This article optimizes the allocation of external current demand among parallel strings of cells in a lithium-ion battery pack to improve Fisher identifiability for these strings. The article is motivated by the fact that better battery parameter identifiability can enable the more accurate detection of unhealthy outlier cells. This is critical for pack diagnostics. The literature shows that it is possible to optimize the cycling of a single battery cell for identifiability, thereby improving the speed and accuracy with which its health-related parameters can be estimated. However, the applicability of this idea to online pack management is limited by the fact that overall pack current is typically dictated by the user, and difficult to optimize. We circumvent this challenge by optimizing the internal allocation of total pack current for identifiability. We perform this optimization for two pack designs: one that exploits switching control to allocate current passively among parallel strings of cells, and one that incorporates bidirectional DC–DC conversion for active charge shuttling among the strings. A novel evolutionary algorithm optimizes identifiability for each pack design, and a local outlier probability (LoOP) algorithm is then used for diagnostics. Simulation studies show significant improvements in diagnostic accuracy for an automotive protocol.

## Introduction

This article examines the problem of optimizing the internal allocation of current among parallel strings of cells in a lithium-ion battery pack such that the Fisher identifiability of these cells' parameters is maximized. By doing so, we improve the accuracy with which unhealthy outlier cells can be detected. This is critical for pack-level diagnostics.

The above research is motivated by three important facts. First, lithium-ion batteries are vulnerable to catastrophic failure modes, such as thermal runaway, when pushed to their charging and discharging limits [13]. Second, in a lithium-ion battery pack, the catastrophic failure of even a single unhealthy cell could cause an entire series string of cells to no longer be usable, or in the worst case, could induce a cascade of catastrophic failures throughout the pack [3,4]. Third, advanced automotive battery management systems often require the accurate knowledge of battery cell parameters, particularly for less-healthy outlier cells [5]. All of these facts point to the importance of fast and accurate diagnostics—the determination of whether or not there is a significantly unhealthy cell in a pack and pinpointing which cell is unhealthy—in advanced lithium-ion battery packs.

This article uses model-based diagnostic algorithms that estimate the parameters of a battery cell model from input/output data and uses the parameter estimates to detect and isolate unhealthy outlier cells. To effectively use model-based approaches for battery pack diagnostics, one must estimate the parameters of the underlying battery model as accurately as possible. This is a challenge for lithium-ion batteries, which are known to suffer from poor parameter identifiability [68]. Poor identifiability makes it difficult to estimate the parameters of a battery model quickly and accurately from experimental data [911]. This challenge can be addressed by optimizing battery tests to maximize their information content. This is primarily achieved through two methods. The first is through the selection of optimal current input profiles from a predefined set [1215]. For example, Marcicki et al. divide the overall experiment into partitions where the selection of optimal trajectories is performed in a piecewise manner [15]. This process decouples the identifiability maximization problem within each division of the experiment and allows for more accurate estimates of the subsets of parameters than if they were all estimated together [15]. The second method for addressing the challenge of poor battery parameter identifiability is to optimize the shape of a single input trajectory [1619]. For example, Forman et al. perform trajectory-shaping identifiability optimization to show that identifiability can be greatly improved before substantial degradation is observed through the creation of a Pareto front of optimal solutions [17].

The literature already shows, both in simulation and experimentally, that optimizing the shape of a lithium-ion battery's input current profile for an identifiability metric such as Fisher information leads to significant improvements in parameter estimation speed and accuracy [19]. However, two critical challenges remain. First, existing research on battery test trajectory optimization for identifiability typically focuses on single battery cells, rather than entire packs. Optimizing pack cycling is a significantly more challenging problem, particularly for battery packs capable of utilizing advanced power electronics to adjust the allocation of total pack current among different cells. Second, the literature also typically assumes that the input current profile acting on a battery cell can be optimized with few constraints. This is unrealistic for online applications, where external pack current demand is typically dictated by a user or supervisory controller.

The overarching goal of this article is to optimize the allocation of current among different parallel strings of cells in a lithium-ion battery pack in a manner that maximizes Fisher identifiability, thereby improving the fidelity of pack-level diagnostics. Modern lithium-ion battery packs can be equipped with power electronics to enable such flexible current allocation among different strings/modules. The literature already explores the use of such flexibility for active charge balancing, with the goal of maximizing pack life [20,21]. Rehman et al. incorporate DC–DC converters in parallel with battery cells to create a battery pack that cycles healthier cells deeper than weaker cells [20]. Li et al. construct an active-balancing pack design through the incorporation of MOSFET switches and multiwinding transformers that achieves high levels of energy transfer efficiency [21]. These research studies highlight the growth in new battery pack architectures that allow for unequal distribution of pack current among cells.

The novelty of our work stems from the fact that we exploit this flexibility for improving battery parameter identifiability. State of charge (SOC) balancing remains important as a constraint in our research. However, our work allows for brief transient departures from a fully balanced state in order to maximize parameter identifiability. This results in significant improvements in the accuracy with which unhealthy outlier cells can be detected. While the link between this body of work and existing literature does not immediately present itself, such studies provide the groundwork to expand upon constrained identifiability optimization through the exploitation of advances in battery pack design. In comparison to the existing body of optimization literature, we provide the new-found ability to optimize trajectories for online-constrained systems that could subsequently be used intuitively to guide the construction of online control algorithms. We demonstrate this ability in a simple case study meant to represent an electric vehicle (EV). While typical EV battery packs found on the market today have one to two strings of cells connected in parallel, we select our pack designs to have six strings in parallel to augment our ability to determine which dynamics increase the pack-level identifiability. The token swapping evolutionary algorithm developed in this article is easily generalizable for future studies that have fewer numbers of strings in parallel.

The remainder of this article is organized as follows: Section 2 provides background information regarding the equivalent-circuit model we use in this study, and the Fisher information metric we exploit for quantifying battery parameter identifiability. Section 3 presents the bulk of this paper's contributions by: (i) formulating a min–max pack-level Fisher identifiability optimization problem taking into account pack current demand constraints, (ii) developing a novel evolutionary algorithm for solving this optimization problem, and (iii) presenting an algorithm for unhealthy cell detection using the local outlier probability (LoOP) criterion. Section 4 demonstrates the above framework for an electric vehicle case study. Two different battery pack configurations are examined in this case study: one that uses simple MOSFET switches to allocate external current among different strings of cells in a battery pack passively, and one that uses bidirectional DC–DC conversion to enable the active shuttling of charge between these strings. Section 5 presents the results of this simulation case study and shows that optimizing battery pack current allocation does, indeed, improve outlier detection accuracy significantly. Finally, Sec. 6 summarizes the article's findings.

## Background

This section presents a second-order equivalent-circuit model of a commercial 18650 LiFePO4 (LFP) battery cell and shows how the Fisher information metric can be used for quantifying the identifiability of this model's parameters. We demonstrate our pack-level identifiability optimization framework for this specific battery model and chemistry, but the framework is equally applicable to other models and chemistries.

### Single-Cell Equivalent-Circuit Battery Model.

Figure 1 shows the battery model used in this article. This is a second-order equivalent-circuit model, where the two energy storage devices are the nonlinear capacitor that produces the OCV as a function of state of charge and the linear capacitor that together with the first resistor represent the transport dynamics of the cell. This model, which consists of linear state-space dynamics and a nonlinear output equation, is given as

Fig. 1
Fig. 1
Close modal
$x˙1=1Qu$
(1)
$x˙2=−1R1C1x2+u$
(2)
$y=VOC(x1)+1C1x2+R2u$
(3)

where the two state variables, x1 and x2, are the SOC of the battery and the amount of charge in the parallel capacitor C1, respectively. Transport dynamics are approximated through the combination of this parallel capacitor and the parallel resistor R1. The charge capacity of the battery cell is Q, and the ohmic resistance is R2. The model's input is the supplied current, u, and the output is the terminal voltage across the battery cell, y. The open-circuit voltage, VOC, is a nonlinear function of SOC. Table 1 presents the values of the four parameters used in this model, and Fig. 2 plots the OCV versus SOC. These parameters and nonlinear OCV–SOC relationship are based on earlier experimental work by Docimo et al. [22].

Fig. 2
Fig. 2
Close modal
Table 1

Experimentally measured parameter values from an 18650 Li-ion battery cell [22]

ParameterValue
Q3921 C
$R1C1$2631 s
C117,327 F
R20.02 Ω
ParameterValue
Q3921 C
$R1C1$2631 s
C117,327 F
R20.02 Ω

### Fisher Information Identifiability Metric.

To quantify the identifiability of a battery model, one must answer two questions. First, is the model structurally identifiable [23]? In other words, is it possible to estimate the model's parameters from input–output data? Second, if the model is structurally identifiable, how accurately can one estimate its parameters? Our focus in this article is on the latter, numerical identifiability question. We quantify numerical identifiability using the Fisher information matrix (FIM), defined as follows [24,25]:
$F=E{(∂∂θlnp(y(t)|θ))(∂∂θlnp(y(t)|θ))T}$
(4)
where θ is a vector of unknown parameters, y is a vector of noisy output measurements, and E is the expectation operator [23,24]. Mathematically, one can interpret this matrix as a measure of the curvature of the log likelihood function around the maximum likelihood estimate of the parameter vector, θ. This interpretation is important, because it leads to the conclusion that a “larger” Fisher information matrix is needed for more accurate parameter identifiability [19]. In fact, it is not possible for any unbiased estimator to determine the parameters of a dynamic system model with better covariance than the inverse of the FIM [11,25]. This result, expressed mathematically below, is often called the Cramér–Rao inequality
$E{(θ̂−θ)(θ̂−θ)T}≥F−1$
(5)
There are several methods in the literature for optimizing the design of an experiment to maximize Fisher information. We adopt the so-called D-optimal experimental design approach, where the goal is to maximize the determinant of the FIM [25,26]. We perform this optimization with respect to the trajectory of the battery's input current, u(t). This makes it necessary to compute the FIM for different input current trajectories, as part of the overall numerical optimization process. This computation is particularly simple in the special case where only the measured battery output voltage is noisy (i.e., the input current is known exactly), and the voltage measurement noise is zero-mean, white, and Gaussian. Under these assumptions, the FIM can be computed as follows [9,10,24,25]:
$F=1σv2STS$
(6)
where $σv2$ is the variance of the voltage measurement noise at every sampling instant, and S is a matrix representing the sensitivity of the battery's output voltage with respect to the unknown parameters. More specifically, every term $Si,j$ of the matrix S is the sensitivity of the output voltage y(t) at the ith sampling time to the jth member of the unknown parameter vector θ, as shown in the following equation [19]:
$Sij=∂y(ti,θ)∂θj$
(7)

## Solution Framework

This section presents the three main pieces of our proposed framework for battery pack-level identifiability optimization and diagnostics. The section begins by formulating the pack-level identifiability optimization problem as a max–min problem, where the goal is to quantify the determinant of the Fisher information matrix for each string of battery cells in a pack separately, then maximize the minimum determinant among these different strings. This optimization is performed subject to the constraints that: (i) the total current demanded of the battery pack at every instant in time must be honored and (ii) no permanent charge imbalance can emerge as a result of identifiability optimization. These constraints motivate the second contribution presented in this section, namely, a novel evolutionary optimization algorithm designed to automatically satisfy the above constraints. Third, the section presents an unhealthy battery cell detection algorithm based on the LoOP metric. We show in Sec. 4 that the fidelity of this outlier detection algorithm improves significantly when battery pack cycling is optimized for identifiability.

### Max–Min Constrained Optimization Formulation.

Consider a battery pack consisting of Ns parallel strings of cells, each string in turn containing Nc cells in series. For example, if there are six parallel modules/strings of battery cells in the pack, each containing 100 cells in series, then Ns = 6 and Nc = 100. Suppose that state of charge within each string is balanced using a dedicated balancing circuit, so each string can be represented for optimization purposes by a single “average” battery cell. Suppose, furthermore, that the total external pack current u(t) can be allocated unequally among the different strings, and let $uk(t)$ be the current allocated to the kth string. Suppose the states of charge of all the battery strings are balanced initially, and suppose that while a temporary imbalance is permissible, the pack must eventually return to a balanced state. Then, one can formulate the problem of optimizing pack current allocation for Fisher identifiability as follows:
$maxu1(t),...,uNs(t) mink{det F1(u1(t))det F2(u2(t))⋮det Fk(uk(t))⋮det FNs(uNs(t))}subject to:$
(8)
$Fk=1σ2[∑i=1NSi1k2∑i=1NSi1kSi2k∑i=1NSi1kSi3k∑i=1NSi1kSi4k∑i=1NSi2kSi1k∑i=1NSi2k2∑i=1NSi2kSi3k∑i=1NSi2kSi4k∑i=1NSi3kSi1k∑i=1NSi3kSi2k∑i=1NSi3k2∑i=1NSi3kSi4k∑i=1NSi4kSi1k∑i=1NSi4kSi2k∑i=1NSi4kSi3k∑i=1NSi4k2]Si1k=∂yk(t)∂θ1|t=i*δt; θ1=1Q k=1,2,...,NsSi2k=∂yk(t)∂θ2|t=i*δt; θ2=1R1C1 k=1,2,...,NsSi3k=∂yk(t)∂θ3|t=i*δt; θ3=1C1 k=1,2,...,NsSi4k=∂yk(t)∂θ4|t=i*δt; θ4=R2 k=1,2,...,Nsx˙1,k=1Quk k=1,2,...,Nsx˙2,k=−1R1C1x2,k+uk k=1,2,...,Nsyk=VOC(x1,k)+1C1x2,k+R2uk k=1,2,...,Nsh1: ∑k=1Nsuk(t)−uapplied(t)=0 ∀ t∈[0...T]h2: SOCk(T)−SOCfinal=0 k=1,2,...,Nsg1: SOCk(t)−SOCmax≤0 k=1,2,...,Nsg2: SOCmin−SOCk(t)≤0 k=1,2,...,Nsg3: 5 Amps−uk,max≤0 k=1,2,...,Nsg4:−30 Amps−uk,min≤0 k=1,2,...,Nsg5: 3.8 V−yk,max≤0 k=1,2,...,Nsg6: yk,min−2 V≤0 k=1,2,...,Ns$
(9)

The above optimization problem statement can be explained as follows. The goal is to maximize the minimum determinant of Fisher information among all the strings of cells. This max–min formulation is important, because it prevents one battery string from gaining an unfair advantage in Fisher information compared to the remaining strings. We perform this optimization by adjusting the instantaneous allocation of current demand among the strings, subject to the constraints that the sum of all string currents must equal the total external current demand. The dynamics of the different battery strings appear as additional constraints, but fortunately our assumption that each string is internally balanced allows us to represent these dynamics using just one representative average cell per string. All strings must start and end at the same states of charge, although temporary imbalance is permitted. To avoid a situation where such temporary imbalance leads to the excessive over-charging or over-discharging of the strings, we enforce a strict bound on state of charge for all strings at all instants in time. We also enforce strict bounds on the C-rate of the cell to stay within safe operating limits provided by the manufacturer. The optimization problem, while conceptually simple, is nonlinear, nonconvex, constrained, and also potentially very large (depending on the value of Ns). This motivates the development of the specialized optimization algorithm below.

### Token Swapping Evolutionary Algorithm.

This section presents “token swapping,” a simple evolutionary algorithm designed specifically for solving the identifiability optimization problem in Eq. (8). The need for this specialized optimization algorithm is twofold. First, the optimization problem in Eq. (8), like many identifiability optimization problems, is nonconvex. This nonconvexity can be explained intuitively in terms of the fact that Fisher information generally improves with more aggressive battery cycling. Finding the globally optimal solution to a nonconvex optimization problem is difficult, and the use of evolutionary optimization is one means of addressing this difficulty. Second, the optimization problem in Eq. (8) is also highly constrained. Two particularly important constraints are: (i) the fact that the total current delivered by the battery pack at every instant in time must equal the external current demand and (ii) the desire to achieve a balanced state of charge among all strings in the pack at the end of the cycling process.

The token swapping algorithm is an evolutionary algorithm that automatically guarantees the satisfaction of the above two constraints. The algorithm quantizes the amount of charge inserted into or removed from any string in a battery pack in terms of “tokens,” each token representing a fixed (and ideally small) amount of charge (in Coulombs or Ampere-hours). Furthermore, the algorithm represents the charge/discharge trajectory for the entire battery pack using a tabular data structure, as shown in Fig. 3. In this tabular data structure, each row represents a battery cell or string, and each column represents a discrete time interval. The number of tokens, positive or negative, occupying each cell in the table represents the total amount of charge removed from or inserted into a given battery cell/string over a given time period, respectively. Assuming the battery pack is initially balanced, the pack is guaranteed to remain balanced at the end of the charge/discharge cycle if (and only if) the sums of the tokens in all table rows are equal. Moreover, the battery pack is guaranteed to meet external current demand if (and only if) the sum of the tokens in each column is commensurate with the amount of charge to be inserted into or removed from the full battery pack over that time interval. Thus, the pack balancing and external current demand constrains translate directly into row and column sum constraints in this tabular data representation.

Fig. 3
Fig. 3
Close modal

The main idea behind the token swapping algorithm can be explained as follows: given a valid token table (i.e., one with the correct row and column sums), one can obtain a different valid token table through the “swapping” process sketched in Fig. 3. In this process, one battery cell/string defers some of its charging/discharging to a later moment in time, while another battery cell/string advances an equal portion of its charging/discharging to an earlier moment in time. As long as the number of tokens swapped forward by one battery cell/string equals the number of tokens swapped backward by the other battery cell/string, the token table remains valid. It is straightforward to show, mathematically, that the entire set of valid token tables can be generated from any member of the set through a finite sequence of such token swaps. In other words, the entire optimization domain is “reachable” from any feasible token table in that domain through a finite number of token swaps. The token swap operator, as described previously, is essentially a “mutation” operator in the language of evolutionary optimization. The token swapping algorithm performs this mutation operator randomly and repeatedly until it converges to the optimal token swapping table. Specifically, the algorithm proceeds as follows:

• Step 1: Given the total number of battery cells/strings in parallel, Ns, select the total number of time divisions, n.

• Step 2: Select the amount of charge represented by one token, q, in Coulombs.

• Step 3: Generate Nt valid token tables, i.e., Nt token tables where, for each table, the row sums are all equal and the column sums correspond to the total amount of charge demanded from the battery pack over the corresponding time division. Ensure that these token tables meet all other optimization constraints. For example, make sure that no token table violates constraints on cell state of charge or charge/discharge rate. Constraints on C-rates automatically translate into bounds on the numbers of tokens in the individual cells in a token table.

• Step 4: Evaluate the Fisher identifiability cost function for each of the Nt token tables. Select the token table that provides the maximum identifiability, and call it the elite (i.e., best) token table. The elite token table will not be subject to random selection: a fact that ensures monotonic convergence to the globally optimal token table.

• Step 5: For every nonelite token table, select two time divisions, two battery cells/strings, and a number of tokens randomly. Perform a random token swap using those two cells, for those two time divisions. Check to see if the resulting token table satisfies all optimization constraints, including constraints on cell SOC and C-rate. If it does, add this table to the population. Perform this process w times per nonelite token table, thereby adding $w(Nt−1)$ token tables to the overall population.

• Step 6: Retain the elite token table, and select population members randomly (either with uniform likelihood or using a roulette wheel selection method) from the nonelite members, until a new population of Nt members is chosen.

• Step 7: Repeat steps 4–6 until convergence.

The above algorithm converges to a globally optimal identifiability-optimizing cycle. Convergence has been fairly slow in the authors' experience, typically requiring 3000 iterations with a population of 100 token tables. However, the algorithm still has the desirable properties of achieving an identifiability-optimizing charge/discharge solution for multiple battery strings/cells in parallel in the presence of nonconvexity and constraints. In the remainder of this paper, we distinguish between “positive” and “negative” versions of the token swapping algorithm. The distinction is simple: positive token swapping refers to an implementation of the above algorithm where the number of tokens in each cell must always remain positive or zero. This corresponds to a battery pack topology where all battery cells must be charged/discharged simultaneously, and it is never possible to charge one cell while discharging another. This is consistent with the first, switching-based battery pack topology examined in this work. In contrast, the negative token swapping algorithm allows some cells in the token table to contain a negative number of tokens while other cells contain positive numbers of tokens. This algorithm enables some battery cells to charge while others discharge. This is consistent with the DC–DC converter topology examined in this article.

### Parameter Outlier Detection Algorithm.

This section presents an algorithm for detecting unhealthy battery cells within a series string of Nc cells. The algorithm furnishes an “outlier score” for each battery cell in the string, quantifying the likelihood that this cell is indeed an outlier. We compute this outlier score based on an established metric from the literature, namely, the LoOP metric [27]. This metric combines probabilistic and density-based approaches for outlier detection [27]. The particular manner in which this work uses the LoOP metric for battery pack-level outlier cell detection is, to the best of the authors' knowledge, a novel addition to the literature. Specifically, we use battery model parameter estimates as inputs to the LoOP algorithm and utilize the LoOP algorithm to assess the degree to which the estimated parameters for a given battery cell deviate from the rest of the series string containing it. Moreover, we show through a detailed case study that optimizing battery cycling for identifiability does, indeed, improve the fidelity of LoOP-based outlier detection.

The LoOP metric from Ref. [27] is outlined below. Suppose that D is a set of parameter estimates for Nc battery cells, containing one vector of estimated parameters per cell. Suppose we select a particular parameter vector, $o∈D$, corresponding to a particular battery cell. Suppose we wish to assess the degree to which this battery cell may be an outlier. Let $b∈D$ be the vector of parameter estimates for an arbitrary neighboring cell. Let d(o, b) denote the Euclidean distance between these two parameter vectors, i.e.,
$d(o,b)=||o−b||2$
(10)
Now let us find the set of s closest neighbors of the battery cell with parameter vector o, where proximity is determined using the above Euclidean distance. Denote this set of close neighbors, together with the parameter vector o, by $B(o)⊆D$. Then, the root mean square distance between o and its neighbors in B(o) is equal to $(Σb∈B(o)d(o,b)2)/|B(o)|$, where $|B(o)|$ is the size of the set B(o). Multiplying this root-mean-square distance by a user-defined factor λ, with the particular choice λ = 3, essentially gives an estimate of how large “three standard deviations” from the parameter vector o might be, if the vector o was truly at the center of the set B. We call this estimate the probabilistic set distance and compute it formally as follows:
$pdist(λ,o,B)=λ×Σb∈B(o)d(o,b)2|B(o)| with o∈D, B(o)⊆D$
(11)
The average battery cell in a given set B(o) will have a probabilistic set distance equal to $Eb∈B(o)[pdist(λ,b,B(b))]$, where E denotes the expectation operator. Cells with a smaller probabilistic distance will be more closely clustered with the rest of the battery string in terms of their parameter values. Cells with a larger probabilistic distance will be further away from the cluster of B(o) cells in terms of parameter values. This simple argument motivates the definition of the probabilistic local outlier factor (PLOF) below. The larger this PLOF, the more likely a given battery cell is to be an outlier
$PLOF(o):=pdist(λ,o,B(o))Eb∈B(o)[pdist(λ,b,B(b))]−1$
(12)
Given the above definition of PLOF, one can compute the LoOP as shown later. LoOP is a statistical estimate of the likelihood that a given battery cell is an outlier relative to its neighborhood, B(o). The closer the LoOP value for a battery cell gets to 1.0, the more likely the given cell is to be an outlier
$LoOP(o)=max{0,erf(PLOF(o)2λE[(PLOF)2])}$
(13)

The diagnostics algorithm used in this article computes the above LoOP metric based on the estimated parameters of different battery cells in a series string. This makes it possible to determine which cells are likely to be unhealthy outliers, provided the parameters of these cells can be estimated accurately.

## Electric Vehicle Case Study

This section applies the identifiability optimization framework from Sec. 3 to an automotive case study. Specifically, we consider an all-electric midsize sedan with a 250-mile range. We begin by developing a simple model of this vehicle's dynamics, and using this model to predict the vehicle's battery pack current demand for a deterministic benchmark automotive drive cycle. We then present two possible reconfigurable battery pack architectures for this vehicle, one relying on MOSFET switches to enable the flexible allocation of current among battery strings, and one relying on bidirectional DC–DC converters to enable the active shuttling of charge among the strings. Finally, we optimize battery current allocation among different strings, for both of the above configurations, for Fisher identifiability, and analyze the degree to which this optimization improves LoOP-based outlier detection. While this study is for a system with online constraints, the optimization problem itself is solved offline. This first approach is meant to highlight the potential for identifiability improvements that lead to diagnostic improvements. This study shows that for a realistic driving profile there is a chance for improvements in diagnostics, and the characteristics of the optimized input trajectories can be used for subsequent real-time control design.

### Electric Vehicle Model and Current Profile.

Figure 4 presents a high-level block diagram of the dynamics of the electric vehicle examined in this case study. The model consists of: (i) a vehicle dynamics submodel that captures the effect of vehicle inertia, rolling resistance, and aerodynamic drag, (ii) a battery pack model that accounts for SOC accumulation, internal battery resistance, and the nonlinear relationship between OCV and SOC, (iii) a “control scheme” that translates driver pedal position into a motor torque command, and (iv) a driver model that uses proportional integral (PI) control to force the vehicle to track a given drive cycle. We use this simple model to translate the vehicle's drive cycle into an overall battery pack current demand. The Appendix presents the block diagrams for the various submodels in this overall vehicle system model. The above model is capable of simulating overall battery pack current demand for different vehicle drive cycles. The particular drive cycle we use in this case study consists of: (i) two federal test protocols (FTP) of the city driving cycle FTP-75, back to back, truncated to furnish a vehicle driving duration of exactly 1 h, followed by (ii) 3 h of vehicle rest. The goal of this benchmark cycle is to assess the degree to which unhealthy outlier cells can be detected by the LoOP algorithm after only 4 h of battery usage, including only 1 h of driving. This 1h driving component of the benchmark is shown in Fig. 5. Detecting and identifying unhealthy battery cells within such tight time constraints are very difficult: a fact that is evident in the results of the case study and motivates our use of identifiability optimization to improve diagnostic accuracy.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

### Electrically Reconfigurable Battery Packs.

Two electrically reconfigurable battery pack designs are examined to determine how varying levels of hardware complexity improve parameter identifiability. The pack architecture diagrams in Fig. 6 show configurations that include six battery cells connected in parallel. Each one of these six cells is modeled as a simplified representation of an entire string of cells. Figure 6(a) outlines the first pack design that incorporates MOSFET switches and adds the ability to split the pack-level current unequally among the six cells through pulse width modulation. Figure 6(b) describes the second design that includes bidirectional DC–DC converters to allow for current splitting and charge shuttling between cells even when this is no external power demand. The cells in both Figs. 6(a) and 6(b) incorporate the equivalent-circuit models described in Sec. 2.1.

Fig. 6
Fig. 6
Close modal

### Electric Vehicle Outlier Detection Study.

Given the above benchmark drive cycle and vehicle model, it is possible to compute an overall vehicle battery pack current demand profile. Given this current demand profile, one can optimize internal current allocation within the vehicle's battery pack for identifiability. This can be done for both the MOSFET-based and DC–DC converter-based reconfigurable battery packs using the positive and negative token swapping algorithms, respectively. The end result is a set of trajectories of string-level currents and states of charge for each of these two configurations. Figures 7 and 8 show these trajectories.

Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal

The main goal of this case study is to address the following question: to what extent do the above optimized current trajectories improve the ability of the LoOP algorithm to detect unhealthy outliers? We address this question through a Monte Carlo case study. This case study examines the battery string with the lowest determinant of Fisher information for each of the above two battery pack configurations. We assume that this string consists of 95 healthy battery cells, plus five unhealthy outliers, parameterized as follows:

• The parameters of the healthy cells are normally distributed around the nominal values presented in Table 1. Specifically, the distribution of each parameter is assumed to be Gaussian, with a mean equal to the nominal value of the parameter in Table 1, and a standard deviation equal to 1.67% of the mean. Thus, a parameter deviation equal to 5% of the nominal value corresponds to three standard deviations. This choice of parameter variability is consistent with experimental research by Baumhöfer et al., in which the cycling of fairly homogenous healthy cells results in $±5%$ capacity variation by the end of the cells' life [28].

• In contrast, the parameters of the five outlier cells are distributed as follows. There are four “univariate” outliers. For each univariate outlier, one particular parameter is 3.33 standard deviations away from the mean, while all other parameters equal their nominal values. For the multivariate outlier, all parameters are 3.33 standard deviations away from the mean. Table 2

Table 2

Distribution of cell parameters from model in Sec. 2.1 in 100-cell series string

Q (C)$τ=R1C1$ (s)C1 (F)R2$(Ω)$
Healthy distribution $N(3921,σQ2)$ $N(2631,στ2)$ $N(17,327,σC12)$ $N(0.02,σR22)$
Univariate outlier #1 3703 2631 17,327 $2.0×10−2$
Univariate outlier #2 3921 2777 17,327 $2.0×10−2$
Univariate outlier #3 3921 2631 18,289 $2.0×10−2$
Univariate outlier #4 3921 2631 17,327 $2.1×10−2$
Multivariate outlier 3703 2777 18,289 $2.1×10−2$
Q (C)$τ=R1C1$ (s)C1 (F)R2$(Ω)$
Healthy distribution $N(3921,σQ2)$ $N(2631,στ2)$ $N(17,327,σC12)$ $N(0.02,σR22)$
Univariate outlier #1 3703 2631 17,327 $2.0×10−2$
Univariate outlier #2 3921 2777 17,327 $2.0×10−2$
Univariate outlier #3 3921 2631 18,289 $2.0×10−2$
Univariate outlier #4 3921 2631 17,327 $2.1×10−2$
Multivariate outlier 3703 2777 18,289 $2.1×10−2$
lists the precise parameter values for these outliers and compares them to the healthy parameter distribution.

We perform Monte Carlo simulation 100 times. For each run of the Monte Carlo simulation, we apply the benchmark and optimized current trajectories to the above string of 100 battery cells and corrupt the resulting cell voltage measurements by zero-mean, Gaussian, and white measurement noise with a 5 mV standard deviation. Using these corrupted voltage measurements together with the string-level current profiles, we obtain least squares estimates of each cells' parameters. We then insert these estimates into the LoOP algorithm for outlier detection. This use of Monte Carlo simulation makes it possible to evaluate the performance of the LoOP algorithm, specifically in terms of its ability to distinguish between healthy cells and unhealthy outliers.

## Results and Discussion

This section presents the results from both token swapping algorithm optimization formulations as well as the results from the outlier detection algorithm for the electric vehicle case study outlined in Sec. 4. The results for both optimization studies are first compared against the benchmark cycle to show the increase in the Fisher information identifiability metric in Table 3. The determinant of the Fisher information matrix increases more than an order of magnitude for the positive token swapping cycle and almost nine orders of magnitude for the negative token swapping cycle when compared against their respective benchmarks. The cost function values of Fisher information have little meaning on their own, but in comparison they present a broad metric of lumped identifiability improvement. The optimized string-level input trajectories and the simulated SOC trajectories from the positive token swapping scheme, which represents the functionality of the MOSFET battery pack design, are shown in Fig. 7. Figure 7(a) shows the optimal fractional components for each of the six battery cells in the pack-level model at each time interval. To better understand these optimization results, the first time division along the x-axis illustrated in Fig. 7(a) as the first tall box shows that cell 1 (box beginning at origin of plot) has the largest fraction of the external current applied to it followed by cell 2 (lighter box located right above cell 1 box) as the second largest fraction. During the second time division, this changes so that cell 2 now receives a much larger fraction of the external current than cell 1 as the lighter box is much larger than the darker box. During the components of the optimized input trajectories where the vehicle is driving, the time intervals are selected as 144 s long (25 time intervals), and there is only one time division for the rest period since there is no external current to split between the cells. The current input is divided into time divisions to create a form that is compatible with the token swapping algorithm. The number of time divisions can be set to any value, but is selected to be 25 for both cases in this article. The SOC trajectories in Fig. 7(b) are provided as another representation of how these fractional internal currents are applied to each cell/string individually, but also how the final SOC constraint is maintained. The optimized input trajectories and the SOC trajectories from the negative token swapping scheme, which represents the DC–DC converter battery pack design, are shown in Fig. 8. Figure 8(a) illustrates the fractional amounts that the six battery cells experience during each time interval. The driving component has time intervals lasting 240 s (15 time intervals) and is followed by the vehicle rest period that has time intervals lasting 1080 s (ten time intervals). This configuration allows the charge/discharge process to be reversed for individual cells which permits the fractional amounts to be positive or negative as long as the fractions sum to 1 (100%) and match the external current demanded from the vehicle. During the driving component, the internal current fractions sum to 1 (100%) to match external current demand, and during vehicle rest they sum to 0 (100% of external current demand). The SOC trajectories show that certain cells primarily discharge so that others can charge at a much higher C-rate and vice versa. Also, by allowing charge shuttling between cells during vehicle rest, parameter identifiability increases substantially.

Table 3

Cost function values for the benchmark and optimal cycles

CycleDeterminant of Fisherinformation matrix
Benchmark$4.4×1040$
Positive token swapping optimal$8.1×1041$
Negative token swapping optimal$3.0×1049$
CycleDeterminant of Fisherinformation matrix
Benchmark$4.4×1040$
Positive token swapping optimal$8.1×1041$
Negative token swapping optimal$3.0×1049$

The values of the determinant of the Fisher information matrix found in Table 3 do not have meaning by themselves, but when the differences between the benchmark and optimal cases are studied, a connection becomes clear. The increase in this optimization function value is directly correlated to the decrease in the overall parameter variance for the best possible unbiased estimator, due to the Cramér–Rao lower bound. The MOSFET pack design only has an order of magnitude improvement in identifiability, which leads to small decreases in estimated parameter variance, while the DC–DC converter pack design has 9 orders of magnitude improvement over the 4 h benchmark cycle. The key components of the optimal signals that increase identifiability are:

• The positive token swapping optimal solution tries to cycle a single string at a time to increase C-rate and produce large voltage transients, which improves identifiability.

• The negative token swapping optimal solution increases identifiability by charging to very high SOC ranges where the OCV versus SOC curve became highly nonlinear.

• This second optimal solution also increases identifiability with very high C-rates and cycling during the vehicle rest period with charge shuttling to increase the depth of discharge achieved by each string.

These insights into the dynamics which improve identifiability agree with the conclusions presented in Refs. [8] and [19], but do require aggressive cycling. Both of these studies show that identifiability is substantially reduced in the middle of the SOC range and therefore pushes the optimized trajectories to low SOC values when possible (positive token swapping result). While aggressive cycling with high C-rates and large depths of discharge may increase the potential for degradation to occur, previous research shows that identifiability can be improved greatly without substantial increases in degradation through a Pareto front comparing the two quantities [17]. Given the above improvements in battery parameter identifiability, one would expect an improvement in the LoOP algorithm's ability to detect outliers for the MOSFET-based pack architecture and an even bigger improvement for the DC–DC converter architecture. To test this hypothesis, Fig. 9 plots the “receiver operator curves” (ROCs) for three scenarios: (i) one where the benchmark current profile is split equally among parallel battery strings, (ii) one where current is split optimally among the strings using a MOSFET-based battery pack configuration, and (iii) one where current is split optimally among the strings using a DC–DC converter configuration. A receiver operator curve is a plot of the LoOP algorithm's true positive rate (i.e., probability of correctly labeling a cell as unhealthy) versus the algorithm's false positive rate (i.e., probability of labeling a healthy cell incorrectly as unhealthy). We generate these receiver operator curves using Monte Carlo simulation, for each type of outlier. A “perfect” receiver operator curve is one that passes through the point (0,1), meaning that the LoOP algorithm can detect outlier cells 100% of the time, while never labeling a healthy cell incorrectly as an outlier.

Fig. 9
Fig. 9
Close modal

Figure 9 generally supports this article's main hypothesis that optimizing battery cycling for parameter identifiability does lead to more accurate LoOP-based outlier detection. More specifically, a detailed examination of Fig. 9 reveals the following insights:

• First, all three battery pack configurations (benchmark, MOSFET, and DC–DC converter) are able to detect the multivariate outlier with near-perfect accuracy. This is visible from the receiver operator curve in Fig. 9(e). This result is not surprising, considering the extreme nature of the multivariate outlier: it lies 3.33 standard deviations away from the nominal values of all battery parameters. This makes it relatively easy to detect this outlier, for all three cases considered.

• Second, the MOSFET-based battery pack architecture does improve outlier detection accuracy over the benchmark for three of the four univariate benchmarks. This is consistent with our hypothesis that, by improving Fisher identifiability, this architecture also exhibits better outlier detection accuracy. The improvements in outlier detection accuracy are modest, consistent with the modest gains in Fisher identifiability attainable with this architecture.

• Third, the MOSFET-based battery pack architecture does not furnish significant improvements in outlier detection accuracy over the benchmark for the first univariate outlier, as shown in Fig. 9(a). This particular outlier is different from the healthy cells in terms of its charge capacity. The benchmark cycle is capable of estimating charge capacity with sufficient accuracy to the point of achieving excellent combinations of true and false positive rates, as shown in Fig. 9(a). This makes it very difficult for the MOSFET-based architecture to achieve better outlier detection accuracies compared to the benchmark for this particular univariate outlier.

• Finally, the DC–DC converter pack architecture achieves very significant improvements in outlier detection accuracy, compared to both the benchmark and MOSFET architecture, for all univariate outlier. This is quite visible throughout Fig. 9, where the DC–DC converter architecture achieves nearly perfect receiver operator curves for all five types of outliers. This dramatic improvement in outlier detection accuracy is not surprising considering the underlying improvement in Fisher identifiability.

Overall, the diagnostics results for these two battery pack architectures highlight two very important facts. First, the incorporation of MOSFET switches, which are the cheaper component and add less complexity to the pack, only provide minor improvements in Fisher information and, subsequently, minor improvements in outlier detection. In contrast, the incorporation of DC–DC converters, which are the much more expensive component and add substantial complexity to the pack, provide the largest improvements in Fisher information and subsequently outlier detection. These tradeoffs in cost/pack complexity and diagnostics accuracy must be taken into account in judging the success of these results.

## Conclusions

This article uses the optimal trajectories created from an input-shaping problem formulation to improve the ability to accurately perform battery pack diagnostics through parameter outlier detection. The first novel contribution of this article is the token swapping evolutionary algorithm that optimizes the fractional input trajectories for a multicell equivalent-circuit battery model given an external current demand profile from an electric vehicle. The second contribution is the implementation of the probability-based outlier detection algorithm that illustrates how increased parameter identifiability will lead to more accurate outlier detection. Both electrically reconfigurable pack designs with optimized trajectories improved the accuracy of outlier detection, or fault isolation, when compared against the conventional battery pack design. The more complex DC–DC converter pack design had the largest improvement in fault isolation/identification accuracy.

## Acknowledgment

These authors gratefully acknowledge the funding support for this project from the ARPA-E AMPED Award No. 0675-1565 (“A Modular, Intelligent, and Reconfigurable Battery Pack Management System”; PI: Hosam K. Fathy).

### Appendix: Electric Vehicle Simulink Model

The input to the EV model presented in Fig. 4 is the FTP-based drive cycle, which is a vehicle speed as a function of time trajectory. This trajectory is passed into the driver block in Fig. 4 where a PI controller translates this vehicle speed profile to a throttle command profile. The throttle command is passed into the control scheme block where it is either passed to the electric motor torque-speed map or the electric generator torque-speed map. These two maps represent the varying levels of torque demand based on the motor speed and throttle command. The torque demanded from the motor and generator is passed to the vehicle dynamics block where the vehicle speed is determined. There is one state equation to govern the vehicle dynamics, which is given as
$U˙=1m(Froad−Froll−Fdrag)$
(A1)

where U is the speed of the vehicle, m is vehicle mass, and Froad is the force from the electric motor applied through the wheels to the road. The rolling resistance force is Froll, and the aerodynamic drag force is Fdrag.

The motor torque is also used to determine the power demand from the battery pack. The torque and motor speed are used to determine the motor efficiency from the map found in Ref. [29]. This efficiency is used to determine the power demand from the battery pack and is scaled to cell-level power demand. The cell-level power is divided by the instantaneous OCV, which is a function of battery cell SOC, to obtain the instantaneous current. This value is compared against the following equation to determine which component of the quadratic solution to implement based on which value is closer to the true current:
$I=−VOC±VOC2+4PRcell2Rcell$
(A2)
where VOC is the OCV, P is the power demand passed into the equation, and Rcell is the ohmic resistance of the battery cell. The true current profile found to be closest to the approximate current from the two quadratic solutions is designated as I. This current profile is used as the external current demand presented in Sec. 4.2. The current is used to solve for the battery SOC with the following equation:
$SOC.=IQcell$
(A3)

where Qcell is the charge capacity of the battery cell. SOC is the state of charge of the cell at any instance in time. This value is passed back to the control scheme block in Fig. 4 and used for the rest of the simulation.

##### Subsystems Contained Within the Electric Vehicle Simulink Block Diagram.

Figures 1014 present the internal block diagrams for all of the subsystem blocks presented in Fig. 4 in Sec. 4.2. These block diagrams illustrate all of the internal dynamics and controllers for the EV model.

Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal
Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal

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