![Chapter 1
Introduction
Abstract
With the development of society, energy security and the environmental pollution
caused by it has become a key issue that all sectors of society are concerned about
and urgently need to be resolved. The deepening of the world energy crisis has led
to the emergence of new energy industries as well as the increasing awareness of
environmental protection. Among them, lithium-ion batteries have developed
rapidly in the field of new energy due to their higher energy density and longer
cyclic life. This chapter briefly introduces the use scenarios and market conditions
of lithium-ion batteries, the common methods of lithium-ion battery state estima-
tion, and their research significance. Then, the development status of the battery
management system is briefly described. At the same time, the estimation methods
of various state parameters are introduced for lithium-ion batteries, which lays a
foundation for subsequent research. The research conclusions and further research
plans are discussed, the content is reviewed, and the system state estimation is
emphasized to achieve the purpose of safety protection and lifetime guarantee.
Keywords: Lithium-ion battery; Energy crisis; Development history; System
state estimation; Filtering algorithm; Equivalent circuit model; Safety
protection
1.1 State of the art
Currently, lithium-ion batteries are important new energy sources in the twenty-
first century, and research in the area of improving and enhancing the technical
performance of lithium-ion batteries through various technologies and methods is
the key to tapping greater potential. Lithium-ion batteries are used in
portable products, such as laptops, notebooks, mobile phones, tablets,
portable digital assistants, cameras, rechargeable lamps, toy cars, and toys that use
rechargeable batteries and robust products such as the electric vehicle, electric
motorcycles, balance cards, scooters, and smart grids, in all aspects of life and all
sectors, agriculture, health, education, industry and in wireless sensor networks and
recently very popular. The key to improving, controlling, monitoring, and mana-
ging the lithium-ion batteries is the battery management system (BMS) [1], as well](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-28-320.jpg)
![as the estimation of the state of charge (SOC), state of health (SOH), state of power,
and other battery parameters are very important research fields, ensuring the safety
and reliability of electronic devices that use the battery as a power source.
The global attention in the direction of the batteries used in electric vehicles
(EVs) is gaining much popularity as they are the sustainable mode of transportation
and the alternatives of the International Combustion Engine-based vehicles [2]. The
EVs effectively overcome the fossil fuel crisis and environmental pollution, con-
sidering the major hurdle for the automobile sector [3]. As the only power supply in
pure EVs, the capacity of the battery pack is of great importance. In this case, the
high specific energy lithium-ion batteries are widely used in EVs with superiority [4].
Lithium-ion batteries have increased in popularity. Due to their advantages such as
lightweight, fast charging speed, high energy density, low self-discharge, and long
lifespan [5]. The usable energy classified as SOC is the same as the available energy
of the vehicle fuel gauge operated by a combustion engine.
BMS is responsible for accurately measuring the status of the battery, ensuring
safe operation, and prolonging the battery life [6]. The accurate SOC estimation of
the lithium-ion battery is a very challenging task because of its high time-variant,
nonlinearity, and complex electrochemical system [6]. An improved Thevenin
equivalent circuit model is proposed, designed, and implemented through experi-
mentation and simulation. The model is achieved by adding an extra RC branch to
the Thevenin model, making it a second-order resistor capacitor. The second-order
Thevenin model has good accuracy, stability, robustness and is very effective for
SOC estimation [7]. This model is used to study and record parameters and estimate
the relationship between voltage, current, SOC, and charge–discharge characteristics.
Experimental data results and simulated results are compared and analyzed to
further appreciate the effectiveness of the improved adaptive extended Kalman
filtering (AEKF) algorithm used [8]. The improved AEKF algorithm is employed
in this work to accurately estimate the SOC of the lithium-ion batteries. The
algorithm estimates the SOC dynamically but easily causes divergence due to the
uncertainty of the battery model adopted and to what extent the system noise is
taken care of. The AEKF algorithm is used for the SOC estimation of the lithium-
ion battery model designed to obtain a more accurate result for the research. It is
worth noting that the proposed SOC estimation using the AEKF algorithm is more
accurate and reliable comparatively than the extended Kalman filtering (EKF)
algorithm.
The safe operation and efficient energy management strategy are essential for
the BMS of batteries, which are based on the SOC estimation accuracy [9].
However, the evaluation of battery SOC is challenging in uncertain and complex
EVs environments [10]. Therefore, to manage the lithium-ion more efficiently and
improve battery performance, it is necessary to obtain the inner state parameters
and accurately make an accurate SOC [11]. SOC is not directly measurable and
requires a particular type of method to be estimated but often uses some specific
models such as the empirical model, the equivalent circuit model, and the elec-
trochemical model [12]. Deterioration of battery cells leading to reduced perfor-
mance is a problem that limits battery life.
2 Battery state estimation: methods and models](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-29-320.jpg)
![Moreover, electrical equivalent circuit (EEC) modeling has been investigated
specifically for applications such as BMS development and vehicle power man-
agement control [13]. A good battery model can estimate the battery storage power
details of the battery and the voltage response to the load [14]. The EECs can
describe these characteristics of lithium-ion batteries [15]. In some cases, modeling
the side reactions is required in terms of battery losses, which can also be realized
with the EECs [16]. A novel approach for identifying parameters requires a high-
order equivalent circuit proposed in this literature with improved mathematical
analysis. The EKF algorithm is proposed to estimate the SOC in real-time solid
data [5]. The validation results show that the proposed methods have good per-
formance in estimation accuracy and uncertainty of the parameters.
Electrical vehicles powered by lithium-ion batteries are more environmentally
friendly than gasoline-powered ones, and this kind of environmentally friendly energy is
gradually widely used in various fields. In such high demand, the battery state is sig-
nificant because it can be estimated. To extend the driving distance, the influencing
factors should be measured in real time for the long-term use to avoid the measurement
error accumulation of the voltage, resistance, and temperature [17]. The highly nonlinear
characteristic also brings lots of difficulties to the battery state estimation. Because the
power batteries operating characteristics are highly nonlinear, it is necessary to establish
an accurate battery model to estimate the accurate state of the battery [18].
To solve this issue, several algorithms are proposed and used to estimate the
battery state, including the current integration, open-circuit voltage (OCV), Kalman
filtering (KF) algorithms, and neural network (NN) [19]. 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 use will
lead to the accumulation and expansion of measurement errors, so it is necessary to
introduce relevant correction factors to correct the accumulated errors. However, to
realize the SOC estimation accurately, the following factors are determined and the
core research is objective, including accurate SOC estimation, understanding the
dynamic characteristics of battery behavior, studying internal reactions and tem-
perature, and monitoring and controlling charge–discharge flow. An effective BMS
is necessary for power lithium-ion batteries.
The above objects are analyzed in various ways to determine a ternary lithium-
ion battery for EVs as the research objective. The entire process is carried out by
the offline process, and the process is mainly established in the laboratory envir-
onment and uses room temperature.
To perform and validate the research objective, the above contents help to
carry out the necessary procedures. The Ampere-hour (Ah) commonly known as
the coulomb-counting method is to perform the first stage. Then, a similar model
can be selected as well as the state-space equation and so on. The high-order
improved equivalent circuit model is used as the significant model to get the
parameters. Furthermore, these parameters are used for SOC estimation, which is
commonly defined as the percentage of the maximum possible charge present
inside a rechargeable battery from 0% to 100%. The detailed contents are briefly
introduced in Figure 1.1.
Introduction 3](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-30-320.jpg)
![The NN method estimates the SOC of the battery by processing the amount of real-
time input and output data of the battery. The KF method obtains the optimal
solution in the sense of minimum variance through continuous iterative operations.
Because of the complex electrochemical reaction inside the battery in use, it often
shows strong nonlinear characteristics.
Also, some defects of the traditional algorithm itself make the above methods
cannot accurately estimate the SOC value of the battery when dealing with the
nonlinear systems, which often have the problems of low estimation accuracy and
large error. In recent years, researchers put forward some improved algorithms
based on traditional algorithms. The complex internal structure of the lithium-ion
batteries makes it show strong nonlinear characteristics in the use process, which
puts forward new requirements for the traditional SOC estimation methods of the
battery. At the same time, the dependence of its estimation on the equivalent model
makes the model selection and construction very important. Therefore, the EEC
and SOC estimation of the battery still need to be further developed under various
conditions.
Research concerning BMS from a global perspective includes those which
display an entire BMS design adopting a distributed structure to reach better scal-
ability and portability [20,21]. Different approaches to designing a BMS depend on
the functionalities desired for the specific application, but most of them focus on
key functions such as SOC estimation [22] and the balancing process [23]. The
improvement is towards the design of intelligent BMS for electric and HEVs in
artificial intelligence applied for the battery state estimation. Therefore, SOC esti-
mation has drawn the attention of many researchers, and many different methods
have been proposed [24,25]. The OCV method, a full charge detector/dynamic load
observer, and robust extended Kalman filtering algorithm are combined. It is dif-
ficult to determine the specific approach when such methods are used. However,
based on the classification [25], two categories are used, namely the direct method
and the indirect method, as well as several sub-categories that summarize the trend
of SOC estimation and appropriate adjustments.
A Thevenin EEC is used in for every single cell in an array of more than
90 series-connected cells to identify the internal resistance of each cell. Two dif-
ferent branches are used in a Thevenin model for charge–discharge that is con-
nected in series n times to represent n cells in a series. According to [26,27], there
are three different EECs of lithium-ion batteries widely adopted because of their
excellent dynamic performance. It shows that the second-order EEC is the most
accurate and has the best dynamic performance and also the most complex. Also,
the discretization equations of each of the three mentioned models are presented
and used in combination with the EKF method to estimate the SOC. The Thevenin
and second-order EEC modeling methods [27] are used for the SOC estimation.
The difference between these models is the way of the SOC equation and calcu-
lation procedure. The parameters of the second-order EEC can be calculated with
different datasets depending on the scenario where the model is going to be used
[28], a comparison between continuous-time and discrete-time equations of the
second-order EEC is made and concludes that discrete-time identification methods
Introduction 7](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-34-320.jpg)
![are less robust due to undesired sensitivity issues in the transformation of discrete
domain parameters.
The application of the EKF algorithm in SOC estimation is also presented in
[29]. It is common to find a combination of the CC and OCV methods with the KF
method to estimate the SOC value [30]. The EKF method [31] is also introduced
for the iterative calculation in combination with CC and OCV parameters. Another
common improvement to the KF algorithm for SOC estimation is the unscented
Kalman filtering (UKF) algorithm, which is used in [31,32] to improve the esti-
mation accuracy. The UKF algorithm is implemented in [33] to estimate SOC using
an improved EEC with a resistance and a capacitor correction factor. This is done
first of all to measure the effect of different current rates and the SOC estimation on
the battery internal resistance and second to identify the impact of different current
rates and temperatures on the battery capacity [34]. The work presented in [35] uses
multiple modeling approaches combined with the EKF algorithm to estimate the
SOC value of the battery. The improved EKF algorithm is implemented in [36] to
be more robust to uncertainties in the system, measurement equations, and noise
covariance. An SOC estimation approach that uses an improvement in the mea-
surement noise treatment is proposed in [37], and an adaptive Kalman filtering
algorithm that can reduce the estimation error is established by correcting the
covariance matrix error in the depicted EKF method. To deal with the variation of
battery parameters due to temperature changes [38], an online approach is proposed
for SOC estimation and parameter updating using a dual square root UKF based on
unit spherical unscented transform.
To obtain a more accurate and reliable SOC, an improved Thevenin equivalent
model is proposed and its parameters are identified. Likewise, an AEKF algorithm
is designed for the SOC estimation of the lithium-ion battery model. The use of the
AEKF algorithm in this research is to estimate the SOC and eliminate or reduce
errors accurately and diligently. The use of the AEKF algorithm is an innovation in
this work coupled with the second-order RC and the other minor features that
would lead to the successful implementation.
This research is centered on accurate state estimation. The focus is specifically
the use of experimental data for parameterization, the modeling, and the imple-
mentation of the circuit for simulation analysis. The study also empathizes with the
implementation of the proposed improved adaptive algorithm for accurate state
estimation. The introductory aspect of the dissertation is made up of the back-
ground which outlines the importance of the research and reasons for the choice of
the direction. The problem statement puts forth the main issue to be tackled, and its
purpose is to reiterate the main goal and objectives of the research. The research
method gives an insight into the methods and techniques used by the researcher to
achieve the main goal and finally emphasizes some literature and key issues as far
as the research area is concerned.
Theoretical and mathematical analysis deals with the introduction of the lithium-
ion batteries specifically and related important areas connected to the study. Battery
modeling techniques and experiments are conducted for parameter identification,
which is introduced into the SOC estimation methods. The mathematical equations
8 Battery state estimation: methods and models](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-35-320.jpg)
![Chapter 2
Mechanism and influencing factors of
lithium-ion batteries
Abstract
This chapter introduces the operating mechanism, influencing factors, key indica-
tors, and some mainstream state estimation methods of lithium-ion batteries. First,
understanding the main composition and internal working principle of lithium-ion
batteries is the prerequisite and basis for other work. Second, internal resistance,
open-circuit voltage (OCV), terminal voltage, current thermal energy, capacity
variation, and temperature characteristics are the main features of the battery,
which can be further used to accurately characterize the battery state. The state of
charge, state of health, state of power, depth of discharge, and cyclic life are the key
indicators of state estimation, while the discharging test, Ampere-hour integral,
OCV, internal resistance, and Kalman filtering are basic state estimation strategies.
Based on the Kalman filter, there are many improved algorithms, such as the
unscented Kalman filter and adaptive Kalman filter, which have achieved good
application effects. Besides, other algorithms such as neural networks, support
vector machines, and some improvement strategies are also introduced. The
advancement of these algorithms has made an important contribution to improving
the whole-life-cycle state estimation effect of lithium-ion batteries.
Keywords: Operating mechanism; Battery characteristics; Influencing factor;
State of charge; State estimation; Kalman filtering; Cyclic life; Temperature;
Open-circuit voltage
2.1 Introduction
With the development of society, the shortage of primary energy and increasing atten-
tion to environmental pollution have promoted the rapid growth of lithium-ion batteries.
The invention of new energy and environmental protection are important research
topics, and the fact that lithium-ion batteries are relatively environmentally friendly in
use has explained the gradual replacement of other relatively polluting batteries.
To improve the function of electronic equipment, especially electric vehicles,
hybrid electric vehicles, smart grids, and UAVs that all use lithium-ion batteries [39].](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-38-320.jpg)
![The research on lithium-ion batteries and their related parameterization is of global
importance. There are several journals and periodicals with published articles and
projects on several important theories and ideas related to lithium-ion batteries and
their parameter estimation, and the number is increasing daily [40]. According to the
global lithium-ion battery market, it has been growing at a compound annual growth
rate of 10.6% since 2016 and is estimated to reach $56 billion by the year 2024,
which means that the demand for lithium-ion batteries is set to be more than double
by this year. This implies that the increasing demand would lead to more improve-
ments and has a lot of broad prospects for more research in the field, as shown in
Figure 2.1.
The battery model studies the relationship between the external characteristics
and the internal state of the battery by establishing a mathematical model. Discrete-
time and state-space forms are also used for the state of charge (SOC) estimation.
The present literature makes mention of electrical equivalent circuit (EEC) models
as being widely used as a foundation for model-based estimation and control
[41,42]. Generally, the equivalent circuit models are selected for the lithium-ion
battery modeling, including the Rint model, Thevenin model, RC model, and
PNGV model [43,44]. These models use RC loops of different orders to model the
polarization characteristics of batteries [45]. Among them, the Thevenin model is
widely used, but as all its components may change with the state of the battery, its
accuracy is not high [46].
For more accurate parameter identification [47,48], a new design using
charge–discharge is being developed, and SOC estimation methods are combined
[49,50] as a means of estimating SOC in the presence of unknown or time-varying
battery parameters. Research in this field either assumes the available precise
SOC–OCV [51,52] or a constant during discharge. The RC parameters are deter-
mined by analyzing the transient state of the battery voltage response [25] under
certain excitations, such as constant current or pulse current experiments. The
voltage source in an EEC typically represents the open-circuit voltage (OCV),
which depends on the SOC [53]. The relationship between SOC and OCV can be
2015 2016 2017 2018 2019
CAGR 10.6% (2016–2024)
2020 2021 2022 2023 2024
56
Figure 2.1 Global lithium-ion battery market size and forecast
12 Battery state estimation: methods and models](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-39-320.jpg)
![identified by charging or discharging the battery with a small current. Parameter
identification based on current–voltage data is addressed [48] by a method that
simplifies the problem of solving a set of high-order polynomial equations to
solving several linear equations and a single variable polynomial equation.
2.2 Operating mechanism
In the global high-tech growth, the lithium-ion battery industry is an excellent
direction. The benefits of lithium-ion batteries are high capacity, high conversion
rate, long lifespan, and no emission. Because of its good electrochemical durability,
high fuel density, long service life, and no maintenance, it is often used in electric
vehicles and various power storage systems. Lithium-ion batteries have been
widely used at present. It primarily encompasses five fields, including transporta-
tion, power storage, mobile communication, modern energy storage, and military
aerospace. It will substitute oil with electricity, minimize emissions of greenhouse
gas, and store electricity in the power grid in the unnecessary conditions of electric
vehicles.
2.2.1 Brief introduction
The battery converts chemical energy into electric energy, which is irreversible in the
primary battery and rechargeable in the secondary battery. Lithium-ion batteries can
be divided into three different categories, such as lithium metal, lithium-ion, and
lithium-ion polymer. Lithium metal batteries are primary batteries, while lithium-ion
and lithium-ion polymer batteries are both rechargeables. The lithium-ion battery
system is a complex system that integrates chemical, electrical, and mechanical
characteristics, so various characteristics must be considered in the design. In parti-
cular, the safety and life attenuation characteristics contained in the chemical char-
acteristics of the battery cell cannot be directly evaluated, nor can they be easily
predicted in a short time. Therefore, the battery technology, group technology, and
battery management system (BMS) technology should be implemented when
designing a battery system, as well as battery protection. Consequently, the reliability
and lifespan of the battery should be considered.
The lithium-ion battery is an advanced battery technology that uses lithium
ions as the key component of its electrochemical reaction. During the discharge
cycle, lithium atoms in the anode are ionized and separated from their electrons.
The lithium ions are small enough to pass through the micro-permeable separator
between the anode and the cathode. This separator recombines with its electrons
and is neutralized electrically. A class of organic compounds known as ether is
used as the electrolyte of lithium-ion batteries. The most common combination of
materials called electrodes is that of lithium cobalt oxide (cathode) and graphite
(anode), which is most commonly found in portable electronic devices such as
cellphones and laptops. Other cathode materials include lithium manganese oxide
and lithium iron phosphate. Lithium manganese oxide is commonly used in hybrid
and electric vehicles. Due to the small size of lithium ions, the batteries can have a
Mechanism and influencing factors of lithium-ion batteries 13](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-40-320.jpg)
![very high voltage and charge storage per unit mass and unit volume. Lithium is the
lightest of all metals, which has the greatest electrochemical potential and provides
the largest specific energy per weight. Rechargeable batteries with lithium metal on
the anode provide extraordinarily high energy densities.
Lithium-ion batteries are facing competition from numerous alternative battery
technologies, most of which are in the development stage [54]. One such alternative
called a saltwater drive battery is developed by the energy equation. They are com-
posed of saltwater, manganese oxide, and cotton, which are made by using abundant,
non-toxic materials, and modern low-cost manufacturing techniques [55]. Because of
this, they are the only cradle-to-cradle-certified battery in the world. The batteries
mentioned in it are all lithium-ion batteries. The cathode substrate of a lithium-ion
battery is lithium-alloy metal oxide [56]. Its negative electrode component is gra-
phite. The essence of charge and discharge is the electrochemical reaction in lithium-
ion batteries. The essential response and working process of lithium-ion batteries are
shown in Figure 2.2.
Figure 2.2 shows that the lithium-ion metal oxide positive electrode material is
emitted during the battery charging phase [57]. The lithium cobalt metal oxide
is transferred through the electrolyte and the separator, and the carbon substrate is
incorporated into the negative electrode-coated frame. The positive electrode enters a
state of prosperity rich in lithium-ion at this stage. On the contrary, the negative elec-
trode enters a lithium-ion lean state [58]. During the discharge process, lithium-ion can
Septum
Electrolyte
Electrolyte
Load
Power
Shell
e-
e-
+ -
Charge
Discharge
Positive
electrode
Negative
electrode
Li+
C
CoO2
Figure 2.2 Schematic structure of a lithium-ion battery
14 Battery state estimation: methods and models](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-41-320.jpg)
![be released from the negative electrode carbon material layer. The carbon content
coating would be deposited in the negative electrode crystal lattice since going through
the separator and the electrolyte [59]. The positive electrode is developing towards a
lithium-rich state. The negative electrode is developing towards a lithium-poor state.
The chemical quality of the electrode continues to recover. The complete equation for
the reaction is achieved by positive and negative lithium-ion electrodes. Each battery is
shown as follows:
P : LiCoO2 ⇄
cd
Li1xCoO2 þ xLi þ xe
N : xLi þ xe
þ 6C ⇄
cd
LixC6
T : LiCoO2 þ 6C ⇄
cd
Li1xCoO2 þ LixC6
8
:
(2.1)
In (2.1), it can be known that the internal resistance, polarization, aging, and
other factors are added in the process of the lithium-ion battery. The embedding
and unpacking going through the separator and traveling in the electrolyte can
impact the calculation of SOC [60]. In the process of model development, these
considerations and SOC estimation need to be solved [61].
2.2.2 Battery composition
Lithium-ion batteries generate electricity through the chemical reaction of lithium. The
battery consists of four key components. The anode, which determines the capacity and
voltage of the battery, is the source of the lithium ions. The cathode allows current to
flow through an external circuit, and when the battery is charged, lithium ions are
stored in the cathode. The electrolyte is composed of salts, solvents, and additives and
acts as the conduit for lithium ions between the cathode and the anode. The separator is
a physical barrier separating the cathode and anode. Common cathode materials
include lithium cobalt oxide (lithium cobaltate), lithium manganese oxide (spinel or
lithium manganate), lithium iron phosphate, lithium nickel manganese cobalt (NMC),
and lithium nickel cobalt aluminum oxide (NCA), as shown in Table 2.1.
Table 2.1 Most common lithium-ion battery compositions
Type Cathode Anode
Lithium cobalt oxide (LCO) LiCoO2 Graphite/hard carbon (LiC6)
Lithium manganese oxide (LMO) LiMn2O4 Graphite/hard carbon (LiC6)
Lithium iron phosphate (LFP) LiFePO4 Graphite/hard carbon (LiC6)
Lithium nickel manganese
cobalt oxide (NMC)
LiNiMnCoO2 Graphite/hard carbon (LiC6)
Lithium nickel cobalt aluminum
oxide (NCA)
LiNiCoAlO2 Graphite/hard carbon (LiC6)
Lithium titanate (LTO) LiMn2O4/
LiNiMnCoO2
Li4Ti5O12
Mechanism and influencing factors of lithium-ion batteries 15](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-42-320.jpg)
![2.2.4 Cycling lifespan
The electrode design can also be strengthened. The capacity of lithium-ion batteries can
be increased by testing the properties and scale of materials used for construction. The
chemical reaction on the electrode takes place inside the lithium-ion batteries. The
internal resistance of the electrodes can reduce the heat generated during use. The cell
potential and lifespan can also be increased [62]. The electrode size should be mini-
mized to improve performance. It has been discovered that there are limitations to the
reduction, such that as if the electrode thickness had fallen below. Then the batteries
could no longer meet the necessary energy requirements. For thinner content sizes, there
are additional trade-offs, including rising processing costs. The electrode scale can be
included as the primary design element in the overall design presented in Table 2.2.
In the design of electrodes, another development field has been formed. By
changing the materials made, the ion journeys may be optimized to maximize overall
conductivity and thermal capacity on the transportation distance or route. In certain
situations, the additional advantage of these improvements is to improve the mechan-
ical quality and the heat efficiency or overall power of the mechanical characteristics.
Gold, graphene, yttrium, and zinc are used in the components applied to the electrodes.
Several various influences affect this mechanism, and further research is ongoing.
2.3 Battery characteristics
After understanding the structural principle of the lithium-ion battery, the para-
meters, and common concepts of the battery that will appear in this work, it is
important to reiterate certain characteristics that affect the battery. Basic para-
meters of the battery include terminal voltage, electromotive force, capacity,
internal resistance, SOC, state of health (SOH), state of power, depth of discharge
(DOD), cyclic life, and self-discharge rate. The characteristics that affect the
parameterization of the lithium-ion battery include the following contents.
2.3.1 State of power
The physical significance of the capacity per unit time of the lithium-ion batteries
is the internal chemical reaction that the battery produces energy output. How
Table 2.2 Life cycle feature of the depth of discharge
Depth of discharge Discharge cycles
100 300–600
80 400–900
60 600–900
40 1,000–3,000
20 2,000–9,000
10 6,000–15,000
Mechanism and influencing factors of lithium-ion batteries 17](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-44-320.jpg)
![much work is done in a certain period is represented by the energy released by a
chemical reaction, which is measured in Watt (W) [63–65]. Depending on the
project, power can be subdivided into actual power and instantaneous power
[66–68]. The battery power equation can be expressed as shown in the following
equation:
P ¼ UI ¼ I E IR
ð Þ ¼ IE I2
R (2.5)
wherein E is the battery electromotive force, R is the total battery internal resis-
tance, and I is a unit of time for the average current.
2.3.2 Internal resistance
Lithium ions move from one pole to the other pole inside the lithium-ion battery
and the factors that hinder the movement of ions constitute the internal resistance of
the battery. Due to the internal resistance of the battery, the terminal voltage of the
battery is lower than OCV in the discharge state. In the charging state, the terminal
voltage of the battery is higher than the OCV value [69–71]. The essence of electric
current is the directional movement of electric charge.
During the movement, the electrons will be resisted by the material itself and
the magnetic field. Internal resistance refers to the resistance of 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 the lithium-ion
battery [72–74]. The internal resistance directly affects the working voltage,
working current, and capacity of the batteries. The internal resistance of the
lithium-ion battery is not constant. In the process of battery charging and dischar-
ging, temperature changes, charge–discharge ratio, charge–discharge time, elec-
trolyte concentration, and the quality of active substances will all cause changes in
the internal resistance of the battery.
2.3.3 Open-circuit voltage
The electrolyte, positive electrode, and negative electrode materials undergo aging
during the cyclic charging and discharging process of lithium-ion batteries and in
their external structure. The period from the battery start state to the battery life end
state is the lifespan of the battery. This is the data of the charge–discharge test by
the manufacturer under the specified standard test environment. Generally, dis-
charge is stopped when its capacity drops to 80% of the rated capacity. The number
of charge–discharge is the total number of continuous charge–discharge cycles of
the battery under this condition.
The OCV value can be measured by connecting a multimeter or voltmeter
directly to the positive and negative ends of the battery. It is the voltage used in the
simulation of the equivalent model in this book. The battery is not an ideal power
source, the electrolyte and electrode material have internal resistance, so the OCV
value is slightly less than the electromotive force value. Considering the internal
resistance, it can be treated as the electromotive force [75–78]. Under actual
operating conditions, the instantaneous closed-circuit voltage of the battery
18 Battery state estimation: methods and models](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-45-320.jpg)
![changes dynamically, which cannot avoid overcharge and over-discharge, causing
damage to the battery system.
2.3.4 Self-discharge current rate
The self-discharge current rate refers to the ratio of the discharge capacity of the
battery to its rated capacity in the no-load state and is used to indicate the con-
sumption rate of the battery capacity. Mainly affected by the battery manufacturing
process, materials, storage conditions, and other factors, it is generally expressed as
the percentage of battery capacity decrease per unit time, and the unit time is month
or year. Therefore, the self-discharge current rate is the leakage current rate, which
is usually used to test the leakage current in the test open-circuit environment. Due
to the low leakage current, the self-discharge current rate of lithium-ion batteries is
usually determined by their monthly values. It can also be regarded as the charge
retention rate of the battery.
2.3.5 Terminal voltage
The terminal voltage refers to the potential difference between the positive and
negative electrodes of the lithium-ion battery, which can be divided into OCV and
operating voltage according to the working conditions of the circuit where OCV is
the terminal voltage of the lithium-ion battery without load and other power sources
[79–83]. At the end of charging, the maximum allowable voltage of the battery is
called the charge cut-off voltage, and the lowest allowable voltage after the end of
battery discharge is called the discharge cut-off voltage. Beyond this limit, the
battery will suffer some irreversible damage, and the cut-off voltage is an important
safety indicator.
Working voltage refers to the potential difference between the positive and
negative electrodes when the battery is in working condition. Generally, due to the
internal resistance of the battery, the working voltage during discharge is lower
than the OCV, and the working voltage during charging is higher than the OCV
value [84–86]. Like the rated voltage classification, according to the working state,
the operating voltage of the lithium-ion battery is also divided into rated voltage,
theoretical voltage, OCV, and working voltage with load. The rated voltage shall be
directly calibrated by the battery manufacturer before delivery.
2.3.6 Current heat energy
There is no ideal power source in the world. No matter it is the current source or the
voltage source, there is some internal resistance. The lithium-ion battery as a power
source is no exception. With a large current through the battery for a long time, it flows
through the resistance to generate a large amount of heat energy, as shown below:
W ¼ PT ¼ I2
RT (2.6)
Resistance will lead to battery energy loss and a continuous increase in the bat-
tery temperature, while the temperature rise will also lead to a rise in the resistance
Mechanism and influencing factors of lithium-ion batteries 19](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-46-320.jpg)
![value, forming a vicious circle, the battery management system will take measures to
deal with heat dissipation after monitoring temperature abnormalities [87–89].
Besides, if the battery is charged for a long time, the high large charge–discharge
current of the battery will be lower than below certain levels, which will lead to
potential safety hazards. It is very important to monitor the current size of the battery
in the charge–discharge process to avoid causing damage to the battery.
2.3.7 Capacity variation
There are different levels of power stored in the battery. When the battery is fully
charged, its capacity is the amount of electricity it contains. Batteries of the same
type are usually evaluated according to the amount of current they can output over
time. Battery capacity refers to the total amount of electricity that can be released in
the full state of the battery. It is represented by the symbol Q, and the unit of
measurement is milli-Ampere-hour (mAh) or Ampere-hour (Ah) [90–92]. The
battery capacity types can be divided into rated capacity, theoretical capacity, and
actual capacity.
The rated capacity is measured by the battery manufacturer and calibrated
directly on the outer surface of the battery. Rated capacity is an important indicator
of battery storage capacity and an indicator of battery life. The rated capacity is
also known as calibration capacity, according to the relevant provisions of the
relevant departments of the state, under certain discharge conditions (temperature,
discharge rate, etc.) to ensure the minimum amount of energy that can be released
by the battery, rated capacity is the Ah capacity indicated by the licensed manu-
facturer, which is one of the important parameters of the battery [93–96]. The
theoretical capacity is the theoretical value of the total charge of the battery cal-
culated by Faraday’s law according to the electrochemical reaction inside the
lithium-ion battery when the internal lithium ions are fully involved in the reaction.
The actual capacity is the maximum capacity that a lithium-ion battery can
release when purchased in the store under certain discharge conditions. It is usually
calculated by the product of the discharge current and the discharge time. The
actual capacity is not necessarily the capacity of the new battery but can also be the
capacity of the used battery. Currently, the actual capacity is less than the actual
capacity of the new battery. The actual capacity is the amount of electricity that the
battery can release when operating in the real environment. The factors that affect
the actual capacity of lithium-ion batteries are very complicated, including the
temperature of the external environment, the materials used to make the battery,
and the service life.
2.3.8 Temperature change
The real-time temperature of the lithium-ion battery is a significant aspect that cannot
be ignored in practical applications. Battery activity is directly proportional to tem-
perature. As the temperature increases, the exchange rate of lithium ions between
electrolytes increases, and the overall activity of the battery also increases, which is
reflected in more battery energy output, increased the usable capacity of the battery,
20 Battery state estimation: methods and models](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-47-320.jpg)
![and improved battery efficiency [97–102]. Under the condition of long-term time and
low-temperature conditions, the activity of anode and cathode material of the battery
decreases. The actual capacity is less than the rated capacity. The charge–discharge
efficiency of the battery decreases and the use efficiency is reduced.
2.4 Critical indicators for battery state estimation
SOC estimation is an obligatory part of precise state estimation and battery man-
agement for lithium-ion batteries. However, some critical influencing factors are
battery related to estimating accuracy. This part of several influences is discussed
elaborately in the portion.
2.4.1 Description of major parameters
There are unknown safety risks in using lithium-ion batteries. It is essential to
consider the critical criteria of lithium-ion batteries to prevent explosions and fires
caused by management failure. To protect against unwanted circumstances, it is
obligatory to measure some critical action. To observe the critical situation, the
core parameters of lithium-ion batteries are necessary to be considered, as shown in
Table 2.3.
In Table 2.3, five key parameters are the basic parameters that deeply affect the
performance of lithium-ion batteries. The battery capacity is a significant factor for
storage capability, which is expressed in mAh. The nominal voltage is the rated
voltage determined by the manufacturer. The end of charge voltage can be obtained
when the battery charge is almost near to be ended. While the batteries are com-
pletely discharged, the end of discharge voltage associates with the lowest voltage
permissible. The self-discharge rate is the capacity consumption rate of a battery
Table 2.3 The major parameters of lithium-ion batteries
Parameters name Parameter definition
Battery capacity The amount of active material determines the battery capacity,
usually expressed in Ah. For example, 1,000 mAh can be
converted into a charge of 3,600 C at a current of 1 A for 1 h
Nominal voltage The electrical potential difference throughout the batteries
between positive and negative electrodes is the nominal battery
voltage
End of charge voltage The active material on the polar plate has reached saturation when
the rechargeable batteries are fully charged. Then, it begins to
charge and the battery voltage will not increase
Discharge end voltage When the batteries are fully discharged, the end of discharge
voltage corresponds to the lowest voltage permissible.
Discharge end voltage is compared to the rate of discharge
Self-discharge rate The amount of the overall power expended when it is not used for
any time. Generally, at an average temperature, the self-
discharge rate of lithium-ion batteries is 5%–8%
Mechanism and influencing factors of lithium-ion batteries 21](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-48-320.jpg)
![when it is unloaded. It is generally expressed by the ratio of its unit monthly dis-
charge capacity to the rated capacity, and the ratio is 5%–8%.
2.4.2 Temperature effects
As the environmental temperature affects the electrochemical reaction process of
lithium-ion batteries, the measurement accuracy of SOC will continue to be com-
promised. The higher the temperature and the higher the degree of the internal
electrochemical reaction and the energy of the battery will also rise [103]. These
factors will generate a large amount of high-temperature gas, which will accelerate
the aging process of the battery, and affect its life cycle and even damage the
battery. Therefore, only by seeking a suitable temperature can the output of the
battery be fully utilized, and the batteries can be charged and discharged
efficiently [104].
The battery capacity indicates how much energy it can hold, which decreases
with the decrease of temperature and increases with the increase of temperature.
Therefore, the battery fails on the cold winter morning, even though it worked the
previous day perfectly. The batteries work in a cold environment most of the year.
In that case, the decreased power of the lithium-ion battery must be taken into
consideration. The typical operating temperature of a battery is 25
C (77
F). At
approximately 30
C (22
F), the battery power will be reduced to 50%. At
freezing, the battery power is decreased by 20%. Capacity is improved at higher
temperatures at 122
F. The capacity of the battery will be around 12% higher.
2.4.3 Charge–discharge current rate
The charge–discharge current can be described by the charge rate and discharge
rate of lithium-ion batteries, which is represented by C. The charge–discharge rate
refers to the ratio of the working current to the rated capacity of the battery during
the charging and discharging process or working process. Within the same dura-
tion, the charge–discharge amount of the battery will vary with different rates of
charge. The battery is charging, discharge mode, the number of cycles, and DOD
will affect the aging process of the battery [105,106].
The C-rates is to regulate the charge–discharge rates of the battery. The rated
charge–discharge rate of the battery is usually 1 C, which means that the fully charged
battery rated at 1 Ah can supply 1 A for 1 hour. The same battery is discharged at
0.5 C should deliver 500 mA for 2 hours, and 2 C should deliver 2 A for 30 min.
Losses at accelerated discharge decrease the time of discharge, and these losses also
impact the time of loading. The C-rate of 1 C is often referred to as 1-h releases such
as the 2-h release of 0.5 C or C/2 and the 5-h release of 0.2 C or C/5. Any high-
performance low-stress batteries can be charged and discharged above 1 C.
2.4.4 Aging degree
The aging of the battery is due to the change in its internal structure, the rise of
temperature, and the batteries dropping. The constant charging and discharge will
lead to a change in the internal structure, influencing the aging degree [107].
22 Battery state estimation: methods and models](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-49-320.jpg)
![During the battery aging process, the SOC value will continue to decrease until the
battery fails to work normally. Generally speaking, the battery charging–discharge
mode, the number of cycles, and DOD can affect the batteries aging [108].
The everyday use of batteries in HEVs implies a paradigm of accurate battery
aging and battery life. The aging of batteries can be divided into two parts,
including the aging of calendars and cycle one. Calendar aging refers to the endless
amount of lost storage ability. In other words, battery capacity is responsible for the
deterioration. Cycle aging is related to the influence of cycles (charge or discharge)
for battery use times. It occurs when the battery is either loaded or unloaded. This is
directly due to the level, the temperature of the usage pattern, and current battery
specifications. As a result, its performance steadily deteriorates during battery life
because its electrochemical components are deteriorating and the performance of
EVs and fuel efficiency deteriorates.
2.4.5 Self-discharge rate
When the battery is not in use, some slow chemical reactions are also taking place
inside it. As the essence of battery charge–discharge is an electrochemical reaction
and the electrochemical reaction inside the battery increases with the rise of tem-
perature, when the temperature is high, the intensity of self-discharge inside the
battery is large, which will lead to the automatic drop of SOC.
2.5 Basic state estimation strategies
SOC is primarily used to describe the remaining capacity of the battery, which
corresponds to the ratio of the remaining capacity and the rated capacity under the
same circumstances following the batteries discharge at a given discharge rate. It is
typically expressed as the percentage (%). The range of its value is 0–1. When
SOC ¼ 0, this means that the battery is completely discharged. When SOC ¼ 1 is
used, this means that the battery is fully charged. Traditional battery SOC predic-
tion techniques include the discharge test method, conductance method, OCV
method, and Ah integration method. Their foundation is relatively simple, and the
implementation process is relatively simple, but the accuracy is often not high or
the adaptability is not strong. New SOC prediction methods for the battery include
extended Kalman filter (EKF), particle filtering (PF), fuzzy logic, and back-
propagation (BP). These methods combine mathematics and computer theory, and
the implementation process is complex, but they can achieve a good estimation
effect and strong adaptability.
2.5.1 Discharging test
The discharge test method is to discharge the target battery continuously at a
constant current until reaching the cut-off voltage of the battery. The time used in
the discharge process is multiplied by the discharge current, which is the capacity
of the battery. This method is generally used as the calibration method of SOC
Mechanism and influencing factors of lithium-ion batteries 23](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-50-320.jpg)
![spring still supplies it; about ten feet above was the movable stage,
sprinkled with sand for the combats, and hence called the arena. A
few feet above the arena was the podium, or seat of the emperor,
vestal virgins, c., protected from the arena by iron bars. Behind the
podium was a double portico, which ran round the whole building.
Fragments of the marble chimeras, with long wings, that
ornamented the seats of the podium have been found.
The three successive tiers were called cavea. Above these was a tier
for the people; above this one for the gods; thus making six in all.
The amphitheatre seated eighty-seven thousand people, and there
was standing room for thirteen thousand more.
The walls standing upon the area, composed of tufa, travertine, and
brick, old material re-used, were built at a period long after the
building was dedicated, when the naval fights being abandoned
there was no longer any occasion for a movable stage or arena as
before. They contained the machinery for the stage above, and for
the lifts or pegmata to send men or beasts from the area to the
arena. Probably these are the walls thus alluded to by Dion Cassius:
He [Commodus] divided the theatre into four parts by two
partitions that cut through diametrically, and by right angles, to the
end that from the galleries that were round about he might with
greater ease single out the beasts he aimed at.](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-60-320.jpg)
![Also, in his life of Septimius Severus, he says: There was a kind of
cloister made in the amphitheatre, in the form of a ship, to receive
them [the wild beasts]. On a sudden there issued out bears, lions,
ostriches, wild asses, and foreign bulls.
The walls before us are of very bad construction, evidently repairs of
a late date: they are the work of either Lampridius, prefect of Rome
under Valentinian III., 425–455, who repaired the steps and
renewed the arena; or of Basilius, who restored the podium and
arena after their destruction by an earthquake in 486—this we learn
from two inscriptions standing at the entrance. Half way, on each
side, two large passages have been discovered choked up with mud:
they were the aqueducts to bring the water for the naumachiæ from
the reservoirs upon the Esquiline and Cælian Hills respectively; from
the small openings in the blind arches the water also poured out
over the top of the dens, thus forming cascades all round. At the
end opposite the present entrance a long passage has been opened,
above the level of the area floor; below this passage is the great
drain, with the remains of the iron grating[6] to prevent large objects
going down: this and the passage were closed by flood-gates on
naval representations, which can be clearly seen in the construction.
On the right and left of this passage, connected with it, but at a
lower level, two dens have been cleared out, 27 yards long by 5
wide, containing six holes in the floor, in the centre of square blocks
of stone, and these holes are faced with bronze, evidently the
sockets into which metal posts were fixed to which the beasts were
chained. On the fragments depicting scenes from the arena, the
animals are shown with a long piece of rope or chain dangling from
their necks, which seems to bear out our idea that they were
attached to posts fixed in these sockets, and that as they were
wanted the chain or rope was cut, and they were free to rush upon
the arena.
The corbels round the front of the line of arches under the podium
are in pairs, and between them the masts were inserted to support
the awning on the inside, as the holes and corbels supported the](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-62-320.jpg)
![semicircle. ('A tabernacle for a shadow in the day-time from the
heat, and for a place of refuge, and for a covert from storm and
from rain,' Isa. iv. 6.) In the centre of the area formed by this
immense colonnade, an Egyptian obelisk, of one solid piece of
granite, ascends to the height of 130 feet; two perpetual fountains,
one on each side, play in the air, and fall in sheets round the basins
of porphyry that receive them. Raised on three successive flights of
marble steps, extending 379 feet in length, and towering to the
elevation of 148, you see the majestic front of the basilica itself. This
front is supported by a single row of Corinthian pillars and pilasters,
and adorned with an attic, a balustrade, and thirteen colossal
statues. Far behind and above it rises the matchless dome. Two
smaller cupolas, one on each side, add not a little to the majesty of
the principal dome.
Five lofty portals open into the vestibule; it is 468 feet in length, 66
in height, and 50 in breadth, paved with variegated marble, covered
with a gilt vault, adorned with pillars, pilasters, mosaic, and bas-
reliefs, and terminated at both ends by equestrian statues, one of
Constantine, the other of Charlemagne.
THE OBELISK
is the only one near its original site, the Spina of Nero's Circus,
which was near the Sacristy, on the left of S. Peter's. An inscription
in the pavement marks the place. Pliny (xxxvi. 14), says: The third
obelisk at Rome is in the Vatican Circus, which was constructed by
the emperors Caius [Caligula] and Nero; this being the only one of
them all that has been broken in the carriage. Nuncorcus, the son of
Sesoses, made it [the original, this is probably a copy], and there
remains [in Egypt] another by him, 100 cubits in height, which, by
order of an oracle, he consecrated to the sun, after having lost his
sight and recovered it. Herodotus says: It was dedicated by Phero,
son of Sesostris, in gratitude for his recovery from blindness. It has
no hieroglyphics, so if this was the original how could they know
who erected it? but it bears this inscription of Caligula—](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-70-320.jpg)
![DIVO. CAES. DIVI. JULII. F. AUGUSTO.—TI. CAESARI.
DIVI. AUG. F.—AUGUSTO. SACRUM.
[To the divine Augustus, son of the divine Julius, and to the divine
Tiberius, son of the divine Augustus.]
S. PETER'S AND THE VATICAN.
View larger image.
The Nuncorcus of Pliny is supposed to stand for Menophtheus, the
king Meneph-Pthah.
Pliny (xvi. 76) gives the following particulars of how it was brought
over:—
A fir tree of prodigious size was used in the vessel which, by the
command of Caligula, brought the obelisk from Egypt, which stands
in the Vatican Circus, and four blocks of the same sort of stone to
support it. Nothing certainly ever appeared on the sea more
astonishing than this vessel; 120,000 bushels of lentils served for its
ballast; the length of it nearly equalled all the left side of the port of
Ostia—for it was sent there by the Emperor Claudius. The thickness](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-71-320.jpg)
![The first inscription relates the gift of olive-yards to provide oil for
the lamps given by Gregory II.
The second, the Bull of Boniface VIII., of the indulgence granted at
jubilee.
The third, Panegyric of Charlemagne on Pope Adrian I.
INTERIOR.
Five portals give access to the edifice, which faces east.
Enter, its grandeur overwhelms thee not.—Byron.
The most extensive hall ever constructed by human art expands in
magnificent perspective before you. Advancing up the nave, you
admire the beauty of the variegated marble under your feet, and the
splendour of the golden vault overhead, the lofty Corinthian pilasters
with their bold entablature, the intermediate niches with their
statues, the arcades with the graceful figures that recline on the
curves of their arches. But how great your astonishment when you
reach the foot of the altar, and, standing in the centre of the church,
contemplate the four superb vistas that open around you; and then
raise your eyes to the dome, at the prodigious elevation of 440 feet,
extended like a firmament over your head, and presenting, in
glowing mosaic, the companies of the just and the choirs of celestial
spirits....
Around the dome rise four other cupolas, small, indeed, when
compared with its stupendous magnitude, but of great boldness
when considered separately; six more, three on either side, cover
the different divisions of the aisles; and six more of greater
dimensions canopy as many chapels. All these inferior cupolas are,
like the grand dome itself, lined with mosaics. Many, indeed, of the
masterpieces of painting which formerly graced this edifice have
been removed [to the Church of S. Maria degli Angeli, see page 265
], and replaced by mosaics, which retain all the tints and beauties of](https://image.slidesharecdn.com/18616107-250604032522-84170e5c/85/Battery-State-Estimation-Methods-And-Models-Energy-Engineering-Shunli-Wang-Editor-73-320.jpg)
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- 6. IET ENERGY ENGINEERING SERIES 212 Battery State Estimation
- 7. Other volumes in this series: Volume 1 Power Circuit Breaker Theory and Design C.H. Flurscheim (Editor) Volume 4 Industrial Microwave Heating A.C. Metaxas and R.J. Meredith Volume 7 Insulators for High Voltages J.S.T. Looms Volume 8 Variable Frequency AC Motor Drive Systems D. Finney Volume 10 SF6 Switchgear H.M. Ryan and G.R. Jones Volume 11 Conduction and Induction Heating E.J. Davies Volume 13 Statistical Techniques for High Voltage Engineering W. Hauschild and W. Mosch Volume 14 Uninterruptible Power Supplies J. Platts and J.D. St Aubyn (Editors) Volume 15 Digital Protection for Power Systems A.T. Johns and S.K. Salman Volume 16 Electricity Economics and Planning T.W. Berrie Volume 18 Vacuum Switchgear A. Greenwood Volume 19 Electrical Safety: A guide to causes and prevention of hazards J. Maxwell Adams Volume 21 Electricity Distribution Network Design, 2nd Edition E. Lakervi and E. J. Holmes Volume 22 Artificial Intelligence Techniques in Power Systems K. Warwick, A.O. Ekwue and R. Aggarwal (Editors) Volume 24 Power System Commissioning and Maintenance Practice K. Harker Volume 25 Engineers’ Handbook of Industrial Microwave Heating R.J. Meredith Volume 26 Small Electric Motors H. Moczala et al. Volume 27 AC–DC Power System Analysis J. Arrillaga and B.C. Smith Volume 29 High Voltage Direct Current Transmission, 2nd Edition J. Arrillaga Volume 30 Flexible AC Transmission Systems (FACTS) Y.-H. Song (Editor) Volume 31 Embedded Generation N. Jenkins et al. Volume 32 High Voltage Engineering and Testing, 2nd Edition H.M. Ryan (Editor) Volume 33 Overvoltage Protection of Low-Voltage Systems, Revised Edition P. Hasse Volume 36 Voltage Quality in Electrical Power Systems J. Schlabbach et al. Volume 37 Electrical Steels for Rotating Machines P. Beckley Volume 38 The Electric Car: Development and future of battery, hybrid and fuel-cell cars M. Westbrook Volume 39 Power Systems Electromagnetic Transients Simulation J. Arrillaga and N. Watson Volume 40 Advances in High Voltage Engineering M. Haddad and D. Warne Volume 41 Electrical Operation of Electrostatic Precipitators K. Parker Volume 43 Thermal Power Plant Simulation and Control D. Flynn Volume 44 Economic Evaluation of Projects in the Electricity Supply Industry H. Khatib Volume 45 Propulsion Systems for Hybrid Vehicles J. Miller Volume 46 Distribution Switchgear S. Stewart Volume 47 Protection of Electricity Distribution Networks, 2nd Edition J. Gers and E. Holmes Volume 48 Wood Pole Overhead Lines B. Wareing Volume 49 Electric Fuses, 3rd Edition A. Wright and G. Newbery Volume 50 Wind Power Integration: Connection and system operational aspects B. Fox et al. Volume 51 Short Circuit Currents J. Schlabbach Volume 52 Nuclear Power J. Wood Volume 53 Condition Assessment of High Voltage Insulation in Power System Equipment R.E. James and Q. Su Volume 55 Local Energy: Distributed generation of heat and power J. Wood Volume 56 Condition Monitoring of Rotating Electrical Machines P. Tavner, L. Ran, J. Penman and H. Sedding Volume 57 The Control Techniques Drives and Controls Handbook, 2nd Edition B. Drury Volume 58 Lightning Protection V. Cooray (Editor) Volume 59 Ultracapacitor Applications J.M. Miller Volume 62 Lightning Electromagnetics V. Cooray
- 8. Volume 63 Energy Storage for Power Systems, 2nd Edition A. Ter-Gazarian Volume 65 Protection of Electricity Distribution Networks, 3rd Edition J. Gers Volume 66 High Voltage Engineering Testing, 3rd Edition H. Ryan (Editor) Volume 67 Multicore Simulation of Power System Transients F.M. Uriate Volume 68 Distribution System Analysis and Automation J. Gers Volume 69 The Lightening Flash, 2nd Edition V. Cooray (Editor) Volume 70 Economic Evaluation of Projects in the Electricity Supply Industry, 3rd Edition H. Khatib Volume 72 Control Circuits in Power Electronics: Practical issues in design and implementation M. Castilla (Editor) Volume 73 Wide Area Monitoring, Protection and Control Systems: The enabler for smarter grids A. Vaccaro and A. Zobaa (Editors) Volume 74 Power Electronic Converters and Systems: Frontiers and applications A.M. Trzynadlowski (Editor) Volume 75 Power Distribution Automation B. Das (Editor) Volume 76 Power System Stability: Modelling, analysis and control A.A. Sallam and B. Om P. Malik Volume 78 Numerical Analysis of Power System Transients and Dynamics A. Ametani (Editor) Volume 79 Vehicle-to-Grid: Linking electric vehicles to the smart grid J. Lu and J. Hossain (Editors) Volume 81 Cyber-Physical-Social Systems and Constructs in Electric Power Engineering S. Suryanarayanan, R. Roche and T.M. Hansen (Editors) Volume 82 Periodic Control of Power Electronic Converters F. Blaabjerg, K. Zhou, D. Wang and Y. Yang Volume 86 Advances in Power System Modelling, Control and Stability Analysis F. Milano (Editor) Volume 87 Cogeneration: Technologies, optimisation and implementation C.A. Frangopoulos (Editor) Volume 88 Smarter Energy: From smart metering to the smart grid H. Sun, N. Hatziargyriou, H.V. Poor, L. Carpanini and M.A. Sánchez Fornié (Editors) Volume 89 Hydrogen Production, Separation and Purification for Energy A. Basile, F. Dalena, J. Tong and T.N. Veziroğlu (Editors) Volume 90 Clean Energy Microgrids S. Obara and J. Morel (Editors) Volume 91 Fuzzy Logic Control in Energy Systems with Design Applications in MATLAB‡ /Simulink‡ İ.H. Altaş Volume 92 Power Quality in Future Electrical Power Systems A.F. Zobaa and S.H.E.A. Aleem (Editors) Volume 93 Cogeneration and District Energy Systems: Modelling, analysis and optimization M.A. Rosen and S. Koohi-Fayegh Volume 94 Introduction to the Smart Grid: Concepts, technologies and evolution S. K. Salman Volume 95 Communication, Control and Security Challenges for the Smart Grid S.M. Muyeen and S. Rahman (Editors) Volume 96 Industrial Power Systems with Distributed and Embedded Generation R Belu Volume 97 Synchronized Phasor Measurements for Smart Grids M.J.B. Reddy and D.K. Mohanta (Editors) Volume 98 Large Scale Grid Integration of Renewable Energy Sources A. Moreno- Munoz (Editor) Volume 100 Modeling and Dynamic Behaviour of Hydropower Plants N. Kishor and J. Fraile-Ardanuy (Editors) Volume 101 Methane and Hydrogen for Energy Storage R. Carriveau and D.S-K. Ting Volume 104 Power Transformer Condition Monitoring and Diagnosis A. Abu-Siada (Editor) Volume 106 Surface Passivation of Industrial Crystalline Silicon Solar Cells J. John (Editor) Volume 107 Bifacial Photovoltaics: Technology, applications and economics J. Libal and R. Kopecek (Editors)
- 9. Volume 108 Fault Diagnosis of Induction Motors J. Faiz, V. Ghorbanian and G. Joksimović Volume 110 High Voltage Power Network Construction K. Harker Volume 111 Energy Storage at Different Voltage Levels: Technology, integration, and market aspects A.F. Zobaa, P.F. Ribeiro, S.H.A. Aleem and S.N. Afifi (Editors) Volume 112 Wireless Power Transfer: Theory, technology and application N. Shinohara Volume 114 Lightning-Induced Effects in Electrical and Telecommunication Systems Y. Baba and V.A. Rakov Volume 115 DC Distribution Systems and Microgrids T. Dragičević, F. Blaabjerg and P. Wheeler Volume 116 Modelling and Simulation of HVDC Transmission M. Han (Editor) Volume 117 Structural Control and Fault Detection of Wind Turbine Systems H. R. Karimi Volume 119 Thermal Power Plant Control and Instrumentation: The control of boilers and HRSGs, 2nd Edition D. Lindsley, J. Grist and D. Parker Volume 120 Fault Diagnosis for Robust Inverter Power Drives A. Ginart (Editor) Volume 121 Monitoring and Control Using Synchrophasors in Power Systems with Renewables I. Kamwa and C. Lu (Editors) Volume 123 Power Systems Electromagnetic Transients Simulation, 2nd Edition N. Watson and J. Arrillaga Volume 124 Power Market Transformation B. Murray Volume 125 Wind Energy Modeling and Simulation, Volume 1: Atmosphere and plant P. Veers (Editor) Volume 126 Diagnosis and Fault Tolerance of Electrical Machines, Power Electronics and Drives A.J.M. Cardoso Volume 128 Characterization of Wide Bandgap Power Semiconductor Devices F. Wang, Z. Zhang and E.A. Jones Volume 129 Renewable Energy from the Oceans: From wave, tidal and gradient systems to offshore wind and solar D. Coiro and T. Sant (Editors) Volume 130 Wind and Solar Based Energy Systems for Communities R. Carriveau and D. S-K. Ting (Editors) Volume 131 Metaheuristic Optimization in Power Engineering J. Radosavljević Volume 132 Power Line Communication Systems for Smart Grids I.R.S Casella and A. Anpalagan Volume 134 Hydrogen Passivation and Laser Doping for Silicon Solar Cells Brett Hallam and Catherine Chan (Editors) Volume 139 Variability, Scalability and Stability of Microgrids S.M. Muyeen, S.M. Islam and F. Blaabjerg (Editors) Volume 145 Condition Monitoring of Rotating Electrical Machines P. Tavner, L. Ran, C. Crabtree Volume 146 Energy Storage for Power Systems, 3rd Edition A.G. Ter-Gazarian Volume 147 Distribution Systems Analysis and Automation, 2nd Edition J. Gers Volume 151 SiC Power Module Design: Performance, robustness and reliability Alberto Castellazzi and Andrea Irace (Editors) Volume 152 Power Electronic Devices: Applications, failure mechanisms and reliability F Iannuzzo (Editor) Volume 153 Signal Processing for Fault Detection and Diagnosis in Electric Machines and Systems Mohamed Benbouzid (Editor) Volume 155 Energy Generation and Efficiency Technologies for Green Residential Buildings D. Ting and R. Carriveau (Editors) Volume 156 Lithium-Ion Batteries Enabled by Silicon Anodes Chunmei Ban and Kang Xu (Editors) Volume 157 Electrical Steels, 2 Volumes A. Moses, K. Jenkins, Philip Anderson and H. Stanbury Volume 158 Advanced Dielectric Materials for Electrostatic Capacitors Q Li (Editor) Volume 159 Transforming the Grid Towards Fully Renewable Energy O. Probst, S. Castellanos and R. Palacios (Editors) Volume 160 Microgrids for Rural Areas: Research and case studies R.K. Chauhan, K. Chauhan and S.N. Singh (Editors)
- 10. Volume 166 Advanced Characterization of Thin Film Solar Cells N. Haegel and M. Al- Jassim (Editors) Volume 167 Power Grids with Renewable Energy: Storage, integration and digitalization A.A. Sallam and B. Om P. Malik Volume 169 Small Wind and Hydrokinetic Turbines Philip Clausen, Jonathan Whale and David Wood (Editors) Volume 170 Reliability of Power Electronics Converters for Solar Photovoltaic Applications Frede Blaabjerg, Ahteshamul Haque, Huai Wang, Zainul Abdin Jaffery and Yongheng Yang (Editors) Volume 171 Utility-Scale Wind Turbines and Wind Farms A. Vasel-Be-Hagh and D.S-K. Ting Volume 172 Lighting Interaction with Power Systems, 2 Volumes A. Piantini (Editor) Volume 174 Silicon Solar Cell Metallization and Module Technology T. Dullweber (Editor) Volume 193 Overhead Electric Power Lines: Theory and practice S. Chattopadhyay and A. Das Volume 194 Offshore Wind Power Reliability, Availability and Maintenance, 2nd edition P. Tavner Volume 198 Battery Management Systems and Inductive Balancing A. Van den Bossche and A. Farzan Moghaddam Volume 199 Model Predictive Control for Microgrids: From power electronic converters to energy management J. Hu, J.M. Guerrero and S. Islam Volume 204 Electromagnetic Transients in Large HV Cable Networks: Modeling and calculations Ametani, Xue, Ohno and Khalilnezhad Volume 905 Power System Protection, 4 Volumes
- 12. Battery State Estimation Methods and models Edited by Shunli Wang The Institution of Engineering and Technology
- 13. Published by The Institution of Engineering and Technology, London, United Kingdom The Institution of Engineering and Technology is registered as a Charity in England & Wales (no. 211014) and Scotland (no. SC038698). † The Institution of Engineering and Technology 2021 First published 2021 This publication is copyright under the Berne Convention and the Universal Copyright Convention. All rights reserved. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publisher at the undermentioned address: The Institution of Engineering and Technology Michael Faraday House Six Hills Way, Stevenage Herts, SG1 2AY, United Kingdom www.theiet.org While the authors and publisher believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them. Neither the author nor publisher assumes any liability to anyone for any loss or damage caused by any error or omission in the work, whether such an error or omission is the result of negligence or any other cause. Any and all such liability is disclaimed. The moral rights of the author to be identified as author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. British Library Cataloguing in Publication Data A catalogue record for this product is available from the British Library ISBN 978-1-83953-529-1 (hardback) ISBN 978-1-83953-530-7 (PDF) Typeset in India by MPS Limited Printed in the UK by CPI Group (UK) Ltd, Croydon
- 14. Contents About the editor xv Foreword xvii Preface xix List of contributors xxi 1 Introduction 1 1.1 State of the art 1 1.2 Application requirements 4 1.3 Research methodology 5 1.4 Research status and direction 6 1.5 Chapter summary 9 Acknowledgment 9 2 Mechanism and influencing factors of lithium-ion batteries 11 2.1 Introduction 11 2.2 Operating mechanism 13 2.2.1 Brief introduction 13 2.2.2 Battery composition 15 2.2.3 Working principle 16 2.2.4 Cycling lifespan 17 2.3 Battery characteristics 17 2.3.1 State of power 17 2.3.2 Internal resistance 18 2.3.3 Open-circuit voltage 18 2.3.4 Self-discharge current rate 19 2.3.5 Terminal voltage 19 2.3.6 Current heat energy 19 2.3.7 Capacity variation 20 2.3.8 Temperature change 20 2.4 Critical indicators for battery state estimation 21 2.4.1 Description of major parameters 21 2.4.2 Temperature effects 22 2.4.3 Charge–discharge current rate 22 2.4.4 Aging degree 22 2.4.5 Self-discharge rate 23
- 15. 2.5 Basic state estimation strategies 23 2.5.1 Discharging test 23 2.5.2 Ah integral method 24 2.5.3 Open-circuit voltage method 24 2.5.4 Internal resistance method 25 2.6 Kalman filtering and its extension 25 2.6.1 Kalman filtering 25 2.6.2 Extended Kalman filtering 26 2.6.3 Unscented Kalman filtering 26 2.6.4 Dual Kalman filtering 27 2.6.5 Adaptive extended Kalman filtering 27 2.6.6 Square root-unscented Kalman filtering 27 2.6.7 Cubature Kalman filtering 28 2.7 Intelligent state estimation methods 30 2.7.1 State observer 30 2.7.2 Monte Carlo treatment 30 2.7.3 Bayesian estimation 30 2.7.4 Support vector machine 32 2.7.5 Particle filtering 33 2.7.6 Neural network 35 2.7.7 Deep learning 39 2.8 Algorithm improvement strategies 41 2.8.1 Bayesian importance sampling 41 2.8.2 Coordinate transformation 42 2.8.3 Binary iteration treatment 43 2.9 Chapter summary 44 Acknowledgment 44 3 Equivalent modeling, improvement, and state-space description 45 3.1 Introduction 45 3.1.1 Application background 46 3.1.2 Modeling principle 46 3.1.3 Modeling types and concepts 47 3.1.4 Model building principle 48 3.1.5 Battery modeling methods 49 3.1.6 Modeling characteristic comparison 50 3.2 Electrochemical modeling 51 3.2.1 Electrochemical modeling 51 3.2.2 Mathematical Shepherd modeling 53 3.2.3 Electrochemical thermal modeling 54 3.3 Electrical equivalent modeling 54 3.3.1 Equivalent circuit modeling 55 3.3.2 Internal resistance modeling 55 3.3.3 Resistance–capacitance modeling 57 3.3.4 Electrical modeling effect comparison 57 x Battery state estimation: methods and models
- 16. 3.3.5 Surface effect modeling 59 3.4 Improved Thevenin equivalent modeling 60 3.4.1 Thevenin electrical modeling 60 3.4.2 Second-order circuit modeling 61 3.4.3 Dynamic high-order equivalent modeling 62 3.4.4 Double internal resistance modeling 64 3.4.5 Improved surface effect modeling 64 3.4.6 State-space description 65 3.4.7 Simulation realization 65 3.5 Improved equivalent circuit modeling 67 3.5.1 Runtime electrical modeling 68 3.5.2 Fractional-order electrical model 69 3.5.3 Improved Thevenin model 70 3.5.4 State-space description 71 3.5.5 Simulation realization 72 3.6 High-order model establishment 73 3.6.1 High-order electrical modeling 73 3.6.2 Ohmic resistance identification 75 3.6.3 State-space expression 76 3.7 Model parameter description 78 3.7.1 Ampere-hour counting 78 3.7.2 Exponential curve fitting 79 3.7.3 Recursive least square 80 3.7.4 Full model parameter identification 82 3.8 Chapter summary 84 Acknowledgment 84 4 Extended Kalman filtering and its extension 85 4.1 Kalman filtering extension strategies 85 4.1.1 Kalman filtering algorithm 86 4.1.2 Extended Kalman filtering 88 4.1.3 Fractional-order adaptive correction 95 4.2 Equivalent circuit modeling 98 4.2.1 Second-order Thevenin modeling 98 4.2.2 Identification procedure design 100 4.2.3 Identification effect verification 102 4.2.4 Corroboration of model parameters 103 4.3 Model parameter identification 104 4.3.1 Recursive least-square calculation 104 4.3.2 Forgetting factor—RLS algorithm 107 4.3.3 Adaptive PSO 107 4.3.4 Parameter extraction results 110 4.4 Fractional experimental test 112 4.4.1 Real-time platform implementation 112 4.4.2 Test step procedure 114 Contents xi
- 17. 4.4.3 HPPC test 115 4.4.4 Capacity tracking experiments 116 4.5 Extended Kalman filtering-based state of charge estimation 117 4.5.1 State of charge determination 118 4.5.2 Application requirements 118 4.5.3 Time-varying correction 119 4.5.4 Simulation interfacing process 120 4.5.5 Pulse-current estimation effect verification 122 4.5.6 Estimation for BBDST conditions 124 4.6 EKF-based state of health estimation 124 4.6.1 Estimation model establishment 124 4.6.2 Model parameter verification 126 4.6.3 State of health estimation for the HPPC test 129 4.6.4 State of health variation for BBDST 131 4.6.5 State of health estimation of dynamic stress test 133 4.6.6 State of health estimation with capacity fade 134 4.7 Chapter summary 137 Acknowledgment 137 5 Adaptive extended Kalman filtering for multiple battery state estimation 139 5.1 Introduction 140 5.2 Iterative calculation strategies 140 5.2.1 Iterative predicting-updating calculation 140 5.2.2 Nonlinear state-space extension 142 5.2.3 Estimation model construction 145 5.2.4 Adaptive extended Kalman filtering 145 5.2.5 Improved adaptive extended Kalman filtering 150 5.3 Parameter identification 152 5.3.1 Test platform construction 152 5.3.2 Parameter identification procedure 153 5.3.3 Parameter varying law extraction 154 5.3.4 Capacity test results 157 5.3.5 HPPC test results 160 5.3.6 Open-circuit voltage tests 161 5.3.7 Combined capacity and HPPC tests 164 5.4 State of charge estimation 165 5.4.1 Simulated estimation results 165 5.4.2 Voltage traction effect 167 5.4.3 Pulse-current estimation verification 168 5.4.4 BBDST estimation results 169 5.5 State of power prediction 170 5.5.1 State of power characteristics 170 5.5.2 SOC-based SOP estimation 171 5.5.3 EEC-based SOP estimation 171 xii Battery state estimation: methods and models
- 18. 5.5.4 Multi-constraint SOP estimation 172 5.5.5 BBDST estimation results 172 5.6 Chapter summary 173 Acknowledgment 173 6 Dual extended Kalman filtering prediction for complex working conditions 175 6.1 Introduction 175 6.2 Aging modeling methods 180 6.2.1 Aging mechanisms 181 6.2.2 Electrochemical aging models 181 6.2.3 Analytical aging models 181 6.2.4 Equivalent circuit aging models 181 6.2.5 Statistical aging models 182 6.2.6 Battery aging model 182 6.2.7 Internal resistance growth 183 6.2.8 Mathematical aging models 183 6.3 Iterative calculation algorithm 185 6.3.1 Cyclic aging expression 185 6.3.2 Rain-flow counting 185 6.3.3 Cyclic charge–discharge variation 187 6.3.4 State of safety analysis 188 6.3.5 Definitions of key points 188 6.3.6 Electrical equivalent circuit modeling 190 6.3.7 Model parameter identification 191 6.3.8 Dual extended Kalman filtering 192 6.4 Parameter test and identification 194 6.4.1 Experimental platform setup 194 6.4.2 Whole-life-cycle HPPC test 194 6.4.3 Capacity characterization test 195 6.4.4 Open-circuit voltage test 196 6.4.5 Recursive least-square method 196 6.5 Complex condition experiment 199 6.5.1 Test platform construction 199 6.5.2 Cyclic aging procedure design 199 6.5.3 Battery aging modeling results 201 6.5.4 Parameter identification results 201 6.5.5 BBDST verification 203 6.5.6 Estimation result verification 204 6.6 Chapter summary 205 Acknowledgment 206 7 Unscented particle filtering of safety estimation considering capacity fading effect 207 7.1 Introduction 207 Contents xiii
- 19. 7.2 Capacity fade modeling methods 208 7.2.1 Capacity fade mechanism 208 7.2.2 Capacity fade modeling methods 209 7.2.3 Capacity fade modeling 210 7.2.4 Mathematical knee point expression 210 7.2.5 Arrhenius model quantification 211 7.2.6 Knee-Arrhenius expression 213 7.3 Estimation modeling methods 214 7.3.1 Equivalent circuit modeling 214 7.3.2 Improved PNGV circuit modeling 215 7.3.3 High-order modeling realization 217 7.3.4 Internal resistance estimation 219 7.3.5 Parameter identification 221 7.3.6 Particle filtering algorithm 222 7.3.7 Unscented Kalman filtering 225 7.3.8 Improved unscented particle filtering 227 7.4 Experimental result analysis 229 7.4.1 Test platform construction 229 7.4.2 Open-circuit voltage characterization 230 7.4.3 Capacity fade modeling effect 232 7.4.4 State of safety evaluation 234 7.4.5 Parameter identification tests 236 7.4.6 State estimation effect verification 238 7.5 Chapter summary 239 Acknowledgment 239 References 241 Index 267 xiv Battery state estimation: methods and models
- 20. About the editor Shunli Wang is a professor at Southwest University of Science and Technology, China, where he heads the New Energy Measurement and Control Research Team. His research focuses on modeling and state estimation research for batteries and multiple generation battery systems. He holds 30 patents, has published more than 100 papers, and won several awards.
- 22. Foreword This book mainly focuses on the accurate state estimation of the lithium-ion battery through experiments by introducing the application of adaptive algorithms. The state-estimation-based management systems are the brains of the battery packs, which are responsible for managing outputs, fast charging, and safe discharging, providing timely and reliable notifications of the battery status. Several modeling methods are investigated for the accurate state estimation of lithium-ion batteries. As it is difficult to estimate the battery state accurately, numerous algorithms and techniques are employed. These battery modeling strategies are very important for the battery system management, which determines the parameters to be identified and how it can be done. Consequently, various battery models are analyzed to simplify the circuitry used in the battery management system, aiming to estimate the state of charge, state of power, state of safety, and state of health with efficient performance, reliable and timely information. Equivalent circuit models are designed to estimate the battery state, according to which experimental approaches are adopted. Data from complex pulse-current tests are used for parameterization. The battery is modeled and simulated with the results and calculations of the experimental data. Improved algorithms are also proposed and analyzed in the quest for accurate state estimation. The idea of improved algorithms is to update the statistical noise covariance parameters, further enhancing the battery state estimation performance. These algorithms reduce the interference of system noise effectively. This work also employs extended methods alongside the traditional algorithms to improve the effectiveness of the adaptive battery state estimation. Results and computations from the experiment and simulation are compared with that from the improved algorithms, which illustrate that the improved algorithms could present good convergence speed. As the adaptive algorithms are stable with high precision accuracy, they can be effectively used for the accurate battery state estimation.
- 24. Preface The lithium-ion batteries have gradually become the preferred power source for new energy supply working conditions, which have the advantages of high energy, small size, and rechargeability. The internal structure is complex, so its state is affected by various complex factors such as current, self-discharge effect, internal temperature, environmental temperature, and battery aging. It makes the state value difficult to be estimated accurately as the accurate state estimation is a symbolic indication of the energy control technique. Through the accurate residual capacity, the battery application strategy can be planned to realize its optimal operation. The parameter extraction is picked in the equivalent circuit modeling, and expanded by its internal resistance. The resistance–capacitance values have been calculated mathematically. Furthermore, the hybrid pulse power characterization experiments are conducted to realize the parameter identification and battery state estimation. Then, the iterative calculation is performed by the open-circuit voltage and extended Kalman filter. In the extended calculation process, the iterative pre- diction and correction strategies are introduced to reduce the initial error. The research motivation and objective of the battery state estimation methods are elaborately discussed as well as the mathematical analysis of estimation approaches. The critical factors of the state estimation approach are also discussed. The coordinate transformation and iterative estimation are presented, according to which various battery modeling types are introduced in detail together with its mathematical analysis. The data collection tactics are extensively incorporated together with parameter identification and experimental verification. The whole content is reviewed by Prof. Zonghai Chen, who provides many insightful and constructive opinions on the publication of this book. It has received scientific support from Mianyang Quality Supervision & Inspection Institute, Sichuan Huatai Electric Co., Ltd, Shenzhen Yakeyuan Technology Co., Ltd, and Mianyang Weibo Electronics Co., Ltd. As battery modeling involves a wide range of aspects, please feel free to contact the authors with the link https://www.researchgate.net/lab/DTlab-Shunli- Wang for effective responses. It is hoped that this book can be served as a com- munication platform to establish contact with readers and promote the development progress of the battery modeling technologies.
- 26. List of contributors Amdadul Haque Southwest University of Science and Technology, Mianyang 621010, China Carlos Fernandez Robert Gordon University, Aberdeen AB10-7GJ, United Kingdom Chunmei Yu Southwest University of Science and Technology, Mianyang 621010, China Dan Deng Southwest University of Science and Technology, Mianyang 621010, China Daniel-Ioan Stroe Aalborg University, Pontoppidanstraede 111 9220 Aalborg East, Denmark Ji Wu Hefei University of Technology, Hefei 230027, China Jialu Qiao Southwest University of Science and Technology, Mianyang 621010, China Jinhao Meng Sichuan University, Chengdu 610065, China Junhan Huang Southwest University of Science and Technology, Mianyang 621010, China Kailong Liu WMG, University of Warwick, Coventry, CV4 7AL, United Kingdom Lei Chen Southwest University of Science and Technology, Mianyang 621010, China Lili Xia Southwest University of Science and Technology, Mianyang 621010, China Long Zhou University of Shanghai for Science and Technology, Shanghai 200093, China Mingfang He Southwest University of Science and Technology, Mianyang 621010, China Monirul Islam Southwest University of Science and Technology, Mianyang 621010, China Peng Yu Southwest University of Science and Technology, Mianyang 621010, China Pu Ren Southwest University of Science and Technology, Mianyang 621010, China Shunli Wang Southwest University of Science and Technology, Mianyang 621010, China Siyu Jin Aalborg University, Pontoppidanstraede 111 9220 Aalborg East, Denmark Weihao Shi Southwest University of Science and Technology, Mianyang 621010, China
- 27. Wenhua Xu Southwest University of Science and Technology, Mianyang 621010, China Xiao Yang Southwest University of Science and Technology, Mianyang 621010, China Xiaoxia Li Southwest University of Science and Technology, Mianyang 621010, China Yanxin Xie Southwest University of Science and Technology, Mianyang 621010, China Yongcun Fan Southwest University of Science and Technology, Mianyang 621010, China Yunlong Shang Shandong University, Jinan 250100, China Yujie Wang University of Science and Technology of China, Hefei 230027, China Chuangshi Qi Southwest University of Science and Technology, Mianyang 621010, China xxii Battery state estimation: methods and models
- 28. Chapter 1 Introduction Abstract With the development of society, energy security and the environmental pollution caused by it has become a key issue that all sectors of society are concerned about and urgently need to be resolved. The deepening of the world energy crisis has led to the emergence of new energy industries as well as the increasing awareness of environmental protection. Among them, lithium-ion batteries have developed rapidly in the field of new energy due to their higher energy density and longer cyclic life. This chapter briefly introduces the use scenarios and market conditions of lithium-ion batteries, the common methods of lithium-ion battery state estima- tion, and their research significance. Then, the development status of the battery management system is briefly described. At the same time, the estimation methods of various state parameters are introduced for lithium-ion batteries, which lays a foundation for subsequent research. The research conclusions and further research plans are discussed, the content is reviewed, and the system state estimation is emphasized to achieve the purpose of safety protection and lifetime guarantee. Keywords: Lithium-ion battery; Energy crisis; Development history; System state estimation; Filtering algorithm; Equivalent circuit model; Safety protection 1.1 State of the art Currently, lithium-ion batteries are important new energy sources in the twenty- first century, and research in the area of improving and enhancing the technical performance of lithium-ion batteries through various technologies and methods is the key to tapping greater potential. Lithium-ion batteries are used in portable products, such as laptops, notebooks, mobile phones, tablets, portable digital assistants, cameras, rechargeable lamps, toy cars, and toys that use rechargeable batteries and robust products such as the electric vehicle, electric motorcycles, balance cards, scooters, and smart grids, in all aspects of life and all sectors, agriculture, health, education, industry and in wireless sensor networks and recently very popular. The key to improving, controlling, monitoring, and mana- ging the lithium-ion batteries is the battery management system (BMS) [1], as well
- 29. as the estimation of the state of charge (SOC), state of health (SOH), state of power, and other battery parameters are very important research fields, ensuring the safety and reliability of electronic devices that use the battery as a power source. The global attention in the direction of the batteries used in electric vehicles (EVs) is gaining much popularity as they are the sustainable mode of transportation and the alternatives of the International Combustion Engine-based vehicles [2]. The EVs effectively overcome the fossil fuel crisis and environmental pollution, con- sidering the major hurdle for the automobile sector [3]. As the only power supply in pure EVs, the capacity of the battery pack is of great importance. In this case, the high specific energy lithium-ion batteries are widely used in EVs with superiority [4]. Lithium-ion batteries have increased in popularity. Due to their advantages such as lightweight, fast charging speed, high energy density, low self-discharge, and long lifespan [5]. The usable energy classified as SOC is the same as the available energy of the vehicle fuel gauge operated by a combustion engine. BMS is responsible for accurately measuring the status of the battery, ensuring safe operation, and prolonging the battery life [6]. The accurate SOC estimation of the lithium-ion battery is a very challenging task because of its high time-variant, nonlinearity, and complex electrochemical system [6]. An improved Thevenin equivalent circuit model is proposed, designed, and implemented through experi- mentation and simulation. The model is achieved by adding an extra RC branch to the Thevenin model, making it a second-order resistor capacitor. The second-order Thevenin model has good accuracy, stability, robustness and is very effective for SOC estimation [7]. This model is used to study and record parameters and estimate the relationship between voltage, current, SOC, and charge–discharge characteristics. Experimental data results and simulated results are compared and analyzed to further appreciate the effectiveness of the improved adaptive extended Kalman filtering (AEKF) algorithm used [8]. The improved AEKF algorithm is employed in this work to accurately estimate the SOC of the lithium-ion batteries. The algorithm estimates the SOC dynamically but easily causes divergence due to the uncertainty of the battery model adopted and to what extent the system noise is taken care of. The AEKF algorithm is used for the SOC estimation of the lithium- ion battery model designed to obtain a more accurate result for the research. It is worth noting that the proposed SOC estimation using the AEKF algorithm is more accurate and reliable comparatively than the extended Kalman filtering (EKF) algorithm. The safe operation and efficient energy management strategy are essential for the BMS of batteries, which are based on the SOC estimation accuracy [9]. However, the evaluation of battery SOC is challenging in uncertain and complex EVs environments [10]. Therefore, to manage the lithium-ion more efficiently and improve battery performance, it is necessary to obtain the inner state parameters and accurately make an accurate SOC [11]. SOC is not directly measurable and requires a particular type of method to be estimated but often uses some specific models such as the empirical model, the equivalent circuit model, and the elec- trochemical model [12]. Deterioration of battery cells leading to reduced perfor- mance is a problem that limits battery life. 2 Battery state estimation: methods and models
- 30. Moreover, electrical equivalent circuit (EEC) modeling has been investigated specifically for applications such as BMS development and vehicle power man- agement control [13]. A good battery model can estimate the battery storage power details of the battery and the voltage response to the load [14]. The EECs can describe these characteristics of lithium-ion batteries [15]. In some cases, modeling the side reactions is required in terms of battery losses, which can also be realized with the EECs [16]. A novel approach for identifying parameters requires a high- order equivalent circuit proposed in this literature with improved mathematical analysis. The EKF algorithm is proposed to estimate the SOC in real-time solid data [5]. The validation results show that the proposed methods have good per- formance in estimation accuracy and uncertainty of the parameters. Electrical vehicles powered by lithium-ion batteries are more environmentally friendly than gasoline-powered ones, and this kind of environmentally friendly energy is gradually widely used in various fields. In such high demand, the battery state is sig- nificant because it can be estimated. To extend the driving distance, the influencing factors should be measured in real time for the long-term use to avoid the measurement error accumulation of the voltage, resistance, and temperature [17]. The highly nonlinear characteristic also brings lots of difficulties to the battery state estimation. Because the power batteries operating characteristics are highly nonlinear, it is necessary to establish an accurate battery model to estimate the accurate state of the battery [18]. To solve this issue, several algorithms are proposed and used to estimate the battery state, including the current integration, open-circuit voltage (OCV), Kalman filtering (KF) algorithms, and neural network (NN) [19]. 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 use will lead to the accumulation and expansion of measurement errors, so it is necessary to introduce relevant correction factors to correct the accumulated errors. However, to realize the SOC estimation accurately, the following factors are determined and the core research is objective, including accurate SOC estimation, understanding the dynamic characteristics of battery behavior, studying internal reactions and tem- perature, and monitoring and controlling charge–discharge flow. An effective BMS is necessary for power lithium-ion batteries. The above objects are analyzed in various ways to determine a ternary lithium- ion battery for EVs as the research objective. The entire process is carried out by the offline process, and the process is mainly established in the laboratory envir- onment and uses room temperature. To perform and validate the research objective, the above contents help to carry out the necessary procedures. The Ampere-hour (Ah) commonly known as the coulomb-counting method is to perform the first stage. Then, a similar model can be selected as well as the state-space equation and so on. The high-order improved equivalent circuit model is used as the significant model to get the parameters. Furthermore, these parameters are used for SOC estimation, which is commonly defined as the percentage of the maximum possible charge present inside a rechargeable battery from 0% to 100%. The detailed contents are briefly introduced in Figure 1.1. Introduction 3
- 31. Figure 1.1 shows the working process step by step that has been carried out. Various experimental methods are integrated, such as the hybrid charge–discharge, constant-voltage charging with different multipliers, constant discharge, and cyclic discharge shelved experiments. It is used to study the lithium-ion battery operating characteristics and analyze the characteristics of the battery response. Besides, its long-term use accuracy in estimating the state of the same battery group can be greatly reduced. The complex effect of factors such as battery temperature, self- discharging current rate, and the varying degree of battery aging have been described elaborately with its internal equation. 1.2 Application requirements The accurate SOC estimation of lithium-ion batteries is one of the most important issues for BMS, especially in EVs and most electronic devices, because it is a critical factor to solve the key issues of monitoring and safety concerns of the batteries. The methods and techniques used for SOC estimation have a great effect on the outcomes and result of analysis. The results greatly depend on appropriate battery modeling and estimation algorithms. The internal resistance and OCV of the battery are influenced by temperature and current detection precision. Therefore, the changes of these parameters are likely to cause a drifting noise in current measurements leading to errors in SOC estimation. Researchers have over the years been using several methods and techniques combining them in the quest to estimate state accurately and improve upon such methods and techniques suc- cessfully. This research focuses on the use of an improved second-order Thevenin Ampere-hour (Ah) strategy Describing remaining energy Indicating the SOC ratio Measuring the discharging current Experimental platform design Using “CT-4616- 5V100” astesting equipment Collaborate with host computer TCP/IP port connect with host computer Pulse-currenttest (HPPC) Selecting the ternary battery as test object Inputing expected parameters into BTS- 7.6 software Saving the real-time data whenfinished Equivalent circuit modeling Picked up a high- ordermodel Investigaing the internal character Parameter identification based on equations Extended Kalman filtering State-space modeling observation by Taylor series Linearization and SOC prediction Updating and correcting SOC value Simulation software design Model design based on high-order modeling Model design based on varying algorithms Softwarecode design based onapproaches Figure 1.1 Major working procedure for SOC estimation approach 4 Battery state estimation: methods and models
- 32. equivalent circuit model and the application of improved algorithms to improve estimation accuracy and to encourage the use of adaptive algorithms in the esti- mation process. EVs and hybrid electric vehicles (HEVs), owing to the benefits of reduced emission and energy conservation, are recognized worldwide as one of the sig- nificant growth paths of automotive industries. Lithium-ion batteries are the necessary power source for EVs due to their high energy and energy density. Therefore, it uses rechargeable batteries to maintain a permanent energy supply— various types of rechargeable industrial batteries, such as lead-acid, Ni–MH, Ni– Cd, and lithium-ion. However, lithium-ion battery production seems to be more suitable than other batteries. In such cases, high power density, high voltages, cri- tical load/unload cycles, and safety are superior. The battery charging status must also be controlled explicitly for battery SOC estimation. The BMS configured algorithms to allow better utilization and longer battery life. The crucial role of the BMS is to control the remaining power. Accuracy in the measurement of the SOC makes it easy for the battery to control adequate elec- tricity, prevents irreversible damage to the internal battery device, and ensures maximum use. Many SOC estimation approaches are available for electrochemical accounting, model based, and black-box often referred to as data oriented. It aims to develop a solution that honors precision and versatility as a target for the deployment of lithium-ion batteries on the BMS. In this idea, an effective SOC estimation algorithm is based on a coulomb-counting algorithm, an accounting approach. It is using a piecewise linear interaction mapping between SOC and OCV to solve this downside approach and increase its accuracy. Experimental data is analyzed and used for parameterization, curve fitting, and simulation of the battery model, while it is used to perfect the estimation and to minimize error. Lithium-ion batteries have become the preferred power source for a lot of electronic devices and require special handling to maintain good SOH and optimum performance. The ability and capability of BMS are very paramount to estimate the battery state parameters accurately and effectively. Research to the improvement of the function, reliability, and performance of the lithium-ion battery technology is important, and any breakthrough in the area would go a long way to improve upon the technology. As SOC estimation is one of the most important states to be iden- tified in a battery, any advancement towards improving estimation accuracy through the invention of improved estimation methods, enhanced performance, operability of the battery, and extending battery lifespan and safety is vital. Research on lithium-ion batteries and their related parameterization is important across the world today to improve especially on electronic devices, and more specifically, the electric vehicle, the HEVs, smart grids, and UAVs. 1.3 Research methodology To achieve the objectives of the research and meet the required standards, several methods and techniques are employed including experiments, tests, and simulations. Introduction 5
- 33. The OCV test, capacity test, and hybrid pulse-power characterization test are employed for parameterization. The data from the tests is used as the experimental result for the figures and analyzes made in this work. To further improve the SOC estimation and make it current and significant, an improved AEKF algorithm is proposed and implemented in the simulation. An experimental research design is a type of research that is used to manip- ulate, control, test, and understand the causal processes of a system model. Process methodology in research is convenient in the study of mechanisms and function- alities of simple to complex systems. The methodology of experimental research is the use of purposeful abstract terms to represent a real object by representing group components and interactions that allow qualitative and quantitative descriptions of the object. These components and interactions can be constructed and described by text and block diagrams and built into a model with other distinctive modules that operate together to achieve a unique output. The research content comprises tests, simulations, and algorithms that need to be performed to achieve the objectives and present appropriate results. An over- view of the research content from the introduction through the experimental stage and simulation to the use of the AEKF algorithm is shown in Figure 1.2. 1.4 Research status and direction The common methods used in SOC estimation of the lithium-ion batteries are the OCV method, Ah integral NN method, and KF method. The OCV method uses the one-to-one correspondence between the OCV voltage and SOC to obtain its value by obtaining the battery OCV, realizing the purpose of accurate state estimation. It is a more traditional method to calculate SOC by the integration of current in time. SOC estimation (Introduction and literature review) Curve fitting (OCV/SOC relationship) Polynomial equation from curve SOC, current, voltage/OCV Accurate SOC estimation Capacity and HPPC test (experimental data) Battery circuit modeling Thevenin ECM Additional RC Proposed battery model SOC estimation Parametarization Module including ECM Capacity, current, voltage, energy, R0, R1, R2, C1, C2 Variation curves and data table Kalman filtering algorithms Simulation (Simulink MATLAB® ) Extended Kalman filter Adaptive extended Kalman filter OCV test (experimental data) Results, analysis and conclusion Parameterization Figure 1.2 Adaptive research approach 6 Battery state estimation: methods and models
- 34. The NN method estimates the SOC of the battery by processing the amount of real- time input and output data of the battery. The KF method obtains the optimal solution in the sense of minimum variance through continuous iterative operations. Because of the complex electrochemical reaction inside the battery in use, it often shows strong nonlinear characteristics. Also, some defects of the traditional algorithm itself make the above methods cannot accurately estimate the SOC value of the battery when dealing with the nonlinear systems, which often have the problems of low estimation accuracy and large error. In recent years, researchers put forward some improved algorithms based on traditional algorithms. The complex internal structure of the lithium-ion batteries makes it show strong nonlinear characteristics in the use process, which puts forward new requirements for the traditional SOC estimation methods of the battery. At the same time, the dependence of its estimation on the equivalent model makes the model selection and construction very important. Therefore, the EEC and SOC estimation of the battery still need to be further developed under various conditions. Research concerning BMS from a global perspective includes those which display an entire BMS design adopting a distributed structure to reach better scal- ability and portability [20,21]. Different approaches to designing a BMS depend on the functionalities desired for the specific application, but most of them focus on key functions such as SOC estimation [22] and the balancing process [23]. The improvement is towards the design of intelligent BMS for electric and HEVs in artificial intelligence applied for the battery state estimation. Therefore, SOC esti- mation has drawn the attention of many researchers, and many different methods have been proposed [24,25]. The OCV method, a full charge detector/dynamic load observer, and robust extended Kalman filtering algorithm are combined. It is dif- ficult to determine the specific approach when such methods are used. However, based on the classification [25], two categories are used, namely the direct method and the indirect method, as well as several sub-categories that summarize the trend of SOC estimation and appropriate adjustments. A Thevenin EEC is used in for every single cell in an array of more than 90 series-connected cells to identify the internal resistance of each cell. Two dif- ferent branches are used in a Thevenin model for charge–discharge that is con- nected in series n times to represent n cells in a series. According to [26,27], there are three different EECs of lithium-ion batteries widely adopted because of their excellent dynamic performance. It shows that the second-order EEC is the most accurate and has the best dynamic performance and also the most complex. Also, the discretization equations of each of the three mentioned models are presented and used in combination with the EKF method to estimate the SOC. The Thevenin and second-order EEC modeling methods [27] are used for the SOC estimation. The difference between these models is the way of the SOC equation and calcu- lation procedure. The parameters of the second-order EEC can be calculated with different datasets depending on the scenario where the model is going to be used [28], a comparison between continuous-time and discrete-time equations of the second-order EEC is made and concludes that discrete-time identification methods Introduction 7
- 35. are less robust due to undesired sensitivity issues in the transformation of discrete domain parameters. The application of the EKF algorithm in SOC estimation is also presented in [29]. It is common to find a combination of the CC and OCV methods with the KF method to estimate the SOC value [30]. The EKF method [31] is also introduced for the iterative calculation in combination with CC and OCV parameters. Another common improvement to the KF algorithm for SOC estimation is the unscented Kalman filtering (UKF) algorithm, which is used in [31,32] to improve the esti- mation accuracy. The UKF algorithm is implemented in [33] to estimate SOC using an improved EEC with a resistance and a capacitor correction factor. This is done first of all to measure the effect of different current rates and the SOC estimation on the battery internal resistance and second to identify the impact of different current rates and temperatures on the battery capacity [34]. The work presented in [35] uses multiple modeling approaches combined with the EKF algorithm to estimate the SOC value of the battery. The improved EKF algorithm is implemented in [36] to be more robust to uncertainties in the system, measurement equations, and noise covariance. An SOC estimation approach that uses an improvement in the mea- surement noise treatment is proposed in [37], and an adaptive Kalman filtering algorithm that can reduce the estimation error is established by correcting the covariance matrix error in the depicted EKF method. To deal with the variation of battery parameters due to temperature changes [38], an online approach is proposed for SOC estimation and parameter updating using a dual square root UKF based on unit spherical unscented transform. To obtain a more accurate and reliable SOC, an improved Thevenin equivalent model is proposed and its parameters are identified. Likewise, an AEKF algorithm is designed for the SOC estimation of the lithium-ion battery model. The use of the AEKF algorithm in this research is to estimate the SOC and eliminate or reduce errors accurately and diligently. The use of the AEKF algorithm is an innovation in this work coupled with the second-order RC and the other minor features that would lead to the successful implementation. This research is centered on accurate state estimation. The focus is specifically the use of experimental data for parameterization, the modeling, and the imple- mentation of the circuit for simulation analysis. The study also empathizes with the implementation of the proposed improved adaptive algorithm for accurate state estimation. The introductory aspect of the dissertation is made up of the back- ground which outlines the importance of the research and reasons for the choice of the direction. The problem statement puts forth the main issue to be tackled, and its purpose is to reiterate the main goal and objectives of the research. The research method gives an insight into the methods and techniques used by the researcher to achieve the main goal and finally emphasizes some literature and key issues as far as the research area is concerned. Theoretical and mathematical analysis deals with the introduction of the lithium- ion batteries specifically and related important areas connected to the study. Battery modeling techniques and experiments are conducted for parameter identification, which is introduced into the SOC estimation methods. The mathematical equations 8 Battery state estimation: methods and models
- 36. for parameterization and the proposed mathematical algorithms are realized for the battery state estimation. Model building and realization introduces the different bat- tery modeling techniques in the area and puts across the proposed circuit model adopted for the research and underscores the choice of the model and its effect on the SOC estimation. Experimental verification includes detailed expression, how each testing pro- cedure is performed with a detailed list of steps and flowcharts. The data collection process for parameter identification results from the conducted experiments pre- sents the results from mathematical calculations and simulations. The comparison and verification of the results are also discussed. Conclusion and further research plan present reflections on the work in its entirety review content and emphasizes the effects of the experiments and designs implemented in the work. A concluding statement on the overall performance of the method used, the difficulties encoun- tered, and some suggestions for further research in the area are also opined. 1.5 Chapter summary This chapter introduces different types of lithium-ion batteries as well as their advantages and disadvantages for applicable occasions. Meanwhile, the working characteristics are analyzed. It provides a basis to choose batteries for different working environments. This chapter briefly introduces the concept, development background, application fields, and working principles of lithium-ion batteries. Meanwhile, the development trend is analyzed. It lays the foundation for the research of lithium-ion batteries. Acknowledgment The work is supported by the National Natural Science Foundation of China (No. 61801407), Sichuan Science and Technology Program (No. 2019YFG0427), China Scholarship Council (No. 201908515099), and Fund of Robot Technology Used for Special Environment Key Laboratory of Sichuan Province (No. 18kftk03). Introduction 9
- 38. Chapter 2 Mechanism and influencing factors of lithium-ion batteries Abstract This chapter introduces the operating mechanism, influencing factors, key indica- tors, and some mainstream state estimation methods of lithium-ion batteries. First, understanding the main composition and internal working principle of lithium-ion batteries is the prerequisite and basis for other work. Second, internal resistance, open-circuit voltage (OCV), terminal voltage, current thermal energy, capacity variation, and temperature characteristics are the main features of the battery, which can be further used to accurately characterize the battery state. The state of charge, state of health, state of power, depth of discharge, and cyclic life are the key indicators of state estimation, while the discharging test, Ampere-hour integral, OCV, internal resistance, and Kalman filtering are basic state estimation strategies. Based on the Kalman filter, there are many improved algorithms, such as the unscented Kalman filter and adaptive Kalman filter, which have achieved good application effects. Besides, other algorithms such as neural networks, support vector machines, and some improvement strategies are also introduced. The advancement of these algorithms has made an important contribution to improving the whole-life-cycle state estimation effect of lithium-ion batteries. Keywords: Operating mechanism; Battery characteristics; Influencing factor; State of charge; State estimation; Kalman filtering; Cyclic life; Temperature; Open-circuit voltage 2.1 Introduction With the development of society, the shortage of primary energy and increasing atten- tion to environmental pollution have promoted the rapid growth of lithium-ion batteries. The invention of new energy and environmental protection are important research topics, and the fact that lithium-ion batteries are relatively environmentally friendly in use has explained the gradual replacement of other relatively polluting batteries. To improve the function of electronic equipment, especially electric vehicles, hybrid electric vehicles, smart grids, and UAVs that all use lithium-ion batteries [39].
- 39. The research on lithium-ion batteries and their related parameterization is of global importance. There are several journals and periodicals with published articles and projects on several important theories and ideas related to lithium-ion batteries and their parameter estimation, and the number is increasing daily [40]. According to the global lithium-ion battery market, it has been growing at a compound annual growth rate of 10.6% since 2016 and is estimated to reach $56 billion by the year 2024, which means that the demand for lithium-ion batteries is set to be more than double by this year. This implies that the increasing demand would lead to more improve- ments and has a lot of broad prospects for more research in the field, as shown in Figure 2.1. The battery model studies the relationship between the external characteristics and the internal state of the battery by establishing a mathematical model. Discrete- time and state-space forms are also used for the state of charge (SOC) estimation. The present literature makes mention of electrical equivalent circuit (EEC) models as being widely used as a foundation for model-based estimation and control [41,42]. Generally, the equivalent circuit models are selected for the lithium-ion battery modeling, including the Rint model, Thevenin model, RC model, and PNGV model [43,44]. These models use RC loops of different orders to model the polarization characteristics of batteries [45]. Among them, the Thevenin model is widely used, but as all its components may change with the state of the battery, its accuracy is not high [46]. For more accurate parameter identification [47,48], a new design using charge–discharge is being developed, and SOC estimation methods are combined [49,50] as a means of estimating SOC in the presence of unknown or time-varying battery parameters. Research in this field either assumes the available precise SOC–OCV [51,52] or a constant during discharge. The RC parameters are deter- mined by analyzing the transient state of the battery voltage response [25] under certain excitations, such as constant current or pulse current experiments. The voltage source in an EEC typically represents the open-circuit voltage (OCV), which depends on the SOC [53]. The relationship between SOC and OCV can be 2015 2016 2017 2018 2019 CAGR 10.6% (2016–2024) 2020 2021 2022 2023 2024 56 Figure 2.1 Global lithium-ion battery market size and forecast 12 Battery state estimation: methods and models
- 40. identified by charging or discharging the battery with a small current. Parameter identification based on current–voltage data is addressed [48] by a method that simplifies the problem of solving a set of high-order polynomial equations to solving several linear equations and a single variable polynomial equation. 2.2 Operating mechanism In the global high-tech growth, the lithium-ion battery industry is an excellent direction. The benefits of lithium-ion batteries are high capacity, high conversion rate, long lifespan, and no emission. Because of its good electrochemical durability, high fuel density, long service life, and no maintenance, it is often used in electric vehicles and various power storage systems. Lithium-ion batteries have been widely used at present. It primarily encompasses five fields, including transporta- tion, power storage, mobile communication, modern energy storage, and military aerospace. It will substitute oil with electricity, minimize emissions of greenhouse gas, and store electricity in the power grid in the unnecessary conditions of electric vehicles. 2.2.1 Brief introduction The battery converts chemical energy into electric energy, which is irreversible in the primary battery and rechargeable in the secondary battery. Lithium-ion batteries can be divided into three different categories, such as lithium metal, lithium-ion, and lithium-ion polymer. Lithium metal batteries are primary batteries, while lithium-ion and lithium-ion polymer batteries are both rechargeables. The lithium-ion battery system is a complex system that integrates chemical, electrical, and mechanical characteristics, so various characteristics must be considered in the design. In parti- cular, the safety and life attenuation characteristics contained in the chemical char- acteristics of the battery cell cannot be directly evaluated, nor can they be easily predicted in a short time. Therefore, the battery technology, group technology, and battery management system (BMS) technology should be implemented when designing a battery system, as well as battery protection. Consequently, the reliability and lifespan of the battery should be considered. The lithium-ion battery is an advanced battery technology that uses lithium ions as the key component of its electrochemical reaction. During the discharge cycle, lithium atoms in the anode are ionized and separated from their electrons. The lithium ions are small enough to pass through the micro-permeable separator between the anode and the cathode. This separator recombines with its electrons and is neutralized electrically. A class of organic compounds known as ether is used as the electrolyte of lithium-ion batteries. The most common combination of materials called electrodes is that of lithium cobalt oxide (cathode) and graphite (anode), which is most commonly found in portable electronic devices such as cellphones and laptops. Other cathode materials include lithium manganese oxide and lithium iron phosphate. Lithium manganese oxide is commonly used in hybrid and electric vehicles. Due to the small size of lithium ions, the batteries can have a Mechanism and influencing factors of lithium-ion batteries 13
- 41. very high voltage and charge storage per unit mass and unit volume. Lithium is the lightest of all metals, which has the greatest electrochemical potential and provides the largest specific energy per weight. Rechargeable batteries with lithium metal on the anode provide extraordinarily high energy densities. Lithium-ion batteries are facing competition from numerous alternative battery technologies, most of which are in the development stage [54]. One such alternative called a saltwater drive battery is developed by the energy equation. They are com- posed of saltwater, manganese oxide, and cotton, which are made by using abundant, non-toxic materials, and modern low-cost manufacturing techniques [55]. Because of this, they are the only cradle-to-cradle-certified battery in the world. The batteries mentioned in it are all lithium-ion batteries. The cathode substrate of a lithium-ion battery is lithium-alloy metal oxide [56]. Its negative electrode component is gra- phite. The essence of charge and discharge is the electrochemical reaction in lithium- ion batteries. The essential response and working process of lithium-ion batteries are shown in Figure 2.2. Figure 2.2 shows that the lithium-ion metal oxide positive electrode material is emitted during the battery charging phase [57]. The lithium cobalt metal oxide is transferred through the electrolyte and the separator, and the carbon substrate is incorporated into the negative electrode-coated frame. The positive electrode enters a state of prosperity rich in lithium-ion at this stage. On the contrary, the negative elec- trode enters a lithium-ion lean state [58]. During the discharge process, lithium-ion can Septum Electrolyte Electrolyte Load Power Shell e- e- + - Charge Discharge Positive electrode Negative electrode Li+ C CoO2 Figure 2.2 Schematic structure of a lithium-ion battery 14 Battery state estimation: methods and models
- 42. be released from the negative electrode carbon material layer. The carbon content coating would be deposited in the negative electrode crystal lattice since going through the separator and the electrolyte [59]. The positive electrode is developing towards a lithium-rich state. The negative electrode is developing towards a lithium-poor state. The chemical quality of the electrode continues to recover. The complete equation for the reaction is achieved by positive and negative lithium-ion electrodes. Each battery is shown as follows: P : LiCoO2 ⇄ cd Li1xCoO2 þ xLi þ xe N : xLi þ xe þ 6C ⇄ cd LixC6 T : LiCoO2 þ 6C ⇄ cd Li1xCoO2 þ LixC6 8 : (2.1) In (2.1), it can be known that the internal resistance, polarization, aging, and other factors are added in the process of the lithium-ion battery. The embedding and unpacking going through the separator and traveling in the electrolyte can impact the calculation of SOC [60]. In the process of model development, these considerations and SOC estimation need to be solved [61]. 2.2.2 Battery composition Lithium-ion batteries generate electricity through the chemical reaction of lithium. The battery consists of four key components. The anode, which determines the capacity and voltage of the battery, is the source of the lithium ions. The cathode allows current to flow through an external circuit, and when the battery is charged, lithium ions are stored in the cathode. The electrolyte is composed of salts, solvents, and additives and acts as the conduit for lithium ions between the cathode and the anode. The separator is a physical barrier separating the cathode and anode. Common cathode materials include lithium cobalt oxide (lithium cobaltate), lithium manganese oxide (spinel or lithium manganate), lithium iron phosphate, lithium nickel manganese cobalt (NMC), and lithium nickel cobalt aluminum oxide (NCA), as shown in Table 2.1. Table 2.1 Most common lithium-ion battery compositions Type Cathode Anode Lithium cobalt oxide (LCO) LiCoO2 Graphite/hard carbon (LiC6) Lithium manganese oxide (LMO) LiMn2O4 Graphite/hard carbon (LiC6) Lithium iron phosphate (LFP) LiFePO4 Graphite/hard carbon (LiC6) Lithium nickel manganese cobalt oxide (NMC) LiNiMnCoO2 Graphite/hard carbon (LiC6) Lithium nickel cobalt aluminum oxide (NCA) LiNiCoAlO2 Graphite/hard carbon (LiC6) Lithium titanate (LTO) LiMn2O4/ LiNiMnCoO2 Li4Ti5O12 Mechanism and influencing factors of lithium-ion batteries 15
- 43. During the charge–discharge process of the battery, while the battery is dis- charging and providing an electric current, the anode releases lithium ions to the cathode, resulting in electron flow from one side to the other, and when charging, lithium ions are released by the cathode and received by the anode. 2.2.3 Working principle The internal chemical reaction of lithium-ion batteries is a basic oxidation–reduc- tion (OXRED) reaction, which is also the operation theory of the battery in the actual process of converting available energy into heat energy through the chemical reaction. The charge–discharge process of the battery is a chemical reaction involving the movement of lithium ions. The positively charged lithium atom undergoes an oxidation reaction when the battery is charged, which loses electrons and becomes a lithium ion. Many lithium ions formed by the oxidation reaction of the positive electrode migrate from the positive electrode to the carbon layer of the negative electrode of the battery. At one end, the capacity of the battery is related to the number of lithium ions generated in the positive electrode reaction, and at the other end, the capacity of the battery is related to the number of lithium ions exchanged through the electrolyte in the negative electrode. During discharge, the negative electrode undergoes an oxidation reaction, and the lithium ions trapped in the negative electrode carbon layer appear and move back to the positive electrode. The higher the discharge power, the more lithium ions return to the positive electrode. More lithium ions are loaded into the anode when the loading power is high. During charging, lithium ions move from the positive electrode to the negative electrode through the electrolyte. The positive and negative reactions of the elec- trode and the whole reaction equation are as follows. The positive electrode reac- tion is LiMxOy ¼ Li 1x ð ÞMxOy þ xLiþ þ xe (2.2) Electrons flow from the positive electrode to the negative electrode, which takes a longer path around the external circuit. The electrons and ions combine at the negative electrode and when the battery is fully charged, the ions stop moving. The negative electrode reaction is represented in the following equation: nC þ xLiþ þ xe ¼ LixCn (2.3) During discharging, the ions flow back through the electrolyte from the negative electrode to the positive electrode. Electrons flow from the negative electrode to the positive electrode through the external circuit. When the ions and electrons combine at the positive electrode, lithium is deposited there. The battery is fully discharged and needs charging when all the ions move back to the positive electrode. The total battery response is shown in the following equation: LiMxOy þ nC ¼ Li 1x ð ÞMxOy þ LixCn (2.4) 16 Battery state estimation: methods and models
- 44. 2.2.4 Cycling lifespan The electrode design can also be strengthened. The capacity of lithium-ion batteries can be increased by testing the properties and scale of materials used for construction. The chemical reaction on the electrode takes place inside the lithium-ion batteries. The internal resistance of the electrodes can reduce the heat generated during use. The cell potential and lifespan can also be increased [62]. The electrode size should be mini- mized to improve performance. It has been discovered that there are limitations to the reduction, such that as if the electrode thickness had fallen below. Then the batteries could no longer meet the necessary energy requirements. For thinner content sizes, there are additional trade-offs, including rising processing costs. The electrode scale can be included as the primary design element in the overall design presented in Table 2.2. In the design of electrodes, another development field has been formed. By changing the materials made, the ion journeys may be optimized to maximize overall conductivity and thermal capacity on the transportation distance or route. In certain situations, the additional advantage of these improvements is to improve the mechan- ical quality and the heat efficiency or overall power of the mechanical characteristics. Gold, graphene, yttrium, and zinc are used in the components applied to the electrodes. Several various influences affect this mechanism, and further research is ongoing. 2.3 Battery characteristics After understanding the structural principle of the lithium-ion battery, the para- meters, and common concepts of the battery that will appear in this work, it is important to reiterate certain characteristics that affect the battery. Basic para- meters of the battery include terminal voltage, electromotive force, capacity, internal resistance, SOC, state of health (SOH), state of power, depth of discharge (DOD), cyclic life, and self-discharge rate. The characteristics that affect the parameterization of the lithium-ion battery include the following contents. 2.3.1 State of power The physical significance of the capacity per unit time of the lithium-ion batteries is the internal chemical reaction that the battery produces energy output. How Table 2.2 Life cycle feature of the depth of discharge Depth of discharge Discharge cycles 100 300–600 80 400–900 60 600–900 40 1,000–3,000 20 2,000–9,000 10 6,000–15,000 Mechanism and influencing factors of lithium-ion batteries 17
- 45. much work is done in a certain period is represented by the energy released by a chemical reaction, which is measured in Watt (W) [63–65]. Depending on the project, power can be subdivided into actual power and instantaneous power [66–68]. The battery power equation can be expressed as shown in the following equation: P ¼ UI ¼ I E IR ð Þ ¼ IE I2 R (2.5) wherein E is the battery electromotive force, R is the total battery internal resis- tance, and I is a unit of time for the average current. 2.3.2 Internal resistance Lithium ions move from one pole to the other pole inside the lithium-ion battery and the factors that hinder the movement of ions constitute the internal resistance of the battery. Due to the internal resistance of the battery, the terminal voltage of the battery is lower than OCV in the discharge state. In the charging state, the terminal voltage of the battery is higher than the OCV value [69–71]. The essence of electric current is the directional movement of electric charge. During the movement, the electrons will be resisted by the material itself and the magnetic field. Internal resistance refers to the resistance of 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 the lithium-ion battery [72–74]. The internal resistance directly affects the working voltage, working current, and capacity of the batteries. The internal resistance of the lithium-ion battery is not constant. In the process of battery charging and dischar- ging, temperature changes, charge–discharge ratio, charge–discharge time, elec- trolyte concentration, and the quality of active substances will all cause changes in the internal resistance of the battery. 2.3.3 Open-circuit voltage The electrolyte, positive electrode, and negative electrode materials undergo aging during the cyclic charging and discharging process of lithium-ion batteries and in their external structure. The period from the battery start state to the battery life end state is the lifespan of the battery. This is the data of the charge–discharge test by the manufacturer under the specified standard test environment. Generally, dis- charge is stopped when its capacity drops to 80% of the rated capacity. The number of charge–discharge is the total number of continuous charge–discharge cycles of the battery under this condition. The OCV value can be measured by connecting a multimeter or voltmeter directly to the positive and negative ends of the battery. It is the voltage used in the simulation of the equivalent model in this book. The battery is not an ideal power source, the electrolyte and electrode material have internal resistance, so the OCV value is slightly less than the electromotive force value. Considering the internal resistance, it can be treated as the electromotive force [75–78]. Under actual operating conditions, the instantaneous closed-circuit voltage of the battery 18 Battery state estimation: methods and models
- 46. changes dynamically, which cannot avoid overcharge and over-discharge, causing damage to the battery system. 2.3.4 Self-discharge current rate The self-discharge current rate refers to the ratio of the discharge capacity of the battery to its rated capacity in the no-load state and is used to indicate the con- sumption rate of the battery capacity. Mainly affected by the battery manufacturing process, materials, storage conditions, and other factors, it is generally expressed as the percentage of battery capacity decrease per unit time, and the unit time is month or year. Therefore, the self-discharge current rate is the leakage current rate, which is usually used to test the leakage current in the test open-circuit environment. Due to the low leakage current, the self-discharge current rate of lithium-ion batteries is usually determined by their monthly values. It can also be regarded as the charge retention rate of the battery. 2.3.5 Terminal voltage The terminal voltage refers to the potential difference between the positive and negative electrodes of the lithium-ion battery, which can be divided into OCV and operating voltage according to the working conditions of the circuit where OCV is the terminal voltage of the lithium-ion battery without load and other power sources [79–83]. At the end of charging, the maximum allowable voltage of the battery is called the charge cut-off voltage, and the lowest allowable voltage after the end of battery discharge is called the discharge cut-off voltage. Beyond this limit, the battery will suffer some irreversible damage, and the cut-off voltage is an important safety indicator. Working voltage refers to the potential difference between the positive and negative electrodes when the battery is in working condition. Generally, due to the internal resistance of the battery, the working voltage during discharge is lower than the OCV, and the working voltage during charging is higher than the OCV value [84–86]. Like the rated voltage classification, according to the working state, the operating voltage of the lithium-ion battery is also divided into rated voltage, theoretical voltage, OCV, and working voltage with load. The rated voltage shall be directly calibrated by the battery manufacturer before delivery. 2.3.6 Current heat energy There is no ideal power source in the world. No matter it is the current source or the voltage source, there is some internal resistance. The lithium-ion battery as a power source is no exception. With a large current through the battery for a long time, it flows through the resistance to generate a large amount of heat energy, as shown below: W ¼ PT ¼ I2 RT (2.6) Resistance will lead to battery energy loss and a continuous increase in the bat- tery temperature, while the temperature rise will also lead to a rise in the resistance Mechanism and influencing factors of lithium-ion batteries 19
- 47. value, forming a vicious circle, the battery management system will take measures to deal with heat dissipation after monitoring temperature abnormalities [87–89]. Besides, if the battery is charged for a long time, the high large charge–discharge current of the battery will be lower than below certain levels, which will lead to potential safety hazards. It is very important to monitor the current size of the battery in the charge–discharge process to avoid causing damage to the battery. 2.3.7 Capacity variation There are different levels of power stored in the battery. When the battery is fully charged, its capacity is the amount of electricity it contains. Batteries of the same type are usually evaluated according to the amount of current they can output over time. Battery capacity refers to the total amount of electricity that can be released in the full state of the battery. It is represented by the symbol Q, and the unit of measurement is milli-Ampere-hour (mAh) or Ampere-hour (Ah) [90–92]. The battery capacity types can be divided into rated capacity, theoretical capacity, and actual capacity. The rated capacity is measured by the battery manufacturer and calibrated directly on the outer surface of the battery. Rated capacity is an important indicator of battery storage capacity and an indicator of battery life. The rated capacity is also known as calibration capacity, according to the relevant provisions of the relevant departments of the state, under certain discharge conditions (temperature, discharge rate, etc.) to ensure the minimum amount of energy that can be released by the battery, rated capacity is the Ah capacity indicated by the licensed manu- facturer, which is one of the important parameters of the battery [93–96]. The theoretical capacity is the theoretical value of the total charge of the battery cal- culated by Faraday’s law according to the electrochemical reaction inside the lithium-ion battery when the internal lithium ions are fully involved in the reaction. The actual capacity is the maximum capacity that a lithium-ion battery can release when purchased in the store under certain discharge conditions. It is usually calculated by the product of the discharge current and the discharge time. The actual capacity is not necessarily the capacity of the new battery but can also be the capacity of the used battery. Currently, the actual capacity is less than the actual capacity of the new battery. The actual capacity is the amount of electricity that the battery can release when operating in the real environment. The factors that affect the actual capacity of lithium-ion batteries are very complicated, including the temperature of the external environment, the materials used to make the battery, and the service life. 2.3.8 Temperature change The real-time temperature of the lithium-ion battery is a significant aspect that cannot be ignored in practical applications. Battery activity is directly proportional to tem- perature. As the temperature increases, the exchange rate of lithium ions between electrolytes increases, and the overall activity of the battery also increases, which is reflected in more battery energy output, increased the usable capacity of the battery, 20 Battery state estimation: methods and models
- 48. and improved battery efficiency [97–102]. Under the condition of long-term time and low-temperature conditions, the activity of anode and cathode material of the battery decreases. The actual capacity is less than the rated capacity. The charge–discharge efficiency of the battery decreases and the use efficiency is reduced. 2.4 Critical indicators for battery state estimation SOC estimation is an obligatory part of precise state estimation and battery man- agement for lithium-ion batteries. However, some critical influencing factors are battery related to estimating accuracy. This part of several influences is discussed elaborately in the portion. 2.4.1 Description of major parameters There are unknown safety risks in using lithium-ion batteries. It is essential to consider the critical criteria of lithium-ion batteries to prevent explosions and fires caused by management failure. To protect against unwanted circumstances, it is obligatory to measure some critical action. To observe the critical situation, the core parameters of lithium-ion batteries are necessary to be considered, as shown in Table 2.3. In Table 2.3, five key parameters are the basic parameters that deeply affect the performance of lithium-ion batteries. The battery capacity is a significant factor for storage capability, which is expressed in mAh. The nominal voltage is the rated voltage determined by the manufacturer. The end of charge voltage can be obtained when the battery charge is almost near to be ended. While the batteries are com- pletely discharged, the end of discharge voltage associates with the lowest voltage permissible. The self-discharge rate is the capacity consumption rate of a battery Table 2.3 The major parameters of lithium-ion batteries Parameters name Parameter definition Battery capacity The amount of active material determines the battery capacity, usually expressed in Ah. For example, 1,000 mAh can be converted into a charge of 3,600 C at a current of 1 A for 1 h Nominal voltage The electrical potential difference throughout the batteries between positive and negative electrodes is the nominal battery voltage End of charge voltage The active material on the polar plate has reached saturation when the rechargeable batteries are fully charged. Then, it begins to charge and the battery voltage will not increase Discharge end voltage When the batteries are fully discharged, the end of discharge voltage corresponds to the lowest voltage permissible. Discharge end voltage is compared to the rate of discharge Self-discharge rate The amount of the overall power expended when it is not used for any time. Generally, at an average temperature, the self- discharge rate of lithium-ion batteries is 5%–8% Mechanism and influencing factors of lithium-ion batteries 21
- 49. when it is unloaded. It is generally expressed by the ratio of its unit monthly dis- charge capacity to the rated capacity, and the ratio is 5%–8%. 2.4.2 Temperature effects As the environmental temperature affects the electrochemical reaction process of lithium-ion batteries, the measurement accuracy of SOC will continue to be com- promised. The higher the temperature and the higher the degree of the internal electrochemical reaction and the energy of the battery will also rise [103]. These factors will generate a large amount of high-temperature gas, which will accelerate the aging process of the battery, and affect its life cycle and even damage the battery. Therefore, only by seeking a suitable temperature can the output of the battery be fully utilized, and the batteries can be charged and discharged efficiently [104]. The battery capacity indicates how much energy it can hold, which decreases with the decrease of temperature and increases with the increase of temperature. Therefore, the battery fails on the cold winter morning, even though it worked the previous day perfectly. The batteries work in a cold environment most of the year. In that case, the decreased power of the lithium-ion battery must be taken into consideration. The typical operating temperature of a battery is 25 C (77 F). At approximately 30 C (22 F), the battery power will be reduced to 50%. At freezing, the battery power is decreased by 20%. Capacity is improved at higher temperatures at 122 F. The capacity of the battery will be around 12% higher. 2.4.3 Charge–discharge current rate The charge–discharge current can be described by the charge rate and discharge rate of lithium-ion batteries, which is represented by C. The charge–discharge rate refers to the ratio of the working current to the rated capacity of the battery during the charging and discharging process or working process. Within the same dura- tion, the charge–discharge amount of the battery will vary with different rates of charge. The battery is charging, discharge mode, the number of cycles, and DOD will affect the aging process of the battery [105,106]. The C-rates is to regulate the charge–discharge rates of the battery. The rated charge–discharge rate of the battery is usually 1 C, which means that the fully charged battery rated at 1 Ah can supply 1 A for 1 hour. The same battery is discharged at 0.5 C should deliver 500 mA for 2 hours, and 2 C should deliver 2 A for 30 min. Losses at accelerated discharge decrease the time of discharge, and these losses also impact the time of loading. The C-rate of 1 C is often referred to as 1-h releases such as the 2-h release of 0.5 C or C/2 and the 5-h release of 0.2 C or C/5. Any high- performance low-stress batteries can be charged and discharged above 1 C. 2.4.4 Aging degree The aging of the battery is due to the change in its internal structure, the rise of temperature, and the batteries dropping. The constant charging and discharge will lead to a change in the internal structure, influencing the aging degree [107]. 22 Battery state estimation: methods and models
- 50. During the battery aging process, the SOC value will continue to decrease until the battery fails to work normally. Generally speaking, the battery charging–discharge mode, the number of cycles, and DOD can affect the batteries aging [108]. The everyday use of batteries in HEVs implies a paradigm of accurate battery aging and battery life. The aging of batteries can be divided into two parts, including the aging of calendars and cycle one. Calendar aging refers to the endless amount of lost storage ability. In other words, battery capacity is responsible for the deterioration. Cycle aging is related to the influence of cycles (charge or discharge) for battery use times. It occurs when the battery is either loaded or unloaded. This is directly due to the level, the temperature of the usage pattern, and current battery specifications. As a result, its performance steadily deteriorates during battery life because its electrochemical components are deteriorating and the performance of EVs and fuel efficiency deteriorates. 2.4.5 Self-discharge rate When the battery is not in use, some slow chemical reactions are also taking place inside it. As the essence of battery charge–discharge is an electrochemical reaction and the electrochemical reaction inside the battery increases with the rise of tem- perature, when the temperature is high, the intensity of self-discharge inside the battery is large, which will lead to the automatic drop of SOC. 2.5 Basic state estimation strategies SOC is primarily used to describe the remaining capacity of the battery, which corresponds to the ratio of the remaining capacity and the rated capacity under the same circumstances following the batteries discharge at a given discharge rate. It is typically expressed as the percentage (%). The range of its value is 0–1. When SOC ¼ 0, this means that the battery is completely discharged. When SOC ¼ 1 is used, this means that the battery is fully charged. Traditional battery SOC predic- tion techniques include the discharge test method, conductance method, OCV method, and Ah integration method. Their foundation is relatively simple, and the implementation process is relatively simple, but the accuracy is often not high or the adaptability is not strong. New SOC prediction methods for the battery include extended Kalman filter (EKF), particle filtering (PF), fuzzy logic, and back- propagation (BP). These methods combine mathematics and computer theory, and the implementation process is complex, but they can achieve a good estimation effect and strong adaptability. 2.5.1 Discharging test The discharge test method is to discharge the target battery continuously at a constant current until reaching the cut-off voltage of the battery. The time used in the discharge process is multiplied by the discharge current, which is the capacity of the battery. This method is generally used as the calibration method of SOC Mechanism and influencing factors of lithium-ion batteries 23
- 51. estimation of battery or in the later maintenance work of battery. It is relatively simple and reliable to use this method without knowing the SOC value of the battery, and the results are relatively accurate, and it is effective for different kinds of batteries at the same time. However, the discharge experimental method has two shortcomings. First, the experiment process of this method requires a lot of time. Second, this method cannot be used to calculate the SOC of the power battery under the working state. 2.5.2 Ah integral method The Ah integral method, also known as the Coulomb counting method, is to calculate the total change of a certain charge–discharge period when the SOC basic value of a certain time is known in advance to obtain the SOC value of the battery at the special time point. Comparing the estimated value with the actual value, the error is obtained. Along with the time extension, the error increases obviously and cannot take the correct response strategy to the error accumulation problem, which cannot guarantee the stability of the test results. This approach has a certain dependence on the initial value. The Ah integral method is widely used in SOC estimation. Its goal is to focus on the external characteristics of the battery system, without considering the complex relationship between the electrochemical reaction and parameters in the battery. The principle of the Ah integral method is shown as follows: SOCt ¼ SOC0 1 C ðt 0 hIdt (2.7) where SOC0 is the initial electric quantity of the battery, SOCt is the electric quantity of the battery at time t, C represents the rated capacity of the battery. The charge– discharge current is described by parameter I, and the discharging direction is the positive direction, and h is the coulomb efficiency coefficient, which reflects the internal electric quantity dissipation of the battery during the charge–discharge process. 2.5.3 Open-circuit voltage method The OCV method estimates SOC according to the approximately linear relationship between SOC and OCV. The actual operation process is described as follows. First, the battery needs to be stationary for a long time to ensure the stability of its OCV and other parameters, and then the SOC value is obtained according to the known approximate linear function relationship. The OCV method has the advantages of a simple method and strong operability. In this way, only after the battery stops working for a while can the OCV be obtained, so it is difficult to monitor SOC in real time. The OCV refers to the battery positive and negative potential difference when the external circuit current is zero and the battery reaches equilibrium after a long time of standing. The OCV of the battery has a relatively fixed mapping relationship with SOC, as shown below: UOC ¼ F SOC ð Þ (2.8) 24 Battery state estimation: methods and models
- 52. In (2.8), the mapping relationship can be expressed in different forms, mainly divided into discrete and continuous forms. Discrete forms, such as the look-up table method, use a series of discrete isolated points to express the one-to-one correspondence between the OCV and SOC that uses a simple piecewise linear method to express the parts between points, which also exposes the coarseness of this method. The continuous form, such as the function method, is to get the OCV– SOC curve through multiple measurements. The continuous relationship expression between discrete points is obtained by function solution or curve-fitting methods. 2.5.4 Internal resistance method For the internal resistance method, there is a certain functional relationship between SOC and internal resistance. The resistance method can also be called the conductivity method. It can analyze the relationship between the conductivity or internal resistance of the battery and SOC from numerous experimental data through a long-term tracking test of the conductivity or internal resistance to predict the SOC of the battery. Since the conductance method only relies on the internal resistance of the battery to estimate SOC, the measurement accuracy of the internal resistance of the battery directly affects the accuracy of the SOC estimation. Therefore, it is necessary to ensure good contact during the measure- ment and try to make the contact resistance zero. In a certain period, the functional relationship between SOC and internal resistance is relatively stable, so the accuracy of the internal resistance method depends entirely on the accuracy of internal resistance measurement. However, the internal resistance of the battery will gradually increase along with the aging pro- cess of the battery, and the fixed function relationship between SOC and internal resistance is no longer applicable, which will affect the accurate estimation of SOC. 2.6 Kalman filtering and its extension 2.6.1 Kalman filtering The Kalman filtering (KF) method is a theory established by the state-space theory in the time domain. It regards the estimated signal as the output parameter of a stochastic linear system under the superposition of white noise. The input–output equation is given by the state and observation equations in the time domain. The KF algorithm is mainly used to estimate the linear time-invariant systems. The recursive linear minimum variance estimation method is used to repair the unob- servable state estimation error by using observable output estimation error of the system, which greatly reduces the noise interference in the data stream and improves the estimation accuracy of the new system. The KF method constructs a set of recursive equations that can describe the char- acteristics of the battery system and carry out the recursive operation. Then, the system state-space expression can be obtained, including signal and noise. SOC is one of the internal states in which a mathematical method is used that is based on the estimation Mechanism and influencing factors of lithium-ion batteries 25
- 53. result of the previous step and the existing measurement data through the optimal regression data processing to get the current optimal estimation results. The advantage of this method is that the dynamic SOC value can be measured with high accuracy, but the KF method requires the high accuracy of the battery model and complex operation. 2.6.2 Extended Kalman filtering The EKF algorithm is used to estimate the nonlinear system, linearize the nonlinear state-space model, and then use the basic KF algorithm to achieve. The nonlinear system is linearized by expanding the state-space equation by the Taylor series and discarding the higher-order terms. EKF is a mathematical method combined with probability theory. The idea is to estimate the state of the system optimally based on the minimum variance. Its basic principle is described as follows. Combining the state-space model of signal and noise, the state variables are recursively pre- dicted according to the state equation, and the current observation value is used for their correction and update. The real-time state variable estimation is realized through the continuous “prediction-update” process. When the KF algorithm estimates the SOC value of the battery, it uses the Ah integration method to calculate the SOC value and uses the measured voltage value combined with the observation equation to continuously modify the SOC value obtained by the Ah integration method. Because the battery is a strong nonlinear system in the working process, and the application object of KF is a linear system, it is necessary to linearize the battery to adapt to KF, that is, the EKF algorithm. When it is used to estimate SOC of the battery, SOC is a component of the state vector, the current is used as control quantity in the input parameter, the output is terminal voltage calculated by the equivalent model, system noise and observation noise are Gauss white noise, and their variance is expressed as Q and R. This method is usually based on the state and measurement equation of the system. The prediction state equation includes the Ah integral method of SOC calculation, and the observation equation represents the equivalent model of the lithium-ion batteries. The accuracy of estimating SOC by the EKF algorithm largely depends on the accuracy of the equivalent model, so it is very important to establish an appropriate equivalent model for the battery. 2.6.3 Unscented Kalman filtering The unscented Kalman filter (UKF) abandons the traditional method of nonlinear functional linearization, adopts the Kalman linear filter framework, and for one- step prediction equations, unscented transform (UT) is used to process the non- linear transfer of mean and covariance. In this algorithm, the UKF uses a fixed number of points to approximate the probability distribution of nonlinear functions. The algorithm does not ignore the errors caused by higher-order terms and does not calculate the Jacobian matrix repeatedly, which reduces computational complexity. The key of this algorithm is UT treatment, in which how to sample has an important impact on the estimation effect. The common sampling methods include symmetric sampling, minimum skewness simplex sampling, and hypersphere 26 Battery state estimation: methods and models
- 54. simplex sampling. The sampling method is generally selected for convenient cal- culation and a good effect. The UKF algorithm approximates the probability den- sity distribution of the nonlinear function. It uses a series of samples to approximate the posterior probability density of the state, instead of approximating the nonlinear function. It does not ignore the high-order terms, so it has higher calculation accuracy for the statistics of nonlinear distribution. 2.6.4 Dual Kalman filtering The essence of SOC estimation using the KF is to combine the OCV method and the Ah integration method, which is used to estimate the system state with the Ah inte- gration method and its feedback with the measurable voltage. Since the KF algorithm largely relies on the accuracy of the equivalent model when estimating SOC, the idea of the dual Kalman filter algorithm is to estimate the equivalent model parameters and system state variables alternately with two KF lines and then adjust their feed- back according to the measurable voltage to obtain more accurate estimation results. In the first stage of Kalman filtering, the internal resistance of the battery model parameter is taken as the state quantity, and after the time update, the internal resistance is measured and updated according to the difference between the actual output voltage and the model output voltage. In the second stage of Kalman filtering, SOC and polarization voltage Up can be used as two-dimensional state vectors for time update and measurement update. 2.6.5 Adaptive extended Kalman filtering To improve the accuracy of the SOC estimation of the lithium-ion battery, the adaptive extended Kalman filtering (AEKF) algorithm is used to estimate the sta- tistical characteristics of noise online based on the improved models. Comparatively, the SOC estimation accuracy of the AEKF algorithm is significantly higher than that of the EKF algorithm, which effectively reduces the noise interference in the SOC estimation process, and has certain reliability and practicability. Although it has higher estimation accuracy and stronger robustness and stability than KF, EKF, and UKF algorithms, it is also based on accurate mathematical models and statistical characteristics of system process noise and observation noise. When the environment around the carrier changes or the motion state changes drastically, the statistical characteristics of process noise and observation noise of the system will change greatly. At this time, the accuracy and stability of the conventional UKF will decrease significantly. In the SOC estimation process of the battery by its standard version, when the working current of the battery changes rapidly with time, the negative determination of the covariance may be encountered in the later stage of the operation. 2.6.6 Square root-unscented Kalman filtering During the calculation process, the covariance Pk of the state variable SOC becomes negative, while the Cholesky decomposition requires that the matrix must be semi- positive qualitative, otherwise the algorithm cannot continue, making the filter invalid because of rounding errors in the numerical calculation. To solve this problem, a new Mechanism and influencing factors of lithium-ion batteries 27
- 55. Another Random Scribd Document with Unrelated Content
- 56. bear-hunt; and the fourth a thank-offering to the goddess of hunting. On the side facing the Colosseum, a bear-hunt, a sacrifice to Apollo, a group contemplating a dead lion, and lastly a consultation of an oracle. Most of these refer to Trajan; we think some refer to Hadrian, because on one of them Antinoüs is represented. On the inside of the arch is a battle-piece, assigned to Constantine by the inscriptions, To the founder of peace, To the deliverer of the city. They are older than his time. Over the side arches are some narrow reliefs referring to Constantine, one of which is peculiarly interesting, as it represents that emperor addressing the people from the Rostra ad Palmam, with some of the principal monuments in the Forum in the background. THE COLOSSEUM. A noble wreck in ruinous perfection.—Byron. The vast amphitheatre erected in the centre of ancient Rome by Vespasian was known to the ancient Romans as the Flavian Amphitheatre. It was begun by the Flavian emperors A.D. 72, and dedicated A.D. 80. It is 157 feet high, and is 1900 feet in circumference, and was built by the captive Jews after the fall of Jerusalem. Originally the upper story was of wood, but this was burned down, and it was rebuilt with travertine stone like the rest of the edifice. Martial tells us that its site was formerly occupied by the artificial lakes of Nero; and Marcellinus (xvi. x. 14) says, The vast masses of the amphitheatre so solidly erected of Tiburtine stone, to the top of which human vision can scarcely reach. All the brickwork we now see are repairs at various dates after the dedication; but there is enough travertine left at different points to show that it was originally built of this stone, as recorded by the historian. For nearly five hundred years it was the popular resort of the Roman populace and their betters. There were eighty arches of entrance, and it held one hundred thousand people, and could be emptied in ten minutes; such were the order kept and regulations observed that there was
- 57. no confusion. It was devoted to the exhibition of wild beasts, their fighting together, gladiators fighting together, or with beasts, and naval fights. On these latter displays the stage or arena was moved, water let in, and naval fights represented in real earnest. Suetonius (Vespasian, vii.), says, He began an amphitheatre in the middle of the city, upon finding that Augustus had projected such a work. Ibid. (Titus, vi.): He entertained the people with most magnificent spectacles, and in one day brought into the amphitheatre five thousand wild beasts of all kinds. The last display was given by Theodoric in 523; and in 555 the lower part was destroyed by a flood from the Tiber, when the whole of Rome was under water for seven days. From then we must date the ruin of the Flavian Amphitheatre—the Romans themselves hastening on the work, using the material for building purposes. Which on its public shows unpeopled Rome, And held uncrowded nations in its womb.—Juvenal. It is held by the Roman Church, on the authority of an inscription found in the Catacombs, that the architect of the Colosseum was one Gaudentius; but that inscription only says that he was employed there. We believe the architect to have been Aterius, whose monument is now in the Lateran, and upon which several buildings are represented of which he was no doubt the architect, also the machine used to raise the stones into their places. He flourished at the end of the first century, and, no doubt, these buildings shown in relief upon his tomb were erected by him, the dates agreeing; for if not, why should they be there represented? First, we have an arch which says on it, Arcus ad Isis. Now if we compare this with the Arch of Constantine, we find it is the same without the attic. Then we have the amphitheatre without the upper story; then an arch (query, Arch of Domitian?). Then another arch with the words, Arcus in Sacra Via Summa: compare this with the Arch of Titus, and, minus the restorations, it will be found to be the
- 58. same. Then there is a temple agreeing with the descriptions of that of Jupiter Stator upon the Palatine. All these buildings were erected or rebuilt about this time, and from being recorded on this monument of the Aterii, tend to show that Aterius was the architect of them. When perfect, the Colosseum consisted of four stories—the lowest, of the Doric order, 30 feet high; the second, Ionic, 38 feet high; the third, Corinthian, about the same height; and the fourth, also Corinthian, 44 feet high. The holes in the cornice with the corbels below them were to receive the masts that supported the velaria on the outside. The numerous holes in the stone were made in the middle ages for the purpose of extracting the iron cramps that kept the stones from shifting. The long diameter is 658 feet, the shorter 558 feet; the arena is 298 feet by 177 in its widest part. The last performance was a bull-fight, held at the expense of the Roman nobles, in the year 1332. Many martyrs are said to have perished in the Colosseum during the persecutions of the early Christians, and among others S. Ignatius, who was brought from Antioch to be devoured by wild beasts. Benedict XIV. consecrated the building to the Christian martyrs, A.D. 1750. In excavating the Basilica of S. Clement, the Rev. Father Mullooly found (1870) the remains of S. Ignatius, and had them carried with great ceremony over the scene on the anniversary of his martyrdom. At the present day there remains sufficient to indicate the construction of the building, though but a small portion of the immense outer shell, which originally both adorned and formed an impenetrable girdle round the whole, has been preserved. In the interior, a great deal of rebuilding has been necessary for its preservation. Vast as the building is, its construction is easily understood; a simple segment of the whole serving to show how all the others succeed
- 59. one another like the cells of a bee-hive. THE COLOSSEUM. View larger image. The upper part was originally of wood only, and was burned, having been set on fire by lightning. The three lower stories only are of the time of the Flavian emperors; the upper story was rebuilt and added in the third century, and only finished in the time of the Gordiani, as is shown by the coins representing it. The imperial entrance was from the Esquiline side, between the arches Nos. 38, 39, which is without number. Commodus constructed an underground passage from the arena to the Palatine, which has not yet been discovered, his so-called passage (on the right in entering) being that by which the dead bodies were carried from the arena. Dion Cassius says: Upon the last day of the sports his helmet was taken off and fell through the door where the dead used to be carried out. The area, basement, or ground-floor, was flooded for the naval fights. Surrounding this were the dens, in front of which was a channel for fresh water for supplying the animals with drink—a
- 60. spring still supplies it; about ten feet above was the movable stage, sprinkled with sand for the combats, and hence called the arena. A few feet above the arena was the podium, or seat of the emperor, vestal virgins, c., protected from the arena by iron bars. Behind the podium was a double portico, which ran round the whole building. Fragments of the marble chimeras, with long wings, that ornamented the seats of the podium have been found. The three successive tiers were called cavea. Above these was a tier for the people; above this one for the gods; thus making six in all. The amphitheatre seated eighty-seven thousand people, and there was standing room for thirteen thousand more. The walls standing upon the area, composed of tufa, travertine, and brick, old material re-used, were built at a period long after the building was dedicated, when the naval fights being abandoned there was no longer any occasion for a movable stage or arena as before. They contained the machinery for the stage above, and for the lifts or pegmata to send men or beasts from the area to the arena. Probably these are the walls thus alluded to by Dion Cassius: He [Commodus] divided the theatre into four parts by two partitions that cut through diametrically, and by right angles, to the end that from the galleries that were round about he might with greater ease single out the beasts he aimed at.
- 61. PLAN OF THE EXCAVATIONS BELOW THE ARENA OF THE COLOSSEUM. View larger image. The emperor having employed himself in shooting from above ... descended afterwards to the bottom of the theatre, and there slew some other private beasts, whereof some made toward him, others were brought to him, and others were shut up in dens. Returning after dinner, he used the exercises of a gladiator, with a shield in his right hand, and in his left a wooden sword. After him fought those whom he had chosen in the morning at the bottom of the theatre.
- 62. Also, in his life of Septimius Severus, he says: There was a kind of cloister made in the amphitheatre, in the form of a ship, to receive them [the wild beasts]. On a sudden there issued out bears, lions, ostriches, wild asses, and foreign bulls. The walls before us are of very bad construction, evidently repairs of a late date: they are the work of either Lampridius, prefect of Rome under Valentinian III., 425–455, who repaired the steps and renewed the arena; or of Basilius, who restored the podium and arena after their destruction by an earthquake in 486—this we learn from two inscriptions standing at the entrance. Half way, on each side, two large passages have been discovered choked up with mud: they were the aqueducts to bring the water for the naumachiæ from the reservoirs upon the Esquiline and Cælian Hills respectively; from the small openings in the blind arches the water also poured out over the top of the dens, thus forming cascades all round. At the end opposite the present entrance a long passage has been opened, above the level of the area floor; below this passage is the great drain, with the remains of the iron grating[6] to prevent large objects going down: this and the passage were closed by flood-gates on naval representations, which can be clearly seen in the construction. On the right and left of this passage, connected with it, but at a lower level, two dens have been cleared out, 27 yards long by 5 wide, containing six holes in the floor, in the centre of square blocks of stone, and these holes are faced with bronze, evidently the sockets into which metal posts were fixed to which the beasts were chained. On the fragments depicting scenes from the arena, the animals are shown with a long piece of rope or chain dangling from their necks, which seems to bear out our idea that they were attached to posts fixed in these sockets, and that as they were wanted the chain or rope was cut, and they were free to rush upon the arena. The corbels round the front of the line of arches under the podium are in pairs, and between them the masts were inserted to support the awning on the inside, as the holes and corbels supported the
- 63. masts on the outside; for we find on examination that those inside are exactly in a line with those outside at the top of the building. These corbels are 29 inches deep, and from them to the level of the area is 10 feet, and to the present surface 11 feet; between each pair of corbels are chases 19½ inches wide, ending on a block of travertine for the masts to rest on, the chases coming down 1½ yards below the corbels, which are 15 feet apart. They probably helped to support the arena, and show what the height of this wooden arena must have been, and that from its vast size it must have had a framework and supports: the numerous holes on the area, in travertine, were for the heels of the supports; one of these, a square one, has remains of the decayed timber in it. In the central passage, resting on the area and extending as far as the excavations, is an ancient wooden framework in a decomposed state. Various suggestions have been made as to its use,—we suppose it to be the framework and joists of the flooring covering the central passage; others, a sort of tramway for running the cages along,—but till the whole space has been cleared out it is impossible to arrive at a correct estimate of its use. Honorius, A.D. 404, having abolished the gladiatorial combats, probably the last display of wild beasts was that given by King Theodoric at the beginning of the sixth century. The soil cleared out in the passage, dens, galleries, and area was found to be composed of mud deposited during a flood or floods by the Tiber, the composition of which may still be seen in parts of the long passage not yet cleared. The most remarkable of these floods, which lasted some days and did immense damage to the city, were those of A.D. 555, 590, 725, 778, 1476, 1530, 1557, and 1598. We may presume, from the nature of the soil, that at some early date, probably A.D. 555, one of those terrible floods reached the Colosseum, and on the waters retiring a great deposit of mud was left, covering the old area floor and filling up the various passages and galleries, and that the authorities, instead of clearing out this
- 64. deposit, added to it to make a solid floor, and used the arena above; for after that date we have no record of its being used, with the exception of the bull-fight. By applying to the custodian, the visitor can ascend to the top, where a most magnificent view is enjoyed, the only way to get a good idea of its size and oval shape, and where the construction of the upper galleries can be studied. It will be seen that the arches forming the tiers of seats have at some date been filled in with brickwork, of the time of Alexander Severus and the Gordiani. The water-courses for keeping the building cool in hot weather can also be traced. The highest wall of all, the inside brick casing of which is partly gone, is built of fragments evidently not originally intended for the purpose for which they are used, corresponding to a great extent with the construction of the walls upon the area. The Colosseum was for a long time used as a quarry, from which several of the palaces in Rome were built. Should the visitor be fortunate enough to see the ruin under moonlight, or when it is illuminated with Bengal lights, he will see it in its grandeur, for it will not bear the brightness of the day.
- 65. SECTION OF SEATS AND ARCHES OF THE COLOSSEUM. View larger image.
- 66. RAMBLE II. THE BRIDGE AND CASTLE OF S. ANGELO—THE TOMB OF HADRIAN—S. PETER'S— THE SACRISTY—THE CRYPT—THE DOME—THE VATICAN—SCALA REGIA— SISTINE AND PAULINE CHAPELS—STANZE AND LOGGIE OF RAPHAEL—THE PICTURE GALLERY—THE MOSAIC MANUFACTORY—THE MUSEUM OF SCULPTURE—THE INQUISITION—PORTA S. SPIRITO—S. ONOFRIO AND TASSO'S TOMB—MUSEUM TIBERINO—THE CORSINI AND FARNESINA PALACES—PORTA SETTIMIANA—VIA GARIBALDI—S. PIETRO IN MONTORIO— PAULINE FOUNTAIN—VILLA PAMPHILI DORIA—S. CECILIA IN TRASTEVERE— CHURCH OF S. CRISOGONO—STAZIONE VII COHORTI DEI VIGILI—CHURCH OF S. MARIA IN TRASTEVERE—PONTE SISTO—FARNESE AND CANCELLERIA PALACES—STATUE OF PASQUINO—CHIESA NUOVA—CIRCO AGONALE— OBELISK—S. AGNESE—S. MARIA DELLA PACE—S. AGOSTINO. IN TRASTEVERE. (Over the Tiber.) THE ROUTE. From the Piazza del Popolo the Via Ripetta leads towards S. Peter's, turning off to the right, past the bridge, by the Via Monte Brianzo. From the Piazza di Spagna we take the Via Condotti to the Via Monte Brianzo and Tor di Nona. At the right-hand end of the latter street is the Apollo Theatre, built on the site of the Tor di Nona prison, where Beatrice Cenci was confined. Passing into the Piazza Ponte S. Angelo, on our left, is the
- 67. Italian Free Church of Gavazzi, and in the Palazzo Altoviti, in front, lived Visconti. We turn to the right over THE BRIDGE OF S. ANGELO, (Ponte S. Angelo,) which is decorated with ten angels standing on the parapet, bearing the instruments of our Lord's passion; and SS. Peter and Paul, an addition made in 1668 by Clement IX. It is the finest bridge in Rome, and was built by Hadrian. TOMB OF HADRIAN, NOW THE CASTLE OF S. ANGELO. (Castel S. Angelo. Permissions required: see page 353.) It was covered with white Paros marble, and decorated with statues of the gods and heroes, the works of Praxiteles and Lysippus, which were hurled upon the heads of the Goths. Erected by Hadrian, A.D. 130. The porphyry sarcophagus, which is supposed to have contained his remains, is now used as the font in the chapel on the left in S. Peter's. Procopius thus describes it: The tomb of the Emperor Hadrian is situated outside the Porta Aurelia. It is built of Parian marble, and the blocks fit close to one another without anything to bind them. It has four equal sides, about a stone-throw in length; its altitude rises above the city walls; on the top are statues of the same kind of marble, admirable figures of men and horses. Lucius Verus, Antoninus Pius, Marcus Aurelius, Commodus, were all buried here. It was first turned into a fortress A.D. 423. Popes John XXIII. and Urban VIII. built the covered way connecting it with the Vatican. One of the barrack-rooms contains frescoes by Pierino del
- 68. Vaga and Sicciolante, another by Giulio Romano. A circular room, surrounded with carved wood cases, once contained the archives of the Vatican. A large iron-bound chest contained the treasury. Some dark cells built in the thickness of the walls are shown as the prisons of Beatrice Cenci (?), Cellini, Cagliostro, and others. Tradition asserts that Gregory the Great saw S. Michael standing over the fortress sheathing his sword as a sign that a pestilence was stayed; to commemorate which the castle is now surmounted by a figure of the archangel in the act of sheathing his sword. This old castle served for a fortress during several ages, and its first cannon were cast out of part of the bronze taken from the roof of the Pantheon. The Borgo Nuovo leads to the Cathedral, passing, on the right, the Church of S. Maria, built on the site of a pyramid to Honorius, 423 A.D., which is represented on the doors of S. Peter's. S. PETER'S. (S. Pietro.) EXTERIOR. Before the era of railways, the traveller in approaching Rome, across the Campagna, was generally electrified by the first glimpse of S. Peter's dome looming in the distance. Then he had full time, in advance of entering the gates of the city, to ponder over all the recollections which the magical word Roma might suggest to him. At present he is rapidly borne into the city, and sometimes before he is aware of having arrived even in its neighbourhood; yet the dome is plainly visible from afar by the railway approach of to-day. Now, as then, the first sight of Rome is always her unequalled cathedral; now, as then, the latter is the great object which the tourist eagerly hastens to visit. The present Church of S. Peter is relatively modern, having been first conceived by Pope Nicholas V. about the year 1450. It is built upon the site of the religious edifice erected in the time of
- 69. Constantine, and consecrated as the Basilica of S. Peter. The old basilica stood on part of the Circus of Nero, and occupies the spot consecrated by the blood of the martyrs slaughtered by order of that tyrant. Tradition supposes that the basilica held possession of the body of the apostle after his crucifixion,—a circumstance which reflected high credit upon it, and dignified its entrance with the appellation of the limina apostolorum (threshold of the apostles). After enjoying the veneration and tributes of all Christendom during eleven centuries, the walls of the old basilica began to give way, and its approaching ruin becoming visible about the year above stated, Nicholas V. conceived the project of taking down the old church, and erecting in its stead a new and more expensive structure. The project was begun, and resulted, after a long series of experiments made by various architects, in the splendid fabric which is now regarded by the world as the chief glory of modern Rome. The work made slight progress until the epoch of Julius II., who resumed the great task, and found in Bramante an architect capable of comprehending and executing his grandest conceptions. The walls of the ancient basilica were then wholly removed, and on the 18th of April 1508 the foundation stone of one of the vast pillars supporting the dome, as we now see it, was laid by Julius with great pomp and ceremony. From that period the work, though carried on with ardour and perseverance, continued during one hundred years to occupy the attention and absorb much of the incomes of eighteen pontiffs. The most celebrated architects of the times displayed their talents in its erection—namely, Bramante, Raphael, San Gallo, Michael Angelo, Vignola, Carlo Maderno, and last, though not least, Bernini, who gave it the finishing touches of ornamentation, and who built the enclosing colonnade. It is estimated that its cost, after completion, was no less than £12,000,000 sterling—a sum representing a far greater value than it does in our day. Colossal statues of Peter and Paul, erected by Pius IX., guard the approach at the foot of the steps on either side. Eustace says: Entering the piazza, the visitor views four rows of lofty pillars, 70 feet high, sweeping off to the right and left in a bold
- 70. semicircle. ('A tabernacle for a shadow in the day-time from the heat, and for a place of refuge, and for a covert from storm and from rain,' Isa. iv. 6.) In the centre of the area formed by this immense colonnade, an Egyptian obelisk, of one solid piece of granite, ascends to the height of 130 feet; two perpetual fountains, one on each side, play in the air, and fall in sheets round the basins of porphyry that receive them. Raised on three successive flights of marble steps, extending 379 feet in length, and towering to the elevation of 148, you see the majestic front of the basilica itself. This front is supported by a single row of Corinthian pillars and pilasters, and adorned with an attic, a balustrade, and thirteen colossal statues. Far behind and above it rises the matchless dome. Two smaller cupolas, one on each side, add not a little to the majesty of the principal dome. Five lofty portals open into the vestibule; it is 468 feet in length, 66 in height, and 50 in breadth, paved with variegated marble, covered with a gilt vault, adorned with pillars, pilasters, mosaic, and bas- reliefs, and terminated at both ends by equestrian statues, one of Constantine, the other of Charlemagne. THE OBELISK is the only one near its original site, the Spina of Nero's Circus, which was near the Sacristy, on the left of S. Peter's. An inscription in the pavement marks the place. Pliny (xxxvi. 14), says: The third obelisk at Rome is in the Vatican Circus, which was constructed by the emperors Caius [Caligula] and Nero; this being the only one of them all that has been broken in the carriage. Nuncorcus, the son of Sesoses, made it [the original, this is probably a copy], and there remains [in Egypt] another by him, 100 cubits in height, which, by order of an oracle, he consecrated to the sun, after having lost his sight and recovered it. Herodotus says: It was dedicated by Phero, son of Sesostris, in gratitude for his recovery from blindness. It has no hieroglyphics, so if this was the original how could they know who erected it? but it bears this inscription of Caligula—
- 71. DIVO. CAES. DIVI. JULII. F. AUGUSTO.—TI. CAESARI. DIVI. AUG. F.—AUGUSTO. SACRUM. [To the divine Augustus, son of the divine Julius, and to the divine Tiberius, son of the divine Augustus.] S. PETER'S AND THE VATICAN. View larger image. The Nuncorcus of Pliny is supposed to stand for Menophtheus, the king Meneph-Pthah. Pliny (xvi. 76) gives the following particulars of how it was brought over:— A fir tree of prodigious size was used in the vessel which, by the command of Caligula, brought the obelisk from Egypt, which stands in the Vatican Circus, and four blocks of the same sort of stone to support it. Nothing certainly ever appeared on the sea more astonishing than this vessel; 120,000 bushels of lentils served for its ballast; the length of it nearly equalled all the left side of the port of Ostia—for it was sent there by the Emperor Claudius. The thickness
- 72. of the tree was as much as four men could embrace with their arms. Suetonius (Claudius, xx.) says: He sank the vessel in which the great obelisk had been brought from Egypt, to secure the foundation of the mole at Ostia. Pliny (xvi. 76), says: As to the one in which, by order of the Emperor Caius, the other obelisk had been transported to Rome, it was brought to Ostia, by order of the late Emperor Claudius, and sunk for the construction of his harbour. Marcellinus says: Subsequent ages to Augustus brought also other obelisks, one of which is in the Vatican. VESTIBULE. Over the entrance outside is a relief of Christ giving the keys to Peter; inside the vestibule is Giotto's (1298) celebrated mosaic, representing our Lord sustaining Peter when he was about to sink whilst walking on the sea. Opposite are the great bronze doors, opened only on special occasions, the work of Antonio Filareto and Simone Donatello in the fifteenth century. The upper panels represent in relief our Saviour and the Virgin, below whom are SS. Peter and Paul; Peter is giving the keys to Pope Eugenius IV. Beneath are the martyrdoms of Peter and Paul: in the former is represented the pyramidal tomb which stood in the Borgo Nuovo, and which was destroyed by Alexander VI. The smaller reliefs represent scenes from the life of the Emperor Sigismund—his coronation, the council of Florence, and his entry into Rome. The framework represents satyrs, nymphs, fauns, Leda and the Swan, Ganymede, the Fox and the Stork, with reliefs of fruit and flowers, and medallions of Roman emperors. The walled-up side door, on the right, is the Porta Santa, which was formerly opened on Christmas- eve of the years of jubilee—every twenty-fifth year.
- 73. The first inscription relates the gift of olive-yards to provide oil for the lamps given by Gregory II. The second, the Bull of Boniface VIII., of the indulgence granted at jubilee. The third, Panegyric of Charlemagne on Pope Adrian I. INTERIOR. Five portals give access to the edifice, which faces east. Enter, its grandeur overwhelms thee not.—Byron. The most extensive hall ever constructed by human art expands in magnificent perspective before you. Advancing up the nave, you admire the beauty of the variegated marble under your feet, and the splendour of the golden vault overhead, the lofty Corinthian pilasters with their bold entablature, the intermediate niches with their statues, the arcades with the graceful figures that recline on the curves of their arches. But how great your astonishment when you reach the foot of the altar, and, standing in the centre of the church, contemplate the four superb vistas that open around you; and then raise your eyes to the dome, at the prodigious elevation of 440 feet, extended like a firmament over your head, and presenting, in glowing mosaic, the companies of the just and the choirs of celestial spirits.... Around the dome rise four other cupolas, small, indeed, when compared with its stupendous magnitude, but of great boldness when considered separately; six more, three on either side, cover the different divisions of the aisles; and six more of greater dimensions canopy as many chapels. All these inferior cupolas are, like the grand dome itself, lined with mosaics. Many, indeed, of the masterpieces of painting which formerly graced this edifice have been removed [to the Church of S. Maria degli Angeli, see page 265 ], and replaced by mosaics, which retain all the tints and beauties of
- 74. the originals, impressed on a more solid and durable substance. The aisles and altars are adorned with numberless antique pillars that border the churches all around, and form a secondary order (Eustace). The variegated walls are in many places ornamented with festoons, wreaths, crosses, and medallions representing the effigies of different pontiffs. Various monuments rise in different parts of the church, of exquisite sculpture, and form very conspicuous features in the ornament of this grand temple. Below the steps of the altar, and, of course, some distance from it, at the corners, on four massive pedestals, four twisted pillars, 50 feet in height, rise and support an entablature, which bears the canopy itself, topped with a cross. The whole is 95¼ feet from the pavement. This brazen edifice—for so it may be called—was constructed of bronze stripped from the dome of the Pantheon, and is so disposed as not to obstruct the view by concealing the chancel and veiling the chair of S. Peter. This ornament is also of bronze, and consists of a group of four gigantic figures, representing the four principal doctors of the Greek and Latin Churches, supporting the chair at an elevation of 70 feet. Under the high altar of S. Peter's is the tomb of that apostle, the descent to which is in front, where a large open space leaves room for a double flight of steps. The rails that surround this space above are adorned with one hundred and twelve bronze cornucopiæ, which support as many silver lamps, burning during the day in honour of the apostle. Upon the pavement of the small area enclosed by the balustrade is the kneeling statue of Pius VI., by Canova. DIMENSIONS. Interior. 613½ feet long. 152½ feet, height of Nave. 87½ feet, width of Nave.
- 75. 33¾ feet, width of Aisles. 197¾ feet, width of Basilica. 446½ feet, length of Transepts. 95¼ feet, height of Baldacchino complete. 139 feet Cupola, interior diameter. 179 feet Cupola high. 277 feet above Floor. 440 feet from Pavement to Base of Lantern. Area. 240,000 square feet. INTERIOR OF S. PETER'S. View larger image. A PROMENADE IN S. PETER'S. On entering, the size of objects may be judged by noticing the cherubs that support the holy water basins; they present no
- 76. extraordinary appearance, but stand by them and their immense size will be appreciated. The first chapel, on the right, contains Michael Angelo's Mary with the Dead Christ; hence it is called the Chapel of La Pietà. It was executed by the great master when only twenty- four, and bears his name across Mary's girdle. This work of art is unfortunately very badly placed for proper observation. Opening out of this chapel are two side chapels, kept closed: in that of the left are kept the relics belonging to the basilica; and in the right, a column, ornamented with flutings and reliefs, and said to be the column against which Jesus leaned when disputing with the doctors. Proceeding up the aisle, on the right, is Fabris's statue of Leo XII.; and opposite, Carlo Fontana's monument to Christina, Queen of Sweden, who died in Rome in 1689, after her abjuration of Protestantism. The chapel beyond contains a beautiful mosaic copy of the Martyrdom of S. Sebastian; the original was by Domenichino. Next is the monument to Innocent XII., supported by Charity and Justice, by Filippo Valle; and opposite is one to the Countess Matilda, by Bernini; the relief is Gregory VII. giving absolution to Henry IV. The Chapel of the Sacrament contains, above the altar, a fresco by Cortona; over the side-altar is a mosaic copy of Caravaggio's Entombment. The principal altar is formed with a model in lapis lazuli and gilt bronze of Bramante's chapel; the original is erected over the spot pointed out as the scene of Peter's martyrdom. Before the side-altar is the bronze tomb of Sixtus IV., with reliefs by Antonio del Pollajuolo; near by is interred Julius II., whose monument, now in S. Pietro in Vincoli, was to have been the grand masterpiece of Michael Angelo. Beyond, on the right, is the monument to Gregory XIII., supported by Religion and Power, with a relief representing the correction of the calendar, the work of Rusconi. Opposite is Gregory the Fourteenth's simple marble urn. The next chapel is named Madonna del Soccorso, containing the monument to Gregory XVI., erected by the cardinals he had made.
- 77. On the left is a mosaic copy of Domenichino's Last Communion of S. Jerome. In the aisle, proceeding on the right, is the monument to Benedict XIV. (with figures of Science and Charity), by Pietro Bracci. Opposite is a mosaic copy of S. Basil Celebrating Mass before the Emperor Valens, after Subleyra's picture. In the transept are mosaic copies of S. Wenceslaus, king of Bohemia, by Caroselli; Martyrdom of SS. Processus and Martinianus, after Valentin; and that of Erasmus, after Poussin. In the aisle, leading out, is Canova's celebrated tomb of Clement XIII. It took eight years to execute. The pope is represented praying: on one side is the genius of Death with inverted torch (the finest piece of sculpture in S. Peter's), and on the other Religion with the cross; at the angles are a wakeful and a sleeping lion. Opposite is a mosaic of S. Peter Walking on the Sea, after Lanfranco. In the next chapel is a mosaic of Guido's S. Michael and Guercino's S. Petronilla. On the left, coming towards the apse, S. Peter Resuscitating Tabitha, from Costanzi's painting; and opposite is the tomb of Clement X., by Ferrata. In the centre of the apse is S. Peter's chair. January 18th is the feast of the chair of S. Peter in Rome. Some remarks on the chair which does duty for S. Peter's may be of interest to our readers. A photograph of this famous object was taken in 1867, when it was last exposed to view, and can be had at any of the shops in Rome. Visitors must be content with looking at the photograph, for the chair itself is not to be seen. At present it is enclosed in the bronze covering which is supported by the four colossal figures of the doctors of the Church—SS. Gregory, Jerome, Ambrose, and Augustin. It is encased in a framework, in which are the rings through which the poles were inserted in order to carry the person seated. This casing, consisting of four posts and sides, is made of oak, and is very much decayed. The straight vertical joints are easily distinguished where the frame is attached to the chair itself, which is composed of dark acacia wood. The front panel is ornamented with
- 78. three rows of square plates of ivory, six in a row, eighteen in all, upon twelve of which are engraved the labours of Hercules, and on the other six, constellations, with thin laminæ of gold let into the engraved lines. Some of the ivories are put on upside down, and had evidently nothing to do with the original chair: they are Byzantine in style, of the eleventh century. The ivory band decorations of the back and sides evidently belonged to the chair, and correspond with its architecture and fit into the woodwork. They are sculptured in relief, representing combats of men, wild beasts, and centaurs. The centre point of the horizontal bars has a portrait of Charlemagne crowned as emperor. In his right hand is a sceptre (broken), and in his left a globe; two angels on either side offer him crowns and palms, they having combatants on each side. The chair is 4 feet 8¾ inches high at back, 2 feet 10½ inches wide, 2 feet 2⅓ inches deep, and 2 feet 1½ inch high in front. Fancy Peter using such a chair as this! It is asserted by the Roman Church that this chair was used by S. Peter as his episcopal throne during his rule over the Church at Rome. Even if we grant, for argument's sake, that he was bishop in Rome, there is no evidence to prove that this was his chair; in fact, every evidence to the contrary. All the primitive episcopal chairs are of marble, and as unlike this one in construction as possible; for it is not an episcopal throne, but a sella gestatoria or cathedra, similar to the chairs introduced into Rome in the time of the Emperor Claudius, mentioned by Suetonius (Nero, xxvi.), and Juvenal (i. 64, vi. 90). It is not unlike in shape the one used to carry the Pope in grand ceremonies in S. Peter's. Some early authors speak of a sella gestatoria which was placed in the baptistery of old S. Peter's by Damasus, and which, formerly on the 22nd of February, was carried hence to the high altar, where the Pope, with much ceremony, was enthroned upon it. The chair which was originally assigned as that of S. Peter was eventually passed on from one chapel to another, till, it is said, that, when Rome was sacked by the imperialists in 1527, they stripped it
- 79. of its ornaments and covering, for the sake of their value; and that beneath they found an old carved wooden chair, with the inscription, There is only one God, and Mohammed is his prophet—which same formula is engraved upon the back of the marble episcopal chair in the Church of S. Pietro in Castello at Venice. In 1558, the feast of the chair of S. Peter was fixed in Rome for the 18th of January, and in Antioch for February 22nd; and in 1655 Pope Alexander VII. placed this chair where it now stands. The present chair is medieval, ninth century, and is unlike early representations in art of the chair used by the Apostle Paul, which we may look upon as episcopal. The ivory diptych of St. Paul (A.D. 400), the property of Mr. Carrand of Lyons, engraved by the Arundel Society, represents Paul seated on a chair, holding in his left hand a roll, the symbol of apostleship, whilst the right hand is raised in the act of blessing Linus, who carries a book in his hand. At the back of the chair is S. Mark, holding a roll in his left hand. The chair is light, and not unlike a modern library one in shape. Later art agrees with the present chair. A fresco at S. Clement's, Rome (1050), represents Peter installing Clement into the Papal chair—a chair, so far as can be seen, not unlike the present one of S. Peter, which was made after the coronation of Charlemagne as Emperor of the Holy Roman Empire (A.D. 800). Upon our right is the tomb of Urban VIII. His bronze statue is by Bernini, with Justice and Charity in marble. On our left is Della Porta's monument to Paul III.; likewise a bronze figure, with Prudence (the Pope's mother, Giovanna Gaetani) and Justice (his sister, Giulia Farnese). Justice is a beautiful figure, but the tin drapery put on to cover its nakedness by Bernini destroys its beauty. It is necessary to re-paint the tin every now and then. There is a deal of this mock modesty in S. Peter's. Turning into the south aisle, on our right, is the tomb of Alexander VIII. The bronze statue is by Arrigo, and the figures of Religion and Prudence by Rossi. The relief represents the Pope canonizing five
- 80. saints. Opposite is the mosaic of S. Peter at the Gate of the Temple. It is said that this scene, here represented, gave to President Lincoln the idea for his proposed motto for the greenbacks. When the commission applied to him for a motto to put upon the notes, he said, I can think of nothing better than what Peter said to the sick man at the gate of the temple—'Silver and gold have I none, but what I have that give I unto thee.' Beyond, upon the right, is a splendid alto-relief by Algardi, representing Leo threatening Attila with the vengeance of Peter and Paul if he should attack the holy city of Rome. It is the largest relief ever executed. A circular marble slab below it marks the tomb of Leo XII. Upon the right, coming down the aisle, is the tomb of Alexander VII., by Bernini. Justice, Prudence, Charity, and Truth surround the kneeling pontiff. A bronze gilt figure of Death supports the marble canopy. The naked Truth was clothed in tin by Innocent XI. Opposite is Vanni's oil-painting, the Fall of Simon Magus. The south transept contains mosaics of S. Thomas by Camuccini, the Crucifixion by Guido, and S. Francis by Domenichino. On the left is the chair of the Grand Penitentiary, where great princes have to make their public confession as pilgrims. Returning to the aisle, on the right is the tomb of Pius VIII., by Tenerani. Our Saviour is blessing the Pope; Peter and Paul are on either side; Justice and Mercy are represented in relief below. Opposite is a mosaic of Ananias and Sapphira after Roncalli. Beyond is the Miracle of Gregory the Great, by Sacchi. Facing us is the tomb of Pius VII., by Thorwaldsen. History and Time support him on either side, with Power and Wisdom below. On the left, nearly opposite, is a mosaic copy of Raphael's Transfiguration. Proceeding down the aisle, on our right, is the tomb of Leo XI., with a relief, by Algardi, representing the abjuration of Henry IV. of France. Opposite is the tomb of Innocent XI., with relief of the raising of the siege of Vienna by John Sobieski, with figures of Religion and Justice, by Monot. On our right is the Chapel of the Choir, decorated by Giacomo della Porta. The mosaic altar-piece of the Conception is after Pietro
- 81. Bianchi. Over the door, in the pier on the left of the chapel, is a niche closed with a wooden sarcophagus; here the body of the Pope is placed till his tomb is prepared. Opposite is the bronze memorial to Innocent VIII. by the brothers Pollaiolo. The spear-head held in the hand of the Pope refers to the spear which pierced our Saviour's side, it being presented to this Pope by the Emperor Bajazet II. On our right is a fine mosaic by Romanelli, the Presentation of the Virgin in the Temple. Beyond, on the left, is Canova's memorial to the last of the Stuarts, who died in Rome, and are buried in the crypt below. It takes the form of an entrance to a tomb, which is guarded by beautiful genii. Over the door are the words—Blessed are the dead that die in the Lord. Above are medallions of the Chevalier S. George, Prince Charlie, and the Cardinal York, the whole being surmounted by the British coat-of-arms, in which is quartered that of France. This monument was erected by George IV. Opposite, over the door leading to the dome, is the monument to Maria Clementina, wife of the Chevalier S. George, whose portrait in mosaic is by Barigioni. Beyond is the baptistery. The font is of red porphyry, which was once the top of the tomb of Otho II., and originally, it is said, of Hadrian. In front is Carlo Maratta's Baptism of Christ in mosaic; upon the left Peter baptizing the jailers in the Mamertine prison, a fiction from Passeri; and opposite is Procaccini's Baptism of the Centurion. This baptistery is said to be on the site of a temple to Apollo, upon what authority we cannot say. The nave has marked in the centre of its pavement the measurement of all the principal churches in the world, whereby it can be seen that S. Peter's is 93 feet longer than S. Paul's, London. The large porphyry circular slab is that upon which the holy Roman emperors were crowned, and where the priest who is made judge of ecclesiastical matters in the Roman Church is ordained. In a niche in each of the piers supporting the vault are colossal statues, 16 feet high, of the founders of the various religious orders; and in the piers of the dome are S. Longinus, the soldier who pierced our Saviour's side, S. Helena, who found the cross, S. Veronica, who wiped his face, and S. Andrew. Above are kept the relics of these saints, which
- 82. are only shown to those who hold the title of a canon of the church. On the spandrels of the arches of the dome are four large mosaics, representing Matthew, Mark, Luke, and John, with their emblems. S. Luke's pen is 7 feet long, and the letters on the frieze are 6 feet high. The great piers are 253 feet in circumference; which space is exactly occupied by the church and house of S. Carlo, in the Via Quattro Fontane. Near the first pier of the right side is the celebrated bronze seated statue of S. Peter, with the keys in one hand, the other raised in the act of blessing, under a canopy erected by Pius IX., whose portrait in mosaic surmounts it. It is asserted by some that this was a statue of Jupiter, supremely good and great, that stood in the Capitoline temple, and that it was altered into S. Peter; others say they recast Jupiter into the Jew Peter. THE SACRISTY is connected with S. Peter's by a long gallery, and is adorned with pillars, statues, paintings, and mosaics. It is entered by passing through a door under the monument to Pius VIII., in the left aisle. There is a very rich collection of church plate and vestments kept in the guardaroba, which visitors should not fail to see. THE CRYPT. Orders must be obtained of Cardinal Ledockowski, Palazzo Cancelleria. It must be visited before 11 A.M. The entrance is at the side of the statue of S. Veronica. It contains the tombs of the early Popes, and also some old bas-reliefs, and some very ancient statues of S. Peter. Adrian IV., the only English Pope, is buried here, and also several distinguished historical characters, including the last of the Stuarts. THE DOME.
- 83. Orders must be obtained of Monsignor Fiorani, in the Sacristy, for visiting the dome, which is only open without an order on Thursdays, between 8 and 10 A.M. It is reached by a winding ascent, the entrance being opposite the Stuart monument. On the platform of the roof the cupolas, domes, and pinnacles are seen to advantage; and hence, by different staircases between the walls of the cupola, the ball is reached. During the ascent, a fine view may be obtained of the lower parts of the church, as well as of the mosaics and stuccoes which embellish the interior of the dome. On reaching the summit, a panoramic view of Rome and the Campagna is had, quite repaying the labour of the ascent. THE VATICAN. From the vestibule of S. Peter's we see, to the fullest advantage, the fine piazza, with the Vatican on our left, which presents very much the appearance of a large factory. Having been erected by different architects in various eras, it has no systematic design, and is, in fact, a collection of palaces built by different Popes. The entrance is at the bend of the colonnade. Permission to visit the Museum, Galleries, Library, c., must be obtained from Monsignor Macchi, at his office, between the hours of 10 and 1, thus enabling a party of five to pay a visit any day, except Saturdays, Sundays, and festas, between 9 and 3, except the Museum of Statues, which is closed every Thursday, when the Egyptian and Etruscan Museums and the Gallery of Tapestries are only open. The galleries are gained by THE SCALA REGIA, built in the pontificate of Urban VIII., from the design of Bernini. The first flight is composed of Ionic columns, the second of pilasters. The ornamental stucco work is from the designs of Algardi. The
- 84. equestrian statue of Constantine is by Bernini. On the first landing, a passage leads to a small flight of steps. At the top, on the right, through a small red baize door, is the entrance to THE SISTINE CHAPEL, built by Sixtus IV. in 1473. It is celebrated for its paintings in fresco by Michael Angelo; the roof alone occupied twenty months in the painting. The Roof.—On the flat part are nine compartments illustrative of—(1) The Separation of Light from Darkness; (2) Creation of the Sun and Moon; (3) Land and Sea; (4) Adam; (5) Eve; (6) the Fall and Expulsion (the figure of Eve is considered to be the most perfect painting of the female form in existence); (7) the Sacrifice of Noah; (8) the Deluge; (9) Noah inebriated. These are bordered by sitting figures of prophets and sibyls: over the altar, Jonah; on the left, Joel, the Sibyl Erithræa, Ezekiel, the Sibyl Persica, Jeremiah and Zechariah; on the right, the Sibyl Lybica, Daniel, the Sibyl Cumæa, Isaiah, and the Sibyl Delphica. In the four corners are—Moses lifting up the Brazen Serpent, King Artaxerxes, Esther and Haman, David and Goliath, Judith and Holofernes. In the arches over the windows, and in the recesses, Genealogy of Christ from Abraham to Joseph. The Walls.—Behind the altar is the great fresco of Michael Angelo, representing the Last Judgment, designed by him when in his sixtieth year, and completed in eight years (1540). At the top is our Saviour, with the Virgin seated on his right, above angels bearing the instruments of the passion. On one side of our Lord are saints and patriarchs, and on the other martyrs. Below, a group of angels sounding the last trump and bearing the books of judgment. On the right is represented the fall of the condemned; Charon ferrying some of them across the river Styx, striking the tumultuous with his oar. The figure in the right-hand corner, representing Midas with ass's ears, is Messer Biagio of Casena, the Pope's master of the ceremonies, who said the nude figures were indecent; on which
- 85. account the Pope ordered Daniele da Volterra to cover them with drapery, which obtained for him the cognomen of Braghettone (breeches-maker). Michael Angelo said, Let the Pope reform the world, and the pictures will reform themselves. And to spite Biagio, he represented him in hell, whereat he complained to the Pope in order to have his figure removed. The Pope replied that as he was in hell he must stop there, as he had no power to release from hell, but from purgatory! On the left, the blessed are ascending to heaven assisted by angels and saints. Between the windows, portraits of the Popes of the time, by the artist of the subject below. The lower part of the walls is painted in imitation of drapery, over which were hung on grand ceremonies tapestries from Raphael's cartoons. On the side walls are scenes from the life of Moses typical of the life of our Lord. On entering, to the right— TYPE. FULFILMENT. Moses and Zipporah going down into Egypt. By Luca Signorelli. Baptism of Christ in Jordan. By Perugino. Moses slaying the Egyptian. Driving away the shepherds. The Lord appearing in the burning bush. By Sandro Botticelli. Our Lord being tempted. By Sandro Botticelli. Pharaoh overwhelmed in the Red Sea. By Cosimo Rosselli. Christ calling Peter and Andrew. By Dom Ghirlandajo. Moses receiving the tables of the law. Destruction of the Golden Calf. By Cosimo Rosselli. The Sermon on the Mount. By Cosimo Rosselli. Destruction of Korah, Dathan, and Abiram, and the sons of Aaron. By Sandro Botticelli. Christ giving unto Peter the keys of the kingdom of heaven (Matt. xvi. 19). By Perugino.
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