, internal resistance, state of charge, depth of discharge, cycle life, and selfdischarging current rate. After that, its circuit characteristics are analyzed.

1.4.1 Internal parameter relationship The terminal voltage can describe the potential difference between the positive and negative electrodes of a lithium-ion battery, which can be divided into open-circuit voltage and working voltage according to the operating conditions of the circuit [32]. Open-circuit voltage is the lithium-ion battery terminal voltage of the no-load and no-power-supply. The whole structure of the internal battery parameters can be described as shown in Fig. 1.7. The open-circuit voltage is the voltage at both ends of the battery when the battery is fully stationary. When a current passes through the battery, the potential deviates from the equilibrium potential, called electrode polarization [33]. Electrode polarization can be divided into concentration polarization and chemical polarization. Due to the influence of the ohmic

Voltage Current Temperature

1.SOC Estimation

3.SOF Estimation

Zero load voltage method Ah segmentation method EKF-online identification

SOC

Time

SOC limit

Current control area Battery temperature

Diagno sis SOC 1. sensors 2. actors Battery 3. network temperature 4. cell Voltage 5. over voltage 6. over current Current 7. SOC 8. loose joints Isolation 9. isolation Leakage gas 10. over or low concentration temperature 11. over temperature Signals in rising rate network 12. consistent

Min

Temperat ure limit Limit 2. Health status assessment

Time Battery temperature

Life prediction SOL SOH

Voltage Current Fault status diagnosis

Fault limit

FIG. 1.7 Internal parameter relationship of the battery management system.

Battery output current capability

23

1.4 Lithium-ion battery characteristics

effect and the polarization effect, the voltage of the battery fluctuates in its application process. Therefore, the acquisition of open-circuit voltage usually needs the battery to be still for a long period [34]. Because of the one-to-one correspondence between the open-circuit voltage and the state of charge [35], its corresponding state value can be obtained if the open-circuit voltage value of the battery is known.

1.4.2 Capacity characteristics Due to the internal resistance of the battery, the working voltage of the discharge is lower than the open-circuit voltage, and the working voltage of charging is higher than the opencircuit voltage. At the end of charging, the highest allowable voltage of the battery is called the charge cut-off voltage, and the lowest allowable voltage after the end of the battery discharge is called the discharge cut-off voltage. Beyond this limit, the battery suffers some irreversible damage and the cutoff voltage is an important safety indicator. The capacity variation in the charge-discharge process can be described as shown in Fig. 1.8. The working voltage refers to the potential difference between the positive and negative electrodes when the battery is working. Due to the internal resistance of the battery, the working voltage is lower than the open-circuit voltage. According to the working state as the rated voltage classification, the lithium-ion battery working voltage is also divided into rated voltage, theoretical voltage, open-circuit voltage, and closed-circuit voltage connected with load [36]. The rated voltage should be directly calibrated by the battery manufacturer before delivery. The theoretical voltage is the value of the positive and negative electrode potential difference after the chemical reaction equilibrium in the battery, represented by E. Its calculation process is described as shown in Eq. (1.3): E ¼ φ +  φ

(1.3)

In the above equation, φ+ is the battery positive electrode potential, and φ is the battery negative electrode potential. Open-circuit voltage can be measured by connecting a multimeter or voltmeter directly to the positive and negative ends of the battery, which is the voltage used in the equivalent model simulation in this book. Because the lithium-ion battery is not an ideal power source, the electrolyte and electrode materials have internal resistances. Consequently, the open-circuit voltage is lower than the electromotive-force value. When the internal resistance is minimal, the open-circuit voltage can be treated as the

C/Ah

T1

Capacity calculation Charging

Charging

Discharging

T2

Capacity calculation

Discharging

0

FIG. 1.8 Capacity measurement of lithium-ion batteries.

Charging

Discharging T/h

24

1. Lithium-ion battery characteristics and applications

Charging stage Capacity Discharging stage

Full capacity

Initial capacity Termination capacity

Charging time

Discharging Discharging Charging time time time

Time

FIG. 1.9 Battery charge-discharge process for capacity determination.

electromotive force value. Consequently, the capacity measurement procedure can be realized as shown in Fig. 1.9. Under working conditions, the instantaneous working voltage changes dynamically. The phenomena of overcharge and overdischarge cannot be avoided, which damages lithium-ion batteries. Therefore, the upper and lower limits of voltage should be set in use, in which a limited cutoff voltage should be reasonable. If the upper limit cutoff voltage is too high, security risks may occur [37]. If the value is too low, the battery may not be fully charged. Similarly, if the lower limit cutoff voltage is too low, battery safety risks may occur. However, if it is too high, the efficiency of the battery power use is too low. According to the study, the discharge limited cutoff voltage can be a little lower when discharging the battery with a higher discharge ratio C and lower ambient temperature conditions. In order to obtain the characteristics and relationship of open-circuit voltage, the battery is subjected to a cyclic discharge shelved experiment at room temperature of 23°C. The specific experimental steps are described as follows. S1: The battery is charged by using the constant-current-constant-voltage method. It is charged with a constant current of 1C to the upper limit cut-off voltage 4.20 V and then charged with this constant voltage using the cut-off current as 0.05C [38]. After charging, the battery is shelved to stabilize its voltage. Considering the capacity of the selected lithium-ion battery, the shelved time should be 30 min or longer. S2: The state-of-charge sampling points can be selected as 1, 0.95, 0.9, 0.85, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.15, 0.1, and 0.05. Because lithium-ion batteries have significant voltage changes in the early and late stages of discharge, in order to obtain a more accurate open circuit voltage of the charge state characteristic curve, the sampling point interval should be small. S3: The constant-current discharging treatment of the lithium-ion battery is carried out with a 1C ratio, and the discharge cut-off voltage is 2.75 V. When the discharging treatment reaches the state sampling point, the discharge is stopped. It is shelved for 30 min, and the

1.4 Lithium-ion battery characteristics

25

FIG. 1.10 Voltage curves under the different discharge current rates. (A) Multipulse current discharge curve. (B) Multiconstant current discharge curve.

terminal voltage at this time point is considered the open-circuit voltage. After that, the discharge continues until the state of charge is 0, and the experiment is finished. The discharge time is set as 4 min. Considering that the selected lithium-ion battery has a small capacity, the shelved time for discharge is set to be 12 min. The open-circuit voltage collection is performed at each sampling point. After the experimental data are obtained, the curve of the open-circuit voltage with time is obtained for discharge. The specific experimental process is as follows. S1: 1.0C, 4 min discharge, 15 cycles. S2: 1.5C, 4 min discharge, 13 cycles. S3: 2.0C, 4 min discharge, 10 cycles. To prevent the battery from overdischarge, the cut-off voltage is set to 3.00 V, and the intermediate shelved time is 12 min. Several kinds of multiplier discharging processes are obtained for comparison, as shown in Fig. 1.10. Wherein, 1.0C, 1.5C, and 2.0C represent different discharging current rates. As can be known from the experimental results, the open-circuit voltage shows a decreasing trend with the discharge time extension of different current rates. The higher the discharging current rate, the more obvious the decreasing trend. Meanwhile, the open-circuit voltage is low when the discharge current rate is high. The effect of varying shelved time is analyzed for the open-circuit voltage of the battery. To study the lithium-ion battery open-circuit voltage variation characteristics of the varying shelved time, the battery discharge shelved experiment with a 1.5C discharge ratio is selected for analysis. The shelved time is set as 5, 8, and 12 min. Then, three different voltage variation curves are obtained, as shown in Fig. 1.11.

FIG. 1.11 Voltage variation to the shelved time in the pulse-discharge process. (A) Original voltage variation. (B) Voltage changing trend after shelving.

26

1. Lithium-ion battery characteristics and applications

Wherein, the voltage change is described under the different shelved and discharge times. As can be seen from the figure, the voltage change trend in the early stage of discharge is large. It is relatively gentle in the later stage. To analyze the variation on open-circuit voltage under the varying shelved time, the open-circuit voltage under the battery shelved is taken. The opencircuit voltage is plotted and compared under three shelved times. U1, U2, and U3 respectively represent the voltage characteristic variation curve of the lithium-ion battery under the shelved time of 5, 8, and 12 min. The abscissa n represents the number of times the battery has shelved in the discharge. As can be known from the figure, at the early stage of discharge, the difference in the open-circuit voltage of the battery of the three shelved-time kinds is not obvious. However, it rises higher when the shelved-time is longer with the increase in the discharge period. The real-time temperature of the lithium-ion battery is an important factor that cannot be neglected. The battery activity is in direct proportion to temperature. As it increases, the exchange rate of the lithium ions between electrolytes increases. The overall battery activity increases, which is reflected more through the energy output, increased usable capacity, and improved efficiency. It is found that the lattice structure stability of the battery anode material is gradually weakened under long-term high temperature. After that, its safety and cyclic charge-discharge life are reduced seriously. On the contrary, the battery activity of the anode and cathode material decreases for the long-time or low-temperature conditions. The capacity is less than the rated value, in which its charge-discharge efficiency decreases as well as its service efficiency. Because the material is reduced, the exchange capacity of the battery internal lithium-ion decreases and the electrolyte transport lithium-ion drops. Consequently, lithium ions will be deposited on both ends of the positive and negative electrodes as well as the electrolyte, causing safety hazards. In a low-temperature environment, reducing the charge and discharge current can reduce the deposition of the lithium ions to reduce hidden dangers. To sum up, a long period of working at high- or low-temperature conditions harms the battery and reduces its service lifespan. The battery capacity refers to the total amount of electricity that can be released in a full state of the lithium-ion battery, generally expressed by the symbol Q, and the unit is mAh shortened from the milliampere hour or Ah shortened from the ampere hour. The battery capacity is an important measure of battery quality, which is closely related to battery life. In the laboratory, a constant-current or constant-voltage discharge method is generally adopted to discharge the battery. By measuring the amount of electric energy released by the battery of the whole discharge process or calculating the product of discharging current and time, this is the storage capacity value, whose unit is Ah or mAh. The battery capacity can be divided into rated capacity, capacity, and theoretical capacity. The theoretical capacity is the initial value of the battery total charge calculated according to the electrochemical reaction against the battery when the internal lithium ions completely participate in the reaction. Because the theoretical capacity is equivalent to its ideal value, its value is larger than the rated capacity. The ideal capacity is the maximum amount of energy that a battery can release under theoretical conditions; all the chemical energy stored inside is converted into electrical energy without loss. In the discharge process, the discharge capacity is a part of the ideal value due to the existence of other side reactions to the battery. The rated capacity is measured by the battery manufacturer and is directly calibrated on its outer surface. It is an important indicator of how much charge is stored in the battery, which is also an indicator of how long the battery lasts. It is also known as calibration capacity, according to the relevant provisions of the relevant departments of the state, under certain

27

1.4 Lithium-ion battery characteristics

discharge conditions such as temperature, discharging current rate, and so on to ensure the minimum amount of electricity that can be released by the battery. Rated capacity is the ampere hour capacity indicated by the licensed manufacturer, which is one of the important parameters of lithium-ion batteries. The capacity is the maximum that a battery can release when it is bought in the store under certain discharge conditions. The capacity of the battery is often calculated by the product of discharging current and discharge time. It is not the capacity of the new battery, but also the used capacity. At this time, the capacity is less than the capacity of the new battery. The capacity is the battery power amount that can be released when it is operating in the measured environment. The factors that affect battery capacity are very complicated, including the external environment temperature, material, and service lifespan. The discharge of a fully charged battery for certain conditions is the integral of discharging current and time. The discharge capacity can be calculated by the constant-current discharging treatment using the alternating current values, as shown in Eq. (1.4): C¼IT ¼

ðT

I ðtÞdt

(1.4)

0

Due to the aging caused by the recycling of the battery materials, there is a large deviation from the battery capacity and the calibration capacity of the factory. The measured discharge capacity is very important to realize its accurate state estimation, so it should be calibrated first. Three lithium-ion batteries are selected, and the experimental object is 1.40 Ah/3.70 V lithium-ion batteries. The experiment is carried out at 23°C. The specific steps are described as follows: (1) The battery is charged with a constant current rate of 1C, and the charging cutoff voltage is set to 4.20 V. It is then converted to a constant-voltage charge with a charge cutoff current of 0.02C to ensure that the battery is fully charged. (2) Due to the small capacity of the selected battery, the shelved time can be selected to be 30 min. (3) A constant discharge experiment is conducted on the battery of a 1C current ratio, and the discharge cutoff voltage is 2.75 V. (4) After the discharge, it is shelved for 30 min. (5) The procedure repeats steps (1) to (4) three times to end the experiment. Taking the unit of capacity as Ah, the experimental results are described as shown in Table 1.4. As can be found from the experimental results, the discharge capacity of the single battery is always less than the charging capacity for the same cycle. This is due to the influence of internal resistance. In the discharge process, the battery itself dissipates a part of the energy, which makes the released energy in the working process always less than the charged battery energy. TABLE 1.4

Volume calibration experimental data. Unit 1

Unit 2

Unit 3

No.

Charging capacity

Discharge capacity

No.

Charging capacity

Discharge capacity

No.

Charging capacity

Discharge capacity

1

1.472

1.388

1

1.624

1.395

1

1.535

1.402

2

1.406

1.294

2

1.532

1.396

2

1.486

1.401

3

1.387

1.390

3

1.508

1.395

3

1.460

1.404

28

1. Lithium-ion battery characteristics and applications

1.4.3 Open-circuit voltage The electrolyte, anode, and cathode materials are aged in the cyclic charge-discharge process of the lithium-ion battery as well as its external structure. The period from the beginning state of the battery to its end-of-life status is the cycle life of the battery. This experimental result is the data obtained from the charge-discharge test conducted by the manufacturer following the stipulated standard test environment. Generally, it is the discharge of the capacity when it becomes 80% of the rated capacity. The number of charge-discharge cycles is the total continuous battery charge-discharge cycle number of this condition. Seven battery cells of the same type are used with a series connection for experimental analysis. The characteristic curve of the relationship between the open circuit voltage and the state of charge can be obtained. And then, its curve fitting treatment can be conducted by introducing the polynomial equation into the mathematical expression process toward the experimental data, as shown in Fig. 1.12. As can be known from the experimental results, the open-circuit voltage of the battery increases along with its state-of-charge increase. In the middle stage of the battery discharge, voltage changes tend to be flat, which is called the battery discharge platform effect. The open-circuit voltage of the battery changes obviously at the beginning and end of discharge. At this time, the slight variation error of the open-circuit voltage of the battery causes a drastic change in the state of charge. Therefore, the method using the open-circuit voltage is not suitable for online battery state estimation to process the experimental data. The seventh-order polynomial of open-circuit voltage toward the state of charge can be fitted as shown in Eq. (1.5): Uoc, k ¼ f ðSk Þ ¼ 75:232  S7k  270:75  S6k + 389:23  S5k 287:21  S4k + 117:89  S3k  27:427  S2k + 3:9258  Sk + 3:2377

(1.5)

wherein Sk is the battery state-of-charge value at the time point of k, and UOC,k is the corresponding open-circuit voltage value. To obtain the open-circuit voltage of the battery characteristics in the charge-discharge experiment, the EBC-A10H battery capacity measuring instrument is used to conduct the cyclic discharge experiment at a room temperature of 23°C. The rated voltage of the lithium-ion battery is 3.70 V. Considering that the battery working voltage slightly exceeds the rated voltage in operation, it is first charged with a constant voltage of 1C and the cutoff voltage is 4.15 V. A continuous discharge experiment can be conducted on the battery by using the multiplier discharge as 1.0C, 1.5C, and 2.0C. The voltage can be measured to obtain its change diagram over time when the battery is shelved. As 29 27

U (V)

U (V)

28

UOC

26 25 24 23 22 0.0

(A)

FIG. 1.12 comparisons.

0.2

0.4

S (1)

0.6

0.8

1.0

(B)

29 28 27 26 25 24 23 22 21 0.0

UOC UOC2 UL_Avr_1C1C UL_Avr_1C02C

0.2

0.4

S (1)

0.6

0.8

1.0

Open-circuit voltage relationship. (A) Long-time shelved voltage curve. (B) Multiple method voltage

1.4 Lithium-ion battery characteristics

29

can be known from the experimental results, the open-circuit voltage changes of different discharge ratios tend to be flat. It is higher at the end of 2.0C than 1.5C, which is also higher than 1.0C. This is because the high voltage discharge time is fast compared with the low opencircuit voltage change. Consequently, the voltage at the end of the discharge rebound is high. The self-discharging current rate refers to the ratio of the battery discharge capacity and its rated power of the condition of no load, which is used to express the consumption rate of the battery capacity. It is mainly affected by factors such as the battery manufacturing process, materials, storage conditions, and so on. Generally, it is expressed as the percentage of the battery capacity decline in unit time for months or years. Therefore, the self-discharging current rate is a leakage, which is used in the testing open-circuit environment. Because the leakage current is very small, the self-discharging current rate of the lithium-ion battery generally refers to its monthly value. It can also be regarded as the battery charge retention rate.

1.4.4 Internal resistance characteristic Lithium ions move from one pole to the other in the battery, and the influencing factors constitute its internal resistance which hinder the movement of ions. Due to the internal resistance of the battery, the battery terminal voltage is lower than the electromotive force and the open-circuit voltage toward the state of charge. In the charging state, the battery terminal voltage is higher than the electromotive force and open-circuit voltage. The essence of electric current is the directional movement toward the electric charge. In the movement, electrons are resisted by the material itself and the magnetic field. Internal resistance refers to the resistance of the current flowing through the battery when the battery is working, which is characterized by internal resistance. Internal resistance is one of the important parameters of a lithium-ion battery. The size of the internal resistance directly affects the working voltage, working current, capacity of the battery, and so on. The internal resistance of a battery is not constant. The quality of active substances causes internal resistance changes in the charge-discharge process, temperature changes, current ratio, discharge time, and electrolyte concentration. The internal resistance includes internal resistance R0 and polarization resistance Rp. The electric field is generated in the battery redox reaction process. Under the action of this electric field, the dielectric generates a significant charge due to the polarization effect. The polarization resistance corresponds to the current change. In the charge-discharge process, the internal reaction process mainly includes the electrochemical polarization and concentration polarization due to the internal resistance generated by the polarization reaction. The nature of the active substances, electrode structure, manufacturing process, and working conditions leads to different polarization resistances. The total internal resistance is equal to the ohmic resistance plus polarization resistance. The calculation formula is described in Eq. (1.6): R ¼ R0 + Rp

(1.6)

The internal resistance characteristic refers to the battery resistance itself to the current in the process of use. Even with the same lithium-ion battery type, the aging degree also shows different sizes of internal resistance due to the internal material composition and its

30

FIG. 1.13

1. Lithium-ion battery characteristics and applications

Cyclic discharge test voltage curve. (A) Voltage changing mechanism description. (B) Overall processing

voltage.

application environment. Therefore, it is introduced into the estimation modeling process to describe the various characteristics of the internal resistance. The internal resistance of the battery generally includes ohmic and polarization resistances. The magnitude order is usually in the order of microohms or milliohms. The internal resistance is an important indicator of battery performance, the characteristics of which are influenced greatly in the equivalent modeling process as well as its construction and state estimation. In the cyclic dischargeshelved experiment, the local voltage change curve can be obtained as shown in Fig. 1.13. Wherein the battery is connected with the load and discharges until the terminal time point after the end of discharge, which can be described by the parameters of dUR and dUR1. The voltage at the end of the battery suddenly drops. Due to the internal resistance of the battery itself, it has dramatic voltage changes at the beginning and the end of discharge; this is called the internal resistance effect. In the first stage, the terminal voltage drops rapidly, and then slowly decreases. The voltage rises slowly at the end of discharge, that is, dU1 and dUR1 in the figure, which is caused by the polarization effect. When the battery is fully stationary, its internal balance is reached, and the polarization effect disappears. As for the polarization effect, its resistance is affected by its production process, internal structure, and working conditions. The discharging current and operating temperature have an obvious effect on battery polarization resistance. This is due to its effect on the movement toward the lithium ions inside the battery. The internal resistance is the most sensitive parameter to the temperature that varies greatly at time-varying temperature conditions. As the main reason for the battery degradation performance at low temperatures, its internal resistance is too large. In general, the lithium-ion battery is used as a power source and its internal resistance should be as small as possible. Especially in the case of high-power applications, small internal resistance is a necessary condition.

1.4.5 Power capability variation No matter the current or voltage sources, there is a certain internal resistance and the lithium-ion battery as a power source is no exception. When the battery flows through a large current for a long time, the current flows through the resistance to generate a large amount of heat energy, as shown in Eq. (1.7):

31

1.4 Lithium-ion battery characteristics

W ¼ Pt ¼ I 2 Rt

(1.7)

This leads to a continuous increase in the battery temperature while the temperature rise also leads to an ascension of the resistance value, forming a vicious circle. The management system takes measures to deal with heat dissipation after monitoring temperature abnormalities. Besides, the long-time and large current values also lead to potential safety hazards. Therefore, it is necessary to monitor the current size in the charge-discharge process, in which the current ratio can be used to represent the current rate that is expressed by C. The chargedischarge current rate refers to the ratio of the working current and the battery rated capacity for the charge-discharge working process, which can be estimated at different current rates. The charge-discharge current rate is related to the power supply efficiency that is different from the varying current values.

1.4.6 Coulombic efficiency In the capacity calibration experiment, the battery released capacity is always less than its charging capacity for the discharge process due to its consumption. The concept of the Coulomb efficiency of the lithium-ion battery is proposed. The Coulomb efficiency is usually used to describe the released battery capacity. It refers to the ratio of the discharge capacity after the full charge and the charging capacity of the same cycle. It is usually a fraction of less than 1. Due to electrolyte decomposition, material aging, ambient temperature, and different chargedischarge current rates, the discharge efficiency of the battery is affected. In the later simulation model, the charge-discharge efficiency is added to compensate and correct the capacity value. Combined with the capacity calibration experiment, the calculation of the Coulomb efficiency in different charge-discharge processes is described as shown in Table 1.5. Wherein the Coulomb efficiency is a fraction smaller than 1. This is because the internal resistance of the battery itself consumes some electrical discharge energy so that the total electrical energy released by the battery is always less than the total electrical energy charged. Besides, the experiment verified that when the discharging current is larger, the Coulombic efficiency is smaller, and the internal resistance is larger. Therefore, self-consumption is also increased, the discharge power is reduced, and the Coulomb efficiency is reduced. Because the greater the discharging current, the more intense the electrochemical reaction against the battery, and the greater the internal resistance generated, thus the greater the energy consumption of the battery itself. TABLE 1.5

Coulomb efficiency. Unit 1

Unit 2

Unit 3

No.

Coulomb efficiency

No.

Coulomb efficiency

No.

Coulomb efficiency

1

0.942

1

0.859

1

0.913

2

0.920

2

0.911

2

0.943

3

1.002

3

0.925

3

0.962

32

1. Lithium-ion battery characteristics and applications

1.5 Battery aging behavior In production, lithium-ion batteries can be divided into liquid and Li-polymer types through the internal structure of the batteries. The two batteries have many similarities. The materials used for their positive and negative electrodes are the same. The only reason for the difference between the two batteries is the difference in the electrolyte. The electrolyte of the Li-polymer battery is a colloidal polymer while the electrolyte of liquid batteries is liquid. Due to the difference between electrolytes, a Li-polymer battery has more advantages over liquid batteries in terms of both quality ratio and service lifespan. However, because the working temperature requirement is relatively low, the higher temperature causes serious potential safety hazards, so it is not suitable for new energy vehicles. The batteries use a wide variety of cathode materials, in which the most commonly used materials are lithiummanganate, ternary types, lithium-cobaltate, and lithium-nickelate.

1.5.1 Aging mechanisms The most commonly used lithium-ion battery as a power source is the lithium-ironphosphate battery, but its disadvantages are that there is a big gap among energy density, operating performance at high temperature, low temperature, and charge-discharge technologies. In the face of these technical difficulties, the development of lithium-iron-phosphate batteries is facing great bottlenecks. Lithium iron manganate battery has low energy density, weak stability, and poor storage memory disadvantages in its high temperature operation and cycling application processes. Therefore, lithium-manganate is only used as the positive electrode material in the first generation of international lithium-ion batteries. Multicomponent material batteries are gradually beginning to be understood and are increasingly recognized by the industry because they have comprehensive performance, low cost, and more promising characteristics than the lithium-iron-phosphate and lithiumiron-manganate ones. The ternary lithium-ion battery has become the main technology research route to replace the widely used lithium cobalt batteries; it is used in notebook batteries. The electrochemical principle of the battery charge-discharge is the process of the lithium-ion separating and being sucked into the positive and negative plates. In this process, no other side reactions occur with the reversible reaction, which ensures the safety and reliability of lithium-ion batteries. When the battery is charged, the lithium ions escape from the positive electrode material and pass through the solid-electrolyte-interphase diaphragm to enter the graphite negative electrode. Conversely, when the battery is discharged, the lithium ions are separated from the graphite and pass through the separator to return to the positive electrode. As chargedischarge progresses, they continue to intercalate and deintercalate from the positive to negative electrodes. The single cell of the ternary lithium-ion battery can have a high voltage that reaches 3.70 V, which is at a relatively high level in the relatively mature battery. Besides, the specific energy of the ternary battery is high, generally, 160–190 Wh/kg; some products have exceeded 200 Wh/kg. Meanwhile, the cycle life of the ternary lithium-ion battery is long [39]. After 2000 cycles, its capacity is still 80% of the initial capacity.

1.5 Battery aging behavior

33

FIG. 1.14 Charge-discharge internal and external changes. (A) Charge-discharge schematic. (B) Charging currentvoltage curve.

According to the charge-discharge principle, the battery characteristics are tested using a testing device, from which its closed-circuit voltage and capacity characteristics can be obtained. Under working conditions, the battery charge-discharge current changes, which can be controlled by the charging device. The discharging condition varies greatly depending on the load-type working conditions. Therefore, the discharge experiment is set at different magnifications, in which the current and voltage curves obtained for the charging experiment are described in Fig. 1.14. The charging experiment is performed by a constant-current charging to the upper limit of the battery voltage and then performing constant-voltage charging. At this time, the current gradually decreases until the end condition can be satisfied. When the current reaches 0.05C, the charging process is completed. It can be seen that the battery is in a constant-current charging process before the time point of T. It can be considered a fast-charging phase until the voltage reaches the upper limit of 4.20 V, after which the battery enters a constant-voltage charging stage. The current decreases gradually as time goes by; it enters the trickle charging phase until the current reaches a minimum value of 0.05C before the charging process ends. In the discharge experiment, the constant-current discharge is performed at different magnifications. The discharge is terminated to the cutoff voltage. The changing battery voltage and varying capacity curves of the discharge experiment are shown in Fig. 1.15.

(A)

(B)

FIG. 1.15 The different current rate discharge curves. (A) Voltage curve. (B) Capacity curve.

34

1. Lithium-ion battery characteristics and applications

As can be known from the voltage response curve in the figure, the lithium-ion battery is a nonlinear electrochemical system. To describe its nonlinear characteristics very accurately, an effective method is to analyze the electrochemical reaction mechanism inside the battery. Starting from the working principle of the battery establishes a kinetic partial differential equation for the electrolyte and the electrode. This method can accurately obtain the macroscopic physical quantities, such as voltage and current. It can also largely simulate the microscopic physical quantity state of the battery, such as lithium-ion concentration and electrode current density. However, the method of analyzing the principle is too complicated. Consequently, it needs lots of knowledge in the field of physical chemistry. It is necessary to deeply understand the battery internal reaction mechanism, solve complex mathematical equations accurately, and establish the simulation model effectively, which are all quite difficult. Therefore, a relatively simple circuit model can be built with a model framework to solve the nonlinear characteristic simulation problem of the battery. The lithium-ion battery will experience a slow decline process of the main discharge stage, which shows its capacitance characteristics. Therefore, the dynamic response battery characteristics are simulated by a combined network of circuit components such as resistors and capacitances, thereby realizing the battery characteristic simulation. This method is highly feasible, the high accuracy of which can be achieved within the main capacity range. As can be known from the discharge capacity curve, the targeted battery capacity characteristics can be obtained for the management system design. The discharging curves reach the N line of the rated discharging current rate compared to the other discharging current rates, which indicates that the battery can be discharged. The maximum released capacity is also closest to the nominal capacity of the battery. Thereby, the battery capacity characteristic can be obtained. When it is discharged at the rated discharging current rate, the battery has the most capacity as the rated value. In general, the battery voltage and power rise along with the charging process, as shown in Fig. 1.16. Wherein both the capacity curve and the voltage rise relatively fast of the initial charging process of the lithium-ion batteries. The maximum rising value of the voltage curve is 4.30 V at this time point, in which the voltage no longer rises because of the internal resistance of the battery change. As can be known from the figure, the capacity is always increasing in the

(A)

(B)

FIG. 1.16 Charging voltage and electric quantity characteristic diagram. (A) Voltage variation curve. (B) Charging current curve.

1.5 Battery aging behavior

35

whole charging process. In the early stage, the faster the voltage increases, the faster the capacity increases. On the contrary, the charging current decreases regularly and exponentially with the time extension. The mathematical expression can be obtained as shown in Eq. (1.8): I ðtÞ ¼ I0 eat

(1.8)

wherein, the function I(t) can describe the relationship between the current I and the independent time variable t. Consequently, I(t) is the instantaneous charging current at any time. I0 is the maximum charging current initially bearable at the time point of t ¼ 0. The charging acceptance efficiency is a. The characteristic diagram of the lithium-ion battery charging can be depicted by corresponding function expressions. Along with the time change, I(t) decreases regularly, and the law is exponential. According to the theory developed by Joseph A, when the charging planning strategy diagram is closer to the curve described in the above diagram, the charging speed can reach the fastest and the highest charging efficiency can be obtained. When the curve exceeds, it is easy to cause damage to the battery.

1.5.2 Calendar aging process The discharging characteristics of the lithium-ion battery can be affected by the discharge depth, working environment temperature, and the internal temperature of the battery, thus causing the aging process. The charge-discharge current rate, discharge time, and other factors can also affect the discharging characteristics of the lithium-ion battery. The discharging characteristics change from different discharging current rates. The battery emits a certain battery capacity for a specified time, and the current value required at this time point can be defined as the discharging current rate. The formula is described as shown in Eq. (1.9): Rd ¼ Id =Cr

(1.9)

wherein Rd is the discharging resistance, which can be calculated by the discharging current Id and its rated capacity Cr. The lithium-ion battery should not be discharged too much when discharging because the battery can be heated up easily when the battery charge-discharge rate is high. Once the battery heats a lot, permanent irrecoverable damage will be caused to it. The battery voltage under the relative residual capacity is inversely proportional to the discharging current rate. When discharging at a rate of 0.2C for the rated capacity, although the battery voltage drops to 2.75 V, it can continue to release energy. When discharging at a current rate of 1C, the maximum discharge capacity is only 98% of its rated value. The characteristic curves of the voltage variation on the battery storage capacity of different charge-discharge current rates are described in Fig. 1.17. The drawing of the graph is performed at discharging current rates of 0.2C, 0.5C, and 3.0C, respectively. As can be seen from the figure, the change of voltage relative capacity of the lithium-ion battery is different from different discharging current rates, and the curve decreases faster with a higher rate. As can be seen from the figure, the higher the rate of discharge, the less energy that can be released in the end. Also, it can be seen from the figure that no matter what rate of discharge is used, the final voltage stops at about 2.80 V and cannot continue to drop. If the discharge is continued, the battery will be damaged.

36

1. Lithium-ion battery characteristics and applications

(A)

(B)

FIG. 1.17 Discharging characteristic graph. (A) Voltage curve for different current rate. (B) Discharging efficiency verification.

1.5.3 Temperature effect on aging process The low-temperature performance of the lithium-ion battery is good, and it can still reach about 70% of the initial capacity for 30°C. For the case of discharge less than a 1C rate, the discharge capacity degradation is not obvious for the environment of 20°C. Compared with other types, ternary lithium-ion batteries are less secure. At 80–120°C, decomposition of the solid-electrolyte-interphase septum begins to occur. When the battery temperature is around 120°C, the negative electrode starts to react with the electrolyte. When the temperature rises to 200°C or higher, the positive electrode active material starts to decompose, so the thermal runaway phenomenon occurs. Then, a state can be formed in which the current and temperature can promote each other mutually. When the battery is overcharged, lithium dendrites appear inside the battery. When overdischarged, the negative copper foil begins to dissolve, causing the battery performance to drop sharply. Because of the above various characteristics of lithium-ion batteries, the requirements for the battery management system are more stringent. Consequently, it must be more accurate and reliable, as shown in Fig. 1.18. The working performance of lithium-ion batteries is particularly affected by temperature. The temperature changes have a greater impact on the charge-discharge of the batteries. The lowest temperature for its normal operation is 30°C and the highest temperature is 60°C. Therefore, when it is used as energy supply equipment, the temperature should be paid much

(A)

(B)

FIG. 1.18 Discharging characteristic graph. (A) Charging efficiency variation. (B) Capacity variation and its normalization.

1.6 Lithium-ion battery applications

37

attention. To obtain the most effective and efficient charging method, a reasonable temperature should be selected. As the internal resistance of the battery increases, its power consumption is fast, which leads to the rapid consumption of the battery capacity. Considering the current flowing through the internal resistance, the relationship expression of the lithiumion battery heat generation can be obtained as shown in Eq. (1.10): Q ¼ I 2 R0

(1.10)

wherein Q is the heat generation rate of the lithium-ion battery. The current flowing through the battery is described by I. R0 is the internal resistance of the battery. Under normal room temperature working conditions, the capacity changes in direct proportion to the temperature increase. Once the temperature is too high, the battery releases lots of heat. If the heat rises sharply, it is easy for the battery to be damaged. This not only makes the battery unable to work but can also easily cause a fire when it is serious. The temperature not only affects the normal operation of the battery but also the key parameters in the battery, such as internal resistance. The lower the ambient temperature of the battery, the greater the internal resistance of the battery.

1.6 Lithium-ion battery applications If the manufacturer has more advanced safety management technology, the risk of the system can be kept within the controllable range. It can turn away the risk brought by a single battery. Conversely, the larger-capacity battery packs are used to build a system that is theoretically based on the design of a single battery pack. Capacity and security cannot be obtained at the same time point. The risks and benefits brought by capacity improvement are the obvious knowledge in the industry.

1.6.1 Applications The Tesla Model 3 is fully powered by the 21,700 ternary lithium-ion battery, opening a new phase in cylinder battery capacity improvement. The battery system of the Tesla Model 3 has an energy density of about 300 Wh/kg, which is more than 20% higher than the 18,650 battery used by the original Model S with a 35% increase in unit capacity and a 9% reduction in system cost. As for the related manufacturers and models, there are only a handful of companies in the world that can mass produce batteries. Apart from the 21,700 battery jointly developed by Panasonic and Tesla, Samsung SDI has previously displayed relevant products. It is known that this product has not been mass produced. The Ministry of Industry and Information Technology of the People’s Republic of China issued the “Announcement on Road Motor Vehicle Manufacturers and Products” for new vehicle product announcements, publicizing two products of Nanjing Jinlong and BAIC. Among them, two pure electric van transporters, NJL5040XXYBEV25 and BJ5040XXYCJ06EV, are produced by this company. This automobile company is the first to carry ternary lithiumion batteries. Nickel cobalt aluminate, a three-way positive electrode material, has the advantages of a high capacity that is bigger than 195 mAh/g with a normal voltage of 2.75–4.30 V,

38

1. Lithium-ion battery characteristics and applications

easy processing, and superior storage performance. It is suitable for existing engineering conditions that are expected to be applied to electric vehicles on a large scale to solve the problems of the range and safety of electric vehicles. Because the specific energy is increased constantly, the 18,650 battery faces many challenges in increasing energy density while maintaining the same external size [40]. The supply chain of new materials is not yet mature with the high cost and unstable supply, such as nickel cobalt aluminate and silicon carbon. The new material manufacturing process has high environmental requirements, high investment in fixed assets, and huge energy consumption. With the low capacity of a single battery, packing assembly can be realized through packing technology. Packing technology is a key step in the production of power battery packs. Its importance is becoming more obvious for the continuous expansion of the electric vehicle market [41]. At present, automotive power batteries are composed of the following systems: the battery module, the energy management system, the thermal management system, and the electrical and mechanical systems. However, the pack has high technical requirements and high cost. The polar ear is a raw material of the lithium-ion polymer battery products. The battery is divided into positive and negative poles. The polar ear is the metal conductive body that draws out from the battery positive and negative electrodes. The ears of the positive and negative battery poles are the contact points for the charge-discharge process. The single battery cell is the most suitable for the negative structure of positive single and double ears, which has a significant effect on the energy density. When high energy density and high-power charging are required at the same time, the design space is very small. According to the new national standard, 2.40 Ah of 1C has reached the limit of design. The larger-diameter cylindrical lithium-ion battery becomes an inevitable trend, showing the obvious advantages of the larger size cell compared with the 18,650 in the design of the polar ear and the curvature of the winding. In summary, the size is increased from 18,650 to 21,700 with the following benefits. When the energy density is appropriately raised, conventional materials can be selected with stable and high-cost performance. A multipole ear mechanism can be designed appropriately to reduce internal resistance. With the same energy density, fast-charging graphite can be selected to improve the fast-charging performance. A more effective volume can be obtained by increasing the diameter and height appropriately. The capacity of the single battery cell increases and the proportion of auxiliary components decreases, according to which the packing costs are reduced.

1.6.2 System state estimation In the newly promulgated standard, the original “only 18650 and 32650” in the original draft for comments have been revised to “4 specifications including 21700”. This is because obvious advantages for the 21700 type batteries. However, the path to increased capacity conflicts with the earlier idea that a single cell is equivalent to a high level of security [42]. There are many mature prevention and control methods of thermal safety for small lithium-ion batteries less than 3.00 Ah. The battery modules should be less than 150 Ah such as adding a positive temperature coefficient, which introduces a current interruption mechanism or a pressure sensor. However, the safety control of a large lithium-ion single battery greater than 6 Ah or a module greater than 200 Ah is still a challenge.

1.6 Lithium-ion battery applications

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Compared with the small battery, a large power lithium-ion battery has higher energy. When the thermal safety problem occurs, the consequence is more serious. As the volume of the battery increases, the specific surface area of the battery decreases, and the heat dissipation area per unit volume of the battery decreases. The external surface area and internal surface area are divided into two categories. The reduction of the national standard unit m2/g reduces the heat dissipation area per unit volume. The internal temperature inconsistency of the battery also appears with the upsizing and grouping of lithium-ion batteries. The temperature difference between the battery cells increases the risk of battery thermal runaway, which leads to a series of problems. Whether the main engine factory fully accepts the results, it is willing to give up some security that has a key link between further development of 21,700 [43]. The adjustment and update of the equipment needed for production are influenced by the viewpoint of the equipment supplier. As can be known from the perspective of technology accumulation alone, 18,650 has been studied by enterprises and academic units for many years. There are a lot of public data and information, such as the thermal management models and the thermal runaway prediction. It is a kind of resource for small enterprises with imperfect research and development abilities. When charging, the lattice parameters of the electrode material change, causing the electrode to expand [44]. The expansion force of the electrode acts on the case, causing the battery case to deform. For high-temperature storage, a small amount of electrohydraulic decomposition and gas pressure increase will cause the battery casing to deform due to temperature effects. Among the above three reasons, the expansion of the housing caused by electrode expansion is one of the most important. The important reasons for the expansion of the battery are as follows. As for the formation of the solid electrolyte, interphase gas will be generated, increasing the gas pressure in the battery. When charging, the lattice parameters of the electrode material change, resulting in the expansion of the electrode [44]. The expansion force of the electrode acts on the shell, causing the deformation of the battery shell. For the high-temperature storage, a small amount of electrohydraulic decomposition and gas pressure increase will cause the deformation of the battery shell due to temperature effects. Among the above three reasons, shell expansion caused by electrode expansion is the most important one.

1.6.3 Battery safety protection The swelling problem is a common problem for the square battery that is especially serious when occurring in applications of the large-capacity square lithium-ion battery. The swelling of the battery causes internal resistance increases, local electrohydraulic exhaustion, and even shell rupture, which seriously affects the safety performance and cycle life. The first solution is to adopt a small structure to strengthen the shell. The second solution should be to optimize the alignment of the two electrodes. Through the two methods given above, the problem of square battery swelling can be solved effectively. Fast-charging electric buses were used in batches by the Chongqing Public Transport Group. So far, the first batch of buses has been in operation for 2 years. The core technologies used by these vehicles are micro-macro-powered lithium-titanate batteries and 10 min rapid

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charging technology. It passed the rigorous test of the commercial operation of public transportation, and has performed well overall. Its applications to energy storage have been mass produced by the Chinese Zhuhai Yinlong New Energy Co. It has also been acquired by ALTAIR, a listed company on NASDAQ in the United States with world-leading lithium-titanate technology. The strengthened shell of the battery was designed instead of the original plane shell to bear the pressure from the internal structure of the battery. According to different fixed modes with fixed-length direction and fixed-width direction [45], the effect of strengthening the structure can be observed. Taking the case of fixed width as an example, the deformation data of a shell without a reinforced structure is 4.1 mm under 0.3 mPa pressure. Meanwhile, the deformation amount of the shell with the reinforced structure is 3.2 mm under the same pressure, which is a reduction of more than 20% [46]. The arrangement mode of the cell in the module is different and its thickness direction shape variable is also different. The solution to optimizing the arrangement mode is to choose the arrangement mode with the smallest shape variable. The large square battery has poor heat dissipation performance with the cell volume increase. As the distance between the heating part of the battery and the shell becomes longer, the conduction medium and interface turn thick. It makes heat dissipation difficult and the problem of uneven heat distribution of the cell is obvious. Using the battery charge-discharge equipment named Xinwei CT-3001W-50V120ANTF, an experimental study is carried out with 3.2 V/12 Ah square lithium-ion batteries. In the test, the ambient temperature is 31°C, and the heat dissipation is air cooling.

1.6.4 Battery life guarantee The 12 A current is used in constant-current charging to charge the battery until the charging cutoff voltage is 3.65 V, which is set aside for 1 h after charging to stabilize the battery. Then, the battery is discharged at different multipliers with a constant discharging current rate until the cut-off voltage is down to 2.00 V. The discharge current rate is set as 1C, 2C, 3C, 4C, 5C, and 6C. At the different discharging current rates, the surface temperature of the battery will be higher along with the power ratio increase. The maximum surface temperatures of the battery corresponding to each discharging current rate are 38.1°C, 48.3°C, 56.7°C, 64.4°C, 72.2°C, and 76.9°C, respectively. When the battery is discharged at a 3C rate, the maximum temperature can exceed 50°C. At 6C, the temperature increases to 76.9°C. The length of time is 470 s while the temperature is greater than 50°C. It occupies two-thirds of the entire discharge process, which is unfavorable to continued safe operation of the battery. The ternary lithium-ion battery has become the first choice of manufacturers because of its high energy density and low price. In addition to energy savings, the new energy automobile industry clearly states that the energy density of the battery modules shall be greater than 150 Wh/kg.

1.6.5 Status and trends As a relatively mature and advanced battery type, the lithium-ion battery has been widely welcomed because of its light mass and large storage capacity. In particular, it is in short

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1.6 Lithium-ion battery applications

Upstream industry Raw ma terials

Middle industry Battery materials

Ni,Co,Mn etc.

Cathode material

Graphite

Anode material s

Lithium mine

Electrol yte

Downstream industry App lication fiel d

Diaphr agm Consumer electronics

Manufacturing and packaging

Power battery

Energy storage

FIG. 1.19 Lithium-ion battery industry chain schematic diagram.

supply with the development of mobile phones, smart wearable devices, and new energy vehicles. Its downstream applications mainly include the power battery, 3C battery, and energy storage battery. Because the energy storage battery market has not been opened yet, the scale is relatively small. The lithium-ion battery industry chain is described in Fig. 1.19. Therefore, the industry is still in an acceleration stage for its future development. Because of the current market, the lithium-ion battery demand is very large. Unless fuel cell technology matures and becomes widely available, that could affect its status. With the increasing demand, many companies see an opportunity to launch battery projects. With the influx of major enterprises into the industry, the competitiveness of the market will be increasingly fierce. However, it always follows the survival rules whether in nature or the market. Survival of the fittest is the trend of social development. China is one of the main producers of lithium-ion batteries. According to the “Analysis Report on Market Demand Forecast and Investment Strategic Planning of the Lithium Battery Industry” released by the Qianzhan Industry Research Institute, its productivity reached 5.287 billion. It accounts for 71.2% of total global productivity, ranking first in the world for 10 consecutive years. In 2016, the annual production of lithium-ion batteries reached 7.842 billion, an increase three times from 2.687 billion with a compound annual growth rate of 19.5%. The production history of lithium-ion batteries is shown in Fig. 1.20. Consumer batteries are the largest application area, accounting for 88.4% in 2012 and 49.84% in 2016. But it is still the biggest application field. The power battery is a new force, accounting for 6.94% in 2012, rising to 45.08% in 2016, with huge development potential. Energy storage batteries are still to be developed, accounting for 4.62% in 2012 and 5.08% in 2016. The lithium-ion battery has great potential for cascade utilization, which further increases the proportion of industrial energy storage batteries. It has high energy density, low

42

FIG. 1.20

1. Lithium-ion battery characteristics and applications

Lithium-ion battery production development history.

self-discharging current rate, no memory effect, and no pollution to the environment. Consequently, its performance is better than other types of energy storage batteries. The recycling of waste batteries has always been in an imperfect state, which has a great influence on the policies and personal habits. The littering of the batteries has become a habit. Lithium-ion batteries, though, have less environmental impact than some dry batteries that contain Hg. However, the eliminated number is also not negligible with the rapid development of the lithium-ion battery industry. The energy storage supply of the cascade utilization end brought by the decommissioning of the power battery can be fully absorbed by the market for demand. It is predicted that 79.5 GW of photovoltaic power generation will be added shortly. Considering that the energy storage system equipped with renewable energy power generation accounts for 5%–20% of the new installation of photovoltaic new energy power generation, and the capacity power ratio is two to four times, it can be calculated that the demand for new renewable energy power generation equipped with energy storage system can reach 23.85 GWh. According to the forecast of the Qianchan Industry Research Institute, the potential scale of annual new echelon utilization can reach 33.60 GWh in 2025. Application scenarios can be realized such as the economic demand of the peak cutting and valley filling. The explosive growth can be investigated for distributed photovoltaic installations. The layout acceleration of the electric vehicle energy storage and charging stations has led to the increasing demand for boost utilization. In the future, the growth of consumer electronics is stable at a low level, and the installed increment of energy storage is small. Consequently, the power battery demand is the main driving force for the high growth of the battery industry. Since the 1960s, the industry has developed rapidly, which has promoted human progress. It has also made human beings face problems such as environmental pollution and energy crises. In various fields, the massive consumption of oil has brought serious air pollution to the Earth. Besides, the consumption of oil has also brought an energy crisis. A series of economic and political problems has been caused by the resource crisis. The crisis of resources

1.6 Lithium-ion battery applications

43

affects the rapid development of science and technology, which seriously affects the survival of human beings. According to the survey data, there are hundreds of big cities, but only less than 1% of them can meet the set air quality standards. We have a strong dependence on oil, but pollution caused by oil and other resources has destroyed the environment. The terrible events caused by air pollution are far more than these, as this has a very big impact on plants and people and can easily make people sick [47]. More frightening problems are threatening humanity, such as rising global temperatures and melting glaciers. Resource scarcity and environmental pollution have gradually attracted world attention [48]. In this environment, new energy technologies have achieved sustainable development with the attention and strong support of all countries. It has already raised resources and environmental issues to the height of national strategy. Among all kinds of power batteries, the lithium-ion battery has many outstanding advantages over others, so it has been widely used in various fields and has been gradually studied by more researchers. Advanced measurement and control technology is a prerequisite for the reliable battery application. Consequently, science development has led to the rapid production progress of the high energy density lithium-ion batteries [49, 50]. With more research on lithium-ion batteries, the battery characteristics are understood gradually. It has prominent advantages, such as high voltage, light mass, and high energy density, making it charge quickly. It has developed and has been applied to all aspects of society [51]. In the earliest days of lithium-ion batteries, they allowed implanted pacemakers to operate for longer periods before recharging [52]. With the rapid development and wide application of digital products, it is used in various digital products. Lithium-ion batteries are also used in various storage systems. At present, all countries in the world are actively investing in the research and development of the lithium-ion battery. The United States, as a superpower among developed countries, has made a considerable investment of science and technology as well as funding in this aspect. The government has also issued a series of incentive policies and protective measures promoting its rapid development and research. In terms of lithium-ion battery research, Sony of Japan is an early company in battery research and development compared with other countries and regions. The Japanese government has decided to increase investment in research on electric vehicles. Relevant professional members are organized directly to implement the national plan to research and develop advanced battery technology, which can be applied to the new generation of electric vehicles. Japanese lithium-ion batteries have reached the world-leading level. A resource crisis is a huge crisis that cannot be neglected, and Japan pays special attention to new energy development [53]. As the first country to study electric vehicles, Japan has always been at the leading level. The research and production of lithium-ion batteries has been growing rapidly with social development. At present, there is no doubt that the manufacture of lithium-ion batteries plays an important role in the society development. Its research and manufacturing are still on the rise, making its place in the global rankings continue to grow. New energy and environmental protection issues are important topics of research. It is relatively friendly to the environment in its power supply application, so it has gradually replaced those batteries that have a large impact on environmental pollution. It has become an indispensable energy storage element in

44

1. Lithium-ion battery characteristics and applications

social development [54]. The development plan for the battery industry is to increase lithiumion batteries in the future, so the efforts in promotion can be more comprehensive in the application field with increasing market share [55]. There are more battery manufacturers, increasing space for research and production.

1.7 Conclusion This chapter introduced different types of lithium-ion batteries as well as their advantages and disadvantages for applicable occasions. Meanwhile, the working characteristics were analyzed. This chapter provided a basis to choose batteries for different working environments. This chapter briefly introduced the concept, development background, application field, and working principle of lithium-ion batteries. Meanwhile, the development trend was analyzed. It laid the foundation for the research of lithium-ion batteries.

Acknowledgments The work is supported by the National Natural Science Foundation of China (No. 61801407), the Sichuan Science and Technology Program (No. 2019YFG0427), the China Scholarship Council (No. 201908515099), and the Fund of Robot Technology Used for Special Environment Key Laboratory of Sichuan Province (No. 18kftk03).

Conflict of interest There is no conflict of interest.

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C H A P T E R

2 Electrical equivalent circuit modeling 2.1 Modeling method overview Complex chemical reactions occur in the application process of lithium-ion batteries, and the reaction process is easily affected by the working environment. In the battery state estimation, the offline experimental data fitting function is generally used to determine the values of various parameters in the battery model. However, using this method leads to a large error in the state estimation results. To improve the state estimation accuracy, the online identification of model parameters and their real-time correction should be particularly important. Based on the second-order resistance-capacitance equivalent circuit model, this chapter discusses the identification of model parameters using the forgetting factor recursive least-square algorithm.

2.1.1 Modeling types and concepts The battery modeling methods can be divided into mechanism modeling, experimental modeling, and hybrid modeling methods. The mathematical model is established based on theoretical formulas such as physical formulas and the chemical reaction principle [1, 2]. The controlled object is regarded as a black box, and the model is constructed by experimentally recording the variation law of the target object characteristic parameters [3–6]. Mechanism modeling and experimental modeling are combined to build a model. The battery is a highly complex nonlinear electrochemical energy storage device. It is difficult to describe the interactions and reactions occurring inside the control process of precise formulas [7–9]. Experimentally generated data modeling such as a neural network needs lots of data input and learning, so hybrid modeling is more common. To obtain the battery state conveniently and accurately, it is necessary to select the appropriate battery model to get the appropriate scheme. (1) The internal resistance equivalent model is used, in which the ideal voltage source is introduced to represent the open-circuit voltage. The internal resistance R0 and open-circuit voltage UOC are all functions related to the state of charge levels. Only the positive and negative of the current need to be determined to judge the charge-discharge direction. This model is shown in Fig. 2.4. (2) The resistance-

Battery System Modeling https://doi.org/10.1016/B978-0-323-90472-8.00008-1

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Copyright # 2021 Elsevier Inc. All rights reserved.

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2. Electrical equivalent circuit modeling

capacitance equivalent model can be obtained by using a resistance-capacitance circuit that can effectively describe the surface effect of the circuit [10–13]. The model consists of three resistors and two capacitors, in which C1 is the polarization capacitance of the battery and C2 is the small capacitance produced by the surface effect of the battery, which is shown in Fig. 2.6. (3) The Thevenin equivalent model can be obtained by combining the internal resistance model and the resistance-capacitance model analog circuit, which can take the effects of temperature and polarization into account. This equivalent circuit model is shown in Fig. 2.9. (4). The Partnership New Generation of Vehicles (PNGV) equivalent model is proposed according to the battery test manual, which is shown in Fig. 2.10. Based on the Thevenin model, the circuit is connected to a series with the capacitance Cb. R0 is the internal resistance. Rp is the polarization resistance and Cp is the polarization capacitance. The load current is described by I(t). UL is the terminal voltage. In the above four common equivalent models, although the internal resistance equivalent model is convenient and fast to measure UOC, it does not take into account the transient characteristics of the electrochemical reaction process of the battery and cannot accurately characterize its change process. While the resistance-capacitance equivalent model can better simulate the dynamic battery characteristics and increase the description of the surface effect of the battery, it ignores the effects of temperature and cell polarization. The PNGV model has high accuracy because of considering the self-discharge effect, but the introduction of the series-connected capacitance Cb makes the method prone to cumulative error in a long-time simulation [14]. The Thevenin model can overcome the error of the polarization effect. The steps are short and the principle is clear. It is very suitable for the transient power battery charge-discharge analysis. Compared with the PNGV model, the general nesting logit (GNL) model, and other models, the Thevenin model is simple in structure. It belongs to a nonlinear low-order model and involves fewer parameters, so its accuracy can comply with the requirements of engineering applications.

2.1.2 Comparative equivalent models It is simulated by a common equivalent model, and the current battery state can be estimated accurately. The optimal scheme is obtained by comparing several equivalent models [15]. The internal resistance model is simple and easy to be realized for measurable various parameters, and it has certain versatility. However, the accuracy of the estimated voltage and state is not high enough to reflect the working battery characteristics together with the transition when the charge-discharge process changes. The resistance-capacitance equivalent model uses a controllable voltage source to subtly link the two loops to the UOC, which achieves the voltage balance state. The electrical characteristic models are generally divided into three categories: black-box, electrochemical mechanism, and equivalent circuit models. They are described in Fig. 2.1. The black box model is a linear or nonlinear function that describes the voltage response characteristics of a battery. It shifts the focus onto the battery internal mechanism of the data itself, which determines its flexibility to establish the model structure and parameterization, but lacks the physical meaning of the essence. The difficulty of black-box modeling is that the model performance is very sensitive to the training data quantity. It usually needs to be supported by data-driven algorithms, such as neural networks and support vector regression.

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2.1 Modeling method overview

Equivalent model of Li-ion batteries

Current Voltage

Black box model

Electrochemical model

Equivalent circuit model

(A) Classification

SOC Battery model

Temperature

SOE

FIG. 2.1 Equivalent model classification and structure. (A) Classification. (B) Structure.

SOH(RUL) SOP

(B) Structure

The electrochemical mechanism model is realized by analyzing the reaction mechanism inside the battery, establishing the partial differential equations of the electrode and electrolyte kinetics. The equivalent circuit models can predict the macroscopic physical parameters accurately and simulate the important macroscopic physical quantity distribution of the battery. Therefore, it is very suitable to realize the optimization design and safety analysis of lithium-ion batteries. As the complexity of the electrochemical model is high and the system computing ability may not meet its theoretical requirements, the research mainly focuses on how to simplify the model of the battery application. The commonly used battery equivalent circuit models mainly include the Rint, resistance-capacitance, Thevenin, and PNGV types. The internal resistance model is the simplest equivalent model, but it has not been widely used due to its low precision. The resistance-capacitance equivalent model and the Thevenin model can simulate the above voltage characteristics, but the accuracy of each simulation is different. They belong to the nonlinear low-order models. Their structures are simple and involve fewer parameters, the accuracy of which can meet the requirements of engineering applications. It provides an experimental basis for the modeling simulation and battery management system design. The equivalent model uses circuit components such as capacitors, resistors, and constantvoltage sources to form a circuit network to simulate the dynamic voltage response characteristics of the battery. The relationship between the parameters in this model is direct and obvious, and it generally contains relatively few components. This equivalent model is quantified as some electronic components, which makes the mathematical state-space description easier, so it has been widely used in system simulation and management. The battery electrical circuit models contain a variety of structural frameworks. The following is a brief description of several related concepts. The internal resistance is composed of the electrode material, the electrolyte, the internal resistance of the diaphragm, and the contact resistance of each part. The following are several sets of experimental data plots. The pressure differential curve produced by the polarization of internal resistance to the constant-current phase is described in Fig. 2.2. FIG. 2.2 Constant-current phase polarization effect.

U Concentration polarization process Electrochemical polarization process Ohmic polarization process

t

50

2. Electrical equivalent circuit modeling

FIG. 2.3 Schematic diagrams of the composite pulse experiment. (A) Current characteristics. (B) Voltage characteristics.

I

U5

U

t2

Id Discharge

U1

Charging t

U3

t4 0

t1

U4

t3

U

Discharging U2

charge Ic

(A)

t5

t

t1

t2 t 3

t4

t5

t

(B)

The polarization resistance is caused by the electrochemical polarization reaction phenomenon, including the resistance caused by electrochemical polarization and rich polarization. It is paralleled with the polarization capacitance to form a resistance loop for simulating battery polarization generation, which greatly reduces the dynamic characteristics exhibited in the charge-discharge process. To illustrate the polarization reaction phenomenon, the composite pulse experimental current and voltage characteristics are described in Fig. 2.3. The advantage of a simple combination structure is that it can achieve a fast and inexpensive model, but the simulation accuracy is lower and the combined structure considers a more comprehensive model to improve accuracy. Greater complexity makes it more difficult to identify parameters and realize the iterative state calculation [16]. Therefore, in practice, the advantages and disadvantages of the existing equivalent circuit model should be weighed. Consequently, the appropriate circuit model should be selected by a comprehensive analysis. The comparison of several equivalent circuit models is investigated. The voltage variation in the composite pulse experiment is obtained, in which the mechanism modeling, electrochemical models, and equivalent circuit models can be used [17]. In contrast, the equivalent circuit model does not need an in-depth analysis of the electrochemical reaction against the battery. It describes the open-circuit voltage, direct current internal resistance, and polarization internal resistance of the battery through circuit relationships to achieve the characterization of the external battery characteristics [18, 19]. The model constructed by this design is within the scope of the equivalent circuit model type. Monitoring the battery capacity, the circuit also characterizes the instantaneous response of the internal resistance. The coordination work fully takes into account the steady-state and transient battery characteristics, but it does not take into account the effects of self-discharge and temperature. The Thevenin equivalent model can accurately simulate the chargedischarge process of lithium-ion batteries but does not consider the open-circuit voltage change, overcharge, and self-discharge caused by current accumulation. It is often used for transient analysis of the power battery charge-discharge process, which is not suitable for a long-time simulation. The model is less demanding and easy to be implemented. It has high precision that is more suitable for simulating the dynamic performance of the battery.

2.1.3 Commercial circuit models Thanks to the advantages of lithium-ion batteries, new battery models have been proposed to visually characterize the relationship between external characteristics for the battery

2.1 Modeling method overview

51

voltage, current, and temperature. After that, the internal state quantities such as resistance and electromotive force are used to establish the mathematics. A relational expression calculates the amount of internal state indirectly based on the external characteristics of the battery. The battery equivalent modeling is the basis of the state estimation. Its accuracy directly affects the state estimation result [20]. At present, the battery model can be divided into different types according to the research mechanism and the object. The most common methods are the electrochemical models and equivalent circuit models. The equivalent circuit model uses the ideal voltage source, resistance, capacitance, and other electrical components to form a circuit model to simulate the external characteristics of the power battery. After years of research, power battery equivalent circuit models have been gradually formed, including Rint, PNGV, Thevenin, and GNL models.

2.1.4 Electrochemical model The original intention of the electrochemical model was to design the battery structure. To obtain the internal state of the battery and combine it with its energy generating mechanism, the analysis of macroscopic data is conducted onto the battery together with the internal microscopic particle activity. The model mainly reflects the internal battery chemical reaction status. Electrochemical models can analyze the internal characteristics from a microscopic perspective. To express these characteristics of the battery, electrochemical models usually need to establish multiple sets of complex time-varying partial differential equations. It usually takes a long time to solve these equations.

2.1.5 Equivalent circuit model The equivalent circuit model uses a circuit component to form a specific circuit network to characterize the operational characteristics of the circuit. This model establishes the relationship between the external characteristics exhibited by the battery of operation and the internal state of the battery itself. The equivalent circuit model is more intuitive, easy to process, and moderate in computation. The parameters of the model are easy to identify and suitable for simulation experiments with circuits. Therefore, the equivalent circuit model is widely used in practical engineering applications. The establishment of the battery equivalent model has two methods, theoretical and experimental analysis. Theoretical analysis is based on the understanding of the internal law for the research objective, deducing the dynamic equation of the change law to the object. The experimental analysis needs to collect the input and output signals of the object as well. The advantage is that the equivalent circuit model has high precision and can reveal the evolution mechanism of the battery characteristics deeply. The disadvantage is that the model has high complexity, which is not conducive to its engineering application. The logical structure of the circuit model is described as follows. According to the different establishment mechanism of the battery model, it can be divided into the simple electrochemical model, the intelligent mathematical model, and the equivalent circuit model. The equivalent circuit model is currently widely used due to its clear physical meaning and simple

52

2. Electrical equivalent circuit modeling

mathematical expression [21, 22]. Because the battery characteristics such as current, power, state of charge, and temperature are nonlinear, if the battery modeling needs to fully consider these factors, the computational complexity is increased. The general controller cannot meet the requirements. The key points of the modeling establishment can be obtained by its mathematical analysis as well as its logical structure.

2.1.6 Principle description The battery simulation model is used to verify the accuracy of the parameter settings in the model, so its input is current and the output is the terminal voltage. In the battery management system, both current and terminal voltage are input. The electrochemical model is more complicated and difficult to apply in practical products, so it is mainly used to assist in the design and manufacture of batteries [23]. The intelligent mathematical model is mainly a neural network model, which can theoretically complete battery modeling, but it needs a large amount of data. Its practical application is limited by the training, high technical threshold, and long processing time. Modeling steps The empirical model is a battery dynamics replacement model based on sample data and system identification methods [24]. Model building is mainly divided into seven steps: S1: Selecting appropriate equivalent circuit model. S2: Determining the input, output, and state variables. S3: Writing the state equation. S4: Modeling realization. S5: Parameter identification of the experimental data. S6: Real-time correction according to variables such as temperature, state of charge, and model parameters. S7: Simulations. Model selection According to the needs, different battery equivalent models are selected that have a great influence on battery modeling accuracy. Meanwhile, the accuracy of the model is high when the structure and operation are complicated. The corresponding hardware requirements are improved. In the equivalent model, the equivalent circuit with the resistance-capacitance loop can have higher accuracy. The accuracy is high when the resistance-capacitance order is high.

2.1.7 Parameter identification According to the experimental results of the hybrid pulse-power characteristic test, the curve fitting is performed by the least-square method. Then the relevant parameters are identified. To predict the behavior of the battery, many models have been established, but no model can fully and accurately simulate the dynamic battery behavior of the operating conditions. The equivalent model of the battery can be divided into empirical models and electrochemical models according to the modeling methods. The electrochemical model is a mathematical expression of obtaining dynamic battery characteristics by using the mechanism analysis method.

2.2 Improved internal resistance modeling

53

2.2 Improved internal resistance modeling The internal resistance model is a relatively simple model designed by the Idaho National Laboratory in the United States. It just includes the battery ideal voltage source and the internal resistance.

2.2.1 Theoretical resistance modeling Because the model does not consider the polarization characteristics of the battery, the model accuracy is low. The internal resistance equivalent model of the battery can be considered as the simplest battery equivalent model. Therefore, the internal resistance equivalent model of batteries often appears in some simple battery simulation analyses, but it is rarely seen in practical applications. Its internal equivalent structure is described in Fig. 2.4. This model contains only one internal resistance R0 and an ideal voltage source UOC. However, the internal ohmic resistance and the open-circuit voltage of the battery are also constantly changing due to the complexity of the internal battery electrochemical reaction. Its terminal voltage is described in Eq. (2.1): UL ¼ UOC  ðR0  IL Þ

(2.1)

Then, the equivalent model can be established by using the acquired signals. It is subjected to parameter identification processing [25]. For lithium-ion batteries, the internal physical and chemical changes are very complicated, so the battery dynamic equation derived based on the electrochemical theory is difficult to apply in practice. Therefore, experimental analysis methods are often used in most cases.

2.2.2 Battery model establishment The PNGV and GNL equivalent circuit models can be built by using the Thevenin model, which adds a large capacitance to characterize the battery capacity and the open-circuit voltage to compensate for the defects of the Thevenin model. Consequently, the model obtains better simulation performance that can represent the battery transient response, steady-state voltage, and current characteristics. However, the simulation accuracy of the two models can be still improved in terms of battery capacity as well as operating time. The nonlinear relationship between the open-circuit voltage toward the state of charge is still limited. To this end, a more accurate battery electrical model is proposed that is intuitive and versatile. The improved equivalent circuit model is described in Fig. 2.5. As it is an improved battery equivalent circuit model, the model consists of a constantvoltage source UOC to analog battery open-circuit voltage. R0 is used to analog the internal resistance of the battery. The resistance-capacitance parallel circuit of C1 and R1 can be used R0

I(t)

FIG. 2.4 Internal resistance model.

+

UOC

UL -

RL

54

2. Electrical equivalent circuit modeling

FIG. 2.5 Improved battery equivalent circuit model.

R2

R1 R0 UOC

C1 ULoss

IBat

IL

C2 ILoss

UL

to simulate the electrochemical polarization effect of the battery. The resistance-capacitance parallel circuit composed of C2 and R2 can be used to analog the battery concentration polarization effect. The constant-current source Iloss simulates the battery self-discharge phenomenon. The battery polarization effect can be divided into ohmic polarization, electrochemical polarization, and concentration polarization. The ohmic polarization is conducted corresponding to the internal resistance as a function of the state and temperature. The electrochemical polarization corresponds to the electrochemical capacitance of the battery, which is also known as the double-layer capacitance and electrochemical resistance. It is also known as double-layer resistance, and it varies with the state and temperature. The concentration polarization corresponds to the battery concentration capacitance, which is also known as diffusion capacitance C2. The concentration resistance trends with state and temperature, and is also known as diffusion resistance R2. Its advantage is that it can guarantee better model accuracy and reduce the amount of calculation of the model, but this method must be completely retested for each batch of new batteries. Even if different batches of batteries produced by the same manufacturer have varying parameters, the characteristic parameters of the new batch of batteries should be retested before they can be used. Therefore, the portability of the model becomes a noncritical consideration. As can be seen from the electrical model, the battery terminal voltage UL is given exactly by Eq. (2.2): UL ¼ UOC + Uloss

(2.2)

wherein UOC represents the open-circuit voltage of the battery, which is a function of the state of charge and temperature. If the battery consists of a plurality of the battery cells connected in series, the total voltage can be then obtained as shown in Eq. (2.3): UOC ¼ ncell Ucell ðS, T Þ

(2.3)

wherein the subscript cell represents the number of battery cells connected with the series, and Ucell(S, T) represents the open-circuit voltage of the battery cells. In the charge-discharge process of the battery, the calculated loss voltage can be realized by Eq. (2.4):   ð ð 1 U1 1 U2 IBat  IBat  dt + dt + IBat R0 ðS, T Þ (2.4) ULoss ¼ R1 R2 C1 C2 Among them, ULoss represents the battery loss voltage, the voltage drop caused by the battery polarization effect. IBat indicates the current in the battery, which is not equal to the external current. C1 indicates the double-layer capacitance. R1 represents the double-layer resistance. C2 represents the battery diffusion capacitance. R2 represents the diffusion resistance. U1 represents the loss voltage of the double-layer capacitance. U2 represents the loss

2.2 Improved internal resistance modeling

55

voltage of the diffusion capacitance. R0 represents the internal resistance. T indicates the battery temperature. S represents the battery state of charge. The improved battery model removes the large capacitance that characterizes the battery capacity and performs online calibration by testing the battery test data. The battery capacity directly determines the various parameters of the battery polarization effect. This method does not reduce the calculation accuracy of the model while simplifying the model structure to reduce the calculation amount. Each parameter in the model, except for the selfdischarging current, is a function of the state of charge and temperature. The model considers the self-discharging current to be a function of temperature. It mainly calculates the parameters of a specific state of charge and temperature point, and uses the interpolation algorithm to calculate the battery parameters over the entire state of charge and temperature range. The self-discharging current ILoss of the battery at different temperatures can be approximated by Eq. (2.5): ILoss ¼ I0 ncell Ucell ðS, TÞ

(2.5)

Among them, I0 represents the battery self-discharging current under standard temperature conditions. T indicates the battery temperature. The battery state of charge is an important parameter to characterize its available residual energy. Accurate state estimation is one of the core tasks of the battery management systems. Using the ampere hour integral method to calculate the battery state is the most feasible of the existing state calculation methods of various batteries. However, this algorithm has one of the biggest drawbacks. It needs to know the initial value S0. Therefore, the Kalman filter can be used to calculate the initial state value S0 of the battery. For the battery, the battery current is defined as IL, and the battery selfdischarging current is ILoss. If IL  ILoss > 0, the battery is charging. If IL  ILoss < 0, the battery is discharging. The battery state of charge can be expressed in the form of Eq. (2.6): ð (2.6) S ¼ S0 + ðIL  ILoss Þdt=KN Among them, S0 represents the initial battery state of charge. KN indicates the nominal capacity of the battery. IL indicates the current outside the battery. ILoss represents the selfdischarging current of the battery. The coupling model is a model that interacts with the electrochemical process for the heat-generation and heat-transfer processes. The model describes the external battery characteristics when it is in operation. The equivalent circuit model is a type of battery performance model that uses circuit components such as resistors, capacitors, and voltage sources to form a circuit that simulates the dynamic battery characteristics. The equivalent circuit model has become the most widely used battery modeling type, and it is easy to be analyzed by circuit and mathematical methods.

2.2.3 Internal resistance description The internal resistance model is a more common battery model. It consists of an ideal voltage source UOC in series with an equivalent internal resistance R0. In the internal resistance model, the parameter Uoc represents the electromotive force of the battery, which can be obtained by measuring the terminal voltage of the battery after the battery is in an idle state

56

2. Electrical equivalent circuit modeling

and after sufficient shelved. The resistance R0 characterizes the direct-current internal resistance of the battery, and the battery can be tested when it is connected with the load. The terminal voltage and the current flowing through the battery loop are calculated, according to which the calculation process is described in Eq. (2.7): R0 ¼ ðUOC  UL Þ=I

(2.7)

wherein UL is the terminal voltage of the model. The current flowing through the model is characterized by parameter I. When this model is applied, the accuracy of the model is generally insufficient due to the consideration of the shelved characteristics of the battery, and the dynamic characteristics exhibited inside the battery are neglected, so it is not suitable for battery modeling. The large capacitance Cb describes its ability to store energy. R1 is the polarization resistance and the polarization capacitance is characterized by C1. R2 is the surface resistance. C2 describes the surface capacitance and diffusion effect of the battery. The resistance-capacitance model consists of two capacitances and three resistors, as shown in Fig. 2.6. Taking the voltages U1 and U2 across the two capacitances as the state variables, the terminal current IL is used as the input variable, and the terminal voltage UL is used as the output variable. After that, the equations are listed according to Kirchhoff’s law. The state-space equation of the resistance-capacitance model can be obtained. The battery exhibits polarization and hysteresis effects of the power supply application. This effect is consistent with the characteristics exhibited by the circuit resistance model. Therefore, the equivalent circuit modeling and state-space expression can be performed on the battery. Taking a 2.9 Ah cylindrical ternary lithium-ion battery as an example, mixed pulse-power characteristic tests can be conducted. Regardless of the charging or discharging processes, the value of internal resistance R0 is significantly different at varying S levels for different pulse lengths. The R-value is initialized at S ¼ 0.2 to S ¼ 0.9. The value of the internal resistance R0 of discharge is higher than the R-value in the charging process of the different pulse lengths [26]. It can be known that the internal resistance is not constant when the battery is working. It is affected by factors such as temperature, depth of discharge, charge-discharge current rate, and the number of cycles. The different battery cells of the same type are also related to inconsistency levels and working environments.

2.2.4 Open-circuit voltage characteristics The open-circuit voltage is the terminal voltage of the lithium-ion battery after being shelved for a long time. In the shelved state, the open-circuit voltage has a good mapping relationship to the battery state of charge. Therefore, the open-circuit voltage characteristics of lithium-ion batteries are used to obtain its one-to-one correspondence with the state of FIG. 2.6 Resistance-capacitance battery model.

R2

R1 R0

+ U1 -

I(t)

+ U2 +

UOC

IC 1(t)

C1

IC 2(t) C2

UL

RL -

2.2 Improved internal resistance modeling

57

FIG. 2.7 Open-circuit voltage and state-of-charge calibration.

Start Constant current constant voltage charging 4.2V / 1.5A (1C)

I < 0.075A Shelving for one hour Count= 0 Constant current discharge 1.5A Shelving for 40 minutes

Count++

Count< 10 & UL> 3V End

4.2 4.1 4.0 3.9 3.8 3.7 3.6 3.5 3.4 3.3 0.0

(A)

UOC(V)

UOC(V)

charge [27]. The open-circuit voltage is performed toward the state calibration experiment for lithium-ion batteries using the test device at a suitable time. The test steps for the open-circuit voltage characteristics are described in Fig. 2.7. Before the start of the formal experiment, the battery is fully charged by constant-current and constant-voltage charging. In this experiment, the battery is intermittently discharged at a discharging current rate of 1 C. Furthermore, each cycle discharge time is set according to the discharging current rate and the number of the required relationship-coordinate points. It is necessary to obtain 10 relationship-coordinate points, so each discharge time is 6 min, and each discharge is left after the discharge. The battery is shelved for 40 min, and its internal chemical state is stabilized to obtain its open-circuit voltage. The last discharge experiment likely won’t continue until the expected discharge time. The released capacity can be obtained according to the discharge duration and the discharging current, thereby obtaining the battery state value after the last discharge of the battery. After the end of each discharge for 40 min, the battery voltage is the open-circuit voltage value corresponding to the present state. The least-square method is a mathematical optimization technique that finds the best function matching of the data by minimizing the square of the error. It is suitable for curve fitting. The open-circuit voltage relationship curves toward the state of charge. Then, the relational polynomial can be obtained by fitting with this method. These discrete points are extracted from the laboratory data to obtain the opencircuit voltage toward the state-of-charge relationship scatter plot, as shown in Fig. 2.8. The scatter obtained from the experiment is a point of high trust, so the curve should be passed through every point as much as possible. The curve should reflect the relationship between the open-circuit voltage and the state of charge as accurately as possible. Consequently, the fitting curve can be set in the scatter and the transition to the intervals should be as smooth as possible, so the rate of change should not be too large.

0.1

0.2

0.3

0.4

0.5

S (1)

0.6

0.7

0.8

0.9

1.0

(B)

4.2 4.1 4.0 3.9 3.8 3.7 3.6 3.5 3.4 3.3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

S (1)

FIG. 2.8 Fitting relationship of open-circuit voltage and state of charge. (A) Charging open-circuit voltage. (B) Discharging opencircuit voltage.

58

2. Electrical equivalent circuit modeling

Obviously, the higher the fitting polynomial order, the better the accuracy of the fitting. It is also the computing power limit of the processor under conditions. The fitting polynomial cannot be too high to reduce computational complexity. Considering the above situation comprehensively, after repeated tests to compare the fitting effect, it is found that the fifth-order polynomial has a good fitting effect that is moderately acceptable to the processor. Therefore, the functional relationship of the open-circuit voltage and the state of charge obtained by fitting with a fifth-order polynomial is described in Eq. (2.8): UOC ¼ 1:8670  S5  5:0687  S4 + 5:7087  S3  3:5125  S2 + 1:7145  S + 3:440

(2.8)

wherein UOC represents the open-circuit voltage. It can be seen that the open-circuit voltage is 3.440 V when the battery state is used as the subject of this experiment. The open-circuit voltage toward the state-of-charge relationship provides a calculation basis of the open-circuit voltage in the subsequent battery equivalent circuit modeling, and the experimental parameters provide mathematical support. The relationship can then be used to estimate the battery capacity. The estimated shelved capacity of the battery can be obtained when it is left for a long time. The usual method is to use the look-up table method. Although the calculation workload is saved, the accuracy is not high. The fitting relationship is not adopted, so it is necessary to introduce an efficient and accurate method to estimate the battery state value.

2.3 Thevenin modeling At this stage, researchers have done lots of work in which lots of effective methods and conclusions have been obtained, laying a solid foundation for the safe application of lithium-ion batteries. The state parameter is mainly used to reflect the remaining capacity of the battery. It refers to the ratio of the remaining capacity after discharging at a certain discharging current rate and the rated capacity for the same conditions.

2.3.1 Construction of Thevenin model The dynamic Thevenin model describes the fixed parameters in the classic model as variables with the state of charge and temperature dynamics. Then, the battery packing characteristics are equivalent by using the electromotive force, internal resistance, and polarization resistance. A simplified model of polarization capacitance and the terminal voltage describes the battery performance more accurately. Zhang Qun of Fuzhou University is aiming at the identification of the equivalent model parameters of pure electric vehicle power lithium-ion batteries. The 87 units of pure electric vehicles are connected to series with 84 Ah, which is a unit for measuring the capacity of a power storage device that is also important to measure the battery performance. The indicator, the nickel-cobalt-manganese ternary lithium-ion battery capable of discharging for 84 h of the current 84 A, is selected as the research objective. The equivalent battery model of the second-order resistance-capacitance is selected to identify the absolute error of the largest single cell in the initial open-circuit voltage. The average is 3.62% and the average absolute error of the smallest cell is 3.24%. The data can be used in engineering practice.

2.3 Thevenin modeling Rp

FIG. 2.9 Thevenin model.

I(t)

R0

59

+ UOC

Ip(t) Cp

UL

RL

-

The model improvement is the addition of a resistance-capacitance circuit to characterize the polarization effects of the battery operation compared to the internal resistance model. It can characterize the battery dynamic response. R0 can represent the instantaneous change of the battery voltage response of its charge-discharge process. The resistance-capacitance circuit can reflect the gradual change of the battery voltage after the charge-discharge [28]. Because the equivalent circuit model is not only simple in structure but also able to meet the simulation requirements, the Thevenin battery model is often used in practical applications. It is used to build the simulation model, and the parameters are set by the experimentally identified parameters. Finally, the experiment results are compared with simulation results and the error is within the acceptable range. Its modeling structure is described in Fig. 2.9. In this model, UOC is the ideal voltage source. R0 is the internal resistance of the battery. Rp is the polarization internal resistance of the battery. Cp is the polarization capacitance, in which each parameter is a function of the state of charge and temperature. The battery terminal voltage is described in Eq. (2.9): UL ¼ UOC  ðR0  IL Þ  Up

(2.9)

This model can approximate the external battery characteristics by using one modeling order. In practical applications, the dynamic model calculation results and the battery characteristics still have relatively large errors. Due to the deepening application of various fields for the lithium-ion batteries, the complexity of the management process is increasing, and a more complete battery management system is needed. The battery equivalent model and working characteristics are crucially aimed at the lithium-ion battery pack, designing the equivalent modeling construction scheme, hardware system production, and platform construction. The software part of the system is designed to realize the charge-discharge, state measurement, and estimation. The control adjustment methods are studied and implemented on the hardware system. Combined with a series of experimental tests, the effect is verified for the used various functions and system performance indices. On this basis, the influence of polarization on voltage and state of charge is fully considered. Then, the nonlinear state-space equation is established, according to which the state of charge is estimated. The model uses a series-resistance and a resistance-capacitance network to simulate the battery characteristics. The parameters to be identified with this model include internal resistance R0, polarized internal resistance Rp, and polarized capacitance Cp. The hybrid pulse-power characteristic experimental method is employed for parameter identification. According to Kirchhoff’s law, the equivalent circuit expression can be obtained from the model, as shown in Eq. (2.10): ( UL ¼ UOC  U0  Up (2.10) I ðtÞ ¼ Cp dUp =dt + Up =R0

60

2. Electrical equivalent circuit modeling

Among them, the open-circuit voltage can be represented by the state variable state of charge, and the nonlinear function relationship can be obtained. Using the knowledge of modern control theory, the equivalent circuit model can be discretized. The discrete statespace equation can be obtained by selecting state-space variables, input variables, and output variables in combination with the state-of-charge definition, as shown in Eq. (2.11): 8   k X > S > > + vk ηI ðkÞ=Q0 ,U0,k ¼ UOC, k  R0, k Ik + ½0  1 k < Sk ¼ S0  U1, k (2.11)   0       > Δt=Q Sk w1, k Sk + 1 1 0 > N >   : ¼ + Ik + R1 1  eT=τ1 U1, k + 1 w2, k 0 eΔt=τ1 U0,k + 1 wherein Δt is the sampling interval, which is combined with τ1 ¼ RpCp. ω is the state error. The measurement error is described by the parameter νk. Their variances are Qk and Rk, respectively.

2.3.2 Charge-discharge characteristics Overcharging may cause the internal pressure of the battery to become high, causing the battery to be deformed and leak, which greatly reduces battery operating efficiency. The battery overdischarge electrode may cause irreparable consequences, directly damaging the battery, especially the large current overdischarge; repeated overdischarge has a greater impact on the battery [29]. The discharge cutoff voltage is controlled indirectly by determining the discharging current. The battery test equipment is used to complete the charge-discharge test of the lithium-ion battery. The corresponding battery open-circuit voltage and internal resistance characteristics can be obtained. The running current of the new energy vehicle changes continuously, usually in the range of 60–200 A [30]. Therefore, the current is often set at 1.5 C, 1 C, 0.5 C, and 0.3 C for experiments. The experimental steps are described as follows: (1) The charging experiments are carried out at room temperature (20°C) for 0.35 C, 0.5 C, and 0.8 C, respectively. The constantcurrent is charged until the voltage reached 3.60 V. Then, the constant-voltage charging is continued until the terminal current is reduced. When the current rate reduces to 0.05 C, the charging process is finished. (2) The discharge experiment is performed at room temperature for 1.5 C, 1 C, 0.5 C, and 0.3 C until the constant-current discharge is reduced to 2.00 V. The experimental results show that the required time for charge-discharge decreases to the cutoff voltage when the discharging current increases. The operating voltage of the battery decreases to the end of the discharge. The influencing factors of the complete feasible charge-discharge system mainly include voltage, current, and temperature. The three sequences are different and have different durations; this has different effects on battery performance. Considering the safety and feasibility, if the batteries are used improperly, such as impact, extrusion, high-temperature, and short circuit, they often lead to safety accidents such as smoke, fire, and even explosion. Therefore, the safety of lithium-ion batteries should be improved. An important prerequisite for is to increase its thermal stability [31]. To improve the internal state of the battery production process, it is also very important to use the battery correctly. It is most important to use

2.3 Thevenin modeling

61

the battery with a reasonable charge-discharge method to improve the safety and service lifespan of the battery. The effects on battery performance are different from the time-varying factors. The detailed analysis is expressed as follows. (1) Temperature. At high temperatures, the activity of the active material in the battery increases while the oxygen evolution potential for the positive and negative electrodes decreases. Consequently, the charging reaction speed is fast. The current is large and the required charging voltage is low. At low temperatures, the active material slows down, and the phosphorus dissolution on the two poles becomes difficult. It is difficult to replenish the phosphorus that is consumed in a large amount of charging, which causes the charging current to decrease. The charging acceptance capability is weakened, thereby causing insufficient charging. (2) Current. Normally, the larger the discharging current, the smaller the total discharge of the battery. Under a high-rate discharging current, the internal resistance drop of the battery increases so that the cutoff voltage can be reached faster. When the internal current of the battery is too large, the battery temperature rises, which affects its performance. When the discharging current is greater than the rated current, the internal polarization phenomenon is intensified and the cutoff voltage is reached faster. Then, the battery utilization rate and the cost performance drop suddenly, so even the deposition of the lithium ions inside the battery is caused and the loss is increased, which seriously affects the working ability. (3) Voltage. The high upper limit may cause overcharging and generate a large amount of flammable gas. The battery expands or even explodes. If the lower limit of the cutoff voltage is too low, it causes overdischarge and an irreversible electrochemical reaction occurs, which seriously affects the battery life. With the low upper limit of the cutoff voltage or the high lower limit of the cutoff voltage, the battery may not be fully charged and the energy release may be incomplete, so the utilization rate is not high. The cutoff voltage is also different from time-varying discharging conditions. Generally, the lower limit of the cutoff voltage is set to a lower level in the discharge process of low temperature and large discharging current rate. Thevenin is a common equivalent circuit modeling method that is known as first-order resistance-capacitance. Wherein UOC is the ideal voltage source. R0 is the internal resistance of the power source. Cp and Rp are the polarization capacitance and resistance of the power source. According to Kirchhoff’s current and voltage laws, the parameter relations of the equivalent circuit model can be obtained as shown in Eq. (2.12): UL ¼ UOC  Up  R0 I

(2.12)

The complexity of the model is also low so it has a wide application range; this makes it a good choice for application research and practical development. By solving its differential relationship, the relation between Up and time is obtained. The zero-state response of the circuit is shown in Eq. (2.13):  (2.13) Up ¼ Cet=τ + IRp ¼ IRp 1  et=τ , τ ¼ Rp Cp

62

2. Electrical equivalent circuit modeling

As the equivalent model is one of the main parts in the lithium-ion battery state-of-charge estimation using the extended Kalman filtering algorithm, the recurrence form of Up can be obtained by discretization, as shown in Eq. (2.14):  (2.14) Up ðk + 1Þ ¼ Up ðkÞeΔt=τ + I ðkÞRp 1  eΔt=τ The Thevenin model is used to represent the dynamic response from the battery. R0 represents the transient response of the charge-discharge voltage. The resistancecapacitance circuit can represent the gradual drop or rise of the battery voltage after the charge-discharge process.

2.3.3 State equation establishment The cutoff voltage of the lithium-ion battery is the termination voltage, which refers to the limited operating voltage when the battery is charged or discharged. The voltage rises or falls to the state where the battery is not suitable to continue charge-discharge. According to the state-of-charge changing process, the iterative calculation process is described in Eq. (2.15): ðt (2.15) St ¼ S0  Iηdt=C 0

wherein St is the instantaneous state-of-charge value at the time point of t. S0 is its initial value. The instantaneous current value at the time point of t is described by I. η is used to describe the Coulomb efficiency, the discharge condition of which is equal to 1. According to the integral of discharging current, the state-of-charge value is the percentage of the residual capacity. Then, it is discretized, as shown in Eq. (2.16): Sk ¼ Sk1  ηΔtI k1 =C

(2.16)

wherein the equation is based on the principle of ampere hour integration. The significance of the battery model research can be summarized as follows. The internal electrochemical reaction process of the battery can be obtained under various conditions, which can improve the battery structure design. The battery cost and experiment time can be saved, which is useful to estimate the battery state. The improved design of the battery management system contributes to the advancement of simulation research in electric vehicles.

2.3.4 Mathematical description The high-precision battery modeling provides an important basis to analyze the real-time battery operating conditions. The model is equivalent to a simplified circuit consisting of electromotive force, internal resistance, polarization resistance, polarization capacitance, and terminal voltage. Based on the Rint model, the internal polarization effect is considered and the circuit model is constructed in the battery charge-discharge process [32]. Based on the ideal equivalent model, the model enhances the expression of the working process by considering two nonlinear parameters. The equivalent model structure is obtained by considering

2.3 Thevenin modeling

63

the equivalent capacitance Cp of the plate and the nonlinear contact resistance Rp between the electrolyte and the plate. The Thevenin model reduces the capacitance Cb representing the open-circuit voltage change, and its parameters have the same meaning as the model. It is generally believed that the open-circuit voltage in the model is varied so that the capacitance in the model can be replaced by the equivalent of the battery voltage toward the state of charge. Further, the battery model is obtained. The equations are listed according to Kirchhoff’s law, and the statespace equation of the model is obtained as shown in Eq. (2.17): (   Up∗ ¼ Up = Rp Cp + IL =Cp (2.17) UL ¼ Up  R0 IL + UOC The Thevenin model has simple parameters that can meet the requirements of the battery for wide application. The voltage Up across the capacitance Cp is taken as the state variable. The terminal current IL is taken as the input variable. The terminal voltage UL is taken as the output variable. UOC is the open-circuit voltage. R0 is internal resistance. Rp is the polarization resistance. Cp is the polarization capacitance. The parallel circuit of Rp and Cp describes the polarization process. UL is the closed-circuit voltage after the battery is connected to the external circuit. The model considers the battery polarization process, and its first-order construct is simple relatively. In the process of obtaining the state-space equation, the relationship between the current flowing through the battery polarization capacitance Cp and its two closed-circuit voltages can be obtained according to the operating characteristics of the capacitance component. Its mathematical relationship is described in Eq. (2.18): Ip ðtÞ ¼ Cp  dUp ðtÞ=dt

(2.18)

Based on the analysis of the equivalent circuit configuration, the voltage relationship of the equivalent circuit is obtained according to Kirchhoff’s voltage law of the circuit, as shown in Eq. (2.19):

 (2.19) R0  Ip ðtÞ + Up ðtÞ=Rp + Up ðtÞ ¼ UOC  UL Up(t) is used as the state variable, so that the above two expressions can be combined to describe the comprehensive working state. The voltage at both ends of the equivalent capacitance is set as a state variable, and the calculation process is analyzed. The state-space equation for obtaining the equivalent model is described in Eq. (2.20):   (2.20) R0  Cp  dUp ðtÞ=dt + 1 + R0 =Rp Up ðtÞ ¼ UOC  UL wherein UOC is an open-circuit voltage. R0 is internal resistance. Rp is a polarization resistance and Cp is a polarization capacitance. UL is the closed-circuit voltage after the battery is connected to an external circuit. The parameters in the equivalent model are fixed values, so there is no parameter correction and adjustment capability. After the equivalent model is treated with internal resistance, the influence of the polarization effect is considered. Consequently, the resistance-capacitance parallel circuit of the capacitive device Cp is added, which has good dynamic characteristics.

64

2. Electrical equivalent circuit modeling

2.4 High-order modeling The high-order equivalent circuit model is constructed for power lithium-ion batteries, and it has high accuracy in simulating transient response. It is suitable for high current, step type, and complicated charge-discharge conditions. In this section, the parameters are identified with the hybrid pulse power characteristic test data, and the identification results are analyzed.

2.4.1 Second-order circuit modeling The American Automotive Research Council announced the Partnership for a New Generation of Vehicles, in which the PNGV model was proposed in the battery experiment manual. The standard equivalent circuit models in the freedom car battery test manual have been established as well as the standard method for the parameter identification of the model [4]. The equivalent circuit model of a lithium-ion battery contains two capacitors and two resistors. This resistor model has one capacitance less than the resistance-capacitance equivalent circuit model. The characteristic of this model is that the open-circuit voltage of the battery is considered, in which the current accumulates error over time. Because the circuit model covers the battery characteristics of polarization and internal resistance, the equivalent circuit model is more accurate. The consideration of the load current impact on the open-circuit voltage of the battery is added to the PNGV model based on the Thevenin equivalent modeling method. In this model, UOC is used to indicate the ideal open-circuit voltage of the battery. Cb is the battery capacitor, representing the open-circuit voltage change caused by the accumulation of load current IL. R0 is the internal resistance of the battery. Rp is the battery polarization resistance and its current is Ip. Cp is the polarization capacitance, representing the polarization voltage Up change caused by the load current IL. The equivalent circuit of the PNGV model is shown in Fig. 2.10. The model is a typical linear lump parameter circuit that can be used to predict the battery terminal voltage variation on the hybrid pulse-power characteristic conditions. The battery terminal voltage is described in Eq. (2.21): UL ¼ UOC  R0  IL  Up  Ub

(2.21)

When the battery discharges, the accumulation of current and time causes a state-of-charge change. The open-circuit voltage of the battery also changes, which is reflected in the model of the change in the capacitor voltage. The size of the capacitor characterizes both the battery capacity and the direct-current response of the battery, which makes up for the deficiency FIG. 2.10

PNGV modeling diagram.

R0 Cb UOC

RP

CP

IL(t)

Ip(t)

+ UL

-

RL

2.4 High-order modeling

R0 Cb E

UOC

R1

R2

Up1

Up2

C1

C2

65

FIG. 2.11 Modeling improvement diagram. IL

UL

Rs

of the Thevenin model. To match the curve fitted by the model with the measured voltage curve, a set of resistance-capacitance circuits is added to the improved model to achieve a better matching degree during the curve fitting process. The improved battery modeling method is shown in Fig. 2.11. As shown in the designed structure diagram of the improved circuit model, it introduces a resistance-capacitance series network and a second-order resistance-capacitance parallel network. UL is the battery terminal voltage. The loop current of the battery is described by the parameter IL, in which the discharge direction is positive and negative when charging. Rs is the self-discharge internal resistance of the battery. UOC is the open-circuit voltage of the battery. R0 is the internal resistance of the battery, and its terminal voltage is used to represent the instantaneous voltage drop caused by the battery current. Cb is used to characterize the opencircuit voltage changes due to current accumulation. E is used as the ideal voltage source, which is integrated into Cb to indicate the change of UOC. R1 and C1 are the polarization resistance and capacitance of the battery, respectively. They are connected in parallel to simulate the polarization characteristic process of the battery charge-discharge, which characterizes the rapid electrode reaction to the battery. R2 and C2 are the electrochemical polarization resistance and capacitance, respectively, which characterize the slow electrode reaction to the battery. To express the battery characteristics fully, it is also possible to continue optimizing based on the above typical battery model, such as self-discharge factors and hysteresis characteristics. These are expressed in the form of circuits, but it increases the complexity of the model, inevitably identifying the difficulty. In the specific application, model selection and model optimization can be carried out using typical models for parameter identification. It is realized considering the differences between the model and the targeted improvement. The battery terminal voltage is described in Eq. (2.22): UL ¼ UOC  R0  IL  Up1  Up2  Ub

(2.22)

The battery model includes the electrical model of the battery and the temperature model of the battery. The electrical model is also called the equivalent circuit model and the temperature model together forms the ontology model of the battery. In the battery, the electrical model mainly calculates the battery voltage, self-discharging current, and state of charge. The temperature model mainly calculates its temperature. It is a typical equivalent circuit model [33] that simulates the transient response process with high accuracy. Consequently, it is suitable for the conditions of high current, step type, and complicated charge-discharge.

66

2. Electrical equivalent circuit modeling

To better reflect the dynamic characteristics of the ternary power lithium-ion battery better under step charge-discharge conditions, the polarization circuit of the PNGV modeling is expanded to obtain the improved model. The model uses a double resistance-capacitance instead of the original single resistance-capacitance circuit. Among them, the parallel circuit composed of R1 and C1 has a relatively small-time constant, which is used to simulate the process of rapid voltage change when the current changes suddenly. The parallel circuit composed of R2 and C2 has a relatively large time constant, which is used to simulate the process of slow and stable voltage. The improved circuit model can better characterize the polarization battery characteristics and better simulate the shelved circuit. To express the external characteristics of the battery more comprehensively, the optimization is continued by using the above-mentioned typical battery model, such as self-discharge factors, hysteresis characteristics, etc., which will inevitably increase the complexity of the model and identification difficulty. For model selection and optimization in a specific application, typical models can be used for parameter identification. Then, the targeted improvement can be made according to the difference between the model and the actual situation. The combination of model parameter identification and hybrid pulse-power characteristic test is the standard model in the battery test manual. The capacitance Cb is used to describe the open-circuit voltage changes. The electromotive force of the power supply is represented by UOC as the ideal voltage source. R0 is internal resistance. Rp is the polarization resistance. Cp is the polarization capacitance. Cb is the open-circuit voltage change caused by the power supply current accumulation. The loop current of the model is described by the parameter I, and UL is the battery terminal voltage. The parallel circuit of Rp and Cp is used to reflect the polarization process of the battery, thus greatly reducing the difference brought by the production process and the influence of the working environment. UOC and Cb indicate that there are differences between cells of opencircuit voltage jointly, such as voltage, capacity, and temperature. These differences increase along with the battery aging process [34]. The model does not consider the effects of selfdischarge and temperature on the battery. By analyzing the equivalent circuit model illustrated, the state-space equation of each parameter can be obtained as shown in Eq. (2.23): ð  (2.23) UL ¼ UOC  I ðtÞdt =Cb  R0 IL  Rp Ip The hybrid pulse-power experimental test is used for parameter identification. Four different sampling time points of a, b, c, and d are selected for the mathematical description of the state-space equation. After the matrix description, the relationship between different sample times could be obtained as shown in Eq. (2.24): 2 ða 3 Ia ðtÞdt Ia ðtÞ Ia ðtÞ 7 61 0 72 2 3 6 6 ðb 7 U 3 ULa 6 7 OC I ð t Þdt I ð t Þ I ð t Þ 1 6 76 1=C 7 b b b 6 ULb 7 6 76 b 6 7 ¼ 6 ð0c (2.24) 74 R 7 4 ULc 5 6 7 0 5 Ic ðtÞdt Ic ðtÞ Ic ðtÞ 7 61 ULd 6 7 Rp 6 ð d0 7 4 5 Id ðtÞdt Id ðtÞ Id ðtÞ 1 0

2.4 High-order modeling

t8(U8) t7(U7)

3.0

I (C)

1.5 0.0

-1.5

t0(U0) t1(U1)

t6(U6)

t9(U9) t10(U10)

28.95

t11(U11)

t5(U5) t4(U4) t3(U3)

-4.5 0

20

40

28.80 28.65

t2(U2)

-3.0

FIG. 2.12 Pulse-current charge-discharge voltage variation diagram.

29.25 I UL 29.10

U (V)

4.5

67

28.50 60

80

100

120

140

t (s)

In Eq. (2.24), UOC represents open-circuit voltage. R0 represents internal resistance. Rp represents the polarization resistance. Cb represents the polarization capacitance. Ia, Ib, Ic, and Id represent load current. Ip represents current passing polarization resistance. UL represents the closed-circuit voltage connected with external load. The subscript parameters a, b, c, and d have four different sample times, which are used to represent the state values of the parameters at different time points. The pulse charge-discharge voltage change is described for the lithium-ion battery packs that are shown in Fig. 2.12. The circuit equations of Kirchhoff’s current and voltage laws are written according to the PNGV model. The parameters are identified by combining the internal battery characteristics reflected by the hybrid pulse-power characteristic test. (1) Internal resistance identification In the hybrid pulse-power characteristic test voltage change curve, the straight drop of voltage from U1 to U2 after t1 is due to the presence of internal resistance, and the internal resistance is obtained as shown in Eq. (2.25): R0 ¼ ðU1  U2 Þ=I

(2.25)

In Eq. (2.25), the current value of the hybrid pulse-power characteristic test pulse discharge can be described by parameter I. (2) Surface-effect capacitance identification The surface-effect capacitance Cb represents the open-circuit voltage change caused by the accumulation of power supply current. The calculation process is described in Eq. (2.26): ð t3 (2.26) Cb ¼ I ðtÞdt=ðU1  U3 Þ t1

In the hybrid pulse-power characteristic test, t1 to t2 is a constant-current current parameter I exile process, and t2 to t3 is the shelf stage. Therefore, Eq. (2.26) can be expressed mathematically further as shown in Eq. (2.27): Cb ¼ ðt2  t1 ÞI=ðU1  U2 Þ

(2.27)

68

2. Electrical equivalent circuit modeling

(3) Polarization resistance identification The battery polarization effect starts from t1, in which t1 to t2 is the zero-state response process of the resistance-capacitance circuit. Then, the value of Rp can be obtained as shown in Eq. (2.28): ð t2  E ¼ U2  I ðtÞdt=Cb  IRp 1  et=τ (2.28) t1

(4) Polarization capacitance identification The first-order resistance-capacitance circuit time-constant can be calculated in the hybrid pulse-power characteristic tests. Therefore, the calculation formula for Cp can be calculated as shown in Eq. (2.29):   U1  U4 U1  U4 (2.29) ) Cp ¼ ðt3  t2 Þ= Rp ln τ ¼ Rp Cp ¼ ðt3  t2 Þ= ln U1  U3 U1  U3 Then, the values of R0, Cb, Rp, and Cp can be obtained accordingly.

2.4.2 Internal resistance description At present, the commonly used methods of measuring the internal resistance of lithiumion batteries include the volt-ampere characteristic curve, density, open-circuit voltage, alternating-current injection, direct-current discharge, and hybrid pulse-power characteristic test methods. The comprehensive comparison shows that the hybrid pulse-power characteristic test is relatively simple to implement. It can obtain the internal resistance of the battery and the polarization resistance of different state-of-charge levels accurately [35]. Therefore, the internal resistance of the battery is obtained by the hybrid pulse-power characteristic test method. Then, the experimental current and voltage curves of the hybrid pulse-power characteristic test can be obtained. The hybrid pulse-power characteristic test is to conduct pulse charge-discharge on the tested battery and to detect the instantaneous voltage change in both ends of the battery, to estimate the internal resistance and polarization internal resistance of the battery. Then, the current and voltage changes can be obtained from the single hybrid pulse-power characteristic test. In the experiment, t1 to t2 is a constant discharge time of 10 s. t2 to t3 is the shelf time of 40 s, which is followed by a t3 to t4 constant-current charging time of 10 s. Exiled from t1 time constant power, the voltage of the lithium-ion battery quickly decreases from U1 to U2. Due to the polarization effect, the voltage declines slower and slower. The discharge process is continued, and the voltage slowly decreases from U2 to U3. t2 is the stop time after discharge, and the terminal voltage rises from U3 to U4 rapidly. Then, it is the same as the polarization effect, up from U4 to U5 slower and slower. The charging process is similar. In this experiment, the lithium-cobaltite battery is selected as the experimental object, and its technical parameters are described in Table 2.1. The experimental charge-discharge device is a power cell large-ratio charge-discharge tester CT-4016-5V100A-NTFA. The chamber is produced by Dongguan Bell, with three layers of independent temperature control high and low-temperature chamber, DGBELL BTT-331C.

69

2.4 High-order modeling

TABLE 2.1 Technical parameters of the battery. Factors

Parameters

Length  width  height (mm)

200  80  180

Rated voltage (V)

3.7

Maximum load current (A)

1.5 C

Charge cutoff voltage (V)

4.15

Discharge cutoff voltage (V)

3.0

Operating temperature (°C)

15 to 70

Rated capacity (Ah)

4.0

This experiment is conducted under a constant temperature of 25°C. The identification steps are described below. (1) The lithium-ion battery is recycled, in which the cyclic charge-discharge is conducted three times. It is charged to 4.15 V at 0.2 C, then discharged to 3.00 V at 0.1 C. Another full charge-discharge should be completed after it is shelved for 12 h. The purpose is to make the battery capacity and performance achieve the best. (2) After discharging the lithium-ion battery of S ¼ 1 for 3 min at the rate of 1 C to be S ¼ 0.95, it is shelved for 40 min. Then, the hybrid pulse-power characteristic test is conducted, in which the parameters of open-circuit voltage, current, and time are recorded at the same interval. (3) Step (2) is repeated, in which 5% capacity is discharged in each cycle at S ¼ 0.90, 0.85, 0.80, …, 0. The hybrid pulse-power characteristic test is performed at 0.05. According to the hybrid pulse-power characteristic experimental data onto each state point, the parameter values of the circuit model can be obtained corresponding to the state-of-charge values. The experimental results are described in Table 2.2. According to the experimental analysis in the table, the internal resistance R0 increases slowly with the state-of-charge decrease. The change range is small, and the change rate is slow. The polarization resistance Rp first decreases slowly, then increases sharply when it is lower than 0.15, according to which the acceleration also increases. After a long-shelved time, the influence of polarization effects and ohmic resistance on the terminal voltage of the battery is greatly reduced. At this time, the measured terminal voltage of the battery is open-circuit voltage, which is its electromotive force E. At the stage from S ¼ 1.00 to S ¼ 0.85, the terminal voltage of the battery decreases rapidly. The open-circuit voltage decreases by 0.04–0.06 V every time the state of charge decreases by 0.05. It is stable between 0.85 and 0.15. When it decreases by 0.05, the open-circuit voltage decreases by 0.03 V, in which the voltage fluctuation is small. When it is lower than 0.15, the open-circuit voltage drops rapidly and the voltage fluctuates greatly as the battery discharges. The experimental results show that when it is lower than 0.15, after the deep discharge of the battery, the chemical reaction inside the battery can be violent and the model parameters all change dramatically.

70

2. Electrical equivalent circuit modeling

TABLE 2.2 Model parameters under the different state-of-charge values. S

R0

Rp

Cp

UOC

1.00

6.0661

6.9821

0.2041

4.13

0.95

6.1911

6.7857

0.2101

4.09

0.90

5.9214

6.8321

0.2090

4.05

0.85

5.8625

6.3768

0.2216

3.99

0.80

5.8214

6.2018

0.2166

3.96

0.75

6.0839

6.1911

0.2320

3.92

0.70

6.0196

6.0000

0.2391

3.89

0.65

6.0589

6.0786

0.2333

3.86

0.60

6.0321

6.0679

0.2465

3.84

0.55

6.3286

6.0857

0.2336

3.82

0.50

6.7232

6.0714

0.2347

3.81

0.45

6.8643

6.1893

0.2315

3.79

0.40

7.2929

6.0571

0.2358

3.78

0.35

7.0500

6.2268

0.2313

3.76

0.30

7.8054

6.6268

0.2162

3.74

0.25

8.0339

6.5214

0.2217

3.72

0.20

8.0161

6.7714

0.2114

3.70

0.15

8.2464

6.8786

0.2072

3.69

0.10

7.8679

8.4786

0.1692

3.58

0.05

8.5714

18.5536

0.0766

3.28

2.4.3 Splice equivalent modeling The main task of the battery temperature model is to calculate the battery temperature change of the heat dissipation. Theoretically, the electrochemical model of the battery is the best way to calculate the battery temperature because the chemical reaction inside the battery may be the direct cause of the temperature change. The electrochemical model accurately describes the chemical reaction mechanism inside the battery, so it is also possible to accurately calculate the amount of heat released by the battery of the chemical reaction. The battery temperature calculated by the electrochemical model is closest to the temperature of the battery. However, the establishment of electrochemical models is greatly influenced by the battery manufacturing process. Due to the enterprise secrets and equipment accuracy of the battery manufacturers, the electrochemical model is only suitable for theoretical research that cannot be applied to engineering control.

2.4 High-order modeling

71

The battery temperature model in this section is based on the analysis of the battery electrical model, plus the standard reaction heat composition of the main chemical reactions in the battery. Although the accuracy of the model is reduced, it is still feasible as an engineering research target that does not need high-temperature. In the temperature model, the temperature TBat of the battery is determined by Eq. (2.30): ð TBat ¼ TBat, 0 + Pw dt=ðcBat,0 mBat Þ (2.30) Among them, TBat,0 indicates the initial temperature of the battery, generally taking the ambient temperature. CBat indicates the specific heat capacity of the battery. The specific heat capacity of the battery is generally 800–1000 J/kg K. The test data returned to the battery manufacturer indicate that the specific heat capacity of the battery sample is 854 J/kg K. mBat indicates the battery quality, and the mass of the single cell is 1004 g, so the total mass of the battery sample is 10.04 kg. The total heat absorbed by the battery can be represented by the parameter Pω. In this model, the heat absorbed by the battery is mainly caused by the heat generated by the internal loss voltage, internal chemical reaction, self-discharging current, heat exchanging, and environment. These several parts can be composed, so the total heat absorbed by the battery Pw can be expressed by Eq. (2.31): Pw ¼ PJoule + PMR + PLR  PRad

(2.31)

Among them, PJoule is used to describe the heat caused by the open-circuit voltage, which is generated by the internal loss voltage. The heat PMR is produced by the internal chemical reaction. The heat PLR is caused by the self-discharging current of the battery. The heat PRad is generated by the exchange of the battery and the environment. The mathematical calculation process can be described in Eq. (2.32): 8 PJoule ¼ IBat ULoss > > > > < PMR ¼ CMR IBat (2.32) > PLR ¼ UBat ILoss > > >  4  : 4 PRad ¼ εSBat σ TBat  TAmb Among them, IBat indicates the current in the battery. ULoss represents the battery loss voltage. ILoss represents the self-discharging current of the battery. UBat represents the battery terminal voltage. CMR represents the battery constant, which is 27 mV for lithiummanganate batteries. TAmb indicates the ambient temperature. SBat indicates the area of contact between the battery and the environment. σ is the Steven-Boltzmann constant, which is 5.67  108 W/m2. ε is the scattering rate, the emission ratio constant, for general building materials, generally from 0.7 to 0.9. Due to the uncertainty, the model temporarily is set as 0.8. The ultimate goal of building a battery model is to simulate the calculation of the battery parameters, and the model needs to be used in the embedded processor, so the continuous integral form of the variables in the model is changed to a discrete form of fixed step size. The running step length of the model is set to 100 ms, which not only ensures the accuracy of the model but also ensures that the resource consumption of the embedded processor is not too high. This model is also the standard battery equivalent circuit model specified in the battery

72

2. Electrical equivalent circuit modeling

experiment manual. The model has a clear physical meaning. The ideal voltage source opencircuit voltage represents the open-circuit voltage of the battery. The equations are listed according to Kirchhoff’s law, and the state-space equation is obtained as shown in Eq. (2.33): 8" # " #" # " # > 0 0 Uo Uo 1=Co > >   ¼ + IL > > < Up 0 1= Rp Cp Up 1=Cp (2.33) " # > U > o > > > : UL ¼ ½ 1 1  U + R0 IL + UOC p The resistance R0 is the internal resistance of the battery. The resistance Rp is the polarization internal resistance of the battery. The capacitance Cp is the polarization capacitance of the battery, and the IL is the battery load current. Ip is the polarization current of the battery. UL is the terminal voltage of the battery. The capacitance Co represents the change in the opencircuit voltage caused by the integration of the load current IL with time. The voltage Up on the capacitor Cp can be used as the state variables. The terminal current IL is the input variable, and the terminal voltage UL is used as the output variable.

2.5 Parameter identification algorithms The significance of the battery model is to establish a clear mathematical relationship, which in turn can get important input and output. The various battery working characteristics can be estimated by establishing an accurate battery model, analyzing the feasibility and effectiveness of the battery model by applying specific software, and finally verifying the accuracy of the model by comparing it with the experiment. The main factors considered in the modeling are system-side voltage, battery operating current, state of charge, ambient temperature, internal resistance, and electromotive force. The correlation between the two is significant for the establishment of the battery model. The estimation accuracy mainly depends on the accuracy of the battery model.

2.5.1 Identification overview By studying the properties and parameters of the circuit components in the equivalent circuit model, the purpose of managing the battery can be realized more accurately and efficiently [36]. Lithium-ion batteries are widely used in various fields due to their high voltage, large-capacity, weak self-discharge, small mass, and small volume. Their materials and characteristics research are also in-depth development. The deepening of the battery application of various fields and the enhancement of process complexity needs an increasingly sophisticated management system. The research on battery equivalent model and working characteristics is very important, so it is of great significance to establish an accurate battery model. Lots of work has been done and lots of effective methods and conclusions have been obtained that provide a more solid foundation for the application of lithium-ion batteries. The dynamic model describes the fixed parameters in the classical model as variables that vary from the state of charge and temperature, enabling a more accurate description of the

2.5 Parameter identification algorithms

73

battery performance. Fuzhou University is concerned with the parameter identification of the power battery equivalent model on pure electric vehicles. The 84 Ah nickel-cobalt-manganese ternary lithium-ion battery packs consisting of 87 cells are used as the research objective in the pure electric vehicle. Then, the second-order resistance-capacitance equivalent model is built for its accurate characterization. As can be recognized by the battery equivalent model, the absolute error of the maximum open-circuit voltage can be obtained as 3.62% with the absolute error of the smallest single cell as 3.24%, which can be used in engineering practice. The parameter identification is the processing of experimental data after it is finished. There are many parameter identification methods of the battery model, which can be roughly divided into two categories. One is to calculate the model parameters by the calculation formula, and this method applies to a specific model. One type is model parameters obtained by data fitting, which is applied to most of the equivalent models [37]. The first type can be calculated by referring to the above formula, such as the resistance-capacitance equivalent model. The second method type obtains the experimental data by the hybrid pulse-power characteristic test, in which the pulse charge-discharge current is used to test the dynamic battery power characteristics of the normal operation. The experimental data can be obtained by the hybrid pulse-power characteristic test with equally spaced state of charge, which is followed by curve fitting using multiple linear regression fitting or the least-square method to obtain the parameters [38]. A regression involving two or more independent variables is called multiple linear regression. The principle and calculation process of multiple linear regression should be the same as one-dimensional linear regression. However, the calculation is quite troublesome due to numerous independent variables and statistical software should be used in practice. The least-square method is a classic data-processing method. The famous scientist Gauss proposed the least-square method, which is applied to the calculation of the orbits of planets and comets. When calculating the orbits of planets and comets, the values of these six parameters are estimated to describe the celestial body motion based on the observations obtained by the telescope. When inferring unknown parameters from observation data, the most appropriate value of the unknown parameter should be such a value that multiplies the square of the difference between each observation. The calculated value can be obtained by the sum of the measured values, making its accuracy the smallest. This is the earliest least-square idea. Since then, this method has been used to solve many practical problems. For different purposes, it is modified and various corresponding least-square algorithms have been proposed. In the field of parameter identification, the least-square algorithm is an estimation method that can be used for both dynamic and shelved systems. It can be used for both linear and nonlinear systems, which can be introduced into the offline and online battery state estimation. In a random environment, the observation data is not required to provide information on its probability and statistics when using the least-square method. Its estimation results have quite good statistical characteristics, which are easy to understand. The identification algorithm developed by the calculation principle is relatively simple to implement, which provides a solution to the problem when other parameter identification methods are difficult to be applied. At the same time, the recursive least-square is improved by the least-square algorithm and its improved method can realize the online estimation of the model parameters. This can minimize the influence of temperature, discharge rate, and other conditions on the estimation accuracy of the parameters during the parameter identification process.

74

2. Electrical equivalent circuit modeling

2.5.2 Least-square functional fitting The least-square method is easy to understand and has fast convergence speed; it has been widely used in system parameter identification. For a single-input single-output linear system, its mathematical description is described in Eq. (2.34): 8  1   1  > < Az YðkÞ ¼ B z UðkÞ (2.34) A z1 ¼ 1 + a1 z1 + a2 z2 + ⋯ + an zn > :  1  1 2 n B z ¼ 1 + b1 z + b 2 z + ⋯ + b n z wherein the system input is described by the parameter a and the system output is described by b. Then, the discrete transfer function can be obtained. The input system is described by U(k) and Y(k) is the system output. The discrete transfer function can be obtained as shown in Eq. (2.35):     B z1 1 + b1 z1 + b2 z2 + ⋯ + bn zn ¼ (2.35) G z1 ¼ 1 Aðz Þ 1 + a1 z1 + a2 z2 + ⋯ + an zn The corresponding difference equation is described in Eq. (2.36): yðkÞ ¼ a1 yðk  1Þ  a2 yðk  2Þ  ⋯  an yðk  nÞ +b1 uðk  1Þ + b2 uðk  2Þ + ⋯ + bn uðk  nÞ + eðkÞ

(2.36)

wherein the included parameters are the parameters to be estimated, and e(k) is the vector equation error ϕðkÞ ¼ ½yðk  1Þ,  yðk  2Þ, …,  yðk  nÞ, uðk  1Þ, uðk  2Þ, …, uðk  nÞ (2.37) θ ¼ ½ða1 , a2 , …, an Þ, ðb1 , b2 , …, bn Þ Thus, Eq. (2.36) can be rewritten accordingly. The input and output of the system can be expanded upon n dimensions respectively, where k ¼ 1, 2, …, N + n. Then, the matrix expression can be obtained as shown in Eq. (2.38): 8 > Y ¼ ϕθT + eðkÞ > > 2 3 > > > yðnÞ yðn  1Þ … yð1Þ uðnÞ … uð1Þ > > > 6 yðn + 1Þ < yðnÞ … yð2Þ uðn + 1Þ … uð2Þ 7 6 7 ϕ¼6 (2.38) 7 4 > ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 5 > > > > > yðn + N Þ yðn + N  1Þ … yðN Þ uðn + N Þ … uðNÞ > > > : Y ¼ ½yðn + 1Þyðn + 2Þ…yðn + N ÞT , e ¼ ½eðn + 1Þeðn + 2Þ…eðn + N ÞT The least squares method is introduced to find the parameter value by recursive calculating. When a ¼ b exists, the minimum value can be obtained. Taking the objective function J(θ), it is then described in Eq. (2.39): J ðθ Þ ¼

N X i¼1

ðY  ϕθÞ2 ¼

N X

e2 ðn + iÞ ¼ ðY  ϕθÞT ðY  ϕθÞ

(2.39)

i¼1

Therefore, the first derivative can be obtained for Eq. (2.39). When a ¼ b, the first derivative is zero. The purpose of the least-square method is to find the value recursively. When it exists,

2.5 Parameter identification algorithms

75

the minimum value can be obtained. Otherwise, the first derivative is zero. Then, it can be obtained for Eq. (2.40): i ∂J ∂ h ¼ ðY  ϕθÞT ðY  ϕθÞ ¼ 0 (2.40) ∂θ ∂θ After that, the least-square estimation value of the system can be obtained as shown in Eq. (2.41):  1 ^ θ ¼ ϕT ϕ ϕT Y (2.41) It is necessary to continuously input and output the latest experimental data to improve the accuracy of parameter estimation in the continuous iterative process until a satisfactory accuracy is reached. The specific calculation process is described in Eq. (2.42): 8

 ^ > θ ð k + 1Þ ¼ ^ θðkÞ + Kðk + 1Þ yðk + 1Þ  ϕT ðk + 1Þ^θðkÞ > > > < Kðk + 1Þ ¼ Pðk + 1Þϕðk + 1Þ (2.42) > > Pðk + 1Þϕðk + 1ÞϕT ðk + 1Þ > > : Pðk + 1Þ ¼ PðkÞ  1 + ϕT ðk + 1ÞPðkÞϕðk + 1Þ wherein it is the estimated value of the parameter at the time point of k. θ (k + 1) is the calculated value of the system observation at the time point of k + 1. y(k + 1) is the observation at the time point of k + 1. In each iteration, the algorithm uses the difference between the system observation calculation value and the gain to correct the final estimation value. The initial value can be any value and the larger value is better. Then, I can be initialed as a unit array. Then, the estimated value of the parameter can be obtained at the time point of k. The calculated value of systematic observation is calculated at the time point of k + 1. As for each iteration, the algorithm uses the difference between the system observation calculation value and the gain to correct the final estimation value. The initial value can be any value. Taking a larger value, I can be initialed as the unit array and P(0) ¼ αI. Wherein α is a larger value as far as possible and I can be taken as the unit array. Taking the Thevenin model as an example, the model can be expressed as shown in Eq. (2.43): 8 RP I ðsÞ > > > < Ua ðsÞ ¼ UOC ðsÞ  UL ðsÞ ¼ R0 I ðsÞ + 1 + τs + e (2.43) Ua ðsÞ + a  sUa ðsÞ ¼ b  I ðsÞ + c  s + e > > > : a ¼ τ b ¼ ðR0 + RP Þ c ¼ τR0 wherein e is the measurement noise. Therefore, the discrete system can be obtained with bilinear transformation, which can be expressed as shown in Eq. (2.44): yðkÞ ¼ ayðk  1Þ + buðkÞ + cuðk  1Þ + meðkÞ

(2.44)

To realize the least square principle for parameter identification, the discrete system formula can be expressed as the least-square form, as shown in Eq. (2.45): 8 T < yðkÞ ¼ xðkÞ  θðkÞ (2.45) xðkÞ ¼ ½ yðk  1Þ uðkÞ uðk  1Þ eðkÞ T : θ ðkÞ ¼ ½ a b c m  T

2. Electrical equivalent circuit modeling

FIG. 2.13 The parameter identification

50000

0.0020

R0 Rp1

Cp 40000 30000

C(F)

R(Ω)

0.0015

0.0010

20000

0.0005

10000

0.0000

0 0

3000

6000

9000

12000

15000

3000

6000

9000

12000

15000

18000

t(s)

(B)

4.25

0.01 U1 U2

E 0.00

4.00

3.75

3.50

3.25

0

18000

t(s)

(A)

U(V)

and closed-circuit voltage traction results. (A) Internal and polarization resistance. (B) Polarization capacitance. (C) Simulation voltage. (D) Simulation voltage error. Wherein R0 is the internal resistance of the Thevenin. Rp and Cp are the polarization resistance and capacitance, respectively. U1 is the measured voltages obtained by the experiment, and U2 is the simulation results by the parameters that are identified by using the recursive least square (RLS) method. E is the error of simulation voltage. It can be seen from the figure that this method can identify the model parameters well.

U(V)

76

–0.01 –0.02 –0.03 –0.04

0

3000

6000

9000

12000

t(s)

(C)

15000

0

18000

(D)

3000

6000

9000

12000

15000

18000

t(s)

According to the above recursive least square formula, the estimation result can be obtained. The experimental data under the dynamic stress test working condition can be used for the model parameter identification, and the results can be obtained as shown in Fig. 2.13.

2.5.3 Forgetting factor correction Due to the phenomenon of filtering saturation in the least-square method, the values of gain k and p become smaller as the number of iterations of the algorithm data increases. This makes the calculation ability to the data correction weaker, and the degree of data saturation becomes larger, which eventually leads to a larger error in parameter identification. Therefore, it is considered to add a forgetting factor based on the least-square identification to improve the online estimation capability of the recursive least-square algorithm. The forgetting factor λ (0 < λ < 1) is introduced to weaken the influence of old data and enhance the feedback effect of new data. The improved objective function is described in Eq. (2.46): J¼

N X

λNk ½eðkÞ2

(2.46)

k¼n + 1

The forgetting factor is used to give less weight to the long-running data while the latest observation data take up more weight in the identification process. Then, the recursive formula for the least-square method is modified as shown in Eq. (2.47): 8

 ^ θ ð k + 1Þ ¼ ^ θðkÞ + Kðk + 1Þ yðk + 1Þ  ϕT ðk + 1Þ^θðkÞ > > <

1 (2.47) Kðk + 1Þ ¼ Pðk + 1Þϕðk + 1Þ ϕT ðk + 1ÞPðkÞϕðk + 1Þ + λ > >

 : Pðk + 1Þ ¼ λ1 I  Kðk + 1ÞϕT ðk + 1Þ PðkÞ Among them, the closer λ to 1, the better the simulation result. When the value is greater than 0.9, the simulated value is 0.98. The smaller the value of λ, the smaller the tracking effect of the algorithm, but it causes fluctuations in the algorithm. When λ ¼ 1, it is the standard

2.5 Parameter identification algorithms R2

R1 R0

FIG. 2.14

IL(t)

77

Second-order resistance-capacitance equivalent circuit model.

+ U2 -

+ U1 -

+ UOC

I1(t)

C1

I2(t)

C2

UL

-

RL

recursive least-square algorithm. The second-order resistance-capacitance equivalent circuit model of the battery is given in Fig. 2.14. The forgetting factor least-square parameter identification method is applied to the parameter identification of the battery equivalent model. According to Kirchhoff’s voltage law, the circuit equation can be obtained as shown in Eq. (2.48): Ub ¼ UL + U0 + U1 + U2

(2.48)

wherein Ub represents the open-circuit voltage of the battery. R0 represents the internal resistance. R1 and R2 both represent the battery polarization resistance. The battery model is converted into a least-square mathematical form as shown in Eq. (2.49):   R1 R2 + + R0 IL + UL (2.49) Ub ¼ R1 C1 s + 1 R2 C2 s + 1 The parameters in the second-order resistance-capacitance equivalent circuit model can be identified accordingly. If the time constants are expressed by τ1 and τ2, Eq. (2.49) can be rewritten as shown in Eq. (2.50): τ1 τ2 Ub s2 + ðτ1 + τ2 ÞUb s + Ub ¼ τ1 τ2 R0 Is 2 + ½R1 τ2 + R2 τ1 + R0 ðτ1 + τ2 ÞIs + ðR1 + R2 + R0 ÞIL + τ1 τ2 Us2 + ðτ1 + τ2 ÞUs + UL Then, the above formula can be simplified as shown in Eq. (2.51): aUb s2 + bUb s + Ub ¼ aRi Is 2 + dI s + cIL + aU2s + bUs + UL a ¼ τ1 τ2 , b ¼ τ1 + τ2 , c ¼ R1 + R2 + R0 , d ¼ R1 τ2 + R2 τ1 + R0 ðτ1 + τ2 Þ As for the s calculation, it is initialized as shown in Eq. (2.52): s ¼ ½xðkÞ  xðk  1Þ=T s2 ¼ ½xðkÞ  2xðk  1Þ + xðk  2Þ=T 2

(2.50)

(2.51)

(2.52)

Substituting Eq. (2.52) into Eq. (2.51) for discretization and taking T as the sampling time, the mathematical relationship can be obtained as shown in Eq. (2.53): bT  2a a ½UL ðk  1Þ  Ub ðk  1Þ + 2 ½ U L ð k  2Þ  U b ð k  2 Þ  T2 + bT + a T + bT + a cT 2 + dT + aR0 dT  2aR0 aR0 I ðkÞ + 2 I ðk  1Þ + 2 I ðk  2Þ + T2 + bT + a T + bT + a T + bT + a

Ub ðkÞ  UL ðkÞ ¼

(2.53)

Then, its mathematical relationship can be obtained as shown in Eq. (2.54): Ub ðkÞ  UL ðkÞ ¼ k1 ½UL ðk  1Þ  Ub ðk  1Þ + k2 ½UL ðk  2Þ  Ub ðk  2Þ +k3 I ðkÞ + k4 I ðk  1Þ + k5 I ðk  2Þ

(2.54)

78

2. Electrical equivalent circuit modeling

Among them, the parameters can be obtained by Eq. (2.55): 8 bT  2a a > > k1 ¼ 2 , k2 ¼ 2 , < T + bT + a T + bT + a 2 > aRi > : k3 ¼ cT + dT + aRi , k4 ¼ dT  2aRi , k5 ¼ 2 2 2 T + bT + a T + bT + a T + bT + a

(2.55)

Eq. (2.54) can be substituted for the recursive least-square method. Eq. (2.55) is taken as the directly identified parameters. The circuit modeling parameters of Ri, R1, C1, R2, and C2 are derived from the identification results of these parameters. The values of each parameter can be obtained from Eq. (2.55) as shown in Eq. (2.56): a ¼ k0 k2 , b ¼ k0 ðk1 + 2k2 Þ=T k0 ¼ T2 =ðk1 + k2 + 1Þ (2.56) ) c ¼ k0 ðk3 + k4 + k5 Þ=T 2 , d ¼ k0 ðk4 + 2k5 Þ=T Ri ¼ k5 =k2 Therefore, these equations are the values of each parameter in the equivalent circuit model. In the process of data analysis and processing, the effective data segment is extracted from the original experimental data, which is then analyzed. The parameters of τ1 and τ2 are then calculated as shown in Eq. (2.57):  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2.57) τ1 ¼ b + b2  4a =2, τ2 ¼ b  b2  4a =2 An effective processing method is adopted to obtain the relationship between the internal parameters of the equivalent circuit model and the state of charge. Further, the values of the remaining parameters can be obtained by combining with Eq. (2.56) as shown in Eq. (2.58): ( R1 ¼ ðτ1 c + τ2 R0  dÞ=ðτ1  τ2 Þ, C1 ¼ τ1 =R1 (2.58) R2 ¼ c  R1  R0 , C2 ¼ τ2 =R2

2.6 Experimental analysis The complex battery electrochemical reaction system can be described by the equivalent models through the experiments. In short, the equivalent is to point to the battery system with the same current. The established equivalent model can track the battery voltage effectively. The voltage error of the model is small with a high-precision effect. The model parameter identification is conducted with the state of charge as well as other influencing conditions of the battery capacity, charge-discharge current rate, cell aging degree, and environmental temperature.

2.6.1 Exponential curve fitting The curve fitting method uses a method in which the entire pulse discharge curve is fitted by a formal function of parameters to obtain model parameters [39]. Its adoption rate of data is much higher than that of the point-and-point formula. For curve fitting, only a fitting function expression containing the parameter symbols needs to be obtained [40]. It is mainly for the

79

2.6 Experimental analysis

relatively complicated calculation parameters, such as Rp and Cp in the model. The internal resistance R0 does not need to use curve fitting, which can be directly obtained accordingly. The fitting curve segment should be selected to obtain the zero-state response curve. The model circuit expression can be obtained by Kirchhoff’s voltage law relationship, so the equation is then abstracted to obtain a parameterized expression as shown in Eq. (2.59):   UL ¼ UOC  IR0  IRp 1  et=τ   (2.59) y ¼ a  b 1  ex=c wherein y represents the ordinate variable to the terminal voltage UL. x represents the abscissa variable at the time point of t. The parameters of a, b, and c are three factors corresponding to Eq. (2.59). The range of parameters can be set for the known conditions of the open-circuit voltage, current, and internal resistance. Then, according to the measured terminal voltage and the observation equation, the zero-state response of the state of charge is 0.9. It can be well expressed by curve fitting, which is shown in Fig. 2.15. The parameter identification is performed for each state-of-charge point phase utilizing curve fitting. Then, the internal resistance, polarization resistance, and polarization capacitance are obtained onto the state of charge from 0.1 to 1 as shown in Table 2.3. The identified parameters are obtained and plotted in the coordinate system. Taking the state of charge as the independent variable and the parameter as the dependent variable, the scatter plot can be obtained for R0, Rp, and Cp. The fitted curve is obtained by the leastsquare polynomial fitting method, as shown in Fig. 2.16. The trends of R0, Rp, and Cp are combined with state variation. As can be known from the figure, the internal parameters of the model fluctuate within a certain range of the state change. Therefore, these parameters can be replaced by their average values or the lookup table in the case where the model accuracy is not high. One way to use model parameters is to have a small amount of computation. As for it, more accurate simulation data are needed. Then, the polynomial fit should be required for the curve. The parameter fitting polynomial can be obtained from the experimental results correspondingly.

2.6.2 Polynomial relationship description

U5

2

U6 U7

U1

28.8

U3

-2

28.6 U2 t1 t2 t3

-4

(A)

0

20

40

t4

60

t (s)

t5 t6

80

t7

100

120

UL UL'

3.87

29.0

U4

0

3.88

29.2

U (V)

4

U (V)

I (C)

Taking the lithium-ion battery LFP50Ah as the research objective, the battery test equipment is subbranch source BTS750-200-100-4, and the hybrid pulse-power characteristic test is conducted on the battery according to the American freedom car battery experiment

3.86 3.85 3.84

Equation

a - b * (1-exp( -x/ c))

a

3.87839 (Err= 2.0638E-4)

b

0.03413 (Err= 7.544E-4) 6.86374 (Err= 0.32728)

c

28.4 140

3.83 0.0

(B)

1.6

3.2

4.8

6.4

8.0

9.6

t(s)

FIG. 2.15 Parameter identification by curve fitting. (A) Pulse charge-discharge variation. (B) Curve fitting process and result.

80

2. Electrical equivalent circuit modeling

TABLE 2.3 Parameter identification results. S (1)

R0 (Ω)

Rp (Ω)

Cp (F)

τ (1)

1.0

0.0257

0.01907

359.92344

6.86374

0.9

0.0484

0.02076

330.62331

6.86374

0.8

0.0077

0.02094

327.78128

6.86374

0.7

0.0209

0.02305

297.77614

6.86374

0.6

0.0257

0.01896

362.0116

6.86374

0.5

0.0043

0.01926

356.37279

6.86374

0.4

0.0021

0.01811

379.00276

6.86374

0.3

0.0413

0.02264

303.16873

6.86374

0.2

0.0449

0.02014

340.80139

6.86374

0.1

0.0242

0.01936

354.53202

6.86374

500

0.04 400

0.03

Cp (F)

Rp (W)

FIG. 2.16 Model parameter identification results. (A) Polarized internal resistance. (B) Polarized capacitance.

0.02 0.01 0.00 0.00

(A)

300 200 100

0.15

0.30

0.45

0.60

S (1)

0.75

0.90

1.05

0 0.00

(B)

0.15

0.30

0.45 0.60 S (1)

0.75

0.90

1.05

manual. The single hybrid pulse-power characteristic test working step is to take the cyclic 1 C with a current of 50 A constant-current for 10 s, shelved for 40 s, and so on. The state of charge is 1.0, 0.9, 0.8, …, 0.2, and 0.1 pulse cycles of 10 are all spaced 40 min apart. The current and its corresponding voltage change curves in the hybrid pulse-power characteristic test are described in Fig. 2.17. The curve of the hybrid pulse-power characteristic experimental voltage variation is obtained and the variation characteristics can be extracted to obtain the model parameters. The following four features can be extracted from the analysis of Fig. 2.17A: (1) The discharge starts at t1. The terminal voltage of the battery drops suddenly from U1 to U2, mainly due to the voltage change caused by the ohmic resistance of the battery. (2) As can be known from t2 to t3, the terminal voltage of the battery drops slowly from U2 to U3, which is due to the battery polarization effect. The charging process of the discharging current to the polarized capacitance is the zero-state response of the double resistance-capacitance series loop. (3) As can be known from t3 to t4, the terminal voltage of the battery rises suddenly from U3 to U4, which is also due to the voltage changes caused by the ohmic resistance of the battery. (4) As for the period from t4 to t5, the terminal voltage of the battery rises slowly from U4 to U5, which is the discharge process of polarization capacitance to polarization resistance and the zero-input response of the double resistance-capacitance circuit.

81

2.6 Experimental analysis

3.80

UL (V)

3.75 t1(U1)

3.70

t5(U5)

3.65 t4(U4)

3.60

t2(U2)

3.55

t3(U3)

1.1x106

1.1x106

1.2x106

1.2x106

t (ms)

(A) 4.4

3,80 3,75

4.2

3,70 3,65

UL (V)

4.0

3,60 3,55 1,1x106

3.8

1,1x106

1,2x106

1,2x106

3.6 3.4 3.2 3.0 0.0

(B)

5.0x105

1.0x106

1.5x106

2.0x106

2.5x106

t(ms)

FIG. 2.17

Hybrid pulse-power characteristic experimental voltage curve. (A) Single pulse-power characteristic test. (B) Complete pulse-power characteristic test.

UOC is the open-circuit voltage that is stable at both the positive and negative ends of a battery in a long-shelved state. The experiment shows that the voltage stability of the battery after 40 min is equal to the open-circuit voltage of the battery. Therefore, U1 can be taken as the open-circuit voltage corresponding to the state-of-charge value as shown in Fig. 2.17B. Through the above hybrid pulse-power characteristic test, 10 open-circuit voltage values corresponding to the state-of-charge values of 1.0–0.1 can be obtained directly. To obtain the internal resistance, the battery terminal voltage changes suddenly at the time point of discharge and the time point of stop. Both are caused by internal resistance. Therefore, the internal resistance value can be calculated through Eq. (2.60): R0 ¼ ½ðU1  U2 Þ + ðU4  U3 Þ=ð2I Þ

(2.60)

82

2. Electrical equivalent circuit modeling

Then, the polarization capacitance and resistance can be obtained. The battery terminal voltage drops from U2 to U3 slowly. It is due to the battery polarization effect on the time point from t2 to t3. The charging process of the discharging current to the polarized capacitance is the zero-state response of the double resistance-capacitance series loop. Time-domain analysis of the circuit is carried out. The data onto t2 to t3 are selected to obtain a functional relationship between UL and the time parameter t, as shown in Eq. (2.61):   UL ðtÞ ¼ U2  IRp1 1  et=τ1  IRp2 1  et=τ2 (2.61) According to the data onto t2 to t3, the parameter values of the time constant can be obtained directly. Then, the values of polarization capacitance Cp1 and Cp2 can be calculated according to equations Rp1Cp1 and Rp2Cp2. The parameter identification of the ternary battery equivalent model needs to be completed by experiment. The experimental platform is formed by a ternary lithium-ion battery such as sciatic CFP50Ah, BTS200-100-10-4 as the battery test equipment, a high-low temperature experiment box, and a computer. The experimental platform hybrid pulse-power characteristic test is set up and carried out under different working conditions for the discharge of verification tests. Because a lithium-ion battery generally adopts the method of constant-current charging followed by constant-voltage charging, the state of charge is relatively easy to be estimated. Therefore, only the experiment of the parameter identification of the discharge direction is carried out, namely the hybrid pulsepower experiment. The hybrid pulse-power experimental purpose is to get the different state-of-charge values corresponding to the parameters of the equivalent circuit model. At room temperature of 25°C, the battery is maintained on a full charge and held after the first pulse discharge for 10 s. Then, it is shelved for 40 s, and is again set to the battery pulse charge for another 10 s. The current value of the pulse discharge and the charging current is 1 C in the experiments. Before the discharge to the point where the state of charge decreases to the next state for 10%, a 40-min interval is investigated after each step in the experimental process to let the electrochemistry reaction against the battery be balanced [41]. In these different state-ofcharge point experiments, the response data onto the voltage and current are obtained according to the same procedure in which the discharging current is assumed to be positive. According to the curves in the above two figures, both the current and voltage values are changed suddenly into the time point of t1, reflecting the internal resistance characteristics of the model. As can be known from t2 to t3, the terminal voltage UL value decreases gradually. According to Kirchhoff’s voltage law of the circuit, the polarization capacitance Cp partial voltage makes the terminal voltage decrease gradually. The reverse inductive polarization voltage Up increases gradually from 0 V. Similarly, it is regarded as the zero-input response process of Cp gradually decreasing from t4 to t5. As Up decreases gradually, UL increases and finally equals the open-circuit voltage. At last, the whole battery voltage-current curve of the pulse charge-discharge process and the first-order resistance-capacitance link to the corresponding model. It shows the battery polarization effect and model features on the association, so a simple first-order resistance-capacitance circuit is a zero-input state equation that can complete the established parameter identification. Similarly, when the conditions are gradually changed into the experimental environment, the corresponding model parameter values can be calculated one by one. When the obtained

2.6 Experimental analysis

83

conditional result as well as the independent and dependent variables are large, the variation rules can be obtained for the variables. To describe the change rule of the open-circuit voltage of the model, the open-circuit voltage is mainly affected by the battery state of charge and the temperature of the external characteristics. When the temperature is certain, the open-circuit voltage of the battery is tested for different state points. This is used for curve fitting to obtain the open-circuit voltage changing function. In the equivalent model, there are four model parameters to be identified: open-circuit voltage UOC, internal resistance R0, polarization resistance Rp, and polarization capacitance Cp. Under a certain aging degree and temperature, these four parameters are mainly affected by the state of charge. Through the hybrid pulse-power characteristic test, the curve relationship between the open-circuit voltage and the state of charge can be obtained easily. Then, the internal resistance R0 identification is carried out. In the hybrid pulse-power characteristic test, the discharge started at t1. At this time, the internal resistance R0 is obtained by dividing the voltage difference before and after the mutation by the discharging current. The identification of polarization resistance Rp is analyzed. In the t4 to t5 stage, the battery stands still for 40 s, where the current is zero. Wherein, UL is the voltage of the battery, through the voltage response curve, is the open-circuit voltage of the battery, according to the type of polarization voltage response curves can be obtained Eq. (2.62): UL ¼ UOC + Up (2.62) Up ¼ Up ð0Þet=τ wherein the end of the discharge voltage is used as the initial value of the polarized voltage. The polarization voltage response curves can be obtained by the least-square fitting considering the time parameters in the model. At the start of each hybrid pulse-power characteristic test, the battery is already shelved for a long time. Therefore, the polarization effect of the firstorder resistance-capacitance circuit disappears, observing the voltage response curve. The process of t2 to t3 can be taken as the zero-state response of the first-order circuit, and its expression can be obtained as shown in Eq. (2.63): UL ¼ UOC + IR0 + Up   (2.63) Up ¼ IRp 1  et=τ Using the same way into the time parameters of the model, the polarization resistance results from the curve fitting treatment. The first-order circuit time parameter is divided by the polarization resistance. Then, the polarization capacitance can be calculated. According to the above parameter identification method, the identified results can be obtained as shown in Table 2.4. The change rule of each parameter is obtained through the hybrid pulse-power characteristic test and the current change. Then, the change is obtained in the experimental results of the hybrid pulse-power characteristic test [42]. By analyzing the diagram, the conclusion can be obtained as follows. (1) U1–U2. The voltage transient is caused by the internal resistance R0, and the voltage drops from U1 to U2. (2) U2–U3. The polarization capacitance Cp in the resistance-capacitance network is charged. The voltage drops slowly, forming a zero-state response.

84

2. Electrical equivalent circuit modeling

TABLE 2.4 Model parameter identification results. S (1)

R0 (Ω)

Cp (F)

Rp (Ω)

E (V)

1.000000

0.0012

73,021.9735

0.0062532

4.18

0.897584

0.0012

38,779.0975

0.0016404

4.05

0.795148

0.0012

31,237.4083

0.0019628

3.94

0.692691

0.0016

38,306.3389

0.0016788

3.84

0.590254

0.0016

28,407.8836

0.0011348

3.74

0.487797

0.0012

77,466.2222

0.0017468

3.65

0.385340

0.0016

66,796.6261

0.0015792

3.62

0.282884

0.0012

48,324.2877

0.0017932

3.58

0.180447

0.0016

69,167.8689

0.0015940

3.52

0.077990

0.0016

22,738.2283

0.0002188

3.43

0.000000

0.0024

22,738.2283

0.0002188

3.26

(3) U3–U4. At this time, when it is still, the current changes to 0 suddenly, and the voltage drop on the internal resistance disappears. It makes the transient voltage rise. (4) U4–U5. The polarization capacitance Cp is discharged through the polarization resistance Rp. The voltage rises slowly, and is the zero-input response of the resistance-capacitance circuit. According to the above analysis, each parameter can be identified. The calculation formula for internal resistance is described in Eq. (2.64): R0 ¼ ΔU=I ¼ ðU1  U2 Þ=I

(2.64)

The calculation formula for polarization resistance is described in Eq. (2.65): Rp ¼ ΔU0 =I ¼ ðU5  U4 Þ=I

(2.65)

The zero-input response expression of the resistance-capacitance network is described in Eq. (2.66): Up ¼ U1 et=τ The calculation formula for open-circuit voltage is described in Eq. (2.67):  UOC ¼ U1  Up ¼ U1 1  et=τ The calculation formula for the time constant is described in Eq. (2.68):   U1  U4 τ ¼ ðt4  t3 Þ= ln U1  U3

(2.66)

(2.67)

(2.68)

85

2.6 Experimental analysis

2.6.3 Identified parameter variation The electrochemical knowledge is introduced into the data analysis software, which is used to calculate all parameters of the selected model under the varying current conditions as accurately as possible. The dependent and independent variable values are used in the calculation process according to the given system, which is the parameter identification process of the battery equivalent model. Based on the theoretical analysis, the experiments are carried out. Assuming that the simulation object is a linear equation with known coefficients, its difference equation can be obtained as shown in Eq. (2.69): Zðk + 2Þ ¼ 1:5Zðk + 1Þ  0:7ZðkÞ + uðk + 1Þ + 0:5uðk + 1Þ + vðkÞ

(2.69)

wherein the parameters to be identified are four coefficients and θ ¼ [a, b, c, d]T, where a ¼  1.5, b ¼ 0.7, c ¼ 1, and d ¼ 0.5. The least-square method is used to identify the four coefficients of the equation, and the parameter identification results are compared with known coefficients to verify the effect of the algorithm. The observation sequence is a pulse sequence generated by simulation. The selected forgetting factor is 0.98. The parameter identification result and estimation variance change are described in Fig. 2.18. The simulation results from the parameter b identification and the variation process of the estimated variance. The process of parameter c identification simulation results and estimation variance change. The simulation results from parameter d identification and the variation process of the estimated variance. As can be known from the experimental analysis, the change of parameter values would be more severe in the initial stage of parameter identification. Then, the variance change fluctuates greatly. It is caused by the difference between the selection of initial values of model parameters and the larger deviation from their initial values. As the identification time lengthens, the parameters change more smoothly and the variance change tends to be stable in the continuous iteration process. The parameter

0.0010 0.00108

Rp (W)

R0 (W)

0.0008 0.00104 0.00100

0.0006 0.0004

0.00096

0.0002 0.00092 0

2800

5600

(A)

8400

11200

0.0000

14000

(B)

t (s)

39000

33000

Up (V)

Cp (µF)

36000

30000 27000 24000 21000 2800

(C)

5600

8400

t (s)

11200

14000

0

2800

5600

8400

11200

14000

11200

14000

t (s)

0.010 0.005 0.000 -0.005 -0.010 -0.015 -0.020 -0.025 -0.030

(D)

0

2800

5600

8400

t (s)

FIG. 2.18 Parameter identification results and changing law. (A) Internal resistance. (B) Polarization resistance. (C) Polarization resistance. (D) Polarization voltage.

86

2. Electrical equivalent circuit modeling

0.3

4.2

0.0

4.0

-0.3

3.8

U(V)

I(A)

identification value at this time is more accurate. It can be seen that all parameters approach their measured values effectively. The working characteristics of lithium-ion batteries refer to their various dynamic performances in the working process, which are easy to change due to interference from external factors. Therefore, it is of great significance to fully understand the dynamic battery characteristics for the model construction. Only by fully considering and optimizing the interference caused by external factors can an accurate and reasonable battery model be built. The batteries also show internal resistance differences due to different internal material composition, environment, and aging degree. It is of great significance to study the various characteristics of the internal resistance for battery state estimation and modeling on the working process. In this section, three lithium-ion batteries are selected as experimental objects, and they are subjected to cyclic charge-discharge experiments at the indoor temperature of 23°C. The specific experimental steps are described as follows: (1) The battery is charged with constant-current and constant-voltage. The constant-current charging with the 1 C current rate is conducted with the cutoff voltage of 4.20 V, which is then treated with the constantvoltage charging with the cutoff current of 0.05 C. After charging is completed, shelve the battery to stabilize the battery voltage. Because the capacity of the selected lithium-ion battery is small, the shelved time is set to 30 min. (2) The appropriate state-of-charge sampling points can be selected as 1, 0.95, 0.9, 0.85, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.15, 0.1, and 0.05. Because the voltage changes in the initial and final stages of the battery discharge are obvious, the interval between sampling points is smaller to obtain a more accurate open-circuit voltage characteristic curve. (3) The constant-current discharge sampling is carried out on the battery. The battery is discharged at a constant current rate of 1 C with a discharge cutoff voltage of 2.75 V. When the discharge reaches the state of charge sampling point, the discharging is stopped. Then, the battery is shelved for 30 min. Then the battery continues discharging until it is 0 to end the experiment. The experimental voltage and current curves are described in Fig. 2.19. Wherein the battery terminal voltage suddenly drops or rises at the time point when it discharges or finishes discharging as it is connected with the load. This is because the internal resistance of the battery itself makes the voltage of the battery change drastically at the initial stage of discharging or at the initial stage of finishing discharging, which is called the internal resistance effect. The voltage decreases slowly after the first rapid drop in terminal voltage or rises slowly when the discharge ends, which is caused by the polarization effect. The

-0.6

3.6

-0.9

3.4

-1.2

3.2

-1.5

(A)

3.0 7000

10500

14000 t(s)

17500

21000

(B)

10000

15000

20000

25000

t(s)

FIG. 2.19 The voltage-current variation in the cyclic discharge experiment. (A) Current variation. (B) Corresponding voltage curve.

2.6 Experimental analysis

U1

U2

4.0

U (V )

UL(V)

4.2

3.8 3.6 3.4 0

(A)

10000

20000

30000 t(sec)

40000

50000

4.2 4.1 4.0 3.9 3.8 3.7 3.6 3.5 3.4

(B)

10

FIG. 2.20 Variation voltage curve in cyclic

8

discharge experiment. (A) Whole power-pulse test. (B) Voltage-current curve in charging.

6

I(A)

4.4

87

4 2 0.0

0.2

0.4

0.6

0.8

1.0

S (1)

polarization effect produces an increase in internal resistance. The internal polarization resistance of the battery is affected by its production process, internal structure, and working conditions. The discharging current and the working temperature of the battery have a particularly obvious influence on the polarization resistance, which is caused by its influence on the lithium-ion movement toward the battery working process. When the battery is shelved fully, its internal balance can be reached and the polarization effect disappears. Based on the equivalent circuit model and the offline identification results of the battery parameters, the comparison between the simulated battery terminal voltage value and the experimental voltage data is conducted on the condition that the cyclic discharge is shelved. The experimental results are obtained as shown in Fig. 2.20. Wherein the solid black line is the estimated value according to the constructed model while the dashed red line is the measured battery terminal voltage value. It can be seen from the figure that the overall estimated value has a good tracking effect on the measured value, and the estimated deviation is 0.10 V, which can represent the terminal voltage value of the battery operation. However, the deviation from the estimated voltage increases to the end of the discharge. By analyzing and comparing the voltage values, the prediction value can track the measured battery voltage accurately. The equivalent circuit model characterizes the change of polarization effect and the battery voltage after it stops discharging. The model describes the battery working characteristics effectively with the offline parameter identification results.

2.6.4 Pulse voltage tracking effect In the construction of the battery model, the parameter identification is of great significance. The accurate acquisition of model parameters is directly related to the accuracy of the battery dynamic characterization. Commonly used battery parameter identification methods can be divided into offline and online identification types. The online identification usually uses the input and output data onto the battery working process, then is combined with the least-square identification method to obtain the model parameter value corresponding to the present state. However, it increases the complexity of the program inevitably and is not suitable for some occasions that need the number of program operations. Due to the limitation of computational power, online identification can only use a simple and effective way to simulate the dynamic state of the lithium-ion battery. It has no physical significance and only requires high simulation accuracy. The offline parameter identification is mainly used by battery developers, making the model have a certain physical meaning that is used to study the relationship between the design and performance of lithium-ion batteries.

88

2. Electrical equivalent circuit modeling

Offline identification usually uses a pulse discharge method or other inputted signals as an excitation method to identify model parameters. The specific experimental steps are described as follows: (1) The battery is charged with a constant current at a rate of 1 C, and the charging cutoff voltage is set to 4.20 V. Then, it is converted to constant-voltage charging with a charging cutoff current of 0.02 C. It aims to ensure that the battery works in a saturated state. (2) To obtain a stable battery voltage, the battery is fully shelved after charging is completed. Because the capacity of the selected battery is small, the shelved time is set to 30 min. (3) It is subjected to constant-current discharge at a rate of 1 C for 10 s, and the battery is shelved for 40 s after the discharge is stopped. (4) It is charged with a constant current rate of 1 C for 10 s, and it is then shelved for 40 s after stopping charging. (5) The experimental steps (3)–(4) are a complete hybrid pulse-power characteristic test. The response battery characteristics are explored for the different state-of-charge values. Then, the battery is subjected to constant-current discharge at a 1 C rate for 6 min so that it is reduced by 0.1. A new state is obtained as a premise. After being shelved for 30 min, the hybrid pulse-power characteristic test is continued until the battery capacity is 0. Then, the experimental voltage and current curves can be obtained accordingly. It is described that every time the state of charge for the battery is reduced by 0.1, the hybrid pulse-power characteristic test is carried out. Thus, the dynamic response battery characteristics can be obtained under different state levels. The internal resistance is then identified accordingly. The analysis shows that the battery terminal voltage is suddenly reduced at the beginning of the discharge and the end of the discharge at t1–t2 together with t3–t4 due to the internal resistance of the battery. As a result, the internal resistance of the battery can be obtained as shown in Eq. (2.70): R0 ¼ jU2  U1 j=IL

(2.70)

wherein the corresponding charge-discharge current is described by the parameter IL, and the internal resistance of the different state-of-charge stages is calculated as shown in Table 2.5. The polarization capacitance and resistance can then be calculated. The battery stops discharging at the time point of t3, and the ohmic effect disappears. The battery voltage rises slowly in the phase of t4–t5. At this time, the zero-input effect of the resistance-capacitance loop in the circuit model is that it returns energy to the battery circuit after primary energy storage, resulting in a voltage rise. The battery terminal voltage can be expressed at this time point as shown in Eq. (2.71): U0 ¼ UOC  IL Rp et=τ

(2.71)

wherein Rp is polarization resistance. The time constant τ ¼ RpCp, in which Cp is the polarization capacitance. Then, the parameter fitting can be done, according to which Rp and Cp are obtained for different state-of-charge levels as shown in Table 2.6. TABLE 2.5 Ohm internal resistance corresponding to the state-of-charge variation (mΩ). S

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

R0

259.00

410.00

419.27

329.27

332.14

285.71

274.29

257.14

250.71

89

2.6 Experimental analysis

TABLE 2.6

Polarization resistance and capacitance to the state-of-charge variation.

S

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Rp(mΩ)

20.79

21.34

20.96

17.18

13.74

13.22

14.52

16.29

21.86

Cp(F)

272.95

297.70

346.78

484.11

526.51

564.88

454.05

351.95

200.69

Correspondingly, the identification is changed to first determine the model parameter values. This method is not conducive to real-time state acquisition of batteries, but it is often applied to engineering occasions due to its simple operation. The online identification of parameters increases the complexity of the algorithm. Consequently, the accuracy improvement is not obvious, so the offline identification method is chosen. At a room temperature of 23°C, hybrid pulse-power characterization is carried out on the battery, according to which the battery model parameters are obtained by studying and analyzing the battery working characteristics of its working process. To verify the built-in equivalent circuit model and the battery voltage characterization, the voltage and current can be introduced into the constructed battery equivalent model under the cyclic discharge hold condition. The model is verified by combining the previous parameter identification results. Using the ampere hour integral method, the current battery stateof-charge value is calculated. Combined with the nonlinear function relationship between the state of charge and UOC, the corresponding battery open-circuit voltage value can be further obtained. Substituting UOC into the equivalent model can calculate the battery terminal voltage value. The estimated value is compared with the terminal voltage value, according to which the comparison result and the corresponding error are described in Fig. 2.21. The comparison between the estimated value of the battery terminal voltage and the measured value of the battery discharges is conducted under the experimental conditions of the cyclic discharge hold, as shown in Fig. 2.21A. The black solid line is the estimated value based on the constructed model. The red dotted line is the battery terminal voltage value. The 4.4

U1

0.009

U2

0.006

4.0

Err(V)

UL(V)

4.2

3.8

0.003 0.000 -0.003

3.6

-0.006 3.4

(A)

-0.009 0

10000

20000

30000 t(sec)

40000

50000

4.2

0.010

3.8

0.005

Err(V)

UL(V)

20000

30000 t(s)

40000

50000

U2

4.0

3.6 3.4

0.000 -0.005

3.2

FIG. 2.21

10000

0.015

U1

(C)

0

(B)

-0.010 0

10000

20000

30000 t(sec)

40000

50000

60000

(D)

0

10000

20000

30000 40000 t(sec)

50000

60000

Pulse charge-discharge voltage traction results. (A) Pulse discharge voltage contrast curve. (B) Pulse discharge voltage traction error. (C) Pulse charge voltage contrast curve. (D) Pulse charge voltage traction error.

2. Electrical equivalent circuit modeling

Beijing bus dynamic street test operating current data. (A) Overall current variation. (B) Enlarged current variation.

I (A)

FIG. 2.22

20 10 0 -10 -20 -30 -40 -50 -60 -70

10 0

I (A )

90

-10 -20 -30 -40

0

(A)

2000

4000

6000

t (s)

8000

10000

12000

1200

(B)

1250

1300

1350

t (s)

1400

1450

variation in the simulation error can be obtained as shown in Fig. 2.21B. As can be known from the experimental results, the estimated value has a good tracking effect on the measured value, and the average estimated deviation is 0.10 V. It characterizes the battery voltage value of the working end effectively. As can be seen from the voltage comparison error, the voltage estimation deviation increases to the end of the battery discharge. In this aspect, the battery voltage changes drastically due to the end of the discharge, and the estimation error follows the effect of the simulation. It also shows that the equivalent circuit model has some shortcomings in characterizing the battery operating characteristics.

2.6.5 Modeling accuracy verification To verify the effect of parameter identification and the accuracy of the equivalent model, the circuit model of the ternary lithium-ion battery is built. The accuracy of the model is verified respectively in the hybrid pulse-power characteristic test, constant-current, and the Beijing bus dynamic street test conditions. The equivalent model can be constructed. In the experimental module, the input current condition of IL is initialized, and the state-of-charge values of each time point are calculated by the ampere hour time integration method. Then, it is input into the LOOKUP TABLE to complete the online identification of the four model parameters by the piece-wise linear interpolation method. Finally, the s-function is used to calculate and output the closed-circuit voltage UL at each time point. After that, it is compared with the experimental voltage corresponding to the experimental load current IL to analyze the model accuracy and complete the model verification. The operating current data in the Beijing bus dynamic street test can be obtained as shown in Fig. 2.22. The equivalent model can track the battery voltage in the cases of the simulation values with the same voltage of the current conditions. The voltage of the overall model can follow the change of the voltage accurately in each current mutation point. The simulated voltage has the same change trend with some mutations. The voltage variation deviation exists due to the internal resistance considering the state-of-charge effect, the charge-discharge current rate, and the temperature. The comparison between the simulation and terminal voltage curves is described in Fig. 2.23. Because of ignoring the factors such as charge-discharge current rate and temperature, the equivalent model parameter identification of internal resistance does not reflect the change of the battery voltage accurately. Consequently, it should be conducted by using the simulation model of zero-input and zero-state responses. The equivalent model uses the polarization capacitance together with resistance to the stage of the building model with a piece-wise linear interpolation method instead of the Cp and Rp. The high-precision equivalent model is established by using a relatively simple parameter identification method with the maximum

91

2.7 Conclusion 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

FIG. 2.23

0

2000

4000

6000

t (s)

8000

10000

Err1 Err2

Err(1)

S(1)

(A)

0.012 0.010 0.008 0.006 0.004 0.002 0.000 -0.002 -0.004

S1 S2 S3

12000

(B)

0

2000

4000

6000

t (s)

8000 10000 12000

State estimation results and comparison. (A) The state estimation result comparison. (B) Estimation error

analysis.

error of 0.08 V and an average error of 0.04 V. After obtaining the mathematical description of each parameter of the model, it is necessary to verify the accuracy of the model parameters [43]. The identified parameters are placed in the equivalent circuit model; the same current as the hybrid pulse-power characteristic test is input. The model output voltage response data and the voltage data are compared so that the modeling effect can be verified. The improved model is optimized according to the verification result. The largest block of the figure is the equivalent circuit model. The input includes the current I, internal resistance R0, polarization resistance Rp, polarization capacitance Cp, opencircuit voltage UOC, terminal voltage UL, and load current IL. Besides the current in the input parameters, the other parameters are the internal parameters of the model. They are functions of the state of charge as an independent variable. As it changes, the functional relationship is obtained from the identification of the previous section, which is the current change effect at any time. The real-time state change can be obtained from the ampere hour integration module in the figure. It is connected to the input end of the above-mentioned parameter function. Then, the model parameters are obtained corresponding to the constant change of the current input. Therefore, the control amount of the entire model is the input current value. The terminal voltage is taken as the model response to simulate the lithium-ion battery operation. In the circuit schematic of the model, each circuit component is a time-varying controllable parameter. The input current I acts on a controllable current source as a load. The output terminal voltage UL is obtained using a voltmeter. As shown on the right side of the figure, the validity and accuracy of the parameter identification can be evaluated in the model, when comparing the terminal voltage output of the model with the terminal voltage data onto the battery of the same input current. If the deviation is large, it means that the parameter identification is not accurate enough, or whether the accuracy of the model itself is considered to be defective. In the case of the battery reaction, the model can achieve an accuracy of more than 96% with the maximum error of 0.165 V and the maximum voltage of 4.20 V.

2.7 Conclusion The model output voltage is in good agreement with the measured value, which indicates that the model is equivalent. The rationality of the circuit model also proves the feasibility and reliability of this parameter identification method. There is no divergence in the whole

92

2. Electrical equivalent circuit modeling

process, and the error occurs in the pulse-power characteristic test stage because the battery inputted current suddenly causes the internal chemical reaction against the battery to agglomerate. The enhancement causes the terminal voltage to change rapidly. The model represents the response of the lithium-ion battery, and the error fluctuates within an acceptable reasonable range. Even in the final stage with a low state-of-charge level, the model has a good estimation effect.

Acknowledgments The work is supported by the National Natural Science Foundation of China (No. 61801407), the Sichuan Science and Technology Program (No. 2019YFG0427), the China Scholarship Council (No. 201908515099), and the Fund of Robot Technology Used for Special Environment Key Laboratory of Sichuan Province (No. 18kftk03).

Conflict of interest There is no conflict of interest.

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[15] Q. Zhang, et al., Electrochemical model of lithium-ion battery for wide frequency range applications, Electrochim. Acta 343 (2020) 136094–136114. [16] Q.F. Zheng, et al., A cyclic phosphate-based battery electrolyte for high voltage and safe operation, Nat. Energy 5 (4) (2020) 291–298. [17] C.W. Zheng, Z.Q. Chen, D.Y. Huang, Fault diagnosis of voltage sensor and current sensor for lithium-ion battery pack using hybrid system modeling and unscented particle filter, Energy 191 (2020) 116504–116517. [18] Z.Y. Zhang, et al., Active cell balancing of lithium-ion battery pack based on average state of charge, Int. J. Energy Res. 44 (2020) 2535–2548. [19] X. Zhang, et al., A novel quantitative electrochemical aging model considering side reactions for lithium-ion batteries, Electrochim. Acta 343 (2020) 136070–136084. [20] Q. Zhang, et al., Fractional calculus based modeling of open circuit voltage of lithium-ion batteries for electric vehicles, J. Energy Storage 27 (2020) 100945–100953. [21] Q.X. Yang, et al., State-of-health estimation of lithium-ion battery based on fractional impedance model and interval capacity, Int. J. Electr. Power Energy Syst. 119 (2020) 105883–105894. [22] L. Yang, et al., Supervisory long-term prediction of state of available power for lithium-ion batteries in electric vehicles, Appl. Energy 257 (2020) 114006–114024. [23] F.F. Yang, et al., State-of-charge estimation of lithium-ion batteries using LSTM and UKF, Energy 201 (2020) 117664–117676. [24] Z.W. Xue, et al., Remaining useful life prediction of lithium-ion batteries with adaptive unscented Kalman filter and optimized support vector regression, Neurocomputing 376 (2020) 95–102. [25] D.J. Xuan, et al., Real-time estimation of state-of-charge in lithium-ion batteries using improved central difference transform method, J. Clean. Prod. 252 (2020) 119787–119798. [26] X.M. Xu, et al., State estimation of lithium batteries for energy storage based on dual extended Kalman filter, Math. Probl. Eng. 2020 (2020) 1–11. [27] W. Xu, J.L. Xu, X.F. Yan, Lithium-ion battery state of charge and parameters joint estimation using cubature Kalman filter and particle filter, J. Power Electron. 20 (1) (2020) 292–307. [28] R. Xiong, et al., Online fault diagnosis of external short circuit for lithium-ion battery pack, IEEE Trans. Ind. Electron. 67 (2) (2020) 1081–1091. [29] S. Xie, et al., Influence of cycling aging and ambient pressure on the thermal safety features of lithium-ion battery, J. Power Sources 448 (2020) 227425–227433. [30] Z.B. Wei, et al., A noise-tolerant model parameterization method for lithium-ion battery management system, Appl. Energy 268 (2020) 114932–114944. [31] Y.J. Wang, et al., A fractional-order model-based state estimation approach for lithium-ion battery and ultracapacitor hybrid power source system considering load trajectory, J. Power Sources 449 (2020) 227543–227555. [32] W.W. Wang, et al., Safety performance and failure prediction model of cylindrical lithium-ion battery, J. Power Sources 451 (2020) 227755–227770. [33] S.L. Wang, et al., A novel power state evaluation method for the lithium battery packs based on the improved external measurable parameter coupling model, J. Clean. Prod. 242 (2020) 118506–118519. [34] S.L. Wang, et al., A novel safety assurance method based on the compound equivalent modeling and iterate reduce particle-adaptive Kalman filtering for the unmanned aerial vehicle lithium ion batteries, Energy Sci. Eng. 8 (2020) 1484–1500. [35] R.R. Wang, H.L. Feng, Lithium-ion batteries remaining useful life prediction using Wiener process and unscented particle filter, J. Power Electron. 20 (1) (2020) 270–278. [36] L.M. Wang, et al., Instantaneous estimation of internal temperature in lithium-ion battery by impedance measurement, Int. J. Energy Res. 44 (2020) 3082–3097. [37] L.B. Wang, et al., Deformation and failure behaviors of anode in lithium-ion batteries: model and mechanism, J. Power Sources 448 (2020) 227468–227476. [38] H.Y. Wang, et al., Remaining useful life prediction for Lithium-ion batteries using fractional Brownian motion and fruit-fly optimization algorithm, Measurement 161 (2020) 107904–107913. [39] L. Ungurean, M.V. Micea, G. Carstoiu, Online state of health prediction method for lithium-ion batteries, based on gated recurrent unit neural networks, Int. J. Energy Res. 44 (2020) 6767–6777. [40] N. Togasaki, et al., Prediction of overcharge-induced serious capacity fading in nickel cobalt aluminum oxide lithium-ion batteries using electrochemical impedance spectroscopy, J. Power Sources 461 (2020) 228168–228176.

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[41] Y. Tian, et al., A combined method for state-of-charge estimation for lithium-ion batteries using a long short-term memory network and an adaptive cubature Kalman filter, Appl. Energy 265 (2020) 114789–114803. [42] J.P. Tian, et al., Online simultaneous identification of parameters and order of a fractional order battery model, J. Clean. Prod. 247 (2020) 119147–119160. [43] G. Thenaisie, C.H. Park, S.G. Lee, A real-time entropy estimation algorithm for lithium batteries based on a combination of Kalman filter and nonlinear observer, IEEE Trans. Ind. Electron. 67 (9) (2020) 8034–8043.

C H A P T E R

3 Electrochemical Nernst modeling 3.1 Nernst modeling and improvement To reduce the computational complexity of the battery equivalent model and improve the simulation accuracy, the simplified empirical model is also widely used.

3.1.1 Model building process Lithium-ion batteries are used increasingly, and researchers have conducted a comprehensive study of the battery models based on the needs of the battery applications. Battery models can be divided into four categories: electrochemical, thermal, coupled, and performance. The first model is based on the electrochemical theory that describes the battery reaction process mathematically. The thermal model can describe the battery heating generation and transformation processes effectively [1–5]. The Shepherd model describes the battery electrochemical characteristics of the voltage and current effectively. After that, the simplified empirical equation model is also constructed [6–13]. Consequently, the battery state value can be obtained from a given closed-circuit voltage. The mathematical description is described as shown in Eq. (3.1). UL ðkÞ ¼ E0  R  I ðkÞ  K1 SðkÞ

(3.1)

Wherein UL(k) is the output voltage at the time point of k. E0 is the open-circuit voltage at S ¼ 100%. R is the internal resistance of the battery. K1 is the state-of-charge parameter that does not have physical meaning when the current is I. The proportional coefficient S(k) is the state value at the time point of k. Furthermore, the Nernst model is constructed by conducting the above research to describe the lithium-ion battery working characteristics accurately. The mathematical expression equation is described as shown in Eq. (3.2). UL ðkÞ ¼ E0  R  I ðkÞ  K2 ln ½SðkÞ + K3 ln ½1  SðkÞ

(3.2)

Among them, there is no physical sign of the state parameter proportional coefficient when K2 and K3 are current I. In summary, a new composite electrochemical polarization model is

Battery System Modeling https://doi.org/10.1016/B978-0-323-90472-8.00002-0

95

Copyright # 2021 Elsevier Inc. All rights reserved.

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3. Electrochemical Nernst modeling

constructed to refine the ohmic, polarization, and charge-discharge internal resistance characteristics of the Nernst model. It reflects the dynamic characteristics of the battery by conducting the step-by-step charge-discharge tests. The model structure is described in Fig. 3.1. Wherein R0 is the internal resistance of the battery. This parameter is used to characterize the transient voltage drop across the positive and negative terminals caused by the ohmic effect in the battery charge-discharge process. Rp characterizes its polarization resistance. Cp is hthe polarization capacitance. The parallel circuit composed of Rp and Cp reflects the generation and elimination process of the battery polarization effect. Rd is the discharge internal resistance to the discharging period, which characterizes the difference in internal resistance exhibited by the battery discharge. To simplify the description process of the state-space equation, Rcd is used to indicate the internal resistance Rc and Rd in the charge-discharge process. When the battery is in the discharging process, the value of Rcd is set to Rcd ¼ Rd, which is set to be Rcd ¼ Rd in the charging process. The open-circuit voltage expressed by UOC describes the state of charge accurately by using the Nernst model. Experiments show that the Nernst model can provide a better fitting effect of the whole charge-discharge process. The equivalent mechanism of each part in the model can be described as follows: (1) The electromotive force in the model is derived from the empirical equation. The value of the equation function is represented by UOC. The ohmic effect of the battery is characterized by the series of internal resistance R0. (2) The first-order resistance-capacitance parallel circuit is used to characterize the polarization effect. (3) The parallel circuit of the resistors Rd and Rc with reverse diodes can be improved and added to characterize the difference between the internal charge-discharge resistance. The accuracy of the description of the working state is further improved.

3.1.2 Parameter identification strategies The open-circuit voltage UOC is the stable voltage difference between the positive and negative electrodes of a battery in a long-shelved state. As can be known from the experimental results, the battery terminal voltage is equal to the open-circuit voltage after it is shelved for 1 h. Therefore, it is exiled from 6 min with a constant 1 C ratio, then shelved for 1 h [14–17]. At this time, the voltage at both ends of the battery corresponds to the open-circuit voltage of the battery of the state of charge. The pulse-current testing voltage variation can be obtained as shown in Fig. 3.2. Wherein the sudden changes in the battery terminal voltage at the start and stop points of discharge are caused by the internal resistance. Consequently, the internal resistance can be calculated through ohmic law as shown in Eq. (3.3). R0

Rp

Rd Rcd

UOC K0+K1LnS+K2Ln (1-S)

Rc

Cp I(t)

FIG. 3.1 Composite electrochemical polarization model structure.

UL

RL

97

3.1 Nernst modeling and improvement

3.675

3.80

3.670

UL (V)

UL (V)

3.75 t1(U1)

3.70

t5(U5) t4(U4)

3.65 3.60

3.665 3.660

t2(U2)

3.55

3.655

t3(U3) 1.1x10

6

1.1x10

6

6

1.2x10

6

1.2x10

1145900

t (ms)

1146600

1147300

1148000

t (ms)

(a)

(b)

FIG. 3.2 The pulse-current testing voltage variation. (A) Pulse charge-discharge curve. (B) Voltage recovery curve.

R0 ¼ ½ðU1  U2 Þ + ðU4  U3 Þ=2I

(3.3)

The improved model is a typical double-RC circuit model. In the hybrid pulse-power characteristic test of the lithium-ion battery, when the battery is in the stage of pulse discharge, the current direction of the modified model can be described by Ib as shown in Eq. (3.4). UL ¼ UOC ðSÞ  I ðtÞR0  Us

(3.4)

The current direction at this time point is set to be positive. Kirchhoff’s voltage law and Kirchhoff’s current law equations are listed according to the reference direction of voltage and current. The direction of loop winding is described as shown in Eq. (3.5). dUs I ðtÞ Us ¼  , dt Cs Rs Cs

dUL I ðtÞ UL ¼  dt CL RL CL

(3.5)

Wherein the terminal voltage of the parallel circuit is described by Us composed of Rs and Cs. UL is the terminal voltage of the parallel circuit composed of RL and CL. According to the current and voltage curves of the hybrid pulse-power characteristic test, the battery is first exiled by the constant-current treatment for 10 s of t1–t3 and shelved for the rest of the time. By timedomain analysis of two series-connected resistance-capacitance circuits, the terminal voltage of the parallel circuit of the resistance-capacitance network can be obtained, as shown in Eq. (3.6). 
 Rs I ðtÞ 1  eðtt1 Þ=τs t2 < t < t3 (3.6) Us ¼ Us ðt2 Þeðtt4 Þ=τs t4 < t < t5 Wherein τs is used for the time constant of the parallel connection circuit of Rs and Cs. By the time-domain analysis of two series-connected resistance-capacitance circuits, the terminal voltage of the parallel circuit of the resistance-capacitance network can be obtained, as shown in (3.7). 
 RL I ðtÞ 1  eðtt2 Þ=τs , t2 < t < t3 (3.7) UL ¼ UL ðt2 Þeðtt4 Þ=τL , t4 < t < t5

98

3. Electrochemical Nernst modeling

Then, τL is used for RL and the time constant for the parallel connection circuit of CL. Then, the calculation processes can be obtained as shown in Eq. (3.8). τs ¼ Rs Cs ,τL ¼ RL CL

(3.8)

In the discharge process, the polarized capacitances of Cs and CL are charged, and the voltage of the resistance-capacitance parallel circuit increases exponentially. After the battery enters the shelved state from the state of charge, the capacitance Cs and CL discharge to the resistance in parallel, respectively. The voltage decreases exponentially. The values of R and C in the model are related to the battery state of charge [18, 19]. It is used to curve fit the data obtained in the hybrid pulse-power characteristic test. The values of Rs, RL, Cs, and CL in the improved model can be obtained by using the undetermined parameters. The specific methods are described as follows. First, the battery terminal voltage at different state-of-charge values and the same stage of U4–U5 can be taken out from the hybrid pulsepower characteristic experimental data. When the state-of-charge value is 0.4, its terminal voltage curve is described as shown in Fig. 3.3. According to the characteristics, the change of terminal voltage at this stage is the discharge process of polarization capacitance to polarization resistance and the zero-input response of the double resistance-capacitance loop. In this circuit stage, Ib is 0. Therefore, the output equation of the battery terminal voltage can be obtained in this process, as shown in Eq. (3.9). UL ¼ UOC  IRs et=τs  IRL et=τL  IR0

(3.9)

To simplify the parameter identification process, the equation shown in Eq. (3.9) is replaced with coefficients as shown in Eq. (3.10). UL ¼ f  aect  bedt

(3.10)

U (V)

Among them, the parameters are taken as undetermined for f, a, b, c, and d. The experimental data are used to conduct the curve fitting effect of the single exponential and double exponential with Eq. (3.10) as the objective. The terminal voltage fitting and its error curves are obtained for the polarization reaction disappearance process. The maximum single exponential fitting error is up to 0.01 V while the double exponential fitting error is less than 0.003 V. According to the fitting results of the double exponential curve, each parametric equation can 4.2 4.0 3.8 3.6 3.4 3.2 3.0 2.8 2.6 2.4

LiMnO4 LiFePO4

0

20

40

60

S (%) FIG. 3.3 Different battery terminal voltage variation and comparison.

80

100

3.1 Nernst modeling and improvement

99

be obtained by comparing Eq. (3.9) and Eq. (3.10). The mathematical calculation process can be obtained as shown in Eq. (3.11).  UOC  IR0 ¼ f ,Rs ¼ a=I, RL ¼ b=I (3.11) Cs ¼ 1=ðRs cÞ, CL ¼ 1=ðRL dÞ Therefore, the double exponential fitting can achieve a better fitting effect than a single exponential fitting. The parameters in the improved model are identified by the double exponential fitting. Cb in the modified model is used to represent the change of open-circuit voltage caused by the load current change. Its addiction treatment enables the model to characterize the steady-state lithium-ion battery characteristics well, and the calculation process is described as shown in Eq. (3.12). ð U100%S udu ¼ Qn UOC (3.12) W Cb ¼ Cb U0%S

Wherein W_Cb is the electric quantity stored by the energy storage capacitance Cb. The opencircuit voltage of the battery is described by UOC corresponding to the different state-ofcharge values. Qn is the rated capacity of the battery.

3.1.3 State-space description The open-circuit voltage UOC has a mathematical function of the battery state of charge that is initialed as UOC ¼ f(S). After measuring the specific data onto the open-circuit voltage corresponding to the battery state of charge, UOC and S are performed by the composite model of polynomial fitting. This method solves the core problem, in which the small variation of the high-order polynomial fitting coefficient brings a large estimation error. The fitting coefficient is small and the initial value is easy to be determined with good practicability [20–30]. According to the special working conditions of the lithium-ion battery drone, the three constants K1, K2, and K3 can fit the data well through different combinations of tests. The structural analysis of the circuit is based on Kirchhoff’s circuit law, and the state-space equations are constructed together with its fitting expression as shown in Eq. (3.13). 8 > < UL ¼ UOC  Up ðkÞ  IL ðkÞR0  Rcd dUp ðkÞ=dk ¼ IL ðkÞ=Cp  Up ðkÞ=Cp Rp (3.13) > : UOC ¼ K1 + K2 + ln ½SðkÞ + K3 ln ½1  SðkÞ Wherein k is the time point, at which the state-of-charge estimation is located for the battery drone. UL(k) is the battery closed-circuit voltage value at the time point of k. R0 is the internal resistance of the battery drone, and IL(k) is the output current of the battery. Up (k) is the voltage value of the polarization resistor at the time point k. Rp is the polarization resistance. Cp is the polarization capacitance. The ideal voltage source is equivalent to the Nernst model, and its parameter value is represented by UOC, which characterizes the open-circuit voltage of the battery. Meanwhile, the resistance ohm R0 is used to characterize the internal resistance of the battery. The parallel circuit of Rp and Cp can reflect the generation of polarization, which greatly

100

3. Electrochemical Nernst modeling

reduces the polarization process of the battery [31]. The above state-space equations are analyzed, and the first-order backward difference is substituted for the differentiation of Up(k) in Eq. (3.13). The discretized equivalent differential expression is obtained as shown in Eq. (3.14). dUp ðkÞ Up ðkT Þ  Up ½ðk  1ÞT  Up ðkÞ  Up ðk  1Þ ¼  dk T T

(3.14)

Wherein the voltage values across the polarization resistance to the time points k and k-1 can be described by Up(k) and Up(k-1), respectively. T is the battery parameter measurement period, the signal sampling time interval. It is determined by the current versus time integration method. As by the Coulomb efficiency, the ratio of the discharge capacity is to the charge capacity for one charge-discharge process and its value is defaulted to be 1.00. Inserting Eq. (3.14) into Eq. (3.13), the calculation method of obtaining Up(k) is described as shown in Eq. (3.15). 
 (3.15) Up ðkÞ ¼ Cp Rp UL ðkÞ  UL ðk  1Þ + ðR0 + Rcd Þ IL ðkÞ  IL ðk  1Þ + Rp IL ðkÞT =T To obtain the calculation expression of UOC, Up(k) is inserted from Eq. (3.15) into Eq. (3.13). Then, UOC is replaced from the state equation, whose results are transformed into a simplified discrete form as shown in Eq. (3.16). ( UOC ðkÞ ¼ K0 + K1 ln SðkÞ + K2 ln f1  SðkÞg (3.16) UL ðkÞ ¼ a1 + a2 UL ðk  1Þ + a3 ln ½SðkÞ + a4 ln ½1  SðkÞ + a5 IL ðkÞ + a6 IL ðk  1Þ Wherein the coefficients can be described in the discrete state-space equation by a1, a2, …, and a6. Their values can be solved by the least-square parameter identification. The expanded forms are described as shown in Eq. (3.17). 8 Cp Rp TK1 TK2 TK3 > > , a2 ¼ ,a3 ¼ , a4 ¼ , > a1 ¼ > < T + Cp Rp T + Cp Rp T + Cp Rp T + Cp Rp (3.17)   > > Cp Rp ðR0 + Rcd Þ + T R0 + Rp + Rcd Cp Rp ðR0 + Rcd Þ > > a5 ¼ , a6 ¼ : T + Cp Rp T + Cp Rp Wherein T is the battery parameter measurement period, in which the sampling time of the experiment is 1 s. At this time, the parameter calculation of the model is described as shown in Eq. (3.18). 8 a1 a3 a4 > > K ¼ , K2 ¼ , K3 ¼ , > < 1 1  a2 1  a2 1  a2 (3.18) > a6 a2 a5 + a6 a22 > > ¼  R , R ¼ , C ¼  R 0 p p : cd a2 a2 a5 + a6 a22  a2 In the battery experiment process, the internal resistance of the battery is measured by the tester AT520B. The measurement range of the internal resistance test equipment is 0.01 mΩ–300.00 Ω and the accuracy is 0.50%. The charge-discharge internal resistance Rcd in the model can be obtained by combining the parameter identification results of

3.1 Nernst modeling and improvement

101

Eq. (3.18). Using the state space equation in the discrete form of the model, a group of N input-output six-tuples can be obtained as shown in Eq. (3.19). fUL ðkÞUL ðk  1ÞSðkÞSðk  1ÞIL ðkÞIL ðk  1Þg

(3.19)

Using the least-square estimation, the parameters in the model are solved in a closed form. The specific methods can be described as follows. First, the dependent variable vector Y can be designed as shown in Eq. (3.20). Y ¼ ½UL ð1ÞUL ð2Þ⋯UL ðN ÞT

(3.20)

Second, the independent variable matrix X can be obtained as shown in Eq. (3.21). X ¼ ½βð1Þβð2Þ⋯βðN ÞT

(3.21)

Wherein the single parameter can be defined as shown in Eq. (3.22). βðkÞ ¼ ½1UL ðk  1Þ ln ½SðkÞ ln ½1  SðkÞIL ðkÞIL ðk  1Þ

(3.22)

Finally, the coefficient matrix A can be initialed as shown in Eq. (3.23). A ¼ ½a1 a2 a3 a4 a5 a6 T

(3.23)

The coefficient matrix A is the parameter that should be obtained. By using the least-square estimation theory and combining the dependent variable matrix Y as well as the independent variable matrix X, the coefficient matrix A can be calculated as shown in Eq. (3.24).  1 (3.24) A ¼ XT X XT Y ¼ X1 Y The relevant experimental data are used to determine all the parameters in the composite electrochemical polarization model. Then, the quadratic matrix is used to discretize the spatial equation in the continuous state, so that the system can converge fast after the algorithm engine is introduced, preventing abnormal data from causing model interference. The statespace equations are described as shown in Eq. (3.25). 8 2 R +R +R 3 R1 1 S e > " # " # >   > > 6 τ1ðRs + ReÞ τ1ðRs + Re Þ 7 I U1 > > 7 ¼6 > > 4 < R2 R2 + Rs + Re 5 Uoc U2   (3.25) τ2 ðRs + Re Þ τ2 ðRs + ReÞ > > " # >

> > U1 > Rs Rs RsRe Rs > >  I+ Uoc  U ¼  : Rs + Re Rs + Re U2 Rs + Re Rs + Re Wherein τ1 is the time constant of R1 and C1. τ2 is the time constant of R2 and C2. In the control theory of the system, the mapping relationship from the S domain to the Z domain is the discretization basis of the general system. Because of its simple structure, only a mathematical model can be built. It needs to be transformed from the time domain when used. To reduce the computational complexity, the state-space equation of the continuous system is discretized by the quadratic matrix in this research, which makes it easier to be realized through the experimental platform. The general state equation of the continuous system is described as shown in Eq. (3.26). ^ ðtÞ ¼ FðtÞXðtÞ + BðtÞUðtÞ + GðtÞW ðtÞ X

(3.26)

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3. Electrochemical Nernst modeling

Wherein the state vector is the X(t) and its derivative concerning time expression is characterized by Xk. The time input vector is described by U(t), which is the coefficient matrix of the differential state equation. B(t) is the input control matrix and G(t) means the noise distribution matrix. Then, the mathematical expression of the required parameters can be obtained as shown in Eq. (3.27). 8 N X > 1 > > ϕk + 1 ¼ Fðtk Þt1 Δtt1 > > N > > i¼1 > > < N X 1 (3.27) Fðtk Þt1 Bðtk ÞΔtt1 Γk + 1 ¼ > N > I¼1 > > > N X > 1 > > > γ ¼ Fðtk Þt1  Gðtk ÞΔtt1 : k + 1κ N i¼1

Through the discrete transformation of the obtained equations, the discrete form of the equivalent model circuit can be obtained as shown in Eq. (3.28). 2 3 2 32 3 2 3 ηΔt=C 0 Sk + 1 1 0 0 Sk 4 U1k + 1 5 ¼ 4 0 a b 54 U1k 5 + 4 e (3.28) f 5½ ik Uock T + wk g 0 c d U2k + 1 U2k d In the equation, all the small parameters can be calculated as shown in Eq. (3.29).



8 e f a b > > , Γk + 1¼ < Φk + 1 ¼ g h c d (3.29) > > : Uk + 1 ¼  Rs U1κ  Rs U2κ  Rs Re Iκ + Rs Uocκ Rs + Re Rs + Re Rs + Re Rs + Re Wherein the state-space variable is Xk. The control variable is Ik. The observation variable is W. The mathematical expression is described as shown in Eq. (3.30).
T (3.30) Xk ¼ Sk , Up1,k , Up2, k , Wk ¼ ½W1, k , W2, k , W2, k T The parameters refer to the battery state of charge and the terminal voltage of two resistance-capacitance parallel circuits for the sampling time point k, respectively.

3.2 Modeling realization 3.2.1 Simulation modeling structure The traditional lithium-ion battery simulation system model is given priority with code; the pure circuit way is used to improve the traditional battery simulation model. According to the modified equivalent circuit modeling structure of parameter validation, improved concrete modeling structure can be realized in the establishment system. The improved simulation model mainly includes three subsystems, including the state-of-charge updating, parameter updating, and terminal-voltage outputting subsystems. Among them, the battery state updating subsystem is calculated by ampere hour integral treatment. The model parameter updating subsystem is composed of six fitting polynomial functions of R0 (S),

103

3.2 Modeling realization

Gathering module Charger module Li-ion battery

Discharge module

Processing unit

Upper computer

Human-computer interface

FIG. 3.4 The charge-discharge control system structure.

Rs (S), Cs (S), RL (S), and CL (S). Consequently, the mathematical relationship can be obtained for the open-circuit voltage toward the state of charge. The terminal voltage output subsystem of the battery consists of an improved circuit model. The structure of the battery terminal voltage output subsystem is designed. Besides the three subsystems, there are two signal builder modules in the entire simulation system model: the current module and the voltage module. The current module is used as the only input of the whole simulation system. It stores the charge-discharge current data onto the hybrid pulse-power characteristic test. The voltage module is the experimental data onto the battery test. It is used to show the comparison between the simulated and measured values of the parameter verification process visually. The overall hardware design framework is based on the design charge-discharge system. The buck and rectification effects are analyzed in detail through the implementation of the battery charge-discharge scheme. The conclusion of the working battery characteristics is obtained. The feasibility of the inspection scheme is tested for physical test accuracy. The charge-discharge system control structure is described in Fig. 3.4. The discharge module is designed to be shunted in parallel with four 11-Ω resistors. By changing the number of resistors added to the main circuit to adjust the amount of discharge time, it is easy to save time; the discharging current rate can be controlled by adding a resistor. If the discharge ratio is 1 C when a resistor is connected, the two resistors are connected parallel to 2 C. The three resistors are connected parallel to 3 C, in which the stable operation state of the single-chip microcomputer is above 3.24 V. Then, a more accurate method is realized for the state-of-charge estimation. It is adopted in the freedom car battery experiment manual as shown in Eq. (3.31). S ¼ Qc =QI

(3.31)

Wherein the present voltage can be expressed by U and S should be at most 1, so the ratio of Qc to QI is the present state-of-charge value. The error is found to be about 0.10 V by measurement. The physical object can complete the estimation display of the battery state of charge and the current-voltage measurement display substantially.

3.2.2 Characteristic description The experiment uses a mixed pulse-power performance test to record the battery state value and the open-circuit voltage of the analog circuit. The battery model is verified and analyzed by conventional charging and discharging tests. Then, the internal parameters of the circuit are obtained by using the state equation of the equivalent analog circuit [32–35].

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3. Electrochemical Nernst modeling

P2 P3

P1 S2 P4 S3

S1

0

500

1000

1500

2000

2500

3000

P2 P3

4.2 4.0 3.8

P1 S2 P4 S3

S1

3.6 3.4

0

3500

500

1000 1500 2000 2500 3000 3500

t (10s)

t (10s)

(a)

31 30 29 28 27 26 25 24 23 22 21 20

U (V)

31 30 29 28 27 26 25 24 23 22 21 20

U (V)

U (V)

The hybrid pulse-power characteristic experimental process is described as follows. First, the battery is discharged for 10 s. Then, it is recharged for 10 s after it is charged for 40 s. After that, the battery is charged again with 40 s. The whole process can be realized by the intermittent constant current discharge rate of 1 C for the battery. In the cycle test, the battery is subjected to a composite pulse experiment at an equally spaced state point, and the state of charge is 0.1, 0.2, …, 0.9. The experimental interval between adjacent pulses is 1 h. The battery is discharged at a 1 C current for 6 min, in which the state value decreases by 0.1. In the single hybrid pulse-power characteristic cyclic test, the current input and voltage response curves are described as shown in Fig. 3.5. Wherein the jumped 10 s discharge is the result of the ohmic resistance balance voltage, which is an important part of the experimental investigation. The hybrid pulse-power characteristic test is carried out on the model. Through the analysis of the experimental results, the voltage parameter value of the battery equivalent circuit can be obtained at a certain state-ofcharge node. Its relationship with UOC can be obtained accordingly. Then, the characteristic equations are used to derive various parameters in the model. The charging mode is set as constant-current and constant-voltage first. The experiment is conducted with the charging voltage of 4.20 V and the nominal charging current of 0.25 C to 1 C for the 18,650 lithium-ion batteries. The charging current and voltage curves are described in Fig. 3.6.

(b)

P3

P2

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U (V)

3.90

P1

3.75 3.60

S2

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P4

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S3

3.30 3.15 0

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(b)

4.2 4.1 4.0 3.9 3.8 3.7 3.6 3.5 3.4 0.0

10 8 6

I(A)

4.20

U (V)

FIG. 3.5 Packing current input and voltage response variation. (A) Charge-discharge voltage characteristics. (B) Packing-cell voltage comparison.

4 2 0.2

0.4

0.6

0.8

1.0

S (1)

FIG. 3.6 Charging process using the constant-current to constant-voltage method. (A) Varying cell-to-cell voltage changes. (B) Charging voltage and current variation.

105

29 28 27 26 25 24 23 22 21 1.0

4.2

FDRV

UL_DisCha_1C

1.0C 0.5C 0.3C 0.2C

4.0

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3.8

U (V)

U (V)

3.2 Modeling realization

FDRV

3.6 3.4 3.2 3.0

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0.0

0

(a)

1

2

3

4

5

6

7

t (h)

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(b)

FIG. 3.7 Discharge voltage variation on different magnifications. (A) Voltage characteristics for 1 C rate. (B) Voltage variation for different currents.

The battery discharging experiment is carried out with different discharging current rates. The voltage variation at each discharging current rate for battery packs and cells can be described as shown in Fig. 3.7. The observation curve can be obtained accordingly. When the discharging current rate is larger, the discharge amount is less. Then, the discharging current rate is getting faster. As the state of charge decreases, the discharge speed is accelerated. The calibration experiment of open-circuit voltage and state of charge is the intermittent discharge experiment. When it is released for 0.1, it should be shelved for 45 min; the next experimental test follows. The battery voltage in this state is taken as the open-circuit voltage. And finally, the parameter identification can be realized. The relationship between the open-circuit voltage and the state of charge for both battery cells and packs is described in Fig. 3.8. Under the same state condition, the magnitude of the voltage drop of the battery increases significantly as the discharging current rate increases. Due to the constraints on various aspects, this is not absolute and the general direction can be judged accordingly. The state of charge is between 0.8 and 0.2, in which the voltage value is relatively stable. When the current continues to increase or decrease, the line moves closer to the two levels gradually. This is one of the important battery charge-discharge characteristics.

29 28

U (V)

U (V)

27

UOC

26 25 24 23 22 0.0

(a)

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0.4

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S (1)

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4.152 4.150 4.148 4.146 4.144 4.142 4.140 4.138 4.136

CV_1 CV_2 CV_3 CV_4 CV_5 CV_6 CV_7

0

(b)

50

100

150

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250

300

350

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t (10s)

FIG. 3.8 Open-circuit voltage variations. (A) Voltage variation to the battery state. (B) Varying voltage toward the shelved time.

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3. Electrochemical Nernst modeling

3.2.3 Testing procedure design The hybrid pulse-power characteristic test is a battery performance test method that is described in the freedom car battery test manual for power-assist hybrid electric vehicles. Through hybrid pulse-power characteristic tests, effective identification can be performed for the model parameters. The hybrid pulse-power characteristic experimental process is described as follows. First, the lithium-ion battery is discharged at 1 C for 10 s. It is then charged with 0.75 C for 10 s after being shelved for 40 s. Then, the lithium-ion battery is discharged at 1 C for 6 min after being shelved for 40 s. It is then shelved for 1 h to continue cycling. In the cyclic test, the battery is subjected to a pulse-power characteristic test of an equally spaced state point, in which the state of charge varies from 0 to 1. At equally spaced points, the battery needs to be left for a longer period between adjacent pulse tests to restore the battery to electrochemical and thermal equilibrium. The entire experiment primarily consists of a single repeated charge-discharge pulse test, according to which the hybrid pulse-power characteristic test procedure is designed as shown in Fig. 3.9. All the model parameters can be obtained through the hybrid pulse-power experimental test, but it takes a long time, around 12 h. It requires great consumption of the experimental equipment and operators. Most of the time of the hybrid pulse-power characteristic test is spent on the repeated shelved stage. This experiment uses two test channels to solve the long-time testing problem. Two lithium-ion batteries of similar capacity are selected to conduct the hybrid pulsepower characteristic test at the same time, making batteries of similar capacity for the same batch [15–18, 36–38]. This is used to ensure the consistency of the quantities that affect the experimental results, except for the experimental quantities. Each channel has only five cycles for the pulse-power characteristic test. The first channel has a pulse-power characteristic test

Start CC-CV charging 4.2 V/1.5 A(1 C) N I ð , k Þ  f X , k + X  X f X > k k k k < ∂Xk Xk ¼X^ k  (4.20)   ∂hðX , kÞ   > k >  ^ ^ > Xk  X k : hðXk , kÞ  h Xk , k + ∂Xk Xk ¼X^ k The new state transition matrix and observation driving matrix can be obtained by using Taylor series expansion and ignoring the high-order term, to update the linear space equation. Based on the space equation of the linear system, the principle of extended Kalman filtering is introduced. As for Eq. (4.20), the coefficients are described by using Eq. (4.21):  8   ∂f ðXk , kÞ > ^k ^ k , k  Ak X > ¼ , Bk ¼ f X A > k  < ∂Xk Xk ¼X^ k  (4.21)   > ∂hðXk , kÞ > ^ ^ > , Dk ¼ h X k , k  Ck X k : Ck ¼ ∂Xk xk ¼X^ k This method is based on the system state and measurement equations. The prediction state equation includes the ampere hour integral method of the state calculation. The observation equation represents the equivalent model of the lithium-ion batteries. The state estimation accuracy of the extended Kalman filtering algorithm largely depends on the equivalent model, so it is very important to establish an appropriate lithium-ion battery equivalent model. Then, Eq. (4.20) can be discretized as shown in Eq. (4.22): ( Xk + 1 ¼ Ak Xk + Bk Zk + Wk (4.22) Yk ¼ Ck Xk + Dk + V k The state-space equation is shaped and contains the same matrix meaning, in which Dk is also the system observation matrix. The recursive process is obtained by applying the

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4. Battery state estimation methods

equation to the discretized model. The initial filtering state and its variance can be described as shown in Eq. (4.23):  X ð 0Þ ¼ E ½ X ð 0Þ  (4.23) Pð0Þ ¼ Var½Xð0Þ The system state equation is used to obtain the estimated value of the state variable and the average square error at the time point of k + 1. The status and average square error of the present time are obtained as shown in Eq. (4.24): 8  

< yi ¼ f netj ðj ¼ 1, 2, 3, …, mÞ n X ¼ wij xi ðj ¼ 1, 2, 3, …, lÞ net > j : i¼0

V4

x1

Wn

x2

xn

Output layer Input layer Hidden layer

FIG. 4.8 The three-layer neural network structure diagram.

(4.28)

4.4 Machine learning algorithms

151

When the output of the network is different from the expected output, the error can be generated that is defined in Eq. (4.29): l 1 1X E ¼ ðd  oÞ2 ¼ ð dk  o k Þ 2 2 2 k¼1

(4.29)

The training process of the calculation algorithm is modifying the weight reduction error of the network continuously. The gradient descent method is used to update the parameters. Bringing Eq. (4.27) into the above equation gives the relationship between the hidden layer and the input error, as shown in Eq. (4.30): 2 0 13 2 l l m X X 1X 1 4 dk  f @ ðdk  f ðnetk ÞÞ2 ¼ wjk yi A5 (4.30) E¼ 2 k¼1 2 k¼1 j¼0 If the partial derivative is bigger than 0, it should change according to the opposite direction of the partial derivative. If the partial derivative is less than 0, this direction change can be then calculated accordingly. Further, each variable is brought into the above equation and simplified, according to which the functional relationship is obtained, as shown in Eq. (4.31): 8 2 2 0 13 2 !392 = l m l < m n X X X X   1X 1 4dk  f @ wjk f netj A5 ¼ dk  f 4 wjk f wij xi 5 (4.31) E¼ ; 2 k¼1 2 k¼1 : j¼0 j¼0 i¼0 As can be known from Eq. (4.31), the error of the network is closely related to the weight of each layer. Meanwhile, a learning rate η is set, in which the learning rate cannot be too fast nor too slow. Too fast may lead to crossing the optimal solution. The algorithm efficiency may be reduced if it is too slow. Its parameter expression is described as shown in Eq. (4.32): 8 ∂E > > > < Δwjk ¼ η ∂w ðj ¼ 1, 2, 3, …, lÞ jk (4.32) ∂E > > > Δvij ¼ η ði ¼ 1, 2, 3, …, n; j ¼ 1, 2, 3, …, mÞ : ∂wij wherein the negative sign indicates a gradient decrease in which η is a constant. It reflects the learning rate varying from 0 to 1.

4.4.4 Deep learning for life prediction The remaining useful life is defined as the time when equipment performance degrades to the failure threshold for the first time or the first arrival time [53–59]. The remaining-usefullife prediction methods of batteries can be divided into three categories: model-based methods, data-driven methods, and hybrid methods. Model-based methods aim at relating the observable quantities of the indicators with interest by building either a detailed electrochemical model of the degradation processes affecting the battery life or equivalent electrical circuits of the battery. The data-driven methods aim at mapping the relationships between the accessible observations and the hidden indicators by some approximating, general model

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adaptively built based on available data. Hybrid methods aim at combining two or more model-based or data-driven methods to improve the prediction performance. Deep learning is playing an increasingly important role in our lives. It has already made a huge impact on several areas, such as cancer diagnosis, precision medicine, self-driving cars, predictive forecasting, and speech recognition. The painstakingly handcrafted feature extractors used in traditional learning, classification, and pattern recognition systems are not scalable for large datasets. In many cases, depending on the problem complexity, it can also overcome the limitations of earlier shallow networks. Consequently, it prevents efficient training and hierarchical representations of multidimensional training data. The deep neural network uses multiple deep layers of units with highly optimized algorithms and architectures. It is perhaps the most significant development in the field of computer science in recent times, the impact of which has been felt in nearly all scientific fields. It has already disrupted and transformed businesses and industries, making it a race among the world-leading economies and technology companies for it. There are already many areas where it has exceeded human-level capability and performance, such as predicting movie ratings, decisions to approve loan applications, and time taken by car delivery. It has the potential to improve human lives. Deep learning research has many fields, including deep learning network architectures, deep learning algorithms, optimization, and the latest implementations and applications. Machine learning is witnessing its golden era as deep learning slowly becomes the leader in this domain. This uses multiple layers to represent the abstractions of data and build computational models. It includes generative adversarial networks, convolutional neural networks, and model transfers, which have completely changed our perception of information processing. However, there exists an aperture of understanding behind this tremendously fast-paced domain because it was never previously represented from a multiscope perspective. The lack of core understanding renders these powerful methods as black-box machines that inhibit development at a fundamental level. Moreover, deep learning has repeatedly been perceived as a silver bullet to all stumbling blocks in machine learning. In recent years, machine learning has become more and more popular and has been incorporated into a large number of applications, including multimedia concept retrieval, classification, recommendation, and network analysis. Among various machine learning algorithms, deep learning is also known as representation learning, and is widely used in these applications. The explosive growth and availability of data and the remarkable advancement in hardware technologies have led to the emergence of new studies in distributed deep learning, which has its roots in conventional neural networks and significantly outperforms its predecessors. Deep learning utilizes graph technologies with transformations among neurons to develop many-layered learning models. Many of the latest deep learning techniques have been presented that demonstrate promising results. Traditionally, the efficiency of machine learning algorithms highly relied on the goodness of the representation of the input data. A bad data representation often leads to lower performance compared to a good data representation. Therefore, feature engineering has been an important research direction in machine learning for a long time. It focuses on building features of raw data and has led to lots of research studies. Furthermore, many different kinds of features have been proposed and compared at the present stage. However, once a new feature with better performance is proposed, it will become a trend in years to come.

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4.4 Machine learning algorithms

Comparatively, the deep learning algorithm performs feature extraction in an automated way, which allows researchers to extract discriminative features of minimal domain knowledge and human effort. These algorithms include a layered architecture of data representation, where the high-level features can be extracted from the last layers of the networks while the low-level features are extracted from the lower layers. These kinds of architectures were originally inspired by artificial intelligence simulating its process of the key sensorial areas in the human brain. With great success in many fields, it is now one of the hottest research directions in machine learning. An overview of deep learning can be obtained from different perspectives, including history, challenges, opportunities, algorithms, frameworks, applications, and parallel and distributed computing techniques. Consequently, machine learning is widely used for remaining-useful-life estimation. Its whole structure is shown in Fig. 4.9. Deep learning is considered a huge research field. The topmost challenge is to train the massive datasets available. As the datasets become bigger, more diverse, and more complex, the deep learning algorithm has been in its path to be a critical tool for catering the big data analysis. Challenges and opportunities in key areas are raised that require first-priority

CYBE R SYSTEM: CLOUD-BA SED BA TTER Y MODELING AND MONITORING Battery database Deep learning algorithm ( Data mining ) Cloud-based battery model

Actuate Comm and

Sensed Data Comm unication Network

Sensor and Actuator Node

Sensor And Actuator Network

Sensor and Actuator Node

Sensor and Actuator Node

Data Transmission Module

BMS IN VEHICLE

Battery Management System Terminal Voltage Current

Battery Pack

Cell voltage

Sampling Circuit

Temperature Control Signal

PHYSICAL WORLD: BATTERY AND VEHICLES FIG. 4.9 Machine learning application for remaining-useful-life estimation.

Main control system

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4. Battery state estimation methods

attention, including parallelism, scalability, power, and optimization. To solve the aforementioned issues, different kinds of deep networks are introduced into different domains such as recurrent neural networks and convolutional neural networks. In addition, the popular deep learning tools are also introduced and compared, including Caffe, TensorFlow, Theano, Torch, and optimization techniques in each tool.

4.5 Conclusion This chapter introduces state estimation methods of lithium-ion batteries that are suitable for different working conditions. A further understanding of the working environment and battery state estimation can be obtained by using algorithms such as extended Kalman filtering, particle filtering, machine learning, support vector machine, neural network, and deep learning. When these algorithms are applied, the modeling accuracy can improve state estimation accuracy. In recent years, researchers have also proposed some fusion algorithms that combine the advantages of the above algorithms, in which the errors are corrected to improve the estimation accuracy. In the condition of ensuring accuracy, the calculation amount can be reduced by taking fewer sampling members.

Acknowledgments The work is supported by the National Natural Science Foundation of China (No. 61801407), the Sichuan Science and Technology Program (No. 2019YFG0427), the China Scholarship Council (No. 201908515099), and the Fund of Robot Technology Used for Special Environment Key Laboratory of Sichuan Province (No. 18kftk03).

Conflict of interest There is no conflict of interest.

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C H A P T E R

5 Battery state-of-charge estimation methods 5.1 Introduction The lithium-ion battery has gradually become the preferred power supply for new energy vehicles because of its advantages of high energy, small size, and rechargeability. Over time, it has been observed that reasonable use of the battery can prolong its service lifespan a well as create energy savings and emission reductions while avoiding unnecessary losses. The internal structure of the lithium-ion battery is complex, so the level for the state of charge is affected by various complex factors such as discharging current, self-discharge effect, internal temperature, external environment temperature, and battery aging [1–4]. This makes it difficult to accurately estimate the state value. At present, state estimation includes several methods. The relationship curve between the open-circuit voltage and the state of charge is obtained by a fixed discharging current rate. Then, the corresponding state value of the relation curve is searched by the known open-circuit voltage [5–7]. This method is called the open-circuit voltage method. Although the state value can be measured by this method, the battery must be shelved for more than 1 h before measurement. Moreover, the battery itself is vulnerable to the influence of temperature and its quality. The state under the different conditions of the same open-circuit voltage is quite dissimilar, so it is not suitable for its estimation in operation. The discharge experiment method uses one discharge to discharge the battery of a constant current until the cut-off voltage. By the product of discharge time and discharging current, the state value can be calculated directly. The measurement error of the state value is small, but it cannot be measured in real time and takes a long time, so it is not suitable for large-scale applications. The ampere hour integral method uses the accumulated charge-discharge capacity to estimate the state-of-charge value of the battery. Meanwhile, the state value can be modified by the charge-discharge current rate and temperature. This method cannot only predict the state

Battery System Modeling https://doi.org/10.1016/B978-0-323-90472-8.00009-3

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5. Battery state-of-charge estimation methods

value accurately but also realizes the real-time state estimation [8–10]. It is simple and the principle is easy to understand. The essence of the ampere hour integral method is a superposition process. If the initial value cannot be measured accurately enough, similar deviations appear in the latter results in turn. The high requirements for the preliminary value limit its application as well. The calculation formula does not consider the current collection, selfdischarge effect, aging condition, and charge-discharge current rate of the battery. Long-term integration processing produces a cumulative effect, and the extent of the state deviation from the measured value of the estimation processing does not decrease but increases as the error continues to expand. The Kalman filtering algorithm is introduced into the iterative calculation process that is combined with the open-circuit voltage method of the correction. The curve relationship between the open-circuit voltage and the state of charge is fitted. This is followed by prediction, measurement, and correction. According to the principle of minimum average square deviation, the optimal state-of-charge estimation is conducted by the dynamic system state calculation. Then, the state variables are updated and obtained under the measurement conditions [11–20]. It can correct the predicted value and reduce the error. Because noise is effectively introduced into the application of state prediction and observation equations, it can achieve the goal of minimizing the state estimation error as well as the real-time correction. It is proposed to construct the linear state model. As the working characteristics of the lithium-ion battery are nonlinear in most cases, this must be taken into account in the algorithm. Meanwhile, this situation is also the main reason that it is greatly limited to the practical application. Consequently, the extended algorithm is proposed, the characteristics of which are to expand the nonlinear functions of open-circuit voltage and state of charge into the Taylor series around the prediction value of the state and omit the terms above the second-order, so an approximate line can be obtained in the model [21–25]. When the filtering and prediction errors are small, they can realize the real-time state estimation effectively.

5.2 State-of-charge estimation methods 5.2.1 Calculation algorithm comparison The lithium-ion battery includes a very complex electrochemical reaction process. Factors will affect the battery state estimation such as the discharge rate, ambient temperature, cycle times, self-discharge rate, and so on. At the same time, these factors will occur as the number of cycles increases, thereby increasing the difficulty of the battery state-of-charge estimation accordingly. At present, the estimation methods mainly include the open-circuit voltage method, the ampere hour measurement method, the impedance method, the Kalman filter method, and the neural network method. These methods are introduced and analyzed in detail. (1) Ampere hour integral method The expression of the ampere hour integral method can be described as shown in Eq. (5.7).

5.2 State-of-charge estimation methods

SðtÞ ¼ Sðt0 Þ 

159

ðt

ηI dτ t0 3600CN

(5.1)

Wherein S (t) represents the state-of-charge value at the time point of t. S (t0) represents the battery state of charge in the initial state of the measurement. η is the battery Coulomb efficiency. CN is the battery capacity and I is the discharging current. The ampere hour measurement method estimates the battery state-of-charge value by accumulating the discharged power of the battery, which corrects the estimated value based on factors such as discharge rate and temperature. The ampere hour measurement method has the following shortcomings. First, it needs to give an initial value. Second, the current measurement error will accumulate over time, making the estimation error larger and larger. Eventually, the estimated and actual values may generate deviation. Therefore, it is often used in combination with other methods. The use of high-precision sensors can solve the current measurement error problem, but this will lead to a significant increase in cost. (2) Impedance analysis method The impedance analysis method measures the internal AC impedance of the battery by exciting the battery of different frequencies, which calculates the state-of-charge value based on the static model of the AC impedance and the remaining power. The impedance analysis method only considers the factors of AC impedance and discharge rate, and does not involve factors such as the number of battery cycles and the ambient temperature. The final estimation accuracy is not high. Besides, the AC impedance of the battery varies greatly from different frequency currents. (3) Kalman filter method The battery is regarded as a power system in the Kalman filter method, and the state of charge is regarded as an internal state of the system. The key to the Kalman filter method is to establish an accurate equivalent battery model. Then, the state equation is established according to the model. The Kalman filter method has two main disadvantages. First, an accurate equivalent model needs to be established to obtain accurate estimation results, and the accuracy of the equivalent model is proportional to the complexity. Second, a large number of matrices in the algorithm operation leads to a large amount of calculation. On this basis, the extended Kalman filtering based on the nonlinear model and the unscented Kalman filtering algorithms are developed to estimate the battery state of charge. The estimation accuracy based on the Kalman filter theory is relatively high, but the Kalman filter estimation must rely on an accurate battery model to achieve highprecision estimation. (4) Neural network method The neural network method uses the nonlinear imaging characteristics of the neural network to estimate the battery state. When building the neural network model of the battery, the internal parameters of the battery need not be included in the considerable range. The input parameters should be measured as well as the output parameters, such as current, voltage, temperature, and so on. The mapping relationship between them

160

5. Battery state-of-charge estimation methods

should be trained and determined by trial and error. The advantage of the neural network is that it is suitable for all kinds of batteries. The disadvantage is whether the selection of model input variables is reasonable, and this has a great impact on the calculation accuracy. Besides, it requires a large number of training samples. (5) Open circuit voltage method As for the open-circuit voltage method, there is a critical question to estimate the state of charge of continuous function. When the open-circuit voltage condition is known, this independent variable can be taken as a function of the dependent variable, which can be used to calculate the battery state-of-charge value according to its nonlinear function relations. The general nonlinear function is a high-order polynomial. The dependent variable can be expressed by the given open-circuit voltage when the corresponding relationship function is constructed directly.

5.2.2 Coordinate transformation One idea is to exchange the horizontal and vertical coordinates of the coordinate system, which is used to make open-circuit voltage an independent variable. The state of charge is also a dependent variable before solving the fitting curve and polynomial function through discrete points. Then, the curve fitting is performed to obtain the open-circuit voltage curve and its polynomial functional relation. This way is feasible as it can be realized without too much calculation treatment for the nonlinear curve fitting degree. The open-circuit and closed-circuit voltage variations are shown in Fig. 5.1. The strong nonlinear relation between the open-circuit voltage toward the state of charge can be described by Fig. 5.1A. The direct polynomial fitting effect is good, obtaining the open-circuit voltage relationship to the battery state. The conversion coordinates the great divergence after fitting in the same order, as shown in Fig. 5.1B. As can be known from the comparison of the fitting effect before and after a coordinated transformation, this method is not highly feasible for direct coordinated transformation, even if the expected effect is achieved under some circumstances and the fitting curve converges. It can only correspond to a particular kind or even a batch of batteries, which is not applicable universally.

4.1

U_OC

4.0

U (V)

U (V)

3.9 3.8 3.7 3.6 3.5 3.4 3.3 0.0

(A)

0.2

0.4

0.6

0.8

S (1)

10

4.2 4.1 4.0 3.9 3.8 3.7 3.6 3.5 3.4 0.0

1.0

(B)

8

U

6

I(A)

4.2

4 2 0.2

0.4

0.6

0.8

1.0

S (1)

FIG. 5.1 Voltage variation toward the varying state-of-charge conditions. (A) Open-circuit voltage relationship. (B) Voltage-current corresponding curve.

161

5.2 State-of-charge estimation methods

In addition, individual inconsistencies of the battery cells can cause different curves. Therefore, the abrupt conversion to coordinate is inconsistent with the internal battery working principle.

5.2.3 Binary iterative algorithm The establishment of the battery equivalent model is the basis of accurate battery state estimation. The battery model can describe the relationship between the state and various battery parameters accurately. Meanwhile, the influence should be considered for temperature, aging degree, and other factors. The ternary lithium-ion battery is taken as the research objective. It is combined with the extended Kalman filtering algorithm, and the state-of-charge estimation can be realized as shown in Fig. 5.2. Through the complete composite pulse-power experiment, the battery model parameters are tested for different state-of-charge points, and an accurate battery equivalent model is established. On this basis, the algorithm is used to estimate the state value. To make the equivalent model more accurate in the battery characterization, this chapter is based on the secondorder resistance-capacitance equivalent circuit model.

Start Terminal voltage

Initial S S value at T=k Calculate the S estimation value at T=k+1

Current

Calculate the terminal voltage at T=k+1

Terminal voltage

Calculate terminal voltage error Calculate corrected gain

Output S

Filter gain

Corrected S value at T=k+1 N

Stop? Y

End

FIG. 5.2 Binary iterative state-of-charge estimation procedure.

162

5. Battery state-of-charge estimation methods

5.2.4 Extended Kalman filtering The Kalman filter algorithm gives high optimization estimation accuracy in a linear system, but as battery SOC estimation is a typical nonlinear system, it is difficult to use the traditional Kalman filter algorithm in the battery management system (BMS). The EKF algorithm is one of the most common SOC estimation algorithms. This algorithm not only solves the state estimation problem about a nonlinear system but also ensures fast correction and therefore achieves adaptive target tracking in the event of a relatively big error of the initial value. MILS and EKF are combined to improve the robustness of BMS and the accuracy of SOC, to achieve an accurate estimation of lithium-ion battery status. The algorithm iteration process is described in Fig. 5.3. To use the EKF algorithm, SOC, U1, and U2 are used in the state equation as the state variables. The current I is used as the system input, and UOC is used as the system output, according to which the expression of the state equation and the observation equation is constructed, as shown in Eq. (5.2).  xk ¼ Ak1 xk1 + Bk1 xk1 + wk1 (5.2) yk ¼ Ck xk + Dk1 uk + vk where x is the system state variable, u is the system input, and y is the system output. wk and vk are the system noise, with the covariance matrices being Q and R, respectively. The A, B, C, and D matrix of the above-mentioned formula is expressed as shown in Eq. (5.3). 2 3 8 1 0 0  T > > > dUOC ðSOCÞ 7 < A ¼ 6 0 exp ðT=τ Þ 0 , 1, 1 4 5 , Ck ¼ 1 k dðSOCÞ (5.3) > 0 0 exp T=τ ð Þ 2 > > : Bk ¼ ½ηT=QN , R1 ½1  exp ðT=τ1 Þ, R2 ½1  exp ðT=τ2 Þ, Dk ¼ R0 The EKF algorithm uses the first-order Taylor expansion of the open-circuit voltage equation to transform the nonlinear system into a linear system, and its calculation is shown in Eq. (5.4). dUOC ðSOCÞ K1 K2  ¼ SOC 1  SOC dðSOCÞ

(5.4)

The parameter values estimated in real time by the MILS algorithm and the EKF algorithm are combined to realize the iterative calculation of the SOC values. The calculation of the state prediction equation and the state covariance prediction equation is shown in Eq. (5.5).  ^ xk + 1=k ¼ Ak ^xk + Bk uk + wk (5.5) Pk + 1=k ¼ Ak Pk ATk + Qk where Pk is the covariance matrix. To obtain the optimal filter gain matrix Kk +1, the optimal state matrix xk +1, and the optimal covariance matrix value Pk +1, the above prediction equation needs to be updated over time; the updated equation is shown in Eq. (5.6). 8  1 < Kk + 1 ¼ Pk + 1=k CTk+ 1 Ck + 1 Pk + 1=k CTk+ 1 + Rk + 1 (5.6) x ¼^ xk + 1=k + Kk + 1 yk + 1  ^yk + 1 : k+1 Pk + 1 ¼ ðIk  Kk + 1 Ck + 1 ÞPk + 1=k

5.2 State-of-charge estimation methods

163

FIG. 5.3 The EKF algorithm iteration flowchart.

By combining with the above-mentioned EKF optimal iteration process, the MILS algorithm is used to identify the parameters of the E-DCP model and to achieve state estimation of the BMS, improving the SOC estimation precision. Then, the binary iterative method is used to estimate the shelved state-of-charge value, which uses open-circuit voltage toward the state to fit the monotonically increasing function of a polynomial function and starts from

164

5. Battery state-of-charge estimation methods

Start S scope initialization Take the intermediate S Input measuresUOC >4 .2V

k k k > > > i¼1 > > > Ni > h   ih   iT X > > < Pyy ¼ ^k h e ^k Wi h e ξi , Zk1  Y ξi , Zk1  Y (5.10) i¼1 > > > N h   i h   i i > X T > > ^ h e ^ > Wi h e ξi , Zk1  X ξi , Zk1  X > Pxy ¼ k k > > > i¼1 > >  1 : Kk ¼ Pxy Pyy + Pv Due to the complex internal structure of the lithium-ion battery, the estimation data often show strong nonlinearity in the working process, which makes the traditional state estimation algorithm difficult to obtain the real-time and accurate state of the battery. The extended Kalman filtering algorithm is introduced to solve the problems of large error and slow convergence speed caused by the nonlinear battery characteristics in the state-of-charge estimation process. Considering the state prediction, the error covariance is then corrected, in which the expression is described as shown in Eq. (5.11). 8  

< k1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i xk1 ¼ ^ xk1 + ðn + kÞpk1 i , i ¼ 1, 2⋯n (5.14) > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : i xk1  ðn + kÞpk1 i ,i ¼ n + 1, n + 2⋯2n xk1 ¼ ^ The unscented Kalman filtering eliminates errors caused by the linearization of the statespace equations. The square root is obtained by the Cholesky decomposition method, but a necessary condition is that the decomposed matrix must be a positive definite matrix. After the sigma points are obtained, the system state quantity and variance matrix need to be predicted, as shown in Eq. (5.15). 8 2n X > > i > ωm > xik1 ¼ AK1 xik1 + Bk1 uk1 ;^xk ¼ i xk1 < i¼0 (5.15) 2n X >



 T > c i i > > ωi xk1  x^k xk1  x^k + Qk : Px, k ¼ i¼0

ωi and ωi are used as weight factors. Qk is the covariance matrix of process noise. To avoid the matrix not being a positive definite matrix, the square root decomposition method is used to replace the covariance matrix. The square root of covariance matrix Sk can be used instead of Pk to participate in the iterative operation, as shown in Eq. (5.16). 8 h pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi i > xk1 , ^ xk1 + n + kSk , ^xk1  n + kSk xk1 ¼ ^ > > < pffiffiffiffiffic  i  pffiffiffiffiffiffi (5.16) ωi xk1  ^xk , Qk S∗k ¼ qr > > > ∗ 0

: Sk ¼ cholupdate Sk , xk1  ^xk , ωc0 c

m

5.2 State-of-charge estimation methods

169

Wherein Sk is the square root of the covariance matrix Pk. xk-1 is the sigma point sets obtained by unscented transformation. The square root of the covariance matrix can be used to solve the problem. The matrix cannot be decomposed because it is not positive definite, as shown in Eq. (5.17). 8 i > < yk1 ¼ Ck1 xk1 + Dk1 uk1 2n X (5.17) i ^ > y ¼ ωm : k i yk1 i¼0

The resampled values are brought into the system observation equation to update the observation variables. The variance matrix at the time point of k can be calculated by using the predicted observation value and the observation value calculated by the weighting factor, as shown in Eq. (5.18).  pffiffiffiffiffi pffiffiffiffiffic  i 8 ∗ ωi yk1  ^yk , Rk Syy ¼ qr > > > > < Syy ¼ cholupdate S∗yy , y0  ^y , ωc 0 k1 k (5.18) 2n > X



 > T > c i i > ωi xk1  ^xk yk1  ^yk : Pxy ¼ i¼0

Syy is the variance matrix of the output variable at time k. Pxy is the covariance matrix of the state quantity and the observed quantity at time k. The Kalman gain should be calculated as shown in Eq. (5.19).   (5.19) Kk ¼ Pxy = STyy Syy The Kalman gain is a weight relation between the predicted value and observed value. Finally, the Kalman gain is used to update the state matrix and error covariance to complete an iterative calculation, as shown in Eq. (5.20). 8   xk + Kk yk  ^yk > < xk ¼ ^ (5.20) Uk ¼ Kk Syy > : Sk ¼ cholupdatefSk1 , Uk ,  1g A complete iteration operation is completed here, and the state matrix and covariance are brought into the next iteration operation. A repeated iterative process to update the state matrix can make the value of the state constant close to the real value. The algorithm process of unscented Kalman filtering is described in Fig. 5.7. The algorithm has two processes: time update and status update. Compared with the extended Kalman filtering, it has the advantage of higher accuracy, but it also brings more computation. Similar to the classical Kalman filtering algorithm, its accuracy largely depends on the modeling accuracy.

5.2.7 Cubature Kalman filtering In the SOC estimation process of the lithium-ion battery, there is a certain error between the estimated value and the measured value. In addition to capacity attenuation, there are also the two following problems.

170

5. Battery state-of-charge estimation methods

Time update State equation SOC at the time point of t

Error covariance equation

Take Sigma point

Kalman gain

Observation equation Status update Status update Error covariance update

FIG. 5.7 Processing flowchart of the unscented Kalman filtering algorithm.

(1) There is noise during the operation of the lithium-ion battery. Due to the interference in temperature as well as electromagnetic and other factors, the current has zero-mean Gaussian noise. At the same time, noise interference also exists in the observation process. (2) The estimation method is different. The filtering methods of nonlinear systems are mainly EKF, UKF, and CKF. EKF is to linearize the nonlinear function. EKF is simple and easy to implement, but there are truncation errors. UKF determines the probability density function of the sigma point approximating the system state through UT transformation to achieve the purpose of state estimation. The accuracy of UKF is high, but it is computationally intensive and difficult to apply. Unlike the UT transformation, the CKF algorithm uses spherical radial criteria to determine the cubature point. The cubature point weights are equal in size and appear symmetrically, so there is no need to set parameters in advance, and the application is simple. At the same time, the weight value of the cubature point is always positive, thus ensuring numerical stability. Its calculation amount is smaller, and the filtering accuracy is higher. Through the characteristic law of the second-order circuit, the system state equation can be obtained. Then, the system state equation of unscented Kalman filtering is described as shown in Eq. (5.21). 2 3 T 3 2 3 3 2 T=C 0 0 2 1 6 7 Up1 ðkÞ p1 Up1 ðk + 1Þ Rp1 ∗Cp1 6 7 76 7 6 7 6 7 6 T (5.21) 74 Up2 ðkÞ 5 + 4 T=Cp2 5I ðkÞ + wðkÞ 4 Up2 ðk + 1Þ 5 ¼ 6 07 0 1 6 4 5 ∗C T= R p2 p2 Sðk + 1Þ SðkÞ  QNOW 0 0 1 Wherein the mathematical relationship is used to express the relationship between Uocv and SOC, as shown in Eq. (5.23). Uocv ¼

∂Uocv S ∂S

(5.22)

5.2 State-of-charge estimation methods

Then, the calculation process can be expressed by Eq. (5.23). 2 3   Up1 ðkÞ ∂Uocv 6 7 Ut ðkÞ ¼ 1  1 4 Up2 ðkÞ 5  I ðkÞ  R0 + vðkÞ ∂S SðkÞ

171

(5.23)

In Eq. (5.23), T is the sampling period. The maximum capacity of the battery can be obtained from the constant current discharge. w(k) and v(k) are the state noise and observation noise, respectively. The equivalent circuit model of the lithium-ion battery can be expressed by the state space shown in Eq. (5.24). ( xðkÞ ¼ f ½xðk  1Þ, iðkÞ + wðkÞ (5.24) yðkÞ ¼ g½xðkÞ, iðkÞ + vðkÞ Among them, x and y are the state quantity and observation, respectively. f and g represent nonlinear functions. w and v represent process noise and observation noise. The calculation steps of the volume Kalman filtering algorithm are described as follows. The first step of the algorithm should be to initialize the state variables, error covariance, process noise, and measurement noise. ( pffiffiffiffiffiffiffiffiffi ξi ¼ m=2½δi (5.25) ωi ¼ 1=m, i ¼ 1, 2⋯, m Then, the volume point is calculated according to the mathematical description as shown in Eq. (5.26). ( Sk1jk1 ST k1jk1 ¼ Pk1jk1 (5.26) ^ k1jk1 Xi, k1jk1 ¼ ξi Sk1jk1 + X ξi is the calculated volume point, which is the key to the volume Kalman filter. m is the dimension of the state vector. ωi is the calculation factor, which is related to the dimension of the state vector. The next step is a one-step forecast, as shown in Eq. (5.27).

 8 Xi, kjk1 ¼ f Xi, kjk1 , I ðkÞ > > > > > m X > >

> > m     X > T > > ^ kjk1 Xi, kjk1  X ^ kjk1 + Qk1 > Xi, kjk1  X : Pkjk1 ¼ wi i¼1

Xk j k-1 is the propagation volume point, X’k j k-1 is the state prediction value calculated through the propagation volume point, and Pk j k-1 is the error covariance prediction value calculated.  Skjk1 ST kjk1 ¼ Pkjk1 (5.28) ^ k1jk1 Xi, k1jk1 ¼ ξi Sk1jk1 + X The next step is to get the generated sampling points and covariance prediction, which is shown in Eq. (5.29).

172

5. Battery state-of-charge estimation methods

8 m

X  > > ^ kjk1 ¼ wi >Y g Xi, k1jk1 , I ðkÞ > > > > i¼1 > > > m  < X  T   ^ kjk1 g Xi, k1jk1 T ^ kjk1  Y g Xi, k1jk1 Y Pxy ¼ wi > > i¼1 > > > m  > X >   T > ^ kjk1 Y ^ T kjk1 + Rk > g Xi, k1jk1 g Xi, k1jk1  Y > : Pyy ¼ wi

(5.29)

i¼1

Xi,k j k-1 is the recalculated volume point, Yk j k-1 is the measured predicted value, Pyy is the measurement error covariance, and Pxy is the cross-covariance. State prediction can be performed after covariance prediction is completed; this is shown in Eq. (5.30). 8 Gk ¼ Pxy Pyy 1 > > <   ^ kjk1 ^ kjk1 ¼ X ^ kjk1 + Gk Yk  Y (5.30) X > > : Pkjk ¼ Pkjk1  Gk Pyy GTk Wherein Gk is the gain, and Xk j k-1 is the updated state value. Pk j k is the updated state error. The iterative operation is completed in which the state matrix and covariance are brought into the next iteration operation. A repeated iterative process is used to update the state matrix that can make the value of the state constant close to the real value.

5.3 Iterative calculation and modeling The battery state of charge is affected by temperature, charge-discharge, self-discharge, aging, and other factors. The internal chemical reaction is variable with highly nonlinear characteristics, which makes the state estimation difficult. Kalman filtering is an optimal autoregressive data processing algorithm that can improve the state estimation accuracy by the real-time optimal estimation of state variables.

5.3.1 Equivalent circuit modeling In order to describe the battery hysteresis, polarization, and self-discharge effects, Thevenin modeling is employed as the equivalent circuit method. The model structure and the resistance are moderate. The capacitive parallel circuit can characterize its dynamic characteristics, and the parameter identification is moderately difficult, which can fully meet the accuracy requirements of the battery equivalent model. There is only a first-order RC circuit in the model, so the structure is simple and computationally small. It can reduce the load of the processor during the later algorithm migration process, which can improve the operating speed of the algorithm. It is also a fast, real-time, and stable system that guarantees accurate state monitoring. The equivalent circuit model is described as shown in Fig. 5.8. According to the model, the discharge direction is taken as the positive direction. Considering the voltage across the polarization capacitance, the mathematical relationship can be obtained as shown in Eq. (5.31).

173

5.3 Iterative calculation and modeling

Rp

I(t)

R0 + E(t)

Up(t) Cp

UL

RL

-

FIG. 5.8 Thevenin modeling for lithium-ion batteries.

8 UL ¼ UOC  Up  IR0 > > > < Up dUp I ¼  Cp R dt > p > > : t=τ Up ¼ Ce + IRp , τ ¼ Rp Cp

(5.31)

Wherein the differential equation can be obtained accordingly. The current is supposed to be 0 in a certain period corresponding to the battery shelved state. The circuit has a zero-input response. At this time, Up is used to get an initial state Up0, and the expression of the polarization voltage can be rewritten as described in Eq. (5.32). Up ¼ Up0 et=τ

(5.32)

Assuming that the initial state of Up is 0, the circuit has a zero-state response and the equation can be rewritten accordingly. The discretization of equations yields a recursive form of Up as shown in Eq. (5.33). 8   t=τ > < Up ¼ I  Rp 1  e Upðk + 1Þ ¼ UpðkÞ eΔt=τ (5.33)   > :U Δt=τ Δt=τ + Rp I ðkÞ 1  e pðk + 1Þ ¼ UpðkÞ e From the above two discrete recursive forms, the discretization form of Up is exercised. Discrete forms of state and output equations are well suited to the algorithm iteration and program implementation, providing a reliable basis for subsequent algorithm research.

5.3.2 Parameter identification For the Thevenin model, the parameters to be identified are the ohmic internal resistance R0, the polarization internal resistance Rp, and the polarization capacitance Cp. These parameters need to be identified through experiments. Here, the ternary lithium-ion battery is selected for testing. The nominal capacity of the battery is 72 Ah with the measured capacity of 68 Ah. The hybrid charge-discharge experiment is performed on the battery at 27°C. It obtains battery model parameters by analyzing the operating characteristics during operation. The cyclic charge-discharge process of the experiment can be described as shown in Fig. 5.9. The figure also reflects the transient characteristics and steady-state characteristics of the lithium-ion battery. When the pulse discharge starts, the battery voltage will drop momentarily, Then the voltage will slowly decline with time during the discharge. When the

174

5. Battery state-of-charge estimation methods

4.2

U (V)

4.1

4.0

3.9

-5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

t(s)

FIG. 5.9 Voltage response curve.

discharge ends, the battery voltage will rebound immediately. During the period of suspension, the voltage gradually rises and tends to be stable. The charging process is opposite to the response to the discharging process voltage. As can be known from the five points of A, B, C, D, and E, the basis of taking the point can be described as follows. The sudden drop of the discharge starting voltage is the action of the ohmic internal resistance R0 until the same discharge ends. The rapid rise of voltage is also the effect of R0. The phenomenon of voltage drop during discharge can be explained by the RC circuit in the model. Therefore, the ohmic internal resistance of the battery can be obtained from the AB and CD segments. The values of the polarization internal resistance and capacitance of the RC circuit can be obtained from the BC and DE segments. Then, the calculation formula can be obtained for the model parameters. The calculation process can be described as shown in Eq. (5.34). R0 ¼

jΔUAB j + jΔUCD j 2I

(5.34)

Wherein Δ UAB is the voltage difference of the AB segment, Δ UCD is the voltage difference of the CD segment, and I is the discharge current. The DE segment voltage response is analyzed corresponding to the zero-input response to the model, and the mathematical relationship can be obtained as shown in Eq. (5.35).  UD ¼ UA  Up0 e0=τ (5.35) UE ¼ UA  Up0 eðtE tD Þ=τ Wherein UA represents the voltage at point A, which is the open-circuit voltage at the current stage. When the point D is zero, the time for point D to point E is tE–tD. UD and UE represent the voltages at the time points of D and E, respectively. Solving the system of equations yields a formula for the time constant τ of Eq. (5.36).    UA  UE (5.36) τ ¼ ðtE  tD Þ= ln UA  UD

5.3 Iterative calculation and modeling

175

After getting the time constant τ, the parameter Rp or Cp can be calculated, and the other can be derived by Eq. (5.37). τ ¼ Rp C p

(5.37)

The voltage response of the BC segment corresponds to the zero-state response of the model, and the mathematical relationship can be obtained as shown in Eq. (5.38). 8   t t <  Cτ B UC ¼ UA  IR0  IRp 1  e (5.38) : UB ¼ UA  IR0 Wherein UC represents the voltage at the time point C. When point B is zero, the time from point B to point C is tB–tC. The polarization resistance Rp can be calculated as Eq. (5.39). UB  UC  Rp ¼  tC tB I 1  1  e τ

(5.39)

Then, the key points in the pulse test of each stage can be obtained, according to which the model parameter values of R0, Rp, and Cp can be calculated.

5.3.3 Kalman filtering algorithm There are many commonly used state estimation methods. The error of the ampere hour integral method is large. The open-circuit voltage method is mainly used for offline estimation. The high-precision estimation methods include the neural network, fuzzy inference, and Kalman filtering methods. The extended Kalman filtering algorithm can linearize the Taylor series expansion of the first-order nonlinear functions that neglect the remaining high-order terms, thus transforming the nonlinear problems with linear problems, which can get higher accuracy. The essence of the Kalman filtering algorithm to estimate the battery state of charge is to calculate the state by the ampere hour integral method. It is also used to modify the state by the ampere hour integral method of the measured voltage values. The estimation process includes the time and measurement update. The time update, also known as the prediction, is a process of predicting the present state variables and providing a prior estimation of the next time point. The measurement-update process, also known as the correction process, is a process of feedback on the observed values and correcting the deviation. Its process is described as shown in Fig. 5.10. When using the Kalman filters to estimate the battery state, it is necessary to establish an appropriate equivalent battery model. Its accuracy depends on the accuracy of the battery model. When it is used to estimate the battery state, the battery is considered as a power system and the state of charge is the system state [55–64]. The battery charge-discharge current rate is regarded as the input of the system, and the terminal voltage can be regarded as the output. By comparing the observed value of the terminal voltage and the error of the estimated state, the system state is updated constantly to obtain the minimum variance estimated

176

5. Battery state-of-charge estimation methods

Initial state value S value at k Filter input

Calculate S value at k + 1

Current measurement

Calculate estimated voltage at k + 1

Measured voltage value at k + 1

k + 1 time estimation error

Calculate filter gain

Correction bias Modify the estimated S at k + 1 Get the estimated S value at k + 1 Filter output

FIG. 5.10

Processing flowchart of the Kalman filtering algorithm.

state [65]. It is an optimal autoregressive data processing algorithm that is suitable for both stationary and nonstationary processes. It has good real-time performance that is easy to be realized. However, this algorithm has a strong dependence on the model, which can only be applied to the linear system. Its state equation and observation equation are described as shown in Eq. (5.40).  Xk ¼ Ak1 Xk1 + Bk1 Zk1 + Wk1 (5.40) Yk ¼ Ck1 Xk + Dk Zk + Vk Among them, the state variable at the time point of k is described by the parameter Xk, and Yk is the systematic observation variable at the time point of k. Zk is the system input, as the control variable. Ak is the transfer matrix of state x from the time point of k-1 to the time point of k. Bk is the input matrix. Ck is the measurement matrix. D is the feedforward matrix. Wk is the noise of the system state equation as processing noise, and its variance is Qk. Vk is the noise of the measurement equation as the observation noise, whose variance is Rk. Its key step can be realized to establish a reasonable state equation and observation equation, so it is necessary to establish an accurate battery model. However, its effect applied to the nonlinear system is not ideal due to the strong nonlinearity of the battery system. Therefore, it is proposed to linearize the nonlinear system through the Taylor series expansion. After linearization, the Kalman filtering algorithm can be used to estimate the state variables.

5.3.4 Extended Taylor series expansion The extended Kalman filtering algorithm is a linearization process of the nonlinear system. It estimates the value of the next time point according to the previous time point. The state variables are updated constantly with the input and output observations of the system to achieve the optimal estimation [12, 24, 66–70]. When it is used to estimate the battery state, the processing and observation noises are required to be white noise with approximate Gaussian distribution. This is also a limitation of all Kalman filtering methods. In this case,

5.3 Iterative calculation and modeling

177

Extended Kalman filter

State calculation process

Variance calculation process

Filtering update

Time update

Kalman gain matrix update State estimation measurement update

FIG. 5.11 The processing and updating processes of extended Kalman filtering.

the covariance of processing and observation noises can be easily controlled. In the process of estimation, the Taylor series expansion algorithm is implemented to expand the system model of the battery, then, a first-order linear model is left after removing the high-order terms. After the linearization model is obtained, the battery state is estimated. The algorithm flowchart is described as shown in Fig. 5.11. At the time point of k, the state estimation covariance of the previous time point of k-1 and the estimation error can be obtained. The latter time should be a function of the preceding time for the mathematical model through the state value at the time point of k-1 to predict the state value at the time point of k in advance. Therefore, the prior prediction value of the present time can be obtained by combining the system state and error covariance of the previous time point along with the Kalman gain. The two parameters written in the model are measured before and after the time point of k, according to which the measured values can be obtained, as shown in Fig. 5.12. The posterior state estimation is used in the previous time point, so the value should also be used in the advance prediction of the next step. According to the observation function of the mathematical model, it can be known that the observed value is a function of the estimated value. In reality, the estimated value cannot be obtained. In the absence of noise and linearization error, the value can be calculated and processed by the mathematical model that can be directly measured by physical means without measurement noise [71, 72]. The measured value is equal because there is a certain error between the linearization error of the mathematical model plus the prior estimated value and the estimated measured value. There is a certain

FIG. 5.12 State-space processing procedure for state and estimation equations.

178

5. Battery state-of-charge estimation methods Unscented transform

S(k-1) P(k-1)

Multiply by the corresponding weight

State Equation

Ssigmapre

Ssigma

Unscented transform again

Observation equation

Corresponding to 3 weights Wc and Wm Iteration

P(k)

Kalman gain Kk

Multiply by the corresponding weight

ULsigmapre

S(k)

ULpre Pzz

FIG. 5.13

Ssigma1

Spre

Pxz

Double unscented transform extended Kalman filtering process.

error between all the measured values obtained by the mathematical model and the measured value, as shown in Fig. 5.13. The extended Kalman filtering method is improved based on the ordinary calculation process, which is a linearization technology. It means to linearize the estimated parameters, then carry out the linear calculation. Thus, the battery state can be estimated, in which the system and processing noises are generally approximated as white noise, which accords with Gaussian distribution. It linearizes noise expectation and covariance in the estimation process. This algorithm uses a linear state equation to estimate the present state of the system through the input and output data onto the system. Because this algorithm estimates at the time domain completely, the calculation is small without the mutual conversion to the time domain and the frequency domain, in which the real-time estimation effects are quite good. The classical algorithm is suitable for a linear system while the battery system is nonlinear. It uses the Taylor series expansion to get the approximate linear space equation. It then uses the Kalman filtering algorithm to estimate the present state, which is suitable for the discrete nonlinear system. The expression and observation equations of the discrete nonlinear system are described as shown in Eq. (5.41). ( Xk + 1 ¼ f ðXk , kÞ + W k (5.41) Zk ¼ hðXk , kÞ + V k The first part of Eq. (5.41) represents the state equation and the second part represents the observation equation. k is a discrete-time point. Xk+1 is the n-dimensional state vector. Zk is the m-dimensional observation vector. Wk and Vk are independent Gaussian white noise. To apply the Kalman filter, the first-order Taylor series expansion of nonlinear functions f () and h () is carried out around the estimated value. Consequently, the results are described as shown in Eq. (5.42). 8   ∂f ðX , kÞ   > k >  ^ ^k > Xk  X < f ðXk , kÞ  f Xk , k + ∂X  ^ k X ¼X (5.42)   ∂hðX , kÞ k k   > k >  ^ ^ > ð , k Þ  h X , k + X  X h X k k k k : ∂Xk Xk ¼X^ k

5.3 Iterative calculation and modeling

Wherein the values of Ak, Bk, Ck, and Dk can be calculated, as shown in Eq. (5.43).  8   ∂f ðXk , kÞ > ^k ^ k , k  Ak X > ¼ , B ¼ f X A k k >  < ∂Xk Xk ¼X^ k    > ∂hðXk , kÞ > ^k ^ k , k  Ck X > , D ¼ h X : Ck ¼ k ∂Xk Xk ¼X^ k Furthermore, Eq. (5.41) can be linearized as shown in Eq. (5.44). ( Xk + 1 ¼ Ak Xk + Bk + wk Zk ¼ Ck Xk + Dk + vk

179

(5.43)

(5.44)

The recursive process of the extended Kalman filtering is obtained by applying the equations to the linearized Eq. (5.44) which is described as shown in Eq. (5.45).    8  ^ k , P^ ¼ Ak P^k AT + Qk + 1 ^ > ¼ f X X > k+1 k+1 k > > > >  1 >   < Kk + 1 ¼ P^k + 1 CTk+ 1 Ck + 1 P^k + 1 CTk+ 1 + Rk + 1 (5.45) >

   >   >X ^k+1 ¼ X > > k + 1 + Kk + 1 Zk + 1  h Xk + 1 > > :^ Pk + 1 ¼ ½I  Kk + 1 Ck + 1 P k+1 Wherein P is the average square error and K is the Kalman gain. The parameter I should be initialed as the n  m unit. Q and R are the variances of W and V, respectively, which generally do not change along with the system. The initial state value is X(0) ¼ E[X(0)], and its variance is P(0) ¼ Var[X(0)]. The calculation steps for the time point of k + 1 are described as follows. First, the state and mean-square error of the present time can be estimated by the last time point to obtain the prior state as well as its average square error. Then, the Kalman gain of this time point is calculated. Finally, Kk +1 is used to modify the prior state to get the present state and the prior average square error is modified to get the current average square error.

5.3.5 Estimation model construction The battery state estimation expression is described as shown in the first part of Eq. (5.46). In the measurement process, the current parameters are easy to be detected directly, so the calculation process of the battery state can be expressed in the second part of Eq. (5.4). S ¼ Qt =Qn  100% ) SI ¼ QIt =QIn  100%

(5.46)

In the above expression, SI is the state-of-charge value when the current condition is set to be I. Qt is the remaining electricity, and Qn is the rated capacity. The improved iterative calculation can then be obtained, as shown in Fig. 5.14. After discretization, the technical implementation of the battery discrete-time state estimation can be expressed as shown in Eq. (5.47). ðt ηi I ðtÞ η Δt dτ ) Sn + 1 ¼ Sn  i In (5.47) SðtÞ ¼ Sð0Þ  Q Qn n 0

180

FIG. 5.14

5. Battery state-of-charge estimation methods

Improved Kalman filtering iterative calculation.

Wherein is the present state-of-charge calculation method of simple calculation characteristics. For the continuous-time state estimation realization process, it can be expressed by the mathematical expression of the first part of the formula. In practical application, the discrete state estimation model should be applied. The discrete state estimation model is convenient for the control of a digital system.

5.3.6 Iterative prediction and correction Through parameter identification, the functional relations of resistance, capacitance, voltage, and state of charge are obtained. Then, the circuit module is built in Simulink. In the process of simulation, submodule packing processing is carried out on the module to optimize the interface for better simulation. The measured and estimated voltages are obtained with comparison by simulating after the model construction is completed, according to which the iterative calculation procedure can be obtained. The error in the parameter identification process can be studied by changing the value of the parameters, and the optimal simulation model can be obtained by modifying the formula. The establishment of an equivalent model of the lithium-ion battery is the basis of accurate battery state estimation. After the main parameters in the model are obtained, the state-space equation is obtained according to the relationship between voltage and current, as shown in Eq. (5.48).

5.3 Iterative calculation and modeling

8 EðtÞ ¼ UL ðtÞ + R1 I ðtÞ + Up ðtÞ > > > < I ðtÞ ¼ uðtÞ=R + C  dU =dt 2 p ðt > > > : SðtÞ ¼ Sðt0 Þ  ηI ðtÞdt=Qn

181

(5.48)

t0

With the ternary battery as the research objective, the composite pulse-power experiment is conducted to test the battery model parameters at different state points and establish an accurate equivalent battery model. By combining the three formulas, the state equation can be obtained, as shown in Eq. (5.49). 8 xðkj k  1Þ ¼ Ak1 xðk  1Þ + Bk1 Ik1 + wk > > < ! ! t=Qn 1 0 (5.49) > ,Bk ¼   > : Ak ¼ t=τ t=τ R2 1  e 0 e The accurate battery model describes the relationship between the state of charge and various parameters. Meanwhile, the influence of temperature, aging degree, and other factors should also be considered. The observation equation is described as shown in Eq. (5.50). yk ¼ hðxk1 , Ik1 Þ + vk ¼ UOC  Ri Ik  Upk + vk

(5.50)

Then, the state-space equation can be obtained accordingly, as shown in Eq. (5.51). 8       ηΔt=Q Sk Sk + 1 1 0 > >   ¼ Ik + Wk > p + < Up Rp 1  eΔt=τ 0 eΔt=τ Uk k+1 (5.51)   > Sk > > Uk ¼ ½ ∂UOC ðsÞ=∂s 1  p  R0 Ik + Vk : Uk After the first-order Taylor linearization of the obtained equation, the values of Ak, Bk, and Ck are obtained, as shown in Eq. (5.52). ! !   t=Q0  1 0 ^ k ¼ ∂UOC 1  ^k ¼ ^k ¼   (5.52) , B , C A  ∂s R2 1  et=τ 0 et=τ xk ¼^xk The first step is state prediction. The predicted value at the time point of k in the calculation is described as shown in Eq. (5.53). xðkj k  1Þ ¼ Ak1 xðk  1Þ + Bk1 Ik1

(5.53)

The subsequent step is the prediction of covariance. By calculating the estimation error of x (k j k-1), the covariance matrix of the corresponding x (k j k-1) is obtained, as shown in Eq. (5.54) Pðkj k  1Þ ¼ Ak1 P^k1 ATk1 + Qk

(5.54)

The third step should be used to calculate the Kalman gain, which is obtained at the time point of k as shown in Eq. (5.55).  1 (5.55) Kk ¼ Pk CTk Ck Pk CTk + Rk

182

5. Battery state-of-charge estimation methods

P is used to update the status. According to the open-circuit voltage UOC(k) obtained by real-time measurement, the optimal estimation value of the existing state is estimated as shown in Eq. (5.56). ^ xk ¼ xðkj k  1Þ + Kk ½UL ðkÞ  Ck xðkj k  1Þ

(5.56)

The fifth step is conducted to update the noise covariance. Moreover, the charge-discharge efficiency ŋ itself is obtained in advance by the charge-discharge experiment. However, the current of the power battery is irregular when it is used in the discharge, which may exceed the set data and estimation ranges from the experiment, resulting in error. The noise covariance is updated according to the Kalman gain and the noise covariance of the previous time point, as shown in Eq. (5.57). P^k ¼ ðE  Kk Ck ÞPk

(5.57)

If the battery linearization is not tenable, the performance of the filter is reduced by using this algorithm, so that the result is divergent. Therefore, if you want to improve the accuracy and avoid the linearization error, you need to consider other algorithms. Therefore, there are still many areas to be improved upon in the state estimation method of the lithium-ion battery. The second-order resistance-capacitance equivalent circuit model is used to simulate the battery terminal voltage. The estimation results show that the model can correct the error of the initial value well and the estimation effect is good. The iterative state-of-charge calculation process is described in Fig. 5.15. The five steps should be cycled continuously in the calculation process. The estimated state should be updated constantly so that the estimated value is closer to the measured value of the updating process.

Start The initial value of S The optimal prediction

Calculate the prediction value of S at time k Find the covariance matrix at time k Calculate the Kalman gain at time k

Actual terminal voltage

Calculate the modified estimation of S at time k

The Kalman gain

Update the error of the optimal estimate at time k The output of S End

FIG. 5.15

The structural state estimation process based on Kalman filtering.

183

5.4 Experimental result analysis

5.4 Experimental result analysis According to the equivalent model of the ternary lithium-ion battery, the simulation model is built and the extended Kalman filtering algorithm is input into the S-function block to realize the accurate iterative state estimation of the ternary lithium-ion batteries.

5.4.1 Pulse-power characteristic test The hybrid pulse-power characteristic test method exhibits a change in battery voltage as a function of the battery charge-discharge conditions, which reflects the dynamic battery characteristics of operation. The key parameters required for the equivalent circuit model can be easily determined by the hybrid pulse-power characteristic test. The hybrid pulse-power characteristic test detects the internal ohmic resistance of the battery by dynamically changing its terminal voltage at the beginning and end of the charge-discharge treatment. After that, the internal voltage is slowly changed to detect and determine the internal polarization resistance together with capacity values. The specific experimental steps are described as follows. S1: The battery is charged with a constant current of 0.3C. The cut-off voltage of the charging is 3.65 V. When the voltage is charged to this level, the voltage continues to 3.65 V and the procedure turns to the constant-voltage charging. When the battery current decreases to 0.05 C, the battery is already full. To stabilize the voltage, the battery should be shelved for 1 h. S2: The capacity releases can be realized by 1 C constant-current discharge, discharge 10 s, and shelved for 40 s. S3: The battery is charged with 1 C constant-current, charge for 10 s, and shelved for 40 s. It is then discharged for 6 min using 1 C constant current discharge, and is shelved for 40 min after that. S4: After performing S2 and S3 in sequence, a complete cycling test of the hybrid pulse-power characteristic description can be completed. The hybrid pulse-power test renderings are described in Fig. 5.16. At the beginning of the main discharge and the resting stage, the curve overlaps. At the end of discharge, the curve overlaps due to the polarization effect, so that a difference occurs. The reason is that the environment is different. Besides, the resistance and capacitance can be 80

4.4

40

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FIG. 5.16 Pulse-power test renderings for the battery cells. (A) Overall voltage variation law. (B) Pulse current experimental design.

184

5. Battery state-of-charge estimation methods 4.4 U1

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FIG. 5.17 Coincidence curve of measured and modeling voltage. (A) Voltage traction for whole test process. (B) Voltage traction error variation curve.

estimated, in which there is an error with the resistance and capacitance. The completed hybrid pulse-power experimental curve of voltage can coincide as shown in Fig. 5.17. Wherein U1 is the voltage variation obtained by the experimental test, and U2 is the estimated voltage. There is a deviation from the simulation results and the hybrid pulse-power characteristic test curve difference; the model curve has no measured voltage polarization effect. When the internal temperature of the battery rises, the resistance and capacitance will change along with it. After that, the local amplification cycle can be realized as shown in Fig. 5.18. The experimental result is obtained from the hybrid pulse-power characteristic experimental voltage of the model simulation voltage, in which the different curves show the error times in one period. The coincidence curve of the voltage and current variation can be described as shown in Fig. 5.19. Furthermore, the comparison verifies whether the error lies in a charging period with a resting period or a discharging period, as shown in Fig. 5.20. Wherein U1 is the voltage variation obtained by the experimental test, and U2 is the estimated voltage. The electrochemical properties of lithium-ion batteries can be represented by some energy storage function of electronic components. The general battery equivalent circuit contains the voltage source, resistance, capacitance, and energy storage element. These components can build several classical equivalent circuit models of lithium-ion batteries, such as the Rint and resistance-capacitance models. A comprehensive analysis is conducted for the 1.0

80 60

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FIG. 5.18

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Coincidence curve of one-cycle current and modeling state estimation. (A) Enlarged one-cycle current variation. (B) State estimation processing result.

185

5.4 Experimental result analysis 4.2

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FIG. 5.19 Coincidence curve of the voltage-current variation in the pulse charging process. (a) Corresponding voltage variation. (b) Current variation design. 1.0

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FIG. 5.20 Differential voltage-state coincidence curve. (A) State variation. (B) Enlarged current variation. (C) Voltage traction. (D) Voltage traction error.

pros and cons of the above several kinds in the equivalent circuit. The battery Rint and resistance-capacitance equivalent models are applied to extend Kalman filtering to estimate the battery state. The internal resistance model accuracy is not in conformity with the requirements. The dynamic battery effect is reflected at the same time with less amount of calculation, in which the model integrates the advantages of the choice. Compared with the Rint model, the Thevenin model has one more RC network, which increases the description of the battery polarization effect. By modifying the functional relation of the input resistance and capacitance of the equivalent circuit model, the specific influence position on the output waveform can be analyzed for ohmic resistance, polarization resistance, and polarization capacitance. At the early stage of discharge, the current and voltage

5. Battery state-of-charge estimation methods 0.095 0.090 0.085 0.080 0.075 0.070 0.065 0.060 0.055

(A)

R0 (Ω)

R0 (Ω)

186

0.0

FIG. 5.21

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S (1)

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(B)

R0 R0'

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S (1)

The ohmic resistance changing curve. (A) Ohmic resistance results. (B) Different results comparison.

0.018

14000

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Cp (F)

Rp (Ω)

suddenly changed, so the response characteristic is mainly determined by the battery ohmic resistance. When the parameter value of the ohmic resistance is large, the internal resistance parameter error between the measured voltage and the model voltage increases. The model voltage drops more slowly because of the resistor value increase. The voltage remains unchanged and the reaction time increases, as shown in Fig. 5.21. After the voltage mutation stage, the voltage at both ends of the battery does not become flat immediately after the voltage drop, which is the battery polarization effect. By increasing the polarization capacitance 10 times, the curve of changing the polarization capacitance parameter is obtained, as shown in Fig. 5.22. Wherein Rp is the polarization resistance and Cp is the polarization capacitance. S is the state-of-charge level, which varies from 0 to 1. These experimental results are obtained by conducting the experimental analysis. The experimental battery is the ternary lithium-ion battery composed of lithium nickel manganese cobalt oxide with a capacity of 60 Ah. According to the experimental analysis, the change of the polarization capacitance value is consistent with the theory. The parameter identification can be completed after establishing the functional relationship between the internal model factors and the state of charge levels. After that, the voltage through the model and the measured voltage contrast can be concluded as well as its error curve. The parameter identification error can be influenced by the ohmic internal resistance, the polarization capacitance, and the polarization resistance. Thus, it should be used to obtain the model parameters and functional relation. Similarly, the polarization resistance can be obtained to change the battery parameter.

0.015 0.014

(A)

FIG. 5.22

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Changing polarization capacitance parameter curve. (A) Polarization resistance results. (B) Polarization capacitance results.

187

5.4 Experimental result analysis

5.4.2 Estimation features and comparison Ordinary Kalman filtering is used to obtain the dynamic estimation of the target using the minimum average square error criterion in the case of linear Gaussian. For the highly nonlinear working characteristics of large systems, the extended algorithm is proposed that performs a Taylor series expansion and discards higher-order components. Then, a linear approximation can be achieved with the nonlinear relationship [73]. The high-order truncation error between the second order and above exists. The method of approximating the posterior probability density of the state without Kalman filtering and eliminating the step of solving the Jacobian matrix reduces the calculation amount, which improves its system stability, accuracy, and filtering performance. Then, real-time online prediction can be realized to obtain the battery state value with high accuracy and other advantages. Combined with the extended Kalman filtering-based state estimation and experimental results, the iterative calculation needs a relatively high initial state that is relative to it. When there is a large deviation between the initial parameter value of the algorithm and the experimental values, this algorithm converges slowly. To verify the feasibility of the above estimation algorithm, the state estimation model is established, according to which the data obtained by the experiment and constant-current operating conditions are analyzed. As the system parameters, they are input into the model and simulated. The obtained results are described as shown in Fig. 5.23. Wherein one curve line is obtained by extended Kalman filtering, and the other is not filtered. As can be known from the experimental results, the established state estimation model can achieve the state-of-charge estimation purpose with small errors. Compared with the traditional algorithm, it performs two matrix decompositions in each iteration and calculates multiple sigma points with more computational complexity. Therefore, the extended Kalman filtering state-ofcharge estimation is realized, which is suitable for real-time online state estimation.

5.4.3 Thermal influencing effect Because the lithium-ion battery voltage platform of the low-temperature discharge is lower, it will cause a significant decrease in discharge capacity for low temperatures. However, this loss will be automatically compensated for in the progress of the normal temperature charge-discharge, which is a reversible loss. But for low-temperature charging, a 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4

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FIG. 5.23 The estimation results comparison. (A) Estimation results. (B) Particle enlarged results.

88

188

5. Battery state-of-charge estimation methods

temperature that is too low or a current rate that is too high will cause the irreversible formation of lithium dendrites and the irreversible capacity loss of the battery, which affects the battery safety performance. As the temperature drops, the reaction rate of the electrode also drops. Assuming that the battery voltage remains constant and the discharging current decreases, the power output of the battery will also decrease. Among all environmental factors, the temperature has the greatest impact on battery charge-discharge performance. The electrochemical reaction is related to the ambient temperature against the electrode and electrolyte interfaces. The interface is regarded as the battery heart. If the temperature rises, the output power of the battery will rise as well. Temperature also affects the transfer speed of the electrolyte. When it rises faster, the transfer temperature drops. If the transfer slows down, the battery charge-discharge performance will also be affected. But if the temperature is too high, exceeding 45°C, it will destroy the chemical balance in the battery and cause side reactions. If the lithium-ion battery is used in a low-temperature environment for a short time, or it is not low enough, it will temporarily affect the battery capacity but will not cause permanent damage. However, if it is used in a low-temperature environment for a long time, or in an ultralow temperature of 40°C, the battery may be “frozen,” causing permanent damage. Besides, charging batteries of low temperatures will precipitate metallic lithium on the battery anode surface, and this process is irreversible. This will cause permanent damage to the battery and reduce the safety of the battery. Therefore, most battery devices have protection devices that make it impossible to charge at low temperatures. Temperature is one of the most important parameters controlled in the power supply system of electric vehicles, and it is also the most important parameter that affects battery performance. In all battery testing systems, the temperature must be monitored because the temperature has a relatively large impact on battery performance, including internal resistance, charging performance, discharging performance, safety, lifespan, and so on. As the temperature decreases, the average discharging voltage and the lithium-ion battery capacity will be reduced. Especially when the temperature is below 20°C, the discharging capacity and voltage will drop fast. The battery capacity change corresponding to the temperature variation can be described as shown in Fig. 5.24. From an electrochemical point of view, the solution resistance and solid electrolyte interphase film resistance change little over the entire temperature range and have little impact on 50

Q (Ah)

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T (°C) FIG. 5.24

The battery capacity change corresponding to the temperature variation.

189

5.4 Experimental result analysis

the low-temperature performance of the battery. The charge transfer resistance increases significantly with the decrease of temperature. Furthermore, it varies along with the temperature in the entire temperature range. The changes are significantly greater than the solution resistance and the solid electrolyte interphase film resistance. This is because as the temperature decreases, the ionic conductivity of the electrolyte decreases as well as the solid electrolyte interphase film resistance. The electrochemical reaction resistance increases, resulting in an increase in ohmic polarization, concentration polarization, and electrochemical polarization at low temperatures. The discharge curve of the battery shows that the average voltage and discharge capacity both decrease as the temperature decreases. The best operating temperature for lithium-ion batteries is 0–35°C. A low-temperature environment will reduce the battery’s activity, weaken its discharge capacity, and shorten the usage time. If the battery is in a low-temperature environment for a short time, this damage is only temporary and will not damage the battery capacity. When the temperature rises, the performance will recover.

5.4.4 Time-varying condition influence

20 10 0 -10 -20 -30 -40 -50 -60 -70

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I (A)

I (A)

The model parameters at different state points are verified by the above hybrid pulsepower characteristic tests. The parameter value obtained by the parameter identification is selected as the parameter value of the model, and the terminal voltage after the shelved period is measured by the hybrid pulse-power characteristic test. The open-circuit voltage is selected as the direct-current voltage source input value in the simulation model; it is controllable in the simulation model. According to the equivalent structure diagram of the model, the simulation model diagram can be realized. The current source can simulate the charge-discharge current values obtained in the hybrid pulse-power characteristic test. The magnitude of the charge-discharge current can be given by the pulse. The waveform diagram begins to discharge at 1 C for 10 s, then rests for 40 s. Each set of parameters is given a pulse of time verification, which simulates the discharge process of the battery state of charge from 1 to 0. The pulse waveform for the complex working condition is shown in Fig. 5.25. By comparing and analyzing different equivalent circuit models of lithium-ion batteries, the model is considered comprehensively based on the model. Then, the key parameters in the equivalent circuit model are identified by performing experiments on the battery study.

-10 -20 -30 -40

0

(A) FIG. 5.25

2000

4000

6000

t (s)

8000

10000

12000

1200

(B)

1250

1300

1350

1400

1450

t (s)

Pulse current waveform for experimental test. (A) Overall experimental test procedure. (B) Single pulsepower variation.

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5. Battery state-of-charge estimation methods

The adaptive Kalman filtering method has a better follow up to the state estimation of the lithium-ion battery as well as small fluctuation and small error. The error is much smaller than the extended Kalman filtering method, in which the estimated value is very close to the reference value. The overall effect is also better. When estimating the battery state, the adaptive Kalman filtering method continuously updates the covariance, the average of the processing noise, and the measurement noise. It continuously updates the estimated battery state value. To further demonstrate the accuracy and reliability of the adaptive Kalman filtering method, it is compared with the measured value by simulating the same model simultaneously in the simulation based on the original experiment. After many experimental tests, the estimated curves are obtained for the battery state. The initial state value is 0.75. In the noise-driven matrix, the forgetting factor b is equal to 0.95. The observed noise matrix Q is [0.001, 0; 0, 0.001], and the system noise R is 0.00001. A comparison of the predicted state can be obtained as shown in Fig. 5.26. Wherein S1 describes the estimated battery state value by using the adaptive Kalman filtering method. S2 describes the measured battery state value. As can be known from the experimental results, the estimation effect fits the measured state curve well. Only at the beginning of the experiment is the error large because the convergence estimation itself needs a certain amount of time, and the effect is much better at the end of the experiment. To analyze the error effect more intuitively, the error curve is compared with the results obtained by the extended Kalman filtering method. The results are shown in Fig. 5.27. Wherein S1 describes the estimated battery state value by using the adaptive Kalman filtering method. S2 describes the estimated battery state value by the extended Kalman filtering calculation. S3 describes the measured battery state value by using the ampere hour integral method. Err1 describes the estimation error by comparing S1 and S3. Err2 describes the estimation error by comparing S2 and S3. By analyzing the comparison between the measured and predicted states, the predicted value can follow the state accurately. The followability and real-time performance of the algorithm meet the requirements, in which the error of the algorithm is within 0.03%. It can be further seen from the above figure that the effect of the adaptive Kalman filter on estimating the battery state is significantly better than that of the extended Kalman filter. To verify the estimation effect, the time is reduced corresponding to the interval width of the time axis coordinates. 0.05

1.0

S1 S2

0.00

Err (1)

S(1)

0.8 0.6 0.4

-0.05 -0.10 -0.15 -0.20

0.2

(A)

-0.25 0

2000

4000

6000

t (s)

8000

10000

12000

(B)

0

2000

4000

6000

8000

10000

12000

t (s)

FIG. 5.26 Comparison of the predicted state results. (A) Estimation result and comparison. (B) Estimation error variation trend.

191

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

0.012 S1 S2 S3

0.008 0.006 0.004 0.002 0.000 -0.002

0

(A)

Err1 Err2

0.010

Err(1)

S(1)

5.4 Experimental result analysis

2000

4000

6000

8000

10000

12000

t (s)

-0.004

(B)

0

2000

4000

6000

8000

10000

12000

t (s)

FIG. 5.27 Algorithm error analysis compared with ampere hour calculation. (A) Estimation effect and comparison. (B) Estimation error comparison.

5.4.5 Complex current rate verification The Beijing bus dynamic stress test is used to test the LFP50 Ah ternary lithium-ion battery. The battery charge-discharge equipment is the BTS750–200–100-4 battery testing equipment provided by Shenzhen Yakeyuan Technology. The steps are set and the experiments are carried out as shown in Table 5.1. The test is the working condition obtained from the measured data collection of Beijing buses. Ph (kW) is the battery output power of the measured bus under the conditions of starting, accelerating, and taxiing. As the 50 Ah battery is tested, the Ph is reduced to obtain the data onto Pc (W). As can be seen from the table, the complete working time is 300 s. The 20 working-time tests are carried out on the battery. Its working time data can be obtained as shown in Fig. 5.28. The current and voltage data onto the experiment are conducted as shown in Fig. 5.28A and B. The current and voltage data onto the experiment are investigated under the 20 test conditions, as shown in Fig. 5.28C and D. Because the experimental condition is constant power discharge, the overall voltage curve of the battery gradually decreases along with the number of cycles and the current curve also increases gradually. The second-order resistance-capacitance equivalent circuit model is established by the parameter identification with hybrid pulse-power characteristic test data. To verify the validity of the model, the model data are compared and analyzed with other battery condition data. The test is used to verify the modeling effect, and the constant power discharge at a certain time is used to simulate various working conditions. The current value in the experimental data obtained by the test equipment is taken as the input controlled parameter. The terminal voltage is obtained through the simulation model and the experimental terminal voltage is compared. Then, the results are obtained as shown in Fig. 5.29. Wherein U1 is the curve of the measured terminal voltage data obtained by the test equipment. U2 is the output terminal voltage curve obtained by the simulation model under the input current condition. As can be seen from the figure, the variation trend of the simulation curve is similar to that of the test curve, and can better simulate the battery discharging characteristics effectively. According to the established second-order resistance-capacitance equivalent circuit model and the extended Kalman filtering algorithm, the state estimation is carried out for the Beijing bus dynamic street test working condition. The error between

192

5. Battery state-of-charge estimation methods

TABLE 5.1 Beijing bus dynamic street test operating parameters. Ph (kW)

Pc (W)

Single-step (s)

Cumulation (s)

Working condition

37.5

69

21

21

Start

72.5

135

12

33

Speed up

4.5

9

16

49

Slide

15

27

6

55

Brake

37.5

69

21

76

Speed up

4.5

9

16

92

Slide

15

27

6

98

Brake

72.5

135

9

107

Speed up

92.5

174

6

113

Urgent to accelerate

37.5

69

21

134

Speed up

4.5

9

16

150

Slide

15

27

6

156

Brake

72.5

135

9

165

Speed up

92.5

174

6

171

Urgent to accelerate

37.5

69

21

192

Speed up

4.5

9

16

208

Slide

35

66

9

217

Brake

15

27

6

229

Brake

4.5

9

71

300

Parking

the initial value and the measured value is randomly given in advance, exceeding 20%, so that the state estimation could be carried out. The obtained results are described in Fig. 5.30. Wherein Fig. 5.30C is the estimation result. S1 is the measured state value, and S2 is the state estimation using the extended Kalman algorithm. Fig. 5.30D is the error curve obtained by subtracting two estimated state curves. As can be seen from the figure, the error of the state estimation is < 0.014 based on the established second-order resistance-capacitance model, which can correct the initial error effectively and has a strong correction function. In this experiment, the parameter values are all substituted for the algorithm in the way of average values, including R0, Rp, and Cp. The test object is an AVIC lithium-ion battery of 50 Ah. The algorithm of estimating the battery state is validated with 0.5 C constant-current discharge by combining Eqs. (5.48), (5.49), (5.54), and (5.55) to (5.46). The processing effect is described as shown in Fig. 5.31. Wherein the red curve is the measured state of charge. The black curve is the state curve estimated by extended Kalman filtering. The error analysis results are obtained for the estimation and the measured value of the state. It can be observed in the experimental results that

193

5.4 Experimental result analysis

4

U (V)

I (A)

2 0

-2 -4 0

(A)

1000

2000

3000

4000

5000

(B)

t (s) 1.0

U1 U2

0

1000

2000

3000

4000

5000

6000

t (s)

0.010

S1 S2

0.8

ΔS

0.008

0.6

S (1)

S (1)

6000

30 29 28 27 26 25 24 23 22 21

0.4 0.2

0.006 0.004 0.002

0.0

0.000 0

1000

2000

(C)

3000

4000

5000

6000

0

1000

2000

3000

t (s)

(D)

t (s)

4000

5000

6000

FIG. 5.28 Beijing bus dynamic street test experimental data. (A) Whole procedure current waveform. (B) Corresponding voltage variation. (C) State estimation result and comparison. (D) Estimation error changing curve.

4

U (V)

I (A)

2 0

-2 -4

(A)

0

1000

2000

3000

4000

1000

(B)

2000

3000

4000

5000

t (s) 0.014

S1 S2

0.8

ΔS

0.012 0.010

0.6

S (1)

S (1)

U1 U2

0

5000

t (s)

1.0

0.4

0.008 0.006 0.004

0.2

0.002

0.0

(C)

30 29 28 27 26 25 24 23 22 21

0.000 0

1000

2000

3000

t (s)

4000

5000

(D)

0

1000

2000

3000

4000

5000

t (s)

FIG. 5.29 Comparison diagram of the estimation results. (A) Current variation waveform. (B) Corresponding voltage change waveform. (C) State estimation results and comparison. (D) Estimation error variation.

194

5. Battery state-of-charge estimation methods

4

U (V)

I (A)

2 0

-2 -4

(A)

0

1000

2000

3000

4000

5000

1.0

S1 S2

0.8 0.6 0.4 0.2 0.0

(C)

U1 U2

0

(B)

1000

2000

3000

4000

5000

6000

7000

t (s)

0.020 0.018 0.016 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000 -0.002

ΔS

S (1)

S (1)

6000

t (s)

30 29 28 27 26 25 24 23 22 21

0

1000

2000

3000

4000

5000

6000

t (s)

0

(D)

1000

2000

3000

4000

5000

6000

7000

t (s)

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4

S (1)

S S'

-20

(A)

0

20

40

60

Err (1)

FIG. 5.30 State estimation result and error curve. (A) Current changing waveform. (B) Voltage variation curve. (C) State estimation comparison. (D) Estimation error curve. 0.04 0.03 0.02 0.01 0.00 -0.01 -0.02 -0.03 -0.04

80 100 120 140 160

t (min)

0

(B)

20

40

60

80

100

120

140

t (min)

FIG. 5.31 State estimation result for the constant current discharging. (A) State estimation results. (B) Estimation error.

the state estimation error is less than 4.00%, which is within the allowable error range. As can be known from the figure, the predicted curve gradually approaches the measured curve with the increase in filtering time and fluctuates near the measured curve. However, the value of error also slightly decreases from the larger error at the beginning and finally fluctuates around 0, showing a trend of convergence. The self-discharging current rates and uneven battery aging degree can reduce the long-term use accuracy of this method to estimate the state of the same battery. The designed state estimation algorithm is adopted under the complex working conditions. The experimental results are obtained as shown in Fig. 5.32.

195

1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

S1 S2

S (1)

S (1)

5.5 Conclusion

0

2000

4000

6000

(A)

8000 10000 12000 14000

t (s)

0.005 0.000 -0.005 -0.010 -0.015 -0.020 -0.025 -0.030 -0.035 -0.040

ΔS

0

(B)

2000

4000

6000

8000 10000 12000 14000

t (s)

FIG. 5.32 Validation of the pulse-current test. (A) State estimation results. (B) Estimation error variation curve.

As can be known from the experimental results, the extended Kalman filtering algorithm is adopted to correct the error estimation when the initial state value error is large; it has strong robustness. Besides, the iterative algorithm for the state-of-charge estimation effect is best in the second half of the simulation. The overall estimation error has decreased trends, and the state estimation has a small-scope fluctuation near the measured value. As can be known from Fig. 5.32B, the maximum error is only 0.036, realizing an accurate battery state estimation. Based on the hybrid pulse-power characteristic test, an accurate equivalent model is established. It is used to build the state estimation model. Combined with the extended Kalman filtering algorithm, a filtering program is written in S-function to estimate the state value. Furthermore, the working condition verification experiment is designed to demonstrate the feasibility and accuracy of the estimation method.

5.5 Conclusion In this chapter, the battery state-of-charge estimation was adopted by considering the advantages of low error, long-time test, polarization effect error, and transient analysis applicable to the power battery charge-discharge process. The proposed algorithm was used to reduce the impact on the nonlinear equation effectively and realize the power battery state estimation. The experimental results are accurate with detailed procedure design. The algorithm complexity is moderate, and has a certain reference value for the rational distribution of power lithium-ion batteries.

Acknowledgments The work is supported by the National Natural Science Foundation of China (No. 61801407), the Sichuan Science and Technology Program (No. 2019YFG0427), the China Scholarship Council (No. 201908515099), and the Fund of Robot Technology Used for Special Environment Key Laboratory of Sichuan Province (No. 18kftk03).

Conflict of interest There is no conflict of interest.

196

5. Battery state-of-charge estimation methods

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C H A P T E R

6 Battery state-of-energy prediction methods 6.1 Overview Electrical vehicles powered by lithium-ion batteries are more environmentally friendly than gasoline-powered ones, and this kind of environmentally friendly energy is gradually being used in various fields. In such high demand, the battery state is particularly important because it can be estimated. To support the electric car driving distance, the influence of such factors as long-term use can lead to a measurement error cumulate expansion due to the voltage, resistance, and temperature [1–4]. The highly nonlinear characteristic also brings to the battery state estimation lots of difficulties. To solve this issue, several algorithms are proposed and used to estimate the battery state, including the current integration, open-circuit voltage, Kalman filtering, and neural network [5–8]. The current integral method neglects the influence of the battery self-discharging current rate, aging degree, and charge-discharge current rate on the battery state. Long-term usage contributes to measurement error accumulation and expansion, so relevant correction factors need to be added to correct the accumulated error. Before the open-circuit voltage measurement, the target battery must be set for more than 1 h, so that the electrolyte inside the battery is evenly distributed to obtain stable terminal voltage. Therefore, the battery state cannot be measured in use. The state estimation accuracy is largely dependent on the accuracy of the battery model. Because the power battery operating characteristics are highly nonlinear, it is necessary to establish an accurate battery model to estimate the accurate state. The initial workload of the neural network is relatively large, and it needs to extract numerous comprehensive target sample data to train the system [9–12]. The input training data and methods can affect the state estimation accuracy to a large extent. Besides, its long-term use accuracy to estimate the state of the same battery group can be greatly reduced under the complex effect of factors such as battery temperature, self-discharging current rate, and the varying degree of battery aging.

Battery System Modeling https://doi.org/10.1016/B978-0-323-90472-8.00005-6

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Copyright # 2021 Elsevier Inc. All rights reserved.

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(1) Various experimental methods are integrated, such as hybrid charge-discharge, constant-voltage charging with different multipliers, constant discharge, and cyclic discharge shelved experiments. These are used to study the lithium-ion battery operating characteristics and analyze the characteristics of the battery response under different conditions. (2) The equivalent circuit model is set up, using the hybrid pulse-power characteristic experiment method of research to identify the equivalent circuit model’s parameters and the circuit model parameters in different phases of the discharge relationship. To establish the dynamic parameters of the equivalent model of batteries, the validation model could fully express the battery working state [13, 14]. (3) A comprehensive analysis is conducted for a variety of battery state estimation algorithms. First of all, the open-circuit voltage method is used to estimate the initial value together with its calibration. Then, it is used for the real-time state estimation considering the high-precision accuracy.

6.2 Iterative algorithm and realization The unscented Kalman filtering algorithm is applied to estimate the battery state, in which the equivalent circuit model is used to record the experimental data onto the pulse chargedischarge of the hybrid pulse-power characteristic test analysis together with its data processing. Then, a more accurate model is obtained.

6.2.1 Equivalent modeling The models for lithium-ion batteries are three types: electrochemical models based on the internal chemical reactions in the battery, neural network models that simulate human brain work, and equivalent circuit models built using electronic components. The electrochemical model is divided into the Shepherd model and the Nernst model. The electrochemical model is complex to apply in practical products. It is mainly used to assist battery design and manufacture. The neural network model is theoretically suitable for battery modeling. However, it needs lots of data onto training that has the characteristics of a high technical threshold and long processing period. The estimation error is large when the sampling data are insufficient, in which the convergence and stability of the algorithm are not guaranteed for the time being. Its popularization and application are in the stage of a simulation experiment. The equivalent circuit model is simple in physical meaning, so the mathematical expression is simple. Its estimation accuracy and stability are optimized in three modeling types. The equivalent circuit model contains many kinds of model structures. The simple and practical model structure shortens the development cycle, which reduces the development cost, but the implementation accuracy is often low. Complex models improve the accuracy of the algorithm; the difficulty of model parameter identification increases accordingly. Therefore, we should consider all factors for the accuracy and complexity requirements in practice, choosing the appropriate battery model. The internal resistance equivalent circuit model regards the battery as an ideal voltage source. However, it is not suitable for long-term simulations and it is often used for the transient analysis of batteries due to the current accumulation problem. The second-order

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resistance-capacitance model is based on the Thevenin model. It has a couple of electrical circuit models added to balance the battery steady-state characteristics and transient characteristics. However, the effects of self-discharge and temperature are not taken into consideration. The PNGV model added a capacitance Cp to describe the current accumulation problem based on the model. It has the advantages of easy implementation, which is suitable for simulating the battery dynamic model but does not consider the charging process. Hence, it is not suitable for long-term simulations. It has a very strong impact on short-term simulations. According to the above introduction, the PNGV model has the advantages of a simple model, high precision, and less difficulty in parameter identification, which satisfies the battery model accuracy requirements. Therefore, it is constructed to explain a series of problems with the battery equivalent process [15]. The model parameter identification combined with the hybrid pulse-power characteristic test is the standard battery model in the battery test manual. Compared with the equivalent circuit, Cb is added to describe the change in open-circuit voltage, and its model structure is described in Fig. 6.1. Wherein UOC is the ideal voltage source. R0 is internal resistance. The parallel circuit using Rp and Cp reflects the generation of the battery polarization process, in which Rp is the polarization resistance and Cp is the polarization capacitance. The internal battery operation state is reflected through the polarization process, thereby reducing the production and working environment influence greatly. Based on the model, the discharge direction is taken as the positive direction. The voltage across the internal polarization resistance is Up. The voltage across the storage capacitance Cb is Ub. The parameter relationship of the model can be obtained according to Kirchhoff’s voltage law, which is described in Eq. (6.1): UL ¼ UOC  Up  Ub  R0 I

(6.1)

A series-connected resistor has the advantages of easy modeling, parameter measurement, and simulation. Its accuracy is not high, so it cannot reflect the battery recharge-discharge characteristics. In the transition process, the first-order electrical circuit models are used to equate the battery circuit of a voltage source and a parallel resistance capacitance. By analyzing each functional relationship, a time-domain relation equation can be established, as shown in Eq. (6.2):
Cb  dUb =dt ¼ I (6.2) Cp  dUp =dt ¼ I  Up =Rp

FIG. 6.1 PNGV electrical equivalent model.

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6. Battery state-of-energy prediction methods

Because the circuit contains two capacitive components of Cb and Cp, the circuit is a secondorder circuit. The resistance-capacitance parallel circuit is used to simulate the dynamic battery charge-discharge process, which can simulate the energy transition process accurately. The second-order differential equation of the circuit model can be obtained by transforming, as shown in Eq. (6.3):        0 0 ub 1=Cb dub =dt ¼ + (6.3) 0 1=Rp Cp up 1=Cp dup =dt According to the above model analysis, the model parameters need to be identified with the correlation experiment, including UOC, Cb, R0, Cp, and Rp.

6.2.2 Mathematical description By the algorithm program design, the error estimation and comparative analysis are conducted between the estimated value of different working conditions. For a nonlinear system, its state and measurement equations are described as shown in Eq. (6.4):
Xk ¼ f ðXk1 , Uk Þ + Wk (6.4) Yk ¼ gðXk1 Þ + Vk wherein k is the time point and f(Xk1, Uk) is the nonlinear system state transformation equation. g(Xk1) is the nonlinear measurement equation. Xk1 is the state variable and Uk is the known input. Yk describes the measurement signal. Wk is the processing noise and Vk is the measurement noise. Assuming that Wk and Vk are uncorrelated Gaussian white noise with zero mean, their covariances are initialed as Qw and Rv, respectively. Then, the specific process can be obtained accordingly. The initial value can be calculated as shown in Eq. (6.5): 8 x ¼ E ð x0 Þ > > > > > > P0 ¼ E ð x 0  x Þ ð x0  x Þ T > >   < yik|k1 ¼ g xik|k1 + vik1 (6.5) > > > 2L 2L h   i > X X > > i i i > ωm ωm > i g xk|k1 + vk1 ¼ i yk|k1 : yk ¼ i¼0

i¼0

The unscented Kalman filtering algorithm is used to estimate the state, which could reduce the system error and correct the initial state deviation effectively [16–19]. Therefore, the proposed adaptive unscented Kalman filtering algorithm performs two unscented transformations at the cyclic data sampling points when dealing with the nonlinear battery state estimation system. Its initial parameter values are set as shown in Eq. (6.6):   8 xh0+ ¼ E X0x , R0+ ¼ EðRi0 Þ > > k ¼ 0, ^ <   T + ¼ E X0x  ^x0+ X0x  ^x0+ Px0 (6.6) h i > > : P + ¼ E R  R + R  R + T R0

0

0

0

0

After obtaining the sigma point set, its nonlinear transformation is carried out directly. Then, the cyclic iterative relationship is established by the probability state density functional

6.2 Iterative algorithm and realization

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calculation. The unscented transformation is used to construct the sigma data point set and its corresponding weight coefficient ω related to the state quantity [20–22]. The extended state variable is set as Xk and the sigma point set through the extended state variable is set as PX, k, which is used for the square root treatment of the matrix, as shown in Eq. (6.7):

( Xk ¼ xTk , wTk , vTk , PX, k ¼ diagfPX, k , Q, Rg (6.7) pffiffiffiffiffiffiffiffiffiffiffiffiffiT pffiffiffiffiffiffiffiffiffiffiffiffiffi PX,k1 ¼ PX, k1 PX,k1 Then, the estimated state quantity of the previous step can be obtained, so the selection method of the sigma data point set is calculated as shown in Eq. (6.8): 8 ^ > ¼0 < Xk1 ¼ Xk1 , i p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ Xk1,i ¼ Xk1 + ðn + λÞPX, k1 , i ¼ 1, 2, …,n (6.8) > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p : ^ k1  ðn + λÞPX, k1 , i ¼ n + 1, 2,…,2n Xk1,i ¼ X In this equation, the column number for the square root of the state matrix is represented by the parameter i. Setting the sigma point number as 2n + 1, the arithmetic about the sigma point weight is described as shown in Eq. (6.9): 8   λ λ > < ωm , ωe0 ¼ + 1  α2 + β 0 ¼ n+λ n+λ (6.9) 1 > ωm ¼ ωe ¼ : , i ¼ 1, 2,…,2n i i 2ð n + λ Þ In this equation, ωm is the weight coefficient of the average estimated state value. ωe describes the weight of the estimated covariance. n is the dimension of the expanded state variable, which is initialed as n ¼ 7. α describes the extent to which the sigma pointing dataset deviated from the estimated state value, as shown in Eq. (6.10): 2

3T ^ k1 X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6^ x n + λÞPX, k1 i 7 ¼6 Xk1 4 Xk1 + pðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

5 ^ k1  X ðn + λÞPX, k1 i

(6.10)

After getting the data onto the above equation, the weight coefficient ωi and the prediction of the sigma points are combined to get the battery state situation. The prediction process of the state data and weight parameter is obtained as shown in Eq. (6.11): 8
h 

X 2n i > > x x > ¼ ¼ E F X , I ωm X > k1 k|k1 i Xk1, i k1, i > < i¼0,1,…,2n i¼0 > 2n h   T >  T i X > e x x > > x P ¼ E x  X  X ω X  X  X +Q ¼ X k k x, k|k1 k|k1 k|k1 k|k1 k|k1 : i k|k1, i k|k1, i i¼0

(6.11) The sigma point scaling parameter is then used to decline the total prediction error. After experimental verification, the best choice should be β ¼ 2. Using Eq. (6.11), the parameters are

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6. Battery state-of-energy prediction methods

set for the next calculation step. The prediction of the sigma points is realized by one step, as shown in Eq. (6.12):   8 x x > ¼ F X , I X , i ¼ 0,1,…, 2n k1 > k|k1, i k|k1, i > > 3T 2 < ^ k|k1 X (6.12) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6^ > Xx > ¼ X + ð n + λ ÞP 5 4 > x, k1 k|k1 > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i : k|k1, i ^ k|k1  X ðn + λÞPx, k1 i According to the data from the product and using the unscented transformation exchange again, the new sigma points can be calculated accordingly [23–27]. To obtain the one-step prediction of the quantity, the sigma data point set obtained in the last step is inserted into the observation equation. The establishment of the sigma data points is described as shown in Eq. (6.13): 8 ¼0 > < xk1 , i p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i x + ðL + λÞPk1 , i ¼ 1, …, L (6.13) xk1 ¼ k1 > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : xk1  ðL + λÞPk1 , i ¼ L + 1, …,2L On this basis, the equivalent circuit model is introduced into the iterative calculation process to characterize the battery operating characteristics. Considering the importance of the early state estimation accuracy for later estimation, the open-circuit voltage is used to calibrate the estimated initial value. Then, the established observation equation can be obtained as shown in Eq. (6.14): n o x yk|k1 ¼ G Xk|k1,i , Ik , k ¼ 1, 2, …,2n + 1 (6.14) The predicted values and covariance matrices are obtained by a weighted sum of one-step predicted values. After that, the observed parameter value can be obtained as shown in Eq. (6.15): 8 2n n   o X > > > yk ¼ E G Xkxjk1, i , Ik ; i ¼ 1, 2, …, 2n ¼ ωm > i ykjk1, i > > > i¼0 > > < 2n X   T (6.15) Pyyk ¼ wci ykjk1  ^yk ykjk1  ^yk + R > > i¼0 > > > 2n > X   T > > > P ¼ wci ykjk1  ^yk ykjk1  ^yk : xyk i¼0

The adaptive unscented Kalman filtering algorithm uses a set of sigma points to get the battery characteristics, which is different from the double Kalman filtering algorithm. The progress of getting the Kalman gain matrix is described as shown in Eq. (6.16): Kk ¼ Pxyk + P1 yy k

(6.16)

According to the above equations, the battery characteristics can be described effectively after running in the unscented Kalman filtering arithmetic. It is then necessary to calculate the battery state and the updated covariance, as shown in Eq. (6.17):

6.2 Iterative algorithm and realization

(

  ^k ¼ X ^ k|k1 + Kk yk  ^y X k Pxk ¼ Pxk|k1  Kk Pyyk Kk T

205

(6.17)

Then, the unscented Kalman filtering algorithm is also used for the output voltage tracking to realize the real-time high-precision state estimation of the lithium-ion battery. The state update equation is described as shown in Eq. (6.18): 8 2L   X > > i i > xi ¼ f x x ¼ ωm , > i xk|k1 k|k1 k < k|k1 i¼0 (6.18) 2L   T X > > c i i > > ωi xk|k1  x xk|k1  x + Qk : Pk|k1 ¼ i¼0

wherein xkj k1 is the estimated value at the time point of k from the time point of k  1. An accurate iterative calculation treatment of the lithium-ion battery is also realized for real-time state monitoring, which greatly influences battery safety control. To solve the problem of difficult real-time estimation and low accuracy of the state monitoring under various working conditions, the operating battery characteristics are studied and analyzed by integrating various experimental methods of different working conditions. Then, the measurement update equation is obtained as shown in Eq. (6.19): 8 2L   T X > c i i > > ¼ ω x  x x  x + Qk P > k|k1 i k|k1 k|k1 > > > i¼0 > > > 2L <   T X Pxy, k ¼ ωci xik|k1  x yik|k1  y (6.19) > > i¼0 > >   > 1 > > > K ¼ Pxy, k Py, k , xk ¼ xk1 + K yk  yk1 > > : Pk=k ¼ Pk=k1  KPxy, k1 KT By repeating the four calculation steps above, the optimal state estimation can be realized, in which the state parameter Xk at the time point of k is obtained as long as the initial conditions X0 and P0 are given. According to the state value at the time point of k  1, the observed value is estimated by taking the state parameters at the time point of k as input.

6.2.3 Iterative calculation procedure The built-in simulation model of the data performance analysis is combined with various working conditions. The experimental results show that the established estimation model can effectively predict the battery state. It has many advantages, such as convergence speed and good tracking performance [28–33]. According to the references, the unscented Kalman filtering algorithm can reduce the estimation error effectively, which has the advantage of high accuracy in the iterative calculation of battery state estimation. According to the mathematical modeling effect analysis, the calculation expression is obtained as shown in Eq. (6.20): EðtÞ ¼ UL ðtÞ + Up ðtÞ + Ub ðtÞ + R0 I ðtÞ

(6.20)

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6. Battery state-of-energy prediction methods

wherein E(t) is the electromotive force of the power supply. Its relationship between the state parameter S(t) can be described by the nonlinear functional curve fitting treatment. UL(t) is the terminal voltage. R0 is internal resistance. Then, S(t) is obtained by combining the ampere hour integration treatment and these parameters, as shown in Eq. (6.21): ðt ηI ðtÞ dt (6.21) SðtÞ ¼ Sðt0 Þ  0 Q The state parameter S(t) is selected as the system state quantity and the terminal voltage UL(t) is used as the observed quantity. I(t) can be used as the system input quantity. As the battery state estimation process is a discrete system, the relationship between S(k + 1) and S(k) is obtained after discretization, as shown in Eq. (6.22): Sðk + 1Þ ¼ SðkÞ  ΔT  I ðkÞ=Q + vðkÞ

(6.22)

The state equation of S(k + 1) and S(k) represents the system state value at the time point of k + 1 and k. Q is the capacity of lithium-ion batteries and ΔT is a unit time that is taken as the sampling time. I(k) can be used to describe the current size. Taking v(k) for noise, the system measurement equation is obtained as shown in Eq. (6.23): UðkÞ ¼ f ½SðkÞ  R0 I ðkÞ  Up ðkÞ  Ub ðkÞ + vðkÞ

(6.23)

In the above equation, U(k) can describe the electromotive force value of the power supply at the time point of k. f[S(k)] represents the nonlinear relationship between the output-voltage and the battery state. R0 is internal resistance. I(k) can be used to describe the current measured at the time point of k. The voltage across the storage capacitance Cb at the time point of k is Ub(k). Up(k) is the polarization voltage value at the time point of k and v(k) is its noise. As for the nonlinear lithium-ion battery state estimation problem, the unscented Kalman filtering uses the forced system linearization and carries out the probability density estimation on the state variables. It is realized with an appropriate sampling strategy based on unscented transformation to obtain the optimal solution.

6.2.4 Parameter initialization strategy The unscented transformation is a kind of solution to the nonlinear battery state estimation problem, and it is an important part of the nonlinear Kalman filtering algorithm. Its idea is realized according to the statistical features of state variables, in which a certain sampling treatment is conducted for the corresponding finite number of sampling data points [34]. The collected data points should have the same mean and covariance value compared with its initial state. Then, these data points are introduced into the nonlinear transformation function, obtaining the corresponding nonlinear function value point set. Consequently, the point set of average and covariance is calculated after the transformation. An asymmetric sampling strategy is introduced into the calculation process. The dimension of state variable x is set to be n. Then, the average and covariance matrix can be calculated respectively by taking y as the

6.2 Iterative algorithm and realization

207

observed variable. Finally, 2n + 1 sampling points can be obtained in the nonlinear system y ¼ f(x) as shown in Eq. (6.24): 8 i x,i ¼ 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n), the QR decomposition is expressed as A ¼ Q  R, where Q is the m  m matrix and R is the m  n upper triangular matrix. The upper triangular part is the transposition of the Cholesky factor of P. As for the Cholesky factor updating, if S ¼ chol(P), the Cholesky decomposition of the matrix is updated as S ¼ Cholupdate (S, u,  v). The Cholupdate function is the update function of Cholesky decomposition. In the square root unscented Kalman filtering, the square root S is passed by instead of the original covariance P.

6.3.2 Calculation algorithm flow The square root unscented Kalman filtering algorithm mainly includes four parts: initialization, sigma point acquisition, time updating, and status updating. The details and the initial value are determined as shown in Eq. (6.31): 8 x0 ¼ Eðhx0 Þ >

: S0 ¼ cholðP0 Þ Initialization is to determine the initial value of the state variable and the initial value P0 of the error covariance. S0 is the Cholesky factor of the covariance. Sigma point acquisition is described as shown in Eq. 6.32: 8 i ^ i¼0 > < xk1 ¼ xk1 , p ffiffiffiffiffiffiffiffiffiffiffiffiffi i (6.32) xk1 + ðn + λÞSik1 , i ¼ 1  n xk1 ¼ ^ > pffiffiffiffiffiffiffiffiffiffiffiffiffi in : i xk1  ðn + λÞSk1 , i ¼ n + 1  2n xk1 ¼ ^ Sik indicates the Cholesky factor of the state variable covariance at the time point of k and column number of i. The average weight ωm and the variance weight ωc are consistent with the unscented Kalman filtering algorithm. According to the state variable at the time point of

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6.3 Improved prediction and correction

k  1 and the value of the input factor, the state variable is predicted in one step by the state equation. Its expression is described as shown in Eq. (6.33): 2n   X ωim xik|k1 xik|k1 ¼ f xik1|k1 , uk1 ) ^xk|k1 ¼

(6.33)

i¼0

The QR decomposition that is performed on the error covariance of the state varies according to the one-step prediction of the sampling point, as shown in Eq. (6.34):
qffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffi 1:2n x1:2n  ^ ¼ qr ω x (6.34) S k|k1 , Qk c xk k|k1 To guarantee the semidefiniteness of the matrix, Sxk represents the square root update value of the state variable error covariance at the time point of k. Considering that the values of α and k may cause a negative value of ω0c , the improved calculation processes are used as shown in Eq. (6.35):


qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    0  0  0 ^ abs ωc xk|k1  xk|k1 , sign ωc Sxk ¼ Cholupdate Sxk , (6.35) According to the one-step prediction result of the state variable in Eq. (6.33), the prediction value of the observed variable can be obtained from the observation equation as shown by Eq. (6.36): 8 2n   X > > i i > ^ ¼ h x , u ¼ ωim yik|k1 , y y k > k|k1 k|k1 k|k1 > > < i¼0  pffiffiffiffiffiio nhpffiffiffiffiffiffiffiffiffiffi (6.36)  1:2n y1:2n  ^ S ¼ qr ω x , Rk > k|k1 c yk k|k1 > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n  >    o > : S ¼ Cholupdate S , abs ω0 y0  ^y , sign ω0 yk

yk

c

k|k1

k|k1

c

wherein Syk represents the square root updated error covariance for the observed variable at the time point of k. The cross-covariance of the state variable and its observed value are described as shown in Eq. (6.37): Pxy,k ¼

2n X

h ih iT ωic xik|k1  ^ xk|k1 yik|k1  ^yk|k1

(6.37)

i¼0

Consequently, its value affects the magnitude of the Kalman gain directly. As its accuracy affects the state estimation effect, its value should be calculated considering multiple influencing factors; the calculation process can be described as shown in Eq. (6.38):  1 (6.38) Kk ¼ Pxy,k Syk STyk In the square root unscented Kalman filtering algorithm, the time update parameter Syk uses the Cholesky decomposition method to calculate the square root of the state variable covariance matrix, aiming to initialize the filtering structure. The state variable update and its error covariance update are described as shown in Eq. (6.39):

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6. Battery state-of-energy prediction methods

(

  ^ xk|k ¼ ^ xk|k1 + Kk yk  ^yk|k1   Sk ¼ cholupdate S xk , Kk Syk ,  1

(6.39)

wherein Yk is the experimental measurement at the time point of k. In the iterative calculation process, the parameters are initialized by calculating the square root state variable covariance matrix by conducting the Cholesky factorization. In subsequent iterations, the propagated and updated factor directly formed the Sigma point, which overcomes the poor stability shortcomings of the unscented Kalman filtering algorithm that ensures the semidefiniteness of the covariance matrix.

6.3.3 Covariance matching As for the lithium-ion battery operation, the external measurable parameter signal measurement is limited and there is an error. Meanwhile, the noise introduced by discretized digital sampling and iterative computation is difficult to reduce greatly, which causes the cumulative error problem of the state estimation process. In the standard unscented Kalman filtering algorithm, the noise variance is often difficult to obtain. Usually, the system and the measurement noise covariances are set to a fixed value, which simplifies the calculation amount. However, the inaccurate noise statistical characteristics can reduce its estimation accuracy and even make the filtering diverge. Aiming at the problem, the adaptive noise covariance matching is introduced, in which the noise covariance matrix is updated automatically and transmitted to make it closer to the measured noise condition. This algorithm is called an adaptive square root unscented Kalman filtering algorithm. In view of the fact that the statistical characteristics of prior noise do not match the actual situation, the accuracy of Kalman filtering is reduced. An adaptive unscented Kalman filtering algorithm is proposed by conducting the on-line estimation of the noise statistical characteristics as well as its innovation sequence. The innovation of the observed variable at time point k is defined, and real-time tracking is realized for the observed noise covariance matrix, which is used to realize the optimal estimation. The innovation is mainly determined by the error, which is the difference between the observed value and the predicted value. The new covariance can reflect the influence of the present time point error well, in which the variance between the previous M innovations is weighted and averaged to obtain the Hk expression. Therefore, it is a real-time estimation covariance function obtained by the window estimation principle, in which M represents the window size. The updating process of the system and observed noises are described as shown in Eq. (6.40): 8 k X > < ^k|k1 , Hk ¼ ek eTk =M e k ¼ yk  y (6.40) i¼kM + 1 > : T T Qk ¼ Kk Hk Kk + Hk , Rk ¼ Hk  Ck Pk Ck wherein the system noise and the observation noise are inseparable from Hk. To improve the estimation accuracy while minimizing the computational complexity, the first three innovations are calculated, that is, M ¼ 3. Because Pk gradually decreases along with time and it eventually tends to 0, the influence of the CkPkCTk part can be neglected.

6.3 Improved prediction and correction

213

6.3.4 Improved correction strategy When calculating the innovation estimation covariance matrix, it is necessary to window average the sample values of the innovation sequence. Consequently, there is a problem with selecting the window function window size. If the window function is too small, there is significant noise in the estimated covariance matrix. On the contrary, if the window is too large, the covariance matrix estimation value reflects the transient characteristics of the battery system. In theory, the size of the window M can take any value from 1 to k. When M ¼ 1, the calculation amount is the smallest, but the algorithm has the worst effect. When M ¼ k, the algorithm works best but the calculation is the largest. Artificially initialing the value of M to 3 still does not give enough good filtering effect and tracking effect. On this basis, the adaptive window factor d is defined to dynamically determine the window size. The advantage of this method is that its determination efficiency is relatively high. It can be quickly converged, which is suitable for real-time filtering. The filtering effect and the calculation amount are considered comprehensively so that the two can reach a suitable balance point. There are no comparative statistical characteristics of the measuring noise. Consequently, the filtering divergence can be avoided effectively, making the estimated value smooth and accurate. The relevant covariance matrix defining the innovation is described as shown in Eq. (6.41):   (6.41) ð1Þ Ek ¼ E ek eTk ¼ Pyy, k ) ð2Þ d ¼ eTk E1 k ek The adaptive window factor d represents the measured residual value, and the expression is realized. The role of d is to dynamically adjust the size of the adaptive window M. The discriminant for determining the size of the adaptive window M is designed. μmin and μmax are the decision thresholds, as shown in Eq. (6.42): 8 < M ¼ 1, d  μmax (6.42) M ¼ k, d  μmin : M ¼ k  ηdμmin , μmin < d < μmax As can be known from experience, the parameter values can be initialed as μmin ¼ 0 and μmax ¼ 1. η represents the convergence rate of the window M, which is any fraction less than 1. k represents the present time. The adaptive window length is a minimum of 1 and a maximum of k. When the innovation correlation matrix is large, the calculated d value is less than 0. It indicates that the difference is large between the predicted value and the measured value. At this time, the window opening window should be increased to make the predicted value closer to the measured value. When M ¼ k is taken and the innovation correlation matrix is small, the calculated d is greater than 1. It indicates that the predicted value follows better. The window opening window should be reduced to reduce the system calculation amount. Taking M ¼ 1, when 0 < d < 1, M ¼ k  ηdμmin takes the appropriate value. Then, the iterative calculation procedure is designed, in which the state matrix and covariance are brought into step-to-step correcting treatment. The repeated prediction-correction process is conducted to update its state matrix, making the constant state approach the measured value. The available capacity is calibrated by taking into account different influencing factors, describing its influencing effect on state estimation obtained by the ampere hour integral. Considering the long-term battery aging process, the factors affecting the maximum usable capacity are divided into two scales. The cyclic aging and capacity calibration

214

6. Battery state-of-energy prediction methods

experiments are designed to explore battery remaining useful life attenuation mechanisms for different temperatures and current rate conditions. Based on the theoretical and experimental analysis, an innovative composite correction factor “F” is proposed. The process of putting forward and calibrating composite correction factors is described as shown in Fig. 6.3. Considering the temperature, state of health, and charge-discharge current influence on specific parameters of the ampere hour integral, the state of health is taken as the main factor that influences its Coulomb and charge-discharge efficiency. Then, the innovative capacity attenuation factor k is proposed to characterize the effect of Coulomb and charge-discharge efficiency on the available battery capacity. Therefore, it is important to calibrate the available battery capacity for specific wide-temperature-range conditions. The available capacity calibration corrects the state estimation error effectively caused by the Coulomb and chargedischarge efficiency changes considering its influence. When calibrated for different temperatures and charge-discharge current influence, it improves the state estimation effect on extreme temperature and current rate conditions. This treatment solves the poor state estimation problem influenced by low temperature and high discharging current rates. After an analysis of its merits and demerits, the combined ampere hour treatment has sufficient accuracy in the estimation process under the influence of extreme temperature, high current rate, and time-varying state of health. Consequently, the available capacity change is analyzed on the state estimation of different temperature conditions. The improved ampere hour integral formula is constructed as shown in Eq. (6.43): ðt I=CT dt (6.43) SðtÞ ¼ ST ðt  1Þ  η t1

wherein S(t) is the battery state for temperature T. ST(t  1) is the predicted value converted from a previous time point of temperature T. CT is battery capacity considering temperature influence. η indicates Coulomb efficiency. This method considers the available capacity change of the temperature variation, which determines the battery state conversion that varies from the previous temperature to the present state conditions, improving the ampere hour integral performance for the low-temperature environment.

FIG. 6.3 Composite correction factor putting forward process.

215

6.4 Experimental results analysis

As the influence of state-of health and current rate on the state estimation cannot be neglected, an adaptive improved integral calculation is realized. The parameter changes are considered under the influence of three main factors: temperature, charge-discharge current, and state of health. Its shifting rule is further analyzed, according to which an improved composite correction factor is proposed for the ampere hour integral process. Its adaptive calculation formula is described as shown in Eq. (6.44): ðt ðt (6.44) SðtÞ ¼ SðT, RÞ ð0Þ  Idt=½k∗CðT, RÞ ¼ SðT, RÞ ð0Þ  Idt=F 0

0

wherein S(T, R)(0) refers to the state converted to present temperature and charge-discharge current from the previous time point by a combined analysis of Fig. 6.3 and Eq. (6.44). In the iterative state calculation process, the predicted value is updated by assignment and conversion, which is also corrected by real-time measured closed-circuit voltage before each cyclic calculation. S(t) is stated at the present temperature and current rate conditions. k is a capacity attenuation factor that is set as 0.8–1.0. C(T, R) refers to the maximum available capacity considering the temperature and current influence. F represents the composite capacity correction factor. Using the improved ampere hour integral formula, the state estimation performance is optimized in the following scenarios. The working conditions of lithium-ion batteries are affected greatly by the temperature, considering environmental factors such as day-night temperature difference, seasonal changes, and regional climate variation. As the average discharging current varies frequently from its power supply application, the internal parameters and various performances of high-power lithium-ion batteries decline gradually along with the increasing cyclic charge-discharge time. Therefore, the capacity composite correcting factor plays a great role in improving its power supply performance.

6.4 Experimental results analysis An experimental test is carried out using a lithium-iron-phosphate battery. The upper limit voltage of charging is 3.65 V, the lower limit of discharge is 2.00 V with the rated capacity of 10.00 Ah. First, the selected second-order resistance-capacitance model needs to be parameterized, in which the parameter identification and state estimation verification experiments are carried out.

6.4.1 Parameter identification The open-circuit voltage identification is conducted at a room temperature of 25°C. The battery is fully charged, and is then shelved for 90 min each time before the 10% discharge. At this time, the closed-circuit voltage is used as the open-circuit voltage. The open-circuit voltage toward the state relationship curve is obtained as shown in Fig. 6.4. Moreover, the response from the resistance-capacitance model to the state is well determined. The model fully considers the state change of the circuit response, which can change

216

6. Battery state-of-energy prediction methods

FIG. 6.4 Open-circuit voltage calibration. (A) Voltage variation of the battery cell. (B) Packing voltage change process.

the equivalent circuit parameters responding to the battery state variation. The relationship of the fitting curve is described as shown in Eq. (6.45): UOC ¼ 5:777S7  24:54S6 + 38:26S5  26:77S4 + 8:009S3  0:8639S2 + 0:2837S + 3:18

(6.45)

The model parameters are influenced by temperature and other environmental factors; this is a first-order resistance-capacitance segment of high accuracy. The parameter identification is conducted as well for other parameters such as R0, R1, C1, R2, and C2. The test curve is described as shown in Fig. 6.5.

FIG. 6.5 Hybrid pulse-power characteristic test curve. (A) Overall test voltage variation. (B) Single pulse test voltage variation. (C) Voltage change in one cycle. (D) Enlarged voltage variation.

6.4 Experimental results analysis

217

R0 is equal to the voltage drop average ratio to the current at the beginning and the end of the discharging process. After pulse discharge, the battery is held on to stabilize its internal state. Consequently, the equivalent resistance-capacitance circuit works in a zero-input response situation. The terminal voltage equation is described as shown in Eq. (6.46): UL ¼ UOC  U1ð0 + Þ et=τ1  U2ð0 + Þ et=τ2

(6.46)

wherein U+)1(0 is the initial voltage value for the discharging end time point resistancecapacitance circuit voltage. U2(0+) is the initial voltage value of the discharging process. Using the least-square fitting method, the pulse discharging voltage curve can be obtained for τ1 and τ2. In the pulse discharging process, the resistance-capacitance circuit has a zero-state response, and the terminal voltage equation can be obtained as shown in Eq. (6.47):     (6.47) UL ¼ UOC  R0 I  1  et=τ1 R1 I  1  et=τ2 R2 I wherein R1 and R2 are also obtained using the least-square method. C1 and R2 can be obtained from τ/R. Finally, the equation of the relationship can be obtained between each modeling parameter and the battery state factor. Using the experimentally derived parameters, the method recursive equation is obtained from the particle filtering algorithm and the observed noise equation. According to the law flow, the current change can be simulated for the discharging process and the state estimation is performed. The experimental curve is described as shown in Fig. 6.6. The simulation results show that the particle filtering algorithm has a good effect on tracking the closed-circuit voltage. Its error is finally stable within 0.004 V, as shown in Fig. 6.7. The algorithm is trained by using the measured data onto the battery. The simulation and measured data tracking effect are compared and it proves that it has good stability. According to the state estimation model of the battery established above, the simulation analysis is carried out in Simulink. Its verification is carried out under the constant-current discharge condition. A full discharge experiment is conducted on the 3.75 V/1400 mAh lithium-ion battery of a 1 C constant discharging current ratio. As can be known from the figure, the unscented Kalman filtering can quickly track the measured state value, although the initial state value is different from the theoretical value, and the algorithm has good estimation stability. The estimation error is described as shown in Fig. 6.8.

FIG. 6.6 State estimation and traction curve. (A) Current variation procedure design. (B) Voltage and state variation.

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6. Battery state-of-energy prediction methods

FIG. 6.7 Voltage tracking error for pulse current discharging. (A) Voltage traction results. (B) Traction error variation curve.

FIG. 6.8 Estimation error under constant-current discharge conditions. (A) Estimation result and comparison. (B) Estimation error variation curve.

As can be seen from the figure, it has a good filtering effect. At the initial stage of estimation, the algorithm converges and the estimation error decreases rapidly. The error between the calculated and measured values is stable within 2%, which proves that the state estimation is of high precision. However, the maximum error of 2% can be realized in the state estimation process, which achieves a high estimation accuracy at the end of discharge.

6.4.2 Pulse-power characteristic test The state estimation error becomes obvious when the battery is used at the end of the discharging process, in which the extensive error is mainly caused by the nonlinear battery characteristics. The battery nonlinearity is strong at the end of the charge-discharge process. The accuracy at the end of the estimation is low for numerous methods. The test procedure for each pulse current test is described as follows. S1: The battery is charged to full capacity, where it equals 1. S2: The setup is shelved for 40 min. S3: It is discharged for 5% of the battery capacity. S4: It is then shelved for 30 min. S5: The cyclic charge-discharge is repeated 11 times from S3 until it equals 0. The experimental open-circuit voltage toward the state relationship can be obtained as shown in Fig. 6.9.

6.4 Experimental results analysis

219

FIG. 6.9 Lithium-ion battery voltage characteristic diagram. (A) Open-circuit voltage changing law. (B) Closedcircuit voltage variation.

The simulation results can be achieved accordingly. The initial simulation state value is set as 0.9, which varies from the theoretical value of 1.0. It is then used to verify its convergence effect in the estimation process. The comparison curve is obtained by unscented Kalman filtering between the theoretical value and the approximate state obtained. The state estimation waveform obtained by the experiment is described as shown in Fig. 6.10. According to the above analysis, the estimated state value is very close to the measured state value and reaches the application error within 5%.

6.4.3 Cyclic intermittent discharge Considering that the battery is always in the state of intermittent discharge in use, the model is further stimulated by the cyclic discharge shelved experimental condition. Real-time current and voltage data are imported into the model for the early cyclic discharging and shelved experimental condition, in which the 1 C discharging current is adapted to discharge the battery. Specifically, it is shelved for 30 min between every 6-min discharge, which is then continued. It is circulated until the closed-circuit voltage equals the cutoff voltage in the discharging process. The experimental results are obtained as shown in Fig. 6.11. Wherein the theoretical calculation value of the battery initial state and its initial estimation state are, respectively, set to be 0.5. And then, the unscented Kalman filter algorithm is introduced for the iterative calculation process. As can be seen from the figure, it has a good convergence and tracking effect. In the initial state estimation stage, the algorithm converges

FIG. 6.10 State estimation comparison. (A) Estimation result and comparison. (B) Estimation error variation curve.

220

6. Battery state-of-energy prediction methods

FIG. 6.11

Voltage traction under cyclic discharge conditions. (A) Corresponding voltage variation. (B) Current changing procedure design.

rapidly, and the time required for tracking down the theoretical value is 120 s. The estimation deviation is described as shown in Fig. 6.12. Wherein the estimation deviation is stable within 2% and the overall performance is good. It proves that the state estimation using the unscented Kalman filtering algorithm has a good convergence and tracking effect. When the battery is in the shelved state, the state estimation error increases. This is due to the lag phenomenon of the battery equivalent circuit model when the battery is in the shelved state. Therefore, state estimation deviation increases.

6.4.4 Packing pulse current test Strict requirements are put forward on the dynamic battery performance, which also brings difficulties to the battery state estimation for complex working conditions. To verify the estimation effect in the model building process under complex application conditions, the model is simulated with the customized experimental data onto the dynamic stress test operating conditions. Meanwhile, the Ah integration method and the extended Kalman filtering algorithm are used for synchronous simulation analysis to analyze its advantages and disadvantages for the same working conditions.

FIG. 6.12 Estimation error under cyclic discharge conditions. (A) State estimation result. (B) Estimation error variation curve.

6.4 Experimental results analysis

221

The specific experimental steps are described as follows. (1) The battery is charged by using the constant-current constant-voltage method. The termination voltage is 4.20 V at the upper limits when it is charged with constant-current treatment for a factor of 1 C. Then, the battery is charged using the constant voltage, and the cut-off charging current is 0.05 C. (2) It is shelved after charging to stabilize the battery voltage. Due to its small capacity, the shelved time can be set as 30 min. (3) The constant electric discharge is carried out at a 0.5 C power rate, and the duration is 4 min. After the discharge stops, it is shelved for 30 s. (4) It is charged with a constant current rate of 0.5 C for 2 min, and is then shelved for 30 s after the charging is finished. (5) The constant electric discharge is conducted at 1 C, lasting for 4 min. Steps (4)–(5) are cycled until the end of the battery discharge process. Then, the experimental voltage data can be obtained, as shown in Fig. 6.13. The voltage and current data obtained from the experiment can be imported into the working platform, according to which the estimation model is analyzed. The extended Kalman filtering and unscented Kalman filtering are used to estimate the battery state value, which is compared with its theoretical value. Then, the experimental results can be obtained as shown in Fig. 6.14.

FIG. 6.13

Battery state estimation under dynamic stress test condition. (A) Voltage change and traction curve. (B) Pulse power current variation.

FIG. 6.14 Estimation effect under dynamic stress test conditions. (A) State estimation result and comparison. (B) Estimation error variation curve.

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6. Battery state-of-energy prediction methods

According to the estimation effect analysis, the battery starts discharging when it is fully charged, and its initial state is considered to be 1. As can be seen from the experimental results, the state estimation shows a trend of fluctuation and a decrease in the whole process. It is because there is an alternating charge-discharge process of the experiment, and the discharge time is longer than the charging time. The initial estimated value of the algorithm is set as 0.95 to verify its convergence in the estimation process and the tracking of the measured value. The statistics of simulation data are described as shown in Table 6.1. Under the dynamic stress test condition, the Ah integration method could not quickly converge to track the measured value of the early estimation stage. Its estimation error is much larger than extended Kalman filtering and unscented Kalman filtering. The error increases together with the estimation time, which is the result of an accumulated error in time. Both algorithms can track the measured state, and its error is stable in the later estimation period. By locally enlarging the state estimation diagram, the experimental results can be obtained as shown in Fig. 6.15. Wherein the local enlargement and estimation error are obtained with high accuracy. Compared with extended Kalman filtering, the unscented Kalman filtering can converge and track the measured value more rapidly, in which the estimation error in the later period is smaller than extended Kalman filtering. The experimental results show that the estimation error of the unscented Kalman filtering is within 2%, which verifies its superiority in the battery state estimation.

TABLE 6.1 State estimation error comparison for different methods. Algorithm

Mean error (%)

Maximum error (%)

Convergence time (s)

Ah

8

13

—

Extended Kalman filtering

0.2

0.3

200

Unscented Kalman filtering

0.1

0.1

4

FIG. 6.15 Partial enlargement estimation under dynamic stress test conditions. (A) Voltage traction. (B) State estimation.

6.4 Experimental results analysis

223

6.4.5 Estimation processing effect The open-circuit voltage can be very accurate, but it needs shelved time to estimate the state value. Consequently, it cannot be realized in the online real-time state estimation. Compared to its relationship at different temperature conditions, different voltage-correction curves are provided for using the same type of lithium-ion battery. The result shows that the characteristic mathematical description is greatly influenced by the temperature change, which cannot be used alone for the correct state estimation. As a consequence, it is suitable to be an additional correction tool for other estimation methods. The open-circuit voltage test can be used for the accurate state estimation for the same battery cells at different temperature conditions. The attached examination of the mathematical temperature modeling is conducted with different charge-discharge current rates, which should be taken into consideration when controlling the battery parameter variables. Considering the upper and lower limits of the temperature, the charge-discharge current rates are not negligible. It should be suitable for battery state estimation in terms of terminal voltage, electromotive force, capacity, internal resistance, and different state conditions. Related studies have been carried out with a temperature of 25°C as the temperature with a varied range of 5–45°C, in which the available capacity and the voltage increase continuously as the working temperature rises. As the temperature continues to rise, the battery available capacity is reduced, reaching a maximum of 25°C. Similarly, the energy absorbed and released by the battery is different from varying charge-discharge current rates for the same temperature. The results are also obtained for the 5°C charging and 25°C discharge experiments. The battery terminal voltage is low as well as the maximum battery releasable capacity when the charge-discharge current rate is large. Wherein the state values can also be different corresponding to the same voltage level. As for the complex condition state estimation effect of the experimental test, the parameter values of R0, Rp, and Cp are all substituted into the algorithm. The improved battery state estimation algorithm is verified by the time-varying current discharge and the combined calculation equations. The state estimation processing effect is described as shown in Fig. 6.16. Wherein the red curve is the theoretical state value of Fig. 6.16A and the black curve is the improved estimated state curve. The error analysis results are described as shown in Fig. 6.16B, in which the red curve is the difference between the estimated state value and its measured value. The maximum state estimation is 3.86%, which is within the tolerances.

(A) FIG. 6.16

(B)

The state estimation processing effect. (A) State estimation effect and comparison. (B) Estimation error variation curve.

224

FIG. 6.17

6. Battery state-of-energy prediction methods

Comparison and partial amplification under simulation conditions.

It can be seen from the figure that the predicted curve approaches the measured curve gradually as the cycling number increases, which fluctuates around the measured curve. The estimation error value also shrinks gradually from the initial large error and fluctuates around 0 eventually, showing a tendency to converge. Based on the construction of the battery equivalent circuit model and its improvement on the corresponding algorithm, the modeling, algorithm, and simulation are combined to verify the modeling accuracy. In the constant-current discharge experiment, the temperature is set as 25°C and the 3.20 V/10 Ah lithium-iron-phosphate battery is discharged completely with a constant current of 1 C. In the stationary period of discharge, the square root unscented Kalman filtering estimation coincides with the theoretical value that follows the theoretical value. Its effect is much better than the unscented Kalman filtering algorithm until the end of the entire discharging process, as shown in Fig. 6.17. Wherein the estimated error curves of the unscented Kalman filtering and square root unscented Kalman filtering algorithm are compared for the battery 1 C constant-current discharge. It estimates the state value with a maximum error of 1.4%, which is significantly better than the estimated error of 2.59% that is obtained under the same discharging condition for the same battery. In the discharging stage, the maximum estimation error of the unscented Kalman filtering algorithm is 0.7%. The estimation error of the proposed algorithm is much lower than 0.3%.

6.5 Conclusion In this chapter, unscented Kalman filtering is improved for its square root algorithm, which is used to estimate the lithium-ion battery state of energy. The estimation effect is good. Based on the previous research, it is used to establish the estimation model. The functions of each component module are analyzed. The model is analyzed with measured experimental data onto various working conditions, such as constant-current discharge, cyclic discharge, and dynamic stress test. Consequently, the model estimation feasibility is verified by comparison. The results show that the proposed algorithm has fast convergence speed, good tracking effect, stable deviation, and accurate overall performance in the battery state estimation.

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Acknowledgments The work is supported by the National Natural Science Foundation of China (No. 61801407), the Sichuan Science and Technology Program (No. 2019YFG0427), the China Scholarship Council (No. 201908515099), and the Fund of Robot Technology Used for Special Environment Key Laboratory of Sichuan Province (No. 18kftk03).

Conflict of interest There is no conflict of interest.

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C H A P T E R

7 Battery state-of-power evaluation methods 7.1 State-space model construction At present, there are many types of battery equivalent circuit models. The vehicles powered by lithium-ion batteries are gradually being used in various fields, so battery management is getting more attention. The Thevenin model is one of the commonly used ones. It has the characteristics of simple operation that can reflect the dynamic and shelved battery characteristics [1–3]. Therefore, this model is used to establish the state-space equation of the battery. The model of the battery is a first-order circuit model; the circuit form is described as shown in Fig. 7.1. Wherein R0 represents the internal resistance of the battery. Rp represents the polarization internal resistance of the battery. Cp is the polarization capacitance of the battery, which is connected with the battery polarization resistance Rp in parallel. UOC represents an opencircuit voltage and Et represents an electromotive force of the battery. As can be known from the model of the battery, Et and It can be obtained as shown in Eq. (7.1):
Et ¼ Ut + R1 It + Uc, t (7.1) It ¼ Uc, t =R2 + C  dUc, t =dt The electromotive force Et is numerically equal to the open-circuit voltage of the battery. Its relationship function of the battery state is Et ¼f(St). Wherein St is the battery state at the time point of t. In the prediction section, the battery state can be obtained by the ampere hour integration method, as shown in Eq. (7.2): ðt (7.2) St ¼ St0  It ηdt=Qn t0

The adaptive Kalman filtering method is used to correct the current value of the battery state through the error covariance while measuring; it also plays an adaptive adjustment role [4, 5]. The current estimation can be corrected by updating the system noise covariance

Battery System Modeling https://doi.org/10.1016/B978-0-323-90472-8.00004-4

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Copyright # 2021 Elsevier Inc. All rights reserved.

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FIG. 7.1 The first-order electrical equivalent model for lithium-ion batteries.

effectively and the error covariance in real time. It can be concluded that the state-space model of the battery is described as shown in Eq. (7.3): 8 #     " ηΔt=Q   > n St 1 0 w1, t < St + Δt h i ¼ + It + Δt=ðRp Cp Þ Δt=ðRp Cp Þ Rp 1  e (7.3) Uc, t + Δt Uc, t w2, t 0 e > : Vt ¼ FðSt Þ  Rt It + Uc, t + vk wherein the state-space variable is taken as the control variable as well as the observed variable and system noise. The covariance is Q. The observed noise is vt and its covariance is R. Then, the coefficient matrix of the state-space model is described as shown in Eq. (7.4): ! !    ∂FðSt Þ 1 0 h ηΔt=Qn i , Bt ¼ 1 (7.4) At ¼ , Ct ¼ Rp 1  eΔt=ðRp Cp Þ 0 eΔt=ðRp Cp Þ ∂S   t

St ¼St

7.2 State estimation structural design The main working principle is to determine the error covariance matrix of the initial state first. After that, the error covariance matrix can be updated at the next time point. Then, the state value can be calculated according to the error covariance, which can be used as the calculated gain for the time point state value for the next one.

7.2.1 Algorithm overview The state estimation is realized by the iterative calculation process in the algorithm. Meanwhile, the error covariance matrix is updated according to the correction steps. Consequently, the processing noise is updated continuously, thereby realizing the accurate battery state estimation [6–10]. The estimation process is followed as shown in Fig. 7.2. The characteristic equation is the same for the Kalman filtering, extended Kalman filtering, unscented Kalman filtering, and adaptive Kalman filtering algorithms. Meanwhile, the order of operation is also the same, and all are based on the Kalman filtering method to make it suitable for the estimation of each environment [11–14]. The adaptive Kalman filtering method adds an adaptive effect based on the extended Kalman filtering so that the estimated value is more closely matched with the measured value.

7.2 State estimation structural design

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FIG. 7.2 Adaptive Kalman filtering framework.

7.2.2 Iterative calculation Adaptive Kalman filtering refers to the use of measurement data onto filtering and continuously determines whether the dynamics onto the system change by the filter itself, which estimates and corrects the model parameters as well as noise statistics to improve the model design then reduces the deviation [15–22]. This method combines the system parameter identification and state estimation organically. The algorithm adds an estimation of the statistical noise properties based on Kalman filtering, as shown in Fig. 7.3. Through the measurement data, the average and variance values of the noise can be estimated in real time. After that, the present state estimation value can be corrected according to the average and variance in the real-time update. It is used to improve the accuracy of the algorithm and avoid the divergence phenomenon. In the linear Kalman filter, the average value of processing and measurement noise is zero. Its definition is described as shown in Eq. (7.5): wk  N ðqk , Qk Þ, vk  N ðrk , Rk Þ

(7.5)

wherein k is discrete-time. The experimental design system noise estimator correlation quantity is calculated as shown in Eq. (7.6):

FIG. 7.3 Adaptive filtering framework.

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8 k X > > > xk + 1  A^ qk + 1 ¼ G ð^ xk  Buk Þ=ðk + 1Þ > > > > i¼0 > > > k  > X > > T T T e > Qk + 1 ¼ G e y L y L + P  AP A GT =ðk + 1Þ > k+1 k+1 k+1 k+1 k+1 k+1 > > < i¼0 k

X  > ^rk + 1 ¼ yk + 1  C^ xk + 1|k =ðk + 1Þ > > > > i¼0 > > > k

> X  > > T T ^k + 1 ¼ > y y  CP C = ð k + 1Þ R > k + 1|k k + 1 k + 1 > > > i¼0 >

 : G ¼ ΓT Γ Γ T

(7.6)

wherein ^ indicates that the statistic is an estimator. Xk is the state of the system at the time point of k. The observation signal is described by Yk corresponding to the state. Zk is the input of the system. A is a state transition matrix. B is the system control matrix. C is the observation matrix and Г is the noise-driven matrix. The values obtained in the equation are arithmetic average values, which are the weighting coefficients of each term. In the time-varying system, the current data have a great influence on the battery state estimation effect. Therefore, the estimator is improved by the exponential weighting. Each formula is multiplied by the different exponential weighting coefficient β. The coefficient satisfies the functional relationship that is described as shown in Eq. (7.7): ! k X βi ¼ 1 (7.7) βi ¼ βi1 b, 0 < b < 1, i¼0

Furthermore, numerical calculations can be used to obtain an iterative calculation expression as shown in Eq. (7.8):   1b i , i ¼ 0, 1, 2, 3, …, k (7.8) β i ¼ dk b , d k ¼ 1  bk + 1 wherein b is a forgetting factor. By replacing each term in the original estimator, the noise estimation requirements of the improved time-varying system are obtained. Based on the linear Kalman filter, the specific steps of the designed adaptive Kalman filter are described as follows. The first step is the initialization of the model parameters. The initial parameter x0 of the battery state and the covariance matrix P0 of the initial state error can be set as shown in Eq. (7.9): h i ^ x0 ¼ E½x0 , P0 ¼ E ðx0  ^x0 Þðx0  ^x0 ÞT (7.9) Next, the state at the time point of k + 1 and the error covariance matrix can be calculated as shown in Eq. (7.10):
^ xk + 1|k ¼ A^xk + Buk + Γ^qk (7.10) ^ ΓT Pk + 1|k ¼ APk AT + ΓQ k

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231

Further, the present state can be obtained in the previous step that is based on the error covariance. Then, the Kalman gain is calculated as shown in Eq. (7.11):

1 (7.11) Kk ¼ Pk + 1|k CT CPk + 1|k CT + Rk According to the observation value Yk +1 of the system, the estimated state and the error covariance matrix at the next time point are updated as shown in Eq. (7.12):
^ xk + 1 ¼ ^ xk + 1|k + Kk yk + 1 (7.12) Pk + 1|k ¼ ðE  Kk CÞPk + 1|k wherein E is the identified matrix. Further, the iterative calculation is continued by updating Qk and Rk accordingly, which returns to the first step until the requirements are met. This method is suitable for nonlinear discrete systems, especially the current operating conditions and statistical characteristics of noise with dramatic changes in operating conditions. To avoid the estimation inaccuracy and the divergence of the filtering process of complex conditions, based on the iterative calculation process, the idea of proportional-integralderivative controlling is introduced to improve the state estimation accuracy of the power battery. The mathematical expression of the quantitative controlled algorithm is described as shown in Eq. (7.13): 8 2 3 k > X > T T < ej + ðek  ek1 Þ5 uk ¼ Kp 4ek + β (7.13) T1 j¼0 Td > > : Δuk ¼ uk  uk1 ¼ Kp ðek  ek1 Þ + K1 ek + Kd ek + Kd ðek  2ek1 + ek + 1 Þ In the above incremental control algorithm, the integral coefficient is k1 ¼Kp  T/ T1  Kd ¼KpTd/T, and it is differential. Kp is the proportional coefficient of the proportional control link. K1 is the proportional coefficient of the integral control link. Kd is the differential control for the proportional coefficient of the link. T is the system sampling period. T1 is the integral time coefficient. Td is the differential practice coefficient. If the adaptive Kalman filtering algorithm is added to the state estimation, the statistical characteristics of the noise can be estimated online. The estimation accuracy can be improved as shown in Eq. (7.14): 2 3 2 32 3 2 3 Sk Sk + 1 Δt=Q 1 0 0

7 6 p1 7 6 76 p1 7 6 Δt=τ1 7 6 Sk + 1 7 ¼ 6 0 eΔt=τ1 6S 7 6 0 7 (7.14) 4 5 4 54 k 5 + 4 Rd1 1  e 5ik + Γwk

 p2 p2 0 0 eΔt=τ2 Rd2 1  eΔt=τ2 Sk + 1 Sk Based on the improved mathematical expression shown in Eq. (7.14) and the equivalent circuit model shown in Fig. 7.1, the extended Kalman filtering algorithm is introduced to estimate the battery state. According to the ampere hour integration method, the terminal voltages Up1 and Up2 can be obtained for the two resistance-capacitance networks, which can be used as the state variables of the battery system. The battery terminal voltage is selected as the observation. After discretization and linearization processing combined with noise effects, the state equation and observation equation of the system are described as shown in Eq. (7.15):
^ xk1 + Bk1 uk1 + Γqk1 xk ¼ Ak1 ^ (7.15) P ¼ Ak Pk1 ATk + ΓQk1 ΓT

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wherein Δt is the sampling time and Q is the rated capacity of the lithium-ion battery. The incremental algorithm has a small amount of calculation, but it causes the proportion integration differentiation operation integral to accumulate and the control amount becomes larger. Therefore, the experiment uses the integral separation and the proportion integration differentiation algorithm with a dead zone. The traditional state estimation method of the battery has the problems of low precision and practicability.

7.3 Calculation procedure design Some unknown noise often interferes with the signal sampling process; these noises can include various noises from the outside world, the sensor, and the signal itself. Frequently, some out-of-range values are collected, in which these interferences are based on adaptive hardware filtering. Kalman filtering not only reduces the hardware costs but is also easy to implement.

7.3.1 Computing framework design Based on extended Kalman filtering, the adaptive Kalman filtering method estimates the dynamic battery state from the measured data onto real-time correction. It estimates and corrects the statistical characteristics of the noise continuously, which can estimate the state of the system accurately. The processing noise covariance Q and the observed noise covariance R are set to be a constant operation of the algorithm. In the application, if the above two noise covariances are inaccurate in the filtering calculation process, it is easy to cause a certain cumulative error and a divergence problem. Moreover, when dealing with some nonlinear systems with high dimensions, the observation of the noise covariance matrix is prone to nonpositive or semipositive definite phenomena in the estimation processes, which may cause filtering divergence. The dynamic model and the observation model are both linear and similar; the results are approximated to the measured value accurately as the linearization model is accurate [23–28]. Besides, the algorithm has a defect. If there is a large error for the assumed initial value and covariance, it also causes divergence of its operation. To solve the noise error problem, an adaptive filtering method is introduced to correct the system noise while estimating the system state. The noise statistics are corrected continuously according to the change of the measured data, according to which the estimation accuracy is improved. After that, the noise influence is reduced and the adaptive value can be obtained. The calculation process is described as shown in Fig. 7.4. The discretization sampling period Ts equals 1 s. The average of the processing noise Wk and the measurement noise Vk are variances. The processing and measurement noises are uncorrelated with unknown Qk and Rk, which are estimated by the adaptive Kalman filtering algorithm. The system is initialized so that k ¼ 0, in which the estimated value X0 of the initial parameter and the covariance of the initial error can be described as shown in Eq. (7.16): h i ^ x0 ¼ E½x0 , P0 ¼ E ðx0  ^x0 Þðx0  ^x0 ÞT (7.16)

7.3 Calculation procedure design

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FIG. 7.4 Computing flowchart of adaptive Kalman filtering.

Then, the state of the k-time uncertainty and the error covariance matrix is time updated by the state at the time point of k  1. The error covariance matrix can be obtained as well, according to which its calculation process is described as shown in Eq. (7.17):
^ xk1 + Bk1 uk1 + Γqk1 xk ¼ Ak1 ^ (7.17) Pk ¼ Ak Pk1 ATk + ΓQk ΓT wherein the matrix AT is the transposed matrix of A. The prior estimation can be realized for the state and its error covariance at the time point of k. Then, the Kalman gain matrix Kk is calculated as shown in Eq. (7.18):

1 (7.18) Kk ¼ Pk CTk Ck Pk CTk + Pk1 If the uncertainty of present state estimation is high, Kk is supposed to be larger because of the enlargement of Pk, and a great update of the system state is conducted effectively. Besides, if the ambient noise is large, Pk 1 becomes larger, which makes Kk smaller. Further, the estimated amount of the next time-point state and the error covariance is continuously updated by using the measurement data. The mathematical expression is described as shown in Eq. (7.19): (

 ^ xk|k1 + yk  C^xk|k1 + Duk + d  rk1 xk ¼ ^ (7.19) Pk ¼ ðE  Kk Ck ÞPk|k1 The state estimation of the next time point is equal to the sum of the a priori estimation of the state at that time point and a weighted correction term. Among them, E is the unit matrix. Due to the new information provided by the measured values, the state uncertainty is usually reduced continuously and Pk is a continuously decreasing process quantity. Finally, the

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measured data are used to estimate the average and variance in the online noise. The updated state is used to continuously replace the present state estimation to achieve an alternate update of the estimated state quantity and noise statistic. The expression is described as shown in Eq. (7.20): 8 qk ¼ ð1  dk1 Þqk1 + dk1 Gð^xk  A^xk1  Buk1 Þ > > > > < Qk ¼ ð1  dk1 ÞQk1 + dk1 G Lke yk e yTk LTk + Pk  APk|k1 AT GT

 (7.20) rk ¼ ð1  dk1 Þrk1 + dk1 yk  C^xk|k1  Duk1  d > > > > : R ¼ ð1  d ÞR + d yk e yTk  CPk|k1 CT k k1 k1 k1 e wherein b is the forgetting factor, 0 < b < 1, taking b ¼ 0.96 and Γ ¼ [0.01 0.01]. As can be known from the above analysis, Qk and Rk are estimated online in real time. The target of the state variable state estimation is continuously corrected to achieve adaptive correction, thereby improving the state estimation accuracy.

7.3.2 Iterative calculation steps Based on linear iterative Kalman filtering, the specific steps of the designed adaptive Kalman filter are described as follows. The first step is the initialization process. The initial state of the system is set as shown in Eq. (7.21): h i ^ (7.21) x0 ¼ E½x0 , P0 ¼ E ðx0  ^x0 Þðx0  ^x0 ÞT Next, the state of the system k and the error covariance matrix are updated, according to which the calculation process is described as shown in Eq. (7.22):
^ xk|k1 ¼ Ak1 ^xk1|k1 + Bk1 uk1 (7.22) Pk|k1 ¼ Ak1 Pk1|k1 ATk1 + Γk1 Qk ΓTk1 Further, the Kalman gain Kk is calculated from the error covariance matrix of the present state obtained in the previous step, as shown in Eq. (7.23):

1 (7.23) Kk ¼ Pk|k1 CTk Ck Pk|k1 CTk + Rk Then, the next time state estimation value and the error covariance matrix are updated according to the observation value of the system as shown in Eq. (7.24): 8 < ek ¼ yk  Ck ^xk|k1  Dk uk ^ xk|k1 ¼ ^xk|k1 + Kk ek (7.24) : Pk|k ¼ ðI  Kk Ck ÞPk|k1 Consequently, the adaptive Kalman filtering method also estimates the covariance of the unknown noise in the algorithm while estimating the battery state, which can make the final estimation effect more stable. Finally, Q and R are updated as shown in Eq. (7.25):  8 < Qk ¼ ð1  dk1 ÞQk1 + dk1 G Kke yk e yTk LTk + Pk  APk|k1 AT GT  (7.25) T : Rk ¼ ð1  dk1 ÞRk1 + dk1 e yk e yk  CPk|k1 CT

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Meanwhile, when estimating the battery state, the method effectively reduces the influence of noise on the estimation effect. Based on theoretical analysis, the algorithm is realized by modeling and simulation, so that the model can be constructed. The system and measurement state calculations are described as shown in Eq. (7.26): ( x_ ðtÞ ¼ f ½xðtÞ, t + g½xðtÞ, twðtÞ, wðtÞ  N ½0, QðtÞ (7.26) zðtÞ ¼ C½xðtÞ, t + vðtÞ, vðtÞ  N ½0, RðtÞ It is developed by using the classic Kalman filtering algorithm. Its idea is the filtering treatment for the Taylor series expansion of a nonlinear system. The initial conditions and the state estimation equation are described as shown in Eq. (7.27):
xðtÞ  N½^ x0 , P0 , cov½wðtÞ, vðtÞ ¼ 0 (7.27) ^ x_ ðtÞ ¼ Fð^ xðtÞ, tÞ + KðtÞ½zðtÞ  Cð^xðtÞ, tÞ Because of ignoring the second-order and higher-order terms, the original system becomes a linear system. The error covariance is described as shown in Eq. (7.28): xðtÞ, t + GðtÞQðtÞGT ðtÞ  KðtÞRðtÞKT ðtÞ P_ ðtÞ ¼ F½^ xðtÞ, tPðtÞ + PðtÞFT ½^

(7.28)

Then, the equations of the extended Kalman filtering algorithm can be obtained by adopting the idea of the classic Kalman filtering algorithm for a filtering system as a linearized model. The gain matrix is described as shown in Eq. (7.29): KðtÞ ¼ PðtÞHT ½^xðtÞ, tR1 ðtÞ

(7.29)

The procedure steps for the battery state estimation are designed accordingly. First, the equations of the state parameters in the circuit are obtained as shown in Eq. (7.30):

 8 < Up ðk + 1Þ ¼ Up ðkÞeΔt=τ + I ðkÞRp 1  eΔt=τ ðt (7.30) : St ¼ S0  Iηdt=C 0

Combined with the extended Kalman filtering state observation equation, the predicted value of the time point k is calculated and the covariance prediction is obtained, as shown in Eq. (7.31):
xðkj k  1Þ ¼ Ak1 xðk  1Þ + Bk1 Ik1 (7.31) Pðkj k  1Þ ¼ Ak1 P^k1 ATk1 + Qk The extended Kalman filter is suitable for a nonlinear system. It is similar to the classical linear Kalman filter with the same algorithm steps and structure. Then, the Kalman gain at the time point of k is calculated as shown in Eq. (7.32):

1 (7.32) Kk ¼ Pk CTk Ck Pk CTk + Rk The difference lies in replacing the system equation of the linear system model with the nonlinear system model equation in the extended Kalman filter. Then, the status and covariance update are obtained as shown in Eq. (7.33): ^ xk ¼ xðkj k  1Þ + Kk ½Uoc ðkÞ  Ck xðkj k  1Þ, P^k ¼ ð1  Kk Ck ÞPk

(7.33)

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FIG. 7.5 The calculation flowchart for the state estimation system.

By repeating the above steps, the estimated state is updated constantly, which is close to the measured value of the iterative calculation process as shown in Fig. 7.5. The partial derivative of the system model is obtained to obtain the matrices A and H in the new extended Kalman filter. In the process of the partial derivative solution, the estimation of the previous time point is used as the reference point in the process of linearization [29–33]. The extended Kalman filter for the nonlinear system is obtained by this modification. The state-space model is described as shown in Eq. (7.34):
zk + 1 ¼ f ðzk , uk Þ + wk (7.34) yk ¼ gðzk , uk Þ + vk The nonlinear functions are expanded by the Taylor series around the estimated value, according to which the above quadratic terms are omitted to obtain its discrete-time value, as shown in Eq. (7.35): 8  ∂f ðzk , uk Þ > > > f ðz , u Þ  f ð^zk , uk Þ + ðzk  ^zk Þ > < k k ∂zk zk ¼^zk (7.35)  > ∂gðzk , uk Þ > > > ðzk  ^zk Þ : gðzk , uk Þ  gð^zk , uk Þ + ∂z  k zk ¼^zk The new linearized state equation is obtained by combining the above equation as shown in Eq. (7.36): ( zk + 1 ¼ AðkÞzk + f ð^zk , uk Þ  AðkÞ^zk + wk (7.36) yk ¼ CðkÞzk + gð^zk , uk Þ  CðkÞ^zk + vk wherein the coefficient parameters are defined without considering the influence of noise as shown in Eq. (7.37):     ^ k ¼ ∂gðzk , uk Þ ^ k ¼ ∂f ðzk , uk Þ , C (7.37) A   ∂z ∂z k

zk ¼^zk

k

zk ¼^zk

7.3 Calculation procedure design

237

The estimation process of the extended Kalman filtering algorithm is designed. Initialization for regulation expressions is realized as shown in Eq. (7.38): h

X + 

T i ^zk,0 ¼ E zk  ^zk+ zk  ^zk+ ^zk+ ¼ E½z0 , (7.38) The state estimation time is updated in the calculation process of k ¼ 1, 2, … and its expression. The update expression of error covariance time is described as shown in Eq. (7.39): X X 

+  X  ^ k1 ^T + ^z, k ¼ A ^z, k1 A ^z z0 , uk1 , w (7.39) k1 0 ¼f ^ It is a linear unbiased recursive filter, which is constantly predicted and corrected in the calculation process. Whenever new data are observed, new predicted values can be calculated at any time, which is very convenient for its real-time processing. The Kalman gain expression is described as shown in Eq. (7.40): X  T X  T h X i1 ^ ^ ^ ^zk C ^zk C v (7.40) Kk ¼ k1 Ck1 k1 + The updated expression of state estimation measurement and the updated expression of error covariance measurement can be obtained as shown in Eq. (7.41): X

  X +  ^k ^zk ¼ I  Kk C ^z ^zk+ ¼ ^z zk , uk , (7.41) k + Kk yk  g ^ k Due to the discharging current rate, temperature, and complex internal chemical reaction, the lithium-ion battery presents a nonlinear state. Based on Kalman filtering, the Jacobian matrix is obtained by using the Taylor formula for linearization, so that the extended Kalman filtering calculation is realized. The established equivalent model of the battery can be expressed by Eq. (7.42):
Xk ¼ AXk1 + BI L, k1 + wk (7.42) UL, k ¼ CXk + DI L, k1 + vk For the state-space model expression shown in Eq. (7.42), the Kalman filtering algorithm is used for state prediction and correction. The Kalman filtering is mainly composed of five formulas, which can be divided into the prediction and correction stages. The recursive relationship between the estimated value of state and covariance in the prediction stage is described as shown in Eq. (7.43): (  ^ k1 + BI L, k1 + w ^ ¼ AX X k k  (7.43) ^ Pk|k1 ¼ AP^k1 AT + Qw According to the last time point, the state estimation of k  1, the state parameter, and its covariance can be calculated directly by the forecast of this time point. The covariance matrix of processing noise is considered as well. The estimated values of the Kalman gain in the correction stage are described as shown in Eq. (7.44):   8  > Kk ¼ P^k|k1 CT CP^k|k1 CT + vk > <  ^  + Kt UL, k  CX ^ ^k ¼ X (7.44) X k k > >   :^ Pk|k1 ¼ P^k|k1  Kt CP^k|k1

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It is then used for calculating the Kalman gain Kk. Then, the state estimation can be realized according to its calculation process to correct the predictive value. In this way, the Kalman filtering algorithm is completed in one iteration, and an iterative estimation is carried out for each observation, with good real-time correction performance.

7.3.3 Algorithm improvement Lithium-ion battery state estimation is an obvious hidden Markov model because of the traditional state estimation dependence on the initial value and the stability of the algorithm [34, 35]. Based on the extended Kalman filtering algorithm and the state covariance matrix square root decomposition, an improved algorithm is proposed to realize initial value adaptive state correction. It can effectively prevent the effect of filtering divergence caused by data beyond dimensions. The algorithm flow of the improved algorithm is described as shown in Fig. 7.6. The main step should be used to set the initial value of state vector X(0 j0) and its initial value of covariance matrix P(0 j 0). The estimated value is calculated at the next time point according to the model, which is then used to calculate the Kalman gain and modify its estimated value. In practical application, there is a divergence problem of filtering, which may be caused by the rounding calculation error of the computer. It makes the calculation values

FIG. 7.6 Improved calculation flowchart of the lithium-ion battery state estimation.

7.3 Calculation procedure design

239

of P(k jk  1) and P(kj k) lose nonnegative qualitative nature, leading to the calculation distortion of Kk and resulting in more error. Its calculation process can be described as shown in Eq. (7.45): 2 3 2 32 3 P11 P12 ⋯ P1n S11 S12 ⋯ S1n S11 0 ⋯ 0 6 P21 P22 ⋯ P2n 7 6 S21 S22 ⋯ 0 76 0 S22 ⋯ S2n 7 6 7 6 76 7 (7.45) 4 ⋯ ⋯ ⋯ ⋯ 5 ¼ 4 ⋯ ⋯ ⋯ ⋯ 54 ⋯ ⋯ ⋯ ⋯ 5 Pn1 Pn2 ⋯ Pnn 0 0 ⋯ Snn Sn1 Sn2 ⋯ Snn For the matrix S, SST is symmetric and nonnegative definite. Therefore, the state covariance matrix can be decomposed into S(kj k), which is nonnegative at any time. The decomposition method of the matrix is described as shown in Eq. (7.46): 9 8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0, 1 ði < jÞ > > u > > 0 , = < i1 X u i1 2 X t (7.46) Sij , Sij ¼ @ Sii ¼ Pii  2A Pii  Sij Sjj , ði  jÞ > > > > j¼1 ; : j¼1 Therefore, an improved algorithm can be obtained correspondingly. First, the initial values X(0 j 0) and S(0 j 0) are set. In the prediction stage, the prediction parameter X(kj k  1) for the time point of k and the decomposition matrix S(kj k  1) can be calculated directly according to the state-space model. The improved calculation process can be obtained as shown in Eq. (7.47):
Xðkj k  1Þ ¼ AðkÞXðk  1j k  1Þ (7.47) Sðkj k  1Þ ¼ AðkÞSðk  1j k  1Þ P(kjk  1) is the corresponding state variable covariance matrix that can be obtained from the optimal predictive value X(k  1 jk  1) at the time point of k  1. Then, the Kalman gain K(k) is calculated in the correction stage as shown in Eq. (7.48):

 8 Fk + R < KðkÞ ¼ Sðkj k  1ÞFk αk , αk ¼ FTkp ffiffiffiffiffiffiffiffi (7.48) 1  αk R : Fk ¼ ST ðkj k  1ÞHðkÞ, γ k ¼ 1  αk R The state estimation is based on the ampere hour integration method and the battery equivalent model [36]. Then, the directly calculated k time point predictive value X(kj k  1) and the covariance decomposition matrix S(k j k  1) can be modified to obtain the optimal estimated value X(k jk). The calculation process is described as shown in Eq. (7.49):
Xðkj kÞ ¼ Xðkj k  1Þ + KðkÞ½UL ðk Þ  HðkÞXðkj k  1Þ  BI ðkÞ (7.49) Sðkj kÞ ¼ Sðkj k  1Þ I  αk γ k Fk FTk Considering the initial X(0j 0) and S(0 j 0), the state vector X(k jk) at each time point can be obtained through the iteration of the above formulas. According to the selected state vector [S UP1 UP2]T, the first term of the output estimated state quantity is the required state estimation. The state estimation is carried out according to the above algorithm. Because P(0 j0) is not negative, at least the state covariance P(k j k) is always nonnegative, to overcome the

240

7. Battery state-of-power evaluation methods

divergence caused by calculation. The system discrete state-space equation is given by the ampere hour time integration process as shown in Eq. (7.50): Sk ¼ Sk1  ηI k1 k1 =QN + ωk1

(7.50)

It contains Kirchhoff’s voltage law relation of the equivalent circuit model in the time domain. The state-space description of the equivalent model is embodied in the observation equation as shown in Eq. (7.51): ULk ¼ f ðSk Þ  Ik R0  upk + υk

(7.51)

wherein S0 represents the state value at the initial time point. St represents the state value at the time point of t. QN represents the rated power of the battery. The current is represented by parameter I. The default discharge direction is positive, indicating the charge-discharge efficiency. The processing noise is Gaussian white noise with an average of 0 and the variance of Q. It represents the internal error distribution in the process of system operation. The voltage of the UL is representative and f is the nonlinear state function toward the open-circuit voltage. R0 and Rp are, respectively, the ohmic resistance and the polarization resistance of the model. τ is the resistance-capacitance time constant of the circuit. Vk is the observation noise and its average value is 0. The Gaussian white noise variance is R. The observation of the error distribution can be then expressed in the calculation process. The extended Kalman filtering implementation process is not to filter the system and observation noises. It depends on these two types of noise information on system state estimation, which is different from improved Kalman filtering methods. Its processing treatment is linearization and its state estimation algorithm process can be obtained as shown in Fig. 7.7. The algorithm needs to set boundary conditions at the beginning. The state vector X is initialized as well as its error covariance matrix P. The state vector must contain the battery state. Other state parameters are related to the established model, including Up. The main part of the algorithm can be divided into four stages: parameter calculation, prediction, linearization, and update. The content of the parameter calculation mainly includes the circuit elements R0, Rp, and Cp in the equivalent model. The significance of state transition matrix A can be seen from the process recursive equation. It is the driving matrix from the state at the previous time point to the state at the next time point as shown in Eq. (7.52):

T (7.52) Xk ¼ AXk1 + Buk1 , X ¼ S, Up The calculation of state transition matrix A is calculated and the input matrix B is the driving matrix of input influence on the state, as shown in Eq. (7.53):     ηΔt=QN 0 1 0

 , B¼ (7.53) A¼ 0 Rp 1  eΔt=τ 0 eΔt=τ The parameter △ t in both equations is the sampling time, and the meanings of other parameters have been given in the theoretical analysis process. Therefore, the formula contains the recursive state calculation of S and Up. In the prediction stage, the prediction of the error covariance matrix in the state prediction is given by Eq. (7.54): Pk=k1 ¼ Ak1 Pk1=k1 AT k1 + Qk1

(7.54)

7.3 Calculation procedure design

241

FIG. 7.7 Detailed battery state estimation flowchart.

wherein Q is the characteristic variance matrix of processing noise. The significance of the error covariance matrix lies in that it reflects the probability distribution of the estimated state of the measured value. The smaller the value of the matrix elements, the more the corresponding state estimation can reflect the measured state. Terminal voltage prediction is to predict the terminal voltage UL of the present time point by Kirchhoff’s voltage law circuit equation of the model according to the predicted value of state, as shown in Eq. (7.55): ULk ¼ f ðSk Þ  Ik R0  Upk

(7.55)

The superposition observation noise is reduced because the observation noise changes from real time and is unknown in the observation process, so only the statistical probability distribution rule can be obtained. For the nonlinear relation, the Taylor series expansion of mathematics is used to obtain the linear expression of the first two terms in the vicinity of the present state estimation Xk. The Jacobian obtained is taken as the state transition matrix C as shown in Eq. (7.56): 8 ∂g > > > < ULk ¼ gðXk , Ik Þ + ∂Xk ðxk  Xk Þ (7.56)     > ∂g ∂g ∂ðUOC  Ro I Þ > > Ck ¼ ∂g ¼ ¼ 1 : ∂Sk ∂Upk ∂Xk ∂Sk

242

7. Battery state-of-power evaluation methods

wherein Xk is the state variable. The content of linearization is the embodiment extension, which extends the range of processing from linear to nonlinear. The linear expression is obtained through a further transformation as shown in Eq. (7.57): ULk ¼ Ck xk + ½gðXk , Ik Þ  Ck Xk 

(7.57)

wherein the last term equation is a nonrandom action term independent of the condition that the present state quantity estimation is known, thus achieving local linearization. In the process of updating, the gain matrix is calculated and the predicted state quantity is corrected by comparing the observed quantity. As can be known from the above steps, the estimated value of state at the time point of k is obtained.

7.3.4 Estimation modeling realization According to the iteration process, a simulation structure of the state estimation of the ternary battery using the extended Kalman filtering algorithm is built. Compared with the simulation structure of model accuracy verification, the input of the terminal voltage data is added in the figure. The system contains six inputs and one output. In the s-function module, the state online accurate estimation model of the battery is constructed according to the iteration program and simulation model. The current IL, the current available capacity Q, and the initial state of the system at a given working condition can be obtained through the ampere hour integration method. The accurate value of the state is taken as a reference to the estimation results. The voltage measurement data UL in response to the IL condition is given to calculate the residual. The four parameters of the identified battery equivalent model are obtained by using the Lookup table module according to piece-wise linear interpolation, which to a certain extent realized the online identification of model parameters. Concerning the timedomain ordinary differential equation of the second-order resistance-capacitance equivalent circuit, the corresponding voltage response equation is solved and discretization is required before modeling to obtain the discretization state-space equation of the second-order resistance-capacitance model, as shown in Eq. (7.58): 8 Ubat ¼ Uoc ðSk Þ  Us ðkÞ  R0 Ik > > <

 Us ðkÞ ¼ et=τs Us ðk  1Þ + 1 + et=τs Rs Ik1 (7.58) > >

 : t=τl t=τl U l ðkÞ ¼ e Ul ðk  1Þ + 1 + e Rl Ik1 Among them, the interpolation parameter state is output and fed back to the Lookup table module through the constructed model because it is impossible to calculate the accurate state using the ampere hour integral during the various working conditions. Next, the simulation module can be built according to this formula. The second-order resistance-capacitance internal circuit is the core part of the whole module, the structure of which is directly used to build the module. It includes the internal resistance, resistance-capacitance parallel structures, controllable voltage source, controllable current source, voltage sensor, current sensor, and input-output interface. The controllable voltage source and controllable current source are the signal interface, which can turn the signal into a material port. The external input voltage

7.4 Experimental analysis

243

sources are converted into a voltage and current source that the circuit can connect to. Voltage and current sensors are also signal transducers, converting physical interfaces into signal interfaces.

7.4 Experimental analysis The simulation system model is established to verify the state estimation effect of the algorithm. The current and voltage data obtained from the experimental test are imported into the model. The simulation part is mainly to write the s-function module of the improved program. Due to the small sampling time of the experimental equipment as 0.01 s, the state value can be obtained by the ampere hour time integration of the current data with high accuracy. The accurate initial state value has a high accuracy, which can be regarded as the measured value compared with the improved algorithm. The obtained value is compared with the measured value, and the observation algorithm is used to estimate the effect.

7.4.1 Parameter identification results The identification of the relationship between the open-circuit voltage and state is conducted. The module input is the current, voltage, and the six polynomial functions obtained according to the model—R0 (S), Rp1 (S), Cp1 (S), Rp2 (S), Cp2 (S), and UOC—so that the output is the state value. In the experimental process, the internal resistance is measured by the tester AT520B, the measurement range of which is 0.01 mΩ to 300.00 Ω with an accuracy of 0.50%. The open-circuit voltage is a very important parameter for estimating the remaining electricity of the battery and indicates the voltage across the positive and negative terminals of the battery when it is not in operation. To measure the accurate open-circuit voltage, the battery must be nonoperational for a long time. Only when the shelved time is extended can the influence of the polarization effect on the battery be eliminated. The experiment needs to measure the open-circuit voltage of the battery of different residual electricity. First, measure the open-circuit voltage when the battery is full. Then, discharge it with a constant current of 1 C for 3 min so that the remaining electricity has been reduced to 0.05 C. At this time, the battery should be shelved. For a period, the voltage across the battery can be stabilized. Then, the open-circuit voltage is measured after stabilization. The experiment is continued until the battery is exhausted. The experiment is carried out in this design of a ternary lithium-ion battery. The open-circuit voltage and the state are nonlinear, so a nonlinear fitting method can be used to fit the relationship between the open-circuit voltage and the remaining electricity. An open-circuit voltage toward the state fit map is described in Fig. 7.8. The parameter identification is conducted for the resistance-capacitance circuit. The battery model has two resistances, internal resistance and polarization resistance. Many factors affect the internal resistance of the battery, such as the chemical reaction between the positive and negative electrodes. The ohmic internal resistance is composed of the electrode material, electrolyte, diaphragm resistance, and contact resistance of various parts. The internal polarization resistance refers to the resistance caused by polarization during an electrochemical

244

7. Battery state-of-power evaluation methods

FIG. 7.8

Open-circuit voltage diagram for varying current conditions. (A) Charge-discharge calculation. (B) Average voltage curve comparison. (C) Curve fitting result for voltage variation. (D) Varying current-rate voltage comparison.

reaction, including the resistance caused by electrochemical polarization and concentration polarization. Then, the electrochemical polarization is generated. When the battery power is changed, the polarization resistance and capacitance of the corresponding battery change along with it. When a voltage pulse test is performed on a plurality of different battery capacities, the following data are obtained, as shown in Table 7.1. As described above, the corresponding parameter values of the model can be calculated according to the hybrid pulse-power characteristic experimental data, and each model parameter value has its corresponding calculation formula. Analyzing the single hybrid pulse-power characteristic test, the parameter calculation formula for the model can be obtained by the experimental results shown in Fig. 7.9. The internal resistance can be calculated as shown in Eq. (7.59): R0 ¼ ½ðU1  U2 Þ + ðU4  U3 Þ=ð2I Þ

(7.59)

The polarization resistance and capacitance can be calculated according to the time constant as shown in Eq. (7.60):   U1  U5 U1  U3  R0 I τ ) ð3ÞCp ¼ ) ð2ÞRp ¼ (7.60) ð1Þτ ¼ ðt4  t3 Þ= ln  ð t t Þ=τ 2 1 U1  U4 Rp  I ½1  e Through the above formula, the parameters of the model can be identified. The model parameter values are different for different state points. The different model parameter values are described as shown in Table 7.2.

245

7.4 Experimental analysis

TABLE 7.1 Voltage parameter values corresponding to the state-of-charge variation. S

U1

U2

U3

U4

U5

1.0

4.107

3.984

3.965

4.083

4.097

0.9

3.984

3.871

3.844

3.959

3.975

0.8

3.880

3.767

3.738

3.854

3.873

0.7

3.782

3.663

3.634

3.756

3.772

0.6

3.682

3.564

3.543

3.663

3.676

0.5

3.632

3.513

3.488

3.614

3.625

0.4

3.599

3.479

3.455

3.578

3.591

0.3

3.550

3.428

3.406

3.529

3.541

0.2

3.463

3.332

3.299

3.435

3.453

0.1

3.417

3.273

3.225

3.380

3.406

FIG. 7.9 Single hybrid pulse-power characteristic test with detailed voltage labels.

TABLE 7.2 Model parameter values corresponding to the state-of-charge variation. S

τ

R0

Rp

Cp

1

44.67356

0.002416

0.002143

20,836.77

0.9

38.65120

0.002416

0.002143

20,836.77

0.8

32.08901

0.002416

0.002143

20,836.77

0.7

41.5180

0.002416

0.002143

20,836.77

0.6

33.06227

0.002416

0.002143

20,836.77

0.5

45.49334

0.002416

0.002143

20,836.77

0.4

39.22216

0.002416

0.002143

20,836.77

0.3

46.02791

0.002416

0.002143

20,836.77

0.2

37.61091

0.002416

0.002143

20,836.77

0.1

34.38934

0.002416

0.002143

20,836.77

246

7. Battery state-of-power evaluation methods

The internal resistance of the electrode material constituting the battery affects its internal resistance. Besides, there are also contact layer resistance and the internal resistance of the electrolyte material inside the battery. The type of battery plays a major role, as do the material of the electrode and the electrolyte. The polarization resistance indicates the time of the battery polarization reaction.

7.4.2 State estimating and voltage tracking The state results are obtained after running the simulation model. The current and voltage data onto the test operating conditions obtained in the experiment are imported from the corresponding module in the model shown in Fig. 7.10. The measured state value of the black line S1 is obtained by the experiment as shown in Fig. 7.10B. The red line S2 is the state estimation result. According to the state estimation result, the battery state used in the experiment decreases 20 times from 1 to 0.65 for the Beijing bus dynamic street test operating conditions. The initial state error is 25%. The absolute value of the error and its average value are taken into consideration to get the average error of 0.45%.

FIG. 7.10

State estimation results for pulse current working conditions. (A) Current variation procedure design. (B) State measurement and calculation. (C) State estimation variation effect. (D) Voltage traction and measured results.

7.4 Experimental analysis

247

7.4.3 Power-temperature variation Different working-condition raw experimental results are used to test the accuracy, stability, and applicability of the algorithms [37]. The constant-current and pulse-current conditions are used respectively compared to the simulation experiment. The comparison is conducted between the ampere hour integral, extended Kalman filtering, and unscented Kalman filtering algorithms. The estimation effect is used to test the feasibility and validity. The verified state is measured by the ampere hour integral of the minimum time interval as the standard. The output effect under each working condition is tested by comparison. Based on experimental and theoretical analyses, the state estimation algorithm is simulated with experimental data based on an accurate battery model. The main steps of simulation are described as shown in Fig. 7.11. The experimental data are adopted to simulate the operation. The data is supposed to be mixed by the random noise with normal distribution. It is a Gaussian white noise with the average zero and variance of R in the effective digit processing. The noise is superimposed on the input data, and the observation of terminal voltage is simulated by this method. The data are introduced into the voltage contrast state estimation update. The algorithm is realized by the equivalent circuit modeling of the forecast and update process input analog observation noise voltage to get the state estimation output. According to the original current data and the ampere hour integration method, the processing noise matrix with variance matrix Q is superposition to obtain the state vector, which is taken as the measured vector. The simulation output and state of the algorithm can be compared. Therefore, the accuracy and stability of the modeling algorithm can be verified. The experimental procedure is designed as shown in Fig. 7.12. For the iterative calculation process, boundary conditions need to be set together with the initial quantity, which includes the initial state amount X0, incorporating initialed state value

FIG. 7.11 The simulation process schematic of the estimation model.

248

FIG. 7.12

7. Battery state-of-power evaluation methods

Experimental procedure schematic design and realization.

S0 and polarization voltage initial values Up0 in the model. In practice, the initial value of the state can record from the battery management system before the state quantity. If the battery is put on hold for a long time, it can be used to improve the state estimation method to get the initial value. The battery state is not obvious before starting the polarization effect, so the polarization voltage can be thought of as the initial value of zero.

7.4.4 Main charge-discharge condition test The initial value of the error covariance matrix P0 can be determined from the X0 error of the initial state. In the application, the initial value should be as small as possible to speed up the tracking speed of the algorithm. Two important parameters are processed noise variance matrix Q and observation noise variance matrix R. As can be known from theoretical formula derivation, the dispose to Q and R plays a key role in improving the estimation effect by using this algorithm. Because it affects the size of the Kalman gain matrix K directly, the value of error covariance matrix P can be described as shown in Eq. (7.61):

 (7.61) Rk ¼ E υ2k ¼ σ 2υ It is the expression of the observation noise variance, which is mainly derived from the distribution of observation error of experimental instruments and sensors, as shown in Eq. (7.62):  2   2  σ ω1,1 σ 2ω1,2 Þ Eðω 1k ω2k E ω1k  ¼ (7.62) Qk ¼ Eðω2k ω1k Þ E ω22k σ 2ω2,1 σ 2ω2,2 It is the relationship between the system noise variance and the processing noise covariance. The two states in the system are generally unrelated and the covariance value is zero.

7.4 Experimental analysis

249

Therefore, the diagonal variance can have a small value. The variance Q of processing noise is mainly derived from the error of the established equivalent model, which is difficult to be obtained by theoretical methods or means. A reasonable value range can be obtained through continuous debugging through simulation, and it is usually a small amount. To verify the applicable range and stability of the algorithm, the input of different working conditions is used to observe the estimation accuracy of the algorithm. In this simulation, two working conditions are used: the constant-current working condition and the Beijing bus dynamic street test working condition. For the constant-current condition, only a constant value needs to be set in the program. To increase the complexity of the condition, two shelved stages are added to the constant-current input list to verify the tracking effect of Ah and the extended Kalman filtering. The following code is generated for the current. The experimental data should be used to simulate the algorithm under its operating conditions. The current input module is realized in the environment under Beijing bus dynamic street test operating conditions. It is the time integration module, whose output is the capacity change of the working condition, and the lower part is the working condition data output module. It outputs the current into the workspace at the minimum time interval of the sampling measurement. The data onto the workspace are a time-series type and the current data need to be extracted. The data extraction and transformation can be performed in the main program, wherein the working condition is the Simulink module. The current is the timeseries data, including the current output. S_Est is the algorithm program module, whose input is the initial S_Est_init and current data onto the estimated state value. The current data are input into the program module of the extended Kalman filtering algorithm based on the model, and the initial values of other parameters have been determined in the program. The program finally outputs the ampere hour time integral and the state estimation curve of the extended Kalman filtering through the minimum time interval of 10 times as the sampling time. The current curve of the experimental test can be obtained as shown in Fig. 7.13. The unscented Kalman filtering algorithm is programmed by writing scripts to realize algorithm simulation to compare the output effect. The specific implementation process conforms to the processing requirements, and the main simulation program can be constructed. The program takes 10 times the minimum time interval as the sampling time and simulates the three methods of the state estimation, including ampere hour time integration, extended

FIG. 7.13

Pulse current operating current curve for the experimental test. (A) Main charge working condition. (B) Main discharge experimental test.

250

7. Battery state-of-power evaluation methods

Kalman filtering, and unscented Kalman filtering, to compare the following situation of the three methods of the state over time. The program simultaneously outputs the state variation curve and estimation error curve with time, so that the tracking effect and error variation on the three methods can be obtained visually. Its implementation process is consistent with script. Different modules are used to replace the code block. The integration advantage of the environment is used in the graphical interface to obtain the same stimulation effect. The operation logic is ensured by the unchanged intuitive presentation of the calculation process. Based on script implementation, the modular simulation is built accordingly. The experiment data are used as the input current, and the state prediction is obtained by the state prediction module. The forecast and model parameters are calculated. After that, the equivalent model can be used to predict the completion status value of Up as well as the output voltage. The input of the algorithm includes the update module, output state correction, and error covariance matrix update as the basis of the forecast. It takes the ampere hour integral result as the minimum time interval of the current data, which is taken as the reference value of the measured state. The estimation effect of ampere hour and extended Kalman filtering is compared when the minimum time interval can be set as 10 times as the sampling period. The stability of the extended Kalman filtering algorithm is evaluated under working conditions. The state variable needs to accept the last estimated value as the basis of the next state prediction, in which the predicted value Up_pre is directly input into the corresponding position of the resistancecapacitance circuit modeling to conduct the terminal voltage prediction of the current polarization voltage. The functions of each part are realized in a more modular way for the complete structure and clear hierarchy.

7.4.5 Pulse-current charge-discharge test The measured parameter value can be obtained, and can be used for the state estimation process. The graph curves in the state estimation result are used as the state value. The estimation algorithm is realized for the conditions of the error. The state estimation results in the complex current simulation test are described as shown in Fig. 7.14. As can be seen from the experimental results, the error accumulation effect of the Ah integral becomes more obvious as time goes by. The deviation from the estimated value and the

FIG. 7.14 Complex current simulation results. (A) State estimation result and comparison. (B) Estimation error and its effect.

7.4 Experimental analysis

251

measured curve becomes larger. This is mainly because the Ah integral method is one-way estimation and there is no feedback correction link, which leads to the phenomenon of error accumulation. The estimation curve always follows the measured value and fluctuates slightly around it, in which the estimation error is filled with good stability. The initial estimation is set to deviate from the measured value to simulate the situation where the initial value is inaccurate. Here, the initial estimation is set to be 0.99 while the measured value is set to be 1. The simulation model is run to verify whether the output waveform of the algorithm is converged on the simulated waveform automatically. The output results are analyzed as well. It can be concluded that the algorithms can automatically adjust when the initial value is wrong. In a short time, the state estimation converges on the measured value with the same accuracy. The simulation effect of the working condition is described as shown in Fig. 7.15. As can be known from Fig. 7.15A, the extended Kalman filtering and unscented Kalman filtering algorithms can also track well the change of the measured value under varying current operating conditions. For the operation of the whole operating condition, the absolute value of the error is small while the estimated value of the Ah integration method is still greatly deviated. When the initial value is set to be inaccurate, the state estimation simulation is run. Here, the initial estimation is set to be 0.95, and the measured initial value is 1. It can be concluded from Fig. 7.15B that the algorithms have a strong self-correction ability, in which the output waveform converges on the curve of the module within a limited period of sampling. According to simulation debugging, the main factor affecting the convergence speed of the algorithm is the initial value of error covariance matrix P, which verifies that the extended Kalman filtering and unscented Kalman filtering algorithms have the ability of adaptive adjustment of the condition of initial value error, in which the state estimation accuracy does not decrease. The convergence speed of the algorithm increases along with P, and the experimental results are shown in Fig. 7.16. According to the comparison of simulation results, the error difference in the stable tracking is not big and the accuracy is roughly the same while the complexity of the extended Kalman filtering algorithm is lower under the same precision. According to the comprehensive analysis, the equivalent circuit model has good stability, accuracy, and fast response

FIG. 7.15 Estimation effect at time-varying current working condition. (A) Estimation effect and comparison. (B) Estimation error variation curve.

252

7. Battery state-of-power evaluation methods

FIG. 7.16 The estimation effect at complex current variation working conditions. (A) Current variation procedure design. (B) State estimation effect and comparison.

ability. Due to its low complexity, the algorithm is suitable for the development of a practical battery management system.

7.5 Conclusion In this chapter, different algorithms were compared, and the extended Kalman algorithm was improved. Based on the model, the adaptive Kalman filtering was used to estimate the battery state. Experimental results show that the estimation effect of this method is better than that of the extended Kalman filtering algorithm. The improved algorithm was used to estimate the battery state. The estimation effect is good, which proves that it can effectively estimate the battery state.

Acknowledgments The work is supported by the National Natural Science Foundation of China (No. 61801407), the Sichuan Science and Technology Program (No. 2019YFG0427), the China Scholarship Council (No. 201908515099), and the Fund of Robot Technology Used for Special Environment Key Laboratory of Sichuan Province (No. 18kftk03).

Conflict of interest There is no conflict of interest.

References [1] Y.Q. Tan, et al., Joint estimation of ternary lithium-ion battery state of charge and state of power based on dual polarization model, Int. J. Electrochem. Sci. 15 (2) (2020) 1128–1147. [2] X.T. Liu, et al., An improved state of charge and state of power estimation method based on genetic particle filter for lithium-ion batteries, Energies 13 (2) (2020) 53–68. [3] X. Lai, et al., Co-estimation of state of charge and state of power for lithium-ion batteries based on fractional variable-order model, J. Clean. Prod. 255 (2020) 158–172. [4] X.T. Liu, et al., State-of-power estimation of Li-ion batteries considering the battery surface temperature, Energy Technol. 6 (7) (2018) 1352–1360.

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[5] S. Xiang, et al., Lithium-ion battery online rapid state-of-power estimation under multiple constraints, Energies 11 (2) (2018) 96–108. [6] M.J. Esfandyari, et al., A hybrid model predictive and fuzzy logic based control method for state of power estimation of series-connected lithium-ion batteries in HEVs, J. Energy Storage 24 (2019) 102–121. [7] M.J. Esfandyari, et al., A new approach to consider the influence of aging state on lithium-ion battery state of power estimation for hybrid electric vehicle, Energy 176 (2019) 505–520. [8] X. Zhang, et al., A novel method for lithium-ion battery state of energy and state of power estimation based on multi-time-scale filter, Appl. Energy 216 (2018) 442–451. [9] S.L. Wang, et al., A new additive 3-isocyanatopropyltriethoxysilane to improve electrochemical performance of Li/NCM622 half-cell at high voltage, J. Power Sources 423 (2019) 90–97. [10] S.L. Wang, et al., Nanostructured Li3V2(PO4)(3)/C composite as high-rate and long-life cathode material for lithium ion batteries, Electrochim. Acta 143 (2014) 297–304. [11] C. Wei, M. Benosman, T. Kim, Online parameter identification for state of power prediction of lithium-ion batteries in electric vehicles using extremum seeking, Int. J. Control. Autom. Syst. 17 (11) (2019) 2906–2916. [12] X. Lei, et al., A novel temperature-hysteresis model for power battery of electric vehicles with an adaptive joint estimator on state of charge and power, Energies 12 (19) (2019) 558–577. [13] H.T. Zhang, M. Zhou, X.D. Lan, State of charge estimation algorithm for unmanned aerial vehicle power-type lithium battery packs based on the extended Kalman filter, Energies 12 (20) (2019) 99–109. [14] S.L. Wang, et al., An adaptive working state iterative calculation method of the power battery by using the improved Kalman filtering algorithm and considering the relaxation effect, J. Power Sources 428 (2019) 67–75. [15] P. Bhowmik, S. Chandak, P.K. Rout, State of charge and state of power management among the energy storage systems by the fuzzy tuned dynamic exponent and the dynamic PI controller, J. Energy Storage 19 (2018) 348–363. [16] W.H. Zheng, et al., State of charge estimation for power lithium-ion battery using a fuzzy logic sliding mode observer, Energies 12 (13) (2019) 256–268. [17] S.L. Wang, et al., A comprehensive working state monitoring method for power battery packs considering state of balance and aging correction, Energy 171 (2019) 444–455. [18] Q. Wang, et al., Power battery state of charge estimation based on extended Kalman filter, J. Renew. Sust. Energy 11 (1) (2019) 152–162. [19] D.H.S. Tan, et al., From nanoscale interface characterization to sustainable energy storage using all-solid-state batteries, Nat. Nanotechnol. 15 (3) (2020) 170–180. [20] A.K. Pati, N.C. Sahoo, A super-twisting sliding mode observer for boost inverter-based hybrid photovoltaicbattery system control, Trans. Inst. Meas. Control. 42 (12) (2020) 2139–2154. [21] M.K.S. Verma, et al., On-board state estimation in electrical vehicles: achieving accuracy and computational efficiency through an electrochemical model, IEEE Trans. Veh. Technol. 69 (3) (2020) 2563–2575. [22] M.S. Wang, et al., State space model of aggregated electric vehicles for frequency regulation, IEEE Trans. Smart Grid 11 (2) (2020) 981–994. [23] M.F. Liu, et al., Study of fast charging battery, Int. J. Mod. Phys. B 34 (1–3) (2020) 153–164. [24] L.H. Jia, et al., A low-power motion estimation architecture for HEVC based on a new sum of absolute difference computation, IEEE Trans. Circuits Syst. Video Technol. 30 (1) (2020) 243–255. [25] R. Carter, et al., Initiated chemical vapor deposition of ultrathin polymer coatings at graphite electrodes for enhanced performance in Li-ion batteries, J. Electrochem. Soc. 167 (6) (2020) 79–91. [26] E.A. Astafev, The measurement of electrochemical noise of a Li-ion battery during charge-discharge cycling, Measurement 154 (2020) 66–79. [27] T. Qian, et al., Deep reinforcement learning for EV charging navigation by coordinating smart grid and intelligent transportation system, IEEE Trans. Smart Grid 11 (2) (2020) 1714–1723. [28] N. Hashemipour, et al., Multi-objective optimisation method for coordinating battery storage systems, photovoltaic inverters and tap changers, IET Renew. Power Gener. 14 (3) (2020) 475–483. [29] G.R. Kalra, et al., A novel boost active bridge-based inductive power transfer system, IEEE Trans. Ind. Electron. 67 (2) (2020) 1103–1112. [30] N. Gauthier, et al., Impact of the cycling temperature on electrode/electrolyte interfaces within Li4Ti5O12 vs LiMn2O4 cells, J. Power Sources 448 (2020) 196–209. [31] X.L. Chen, et al., Operational reliability and economy evaluation of reusing retired batteries in composite power systems, Int. J. Energy Res. 44 (5) (2020) 3657–3673.

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C H A P T E R

8 Battery state-of-health estimation methods

8.1 Equivalent modeling and description The internal resistance is an important aspect of its parameter identification. There are many ways to identify the internal resistance of the battery, such as the alternating-current impedance, volt-ampere characteristic curve fitting method, and so on. The open-circuit voltage and hybrid pulse-power characteristics are analyzed as well.

8.1.1 Equivalent circuit analysis The internal resistance measurement can reach the order of milliohms, in which the battery can be accurately measured under different conditions. To describe the working process of the battery clearly, the experimental variables can be controlled more accurately in the simulation experiment. In the modeling of the lithium-ion battery, the equivalent circuit model is generally selected, which has the advantage that the parameters are closely related and the number is small [1–9]. After that, the mathematical description of its state-space is also relatively easy. For the above considerations, the control variable method is used to control the temperature and the discharging current rate. In the case of minimizing other influence conditions, the equivalent circuit model is used for mathematical modeling. Taking into account the variables controlled, the Rint model contains only one internal resistance; the structure is too simple and it is difficult to characterize the dynamic battery characteristics of the temperature changes [10–14]. The Thevenin model considers the polarization of the battery based on the model, and it can describe the gradual process of the open-circuit voltage. It still lacks the means to describe the temperature. The model adds large capacitance to simulate the capacity of addressing temperature effects. The improved battery electrical equivalent modeling schematic can be described as shown in Fig. 8.1.

Battery System Modeling https://doi.org/10.1016/B978-0-323-90472-8.00007-X

255

Copyright # 2021 Elsevier Inc. All rights reserved.

256

8. Battery state-of-health estimation methods

FIG. 8.1 The improved battery electrical equivalent modeling schematic.

As shown in the designed structure diagram of the improved circuit model, it introduces a resistance-capacitance series network and a second-order resistance-capacitance parallel network. UL is the battery terminal voltage. IL is the current flowing through the load, where the discharge direction is the positive pole and the negative pole when charging. Rs is the selfdischarge internal resistance of the battery. UOC is the open-circuit voltage of the battery. R0 is the ohmic internal resistance of the battery, and its terminal voltage is used to represent the instantaneous voltage drop caused by the battery current. Cb is used to characterize the open-circuit voltage variation due to current accumulation. E is used as the ideal voltage source, which is integrated into Cb for indicating the change of UOC. R1 and C1 are the concentration polarization resistance and capacitance of the battery, respectively. They are connected with parallel to simulate the polarization characteristic process of the battery charge-discharge, which characterizes the rapid electrode reaction against the battery. R2 and C2 are the electrochemical polarization resistance and capacitance of the battery, respectively, which characterize the slow electrode reaction against the battery. The internal resistance at different temperature levels is described in Fig. 8.2.

FIG. 8.2 Internal resistance variation of different temperature conditions.

8.1 Equivalent modeling and description

257

The measurement equipment used in the hybrid pulse-powers characteristic method is not difficult to operate, the cost required for measurement is not high, and high precision can be achieved. However, dynamic integrals are difficult to implement in practical problems, and it is difficult to use the above method to obtain the posterior filtering probability density. Therefore, this method has been limited to practical promotion and needs to be combined with other methods to obtain the posterior probability density. The required posterior probability density is obtained to get an estimation of the required parameters.

8.1.2 Mathematical state-space expression After comparing all the test methods, the hybrid pulse-power characteristic test method is finally selected; it is combined with the existing experimental conditions in the laboratory. Compared with other methods, the hybrid pulse-power characteristic method is convenient for measurement [15–18]. Only the hybrid pulse method can detect the voltage changes in the time point of charge-discharge by Ohm law. The internal parameters of the battery can also pass the detected time-domain relationship to be obtained as shown in Fig. 8.3. The predicted system transition probability density function obtained by the prediction is corrected by the observation value at the time point of k, and finally the state posterior probability density function of the system is obtained. The initial state-space equation of the probability and posterior density is obtained by the Bayesian formula. The previously obtained equation is converted by the probability density formula, which is then converted by the joint probability density formula and the Bayesian formula [19–22]. The observations of each system are independent of each other. This results in a recursive and updated process of the posterior probability density function for the system posterior filtering.

Start 1. Estimate the state result 2. Estimate the coefficient 3. Calculate the gain coefficient N

4. Estimate current optimal estimation value 5. Update prediction coefficient Stop? Y End

FIG. 8.3 Overall state estimation flowchart.

258

8. Battery state-of-health estimation methods

8.2 Particle filtering algorithm The particle filtering algorithm has an important relationship to the particle, and the prediction is based on the particle. It is superior to other methods of predicting and tracking dynamic parameters that can track dynamic parameters more accurately. The probability of random events in the particle filtering is very low, which is equivalent to a prominent advantage of this algorithm.

8.2.1 Bayesian estimation Bayesian estimation is a method of calculating the hypothesis probability by using the prior probability and Bayesian theorem, combined with the new data obtained in practice, to obtain a new probability [23–34]. The random variable x is known together with the prior distribution p(x) of the random variable. When the observation data are not obtained, the random variable x can be estimated according to the prior distribution, and then the data are obtained, according to which the prior probability of the random variable is optimized. The data are corrected and the posterior distribution of the random variable x is obtained. Knowing the state and observation equations, the observation values before the time point k can be obtained by assuming that the initial value of the probability density is known. The prediction equation of the state transition probability density function is obtained to update the equation. With the rapid development of new energy sources, lithium-ion batteries have been widely used due to their long cycle life, no pollution, and no memory effect. Its state estimation is an important parameter of the entire battery system [35–39]. Much research has gone into battery technology today, and the results include state estimation algorithms such as the open-circuit voltage, ampere hour integration, discharge experiment, and Kalman filtering methods.

8.2.2 Monte Carlo treatment The Monte Carlo algorithm is an easy method to implement that can be used to calculate the random numbers of the battery system. The principle is to treat all the integral operations as the mathematical expectation of the random variables, according to which the approximation of the integral operation can then be obtained by the method of random sampling estimation [40–46]. It first randomly samples the affected random variables and then takes the data samples into the function, as shown in Fig. 8.4. The implementation of this method is better because when the random number is sampled, it is necessary to solve the problem for the random variable obeying the corresponding distribution type. Then, the function is substituted to solve the problem, which can improve its computational efficiency. The method is characterized in that it is not affected by the limitation of the problem condition and does not need to be discretized. It is a method to solve the problem directly. Furthermore, the convergence speed of the method has nothing to do with the number of dimensions, and the errors are easy to be determined.

259

8.2 Particle filtering algorithm

HMI of imbalance predict system

Evaluation knowledge acquisition machine

Imbalance anticipation inference engine

Characteristic element interpreter

Function and performance database and management system

Knowledge evaluation rule and its management system

FIG. 8.4 The whole structure of Monte Carlo sampling and mathematical treatment.

8.2.3 Importance sampling The Bayesian importance sampling proposes the concept importance of the probability density, sampling from a known distribution to obtain independent and distributed sampling points identically. Instead of sampling from the posterior probability density, it provides a solution to the difficulty of the sampling method theoretically, as shown in Fig. 8.5. The Bayesian importance sampling method needs to be resampled every time of sampling, which increases the complexity. The weight of each ion needs to be recalculated after each sampling. Then, new observations can be obtained, which also increase the amount of time and calculation. Sequential importance sampling avoids such troubles when obtaining new observation data. It uses the recursive update method to calculate the particle weights, which reduces the calculation amount based on the Bayesian importance sampling.

Measurement Model

W (k)

G V (k+1) X (k+1)

U (k) B

A State Model

Y (k+1) H

X (k)

Delay Unit

AX (k)

FIG. 8.5 The battery state estimation modeling procedure structure.

Z (k+1)

260

8. Battery state-of-health estimation methods

8.3 Estimation modeling process In most systems, it is difficult to extract samples. To make the sampling of the system easier, it is necessary to introduce a parameter that facilitates the random sampling of the reference distribution of the probability density.

8.3.1 Equivalent circuit modeling The premise of implementing the filtering algorithm is to select and build a suitable battery equivalent circuit model. For different research and application requirements, it is necessary to select models with different precision purposes. After weighing the accuracy and complexity, the second-order resistance-capacitance model is built that can be described as shown in Fig. 8.6. Among them, UOC is the open-circuit voltage of the battery that exerts a certain functional relation with the state at a specific temperature. R0 is internal resistance. R1 and R2 are the polarization resistance. C1 and C2 are the polarization capacitances. The resistancecapacitance circuit time-constant τ1 simulates the sudden change of the battery current, and τ2 describes the voltage change of R2 and C2 accordingly. The current flowing through the whole circuit is used as the input of the system. In the case of charge-discharge in the figure, the battery terminal voltage UL is obtained as an observation value. It is concluded that the discretized state-space model and the state equation is described as shown by Eq. (8.1): Sk ¼ f ðSk1 Þ + wk ¼ Sk1 + ηc Ts I ðkÞ=CN + wk

(8.1)

wherein Ts is the sampling interval and wk is the processing noise of the system. The observation equation is described by Eq. (8.2):

 (8.2) ULk ¼ h Sik + vk ¼ Uoc Sik  U1k  U2k  R0 Ik + vk wherein νk is the systematic observation noise. The relationship curve for UOC toward S(k) can be obtained from the calibration experiment of open-circuit voltage along with the variation of battery state:      
 R1 1  eTs =τ1 U1 ðk  1Þ U 1 ðkÞ eTs =τ1 0
 Ik ¼ + (8.3) U 2 ðkÞ 0 eTs =τ2 U2 ðk  1Þ R2 1  eTs =τ2

FIG. 8.6 Second-order battery electrical equivalent circuit model.

8.3 Estimation modeling process

261

wherein U1 and U2 are the terminal voltage of the RC circuits, which can be calculated iteratively from time point k  1 to time point k. Other parameters of the second-order resistancecapacitance model are calculated accordingly. The parameters are identified by the hybrid pulse-power characteristic test, including ohmic internal resistance R0, electrochemical polarization resistance R1, electrochemical polarization capacitance C1, concentration polarization resistance R2, and concentration polarization capacitance C2. The voltage curve is obtained by the least-square method to obtain the corresponding parameters.

8.3.2 Calculation process design The particle filtering algorithm is a statistical filtering method estimated by Monte Carlo and the recursive Bayesian filtering algorithm. It processes the integral operation of Bayesian estimation. The minimum average square error estimation of the system state is obtained. The idea is to collect random samples that are used to adjust the weights and positions of the particles according to the observations to correct the previous empirical condition distribution. Compared with other filtering methods such as Kalman filtering, extended Kalman filtering, unscented Kalman filtering, and particle filtering, it does not have to make any a priori assumptions of the system state that is theoretically applicable to any stochastic system. Applying the particle filtering method can improve the accuracy of the battery state estimation. In the condition of ensuring the same accuracy, the particle sample number of the algorithm can be reduced, thereby reducing the calculation amount. The particle filtering calculation procedure is described as shown in Fig. 8.7. FIG. 8.7 The overall structure of the particle filtering

Start

flowchart.

Initialize k=0, Generating particles Update particle status Update particle weights Calculated estimate N Resample? Y Re-sampling

k=k+1

Predict the next moment state N

Whether to end Y End

262

8. Battery state-of-health estimation methods

According to the illustrated program flow, the iterative calculation process is obtained and described by the following steps. (1) Initialization. The prior probability is used to generate N initialed particles and their weights as shown in Eq. (8.4):  i N  i N (8.4) S0 i¼1 , q0 i¼1 ¼ 1=N (2) The algorithm loop process is described as follows: (1) Update. According to the system update equation, the prior probability sample of the next time point is obtained, according to which the particle weight is updated as shown in Eq. (8.5):  



(8.5) ωik ¼ ωik1 p ULðkÞ Sik ¼ ωik1 ULðkÞ  h Sik , i ¼ 1, 2, …,N (2) Weight normalization can be conducted by using the normalized weight as shown in Eq. (8.6): X N ωik (8.6) ωik ¼ ωik i¼1

(3) The minimum average square estimation is calculated as shown in Eq. (8.7): ^k  S

N X i1

ωik Sik

(8.7)

(4) Resampling. The number of valid particles can be calculated to get a new set of particles as shown in Eq. (8.8): Neff ¼ 1=

N
X i1

ωik

o 2 Neff Ns n i∗ ! S0:k , i ¼ 0, 1, 2, …, N

(8.8)

(5) Prediction. The unknown parameter prediction can be realized by using the state equation. (6) The program ending condition is judged. If not, the time point is set as k ¼ k + 1 and the procedure turns to S1 accordingly. The particle filtering algorithm does not impose any restrictions on the processing noise and the observed noise of the system when estimating the battery state. The established battery state-space model is simulated to realize the particle filter to estimate the battery state. The specific steps are described as follows. First, the state-space model of the lithium-ion battery is established by the process model and the observation model. The state variable of the model is the battery state, and the observed variable of the battery model is the load voltage of the battery, as shown in Eq. (8.9): 8 nI k Δt < + wk , wk Nð0, QÞ, vk N ð0, RÞ x k + 1 ¼ f ð xk , I k , w k Þ ¼ xk  (8.9) ηi ηT ηn Qn : yk + 1 ¼ f ðyk , Ik , wk Þ ¼ k0  RI k  k1 =xk  k2 xk + k3 ln ðxk Þ + k4 ln ð1  xk Þ + vk

263

8.3 Estimation modeling process

Among them, wk is the processing noise of the system. vk is the observation noise of the system. Δt is the sampling period of the system. (1) Initialization As can be known from the initial probability distribution p(x0), N state initial particles are randomly generated to form a new particle. (2) Update the particle status New particle sets are generated, and each particle passes through the state model equation. The matching particles are extracted to generate a new particle set. (3) Particle weighting calculation and normalization After the system obtains new observations, a new particle set is generated by the state equation. Then, the observed value is obtained by the observation equation. The particle weight equation is designed and it is normalized after that as shown in Eq. (8.10): ðyk yi , kÞ2 pffiffiffiffiffiffiffiffiffi xk ¼ e 2R = 2πR ) w∗i

¼1

X N

wi

(8.10)

1

Therefore, the error between the observed value and the predicted value of each particle is calculated to minimize the error. (4) Resampling The new random sample distribution is generated in the previous step to calculate the effective particle. If the effective particle number is less than the threshold of the set effective particle number, high-weight resampling particles can be used and picked by eliminating the low-weight particles. Then, the new particle set can be generated. After sampling, each particle weight of the new particle set is 1/N. Finally, the estimated value can be obtained as shown in Eq. (8.11): xk ¼

N X

wk ðiÞxx ðiÞ

(8.11)

i¼1

(5) Finishing judgment It is judged whether the program is finished. If it is not finished, the time point should be set as k ¼ k + 1. Repeating steps (2)–(4), the recursive estimation of the state quantity Xk can be realized. Particle filtering is an algorithm based on Monte Carlo simulation and approximate Bayesian filtering. The particle filtering algorithm has an important relationship to the particles, according to which the prediction can be realized. It is superior to other methods of predicting and tracking dynamic parameters, which can track dynamic parameters more accurately. The particle filtering is not too high in the probability of the random occurrence of events, which is equivalent to a prominent advantage of this algorithm.

264

8. Battery state-of-health estimation methods

As it can realize the best results from dynamic parameters and the estimation of timevarying systems, the algorithm has been applied to many fields. The algorithm extracts some discrete random particles and then uses them to implement the probability density function instead of the average of the extracted samples, thus eliminating the need for integral operations. In this way, the dynamic parameters of the system can be estimated effectively. It achieves the estimation that is an efficient filtering algorithm. This method has the phenomenon of particle degradation in estimating the battery state. The particle degradation not only increases the amount of computation but also reduces its accuracy. Because the particle filtering algorithm is very important, particle degradation leads to a significant reduction in the effectiveness of the particle, which affects the accuracy of the filtering algorithm. To improve its accuracy, it is necessary to consider the particle degradation problem. A polynomial resampling algorithm is used that is simple to be implemented computationally to reduce the particle degradation and improve the accuracy of the battery state estimation.

8.3.3 Particle degradation expression In the particle filtering algorithm, many particles need to be extracted, which has a great influence on the battery state estimation accuracy. In the calculation process, particle weight imbalance may occur. Some particles have high weights and some have low weights. Such an imbalance seriously affects the estimation accuracy of the algorithm. Many particles have very small weights. In the process of calculation, many small-weight particles that do not work are calculated, which takes a long time and increases calculation complexity. The imbalance of the particle weights causes the particles to vary greatly. If the algorithm calculates the particles of small weights and neglects those with large weights, its accuracy can be reduced greatly. Many particles are not effective against the particles calculated by the algorithm, and this takes time to be calculated, which increases the amount of calculation. In general, the lower the effectiveness of the particle, the more serious the particle degradation. In practical applications, a specific value can be set to indicate the number of effective particles. The number of effective particles is in the algorithm process. If the number of effective particles is low, it does not meet the requirements, and some methods are needed to suppress the particle degradation.

8.3.4 Resampling treatment Resampling is the particle approximation of the posterior probability density of the battery system. In the process of resampling, the particles of high weights are copied, and the particles of low weights are not selected. The copied high-value particles are used to generate a new particle set. The purpose of particle degradation is reduced. There are many commonly used resampling methods such as polynomials and systems. The overall battery state measurement and evaluation structure can be described as shown in Fig. 8.8.

265

8.3 Estimation modeling process

Lithium -ion battery pack

Sensor group measurement data 1

Calculate credibility using different methods

Sensor group measurement data 2

Calculate credibility using different methods

Calculate credibility using different methods

Sensor group measurement data n

Mass function for each measure ment

Combin ation

Information fusion and equilibrium state evaluation results

Calculate credibility using different methods

FIG. 8.8 The overall battery state measurement and evaluation structure.

In the algorithm, resampling can be performed multiple times to improve accuracy, but too much resampling increases the amount of computation and reduces the efficiency of the system. Excessive resampling results in a large reduction or even exhaustion of the sample. Therefore, it is necessary to set a threshold to ensure the number of effective particles of practical applications. Because the polynomial resampling has the advantage of being simple and easy to be implemented, the computational complexity is low and the polynomial resampling is chosen. In the polynomial resampling method, a sample u-U[0, 1] of a particle is randomly extracted from the uniform distribution of [0, 1]. The particle Xk(i) satisfying the formula is copied according to Eq. (8.12): j1 X j¼1

wk ð jÞ  u 

j X

wk ð jÞ

(8.12)

j¼1

The resulting particles are grouped into a new set of particles. The particles are reexamined, in which the particles satisfying the conditions are obtained according to the formula. The weight of each particle is 1/N in the new particle set composed of the new particles. The sampling process is polynomial resampling. In the particle filtering algorithm, the realized steps of the resampling algorithm are described as follows. (1) Prediction New particles of the important density function in the system are extracted and combined with the resulting new dataset. (2) Update The weight value of each particle is calculated accordingly by the normalized processing as shown in Eq. (8.13): X N w∗k ðiÞ (8.13) qðxk j x0:k ðiÞ, yÞ ¼ pðxk j xk1 ðiÞÞ ) wk ðiÞ ¼ w∗k ðiÞ i¼0

266

8. Battery state-of-health estimation methods

(3) State estimation The threshold of the effective particle has been set. At this time, the number of effective particles needs to be calculated, and this is then compared with the set threshold. If the number of effective particles is less than the set threshold, then it needs to be reestablished. After sampling, the particles are reselected and a new set of particles is formed that is based on these selected particles. The weighted average value of the parameter can be obtained as shown in Eq. (8.14): xk ¼

N X

wk ðiÞxx ðiÞ

(8.14)

i¼1

(4) Polynomial resampling With the particle filtering algorithm, when the sample is resampled, it brings some additional random variances in the particle. Therefore, before the resampling, the posterior estimation of the system and other estimates are generally calculated. Resampling is performed, which may cause the sample to depleted. To suppress the sample depletion caused by resampling in the particle filtering algorithm, a parameter α is added that is satisfied according to the idea of the genetic algorithm. The calculation process is described as shown in Eq. (8.15):
α (8.15) wt ðiÞ ¼ wit1 p½zt j xt ðiÞp½xt ðiÞj xt1 ðiÞ=q½xt ðiÞj xt1 ðiÞ, z1:t  wherein α is the annealing factor of the genetic algorithm, which is used to control the influence of the importance weight of the particle filter, thereby reducing the exhaustion of the sample.

8.4 Whole life-cycle experiments The comparison of the proposed estimation method toward the traditional algorithms is conducted out of experiments and some references, in which the same type of lithium-ion battery is used for all the algorithms with a detailed description of the experimental process and results.

8.4.1 Experimental procedure design The lithium cobalt oxide battery is taken as the research objective, which is as HTCNR18650-2200mAh-3.6V. The selected batteries are used in the experimental analysis with the detailed parameters as shown in Table 8.1. At 25°C, the specific experimental implementation steps are designed, including capacity, hybrid pulse-power characteristic, open-circuit voltage, and Beijing bus dynamic stress tests. The open-circuit voltage is divided into two groups of charge-discharge values. The test procedure of the power battery undercharging state is designed as follows. S1: When the battery is discharged to the cutoff voltage at the standard current 0.3C, the terminal

8.4 Whole life-cycle experiments

TABLE 8.1

267

Experimental lithium-ion battery parameters.

No.

Item

Standard

Note

1

Standard capacity

2200 mAh

2200 mA at 1C

2

Capacity range

2150–2250 mAh

0.5C

3

Standard voltage

3.6 V

4

Alternating internal resistance

20 mΩ

5

Charge conditions

Cutoff voltage

4.2  0.05 V

Cutoff current

0.01C

6

Discharge cutoff voltage

2.5 V

7

Cycle characteristic

800 times (100% DOD) 1200 times (80% DOD)

8

Max charging current

6.6 A

9

Max continuous discharge current

6.6 A

10

Pulse discharge current

33 A, 5 s

11

Working temperature

Charge: 0–55°C Discharge: 20°C to 60°C

12

Storage temperature

20°C to 45°C

13

Battery weight

45 g

14

Warranty

2 years

Constant-current charge to 4.2 V at 0.5C, and constantvoltage until 0.01C

No less than 70% of the rated capacity at 1C charge and 3C discharging current rate

Short-term storage