2. The continuous and the discrete time state-space Li-Ion battery model representation

In this section we introduce a generic model capable to describe accurately the dynamics of Li-Ion battery, based on the same set of first-order differential equations in a state-space representation. For simulation purpose, a specific Li-Ion battery model is considered to prove the effectiveness of the proposed SOC estimation strategies. This model can be obtained from the generic model by changing only the values of the model parameters in state-space equations. For our case study we choose the widely-used 2RC - series cells Li-Ion battery equivalent model circuit (EMC) as model-based support, well documented in [1, 8, 11].

#### 2.1. Li-Ion battery terminology

In this subsection we introduce briefly the same terminology as in [2] to introduce the most used terms in this chapter that characterize the Li-Ion battery architecture and its performance. percentage of its rated capacity. More precisely, the SOC can be defined as the remaining capacity of a battery and it is affected by its operating conditions such as load current and

Estimation Techniques for State of Charge in Battery Management Systems on Board of Hybrid Electric Vehicles…

SOC <sup>¼</sup> Remaining capacity

The SOC for a fully charged battery is 100% and for an empty battery is 0%, defined for a

Ahnom <sup>ð</sup><sup>t</sup>

where η is the coulombic efficiency of the charging or discharging battery cycle, Ahnom represents the nominal battery capacity, ið Þτ is the instantaneous value of charging (i tð Þ ≤ 0) or

The relation (3) can be also written as a first order differential equation that further it will be

The SOC is a critical condition parameter for battery management system (BMS), often affected by its operating conditions such as load current and temperature; consequently, an accurate estimation of SOC is very important, since it is the key issue for the healthy and safe operation of batteries. The depth of discharge (DOD) is used to indicate the percentage of the total

The SOH of the battery is defined as the ratio of the maximum charge capacity of an aged battery to the maximum charge capacity when this battery was new. Its life cycle (number of cycles) is given by the number of discharging-charging cycles that the battery can withstand at a specific DOD (normally 80%) before it fails to meet the desired performance criteria. The actual operating life of the battery is affected by the charging and discharging rates, DOD, and by the temperature. The higher the DOD is the shorter will be the cycle life. To attain a higher life cycle, a larger battery is required to be used for a lower DOD during normal operating

The Battery Management System is an integrated battery structure consisting of measurement sensors, controllers, serial communication, and computation hardware with software algorithms such as Proportion Integral Derivative (PID) and adaptive control laws, Kalman filters estimators, adaptive or sliding mode observers designed to decide the maximum charging/ discharging cycles current and the duration from the estimation of SOC and SOH of the

dtð Þ¼� SOC tð Þ <sup>100</sup> <sup>η</sup> � i tð Þ

0

ið Þτ dτ

1

discharging cycle, when discharging battery current is positive, as:

SOC tðÞ¼ 100 1 � <sup>η</sup>

d

battery capacity that has been discharged at time t, defined as:

discharging (i tð Þ ≥ 0) battery current.

used for SOC state estimation:

conditions.

battery pack, as is shown in [11, 13–15].

0 @ Rated capacity (2)

A ð Þ % , ið Þτ ≥ 0 (3)

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59

Ahnom ,i tð Þ <sup>≥</sup> <sup>0</sup> (4)

DOD tðÞ¼ 100 1ð Þ � SOC tð Þ ð Þ % (5)

temperature:

#### 2.1.1. Li-Ion battery characterization architecture terminology

A single battery cell is a complete battery with two current leads and separate compartment holding electrodes (positive (+) and negative (�)), separator, and electrolyte. A few cells are connected in series, parallel or in series - parallel combinations, either by physical attachment or by welding in between cells to form a battery module. Similar, several modules are connected to form a battery pack placed in a single compartment for thermal management. A HEV may have more than one batteries packs placed in a different location of the car.

#### 2.1.2. Li-Ion battery performance terminology

A battery cell is fully charged when its terminal voltage reaches the maximum voltage limit value after being charged at infinitesimal current levels, for example a maximum limit voltage value of 4.2 V at room temperature (25�C). A battery terminal voltage value greater than this limit corresponds to a dangerous over-charging operating condition. Similar, a battery cell is fully discharged when its terminal voltage reaches the minimum voltage limit value after being drained at infinitesimal current levels, for example a minimum limit voltage value of 3.0 V at room temperature. A battery terminal voltage value smaller than this limit corresponds to a dangerous over-discharging operating condition. The capacity of a battery is measured in Ampere-hours (Ah) defined as the total charge that can be discharged from a fully charged battery under specified conditions. The rated Ah capacity is the nominal capacity of a fully charged new battery under the conditions predefined by the catalog specifications of the battery, e.g. the nominal condition could be defined as room temperature 25�C and battery discharging is at 1/25 C-rate. The C-rate is used to represent a charge or discharge rate equal to the capacity of a battery in 1 hour, e.g. for a 6 Ah battery, 1C-rate is equal to charge or discharge the battery at a constant current value of 6 A; in the same way, 0.1C-rate is equivalent to 0.6 A, and 2C-rate for charging or discharging the battery at a constant current value of 12 A. The power density of the battery is an important criterion of battery selection that is defined as the peak power per unit volume of a battery (W/l). In the battery model dynamics, the effect of its internal resistance is significant. It is defined as the overall equivalent resistance within the battery. Also, it is worth to mention that its value varies during the charging and discharging battery cycles, and moreover may vary as the operating condition changes. The peak power according to the U.S. Advanced Battery Consortium (USABC)'s definition is given by:

$$P = \frac{2\text{V}\_{\text{OC}}^2}{\text{\textdegree R}} \tag{1}$$

where VOC is the open-circuit voltage (OCV) and R is the battery internal resistance.

The peak power is defined at the condition when the terminal voltage is 2/3 of its OCV. The SOC of the battery provides an important feedback about the state of health (SOH) of the battery and its safe operation. SOC is defined as battery available capacity expressed as a percentage of its rated capacity. More precisely, the SOC can be defined as the remaining capacity of a battery and it is affected by its operating conditions such as load current and temperature:

2.1. Li-Ion battery terminology

58 New Trends in Electrical Vehicle Powertrains

2.1.1. Li-Ion battery characterization architecture terminology

2.1.2. Li-Ion battery performance terminology

In this subsection we introduce briefly the same terminology as in [2] to introduce the most used terms in this chapter that characterize the Li-Ion battery architecture and its performance.

A single battery cell is a complete battery with two current leads and separate compartment holding electrodes (positive (+) and negative (�)), separator, and electrolyte. A few cells are connected in series, parallel or in series - parallel combinations, either by physical attachment or by welding in between cells to form a battery module. Similar, several modules are connected to form a battery pack placed in a single compartment for thermal management. A

A battery cell is fully charged when its terminal voltage reaches the maximum voltage limit value after being charged at infinitesimal current levels, for example a maximum limit voltage value of 4.2 V at room temperature (25�C). A battery terminal voltage value greater than this limit corresponds to a dangerous over-charging operating condition. Similar, a battery cell is fully discharged when its terminal voltage reaches the minimum voltage limit value after being drained at infinitesimal current levels, for example a minimum limit voltage value of 3.0 V at room temperature. A battery terminal voltage value smaller than this limit corresponds to a dangerous over-discharging operating condition. The capacity of a battery is measured in Ampere-hours (Ah) defined as the total charge that can be discharged from a fully charged battery under specified conditions. The rated Ah capacity is the nominal capacity of a fully charged new battery under the conditions predefined by the catalog specifications of the battery, e.g. the nominal condition could be defined as room temperature 25�C and battery discharging is at 1/25 C-rate. The C-rate is used to represent a charge or discharge rate equal to the capacity of a battery in 1 hour, e.g. for a 6 Ah battery, 1C-rate is equal to charge or discharge the battery at a constant current value of 6 A; in the same way, 0.1C-rate is equivalent to 0.6 A, and 2C-rate for charging or discharging the battery at a constant current value of 12 A. The power density of the battery is an important criterion of battery selection that is defined as the peak power per unit volume of a battery (W/l). In the battery model dynamics, the effect of its internal resistance is significant. It is defined as the overall equivalent resistance within the battery. Also, it is worth to mention that its value varies during the charging and discharging battery cycles, and moreover may vary as the operating condition changes. The peak power

HEV may have more than one batteries packs placed in a different location of the car.

according to the U.S. Advanced Battery Consortium (USABC)'s definition is given by:

where VOC is the open-circuit voltage (OCV) and R is the battery internal resistance.

<sup>P</sup> <sup>¼</sup> 2V<sup>2</sup> OC

The peak power is defined at the condition when the terminal voltage is 2/3 of its OCV. The SOC of the battery provides an important feedback about the state of health (SOH) of the battery and its safe operation. SOC is defined as battery available capacity expressed as a

9R (1)

$$\text{SOC} = \frac{\text{Remaining capacity}}{\text{Rated capacity}} \tag{2}$$

The SOC for a fully charged battery is 100% and for an empty battery is 0%, defined for a discharging cycle, when discharging battery current is positive, as:

$$\text{SOC}(t) = 100 \left( 1 - \frac{\eta}{Ahnom} \int\_0^t i(\tau)d\tau \right) \text{ (\%)}, \text{ i}(\tau) \ge 0 \tag{3}$$

where η is the coulombic efficiency of the charging or discharging battery cycle, Ahnom represents the nominal battery capacity, ið Þτ is the instantaneous value of charging (i tð Þ ≤ 0) or discharging (i tð Þ ≥ 0) battery current.

The relation (3) can be also written as a first order differential equation that further it will be used for SOC state estimation:

$$\frac{d}{dt}(SOC(t)) = -100 \frac{\eta \times i(t)}{Ahnom}, i(t) \ge 0\tag{4}$$

The SOC is a critical condition parameter for battery management system (BMS), often affected by its operating conditions such as load current and temperature; consequently, an accurate estimation of SOC is very important, since it is the key issue for the healthy and safe operation of batteries. The depth of discharge (DOD) is used to indicate the percentage of the total battery capacity that has been discharged at time t, defined as:

$$DOD(t) = 100(1 - SOC(t)) \text{ (\%)}\tag{5}$$

The SOH of the battery is defined as the ratio of the maximum charge capacity of an aged battery to the maximum charge capacity when this battery was new. Its life cycle (number of cycles) is given by the number of discharging-charging cycles that the battery can withstand at a specific DOD (normally 80%) before it fails to meet the desired performance criteria. The actual operating life of the battery is affected by the charging and discharging rates, DOD, and by the temperature. The higher the DOD is the shorter will be the cycle life. To attain a higher life cycle, a larger battery is required to be used for a lower DOD during normal operating conditions.

The Battery Management System is an integrated battery structure consisting of measurement sensors, controllers, serial communication, and computation hardware with software algorithms such as Proportion Integral Derivative (PID) and adaptive control laws, Kalman filters estimators, adaptive or sliding mode observers designed to decide the maximum charging/ discharging cycles current and the duration from the estimation of SOC and SOH of the battery pack, as is shown in [11, 13–15].

#### 2.1.3. Battery management systems: architecture, trends, functions, monitoring, faults detection, hardware and software components

A. Hardware with the following components:

tion; Thermal management.

within the BMS [13].

imbalance, reliability and performance [13]. B. Software with the following components:

tion and Isolation; User interface.

Safety circuitry; Sensor system; Data acquisition; Charge and discharge control; Communica-

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The sensor system block integrates different sensors capable to monitor and measure battery parameters including cell voltage, battery temperature, and battery current. The new proposed BMS in [13] it seems to have a lot of improvements in terms of current safety circuitry designs that can be easily implemented, amongst them is worth to mention the addition of accurate alarms and controls to prevent overcharge, over-discharge, and overheating. Therefore, the external signals such as the current, voltage, and temperature must be measured to improve the capability of state tracking in real life applications [13]. Data acquisition (DAQ) block in conjunction with the data storage block are critical parts for the software in the BMS to analyze

The charge and discharge control block is integrated in hardware architecture of the BMS to implement the charge-discharge protocol. A variable resistor may be necessary to help balance cells or perform internal resistance measurements. In [13] is mentioned also that the "cell balancing control is still a critical design feature with room for improvement in order to equalize the battery pack and estimate the battery status in an efficient way". Most subsystems in a BMS are stand-alone modules, and hence, data transfer throughout the BMS is required. Communication through a CAN Bus is a major way to transfer data between the stand-alone subsystems

Furthermore, the recent development of smart batteries within the context of innovative electronics and artificial intelligence, more data can be collected to communicate with the user and the charger through the microchips incorporated within the battery [13]. In addition, "wireless and telecommunication techniques are gradually being incorporated into charging systems that facilitate communication between the battery and the charger". Finally, the last thermal management hardware module proved during the time that its integration in a BMS is critical to be integrated since the temperature variations between the cells have a great impact on cell

SOC estimation and monitoring; SOH estimation and monitoring; Cell balancing; Fault Detec-

The software compartment integrated in BMS structure is in reality such as an artificial "brain" that controls all hardware operations and examines all sensors data for making decisions and to implement in real-time linear, nonlinear and intelligent SOC, SOH estimators, and also FDI techniques. As is mention in [13], "the switch control, sample rate monitoring in the sensor system block, cell balancing control, and even dynamic safety circuit design should be handled by the software of a BMS". In addition, the software components assure online data processing, and also reliable and robust automated data analysis by controlling battery functions, representing an essential factor to perform successfully the SOC, SOH estimation and fault detection and isolation algorithms. The robust SOC and SOH estimators are design to integrate also a

and build a database for system modeling and estimation algorithms development.

A BMS as an important connector between the battery and the HEV plays a vital role in improving battery performance and optimizing vehicle operation in a safe and reliable manner, as is mentioned in [13]. Nowadays, the trend of the BMSs is a rapid growth of the EV and HEV market, thus it is essential for the automotive industry to develop a comprehensive and mature BMSs. As is stated in [13]"the U.S. Council for Automotive Research (USCAR) and the U.S. Advanced Battery Consortium (USABC) have set minimum goals for battery characteristics for the long-term commercialization of advanced batteries in EVs and hybrid electric vehicles (HEVs)". Furthermore, to increase the market segment of EVs and HEVs automotive industry, the main concerns such as reliability and safety remain constantly for both of them, battery technology (BT) and BMS.

The BMS hardware and software components, the safety circuitry incorporated within the battery packs play an important role to monitor, control, compute and to show continually the safety state, the SOC, SOH, as well as the longevity of the battery.

Moreover, in [13] is emphasized one of the most dangerous situations such as the ignition of a Li-Ion battery during overcharging operating conditions, due to the volatility, flammability and entropy changes. Also, after repeated over-discharging cycles, the battery cell capacity is reduced significantly, due to irreversible chemical reactions, and thus the BMS needs to monitor and control constantly the Li-Ion battery state. Whenever any abnormal conditions happen, such as self-discharge leakage current through the insulation resistance of the battery, well-known as ground insulation resistances of the negative and positive bus of BMS Rp, and Rn respectively, over-voltage or overheating operating conditions are identified, and in a very short time the BMS should notify the user and to execute the preset correction procedures [13, 16]. In addition to these foremost functions, the BMS also monitors the system temperature to provide a better power consumption scheme, and communicates with individual components and operators [13]. Technically, a comprehensive BMS is equipped with the most suitable hardware and software components of the newest generation [14, 15], integrated in the HEVs and EVs structure to accomplish the main following functions: (1) real-time monitoring of battery states by a performing data acquisition system of external signals (i.e. voltage, current, cell temperature etc.); (2) ensure user safety protection, and extend the battery life; (3) using performing and intelligent algorithms (as for example, genetic, fuzzy logic, neural networks and expert systems based on artificial intelligence) has the ability to estimate and monitor the battery internal parameters and states (i.e. the DC resistance, insulation resistances Rp, and Rn, polarization voltage, maximum available capacity, SOC, SOH, etc.); (4) the ability to prevent over-charge or over-discharge of the battery; (5)efficient battery energy utilization, thermal management and SOC cell balancing; (6) delivery of battery status and authentication to a user interface; (7) the ability to communicate with vehicle controller and all other components [13, 16]. In order to achieve these objectives, researchers focus on battery modeling, SOC estimation, consistency evaluation and equalization, such as stated in [16].

Furthermore, to increase the effectiveness of BMS in [13] is proposed a new BMS with the following categories of components:

#### A. Hardware with the following components:

2.1.3. Battery management systems: architecture, trends, functions, monitoring, faults detection,

A BMS as an important connector between the battery and the HEV plays a vital role in improving battery performance and optimizing vehicle operation in a safe and reliable manner, as is mentioned in [13]. Nowadays, the trend of the BMSs is a rapid growth of the EV and HEV market, thus it is essential for the automotive industry to develop a comprehensive and mature BMSs. As is stated in [13]"the U.S. Council for Automotive Research (USCAR) and the U.S. Advanced Battery Consortium (USABC) have set minimum goals for battery characteristics for the long-term commercialization of advanced batteries in EVs and hybrid electric vehicles (HEVs)". Furthermore, to increase the market segment of EVs and HEVs automotive industry, the main concerns such as reliability and safety remain constantly for both of them, battery technology (BT) and BMS.

The BMS hardware and software components, the safety circuitry incorporated within the battery packs play an important role to monitor, control, compute and to show continually

Moreover, in [13] is emphasized one of the most dangerous situations such as the ignition of a Li-Ion battery during overcharging operating conditions, due to the volatility, flammability and entropy changes. Also, after repeated over-discharging cycles, the battery cell capacity is reduced significantly, due to irreversible chemical reactions, and thus the BMS needs to monitor and control constantly the Li-Ion battery state. Whenever any abnormal conditions happen, such as self-discharge leakage current through the insulation resistance of the battery, well-known as ground insulation resistances of the negative and positive bus of BMS Rp, and Rn respectively, over-voltage or overheating operating conditions are identified, and in a very short time the BMS should notify the user and to execute the preset correction procedures [13, 16]. In addition to these foremost functions, the BMS also monitors the system temperature to provide a better power consumption scheme, and communicates with individual components and operators [13]. Technically, a comprehensive BMS is equipped with the most suitable hardware and software components of the newest generation [14, 15], integrated in the HEVs and EVs structure to accomplish the main following functions: (1) real-time monitoring of battery states by a performing data acquisition system of external signals (i.e. voltage, current, cell temperature etc.); (2) ensure user safety protection, and extend the battery life; (3) using performing and intelligent algorithms (as for example, genetic, fuzzy logic, neural networks and expert systems based on artificial intelligence) has the ability to estimate and monitor the battery internal parameters and states (i.e. the DC resistance, insulation resistances Rp, and Rn, polarization voltage, maximum available capacity, SOC, SOH, etc.); (4) the ability to prevent over-charge or over-discharge of the battery; (5)efficient battery energy utilization, thermal management and SOC cell balancing; (6) delivery of battery status and authentication to a user interface; (7) the ability to communicate with vehicle controller and all other components [13, 16]. In order to achieve these objectives, researchers focus on battery modeling, SOC estimation,

Furthermore, to increase the effectiveness of BMS in [13] is proposed a new BMS with the

the safety state, the SOC, SOH, as well as the longevity of the battery.

consistency evaluation and equalization, such as stated in [16].

following categories of components:

hardware and software components

60 New Trends in Electrical Vehicle Powertrains

Safety circuitry; Sensor system; Data acquisition; Charge and discharge control; Communication; Thermal management.

The sensor system block integrates different sensors capable to monitor and measure battery parameters including cell voltage, battery temperature, and battery current. The new proposed BMS in [13] it seems to have a lot of improvements in terms of current safety circuitry designs that can be easily implemented, amongst them is worth to mention the addition of accurate alarms and controls to prevent overcharge, over-discharge, and overheating. Therefore, the external signals such as the current, voltage, and temperature must be measured to improve the capability of state tracking in real life applications [13]. Data acquisition (DAQ) block in conjunction with the data storage block are critical parts for the software in the BMS to analyze and build a database for system modeling and estimation algorithms development.

The charge and discharge control block is integrated in hardware architecture of the BMS to implement the charge-discharge protocol. A variable resistor may be necessary to help balance cells or perform internal resistance measurements. In [13] is mentioned also that the "cell balancing control is still a critical design feature with room for improvement in order to equalize the battery pack and estimate the battery status in an efficient way". Most subsystems in a BMS are stand-alone modules, and hence, data transfer throughout the BMS is required. Communication through a CAN Bus is a major way to transfer data between the stand-alone subsystems within the BMS [13].

Furthermore, the recent development of smart batteries within the context of innovative electronics and artificial intelligence, more data can be collected to communicate with the user and the charger through the microchips incorporated within the battery [13]. In addition, "wireless and telecommunication techniques are gradually being incorporated into charging systems that facilitate communication between the battery and the charger". Finally, the last thermal management hardware module proved during the time that its integration in a BMS is critical to be integrated since the temperature variations between the cells have a great impact on cell imbalance, reliability and performance [13].

B. Software with the following components:

SOC estimation and monitoring; SOH estimation and monitoring; Cell balancing; Fault Detection and Isolation; User interface.

The software compartment integrated in BMS structure is in reality such as an artificial "brain" that controls all hardware operations and examines all sensors data for making decisions and to implement in real-time linear, nonlinear and intelligent SOC, SOH estimators, and also FDI techniques. As is mention in [13], "the switch control, sample rate monitoring in the sensor system block, cell balancing control, and even dynamic safety circuit design should be handled by the software of a BMS". In addition, the software components assure online data processing, and also reliable and robust automated data analysis by controlling battery functions, representing an essential factor to perform successfully the SOC, SOH estimation and fault detection and isolation algorithms. The robust SOC and SOH estimators are design to integrate also a capability assessment of the life status of the battery and to "sets the operating limits according to state-of-the-art algorithms, such as fuzzy logic, neural networks, state-space-based models, and so on "[13]. Most of the soft battery faults are detected through online data processing by means of an intelligent data analysis that provides information about the occurrence of the faults and their persistence by fault warnings, and also indicates abnormal limits of operating conditions. The history of the past measurements data set is recorded and provides the pre-alarm condition before the possible battery faults. In [17, 18] is proposing a real-time diagnosis approach that detects, isolates, and estimate specific sensor faults, such as battery voltage, current temperature, and fan motor. The sensor fault estimation can provide great benefits for enhancing the reliability of BMSs. Also, the sensor fault estimation can provide fault-tolerance capability to the BMS by allowing it to continue in degraded and in safe mode even after the sensor faults occur, as is mentioned in [17]. Failure in Li-Ion batteries can be also attributed to a combination of manufacturing defects, safety component failure, or human operating errors, as is specified in [18]. Furthermore, in [18] are developed some methods for realistic fault injection. Two types of sensor faults are considered in [18], namely an intermittent signal loss due to faulty wiring connections and sensor bias resulting from time or temperature drift. For the fan motor, the only fault considered is a total motor failure, and thus no longer provide cooling to the battery. The occurrence of the realistic battery faults are established based on the preset thresholds. These thresholds are computed as a minimum values that assure the prevention of the false alarms, based on the minimization of the residual error probability when occurs a fault, by studying the probability density functions (PDF)) of the healthy and faulty signals, such is shown in [18]. The PDfs are obtained through extensive experimental data collection or simulations on the system.

in series denote the electrochemical polarization resistances and capacitances, for which T1 = R1C1, T2 = R2C2 can imitate the fast polarization time constant, and the slow polarization time constant for the voltage recovery of the Li-Ion battery respectively {14]. Also, i tð Þ is the battery instantaneous value of the direct current (DC) flowing through the OCV source, and v tð Þ represents the measured battery instantaneous value DC voltage. The main benefit of the RC second order EMC architecture selection is its simplicity and the ability to be implemented

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Also "this choice is due to the early popularity of BMS for portable electronics, where the approximation of the battery model with the proposed EMC is appropriate", as is mentioned in [1, 8]. Related, this approach has been extended easily to Li-Ion batteries for automotive industry and for many other similar energy storage applications [11]. Since the EMC is used in following sections to design and implement in real-time all three proposed nonlinear state estimators, now it is essential to precise the main factors that affects the battery parameters, especially the internal and insulation resistances and their impact on the battery dynamics in a realistic operating conditions environment. The internal resistance of the battery is affected by the following factors: conductor resistance, electrolyte resistance, ionic mobility, separator efficiency, reactive rates at the electrodes, and concentration polarization, temperature effects and changes in SOC. When a battery fails, it is typically since it has built up enough internal resistance that it can no longer supply a useful amount of power to an external load, according to the maximum power transfer between the source and the load. The insulation positive and negative resistances of the battery are denoted by Rn, and Rp respectively, and are shown in the diagram from Figure 1(b). The main objective is to get a good insight on the impact of the insulation

Figure 1. The second order RC equivalent model circuit (EMC) 1(a), and the insulation resistances Rn and Rp 1(b) of Li-

Ion battery, represented in NI Multisim 14.1 editor (see for details [16]).

in real-time applications with acceptable range of performance [1, 8].

Roughly, for many faults scenarios to compute the magnitude of the thresholds we can use also the statistics of the residual errors for healthy system between the measured values data set and its estimated values, such as the mean μ<sup>X</sup> and the standard deviation (σ) calculated in steady state. The threshold can be calculated by addition to the mean μ<sup>X</sup> of residual error approximatively two times the standard deviation 2ð Þ σ . The decision for occurrence of the fault is given by comparison of the residual error and the computed threshold. The tuning of the threshold to avoid the false alarms can be done by well-known trials and error method. The user interface block must display the essential information of the BMS to the users.

#### 2.2. Li-Ion battery EMC description

The equivalent model circuits (EMCs) are simple electric circuits consisting of voltage sources, resistors and capacitors networks, commonly-used to simulate the dynamic behavior of the Li-Ion batteries [1, 8, 11]. The role of the resistor-capacitor (RC) series networks integrated in EMCs is to improve models' accuracy, and also to increase the structural complexity of the models [1, 8]. In this research paper we chose for Li-Ion battery model design and implementation stages an equivalent second order model RC circuit. It consists of a resistor R, two series RC cells (R1C1, R2C2), and an open-circuit voltage (OCV) source dependent of SOC, i.e. OCV (SOC), as is shown in Figure 1(a), in NI Multisim 14.1 editor. In EMC architecture, R denotes the internal ohmic resistance of the Li-Ion battery, the parallel cells (R1, C1), (R2, C2) connected in series denote the electrochemical polarization resistances and capacitances, for which T1 = R1C1, T2 = R2C2 can imitate the fast polarization time constant, and the slow polarization time constant for the voltage recovery of the Li-Ion battery respectively {14]. Also, i tð Þ is the battery instantaneous value of the direct current (DC) flowing through the OCV source, and v tð Þ represents the measured battery instantaneous value DC voltage. The main benefit of the RC second order EMC architecture selection is its simplicity and the ability to be implemented in real-time applications with acceptable range of performance [1, 8].

capability assessment of the life status of the battery and to "sets the operating limits according to state-of-the-art algorithms, such as fuzzy logic, neural networks, state-space-based models, and so on "[13]. Most of the soft battery faults are detected through online data processing by means of an intelligent data analysis that provides information about the occurrence of the faults and their persistence by fault warnings, and also indicates abnormal limits of operating conditions. The history of the past measurements data set is recorded and provides the pre-alarm condition before the possible battery faults. In [17, 18] is proposing a real-time diagnosis approach that detects, isolates, and estimate specific sensor faults, such as battery voltage, current temperature, and fan motor. The sensor fault estimation can provide great benefits for enhancing the reliability of BMSs. Also, the sensor fault estimation can provide fault-tolerance capability to the BMS by allowing it to continue in degraded and in safe mode even after the sensor faults occur, as is mentioned in [17]. Failure in Li-Ion batteries can be also attributed to a combination of manufacturing defects, safety component failure, or human operating errors, as is specified in [18]. Furthermore, in [18] are developed some methods for realistic fault injection. Two types of sensor faults are considered in [18], namely an intermittent signal loss due to faulty wiring connections and sensor bias resulting from time or temperature drift. For the fan motor, the only fault considered is a total motor failure, and thus no longer provide cooling to the battery. The occurrence of the realistic battery faults are established based on the preset thresholds. These thresholds are computed as a minimum values that assure the prevention of the false alarms, based on the minimization of the residual error probability when occurs a fault, by studying the probability density functions (PDF)) of the healthy and faulty signals, such is shown in [18]. The PDfs are obtained through extensive experimental data

Roughly, for many faults scenarios to compute the magnitude of the thresholds we can use also the statistics of the residual errors for healthy system between the measured values data set

approximatively two times the standard deviation 2ð Þ σ . The decision for occurrence of the fault is given by comparison of the residual error and the computed threshold. The tuning of the threshold to avoid the false alarms can be done by well-known trials and error method. The

The equivalent model circuits (EMCs) are simple electric circuits consisting of voltage sources, resistors and capacitors networks, commonly-used to simulate the dynamic behavior of the Li-Ion batteries [1, 8, 11]. The role of the resistor-capacitor (RC) series networks integrated in EMCs is to improve models' accuracy, and also to increase the structural complexity of the models [1, 8]. In this research paper we chose for Li-Ion battery model design and implementation stages an equivalent second order model RC circuit. It consists of a resistor R, two series RC cells (R1C1, R2C2), and an open-circuit voltage (OCV) source dependent of SOC, i.e. OCV (SOC), as is shown in Figure 1(a), in NI Multisim 14.1 editor. In EMC architecture, R denotes the internal ohmic resistance of the Li-Ion battery, the parallel cells (R1, C1), (R2, C2) connected

steady state. The threshold can be calculated by addition to the mean μ<sup>X</sup>

user interface block must display the essential information of the BMS to the users.

and the standard deviation (σ) calculated in

of residual error

collection or simulations on the system.

62 New Trends in Electrical Vehicle Powertrains

2.2. Li-Ion battery EMC description

and its estimated values, such as the mean μ<sup>X</sup>

Also "this choice is due to the early popularity of BMS for portable electronics, where the approximation of the battery model with the proposed EMC is appropriate", as is mentioned in [1, 8]. Related, this approach has been extended easily to Li-Ion batteries for automotive industry and for many other similar energy storage applications [11]. Since the EMC is used in following sections to design and implement in real-time all three proposed nonlinear state estimators, now it is essential to precise the main factors that affects the battery parameters, especially the internal and insulation resistances and their impact on the battery dynamics in a realistic operating conditions environment. The internal resistance of the battery is affected by the following factors: conductor resistance, electrolyte resistance, ionic mobility, separator efficiency, reactive rates at the electrodes, and concentration polarization, temperature effects and changes in SOC. When a battery fails, it is typically since it has built up enough internal resistance that it can no longer supply a useful amount of power to an external load, according to the maximum power transfer between the source and the load. The insulation positive and negative resistances of the battery are denoted by Rn, and Rp respectively, and are shown in the diagram from Figure 1(b). The main objective is to get a good insight on the impact of the insulation

Figure 1. The second order RC equivalent model circuit (EMC) 1(a), and the insulation resistances Rn and Rp 1(b) of Li-Ion battery, represented in NI Multisim 14.1 editor (see for details [16]).

resistances on the battery dynamics, as is described in detail in [16]. Also, in [16] the HEV is considered as a "complex production of mechanical-electrical integration", for which the power supply typically being in the range 100–500 V is obtained by means of several series battery packs. Amongst the BMS hardware devices consisting of high voltage components, the traction battery, electrical motor, energy recycle device, the battery charger and its auxiliary device deal with a large current and insulation [16], thus insulation issue must be under consideration since the design stage. As is stated in [16], the poor working conditions, such as shaking, corrosion, changes in temperature and humidity, "could cause fast aging of the power cable and insulation materials, or even brake the insulation, which would decrease the insulation strength and endanger personnel". Thus, needs to ensure safety operating conditions for personnel are required to evaluate the insulation conditions for entire HEV's BMS.

By defining the following state variables x<sup>1</sup> ¼ vC<sup>1</sup> ð Þt , x<sup>2</sup> ¼ vC<sup>2</sup> ð Þt , and x<sup>3</sup> ¼ SOC tð Þ, representing the polarization voltages of the two RC cells, and considering as battery input u the DC instantaneous current that flows through battery, u ¼ i tð Þ, and as a battery output y, designating the terminal battery DC instantaneous voltage, i.e. y ¼ v tð Þ, the Eq. (6) can be written in a

Estimation Techniques for State of Charge in Battery Management Systems on Board of Hybrid Electric Vehicles…

dx<sup>1</sup> dt ¼ � <sup>1</sup> T1 x<sup>1</sup> þ 1 C1 u

dx<sup>2</sup> dt ¼ � <sup>1</sup> T2 x<sup>2</sup> þ 1 C2 u

dx<sup>3</sup> dt ¼ � <sup>η</sup>

Qnom u

1

same values that are carefully chosen for model validation in [8] as follows

capacitance, R2 = 0.0011 [Ω], C2 = 98246.01 [F] (Farads).

charging cycle, and η = 0.85, for discharging cycle

• K0 = 4.23, K1 = 3.86E-05, K2 = 0.24, K3 = 0.22, K4 = �0.04

y ¼ OCV xð Þ� <sup>3</sup> x<sup>1</sup> � x<sup>2</sup> � Ru

where T<sup>1</sup> ¼ R1C1½ �s , T<sup>2</sup> ¼ R2C2½ �s represent the time constants of the both polarization RC cells, and the OCV combines three additional well-known models, as defined and used in [4, 8],

For simulation purpose, in order to validate the proposed battery model, as well as to prove the effectiveness of the SOC estimation strategies developed in Section 3, we consider as the most suitable nominal values for Li-Ion battery EMC parameters and OCV coefficients the

• The battery internal ohmic resistance (slightly different for charging and discharging cycles), R = 0.0022 [Ω] (ohms), the first RC cell polarization resistance and capacitance, R1 = 0.00077 [Ω], C1 = 14475.24 [F], and the second RC cell polarization resistance and

• The value of the battery capacity, Q = 6 Ah (Amperes hours), the voltage nominal value of the battery, Vnom = 3.6 V (Volts), and the coulombic efficiency, η = 1, for

The OCV xð Þ <sup>3</sup>ð Þt is a nonlinear function of SOC, i.e.x3, similar as in [1, 3, 4, 8–11], increasing the accuracy of the Li-Ion battery EMC model, known as the EMC combined model, one of the most accurate formulation by combining Shepherd, Unnewehr universal, and Nernst models introduced in [4, 8]. The tuning values of model parameters ð Þ K0;K1;K2;K3;K<sup>4</sup> are chosen to fit the model to the manufacture's data by using a least squares curve fitting identification method OCV = h (SOC), as is shown in [3, 4, 8], where the OCV curve is assumed to be the

<sup>x</sup>3ð Þ<sup>t</sup> � <sup>K</sup>2x3ðÞþ <sup>t</sup> <sup>K</sup>3ln ð Þþ <sup>x</sup>3ð Þ<sup>t</sup> <sup>K</sup>4ln ð Þ <sup>j</sup><sup>1</sup> � <sup>x</sup>3ð Þj <sup>t</sup> (8)

http://dx.doi.org/10.5772/intechopen.76230

(7)

65

state-space representation, as follows.

OCV xð Þ¼ <sup>3</sup>ð Þt h xð Þ¼ <sup>3</sup>ð Þt K<sup>0</sup> � K<sup>1</sup>

1. Li-Ion battery ECM parameters:

2. Li-Ion battery characteristics:

3. The OCV coefficients:

given by:

The National Standard (NS) 18384.3-2001, stipulates several safety requirements for HEVs, especially for insulation resistance state, measurement method [16]. According to NS, the insulation state of an EV is evaluated according to the ground insulation resistance of the DC positive and negative bus, Vp, and Vn respectively, as is shown in Figure 1(b) [16]. The definition of traction battery insulation resistance in NS "is the relative resistance to maximum leakage current (in the worst condition) where there is a short between the traction battery and ground (electric chassis)" [16]. Also, "under the conditions that the maximum AC voltage is less than 660 V, the maximum DC voltage is less than 1000 V and the car weight is less than 3500 Kg, the requirements of the high voltage security are as follows: (1) personnel's security voltage is less than 35 V, or the product of the contact current with a person and the duration of time is less than 30 mA s; insulation resistance divided by the battery rated voltage should be more than 100 ΩV�<sup>1</sup> , and preferably more 500 ΩV�<sup>1</sup> ", as is formulated in [16]. Thus, to ensure the insulation security of on-board BMS, it is necessary to detect the insulation resistance and raise an alarm in time. Currently, the insulation measurement methods, such as passive ground detection and active ground detection, include the AC voltage insulation measurement method, and the DC voltage insulation measurement method [16]. The passive ground detection insulation measurement method principle is presented in Figure 1(b) where the presence of leakage current is detected by discovering the resultant magnetic field generated by AC voltage around the mutual inductor.

#### 2.3. Li-Ion battery continuous time state-space representation

By simple manipulations of Ohm's, current and voltage Kirchhoff's laws applied to the proposed Li-Ion battery's EMC can be written the following first order differential equations that are most suitable to capture the entire dynamics of EMC in time domain:

$$\begin{aligned} \frac{d\upsilon\_{\mathbb{C}\_{1}}}{dt} &= -\frac{1}{R\_{1}\mathbb{C}\_{1}}\upsilon\_{\mathbb{C}\_{1}}(t) + \frac{\dot{i}(t)}{\mathbb{C}\_{1}}\\ \frac{d\upsilon\_{\mathbb{C}\_{2}}}{dt} &= -\frac{1}{R\_{2}\mathbb{C}\_{2}}\upsilon\_{\mathbb{C}\_{2}}(t) + \frac{\dot{i}(t)}{\mathbb{C}\_{2}}\\ \frac{d(SOC)}{dt} &= -\frac{\eta}{Q\_{\text{nom}}}\dot{i}(t)\\ \upsilon(t) &= \text{OCV}(SOC(t)) - \upsilon\_{\mathbb{C}\_{1}}(t) - \upsilon\_{\mathbb{C}\_{2}}(t) - Ri(t) \end{aligned} \tag{6}$$

By defining the following state variables x<sup>1</sup> ¼ vC<sup>1</sup> ð Þt , x<sup>2</sup> ¼ vC<sup>2</sup> ð Þt , and x<sup>3</sup> ¼ SOC tð Þ, representing the polarization voltages of the two RC cells, and considering as battery input u the DC instantaneous current that flows through battery, u ¼ i tð Þ, and as a battery output y, designating the terminal battery DC instantaneous voltage, i.e. y ¼ v tð Þ, the Eq. (6) can be written in a state-space representation, as follows.

$$\begin{aligned} \frac{d\mathbf{x}\_1}{dt} &= -\frac{1}{T\_1}\mathbf{x}\_1 + \frac{1}{\mathbf{C}\_1}u\\ \frac{d\mathbf{x}\_2}{dt} &= -\frac{1}{T\_2}\mathbf{x}\_2 + \frac{1}{\mathbf{C}\_2}u\\ \frac{d\mathbf{x}\_3}{dt} &= -\frac{\eta}{Q\_{\text{max}}}u\\ y &= \text{OCV}(\mathbf{x}\_3) - \mathbf{x}\_1 - \mathbf{x}\_2 - Ru \end{aligned} \tag{7}$$

where T<sup>1</sup> ¼ R1C1½ �s , T<sup>2</sup> ¼ R2C2½ �s represent the time constants of the both polarization RC cells, and the OCV combines three additional well-known models, as defined and used in [4, 8], given by:

$$\text{OCV}(\mathbf{x}\_3(t)) = h(\mathbf{x}\_3(t)) = K\_0 - K\_1 \frac{1}{\mathbf{x}\_3(t)} - K\_2 \mathbf{x}\_3(t) + K\_3 \ln\left(\mathbf{x}\_3(t)\right) + K\_4 \ln\left(|1 - \mathbf{x}\_3(t)|\right) \tag{8}$$

For simulation purpose, in order to validate the proposed battery model, as well as to prove the effectiveness of the SOC estimation strategies developed in Section 3, we consider as the most suitable nominal values for Li-Ion battery EMC parameters and OCV coefficients the same values that are carefully chosen for model validation in [8] as follows

1. Li-Ion battery ECM parameters:

resistances on the battery dynamics, as is described in detail in [16]. Also, in [16] the HEV is considered as a "complex production of mechanical-electrical integration", for which the power supply typically being in the range 100–500 V is obtained by means of several series battery packs. Amongst the BMS hardware devices consisting of high voltage components, the traction battery, electrical motor, energy recycle device, the battery charger and its auxiliary device deal with a large current and insulation [16], thus insulation issue must be under consideration since the design stage. As is stated in [16], the poor working conditions, such as shaking, corrosion, changes in temperature and humidity, "could cause fast aging of the power cable and insulation materials, or even brake the insulation, which would decrease the insulation strength and endanger personnel". Thus, needs to ensure safety operating conditions for personnel are

The National Standard (NS) 18384.3-2001, stipulates several safety requirements for HEVs, especially for insulation resistance state, measurement method [16]. According to NS, the insulation state of an EV is evaluated according to the ground insulation resistance of the DC positive and negative bus, Vp, and Vn respectively, as is shown in Figure 1(b) [16]. The definition of traction battery insulation resistance in NS "is the relative resistance to maximum leakage current (in the worst condition) where there is a short between the traction battery and ground (electric chassis)" [16]. Also, "under the conditions that the maximum AC voltage is less than 660 V, the maximum DC voltage is less than 1000 V and the car weight is less than 3500 Kg, the requirements of the high voltage security are as follows: (1) personnel's security voltage is less than 35 V, or the product of the contact current with a person and the duration of time is less than 30 mA s; insulation resistance divided

formulated in [16]. Thus, to ensure the insulation security of on-board BMS, it is necessary to detect the insulation resistance and raise an alarm in time. Currently, the insulation measurement methods, such as passive ground detection and active ground detection, include the AC voltage insulation measurement method, and the DC voltage insulation measurement method [16]. The passive ground detection insulation measurement method principle is presented in Figure 1(b) where the presence of leakage current is detected by discovering the resultant

By simple manipulations of Ohm's, current and voltage Kirchhoff's laws applied to the proposed Li-Ion battery's EMC can be written the following first order differential equations that

> i tð Þ C1

> i tð Þ C2

v tðÞ¼ OCV SOC t ð Þ� ð Þ vC<sup>1</sup> ð Þ� t vC<sup>2</sup> ðÞ� t Ri tð Þ

vC<sup>1</sup> ð Þþ t

vC<sup>2</sup> ð Þþ t

, and preferably more 500 ΩV�<sup>1</sup>

", as is

(6)

required to evaluate the insulation conditions for entire HEV's BMS.

64 New Trends in Electrical Vehicle Powertrains

by the battery rated voltage should be more than 100 ΩV�<sup>1</sup>

magnetic field generated by AC voltage around the mutual inductor.

are most suitable to capture the entire dynamics of EMC in time domain:

R1C<sup>1</sup>

R2C<sup>2</sup>

Qnom i tð Þ

dt ¼ � <sup>η</sup>

2.3. Li-Ion battery continuous time state-space representation

dt ¼ � <sup>1</sup>

dt ¼ � <sup>1</sup>

dvC<sup>1</sup>

dvC<sup>2</sup>

d SOC ð Þ

	- The value of the battery capacity, Q = 6 Ah (Amperes hours), the voltage nominal value of the battery, Vnom = 3.6 V (Volts), and the coulombic efficiency, η = 1, for charging cycle, and η = 0.85, for discharging cycle
	- K0 = 4.23, K1 = 3.86E-05, K2 = 0.24, K3 = 0.22, K4 = �0.04

The OCV xð Þ <sup>3</sup>ð Þt is a nonlinear function of SOC, i.e.x3, similar as in [1, 3, 4, 8–11], increasing the accuracy of the Li-Ion battery EMC model, known as the EMC combined model, one of the most accurate formulation by combining Shepherd, Unnewehr universal, and Nernst models introduced in [4, 8]. The tuning values of model parameters ð Þ K0;K1;K2;K3;K<sup>4</sup> are chosen to fit the model to the manufacture's data by using a least squares curve fitting identification method OCV = h (SOC), as is shown in [3, 4, 8], where the OCV curve is assumed to be the average of the charge and discharge curves taken at low currents rates from fully charged to fully discharged battery [3, 4, 6, 8]. Also, by using low charging and discharging DC currents can be minimized the battery cell dynamics. A simple offline (batch) processing method for parameters calculation is carried out in [3, 4, 6, 8].

2.5. Li-Ion battery equivalent model in ADVISORY MATLAB platform—case study

For EMC validation purpose we compare the results of the tests using a NREL with two capacitors already integrated in an Advanced Vehicle Simulator (ADVISOR) MATLAB platform, developed by US National Renewable Energy Laboratory (NREL), as is shown in [9, 10]. The NREL Li-Ion battery model approximates with high accuracy the Li-Ion battery model 6 Ah and nominal voltage of 3.6 V, manufactured by the company SAFT America, as is mentioned in [9, 10]. Also, for simulation purpose and comparison of the tests results, the EMC battery model is incorporated in a BMS' HEV, and its performance is compared to those obtained by a particular Japanese Toyota Prius, selected as an input vehicle in ADVISOR MATLAB platform, under standard initial conditions and the setup shown in Figure 2.

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67

Among different driving speed cycles for a large collection of cars provided by the ADVISOR US Environmental Protection Agency (EPA), in our case study for Toyota Prius HEV car, the speed profile is selected as an Urban Dynamometer Driving Schedule (UDDS), as is shown in

Figure 2. The setup of the Japanese Toyota Prius HEV' car in ADVISOR MATLAB platform under standard initial

Figures 3 and 4, respectively.

conditions (initial value of SOC of 70%).

#### 2.4. Li-Ion battery discrete time state-space representation

We introduce now the following new notations:

x1ð Þ¼ k x1ð Þ kTs , x2ð Þ¼ k x2ð Þ kTs , x3ð Þ¼ k x3ð Þ kTs , for state variables, u kð Þ¼ u kT ð Þ<sup>s</sup> ,ykð Þ¼ y kT ð Þ<sup>s</sup> for input current profile, and output battery terminal DC samples voltage respectively, commonly used for discrete time description with the sampling time Ts½ �s , similar as in [1, 3, 4, 6, 8–11].

With these notations, a discrete time state space representation of the combined EMC Li-Ion battery model is obtained::

$$
\begin{bmatrix}
\mathbf{x}\_1(k+1) \\
\mathbf{x}\_2(k+1) \\
\mathbf{x}\_3(k+1)
\end{bmatrix} = \begin{bmatrix}
1 - \frac{T\_s}{T\_1} & 0 & 0 \\
0 & 1 - \frac{T\_s}{T\_2} & 0 \\
0 & 0 & 1
\end{bmatrix} \begin{bmatrix}
\mathbf{x}\_1(k) \\
\mathbf{x}\_2(k) \\
\mathbf{x}\_3(k)
\end{bmatrix} + \begin{bmatrix}
\frac{T\_s}{T\_1} \\
\frac{T\_s}{T\_2} \\
\frac{T\_s}{T\_2} \\
\end{bmatrix} u(k) \tag{9}
$$

$$
\begin{bmatrix}
\mathbf{x}\_1(k) & \mathbf{x}\_2(k) & \mathbf{x}\_3(k) & \mathbf{P} \mathbf{W} \mathbf{W} \mathbf{C} \mathbf{O} \mathbf{C} \mathbf{M} \\
\end{bmatrix} \tag{10}
$$

y kð Þ¼ h xð Þ� <sup>3</sup>ð Þk x1ð Þ� k x2ð Þ� k Ru kð Þ¼ OCV SOC k ð Þ� ð Þ x1ð Þ� k x2ð Þ� k Ru kð Þ (10)

Further, a compact discrete time of combined ECM Li-Ion battery state space representation (9) can be written in the following matrix form:

$$\begin{aligned} \mathbf{x}(k+1) &= A\mathbf{x}(k) + Bu(k) \\ \mathbf{y}(k) &= \mathbf{C}\mathbf{x}(k) + Du(k) + \Psi(\mathbf{x}\_3(k)) \end{aligned}$$

$$\mathbf{x}(k) = \begin{bmatrix} \mathbf{x}\_1(k) \ \mathbf{x}\_2(k) \ \mathbf{x}\_3(k) \end{bmatrix}^T,\\ A = \begin{bmatrix} 1 - \frac{T\_s}{T\_1} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & 1 - \frac{T\_s}{T\_2} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & 1 \end{bmatrix},\\ B = \begin{bmatrix} T\_s \\\\ \frac{T\_s}{T\_2} \\\\ -\frac{\eta T\_s}{Q\_{\text{max}}} \end{bmatrix}, \mathbf{C} = \begin{bmatrix} -1 & -1 & -K\_2 \end{bmatrix} \tag{11}$$

$$D = -R\_r \Psi(\mathbf{x}\_3(k)) = K\_0 - K\_1 \frac{1}{\mathbf{x}\_3(k)} + K\_3 \ln\left(\mathbf{x}\_3(k)\right) + K\_4 \ln\left(|1 - \mathbf{x}\_3(k)|\right) \tag{12}$$

where the nonlinear function Ψð Þ x3ð Þk can be further linearized around an operating point to get a linear Li-Ion battery combined EMC, easily to be implemented in real-time. To analyze the behavior of the proposed Li-Ion battery EMC for different driving conditions such as urban, suburban and highway, some different current profiles tests will be introduced in the next subsection.

#### 2.5. Li-Ion battery equivalent model in ADVISORY MATLAB platform—case study

average of the charge and discharge curves taken at low currents rates from fully charged to fully discharged battery [3, 4, 6, 8]. Also, by using low charging and discharging DC currents can be minimized the battery cell dynamics. A simple offline (batch) processing method for

x1ð Þ¼ k x1ð Þ kTs , x2ð Þ¼ k x2ð Þ kTs , x3ð Þ¼ k x3ð Þ kTs , for state variables, u kð Þ¼ u kT ð Þ<sup>s</sup> ,ykð Þ¼ y kT ð Þ<sup>s</sup> for input current profile, and output battery terminal DC samples voltage respectively, commonly used for discrete time description with the sampling time Ts½ �s , similar as in [1, 3, 4, 6, 8–11].

With these notations, a discrete time state space representation of the combined EMC Li-Ion

0 0

y kð Þ¼ h xð Þ� <sup>3</sup>ð Þk x1ð Þ� k x2ð Þ� k Ru kð Þ¼ OCV SOC k ð Þ� ð Þ x1ð Þ� k x2ð Þ� k Ru kð Þ (10)

2 6 4

x1ð Þk x2ð Þk x3ð Þk

Ts C1 Ts T2 � <sup>η</sup>Ts Qnom

u kð Þ (9)

, C ¼ �½ � 1 � 1 � K<sup>2</sup> ,

(11)

Ts C1 Ts T2 � <sup>η</sup>Ts Qnom

<sup>x</sup>3ð Þ<sup>k</sup> <sup>þ</sup> <sup>K</sup>3ln ð Þþ <sup>x</sup>3ð Þ<sup>k</sup> <sup>K</sup>4ln ð Þ <sup>j</sup><sup>1</sup> � <sup>x</sup>3ð Þj <sup>k</sup> (12)

T2 0

Further, a compact discrete time of combined ECM Li-Ion battery state space representation (9)

x kð Þ¼ þ 1 Ax kð Þþ Bu kð Þ y kð Þ¼ Cx kð Þþ Du kð Þþ Ψð Þ x3ð Þk

0 0

, B ¼

T2 0

parameters calculation is carried out in [3, 4, 6, 8].

We introduce now the following new notations:

x1ð Þ k þ 1 x2ð Þ k þ 1 x3ð Þ k þ 1

can be written in the following matrix form:

x kð Þ¼ ½ � <sup>x</sup>1ð Þ<sup>k</sup> <sup>x</sup>2ð Þ<sup>k</sup> <sup>x</sup>3ð Þ<sup>k</sup> T, A <sup>¼</sup>

subsection.

<sup>1</sup> � Ts T1

> <sup>1</sup> � Ts T1

D ¼ �R, Ψð Þ¼ x3ð Þk K<sup>0</sup> � K<sup>1</sup>

0 1 � Ts

0 01

1

where the nonlinear function Ψð Þ x3ð Þk can be further linearized around an operating point to get a linear Li-Ion battery combined EMC, easily to be implemented in real-time. To analyze the behavior of the proposed Li-Ion battery EMC for different driving conditions such as urban, suburban and highway, some different current profiles tests will be introduced in the next

0 1 � Ts

0 01

battery model is obtained::

66 New Trends in Electrical Vehicle Powertrains

2 6 4

2.4. Li-Ion battery discrete time state-space representation

For EMC validation purpose we compare the results of the tests using a NREL with two capacitors already integrated in an Advanced Vehicle Simulator (ADVISOR) MATLAB platform, developed by US National Renewable Energy Laboratory (NREL), as is shown in [9, 10]. The NREL Li-Ion battery model approximates with high accuracy the Li-Ion battery model 6 Ah and nominal voltage of 3.6 V, manufactured by the company SAFT America, as is mentioned in [9, 10]. Also, for simulation purpose and comparison of the tests results, the EMC battery model is incorporated in a BMS' HEV, and its performance is compared to those obtained by a particular Japanese Toyota Prius, selected as an input vehicle in ADVISOR MATLAB platform, under standard initial conditions and the setup shown in Figure 2.

Among different driving speed cycles for a large collection of cars provided by the ADVISOR US Environmental Protection Agency (EPA), in our case study for Toyota Prius HEV car, the speed profile is selected as an Urban Dynamometer Driving Schedule (UDDS), as is shown in Figures 3 and 4, respectively.

Figure 2. The setup of the Japanese Toyota Prius HEV' car in ADVISOR MATLAB platform under standard initial conditions (initial value of SOC of 70%).

Figure 3. The UDDS cycle profile for Toyota Prius car speed test represented on the ADVISOR MATLAB platform.

Figure 4. The corresponding car speed, current profile, exhaust gases emission, and SOC curves to UDDS cycle tests in

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69

Figure 5. The SOC curves for Li-Ion battery EMC and NREL/6 Ah SAFT models in MATLAB R17a (the top graphs), and

its corresponding EMC battery output terminal voltage, Vbatt (the second bottom graph).

ADVISOR MATLAB platform.

Both, the driving UDDS cycle for car speed (mph) profile and its corresponding Li-Ion battery UDDS current profile are represented separately in the same ADVISOR MATLAB platform, as is shown in the first two corresponding top graphs from Figure 4.

Also, the exhaust gases emission and SOC curves, as a result to the same UDDS cycle test on the ADVISOR MATLAB platform, are shown in the last two bottom graphs of Figure 4, accompanied by details in the next section.

The Li-Ion battery EMC and the ADVISOR SOCs, for the same initial condition, SOC = 70% and the same UDDS cycle profile input current test, are implemented in a real-time MATLAB environment, as is shown on the same graph at the top of Figure 5. The SOC simulations reveal a great accuracy between Li-Ion battery EMC and its corresponding ADVISOR NREL model represented on ADVISOR MATLAB platform, as is shown in Figure 4, thus an expected confirmation of the EMC validation results.

The second graph from the bottom in the same Figure 5 depicts the EMC battery output terminal DC instantaneous voltage y ¼ v tð Þ, as a response to the UDDS cycle current input profile test.

The simulation results of Li-Ion battery EMC output voltage show a stabilization of the battery output voltage value at the end of UDDS cycle, i.e. after 1370 seconds.

Estimation Techniques for State of Charge in Battery Management Systems on Board of Hybrid Electric Vehicles… http://dx.doi.org/10.5772/intechopen.76230 69

Figure 4. The corresponding car speed, current profile, exhaust gases emission, and SOC curves to UDDS cycle tests in ADVISOR MATLAB platform.

Both, the driving UDDS cycle for car speed (mph) profile and its corresponding Li-Ion battery UDDS current profile are represented separately in the same ADVISOR MATLAB platform, as

Figure 3. The UDDS cycle profile for Toyota Prius car speed test represented on the ADVISOR MATLAB platform.

Also, the exhaust gases emission and SOC curves, as a result to the same UDDS cycle test on the ADVISOR MATLAB platform, are shown in the last two bottom graphs of Figure 4,

The Li-Ion battery EMC and the ADVISOR SOCs, for the same initial condition, SOC = 70% and the same UDDS cycle profile input current test, are implemented in a real-time MATLAB environment, as is shown on the same graph at the top of Figure 5. The SOC simulations reveal a great accuracy between Li-Ion battery EMC and its corresponding ADVISOR NREL model represented on ADVISOR MATLAB platform, as is shown in Figure 4, thus an expected

The second graph from the bottom in the same Figure 5 depicts the EMC battery output terminal DC instantaneous voltage y ¼ v tð Þ, as a response to the UDDS cycle current input

The simulation results of Li-Ion battery EMC output voltage show a stabilization of the battery

is shown in the first two corresponding top graphs from Figure 4.

output voltage value at the end of UDDS cycle, i.e. after 1370 seconds.

accompanied by details in the next section.

68 New Trends in Electrical Vehicle Powertrains

confirmation of the EMC validation results.

profile test.

Figure 5. The SOC curves for Li-Ion battery EMC and NREL/6 Ah SAFT models in MATLAB R17a (the top graphs), and its corresponding EMC battery output terminal voltage, Vbatt (the second bottom graph).

well documented in [3, 4, 8], excluding the linearization steps required by EKF. In addition, the results of UKF real-time implementation for the majority of similar applications are encouraging, and it seems that the anticipated performance of this approach is slightly superior compared to EKF [3, 4, 8]. Furthermore, the nonlinear UKF SOC estimator can be extended to the applications where the distributions of the process and measurements noises are not Gaussian [19]. Concluding, the implementation simplicity and a great estimation accuracy of the proposed UKF SOC estimator recommend it as the most suitable estimator to be used in almost all similar applications. Explicitly, the nonlinear UKF estimator is an algorithm of "predictor– corrector" type applied to a nonlinear discrete - time systems, such as those described in [3–7]:

Estimation Techniques for State of Charge in Battery Management Systems on Board of Hybrid Electric Vehicles…

where fð Þ: , gð Þ: are two nonlinear functions of system states and inputs; w kð Þ, v kð Þ are zeromean, uncorrelated process, and measurement Gaussian noise respectively. Since the noise injected in the state and output equations are randomly, also the system state vector and the output become random variables, having the mean and covariance matrices as a statistics. The nonlinear UKF SOC estimator has a "predictor-corrector" structure that propagates the mean and the covariance matrix of a Gaussian distribution for the random state variable x kð Þ in a recursively way, in both prediction and correction phases, as is stated in [3–7]. The propagation of these first two moments is performed by using an unscented transformation (UT) to calculate the statistics of any random variable that undergoes a nonlinear transformation [5–7]. As is stated in the original and fundamental work [5], by means of this UT transformation "a set of points, the so-called sigma points, are chosen such that their sample mean, and sample covariance matrix are x and Pxx, respectively". Also, "the nonlinear function gð Þ: is applied to each sigma points generating a "cloud" of transformed points with the mean y and the covariance matrix Pyy". Furthermore, since statistical convergence is not an issue, only a very small number of sample points is enough to capture high order information about the random state variable distribution. The different selection strategies of the sigma points and the UKF algorithm steps in a "predictorcorrector" structure, as well as its tuning parameters are well documented in the literature, and for details we recommend the fundamental work [5]. Since we follow the same design procedure steps for building a nonlinear UKF SOC estimator as in [5–7], we are focused only to the

The simulation results of the real-time implementation of proposed UKF SOC nonlinear estimator in MATLAB R17a environment are shown in Figures 7 and 8. In Figure 7 are presented the simulation results for EMC SOC true value versus EMC SOC-UKF and ADVI-

Also, at the bottom of Figure 7 is shown the EMC battery output terminal true value DC voltage versus EMC-UKF battery output terminal estimated DC voltage. The simulation results reveal an accurate SOC estimation values, and also a very good robustness of UKF estimator to the changes in initial SOC value (guess value, SOCinit = 20%). In Figure 8 are shown the Li-Ion EMC polarization voltages versus Li-Ion EMC -UKF estimates, and in Figure 9 is represented the robustness of EMC-UKF SOC nonlinear estimator to a gradual

implementation aspects.

SOR MATLAB platform estimated values.

x kð Þ¼ þ 1 f xk ð Þþ ð Þ; u kð Þ; k w kð Þ (13)

y kð Þ¼ gxk ð Þþ ð Þ; u kð Þ; k νð Þk (14)

http://dx.doi.org/10.5772/intechopen.76230

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Figure 6. The Li-Ion battery EMC OCV curve during a complete discharging cycle at 1C-rate (top) and the corresponding battery SOC (bottom).

Likewise, the Li-Ion battery OCV for a discharging cycle at 1C-rate (i.e. 6 Ah 1/h = 6A constant input discharging current) is shown in the top of Figure 6, and its corresponding SOC during the same discharging cycle is revealed on the bottom graph of the same Figure 6 respectively.
