COUPLED ELECTRO-THERMAL MODEL
by
ALOISIO HENRIQUE KAWAKITA DE SOUZA B.S.M.E, Universidade Federal de São João del Rei, 2015
A thesis submitted to the Graduate Faculty of the University of Colorado Colorado Springs
in partial fulfillment of the requirements for the degree of
Master of Science
Department of Electrical and Computer Engineering 2020
has been approved for the
Department of Electrical and Computer Engineering by
M. Scott Trimboli, Chair
Gregory Plett
Omid Semiari
May 20th 2020 Date
pled Electro-Thermal Model
Thesis directed by Professor M. Scott Trimboli
ABSTRACT
This thesis presents a novel application of model predictive control (MPC) to the problem of managing lithium-ion cell performance using a highly accurate low-order electro-thermal equivalent circuit model and is experimentally validated via laboratory experiments. The proposed method uses MPC to ensure compliance with cell-level operational limits and has the potential to extend lifetime by mitigating certain mechanisms of cell degradation leading to capacity fade. A five-state electro-thermal model is developed, characterized and validated. Implementation employs a nonlinear Kalman filter for state estimation and MPC for controlling charge/discharge current. The complete method is experimentally validated using a 26650 cylindrical format lithium-iron phosphate (LFP) cell. The new method: (i) develops a fully coupled electro-thermal model of cell dynamics; (ii) incorporates a dynamic hysteresis model to improve accuracy; (iii) employs a nonlinear Kalman filter for accurate state estimation to inform the MPC algorithm; and (iv) utilizes a modified form of MPC which correctly models direct feed-through behavior to characterize ohmic resistance. Based on this MPC framework, a temperature-based maximum power limits algorithm is also developed.
I dedicate this work in memory of my dear late father Aloisio Souza who passed away while I was pursuing this degree. He is the one who encouraged and inspired me to go to graduate school overseas and made this a real possibility. Without him, I would not be the
I would like to thank...
... my advisor, Prof. Trimboli, whose support and guidance has made the completion of this work possible , and also for all the opportunities I was given. His confidence and optimism in me pushed me forward to develop myself more as a student, researcher and engineer than I could possibly have imagined.
...Prof. Plett, for his continuous support.
...Electric Power Systems and the Brazilian government (Science without Borders grant) for funding part of my M.S degree at UCCS.
...Dr. Xavier, for his friendship and collaboration. ...my labmates for helping me in the lab.
...my mom Sonia Kawakita and my brother Alexandre Kawakita de Souza for their unconditional love and support.
...my wife, Jordan Pattee, for her love, patience and support throughout my gradu-ate career. Her support is one of the biggest reasons I keep motivgradu-ated to pursue all my professional and personal goals.
CHAPTER
1 Introduction 1
1.1 Motivation . . . 1
1.2 Lithium-ion batteries . . . 4
1.3 Lithium-ion battery degradation . . . 6
1.4 Battery management systems . . . 11
1.5 Related work . . . 12
1.6 Original work . . . 13
2 Coupled Electro-Thermal Model 16 2.1 Introduction . . . 16
2.2 Electrical equivalent-circuit model . . . 17
2.3 Thermal model . . . 20
2.4 Electro-thermal coupling . . . 22
2.5 Parameterization of the coupled electro-thermal model. . . 24
2.6 Summary . . . 35
3 Kalman Filter 37 3.1 Introduction . . . 37
3.2 Linear Kalman filter . . . 38
3.3 Extended Kalman filter (EKF) . . . 42
3.4 The sigma-point Kalman filter . . . 51
3.5 Summary . . . 58
4 Model Predictive Control 59 4.1 Introduction . . . 59
4.4 MPC stability . . . 79
4.5 MPC tuning parameters . . . 86
4.6 Incorporating feedthrough term . . . 88
4.7 Adaptive control weighting . . . 91
4.8 Summary . . . 94
5 Lithium-ion Battery Charge-Control using MPC 95 5.1 Introduction . . . 95
5.2 Fast charging problem . . . 95
5.3 Charge-control design using MPC . . . 96
5.4 Charge-control architecture . . . 104
5.5 Simulation results and discussion . . . 106
5.6 Experimental validation . . . 109
5.7 Summary . . . 112
6 Temperature-Based State of Power Estimation using MPC 114 6.1 Introduction . . . 114
6.2 Problem definition . . . 115
6.3 MPC-based power estimation method formulation . . . 116
6.4 Bisection Method (BSM) . . . 119
6.5 Simulation results . . . 120
6.6 Summary . . . 131
7 Conclusion and Future Work 134 7.1 Future work . . . 136
7.2 Publications . . . 138
TABLE
2.1 Battery cell specifications provided by the manufacturer. . . 16
2.2 Summary of the CET model equations. . . 25
2.3 Voltage estimation error. . . 31
2.4 Identified parameters. . . 34
3.1 Initial EKF parameters. . . 48
3.2 SPKF weighting constants, h =√3. . . 52 3.3 Initial SPKF parameters. . . 56 5.1 MPC constraints. . . 105 5.2 MPC simulation parameters . . . 106 5.3 Average power. . . 109 5.4 MPC algorithm parameters . . . 111
6.1 Maximum available power constraints. . . 117
6.2 MPC power estimation tuning parameters. . . 120
FIGURE
1.1 Lithium-ion battery price survey [1]. . . 1
1.2 Global EV and internal combustion vehicle(ICE) sales [1]. . . 2
1.3 Energy density of secondary batteries [2]. . . 3
1.4 Nissan Leaf EV available range at various temperatures [3]. . . 4
1.5 Typical electrochemical cell [2]. . . 5
1.6 Schematic of the electrochemical process in a Li-ion cell [4]. . . 6
1.7 (a) Measured SOH curves as a function of time for cycling at a rate of 1C and different temperatures. (b) Arrhenius plot for the aging behavior of 18650 cells cycled at 1C in a temperature range of 20◦C to 70 ◦C. The solid lines correspond to linear fits of the data points of the respective temperature ranges below and above 25◦C[5]. . . . 8
1.8 Temperature effect of capacity loss evaluation during CC cycle tests at 1C: (a) Experimental capacity loss measurements at low temperatures (0 ◦C to 25 ◦C), (b) Experimental capacity loss measurements at high temperatures (25◦Cto 55 ◦C) [6]. . . . 9
1.9 Lithium-ion battery cell-, module-, and pack-level demonstrated by two ve-hicle examples: Tesla Roadster and Nissan Leaf [7]. . . 12
2.1 A123 26650 battery cell. . . 16
2.2 Enhanced self-correcting cell model. . . 17
2.3 Equivalent-circuit lumped thermal model [8]. . . 20
2.4 Coupling between the electrical and thermal models (modified from [9]). . . . 23
2.5 OCV variation during the thermal cycle at SOC= 50%. . . 23
2.6 Temperature coefficient ∂OCV ∂T . . . 24
High Precision Battery Tester, (b) Thermal chamber Cincinnati Sub-zero . . 26 2.9 OCV relationship at 25 ◦C. . . 28
2.10 Input current for dynamic testing at 25◦C, (a) Applied current, (b) Zoom in
UDDS input current. . . 29 2.11 Output voltage of dynamic testing at 25 ◦C, (a) Output voltage, (b) Zoom
in UDDS output voltage. . . 29 2.12 ESC parameters from −25◦Cto 45 ◦C. . . 30
2.13 Validation of the ESC model created, (a) Voltage and estimates at 25◦C,
(b) Voltage estimation error. . . 31 2.14 Input current for thermal model parameter identification, (a) Input current,
(b) Zoomed in input current. . . 33 2.15 Surface and air temperature. . . 34 2.16 Validation of the thermal model, (a) Validation using the identification data,
(b) Validation using an UDDS profile. . . 35 3.1 KF algorithm steps [10]. . . 39 3.2 OCV variation with respect to state of charge. . . 48 3.3 Input data, (a) UDDS input profile, (b) Zoomed in the first UDDS cycle. . . . 49 3.4 Output data, (a) Output Cell voltage, (b) Zoomed-in the first UDDS cycle. . 49 3.5 (a) Surface and air temperature data, (b) Zoomed-in the first UDDS cycle. . . 49 3.6 EKF Tcestimate, (a) Core temperature estimate and error bounds, (b)
Zoomed-in the first UDDS cycle. . . 50 3.7 EKF SOC estimate, (a) SOC estimate and its error bounds (b) Zoomed in
the first UDDS. . . 50 3.8 Approximating statistics using sigma points [10]. . . 53 3.9 Sigma points organized in convenient matrix form [10]. . . 54
of the sigma points [10]. . . 54
3.11 a priori computation [10]. . . 55
3.12 SPKF SOC estimate, (a) SOC and bounds, (b) Zoomed-in the first UDDS. . 57
3.13 SPKF Tc estimate, (a) Core temperature and bounds, (b) Zoomed-in the first UDDS cycle. . . 57
4.1 MPC strategy. . . 59
4.2 Block diagram of MPC with observer [11]. . . 68
4.3 Illustration of constrained optimal solution [11]. . . 71
4.4 Illustration of the constrained optimization problem with inequality con-straints. [11] . . . 74
5.1 Decay of the Lyapunov function. . . 104
5.2 Charge-control architecture. . . 105
5.3 Simulation results, (a) Charging current, (b) Cell voltage, (c) SOC, (d) Core temperature. . . 107
5.4 Charge-control current and voltage zoomed in, (a) Charge current, (b) Cell voltage. . . 107
5.5 Instantaneous power of MPC and OPT simulation results. . . 109
5.6 Laboratory experiment setup schematics. . . 110
5.7 Laboratory experiment setup, (a) Computer, programmable power supply, multimeter and arduino, (b) Battery cell placed inside the thermal chamber.. 111
5.8 Simulation results, (a) Charging current, (b) Cell voltage, (c) SOC, (d) Core temperature. . . 112
5.9 State of charge and voltage zoomed in, (a) State of charge estimate and error bounds, (b) Core temperature estimate and error bounds. . . 113
6.1 (a) Traditional methods. (b) MPC-based method proposed in [12]. . . 116
6.4 Zoomed in maximum discharge power available using MPC and OPT meth-ods. (a) Zoomed in first UDDS cycle, (b) Zoomed in second UDDS cyle. . . . 122 6.5 MPC, OPT and BSM maximum discharge power available estimation within
∆T at k = 3500 s: (a) discharge current; (b) voltage, (c) core temperature, (d) SOC, and (e) instantaneous power.. . . 123 6.6 MPC, OPT and BSM maximum discharge power available estimation within
∆T at k = 4121 s: (a) discharge current; (b) voltage, (c) core temperature, (d) SOC, and (e) instantaneous power.. . . 124 6.7 MPC, OPT and BSM maximum discharge power available estimation within
∆T at k = 4378 s: (a) discharge current; (b) voltage, (c) core temperature, (d) SOC, and (e) instantaneous power.. . . 126 6.8 Zoomed in maximum charge power available using MPC and OPT methods.
(a) Zoomed in first UDDS cycle, (b) Zoomed in second UDDS cyle. . . 127 6.9 MPC, OPT and BSM maximum charge power available estimation within
∆T at k = 4121 s: (a) discharge current; (b) voltage, (c) core temperature, (d) SOC, and (e) instantaneous power.. . . 128 6.10 MPC, OPT and BSM maximum charge power available estimation within
∆T at k = 4500 s: (a) discharge current; (b) voltage, (c) core temperature, (d) SOC, and (e) instantaneous power.. . . 129 6.11 MPC, OPT and BSM maximum charge power available estimation within
∆T at k = 7000 s: (a) discharge current; (b) voltage, (c) core temperature, (d) SOC, and (e) instantaneous power.. . . 130 6.12 Maximum charge/discharge power available computation using MPC with
and without temperature constraint. . . 131 6.13 MPC, OPT and BSM maximum charge power available estimation within
∆T at k = 4121 s: (a) discharge current; (b) voltage, (c) core temperature, (d) SOC, and (e) instantaneous power.. . . 132
Introduction
1.1
Motivation
The electrification of the transport sector has emerged as a promising technology to reduce dependence on fossil fuels and carbon emissions due to the tightening of emission regulations in the last years. Moreover, government incentives, growth of the charging infrastructure deployment and decrease of battery prices have boosted the adoption of electric vehicles (EVs) [13–16]. According to [1], we can expect price parity between electric vehicles and internal combustion vehicles by the mid-2020s, and, more than half of new cars sold in the world are expected to be electric vehicles (EVs) by 2040.
Lithium-ion batteries (LIBs) have emerged as the major technology used in EVs (in-cluding hybrid electric vehicles (HEV), plug-in hybrid electric vehicles (PHEV)) for energy
Figure 1.2: Global EV and internal combustion vehicle(ICE) sales [1].
storage. This preference towards LIBs is mainly due to their superior performance over other battery chemistries (lead acid (PbA), nickel cadmium (NiCd) and nickel metal hy-dride (NiMh)) used in EVs [17–19]. Particular advantages include:
• Higher energy density: LIBs have a very high energy density (100 − 265Wh.kg−1 or
250 −670Wh.kg−1), allowing higher storage of energy in a small space, and as a result, less weight and longer driving range.
• No memory effect: After partial discharge/charge cycles, LIBs do not present dimin-ishing charge capacity.
• High Voltage: Allowing higher power delivery
• Longer cycle life: LIBs can cycle more times without significant loss of capacity (≈20% of capacity loss after 2000 cycles)
Despite the advantages, LIBs have drawbacks that create barriers to commercialization of EVs: Battery life, cost, safety, and time-to-charge are critical factors that influence the consumer’s willingness to buy EVs. In order to extend battery life and guarantee safety, an accurate cell-level estimation of the internal battery temperature, state-of-charge (SOC) and voltage in real time is crucial to good cell management.
When LIBs are operated at high temperatures, the battery capacity and available power decreases and may also cause thermal runaway (which occurs when extremely high
Figure 1.3: Energy density of secondary batteries [2].
temperature triggers an irreversible exothermic reaction that can increase the temperature further) [20]. At low temperatures, the performance of LIB performance decreases notably (see Fig. 1.4). The chemical reactions inside the battery occur more slowly, which accelerates battery aging, consequently causing capacity fade and internal resistance increase [21–23]. Nonetheless, fire hazards can occur by overcharging LIBs, since the additional energy intro-duced to the battery cell can lead to lithium deposition and electrolyte decomposition, which may cause fire or even explosion [24–26]. Overdischarging LIBs may cause capacity loss due to copper dissolution from the anode collector and deposition of copper on the surfaces of the cathode, separator and anode, which hinders the flow and intercalation/deintercalation of lithium-ions during charge and discharge [27, 28].
Apart from the high cost of LIBs, the safety concerns mentioned above add to the high price of EVs due to the need to have a battery management system (BMS). One of the crucial tasks of a BMS is to take voltage, current and temperature measurements and use them to estimate the internal states of the batteries (which cannot be directly measured) and then use these estimates to monitor conditions and prevent the batteries from entering undesirable states of operation, thus guaranteeing safe and reliable operation. Moreover, these cell measurements can also be used in advanced control strategies to mitigate cell degradation in order to extend battery life and increase performance.
Given the background, this dissertation aims to explore temperature-based cell man-agement strategies to ensure safe operation and mitigate cell degradation. First, a coupled
Figure 1.4: Nissan Leaf EV available range at various temperatures [3].
electro-thermal equivalent circuit model is developed and parameterized, then, this model will be used to design a nonlinear Kalman filter estimator, a cell-level fast charging and max-imum power limit estimation algorithms accounting for thermal and electrical constraints. Online experimental results will be also shown in order to validate the control design pro-posed in this thesis.
1.2
Lithium-ion batteries
Traditional electrochemical cells comprise four main components: negative electrode, posi-tive electrode, electrolyte and separator. The negaposi-tive electrode is often a metal or an alloy or hydrogen. The positive electrode is often a metallic oxide, sulfide, or oxygen. The elec-trolyte is an ionic conductor that provides the medium for ion charge transfer between the electrodes. The separator isolates the positive and negative electrodes to avoid self discharge of the cell [2].
During the discharge process, electrochemical potential energy at the negative elec-trode (anode) favors a chemical process of oxidation (losing electrons to theexternal circuit) and release positively charged ions (cations) into the electrolyte. Also, electrochemical po-tential at the positive electrode (cathode) favors a chemical process of reduction (accepting
Figure 1.5: Typical electrochemical cell [2].
electrons from the external circuit) and accept positively charge ions (cations) from the elec-trolyte. During the charge process, cations move from the positive electrode to the negative electrode through electrolyte and electrons move from positive to negative electrode through the external circuit. The resulting potential difference between the terminals is called the cell voltage [2].
It is important to understand that the voltage and current generated by an electro-chemical cell is directly related to the materials used in the electrodes and electrolyte. The electrode potential describes the propensity of a metal or compound to gain or lose electrons in relation to another material. Compounds with negative electrode potential are used for negative electrodes, and compounds with positive electrode potential are used for positive electrodes. The larger the difference between the electrode potentials of the positive and negative electrodes, the greater the cell voltage and the greater the amount of energy that can be produced by the cell [2].
LIBs work slightly differently from the electrochemical cell introduced above. Rather than a redox reaction, they rely on an “intercalation” mechanism. This mechanism involves the insertion of lithium ions into the crystalline lattice of the host electrode without causing crystal structure change. The electrodes comprise open crystal structures (which allow the insertion or extraction of lithium ions in the vacant spaces) and the ability to accept compensating electrons at the same time [2]. As LIBs are cycled, lithium ions migrate
Figure 1.6: Schematic of the electrochemical process in a Li-ion cell [4].
between the positive and negative electrodes. The positive electrode is often a metal oxide with a layered structure, such as lithium cobalt oxide, or a material having a tunneled structure, such as lithium manganese oxide, on an aluminum current collector. The negative electrode material is typically a graphite carbon, also a layered material, on a copper current collector. When a LIB is charged, lithium ions are deintercalated from the positive electrode and intercalated into the negative electrode. The reverse process occurs during the discharge process [4].
1.3
Lithium-ion battery degradation
The goal of this section is to present the need for a constrained control strategy for battery cell management. As previously mentioned in Sect. 1.1, cell degradation can be caused by cell operation at high/low temperatures and when overcharged/overdischarged. Nonetheless, when LIBs operate in these conditions, they can also turn out to be a safety hazard due to the presence of combustible materials and oxidizing agents in LIBs. In order to enable
timely precautions and diagnostic approaches in EV applications, it is extremely important to understand the antecedents, effects and consequences of overdischarge/overcharge on safety performance and thermal stability of LIBs [26]. In this section, we present the major cell degradation mechanisms caused by cell operation at the undesirable states mentioned above.
1.3.1 Degradation caused at high/low temperatures
The effects of temperature in the range of −20◦C to 70◦C on the aging behavior of LIBs
are not the same. Below 25◦C, the degradation rates increase with decreasing temperature,
whereas above 25◦C, degradation is accelerated with increasing temperature. In [5], the
effect of temperature on the aging rate of a NMC cell and a post-mortem analysis were investigated quantitatively. It was found that the dominant degradation mechanism for temperatures below 25◦Cis plating of metallic lithium on the anodes and subsequent
reac-tion with the electrolyte, leading to loss of cyclable lithium and, consequently capacity fade. At temperatures above 25◦C, the increased heat accelerates other degradation reactions,
such as degradation of the cathode and SEI film on the anode, leading to capacity fade and internal resistance increase.
As one can see in Fig. 1.7a, there is a clear trend of rising aging rates with increasing temperature; this is in accordance with the general principle that chemical reactions are accelerated with increasing temperature, which indicates that the change of SOH of the batteries at high temperature is directly related to chemical degradation reactions. At low temperatures, this trend is reversed, one can see higher aging rates with decreasing temperatures. It can also be seen that SOH(%) decreases approximately with the same rate for T = 70◦Cand for T = 0◦C, indicating that the aging behavior at low temperatures is not
consistent with a chemical reaction that is directly responsible for the battery degradation [5]. Referencing Fig. 1.7b, one can see that the difference between the two temperature ranges is clearly visible in the Arrhenius plot1, where two straight lines with different slopes are drawn
1In chemical kinetics, an Arrhenius plot displays the logarithm of a reaction rate constant and is often
used to analyze the effect of temperature on the rates of chemical reactions. For a single rate-limited thermally activated process, an Arrhenius plot gives a straight line, from which the activation energy and the pre-exponential factor can both be determined.
(a) (b)
Figure 1.7: (a) Measured SOH curves as a function of time for cycling at a rate of 1C and different temperatures. (b) Arrhenius plot for the aging behavior of 18650 cells cycled at 1C in a temperature range of 20◦Cto 70◦C. The solid lines correspond to linear fits of the
data points of the respective temperature ranges below and above 25 ◦C[5].
for the two temperature ranges. A change in the slope in an Arrhenius plot is associated with a mechanism change, one can conclude that there are two different aging mechanisms for temperatures below and above 25◦Cas already mentioned.
In [6], the degradation of an LCO prismatic cell was investigated at different temper-atures. It was found that increasing the operating temperature increases the battery cell degradation which includes capacity fading, the effectiveness of the LCO electrode in storing Li-ions, charge transfer rate constant, effectiveness of the graphite electrode in providing its stored Li-ions, increase of the total resistance of electrode/electrolyte and cell impedance. It was also verified that the increase in the degradation rate of irreversible capacity loss of LIB with temperature is due mainly to the formation and modification of the surface films on the electrode and to the structural/phase changes of the LCO electrode.
In [29], the capacity loss caused by charging an LFP cell was investigated at different temperatures in order to develop a temperature-dependent degradation model (see Fig. 1.8). It was found that the loss of capacity occurs in both high and low temperatures. However, no post-mortem analysis was performed to investigate the mechanisms behind the observed loss of capacity.
(a) (b)
Figure 1.8: Temperature effect of capacity loss evaluation during CC cycle tests at 1C: (a) Experimental capacity loss measurements at low temperatures (0 ◦C to 25 ◦C), (b)
Experimental capacity loss measurements at high temperatures (25 ◦Cto 55 ◦C) [6].
1.3.2 Degradation caused by overdischarge/overcharge
Out of all the different forms of battery abuse, overcharge is one of the most important to consider for LIBs. Moreover, LIBs may experience overcharge not only during active charging but also within a battery pack where capacity mismatch between cells in the pack, can cause a difficulty in maintaining an identical SOC for every individual cell [30].
In [30], a series of experiments using NMC cells was conducted to investigate over-charge/overdischarge failure conditions. After post-mortem analysis, it was found that the mechanism of overcharge for LIBs consists of three stages:
Stage (a): During the normal charge phase and even the slight overcharge phase, lithium-ions are extracted from the cathode crystal and transferred to the graphite anode. It was not verified that any side reaction occurred inside of the battery; thus the battery remained stable due to the excess capacity of electrode materials at this stage.
Stage (b): When the battery is severely overcharged and the cathode potential in-creases, the metal on the cathode oxidizes from its metallic form to the metallic ion. Due to the concentration difference between the cathode and anode, the metal on the cathode diffuses to the anode. Consequently, the metallic lithium starts to deposit onto the anode
surface after the surface layer inside the anode is fully intercalated with lithium, causing an increase of the SEI layer and internal resistance.
Stage (c): As the battery continues to be charged, lithium plating occurs due to the continuous deposition of lithium ions onto the surface of the anode. In the meantime, the metallic ions transferred from the cathode to the anode are reduced to metallic form causing the formation of metallic dendrites. This causes an internal short-circuit followed by reactions between the cathode and anode materials, cathode decomposition, electrolyte decomposition, leading to gas generation. Once the inner pressure of these gasses exceeds the threshold, the battery fails.
The mechanism of overdischarge was also investigated in [30] and similarly divided in three stages
Stage (a): In the initial phase of discharge and slight overdischarge, the voltage applied forces the lithium ions to be extracted from the anode and transferred to the cathode. These lithium-ions intercalate into the cathode crystal. It was not verified that any side reaction occured inside the battery in this stage.
Stage (b): When the battery is severely overdischarged, the copper foil (current collec-tor) on the anode oxidizes to Cu+and continues to be oxidized to Cu2+. Then some of the
Cu2+ ions diffuse to the cathode side due to the difference of potential between the cathode
and anode. Due to the excessive loss of lithium ions in the anode, the SEI layer decomposes. On the other hand, the metallic lithium deposits onto the cathode surface.
Stage (c): As lithium-ions continue to deposit on the surface of the cathode, lithium plating occurs. The Cu2+ previously transferred to the cathode reduce to Cu+ ions and Cu
metal to form metallic dendrites. The dendrites in this stage grow continuously and can penetrate the separator causing an internal short circuit.
In [31], the failure mechanism of a commercial LFP cylindrical cell in the overcharge failure process was investigated and postmortem conducted. It was shown that the cell failed quickly after 10 cycles of 10% overcharging with increased surface temperature. It was also verified that the anode potential became more negative at the end of the overcharge process than during the normal charge process, while the cathode potential became more positive during the discharge process. It was concluded that the impurities containing iron in the raw
material formed dendrites leading to internal short-circuits because of the wider reduction potential range in the overcharge process.
The influence of overcharge/overdischarge on the impedance response of an LCO cell was investigated in [24]. It was found that the magnitude of the impedance increased greatly when the battery is overcharged or overdischarged. It was also shown that after being overcharged, the battery returned to a potential within the normal potential range before being overcharged. However, the impedance remained large. Such behavior may be attributed to the sensitivity of impedance to transport and kinetic limitations. On the other hand, the superposition of impedance curves showed that there were no changes besides of a change in the ohmic resistance.
1.4
Battery management systems
In an EV application, a battery pack comprises multiple battery modules, each of which usually contains 4 to 12 battery cells wired in series (many can be wired in parallel). There-fore, an EV battery pack may have hundreds (or thousands) of battery cells and every cell should be individually monitored to guarantee safety and efficient operation which requires a special embedded system called the BMS.
In order to extract the maximum efficiency from a battery pack safely, the battery cells should be completely charged and discharged at the same voltage, and at the same time to avoid SOC mismatch between the battery cells and extend battery life. Other BMS functions include [32]:
a) Sensing and high voltage control: measure voltage, current, temperature; control contactor, pre-charge; ground-fault detection, thermal management.
b) Protection against: overcharge, overdischarge, overcurrent, internal short circuit, overheating.
c) Interface: range estimation, communications, data recording, reporting.
d) Performance management: SOC estimation, power-limit estimation, cell balancing e) Diagnostics: abuse detection, state of health (SOH) estimation, state of life (SOL) estimation
Figure 1.9: Lithium-ion battery cell-, module-, and pack-level demonstrated by two vehicle examples: Tesla Roadster and Nissan Leaf [7].
1.5
Related work
The application of model predictive control (MPC) to a fast charge problem and maximum power limit computation was previously introduced in our research group by Xavier [12,33]. In his master’s thesis [33], MPC was applied to a fast charging problem using an equivalent circuit cell model (Thévenin model) incorporating a non-zero feedforward D matrix and electrical constraints such as voltage and SOC. In his PhD thesis [12], he developed a new approach using MPC as a “smart sensor” to estimate both the available discharge power and charging power that the can be provided by the battery pack using equivalent circuit model and physics-based model accounting for electrical and electrochemical constraints, respectively. Subsequently, also in our research group, Florentino [34] applied MPC to an active balancing architecture with DC-DC converters using extended Kalman filter to estimate the internal state of the batteries. In his work, the extended Kalman filter not only estimates the states but also computes the linearized state space matrices at each time step around the SOC operation point. These linearized state space matrices are then reused by a linear MPC algorithm in it prediction step in order to reduce computational burden.
In other literature, a state-feedback nonlinear MPC (NMPC) scheme based on a macro-homogeneous 1-D electrochemical model is proposed to solve the minimum time charging problem accounting for SOH, electrochemical and thermal constraints in [35]. An NMPC framework to solve a battery trajectory optimization problem online accounting for electrical constraints is proposed in [36]. In [37], an approximation of a P2D model is developed and incorporated into an MPC framework to minimize battery charging time accounting for current, SOC and temperature constraints was presented. In [38], a quadratic dynamic matrix control predictive approach is adopted to perform an optimal charge while taking into account both input and output constraints. An optimal charging control problem was formulated for SOC and SOH reference tracking based on the MPC algorithm using a reduced order model was proposed in [39].
In [40], the power prediction problem is formulated in the framework of economic NMPC where the maximum power is adopted directly in the objective function taking into into account input/state constraints. Model-based power estimation for lithium batteries using equivalent circuit models were derived in [41,42]. Nonetheless, a comprehensive review of established methods for battery power estimation can be found in [43].
1.6
Original work
The objective of this thesis is to present lithium-ion cell management strategies that account for electrical and thermal constraints in order to guarantee safe operation and mitigate cell degradation caused at high temperature and by overcharge/overdischarge. An electro-thermal equivalent circuit model is developed, parameterized and validated in a temperature range from −25◦Cto 45◦C. This model is later used in an extended Kalman filter framework
based on the measurements of voltage, current and surface temperature. The electrical parameters of the model are then linearized around the temperature and SOC operation point at each time step using the Kalman filter estimations. Subsequently, the linearized state space matrices are reused by the linear MPC algorithm at the prediction step. A linear MPC algorithm is here used to solve the fast charging problem and maximum available
charge/discharge power computation taking into account SOC, voltage, current and internal temperature constraints.
1.6.1 Organization of thesis
The following chapters present the original work in this thesis, and are structured as follows: Chapter 2: In this chapter we introduce the electro-thermal equivalent circuit model that will be used throughout this thesis. The model comprises two sub-models, a three-state equivalent-circuit electrical model and a two-state thermal model which are coupled through a heat generation term and temperature dependence of electrical parameters. The proposed model can be parameterized individually, which reduces the complexity of the characterization process. The electrical model is the enhanced self-correcting model proposed by Plett in [2] and will be parameterized in a temperature range from −25◦Cto 45◦Cusing experimental data and the identification methodology from [2]. The thermal model is a two-state lumped model proposed and validated in [9] which captures the surface and core temperature of a cylindrical LiFePO4/graphite (A123
26650) lithium-ion battery. This model is parameterized applying current pulses and using the surface temperature measurements.
Chapter 3: This chapter will focus on introducing both the linear and nonlinear Kalman filter. Since our model is a nonlinear cell model, the extended Kalman filter and sigma point Kalman filter will be applied to our battery model developed in Chapter 2. Chapter 4: This chapter will introduce the theory behind model predictive control and its
modified version that accomodates a direct feedthrough term as proposed in [33]. Chapter 5: Based on the implementation of Kalman filter algorithm on the coupled
electro-thermal model and the modified MPC algorithm, presented in Chaps. 3 and 4, respec-tively, we propose in this chapter a charge-control architecture that accounts for SOC, voltage, core temperature and current constraints. The results of this methodology is then compared to the CCCV charging protocol and to a maximum power optimized
charging scheme. An experimental setup is also presented in this chapter in order to validate the fast charging algorithm presented here.
Chapter 6: In this final chapter, the optimal fast charging framework proposed in Chap. 5 is then applied to the maximum charge/discharge power computation problem pro-posed in [12]. The results of this methodology is then compared to the bisection method proposed by Plett in [10] and to a maximum power optimized solution. Finally, Chap. 7 summarizes this thesis and suggests possible avenues of future research.
Coupled Electro-Thermal Model
2.1
Introduction
In this chapter, a coupled electro-thermal equivalent circuit model (CET model) will be de-veloped and characterized for a cylindrical cell (A123 26650 model ANR26650m1-b), which is a lithium iron phosphate battery that uses LiFePO4 as its cathode material and graphite
as its anode material. The Table 2.1 displays the cell specifications provided by the manu-facturer.
Figure 2.1: A123 26650 battery cell.
Specification Cell
Maximum Voltage 3.65 V
Minimum Voltage 2.0 V
Nominal Voltage 3.3 V
Operating Temperature −45◦Cto 60◦C
Table 2.1: Battery cell specifications provided by the manufacturer.
The terminal voltage, SOC and hysteresis estimation are captured by a three-state equivalent circuit. The thermal model comprises a two-state model that estimates the core and surface temperature [9]. The three-state equivalent circuit model adopted in this work is the enhanced self-correcting (ESC) model [2]. The ESC model parameters are SOC,
current direction and temperature dependent, while the parameters of the thermal model are constant. The parameterization of both models are done individually and will be described in this chapter.
2.2
Electrical equivalent-circuit model
Equivalent-circuit models are empirical models that are represented by electric circuit devices (e.g., resistors and capacitors), which emulate the battery cell behavior. This modeling methodology relies on empirical system identification techniques and experimental data. The electrical equivalent-circuit model adopted in this thesis is the enhanced self-correcting model proposed by Plett in [2]. This model comprises of four elements as shown in Fig. 2.2.
Figure 2.2: Enhanced self-correcting cell model.
The first element is the open circuit voltage (OCV), which is the difference of electrical potential between the negative and positive electrode when the battery is disconnected from an external circuit. The OCV is a function of SOC(z(t)) and temperature and is computed from a lookup table that is created using the lab data. The SOC is defined as z(t) = 100%, when the cell is fully charged and z(t) = 0%, when the cell is fully discharged. The total amount of charge removed when discharging a cell from z(t) = 100% to z(t) = 0% is defined as the total capacity Q (measured in Ah or mAh). Therefore, making the sign of i(t) positive on discharge, the SOC is modeled as
˙z(t) = −i(t)Q . (2.1) z(t) = z(t0) − 1 Q t ˆ t0 i(τ )dτ. (2.2)
Sampling with a sampling period △t and including an efficiency factor η(t), the discrete time equation is given by
zk+1 = zk−
ikηk△t
Q . (2.3)
The efficiency factor η(t) is called “coulombic efficiency” and is modeled as ηk≤ 1 on
charge, and ηk= 1on discharge. The resistance R0 in the circuit from Fig. 2.2, models the
voltage drop when the battery cell is under load. R0 also implies that power is dissipated
by heat. The cell’s voltage drop is given by
vR0,k= R0ik. (2.4)
The parallel resistor-capacitor (RC circuit) models the diffusion voltage, which emulates the diffusion of lithium-ions in the electrode from a higher concentration region to a lower concentration region [2]. This phenomenon can be visualized when a cell is allowed to rest after applying a charge/discharge current; In this case, its voltage decay gradually to OCV. The discrete time equation of the RC circuit is given by (for the derivation see [2]) Eq. (2.5) and the diffusion voltage by Eq. (2.6).
iR1,k+1= exp −R△t 1C1 iR1,k+ 1 − exp −R△t 1C1 ik. (2.5) vC1,k = R1iR1,k. (2.6)
It is important to mention that the diffusion voltage phenomenon can be modeled using more than one parallel resistor-capacitor pair. As mentioned before, when the cell rests after applying a discharge/charge current, the voltage is expected to decay to OCV, however, this does not occur. For each SOC point, there is a range of possible stable “OCV” values [2]. It is noteworthy that the hysteresis voltages changes when SOC changes and diffusion voltages
changes with time. The discrete time equation that models the hysteresis state is given by Eq. (2.7) and the hysteresis voltage by Eq. (2.8) [2].
hk+1 = exp − ηkikγ△t Q hk− 1 − exp(− ηkikγ△t Q )sgn[ik]. (2.7) vh,k = M hk (2.8)
where sgn[ik] is the sign of the input current ik when ik6= 0 and, if ik= 0, sgn[ik] retains
the previous value. The parameter γ is a positive unitless constant which tunes the rate of decay and M is the maximum hysteresis polarization voltage and has units of volts. The hysteresis state is unitless and is always −1 ≤ hk≤ 1. It is important to note from Eq. (2.7)
that the hysteresis state depends on the direction of the current, whether the cell is charging or discharging. Finally, the output cell voltage is calculated by combining the OCV voltage and Eqs. (2.4), (2.6) and (2.8), yielding
yk=OCVk− M0sgn[ik] + M hk−
X
RjiRj,k− R0ik. (2.9)
where M0 is a parameter that models the instantaneous hysteresis voltage. Now that all
the dynamic equations of the ESC model have been presented, a state-space system can be developed as [2]: zk+1 iRj,k+1 hk+1 = 1 0 0 0 ARC 0 0 0 AH,k zk iR,k hk + −ikηk△t Q 0 BRC 0 0 (AH,k− 1) ik sgn[ik] (2.10)
where ARC, BRC and AH,k are defined as
ARCj = exp−R△t1C1 0 . . . 0 exp−R△t 2C2 ... ... , (2.11)
BRCj = 1 − exp−R△t1C1 1 − exp−R△t2C2 ... , (2.12) AH,k= exp(− ηkikγ△t Q ) (2.13)
and the output equation is given by Eq. (2.9).
2.3
Thermal model
A simplified two-state thermal model is adopted to capture the core and surface temperature of a cylindrical cell. The lumped thermal model was developed and validated in [8,9]. Based on the classic heat transfer problem, it is assumed that the heat generation is located at the core and the heat flux is zero in the center, as shown in Fig. 2.3. In [44], in order to assess the validity of the assumption of surface temperature uniformity, infrared imaging of the lateral and terminal surfaces were carried out. It was shown that the temperature difference between the different surfaces was within 2◦C.
Figure 2.3: Equivalent-circuit lumped thermal model [8]. The continuous time two-state lumped model is defined as [9]
CcT˙c = Q +
Ts− Tc
Rc
CsT˙s= Tf − Ts Ru − Ts− Tc Rc (2.15) where Ts is the surface temperature, Tc is the core temperature and Tf is the coolant
flow temperature. In this model, the temperature variation along the battery height is neglected, assuming homogeneous conditions. The heat generation Q in the battery core is approximated as a concentrated source of joule loss. The thermal conduction resistance Rc
is a lumped parameter that aggregates the conduction and contact thermal resistance across the compact and inhomogeneous material and models the heat exchange between the core and the surface of the battery. The heat exchange between the surface and the surrounding coolant is modeled by a convection resistance Ru. In some EV applications, the coolant
flow rate is adjustable to control the battery temperature, which makes the value of Ru
change when the coolant flow changes. However, in this work the coolant flow rate is kept constant (air ambient) in order to make Ru constant. The parameter Ccis the heat capacity
of the jelly roll inside the cell and models the rate of temperature change of the core. The parameter Cs is the heat capacity of the battery casing and models the rate of change of
the surface temperature.
Sampling with a sampling period △t, the thermal model in discrete time is given by the following equations [45]:
Tc,k+1= 1 −R△t cCc Tc,k+ △t RcCc Ts,k+△t Cc Qk (2.16) Ts,k+1= △t RcCs Tc,k+ 1 −R△t cCs − △t RuCs Ts,k+ △t RuCs Tf,k. (2.17)
The validation of the core temperature estimate using the above thermal model was carried out in [9] and it showed the capability of the parameterized model to predict the correct battery core temperature through the insertion of a thermocouple into the core of the battery and comparing the estimate to the measured temperature.
2.4
Electro-thermal coupling
The electro-thermal coupling between the electrical model and the thermal model is made through the heat generation term Q from Eq. (2.14). An expression for the heat generation in a lithium-ion battery was derived in [46] as
Q = I(V − Uavg) + IT∂U
avg ∂T − X i △Hiavgri− ˆ X ( ¯Hj− ¯Hjavg) ∂cj ∂tdv j (2.18) where I is the cell current, V the cell voltage, Uavg the equilibrium potential (OCV), T
the temperature, △Hi the variation of enthalpy of a chemical reaction i, ri the rate of
reaction i, Hi the partial molar enthalpy of species j, cj its concentration, t the time, and
v the volume [44]. The first term models the irreversible heat generation, which is the joule heating and energy dissipated in the electrode over-potentials. The second term models the entropic heat, which is a reversible heat generation term, and can be either positive or negative (depending on the current direction). The third term is the heat generated or consumed by any chemical reaction and can also be either positive or negative. The last term represents the heat of mixing, which is generated during the formation and relaxation of concentration gradients within the cell. In [44] it was verified from the experimental measurements that the third and last term can be neglected due to its small contribution to the total heat generated. Considering the first and second terms, and making Uavg =OCV,
Eq. (2.18) can be rewritten as Qk= ik yk− OCVk+ ikTc,k ∂OCVk ∂T . (2.19)
The electro-thermal coupling also occurs through the temperature dependent parameters and OCV in the ESC model, which are determined in each time step by the estimation of the core temperature Tc in Eq. 2.14.
The reversible term can be determined though OCV experiments at various temper-atures using the potentiometric method [47]. In this method, the cell is discharged to a desired SOC, and after a relaxation, the OCV goes to equilibrium, then the cell is submit-ted to a step by step temperature variation, during which the OCV is measured. The results of the potentiometric method comprise of a curve of the corresponding OCVs as functions of temperature and a function of the OCV versus temperature plot [47].
Figure 2.4: Coupling between the electrical and thermal models (modified from [9]).
Figure 2.5: OCV variation during the thermal cycle at SOC= 50%.
Figure 2.5 shows the OCV variation during the thermal cycle at SOC = 50%. The temperature coefficient is determined by fitting the OCV curve to a function V (t, T ) = A+Bt+Ct2, where A, B and C are constants. B corresponds to the temperature coefficient
∂U/∂T. The temperature coefficient ∂U/∂T is presented in Figure 2.6. As one can see the temperature coefficient is negative up to 40%, and becomes positive for higher SOC values. We can also verify that there is abrupt variation between SOC = 70% and SOC = 80%, according to [44], this variation probably corresponds to the transition between the first and
second stages of graphite, what indicates an excess of graphite over the positive electrode material.
Figure 2.6: Temperature coefficient ∂OCV ∂T .
Now that the coupling term has been presented, we can write the full CET model. First, the coupling term Q will be rewritten in a different form. From Eq. (2.9), we can write
OCVk− yk= −Mhk+ M0sgn[ik] +
X
RjiRj,k+ R0ik. (2.20)
Then we can rewrite Qk by substituting Eq. (2.20) into Eq. (2.19), yielding
Qk= ik M0sgn[ik] − Mhk+ X RjiRj,k+ R0ik+ Tc.k ∂OCVk ∂T . (2.21)
Finally, the CET model equations are summarized in Table 2.2.
2.5
Parameterization of the coupled electro-thermal model.
As previously mentioned, the electrical model and thermal model can both be identified independently. First, the identification methodology of the ESC model developed in [2] will be introduced. Subsequently, the validation of the identified ESC model at different temperatures will be presented. Then, the thermal model is parameterized using the heat generation term and current, voltage and surface temperature measurements.
State of charge zk+1= zk−ikηQk△t Hysteresis hk+1= exp − ηkikQγ△t hk− 1 − exp(− ηkikQγ△t )sgn[ik] R-C pair iR1,k= exp −R△t1C1 iR1,k+ 1 − exp−R△t1C1 ik Core Temperature Tc,k+1= 1 −R△tcCc Tc,k+R△tcCcTs,k+△tCcQk Surface temperature Ts,k+1=R△tcCsTc,k+ 1 −R△tcCs− △t RuCs Ts,k+R△tuCsTf,k
Table 2.2: Summary of the CET model equations.
2.5.1 Parameterization and validation of the ESC Model
The parameterization of the ESC model can be divided in two separate parts: First, we have to identify the OCV relationship through experiments where the cell is very slowly charged/discharged while measuring cell voltage and accumulated ampere hours. These experiments are conducted at different temperatures spread over the operational range of the battery cell. At each temperature, the cell is charged/discharged respecting the voltage limits (vmin and vmax). The second part consists of determining the dynamic relationship
while exercising the cell with a profile of current versus time repeated over the entire SOC range and temperature of the cell [2]. This work is focused on EV applications, therefore, a drive cycle that is used to evaluate emissions standards for light-duty vehicles called UDDS (urban dynamometer drive schedule) is used here. The UDDS drive cycle is converted to a current profile using the mechanical characteristics of a first-generation Chevrolet Volt. The methodology for parameterizing the ESC model parameters used in this work is based on the methodology presented by Plett in [2] and the ESCmodel toolbox [48]. Figure 2.7 shows the overall process for creating an ESC cell model using the test data and the ESCmodel toolbox. The purple boxes represent the laboratory processes, the yellow boxes represent the data files and the green boxes represent the MATLAB functions of the ESCmodel tool-box. The OCV testing and dynamic testing are both performed using the same equipment. The battery is placed inside of a Cincinnati Sub-zero thermal chamber in order to have a controlled enviroment, and, the experiments are performed using an Arbin Instruments
BT-2000 High Precision Battery Tester, which is a multi-channel battery testing system for charge/discharge and simulate custom current profiles. The user interface to control the battery tester is via a computer with an Arbin application called MITS Pro.
Figure 2.7: Creation process of the ESC cell model [2].
(a) (b)
Figure 2.8: OCV testing and DYN testing equipment, (a) Arbin Instruments BT-2000 High Precision Battery Tester, (b) Thermal chamber Cincinnati Sub-zero
The OCV testing comprises four steps: • OCV test script #1 (at test temperature)
1. Soak the fully charged cell at the test temperature for at least two hours to ensure a uniform temperature throughout the cell.
2. Discharge the cell at a constant-current rate of C/30 until cell terminal voltage equals manufacturer-specified vmin
• OCV test script #2 (at 25◦C)
1. Soak the cell at 25 ◦C for at least two hours to ensure a uniform temperature
throughout the cell.
2. If the cell voltage is below vmin, then charge the cell at a C/30 rate until the voltage is equal to vmin. If the cell voltage is above vmin, then discharge the cell
at a C/30 rate until the voltage is equal to vmin.
• OCV test script #3 (at test temperature)
1. Soak the cell at the test temperature for at least two hours to ensure a uniform temperature throughout the cell.
2. Charge the cell at a constant-current rate of C/30 until the cell terminal voltage equals vmax.
• OCV test script #4 (at 25◦C)
1. Soak the cell at 25 ◦C for at least two hours to ensure a uniform temperature
throughout the cell.
2. If the cell voltage is below vmax, then charge the cell at a C/30 rate until the
voltage is equal to vmax. If the cell voltage is above vmax, then discharge the cell
at a C/30 rate until the voltage is equal to vmax.
Once the OCV testing data are collected, the function processOCV.m is used to determine the Coulombic efficiency and temperature-dependent OCV relationship. The details about how the Coulombic efficiency and OCV relationship are determined through the function processOCV.m is found in [2]. The OCV relationship at 25◦C is shown in Fig. 2.9. The
OCV testing was performed at different temperatures in the range from −25◦Cto 45◦Cin
intervals of 5◦C.
The dynamic relationship is determined while exercising the cell with a current profile that represents the final application. The dynamic testing comprises three steps [2]:
Figure 2.9: OCV relationship at 25◦C.
• Dynamic test script #1 (at test temperature)
1. Soak the fully charged cell at the test temperature for at least two hours to ensure a uniform temperature throughout the cell.
2. Discharge the cell using a constant current at a C/1 rate long enough to deplete about 10% of capacity (helping ensure we avoid over-voltage conditions during charging portions of the profile).
3. Execute dynamic profiles over the SOC range of interest, nominally from 90% SOC down to 10% SOC.
• Dynamic test script #2 (at 25◦C)
1. Soak the cell at 25 ◦C for at least two hours to ensure a uniform temperature
throughout the cell.
2. If the cell voltage is below vmin, then charge the cell at a C/30 rate until the
voltage is equal to vmin. If the cell voltage is above vmin, then discharge the cell
at a C/30 rate until the voltage is equal to vmin. A follow-on dither profile can
be used to eliminate hysteresis to the greatest degree possible. • Dynamic test script #3 (at 25◦C)
1. Charge the cell using a constant current at a C/1 rate until voltage is equal to vmax. Then, maintain voltage constant at vmax until current drops below C/30.
A follow-on dither profile at the end can be used to help eliminate hysteresis. 2. Voltage and current are recorded every second. Cell temperature data may also
be collected.
(a) (b)
Figure 2.10: Input current for dynamic testing at 25 ◦C, (a) Applied current, (b) Zoom in
UDDS input current.
(a) (b)
Figure 2.11: Output voltage of dynamic testing at 25 ◦C, (a) Output voltage, (b) Zoom in
As an example, Fig. 2.10 depicts a constant discharge current followed by several UDDS cycles used in the dynamic test script 1 to fully discharge the battery at 25 ◦C. Fig. 2.11
shows the output cell voltage data from the dynamic test script 1. The lab data generated by the dynamic testing 1 and 2 are processed by the function processDynamic.m, which runs a system identification to identify all the ESC model parameters. Details about how these parameters are obtained are described in depth in [2]. Fig. 2.12 depicts the identified ESC model parameters at different temperatures. The ESC model parameters are obtained by cubic spline interpolation from a lookup table. Table 2.3 shows the estimation error for the voltage estimation of script 1 of the dynamic testing using the ESC model created using the methodology illustrated in Fig. 2.7
Figure 2.12: ESC parameters from −25◦Cto 45 ◦C.
2.5.2 Parameterization and validation of the thermal model
Due to the complexity of measuring the core temperature, a parameterization methodology based on the measurements of the surface temperature, coolant flow temperature, voltage and current was developed in [8]. This methodology is used here to identify the conduction resistance Rc, convection resistance Ru and the core heat capacity Cc. The surface heat
capacity Csis assumed to be known because it can be easily calculated based on the specific
(a) (b)
Figure 2.13: Validation of the ESC model created, (a) Voltage and estimates at 25◦C,
(b) Voltage estimation error.
Temperature (◦C) Error (mV) −25 76.61 −15 100.99 −5 20.21 5 9.33 15 7.17 25 8.92 35 5.28 45 5.71
Table 2.3: Voltage estimation error.
is equal to 4.5JK−1. In order to identify the lumped parameters of the thermal model, the
model equations need to be rearranged according to the following discrete-time parametric model
zk= θTφk. (2.22)
In Eq. (2.22), z is the observation, θ is the parameter, and φ is the regressor, which consists of measured signals. The parametric model can be derived from Eqs. (2.16) and (2.17) by combining the two equations and replacing the unmeasured core temperature Tc
with the measured signals I, V, Ts and Tf. The parametric model obtained is [45]
Ts,k+1= 2 −R△t cCs − △t RcCc − △t RuCs Ts,k+1+ △t RcCs + △t RcCc + △t RuCs − △t2 RcCcRuCs − 1 Ts,k
+ △t RuCs Tf,k+1− 1 − △t 2 RcCcRuCs Tf,k+ △t 2 RcCsCc Qk. (2.23)
Assuming that Tf is regulated to be constant, Tf,k+1 becomes equal to Tf,k, and Eq.
(2.23) can be reduced to Ts,k+1= 2 −R△t cCs − △t RcCc − △t RuCs Ts,k+1 + △t RcCs + △t RcCc + △t RuCs − △t2 RcCcRuCs − 1 Ts,k + △t 2 RcCcRuCs Tf,k+ △t 2 RcCsCc Qk. (2.24)
By using the following notation α = 2 − △t RcCs − △t RcCc − △t RuCs β = △t RcCs + △t RcCc + △t RuCs − △t2 RcCcRuCs − 1 (2.25) γ = △t 2 RcCcRuCs δ = △t 2 RcCsCc
we can write Eq. (2.24) in the form of Eq. (2.22), yielding Observation : z = Ts,k+2
Regressors : φ = [Ts,k+1, Ts,k, Tf,k, Qk]T
Parameters : θ = [α, β, γ, δ].
Based on the parametric model, the least squares method can be applied to estimate the lumped parameters
ˆ
θ = φTφ−1φTz. (2.26)
where ˆθ is the estimate of θ. Once the lumped parameters are estimated, the estimates of ˆ
Ru, ˆRc and ˆCc can be determined as
ˆ Ru = ˆ α ˆ β, ˆ Rc= △t 2 ˆ RuCˆs2 − △t 2 ˆ Csˆγ − ˆβ ˆRuCˆs2 −β△tˆ Cs −△tˆ Cs ! ,
ˆ Cc = △t 2 ˆ RcCˆsαˆ .
In order to identify the thermal model parameters using the methodology here pre-sented, lab experiments were performed. The battery cell was placed inside of the thermal chamber regulated to 25 ◦C with a thermistor mounted on its surface for measurement. In
our experimental setup, the air temperature inside the chamber is considered as the coolant flow temperature in our thermal model. In this way, a second thermistor was placed in-side the thermal chamber next to the battery cell to capture the surrounding temperature. Symmetrical and periodic current pulses with 2 Hz frequency and magnitude of 20 A were applied to the battery using the Arbin battery tester in order to raise the temperature of the battery cell as in [44]. Before the current pulses were applied, a 2.5 A constant current was applied for 30 minutes in order to bring the battery SOC to 50%. Consequently, the current pulses are symmetrical in order to hold the battery SOC at an average of 50%. The 2 Hz frequency was selected in [44] from an analysis of the impedance spectra which showed that this frequency is intermediate between the low frequency of the interfacial part and the high frequency of the diffusion part [44]. Fig. 2.14 depicts the input current pulses used to raise the temperature of the battery cell. The current was applied for one hour and thirty minutes and then turned off to allow for temperature relaxation. Fig. 2.15 shows the output data of the experiment which consist of surface and air temperature measurements.
(a) (b)
Figure 2.14: Input current for thermal model parameter identification, (a) Input current, (b) Zoomed in input current.
Figure 2.15: Surface and air temperature.
Finally, the thermal model parameters are given in Table 2.4. It is important to note that the convection resistance Ru is strongly dependent on the convective cooling through
the coolant air and the surrounding temperature of the battery cell. Therefore, for each application or scenario, Ru will vary.
Parameters Value
Conduction resistance Rc 7.4013 KW−1
Convection Resistance Ru 2.0751 KW−1
Core heat capacity Cc 44.07 JK−1
Surface heat capacity Cs 4.5 JK−1
(a) (b)
Figure 2.16: Validation of the thermal model, (a) Validation using the identification data, (b) Validation using an UDDS profile.
In order to validate our thermal model, a simulation using the identification lab data was executed and is illustrated in Fig. 2.16a. A second simulation was executed to validate the model using experimental data for which the input current was a UDDS profile. Both simulations presented very small error compared to the truth data. One can also see a significant temperature difference between the core and the surface of the battery cell, which was also observed in [44] a temperature difference of 10◦C or more.
2.6
Summary
In order to design battery management systems we need to have battery models that simulate the battery dynamics and provide information on the internal states. In this chapter, we developed a coupled electro-thermal equivalent circuit model, which is an empirical model that provides voltage, SOC, hysteresis and internal temperature information. The model uses electrical circuit components to emulate the electrochemical behavior of a cylindrical cell. Despite the fact that this model cannot provide electrochemical information (e.g. lithium concentration, electrode potential, side reaction overpotential), it will be shown in the next chapters that this model is simple, reliable and sufficiently robust to be used in constraint cell management strategies aiming to mitigate cell degradation by setting constraints on voltage, SOC and internal temperature. In the next chapter, an extended
Kalman filter will be developed in order to estimate the internal states that cannot be directly measured (e.g. SOC, internal temperature) based on the measurements of current, voltage and surface temperature.
Kalman Filter
3.1
Introduction
The performance of EVs is highly dependent not only on the characteristics of the ideal batteries but also on cell degradation, operating temperature, charge/discharge rate and charge/discharge cycles. Accurate estimation of the internal battery states is extremely important in order to maximize battery performance, extend battery life and ensure safe operation. In Chap. 2 we developed a coupled electro-thermal model. However, the model itself is not enough to guarantee accurate estimates due to changes in the battery char-acteristics (e.g. cell degradation) over time, measurement noise, measurement errors and unknown initial states of the battery pack when the EV is initialized. Importantly, there are internal battery states that cannot be directly measured, e.g., SOC, hysteresis and core tem-perature to name a few. In this context, an optimal estimator can be used to estimate these hidden states in real time using the current, voltage and temperature measurements. The Kalman filter (KF) algorithm is the estimator adopted in this work. A KF is a model-based estimator that uses the noisy measurements and also compensates for other inaccuracies to estimate the present value of the time-varying states of a dynamic system by estimating a joint probability distribution over the variables at each time step [49]. The Kalman filter theory was first developed in 1960 by Rudolf E. Kalman in [50].
This chapter aims to present Kalman filter theory and apply this to the CET model developed in Chap. 2. First, the linear Kalman filter theory will be presented. Since our battery model is nonlinear, the extended Kalman filter (EKF) will be presented subsequently. Finally, at the end of this chapter, we will present results of the application of the EKF algorithm to our battery model. The next chapter will show how the EKF can be integrated with our MPC algorithm in order to provide accurate estimation of the present state and also
the linearized state space matrices, which can reduce the computational burden significantly. The derivation of the Kalman filter presented here is based on [49] and [10].
3.2
Linear Kalman filter
Consider the following dynamic system represented in “state-space” form
xk = Ak−1xk−1+ Bk−1uk−1+ wk−1 (3.1)
zk= Ckxk+ Dkuk+ vk (3.2)
where wk and xk ∈ Rnx and, vk are assumed to be uncorrelated white Gaussian random
process, zero-mean and covariance matrices with following values
E[wnwTk] = Σw˜, n = k; 0, n 6= k E[vnvkT] = Σv˜, n = k; 0, n 6= k
and E[wkxT0] = 0 for all k. The goal here is to compute xk based on the model equations
conditioned to all the noisy measurements until and including time k. This estimation is known as a posteriori estimate and is denoted by ˆx+
k :
ˆ
x+k = E[xk| y1, y2,..., yk]. (3.3)
The estimate of xkwhich takes all the noisy measurements up to and not including time
step k is known as the a priori estimate. The a priori estimate is obtained by computing the expected value of xk conditioned on all of the noisy measurements up to and not including
time step k. The a priori estimate is denoted by ˆx− k :
ˆ
x−k = E[xk| y1, y2,..., yk−1]. (3.4)
However, there is a series of steps between the computation of the a priori estimate and the a posteriori estimate. The KF algorithm is a recursive algorithm which comprises six steps. The KF steps are illustrated in Fig. 3.1 and are summarized here.
Figure 3.1: KF algorithm steps [10].
In this step, the present state is predicted given only past measurements. The a priori estimate is computed by substituting Eq. (3.1) to Eq. (3.4), and noting that wk−1 is zero
mean, yielding
ˆ
x−k = Ak−1xˆ+k−1+ Bk−1uk−1. (3.5)
• Step 1b: Prediction-error time update.
This step estimates the error covariance of the state prediction computed in step 1a. The error is computed as
˜
x−k = xk− ˆx−k, (3.6)
substituting Eqs. (3.1) and (3.5) into E.q (3.6) yields ˜
x−k = Ak−1x˜+k−1+ wk−1. (3.7)
Then, the covariance of the prediction error is given by Σ−x˜ k = E[(˜x − k)(˜x − k) T], (3.8)
substituting Eq. (3.7) into Eq. (3.8) and noting that the cross terms become zero because the white process noise wk−1 is not correlated with the state at time k − 1, yields
Σ−x˜ k = Ak−1Σ + ˜ xk−1A T k−1+ Σw.˜ (3.9)
In this step, we are propagating the state covariance through the model equations forward in time. The last term of Eq. (3.9) perturbs the trajectory away from the known trajectory based only on the present input by adding uncertainty to our estimate.
• Step 1c: Predict system output.
The best guess of the system output is computed in step 1c. In this step, ˆzk is computed
by the system output Eq. (3.2) and given only past measurements (a priori estimate) as ˆ
zk= E[Ckxk+ Dkuk+ vk| Yk−1]
ˆ
zk= Ckxˆ−k + Dkuk, (3.10)
noting that E[vk] = 0.
• Step 2a: Kalman gain update.
Step 2a is the most important step in the KF algorithm. In this step, the Kalman gain is computed based on several covariance matrices:
Lk = Σ−x˜˜zkΣ
−1 ˜
zk. (3.11)
The covariance matrix Σ− ˜
x˜z,k indicates how the measurements are coupled to the
indi-vidual states within ˆxk and also indicates relative need for correction to ˆxk. To find Σ−x˜˜zk,
we first need to derive Σ− ˜
z,k by calculating the output prediction error as
˜
zk= zk− ˆzk
˜
zk= Ckx˜−k + vk.
Then, the covariance matrix Σz˜kis computed as
Σz˜k = E[(Ckx˜ − k + vk)(Ckx˜ − k + vk) T] Σz˜k = CkΣ − ˜ xkC T k + Σv˜, (3.12)
where the term Σ˜v indicates the uncertainty in the sensor reading due to sensor noise. Now,
we can compute Σ− ˜ x˜z,k as
![Figure 1.1: Lithium-ion battery price survey [1].](https://thumb-eu.123doks.com/thumbv2/5dokorg/4335090.98386/14.918.270.706.744.1007/figure-lithium-ion-battery-price-survey.webp)
![Figure 1.6: Schematic of the electrochemical process in a Li-ion cell [4].](https://thumb-eu.123doks.com/thumbv2/5dokorg/4335090.98386/19.918.286.679.162.557/figure-schematic-electrochemical-process-li-ion-cell.webp)
![Figure 1.9: Lithium-ion battery cell-, module-, and pack-level demonstrated by two vehicle examples: Tesla Roadster and Nissan Leaf [7].](https://thumb-eu.123doks.com/thumbv2/5dokorg/4335090.98386/25.918.172.771.154.513/figure-lithium-battery-demonstrated-vehicle-examples-roadster-nissan.webp)
