INTERVAL ESTIMATION FOR STATEOFCHARGE AND TEMPERATURE IN BATTERY PACKS WITH HETEROGENEOUS CELLS
An interval observer based on an equivalent circuitthermal model for lithiumion batteries is presented. State of chargetemperaturedependent parameters are considered as unknown but bounded uncertainties in a single cell model. A parallel and a series arrangement of five cells are used for observer design, where cell heterogeneity is accounted for through the uncertainty bounding functions.
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The disclosure relates to estimating the operation of multiple lithiumion batteries connected in series and in parallel, including estimating the state of charge and temperature of the batteries.
BACKGROUNDLithiumion (Liion) batteries play a key role in achieving energy sustainability and reduction in emissions. Liion batteries benefit from high energy density, which has motivated their wide use in a variety of applications including electric vehicles and grid energy storage. In recent years, a substantial body of research on realtime control and estimation algorithms for batteries has emerged. However, safe and efficient operation of batteries remains a challenge, especially as the performance requirements of these devices increase.
Energy storage systems for electric vehicles and largescale grid applications often require hundreds to thousands of cells connected in series and in parallel to achieve the demanded power and voltage. A battery pack's instantaneous power capability is crucial for onboard management and safe operation. However, realtime state of charge (SOC) estimation for a battery pack is a very intricate task due to (i) limited measurements, (ii) complex electrochemicalthermalmechanical physics, and (iii) high computational cost.
Different battery models for state observer design have been proposed in the literature, which can be classified into electrochemical white box models, equivalentcircuit gray box models, and data driven black box models, sorted from low to high physical interpretability. Although each modelling framework has their own merits and drawbacks, equivalent circuit models (ECMs) provide a reasonable tradeoff between model complexity and accuracy. ECMs can be made more accurate by increasing the system order to account for additional electrochemical phenomena, like charge transfer, but at the expense of computational cost that grows polynomially with the number of states and linearly with the number of cells.
An important fact often ignored during battery modeling is the timevarying electrical parameters. In practice, internal parameters, e.g. resistances and capacitance, are nonlinearly dependent on the cell's temperature and SOC. Therefore, it is important to model cell temperature and its coupling with the electrical dynamics. Highfidelity temperature models have more accurate temperature predictions, but suffer from high computational cost, rendering them of little use for onboard thermal management systems. First principlesbased twostate thermal models for the cell's core and surface temperatures provide a balanced tradeoff between computational efficiency and fidelity.
Lumped equivalent circuitthermal models with temperature dependent parameters have been studied and used for state estimation via an adaptive observer. Existing techniques for battery pack state estimation include Luenberger observers, Kalman filters (KFs), unscented Kalman filters (UKFs), and sliding mode observers, among others. However, all the previously mentioned techniques require a state observer for each cell within the pack, which makes them computationally intractable for large packs.
In the stochastic estimation/filtering framework, the process and measurement noises are often assumed to be Gaussian. The system characteristics, e.g., mean and variance, are required by filtering algorithms. Nonetheless, the statistical and calibration procedures to obtain these characteristics are often tedious. In contrast, interval estimation assumes that the measurement and process noises are unknown but bounded. The interval observer literature derives a feasible set for state estimation at every time instant. In a battery pack that consists of thousands of cells, executing state estimation algorithms based on highly nonlinear and coupled dynamics for every single cell in real time becomes intractable. The interval observer benefits from its scalability by deriving only upper and lower bounds that enclose all unmeasured internal states for all cells in a pack. Previously, only Perez et al. in “Sensitivitybased interval PDE observer for battery SOC estimation” (American Control Conference, pp. 323328, 2015), had designed a sensitivitybased interval observer for single cell SOC estimation from an electrochemical perspective, but without provable observer stability and inclusion properties.
SUMMARYThe internal states of Liion batteries, notably SOC, need to be carefully monitored during battery operation in order to prevent dangerous situations. SOC estimation in an electrically and thermally coupled parallel connection of cells is a particularly challenging problem because cells in parallel yield a system of differential algebraic equations, which are more difficult to handle than ordinary differential equations (e.g. a series string of cells). For a large battery pack with thousands of cells, applying an estimation algorithm on each and every cell would be mathematically and computationally intractable. The present application tackles these issues using an interval observer based on a coupled equivalent circuitthermal model, and considers cell heterogeneity as well as statedependent parameters as unknown, but with bounded uncertainty. The resulting interval observer maps bounded uncertainties to a feasible set of state estimation and is independent of the number of cells in connection (in parallel or in series). Stability and inclusion of the interval observer are proven and validated through numerical studies.
A robust interval observer, using the measurements, to determine the set of admissible values for cell SOC and temperature at each time instant, when the plant model is subject to bounded uncertainties in the parameters and states' initial conditions, is used.
In the present application, heterogeneous battery cells (where the cells are heterogeneous due to variations in health, temperature, and/or charge levels) in parallel and in series connection are both considered and analyzed, based on an interval observer design, given uncertain model parameters, initial conditions, and measurements. The approach is based upon the theory of interval observers (a.k.a. setmembership estimation). Interval observers provide an elegant approach to directly address the heterogeneity and scalability problems. Namely, heterogeneity is handled by conceptualizing parameters and inputs as existing within some membership set. Scalability is handled by designing a single observer, which estimates the range of SOC and temperature values across the battery pack.
Liion batteries are vulnerable to overcharge and overdischarge. Overcharge can lead to lithium deposition and electrolyte solvent decomposition, resulting in fire or even explosion. This issue is particularly critical for battery packs in plugin hybrid electric vehicles (PHEVs) and battery electric vehicles (BEVs), since those batteries are usually charged to high SOC. In the meantime, the charging (and discharging) capability of a pack is determined by the cells with highest (lowest) SOC. Thus, to prevent overcharging, the cells with lower SOC cannot be fully charged after the cells with higher SOC have been fully charged. As a result, cells are not fully utilized when they are not balanced. In order to prolong the battery life cycle and increase the stack utilization, it is recommended that all the cells in a pack have the same SOC during battery operations. Cell equalization is performed based on the knowledge of the best and worst SOCs in a pack.
In realworld applications, e.g. BEVs, it is desirable to charge the battery pack as fast as possible. However, charging the battery with high current will significantly damage the Liion cells which could lead to catastrophic failure. Ensuring safe operating constraints is a basic requirement for batteries. For instance, while charging, SOC, core and surface temperatures (along with other important variables) of every single cell needs to be maintained within an operational bound to ensure pack safety. The interval observer provides such information that can be utilized to infer whether these variables violate the safe constraints.
Embodiments disclosed herein allow the estimation of SOC and temperature during operation of the batteries, followed by appropriate charging of the batteries within safe limits.
The application will be better understood in light of the description which is given in a nonlimiting manner, accompanied by the attached drawings in which:
Reference throughout this specification to “one embodiment” or “an embodiment” means that a particular feature, structure, material, or characteristic described in connection with the embodiment is included in at least one embodiment of the application, but do not denote that they are present in every embodiment.
Thus, the appearances of the phrases “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily referring to the same embodiment of the application. Furthermore, the particular features, structures, materials, or characteristics may be combined in any suitable manner in one or more embodiments.
Throughout, the symbol I_{dn }denotes the identity matrix with dimension n×n. For a matrix AϵR^{n×n}, ∥A∥_{max}=max_{i;j=1;2 . . . ;n}Ai,j (the element wise maximum norm). The relation Q>0 (Q<0) means that the matrix QϵR^{n×n }is positive (negative) definite. The inner product between x, yϵR^{n }is given by <x, y>=Σx_{i}y_{i}, i=1 to n.
The embodiments described herein are widely applicable to energy storage systems (e.g. Liion batteries) comprising hundreds or thousands of dissimilar cells arranged in series and in parallel. Knowledge of these largescale systems is limited by the amount of sensing locations and cell heterogeneity. One of such examples, among many others, is the high energy pack module developed by Saft, which integrates 40,000 single cells of 18650 type. The module constitutes 20 strings of cells connected in parallel, where each string is made of 400 blocks in a series arrangement. Additionally, each block includes 5 cells in parallel. An interval observer aims to estimate the range of SOC values across the battery pack, with guaranteed provable inclusion and stability properties, which will be discussed in the subsequent sections.
This section reviews an ECM coupled with a twostate thermal model for a single battery cell, which is then electrically and thermally interconnected with other cell models to form a parallel arrangement of cells.
Single Battery Cell
The ECM for a single cell, shown in
dz_{k}(t)/dt=(1/Q_{k})I_{k}(t); (1)
dV_{c,k}(t)/dt=−(1/R_{2,k}(z_{k},T_{k})C_{k}(z_{k},T_{k}))V_{c,k}(t)+(1/C_{k}(z_{k},T_{k}))I_{k}(t) (2)
V_{k}(t)=OCV(z_{k}(t))+V_{c,k}(t)+R_{1,k}(z_{k},T_{k})I_{k}(t) (3)
where z_{k}(t) represents the SOC for the kth cell, V_{c;k}(t) denotes the voltage across the RC pair for the kth cell, V_{k}(t) represents the voltage at the output of the kth cell, and I_{k}(t) represents the current flowing across the R_{1 }and the RC pair in the kth cell. The symbol T_{k }is the cell temperature that will be defined later. The electrical model parameters, namely, R_{1,k}, F_{2,k}, and C_{k}, are dependent on SOC and cell temperature, and such dependence can be explicitly characterized via an offline experimental procedure for a cell of interest (for example, for a LiFePO4/Graphite cell). The output eq. (3) for the kth cell provides the voltage response characterized by a nonlinear open circuit voltage (OCV) as a function of SOC, voltage from the RC pair, and voltage associated with an ohmic resistance R_{1;k}. Herein, positive current is specified for charging and negative current is specified for discharging.
In a twostate thermal model of the kth cell, the model states are cell core temperature and surface temperature:
C_{c}dT_{c,k}(t)/dt={dot over (Q)}_{k}(t)+((T_{s,k}(t)−T_{c,k}(t))/R_{c} (4)
C_{s}dT_{s,k}(t)/dt=((T_{f,k}(t)−T_{s,k}(t))/R_{u}−((T_{s,k}(t)−T_{c,k}(t))/R_{c} (5)
{dot over (Q)}_{k}(t)=I_{k}(t)(V_{k}(t)−OCV(z_{k}(t))) (6)
T_{k}(t)=½(T_{s,k}(t)−T_{c,k}(t)) (7)
where T_{c,k }and T_{s,k }are the core and surface temperatures for the kth cell. Symbols R_{c}, R_{u}, C_{c}, and C_{s }represent heat conduction resistance between core and surface, convection resistance between ambient and surface, core heat capacity, and surface heat capacity, respectively. In this framework, symbol {dot over (Q)}_{k}(t)≥0 is the internal heat generation from resistive dissipation. Note that the electrical model, as represented by eqs. (1)(3), and the thermal model, as represented by eqs. (4)(7), are coupled via internal heat generation {dot over (Q)}_{k}(t) in a nonlinear fashion.
The measured quantities for the coupled electricalthermal model (1)(7) are the cell voltage and surface temperature:
y_{k}(t)=[V_{k}(t),T_{s,k}(t)] (8)
Parallel Arrangement of Battery Cells
For a block of m cells in parallel (shown in
OCV(z_{i})+V_{c,i}+R_{1,i}I_{i}=OCV(zj)+V_{c,j}+R_{1,j}I_{j }i,jϵ{1,2, . . . m}, i≠j (9)
Similarly, Kirchhoff s current law poses the following linear algebraic constraint with respect to cell local currents,
ΣI_{k}(t)=I(t). k=1 to m (10)
where I(t) is the measured total current, and I_{k}(t) represents the local current for cell k. It is worth highlighting that eq. (9) imposes (m−1) nonlinear algebraic constraints with respect to differential states and local currents, whereas eq. (10) imposes one algebraic constraint with respect to local currents. When only the total current is measured, the local cell currents are unknown. Hence, the system of differentialalgebraic eqs. (1)(10) must be solved such that the algebraic eqs. (9) and (10) are fulfilled for all t. This is realized by augmenting the local currents to the differential state vector to form a nonlinear descriptor system.
The cells are thermally coupled through coolant flow and heat exchange between adjacent cells.
For cell k, where k={1, 2, . . . , m},
C_{c}dT_{c,k}(t)/dt={dot over (Q)}_{k}(t)+((T_{s,k}(t)−T_{c,k}(t))/R_{c} (11)
C_{s}dT_{s,k}(t)/dt=((T_{f,k}(t)−T_{s,k}(t))/R_{u}−((T_{s,k}(t)−T_{c,k}(t))/R_{c}+(T_{s,k1}(t)+T_{s,k1}(t)−2T_{s,k}(t))/R_{cc} (12)
T_{f,k}(t)=T_{r,k1}(t)+(T_{s,k1}(t)−T_{r,k1}(t))/R_{u}C_{r} (13)
{dot over (Q)}_{k}(t)=_{k}(t)(y_{k}(t)−OCV(z_{k}(t))) (14)
T_{k}(t)=(T_{s,k}(t)+T_{c,k}(t))/2 (15)
where T_{r,k }is the coolant flow temperature at the kth cell, and R_{cc }denotes heat conduction resistance between adjacent battery cell surfaces. Heat conduction between battery cells is driven by the temperature difference between cell surfaces, and this process is described by the third term on the right hand side of eq. (12). Inside the block of m cells in parallel, it is assumed that the coolant flows through individual cells, and the coolant flow temperature at the kth cell is determined by the flow heat balance of the previous cell, as illustrated in eq. (13). Here it is also assumed that all the battery cells have the same thermal parameters.
Analysis for Cell Heterogeneity
In this section, the heterogeneity for cells in parallel via an openloop simulation study is presented. Without loss of generality, two LiNiMnCoC_{2}/Graphite (NMC) type cells with 2.8 Ah nominal capacity in parallel are considered. In this embodiment, the cells have identical SOCOCV relationship, and the heterogeneity arises from:

 difference in SOC initialization
 difference in electrical parameters due to SOC variation
 unevenly distributed currents due to parameter variation
 difference in temperature due to currents variation
A transient electric vehiclelike charge/discharge cycle generated from the urban dynamometer driving schedule (UDDS) is applied. Specifically, this total applied current (summation of local currents) is plotted in
Two cases are examined here. In the first case, the cells are initialized at the same SOC, z_{k}(0)=0.25 for kϵ{1,2}. Since Cell 2 has larger resistance, its local current is smaller in magnitude relative to the local current of Cell 1, while the summation of the local currents equals to the total applied current for all t, as shown in
In a battery pack composed of hundreds or thousands of cells, executing state estimation algorithms based on a highly nonlinear and coupled model consist of differentialalgebraic equations for every single cell in realtime. This would be mathematically intractable and involve tremendous computational burden. This motivates the present study on interval observers that generate an upper and a lower bound to enclose all z_{k}(t) trajectories, thus reducing computation and design complexity.
Consider the following nonlinear model dynamics:
dx/dt=f(x)+B(θ)u+δf(x,θ) (16)
y=h(x)+δh(θ)u (17)
where xϵR^{n }is the state vector, and uϵR and yϵR are the system input and output, respectively. The considered system is singleinputsingleoutput (SISO). The functions f(x) and h(x) are deterministic and smooth, and δf is uncertain and assumed to be locally Lipschitz continuous with respect to x. It is noted that the nominal terms f(x) and h(x) can be freely assigned by the designer via the modification of δf and δh. The initial conditions for the states belong to a compact set x_{0}ϵ[
From eqs. (16)(17) by setting B=0, δf=0, δh=0, one obtains
dx/dt=f(x) (18)
y=h(x) (19)
A coordinate transformation obtained from the locally observable nominal system (18)(19) is then utilized to transform the original uncertain system (16)(17) into a partial linear system.
In particular, by denoting the gradient of a scalar field h by dh, and the Lie derivative of h along a vector field f is defined by the inner product L_{f}h(x)=<dh(x), f (x)>. Highorder Lie derivatives are computed with the iteration L_{f}^{k}h(x)=L_{f}(L_{f}^{k1 }h(x)) where L_{f}^{0 }h(x)=h(x). The local observability matrix around an equilibrium point for the states x=x_{e }is given by
The nominal system (18)(19) is locally observable around x=x_{e }if the matrix O(x_{e}) is of rank n.
If the system (18)(19) is locally observable, then the vectors h(x), L_{f}h(x), . . . , L_{f}^{n1}h(x) form the new coordinate for the sates in a neighborhood pf x defined by
is a local diffeomorphism. The coordinate transformation obtained from the locally observable nominal system (18)(19) is then utilized to transform the original uncertain system (16)(17) into a partiallinear expression
dξ/dt=A_{0}ξ+δA(θ)ξ+b(ξ,θ) (20)
y=Hξ+v(θ,t) (21)
where v(θ, t)=δh(θ)u. The matrix A_{0}ϵR^{n }is deterministic and the matrix δA(θ)ϵR^{n }represents the uncertain part inherited from the uncertain nonlinear system (16)(17). Symbol b(ξ, θ) indicates a lumped uncertain nonlinear function.
Interval Observer for Batteries
In this section, the interval observer design introduced previously is applied to the Liion battery state estimation problem. Four scenarios are examined: (i) interval observer for SOC of a single battery cell with temperature and SOC dependent electrical parameters; (ii) interval observer for SOC of electrically and thermally coupled cells in parallel, with SOC and temperaturedependent electrical parameters; (iii) interval observer for SOC of electrically and thermally coupled cells in series, with SOC and temperaturedependent electrical parameters, and (iv) temperatureenhanced interval observer for SOC and temperature of electrically and thermally coupled cells in series, with SOC and temperaturedependent electrical parameters.
SOC Interval Observer for a Single Battery Cell
It is hereby assumed that the input current, terminal voltage and surface temperature of the kth single cell are experimentally measured. Ideally, a deterministic state observer is supposed to be proposed and analyzed for the coupled nonlinear electricalthermal system (1)(8) to reconstruct the internal states, i.e. z_{k}(t), V_{c;k}(t), and T_{c;k}. However, this approach becomes intractable due to the nonlinear coupling between electrical and thermal models, the nonlinear dependence of electrical parameters on internal states, as well as nonlinear voltage output function. To tackle this issue, the present embodiment suppresses the electrical parameters' dependence on the internal states, and treats these parameters as uncertain. Specifically, θϵΘ⊂=R^{4}, where θ=[R_{1,k }R_{2,k }C_{k }Q_{k}]^{T }The objective is to design a robust interval observer, using the measurements, to determine the set of admissible values for cell SOC at each time instant, when the plant model is subject to bounded uncertainties in the parameters and states' initial conditions.
Let τ_{k}=1/(R_{2,k}C_{k}), and consider a known nominal value τ_{k,0 }such that
τ_{k}=τ_{k,0}+δ_{τk} (22)
where τ_{k,0 }is a deterministic scalar and δ_{τk }represents the uncertain component. The single cell electrical system of eqs. (1)(3) can thus be formulated in terms of uncertain system of eqs. (16)(17) with:
It is assumed that the following upper and lower bounds are imposed on the uncertain parameters,
Q_{k}ϵ[Q_{k},Q_{k}
so that Θ is a fourdimensional polytope. The local observability matrix for the nominal system is then given by
whose rank is 2 if and only if the first derivative of the OCV function with respect to SOC is nonzero around an equilibrium point x_{1}=x_{1,e }and τ_{k,0}≠0, i.e.,
dOCV/dx_{1}(x_{1,e})≠0, τ_{k,0}≠0 (26)
which aligns with existing results on local observability for battery models. Hence, the coordinate transformation based on Lie algebra
transforms the system (16), (17) with (23) to the nonlinear parametervarying system (20)(21), with
An interval observer is designed according to Stanislav Chebotarev et al. “Interval observers for continuoustime LPV systems with L1/L2 performance”, Automatica, 58:8289, 2015. The bounding functions δA
SOC Interval Observer for Battery Cells in Parallel
As opposed to having one interval observer for each single cell in the preceding discussion, the present design is generalized for a cluster of battery cells connected in parallel. One practical advantage for using an interval observer for a group of cells is scalabilty. An interval observer, composed of only two dynamical systems estimating upper and lower bounds that all trajectories of unknown internal states live in, significantly reduces computation and design effort. For instance, consider two cells in parallel as studied previously and
The interval observer design for parallel cells inherits the essence of the design for single cells. The only difference is to compute a single set of bounding functions that bound uncertainties from each cell in the parallel configuration.
A crucial step in designing interval observers for cells in parallel is to find the bounding functions for the uncertainties. Namely, the bounding functions are closely associated with the instantaneous bounds on the local currents. However, unlike the single cell scenario, the local currents of parallel cells are not available for measurement. Here, it is assumed that appropriate bounds on the local currents are given.
The width/tightness of the estimated intervals is dependent on the magnitude of model uncertainties, and the knowledge of the uncertainties when defining the bounding functions.
SOC Interval Observer for a Battery Cells in Series
According to the above, for a single cell modelled by ECM
dz_{k}/dt=(1/Q_{k})I(t); (30)
dV_{c,k}/dt=−(1/R_{2,k}C_{k})V_{c,k}+(1/C_{k})I(t) (31)
y_{k}=OCV(z_{k})+V_{c,k}+R_{1,k}I(t) (32)
where kϵ{1, 2, ,N} and N is the total number of cells connected in series (see, for example,
dx_{k}/dt=f(x_{k})+δf(x_{k},θ_{k})+B(θ_{k})u (33)
y_{k}=h(x_{k})+δh(θ_{k})u, (34)
with
h(x_{k})=OCV(x_{k,1})+x_{k,2},δh(θk)=R_{1,k }and τ_{k}=1/R_{2,k}C_{k}=τ_{0}+δτ_{k} (35)
where τ_{0 }is a user defined nominal value (assumed greater than zero), δτ_{k }is an uncertain parameter, and θ_{k }is the uncertain parameter vector θ_{k}=[R_{1,k }R_{2,k }C_{k }Q_{k}], for all kϵ{1, 2, ,N}.
By setting B=0, δf=0 and δh=0, in eqs. (33)(34), one obtains an uncertainty free nominal system
dx_{k}/dt=f(x_{k}) (36)
y_{k}=h(x_{k}) (37)
Using the transformation of coordinates Φ(x_{k}), where
obtained from the nominal locally observable system (36)(37), the original uncertain system (33)(34) can be transformed into the following dynamical equations:
dξ_{k}/dt=A_{0}ξ_{k}+δA(θ_{k})ξ_{k}+b(ξ_{k},θ_{k},u) (39)
y_{k}=Hξk+δh(θ_{k})u (40)
where
Temperature Enhanced SOC Interval Observer—Coupled with an Interval Observer for Battery Core and Surface Temperature
The temperature dynamics for a cell can be described by (where the symbols are the same as in Eqs. (4)(6))
C_{c}dT_{c}(t)/dt={dot over (Q)}(t)+((T_{s}(t)−T_{c}(t))/R_{c} (41)
C_{s}dT_{s}(t)/dt=(T_{f,}−T_{s}(t))/R_{u}−((T_{s}(t)−T_{c}(t))/R_{c} (42)
{dot over (Q)}(t)=I(t)[y(t)−OCV(z_{k}(t))] (43)
y_{T}(t)=T_{s}(t) (44)
which is then rearranged into
dx(t)/dt=Ax(t)+Bu(t)+D{dot over (Q)}(t) (45)
y(t)=Cx(t) (46)
where
In the design of an interval observer for the above thermal model, it is assumed that the thermal parameters R_{c}, C_{c}, R_{u}, and C_{s }are constant, i.e., no parametric uncertainties in the model. Here, it is the internal heat generation term, i.e. {dot over (Q)}(t), that is treated as the uncertainty entering the model. From Eq. (43), since I(t) is measured, the max and min of voltages (
The thermal model (45)(46) can be reformulated as
dx/dt=Ax+Bu+D{dot over (Q)}(t)+K(y_{T}−Hx)=(A−KC)x+Bu+Ddq/dt+Ly_{T} (48)
First, K=[K_{1 }K_{2}]^{T }needs to be chosen such that the matrix (A−KC) is Hurwitz and Metzler. Since
L1≤1/R_{c}C_{c }is required. In addition, since matrix A is itself Hurwitz, the choice of a positively small K_{1 }and a positively large K_{2 }would enforce the matrix (A−KC) to be Hurwitz and Metzler.
The interval observer is then
d
dx/dt=(A−KC)x+Bu+D{dot over (Q)}(t)+Ly_{T} (51)
The dynamics for the estimation error ē=
dē/dt=(A−KC)ē+D(
de/dt=(A−KC)e+D({dot over (Q)}(t)−{dot over (Q)}(t))+K(y_{T}−y_{T}) (53)
where ē≥0 and e≥0 according to monotonic system theory.
Thus, an interval observer for core temperature can then be designed, by treating the heat generation {dot over (Q)}_{k}(t) uncertain, and only using the max and min surface temperature information. As shown in
Results
In order to validate the interval observer design, numerical studies are carried out on NMC battery cells modeled with a lumped electricalthermal model (1)(10). The statedependent electrical model parameters are taken from Xinfan Lin et al. “A lumped parameter electrothermal model for cylindrical batteries”, J. of Power Sources, 257:111, 2014. The total current fed to the battery is a UDDS driving cycle. The interval observer is used to estimate the lower and upper bounds on the internal states from only total current and voltage measurements. Three scenarios are considered. First, the state estimation of a single battery cell is tested, which accounts for uncertainties linked to SOC and temperature dependent parameters. Then, the same observer is used to estimate the state interval for a parallel arrangement of five cells, which involves uncertainty due to cell heterogeneity as well as SOC and temperature dependent parameters. Last, a temperatureenhanced interval observer is used to estimate the state interval for a series arrangement of five cells, which involves uncertainty due to cell heterogeneity as well as SOC and temperature dependent parameters.
Interval Observer for Single Battery Cell
A single cell and design of the interval observer according to the previous section is considered. As shown in
Internal Observer for Battery Cells in Parallel
A parallel arrangement of five cells, which differ in their initial SOCs and model parameters is considered. The interval observer is designed according to the previous sections. The initial SOCs are 20%, 30%, 34%, 37%, and 49%, and the initial bounds (interval observer) on SOCs are 15% and 54%.
Temperature Enhanced Interval Observer for Battery Cells in Series
The interval observers described above (electrical model enhanced by the thermal model) were used for determining the SOC and core temperature of five sells in series.
An interval observer based on an equivalent circuitthermal model for lithiumion batteries has been presented. The SOCtemperaturedependent parameters are considered as unknown but bounded uncertainties in the single cell model. Then, both parallel and series arrangements of five cells are used for observer design, where cell heterogeneity is now accounted for through the uncertainty bounding functions. Given that the reduced nominal battery model is locally observable, the original uncertain model can be transformed into a partial linear form for monotone systems, which enables interval estimation. By properly choosing the observer gains, the state matrix of the estimation error is Hurwitz and Metzler, which guarantees stability and inclusion of the state bound estimates. Furthermore, a temperatureenhanced interval observer is designed to simultaneously estimate the SOC and the core temperature of five battery cells. A major feature of the present estimation approach is its scalability, since the number of states of interval observers is independent of the number of cells. Simulation results showcase the effectiveness of the interval observer design.
Numerous modifications and variations of the embodiments presented herein are possible in light of the above teachings. It is therefore to be understood that within the scope of the claims, the disclosure may be practiced otherwise than as specifically described herein.
Claims
1. A method for operating batteries belonging in a battery pack comprising a plurality of cells of batteries, each of the cells being represented by an RC pair comprising a resistor with resistance R2 and a capacitor with capacitance C, the RC pair being in series with a resistor with resistance R1, the method comprising: where ξ = ( ξ 1 ξ 2 ) = ( OCV ( x 1 ) + x 2  τ k, 0 x 2 ). A 0 = [ 0 1 0  τ k, 0 ], δA ( θ ) = [ 0 δτ k / τ k, 0 0  δτ k ]. b ( ξ, θ ) = ( ( 1 / Q k ) φ ( ξ 1 + ξ 2 / τ k, 0 ) + 1 / C k  τ k, 0 / C k ) I, H = [ 1, 0 ]. x = ( T c T s ), A = [  1 / R c C c 1 / R c C c 1 / R c C c  ( 1 R u C s + 1 R c C s ) ], B = ( 0 1 / R u C s ), D = ( 1 / C s 0 ), C = [ 0, 1 ], u = T f;
 estimating a state of charge (SOC) of each of the cells of the batteries belonging in the battery pack based on a coupled electricalthermal model, wherein
 the SOC of each battery in each of the cells is derived from the nonlinear differential equation system: dξ/dt=A0ξ+δA(θ)ξ+b(ξ,θ) dy/dt=Hξ+v(θ,t)
 φ(⋅)=dOCV/dx(OCV−1(⋅)), v(θ, t)=δh(θ)u, where
 u=Ik(t)ϵR is a system input, where Ik(t) is a current flowing in the kth cell, yϵR is a system output, xϵRn is a state vector, having a state x1=zk and a state x2=Vc,k, where zk(t) represents the SOC for the kth cell, and Vc,k(t) represents a voltage across the RC pair in the kth cell, t represents a time variable, OCV is an open circuit voltage across the kth cell, τk=1/R2,kCk=τk,0+δτk where τk,0 is a deterministic scalar and δτk represents an uncertain component, {dot over (Q)}k(t) is an internal heat generation from resistive dissipation, Ck is the capacitance of the capacitor in the RC pair in the kth cell, I is an identity matrix, δh(θ)=R1,k, R1,k is R1 in the kth cell, R2,k is R2 in the kth cell, where
 unknown quantities Qk, Ck, δτk and R1,k have values belonging within intervals [Qk, Qk−], [Ck, Ck−], [δτk, δτk−], and [R1,k·R1,k−], respectively, with set upper and lower limits;
 estimating a surface temperature Ts and a core temperature Tc for each of the cells, wherein
 the Ts and the Tc of each of the cells is derived from the nonlinear differential equation system: dx(t)/dt=Ax(t)+Bu(t)+D{dot over (Q)}(t) y(t)=Cx(t)
 where
 where Rc, Ru, Cc, and Cs represent heat conduction resistance between core and surface, convection resistance between ambient and surface, core heat capacity, and surface heat capacity, respectively, and Tf is a cool ant flow temperature at a cell; and
 adjusting charging of the batteries based on the estimated SOC, Ts, and Tc of the batteries.
2. The method according to claim 1, wherein
 the battery pack comprises a string of a plurality of cells connected in series.
3. The method according to claim 2, wherein
 the battery pack comprises a plurality of strings of cells connected in parallel.
4. The method according to claim 1, wherein
 the battery pack comprises 20 strings of cells connected in parallel, each string comprising 400 blocks of cells connected in series, each of the block comprising 5 cells connected in parallel.
Type: Application
Filed: Jun 24, 2020
Publication Date: Dec 30, 2021
Applicants: TOTAL S.A. (Paris La Défense), University of California  Berkeley (Berkeley, CA), Université Libre de Bruxelles (Bruxelles)
Inventors: Dong ZHANG (Albany, CA), Preet GILL (El Sobrante, CA), Scott MOURA (Berkeley, CA), Luis COUTO (Oxford), Sebastien BENJAMIN (Leognan), Wente ZENG (San Francisco, CA)
Application Number: 16/911,179