Reduced Order Model of Common Battery Pack Architectures for Assessment of Cell Parameter Variation Propagation

Nowadays, the series-parallel (SP) and the parallel-series (PS) configurations represent the main architectures considered for designing a battery pack. In both architectures, cell-to-cell parameters’ variations due to manufacturing tolerances, thermal gradients, and cell degradation can strongly impact the overall performance of battery packs. However, the effect of the parameter variation on the pack performance changes depending on the architecture and the number of cells. This manuscript aims at providing a method for the assessment of the impact of cell-to-cell parameters’ variations due to potential different aging and thermal conditions for the cells in a generic SP and PS pack architecture. For this purpose, a generalized reduced order equivalent circuit model for the battery pack is defined, leading to a set of steady-state equations for easily and systematically calculating the current and voltage imbalances due to capacity and internal resistance variations independently from the pack architecture and the number of cells. A parametric analysis is reported for quantitatively evaluating the severity of the imbalances due to the cell-to-cell parameters’ variations for both battery pack architectures. The results of the comparative analysis demonstrate that PS architecture is more impacted by parameters’ variations with respect to the SP one, since a higher current imbalance is always obtained. The difference in the severity of the current imbalance in the pack architectures strongly increases as the number of series-connected cells rises. Moreover, among the cell parameters’ variations, a major impact of the capacity imbalance on the reduction of the overall performance of the battery pack is observed.


I. INTRODUCTION
Electrochemical energy storage systems are widely adopted in different applications enabling transport electrification and supporting the integration of renewable energy sources in grid-connected power systems [1], [2], [3]. Among the different technologies, lithium-ion batteries (LIBs) are characterized by high specific power/energy density, low The associate editor coordinating the review of this manuscript and approving it for publication was Hao Wang . self-discharge and high lifetime [1], resulting in one of the best commercially-available solutions for energy storage systems. Different aspects need to be taken into account for the optimal design of lithium-ion battery packs with the aim of achieving high performance while ensuring safe operating conditions for the cells. These include: • the development of control systems for monitoring and managing the charge/discharge process (sensors, battery management systems). According to the capacity and voltage requirements of the load, a large number of cells in series and/or parallel connection may be required, leading to complex interconnections of cells in a pack. According to the standard IEC 60050, two main battery pack architectures are commonly adopted, namely parallel-series (PS) and series-parallel (SP) configurations, as shown in Fig. 1. Traditionally, automotive battery packs are assembled in PS configuration. This is mainly driven by the possibility of reducing the cost of the Battery Management System (BMS). For example, Tesla Model S is equipped with a battery pack that has 16 modules connected in series where each module is a PS architecture (6 cells connected in parallel and then 74 units connected in series, 6p74s) [8]. The SP architecture is found in fewer automotive applications, among these the Honda Fit EV [9]. However, the SP configuration is drawing increasing attention especially in space and aviation applications [10], [11] since it can potentially increase reliability and diagnosability [12], [13]. However, a systematic comparison between these two different battery pack architectures has not been fully investigated in literature due to the lack of tools for a comprehensive analysis. This manuscript's objective is to provide a method for quantitatively assessing the impact of cell-to-cell parameters' variations and, thus, properly comparing the different imbalance conditions between the PS and SP configurations.
It is important to highlight that the overall performance of the battery pack is strongly affected by cell-to-cell parameters' variations due to manufacturing tolerances, thermal gradients and cell degradation [14], [15], [16]. Cell-to-cell parameters' variation usually causes capacity and/or resistance changes, leading to voltage and current imbalances among the cells in the pack. At the pack level, these voltage and current imbalances can cause different issues. Voltage imbalance can lead to a reduction of the efficiency of the battery pack, as an example one cell could reach the minimum voltage early, thus limiting the pack's effective available energy. Current imbalance can lead to uneven heat generation and potential fire hazards. Both voltage and current imbalances can induce uneven aging between the cells, causing in some cases a positive feedback due to resistance and temperature increases, thus increasing aging condition [17]. The voltage imbalance can also cause uneven State of Charge (SoC) conditions and requires long and inefficient cell balancing. However, the severity of these imbalances is different depending on the battery pack architecture and the number of cells, therefore it results fundamental to quantitatively evaluate the magnitude of these imbalances to properly design the battery pack, selecting the interconnection architectures and related component sizing (bus bar, wiring, protection) [13].
Due to the large number of cells in a battery pack and related electro-thermal interconnections among the cells and/or modules, the analysis of the cell-to-cell parameters' variation in a battery pack becomes a complex matter. Usually, this issue is approached by using computationally expensive models to represent the behavior of a battery pack [18], [19] and considering specific architectures with a defined number of cells in series and/or in parallel. As matter of fact, the available models are representing the behaviour of all the cells in the pack using equivalent circuit models (ECMs) or electrochemical models, and interconnection equations based on voltage and current Kirchhoff's laws (KVL, KCL). To limit the model complexity and the number of states, a limited number of cells is usually considered for pack analysis. Based on the existing literature, there is a lack in evaluating the impact of the cell parameters' variations on the overall battery pack considering realistic operating conditions and realistic architectures. In particular, the studies on parameter variation effects on battery packs are mainly focused on specific operating conditions and small size pack architectures, resulting in three categories: series-connected string, parallel-connected string and mixed configurations (PS and SP).
A cycle life model that allows for assessing the temperature variation of LIBs during cycling has been developed and applied to series-connected LIBs in [15]. The simulation result shows that the capacity degradation of the battery pack increases with the increase of the average temperature of the cells in the battery pack as well as the temperature difference. Researchers in [20] analyze the impact of the consistency of parameters such as capacity, internal resistance, and SoC on the energy utilization efficiency of the battery pack. The results show that SoC variations lead to the most significant impact. Hence, based on long-term battery data, an evolution model for the SoC and open circuit voltage characteristic is established and used for predicting battery pack energy utilization efficiency and for diagnosis of battery aging status. A pack model with 96 cells connected in series is used to study the cell parameters' variation in [21] and it is found that temperature, self-discharge rate and coulombic efficiency are the major factors that affect battery pack inconsistency. In [22], a series connected battery pack with 96 LiFePO 4 cells was adopted to investigate the capacity fade during a cycle life test. Each cell in the battery system model is represented by its own electro-thermal cell model. When conducting the simulation, cells are set with different initial capacities, initial internal resistances, aging rates, and thermal conditions to simulate cell-to-cell parameter variation scenarios. The simulation results show that the discrepancies in cells aging status are mainly due to different cell loading and different thermal condition.
As for parallel-connected cells, the principles of current distributions are explored in [23] considering two parameter variation cases: resistance or capacity variation only. It is found that a resistance imbalance is more likely to cause noticeable non-homogeneous current distributions. Reference [24] presents that the performance of the battery module is affected by the resistance of the inter-cell connection plates (ICCP) and the position of the battery module posts (BMP). ICCP in the parallel-connected cells affects the pack performance and causes uneven current flow. The cell directly connected to the BMP presents the lowest terminal voltage and SoC during the discharge process. The experimental and modeling results in [16] demonstrate that a 20% internal resistance imbalance between two parallel-connected cells can cause around 40% reduction in cycle life when compared to two parallel-connected cells with identical or very similar internal resistance. The studies in [25] show that temperature difference between parallel-connected cells can lead to unbalanced discharging and aging, then the capacity loss rate increases approximately linearly as the temperature difference increases.
As for mixed architectures, the authors in [26] select a 6S5P battery pack as an example to study the asymmetry its cycle life due to thermal gradients. In [27], the authors explore the discharge characteristics of SP and PS battery packs with nonuniform cell parameters and find that the discharge capacity varies with different cell connections and different numbers of nonuniform cells. It is found that most works regarding cell-to-cell parameters' variations have been carried out for a battery pack with a fixed configuration or fixed number of cells. There are few literature contributions that analyze the effect and comparison of cell-to-cell parameters' variations of different pack architectures considering variable numbers of cells. The investigation on cell performance and inconsistency evolution of generalized series and parallel battery modules is presented in [17]. As result, the cell performance in series modules is self-divergent while self-convergence in parallel modules. However, the analysis in [17] is still limited to single series and parallel connections instead of PS and SP architectures. A low-computation generalized equivalent circuit model (GECM) for both battery pack architectures is proposed in [14], representing a useful tool to evaluate the degree of current/voltage imbalance under cell-to-cell variations. The paper reports some preliminary analyses regarding the effect of cell-to-cell parameters' variations on different battery pack architectures.
This manuscript proposes a generalized reduced order modeling approach capable of assessing the overall performance of a battery pack experiencing current and voltage imbalances among the cells due to parameters' variations with low computational costs without limitations on the number of cells connected in series and parallel. The GECM presented in [14] is extended including the effect on the open circuit voltages of the cells due to capacity, SoC and resistance imbalances. Then, steady-state model equations are achieved, which allow for defining quantitatively the impact of cell-to-cell parameters' variations at pack level. Therefore, the proposed generalized modeling approach is adopted for performing a detailed parametric analysis that aims at assessing the different severity of the voltage and current imbalances among the cells for realistic pack architectures (PS and SP configurations). In particular, many case scenarios are reported to investigate the impact of capacity and/or internal resistance imbalances due to potential aging and/or high/low temperature conditions on the cells' voltage and current distributions in the two pack architectures. Moreover, the parametric analysis also takes into account the impact of the size of the battery pack in terms of cells in series and parallel connections, thanks to the capability of the proposed modeling approach to be applied for large-scale battery packs with low computational costs.
Section II reports the traditional modeling approach for SP and PS architectures based on the KVL and KCL for describing the electrical behavior of the cells. Moreover, a comparative analysis of SP and PS configurations is illustrated with the aim of assessing qualitatively their main advantages and disadvantages in case of imbalances due to cell-to-cell parameters' variations. A quantitative comparison is also reported considering 3P3S and 3S3P architectures as case studies in order to highlight the different severity of the current and voltage imbalances due to parameters' variations occurring among the cells. Section III presents the GECM adopted from [14] and extended in this publication for fully describing the behavior of the complex electrical interconnection of cells in a battery pack architecture. Then, several assumptions are considered for achieving low-computational steady-state model equations capable of directly correlating the amplitude of the imbalance conditions with the severity of the cell-to-cell parameters' variations and the size of the battery pack in terms of series and parallel connections. Therefore, the model errors introduced by these assumptions are investigated in order to validate the proposed generalized modeling approach. Section IV presents the results achieved by adopting the proposed steady-state model equations for analytically comparing the severity of the current and voltage imbalances occurring among the cells in generic PS and SP configurations, including the impact of the number of series and parallel connections. The conclusions are then outlined in section V.

II. QUALITATIVE COMPARISON OF BATTERY PACK ARCHITECTURES
Battery packs consist of a large number of cells properly connected in series and/or in parallel to meet the voltage and capacity requirements of the specific application. According to the standard IEC 60050, two main battery pack architectures can be identified, including parallel-series (PS or M PN S) and series-parallel (SP or N SM P) configurations, as shown in Fig. 1. In detail, considering N the number of cells in series and M the number of cells in parallel, the SP architecture aims to connect the cells in series first and then in parallel, while the PS one vice versa, despite both configurations are composed by N ·M cells. In this publication, a module consists of M parallel-connected cells for the PS configuration and N series-connected cells for the SP configurations. Table 1 shows a comparison between SP and PS configurations considering the number of physical interconnections between the cells, sensing, protections and BMS requirements. The PS configuration requires more cell interconnections, resulting in the increase of packaging complexity and connection impedance [14], and a lower number of voltage sensors, BMS board channels and balancing circuits for cell voltage monitoring and cell balancing [28], [29]. For this reason, PS configuration is mostly used in automotive applications allowing for cost and complexity reduction. As matter of fact, even if N ·M current sensors would be required to properly diagnose and isolate faults in PS modules in real-world scenarios [12], current sensors are avoided at module level. This results in the adoption of only one current sensor for the PS architecture. On the other hand, SP configuration is usually equipped with a less number of protection fuses, but more current sensors compared to PS configuration since a current sensor is typically installed for each string/module. Fuses are generally adopted as protection devices that prevent over-current conditions in battery cells, modules and packs [13]. Note that PS battery architecture needs a greater number of fuses to allow cell-level protection, while in the SP fuses are needed in each module.
Another important aspect to consider when comparing the two architectures is the effect of the cell-to-cell parameters' variations on the pack performance. In order to accurately reproduce and simulate the current and voltage dynamics of a battery pack, proper modeling approaches are needed. These models are usually based on solving the KCL and KVL equations to represent the battery pack interconnections, while ECMs or electrochemical models are used to reproduce the behaviour of each cell [18], [19], [30], [31]. Thermal models can be included as well to model the thermal behaviour of the cells in a pack. However, these models become very difficult to manage when a large number of cells is considered, increasing the number of required equations and states, thus the related computational requirements. Though these models can estimate and predict the overall battery pack performance when cell-to-cell parameters' variations are considered, they still have some limitations such as high computational costs. The following section reports a traditional approach for modeling battery packs.

A. TRADITIONAL MODELING APPROACH FOR BATTERY PACKS
ECMs are usually adopted for system-level analysis for describing the cell terminal voltage and heat generation [32], [33]. The zero-order ECM has been considered since it allows for an easy and intuitive explanation of the phenomena happening in the pack due to cell-to-cell parameter variations. Therefore, all the cells have been modeled with a zero-order ECM, which includes the open circuit voltage V OC,ij in series with the internal resistance R ij . Then, a Coulomb counting method has been adopted for calculating the state of charge of the cells (SoC ij ). Note that the analysis presented in this work can be extended to higher-orders ECMs, however they do not add any relevant contributions to the results since the analysis is conducted considering steady-state conditions. The equations allow for describing the electrical behavior of the generic ij th cell when zero-order ECMs for the cells are adopted: where V ij is the cell terminal voltage, I ij is the cell current (positive in discharge), SoC ij (0) is the initial state of charge and Q ij is the cell capacity. According to Fig. 1 and considering a load current I L , KCL and KVL for the PS architecture can be applied at both module (representing the parallel connection of M cells) and pack levels as follows: PS architecture -Pack level where V i and I i are the voltage and the current of each i th module, respectively.
At the module level, all the M parallel-connected cells of each i th module are characterized by the same terminal voltage (V ij = V i , ∀i = 1, . . . N ) and all the cells partially contribute to the load current I L . Indeed, a lower I ij occurs as M increases, leading to a lower charge/discharge C-rate for the cells. Cell-to-cell parameters' variations may affect the current distribution within the module.
At the pack level, all the modules are subject to the same current I L , while the pack terminal voltage V is equal to the sum of the terminal voltages V i of each i th module. Cell-tocell parameters' variations may affect the voltage distribution among the modules of the pack.
Likewise, KCL and KVL for the SP architecture at module (representing the series connection of N cells) and pack levels result as follows: SP architecture -Module level I 1j = · · · = I ij = · · · = I Nj = I j ∀j = 1, . . . M At the module level, all the N series-connected cells of each j th module are subject to the same current I j and the terminal voltage of the module (V j ) is equal to the sum of the cells' terminal voltages V ij . Cell-to-cell parameters' variations may affect the voltage distribution within the module. At the pack level, all the modules are characterized by the same terminal voltage V and the load current (I L ) results to be equal to the sum of the module currents (I j ). Cell-tocell parameters' variations may affect the current distribution among the modules of the pack.
Ideally, all the cells have the same open circuit voltage and internal resistance. Therefore, (3) and (6) for PS architecture as well as (8) and (9) for SP architecture correspond to a uniform distribution of currents and voltages among the cells. However, cell-to-cell parameters' variations can lead to voltage and current imbalances among the cells that limit the overall pack performance.

B. QUALITATIVE COMPARISON BETWEEN SP AND PS ARCHITECTURES
The different cell connections in PS and SP configurations result in different current and voltage distributions, which can be observed from the module-level and pack-level modeling. In the ideal case, when no parameters' variations occur, the current and voltage splits among the cells are equal regardless of the pack configuration if the same number of series (N ) and parallel (M ) connections is considered. Differently, in case of cell-to-cell parameters' variations, current and voltage imbalances occur and their amplitudes depend on both the parameter variation magnitude and the specific pack architecture. These imbalances can then cause uneven heat generation in the pack, aging and reduced efficiency. In the following analysis, according to Fig. 1, cell 11 in Module 1 is assumed to be affected by parameters' variations, and a constant load current I L is considered.
In the PS configuration, according to the module-level KCL reported in (3), all the cells provide the same current contribution to the load current (I ij = I L /M ). Differently, if cell 11 is affected by parameters' variations, a current imbalance between cell 11 and the related parallel-connected cells (I 1j , ∀j ≥ 2) occurs in Module 1, the severity of which depends on the magnitude of the parameter variation itself. A thermal imbalance can also be observed within Module 1 along with the current imbalance. If cell 11 is aged and has a higher internal resistance, the parallel-connected cells in Module 1 will bear more current in order to maintain the same voltage among cells due to the parallel connection, strongly accelerating degradation. Consequently, the capacity of this module will be limited by the weakest cell. However, it is important to highlight that the impact of the current and thermal imbalances is limited to Module 1, thus the other modules (Module i, ∀i = 2, . . . N ) are not affected by cell 11 , according to the pack-level KCL reported in (5). Differently, with reference to the pack-level KVL reported in (6), the voltage of the whole battery pack is affected by the voltage of Module 1, which is strongly related to the behavior of cell 11 . Indeed, since all the cells are connected in parallel in each module, they share the same voltage according to the module-level KVL reported in (4), thus the voltage of Module 1 is determined by cell 11 and results different from the voltages V i of the other modules (Module i, In the SP configuration, the influence of cell 11 is spread to the whole battery pack. Indeed, since the cells in a module are connected in series, they have the same input current, the amplitude of which is determined by cell 11 in Module 1 according to the module-level KCL reported in (7). This leads to module current imbalances between I 1 and I j (∀j = 2, . . . M ) according to the pack-level KCL reported in (9). Different current splits among the modules will result in different thermal distributions due to the different heat generation. Since all the modules are connected in parallel, both the voltages of Module 1 and the whole battery pack will be affected by cell 11 according to the module and pack levels KVL reported in (8) and (10), respectively. If cell 11 is aged with a higher resistance among the series-connected cells, the overall resistance of Module 1 is higher compared to the other modules/strings (Module j, ∀j = 2, . . . M ). In particular, since the load current I L is constant, the latters will bear more load current in order to maintain the same voltage, accelerating the degradation process of the cells in these strings.

C. EFFECT OF CURRENT AND VOLTAGE IMBALANCES IN SP AND PS ARCHITECTURES
In order to quantitatively evaluate the different severity of the imbalance conditions with respect to the PS and SP pack architectures, a comparison between 3P3S and 3S3P configurations has been carried out as case study. This represents a suitable comparative analysis since both battery packs will have the same nominal voltage and capacity. Table 2 summarizes the main technical specifications of the 3-Ah EFEST IMR18650 cell adopted for the comparative analysis. A zero-order ECM has been experimentally calibrated for the EFEST cells, including SoC, temperature and charging/discharging C-rate dependencies for all the cell parameters.
The modeling equations illustrated in section II-A have been implemented in MATLAB ® with the aim of reproducing the electrical behavior of the cells in 3P3S and 3S3P configurations in case of parameters' variations. Three different operating conditions have been carried out as case studies in terms of parameter variation, including only one cell affected by 5% capacity decrease, 20% internal resistance increase and both variations. Fig. 2 shows the cells' currents and open circuit voltages achieved for both pack architectures when a load current equivalent to a C-rate of C2 for each cell is applied (I L = 4.5A) and an initial SoC = 80% is considered for all the cells. Ideally, when no parameters' variations occur, all the cells provide the same contribution to the load current regardless of the pack configuration, thus no current and open circuit voltage imbalances among the cells can be observed. Differently, when parameters' variations occur, different imbalance conditions result, leading to a lower usable capacity of the whole battery pack. It is important to highlight that the three different operating conditions considered as case studies for the capacity and/or resistance imbalances among the cells have been selected assuming reasonable maximum cell-to-cell parameters' variations within the battery pack, equal to 5% for the capacity and 20% for the internal resistance [34]. Indeed, larger imbalance conditions would represent an inappropriate and unrealistic real-world implementation of a battery pack, due to an incorrect selection of the cells during the pack assembly stage (different aging conditions) or improper design of the heating/cooling system (non-uniform thermal gradients).
Note also that in this paper a cell is defined as unhealthy when its parameters are different from the nominal ones in terms of capacity and/or internal resistance. This may occur due to high or low cell temperatures, manufacturer tolerances, or aging conditions. As possible to notice in Fig. 2, regarding the cells' currents, three contributions are highlighted for the PS configuration, which correspond to the cell affected by parameters' variations (cell 11 ), the parallel-connected healthy cells inside the affected module (cell 12 , cell 13 ) and the healthy cells that compose the other modules (cell 21  Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. of cell 11 may lead to imbalance conditions both inside and outside the affected module.
In order to quantitatively evaluate the severity of the imbalance conditions, Fig. 3 illustrates the differences among the current and the open circuit voltage imbalances occurring in the 3P3S and 3S3P configurations in case of the three parameter variation scenarios previously described. In detail, according to the parameters defined in Fig. 1 and considering cell 11 the one affected by parameters' variations, the imbalance conditions occurring both inside and outside the affected module have been evaluated for each pack configuration, resulting: PS architecture -Open circuit voltage imbalance PS architecture -Current imbalance I out = I 11 − I ij ∀i = 2, 3 , ∀j = 1, 2, 3 SP architecture -Open circuit voltage imbalance V OC,out = V OC,11 − V OC,ij ∀i = 1, 2, 3 , ∀j = 2, 3 SP architecture -Current imbalance where V OC,in and I in represent respectively the imbalance conditions in terms of open circuit voltage and current occurring between cell 11 and one of the cells inside the affected module, whereas V OC,out and I out represent respectively the imbalances occurring between cell 11 and one of the cells outside the affected module.
On the basis of the results achieved from this comparative analysis ( Fig. 2 and Fig. 3), it can be noticed the different severity of both current and open circuit voltage imbalances with respect to the pack architecture. In detail, a negative current imbalance is always observed regardless of the specific parameter variation occurring to cell 11 , thus resulting in a lower contribution to the load current of this cell in a PS configuration according to (13) and (14), as well as of the series-connected cells (cell 21 , cell 31 ) in a SP configuration according to (17) and (18). In addition, regardless of the specific parameter variation occurring to cell 11 , the SP configuration presents a lower current imbalance among the cells with respect to the PS configuration. A different variability of the current imbalance is also observed in the three operating conditions, since it increases or decreases over time in case of capacity imbalance (Fig. 3(a)) or internal VOLUME 11, 2023 resistance imbalance only (Fig. 3(b)), respectively, while it results almost constant when both parameters' variations are considered (Fig. 3(c)). Note that a zero current imbalance inside the module ( I in = 0) occurs for each operating condition in a SP configuration since the series-connected cells of a module are characterized by the same current.
On the other hand, a larger imbalance condition in terms of open circuit voltage can be noticed for the SP configuration, except in case of internal resistance imbalance only (Fig. 3(e)), where a zero voltage imbalance also occurs inside the affected module due to the series connection. Moreover, negative open circuit voltage imbalances result when the parameter variation affects the capacity (Fig.s 3(d) and 3(f)) of the unhealthy cell, whereas a positive imbalance results in case of internal resistance imbalance only. It can be also observed that the open circuit voltage imbalance increases in absolute value continuously over time, leading to high imbalance conditions in the battery pack and thus a strong reduction of its performance.
Note that the severity of the imbalance conditions strongly depends on the pack architecture in terms of number of cells in series and parallel connection as well as on the number and the distribution of the unhealthy cells inside the battery pack. An in-depth comparative analysis considering large-scale battery packs or numerous imbalance conditions cannot be properly operated by means of traditional modeling equations due to the increasing computational costs and number of variables involved. Therefore, it becomes fundamental the development of generalized models that allow for rapidly evaluating the impact of different imbalance conditions inside the battery pack and performing suitable comparative analyses between PS and SP configurations. More details related to the modeling approach adopted by the authors in this article are accurately illustrated in the following section.

III. METHOD: MODELING CELL-TO-CELL PARAMETERS' VARIATIONS IN BATTERY PACKS
In order to evaluate the severity of the imbalance conditions in both PS and SP architectures, the GECM proposed in [14] has been adopted, which allows for reducing complex pack architectures to parallel equivalent strings representing the healthy and unhealthy portion of the pack, as shown in Fig 4. ECMs model the electro-thermal behavior of a single cell, the proposed reduced-order GECM aims at modeling the whole battery packs, which are usually composed of a large number of cells in series and parallel connections, with a low computation cost. In particular, the GECM allows for achieving a closed form mathematical model for pack analysis, which integrates the model equations of an ECM with the ones characterizing the pack interconnections through Kirchhoff's laws and Thevenin's theorem. Note that the model proposed in [14] is here extended by taking into account the variability of the open circuit voltage between one or more unhealthy and healthy cells inside the battery pack architecture. Indeed, the different current splits occurring due to parameters' variations lead the cells to charge or discharge with different C-rates, resulting in uneven distributions of the states of charge (SoCs) and consequently open circuit voltage imbalance conditions, which have not been properly addressed in [14].
Figs. 4(a) and 4(b) depict the PS and SP pack configurations when cell 11 is unhealthy (orange color) while the other cells in the pack are healthy (green color). Depending on the pack architecture, the presence of an unhealthy cell contributes differently to the voltage and current imbalances among the cells. Indeed, cell 11 will affect only the M parallelconnected cells in a PS configuration and the entire battery pack in a SP configuration. Therefore, a lower impact will be observed in a PS architecture in terms of number of cells involved in the imbalance condition. However, the magnitude of the imbalance cannot be assessed without further analysis.
Both pack configurations in Fig. 4(a) and Fig. 4(b) can be simplified by separating the healthy portion of the pack (E n ) from the one affected by parameters' variations (E a ) into GECMs using Thevenin's theorem, as shown in Fig. 4(c) and Fig. 4(d), where orange indicates the unhealthy portion of the pack while green indicates the healthy portion. The parameters of the GECMs are defined for both PS and SP architectures in Table 3, where Q E a , R E a , I E a and V OC,E a are respectively the capacity, internal resistance, current and open circuit voltage of the unhealthy portion of the pack whereas Q E n , R E n , I E n and V OC,E n are the same parameters for the healthy portion of the pack. However, it is important to highlight that the parameters of the GECMs refer to single cells in the PS configuration (since the imbalance condition is contained inside the module), while they refer to a string in the SP configuration. In particular, for the PS architecture, the unhealthy and healthy portions of the pack are represented by the affected cell and a generic parallel-connected cell, respectively. Differently, for the SP architecture, the unhealthy portion of the pack corresponds to the string in which the affected cell is contained, whereas the healthy portion is represented by a generic parallel-connected string. Table 3 summarizes the parameters of the GECMs for PS and SP architectures considering only one cell affected by parameters' variations. The subscript a and n highlight the parameters of the unhealthy cell (Q a , R a , V OC,a ) and those of a generic healthy cell (Q n , R n , V OC,n ), respectively. Note that the evaluation of the impact of the parasitic resistances due to the electrical interconnections among the cells has not been considered in this manuscript, which mainly focuses on the quantitative evaluation of the severity of the impact of the cell-to-cell parameters' variations in the two pack configurations (PS and SP). However, the parameters of the proposed GECM may be easily modified by adding also the contribution of the parasitic resistances. In particular, depending on the pack configuration, they can be integrated in the parameters of the GECMs (Table 3) as an additional resistance contribution to the equivalent resistances of the unhealthy (R Ea ) and healthy (R En ) portions of the battery pack.
Regardless of the pack architecture, the following KVL and KCL equations can be considered for the GECM: According to these equations, the load current I L is distributed between the unhealthy and healthy portions of the battery pack as follows: Consequently, the current imbalance ( I E ) between the generic unhealthy and healthy string of the pack can be yielded: As result, I E strongly depends on the open circuit voltage and internal resistance imbalances as well as the number of parallel-connected cells/strings and the amplitude of the load current. Considering the following assumptions: • linearized characteristic for the SoC-V OC curve of the cells, which represents a good approximation for the majority of the cell chemistries in the typical SoC operating range of 20 − 80%; • constant internal resistance for the cells, thus no variability with respect to the SoC;

the analytical equations can be yielded for both current ( I ss E ) and open circuit voltage ( V ss
OC,E ) imbalances at steady-state as follows: More details related to the calculation and derivation process for achieving these steady-state model equations can be found in [14]. These model equations allow for directly defining the severity of the imbalance conditions on the basis of the equivalent capacities (Q E a , Q E n ) and internal resistances (R E a , R E n ), resulting suitable for analysis on large-scale battery pack with a reduced computational cost compared to the adoption of traditional ECMs for all the cells involved. In particular, it can be noticed that I ss E mainly depends on the imbalance between the equivalent capacities, whereas V ss OC,E is also dependent on the internal resistance imbalance. Moreover, the severity of the imbalance conditions decreases as the number of the cells in parallel increases, highlighting the strong correlation with the pack architecture.

A. MODEL VALIDATION
In order to validate the proposed modeling approach and the effect of the related modeling approximation, the GECM has been implemented in MATLAB ® considering both linearized and experimentally-calibrated zero-order ECM of the cells, which implement a linearized or an experimentallycalibrated SoC-V OC characteristic, respectively. Note that the experimentally-calibrated ECM also takes into account the variability of the internal resistance with respect to the SoC, while a constant value is considered for the linearized ECM (equal to the value at 50% SoC and 23 • C, as for Table 2). The model equations (21) and (22) have been adopted for describing the current split between the healthy and unhealthy portions of the battery pack over time with the aim of validating both current and open circuit voltage imbalances estimated at steady-state by means of (24) and (25). Therefore, the goodness of the proposed modeling approach has been evaluated by comparing the performance achieved with the linearized ECM and the experimentally-calibrated one in different case scenarios. In particular, four different case studies have been analyzed and summarized in Table 4, including capacity and/or internal resistance imbalances, which can result from an unhealthy condition caused by aging, high and/or low cell temperature conditions. Indeed, aging conditions for the cells always involve a reduction of the usable capacity and usually an increase of the internal resistance. Similar effects are observed when the cells operate at low temperatures, whereas an increase in the usable capacity and a decrease in the internal resistance occur at high operating temperatures. Figure 5 shows the results for this analysis evaluating the behavior of cells' currents and open circuit voltages for the different case studies, whereas Fig. 6 reports the related voltage and current imbalance conditions. In both figures, all the results named with the subscripts lin and real represent respectively the ones achieved by implementing the linearized and the experimentally-calibrated ECMs, whereas the results named with the superscript ss represent the steadystate values. Note that these results can be generalized with 96702 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  respect to the specific pack architecture. Indeed, as previously illustrated in Fig. 2, the same behaviour of the cells' current is observed for both PS and SP configurations, with different amplitudes depending on the number of series and parallel connections of the cells. As matter of fact, a generalized pack architecture with M > 2 has been considered for the performance comparison. As result, a good agreement is achieved between the model with realistic battery pack parameters and the proposed GECM.
In case 1, when only capacity imbalance is considered, the cells initially provide the same current (I Ea = I En = I L /M ). Then, due to the capacity difference, the cells gain a SoC imbalance, leading to different V OC . This imbalance is negative and increases over time as shown by V OC,E in Fig. 6(d).
Consequently, a different current split is observed between the unhealthy and healthy portions of the battery pack (I E a < I E n ), leading to a negative imbalance I E , as reported in Fig. 6(a). Nonetheless, according to (24) and (25), a steadystate condition is achieved for both I E and V OC,E , which result to be equal to I ss E and V ss OC,E , respectively. In case 2, when only internal resistance imbalance is considered, the cells initially provide a different contribution to the load current (I E a < I E n ) due to the different voltage drops across the internal resistances of the cells, as illustrated in Fig. 5(b). However, according to (24) and (25), no current imbalance occurs at steady-state (I E a = I E n = I L /M ) even if a V ss OC,E is observed in Fig. 6(e). In this case, a positive V ss OC,E can be noticed. In case 3, when both capacity and internal resistance imbalances are considered, a current imbalance occurs between the unhealthy and healthy portions of the battery pack both initially and at steady-state, as shown in Fig. 5(c). Indeed, the effects of the parameters' variations are combined: the internal resistance imbalance leads to the initial different contribution from the cells to the load current (I E a < I E n ) as shown in case 2, then the current imbalance increases in absolute value over time due to the effect of the capacity imbalance, which does not allow for extinguishing the current imbalance at steady-state, as previously shown in case 1. Consequently, a negative I ss E is observed. Moreover, as for case 1, a negative V ss OC,E can be noticed, as illustrated in Fig. 6(f).
The results related to case 4 have not been reported in Fig. 5 and Fig. 6 since they are similar to those achieved for case 3. . Imbalance conditions of a battery pack under constant current discharge by adopting the GECM with the linearized and the experimentally-calibrated cell model and considering only Q E a < Q E n (a-d), only R E a > R E n (b-e) and both Q E a < Q E n and R E a > R E n (c-f).
In particular, unlike the current split that occurs for case 3, the unhealthy portion of the battery pack has to provide the highest contribution to the load current in case 4, resulting in a positive current imbalance at steady-state ( I ss E > 0) and thus in a mirrored condition with respect to the results achieved for case 3.
According to the numerical results shown in Fig. 5 and Fig. 6, it can be noticed that the current imbalance is always lower or equal to zero ( I ss E ≤ 0), thus the unhealthy portion of the battery pack always contributes less to the current load than the healthy portion, leading to different operating conditions in terms of aging and temperature for the cells. On the other hand, the open circuit voltage imbalance is negative ( V OC,E < 0) in cases 1 and 2, when a capacity imbalance occurs. Therefore, despite the lower contribution to the load current (I E a < I E n ), the effect of the capacity imbalance leads to a faster discharge of the unhealthy portion of the battery pack and consequently a negative increase of the open circuit voltage imbalance The numerical results highlight the goodness of the model equations adopted since a similar behavior is achieved by considering either the linearized or the experimentallycalibrated ECM for the cells. The only difference that can be noticed over time in Fig. 5 and Fig. 6 is related to the nonlinear behaviour of the SoC-V OC characteristic and the variability of the internal resistance with respect to the SoC considered for the experimentally-calibrated ECM. In order to quantitatively evaluate the misalignment between the results achieved by adopting the linearized or the experimentally-calibrated ECMs for the cells, the errors obtained in each case study for all the main parameters have been calculated, including the amplitude of both contributions to the load current (I E a , I E n ) as well as the current and open circuit voltage imbalances ( I E , V OC,E ). In particular, a root mean square percentage error (RMSPE) has been adopted as a performance parameter for the cells' currents and open circuit voltages, whereas a root mean square error (RMSE) has been considered for the imbalance conditions. Moreover, the impact of M on the errors' amplitudes has been also evaluated by considering M ranging from 3 to 9. Table 5 and Table 6 summarize respectively the RMSPEs and RMSEs obtained in each case study by comparing the numerical results achieved with the linearized and the experimentally-calibrated ECMs. According to the RMSPEs, higher contributions result for the current of the unhealthy portion of the battery pack (I E a ). This is due to the fact that only one cell is considered unhealthy in the pack, thus it experiences a larger current amplitude with respect to the other parallel-connected cells or strings. Moreover, the amplitude of I E a rises as the number of parallel connections (M ) increases, still resulting in RMSPEs always lower than 2.5%. Conversely, the RMSPEs decrease as M increases for both the current of the healthy portion of the battery pack (I E n ) and the open circuit voltages (V OC,E a , V OC,E n ). Regarding the current and voltage imbalances ( I E , V OC,E ), the results reported in Table 6 highlight RMSE values lower than 8.90 mA and 1.20 mV for I E and V OC,E , respectively. As result, the low amplitudes of the RMSPEs and RMSEs achieved confirm the applicability of the linearized ECM for the cells and consequently the goodness of adopting the steady-state model equations 24 and 25 for pack analysis.

IV. NUMERICAL EXPERIMENTS AND COMPARISON BETWEEN DIFFERENT PACK ARCHITECTURES
In order to quantitatively evaluate the difference between the amplitudes of voltage and current imbalances occurring in both PS and SP architectures, a comparative analysis has been carried out at steady-state. In detail, according to (24), the current imbalance at steady-state has been analytically defined for each pack architecture by substituting the corresponding parameters of the GECM with those reported in Table 3, resulting: I L (27) where I ss E,PS and I ss E,SP represent the current imbalances at steady-state for the PS and SP configurations, respectively. Then, the comparison has been performed in case of parameters' variations by evaluating the ratio ( I ss E,ratio ) between the current imbalances occurring in both pack architectures: where Q ratio represents the ratio between the capacity of the unhealthy cell and that of a generic healthy cell: It is important to highlight that I ss E,ratio can be properly evaluated only in case of capacity imbalance among the cells (Q ratio ̸ = 1), since the absence of imbalance conditions inside the battery pack (Q a = Q n ) would result in an undetermined form for (28). However, according to (26) and (27), it is still possible to assert that there is no current imbalance for both pack architectures if all the cells are healthy ( I ss E,PS = I ss E,SP = 0). I ss E,ratio only depends on the capacity imbalance between the unhealthy and healthy portions of the battery pack as well as on its size in terms of number of cells in series N and parallel M . As matter of fact, the amplitude of the load current I L and the severity of resistance imbalances among the cells do not affect I ss E,ratio . Since N and M have to be greater than 1 for properly comparing the PS and SP architectures, it results that I ss E,ratio > 1 in all the operating conditions, thus the amplitude of the current imbalance occurring in the PS configuration is always greater than that occurring in the SP one ( I ss E,PS > I ss E,SP ). Fig. 7 shows a quantitative analysis related to the variability of I ss E,ratio with respect to N , M and the severity of the capacity imbalance (Q ratio ). It can be observed that I ss E,ratio increases as either N or the capacity imbalance rises, however the impact of N results to be more relevant. Moreover, the variability of I ss E,ratio with respect to M strongly depends on the severity of the capacity imbalance conditions. Indeed, increasing M , I ss E,ratio increases for Q ratio > 1 and decreases for Q ratio < 1.
Similarly to the operations previously performed for the current imbalance, according to (25), the open circuit voltage imbalance at steady-state has been analytically defined for each pack architecture by substituting the corresponding parameters of the GECM with those reported in Table 3, resulting: Differently from I ss E,ratio , it is important to highlight that V ss OC,E,ratio can be evaluated even when no capacity imbalance occurs among the cells (Q ratio = 1), since V ss OC,E,PS and V ss OC,E,SP depend on the internal resistance imbalance as well. Therefore, assuming Q ratio = 1, (32) exists as long as an internal resistance imbalance occurs among the cells (R a ̸ = R n ). However, as it is possible to notice from (32), the dependency from the resistance imbalance (R a , R n ) disappears by evaluating the ratio between the open circuit voltage imbalances V ss OC,E,ratio , which only depends on Q ratio and the battery pack size in terms of series (N ) and parallel (M ) connections. Moreover, the amplitude of the current load I L does not affect V ss OC,E,ratio as well. Since N and M have to be greater than 1 for properly comparing the PS and SP architectures, it results V ss OC,E,ratio > 1 in case of capacity imbalance greater than 1 (Q ratio > 1), while V ss OC,E,ratio < 1 in case of capacity imbalance lower than 1 (Q ratio < 1). Hence, the amplitude of V ss OC,E,ratio occurring in the PS configuration is greater or lower than that occurring in the SP architecture depending on the severity of the capacity imbalance. Moreover, when there is no capacity imbalance among the cells (Q a = Q n ), the same open circuit voltage imbalance is achieved ( V ss OC,E,PS = V ss OC,E,SP ) only if an internal resistance imbalance occurs (R a ̸ = R n ). Fig. 8 shows a quantitative analysis related to the variability of V ss OC,E,ratio with respect to N , M and the severity of the capacity imbalance (Q ratio This comparative analysis demonstrates the PS architecture to be more impacted by parameters' variations with respect to the SP one, since a higher current imbalance is always obtained. Consequently, it results in a larger contribution to the load current for the unhealthy cells, leading to thermal gradients inside the battery pack and thus non-uniform aging conditions. Moreover, in case of parameters' variations due to high temperature conditions for the cells (case 4), a higher open circuit voltage imbalance is  achieved for the PS architecture, resulting in a reduction of the whole battery pack performance due to the lower usable capacity. It is important to highlight that the same open circuit voltage imbalance results for both pack architectures when only internal resistance imbalance occurs (case 2). In detail, according to (30) and (31), the amplitude of the open circuit voltage imbalance can be yielded as follows: Therefore, a high impact on the open circuit voltage imbalance can be achieved depending on the severity of the internal resistance imbalance. This effect is mitigated as the number of parallel-connected cells/modules increases. On the basis of the results achieved from the comparative analysis, the adoption of the proposed reduced order GECM and the related steady-state model equations allow for a-priori definition of the severity of the imbalance conditions in a battery pack, providing a suitable approach for analysis on large-scale battery pack with a low computational cost. The proposed generalized modeling of battery packs can strongly improve the pack design phase, including the sizing of the protection devices for the cells according to the pack specification and maximum acceptable variation of the cell parameters [13] as well as the optimization of the cooling system for managing and mitigating the effects of the imbalance conditions due to the parameters' variations. In addition, a real-time estimation of the state of health of the cells could be potentially performed by extracting the Q ratio directly from the current imbalances measured in the battery pack. Therefore, it could be adopted for assessing and predicting imbalance conditions for the cells, including aging distribution in the battery pack.

V. CONCLUSION
In this manuscript, a method for assessing the voltage and current imbalances occurring in case of parameters' variations among the cells in the SP and PS pack architectures has been proposed. Different aging and temperature conditions have been considered for assessing the electrical behavior of the cells. A reduced order of the pack has been developed in order to quantitatively evaluate the severity of the imbalances due to the cell parameters' variations in both pack architectures. A detailed comparative and parametric analysis has been illustrated, highlighting the variability of current and open circuit voltage imbalances depending on the severity of the parameter variation as well as the specific battery pack size in terms of number of cells in series and parallel. The comparison results demonstrate the PS architecture to be always characterized by a higher current imbalance with respect to the SP architecture. Moreover, both architectures show a high impact of the capacity imbalance on the open circuit voltage imbalance. The internal resistance of the cells also contributes to the severity of the imbalance conditions. However, the cells' capacity affects the amplitude of both current and open circuit voltage imbalances, whereas the cells' internal resistance only contributes in increasing the open circuit voltage imbalance among the cells. This leads the capacity imbalance to heavily impact the reduction of the overall performance of the battery pack. The results of this analysis show the trade-off between the two architectures and provide guidance to the battery pack designer for the architecture selection.
Despite the consistency and experimental validity of the proposed reduced order model for pack analysis have been widely described throughout the manuscript, experimental tests on SP and PS battery packs prototypes could be also carried out in future works as additional validation. Likewise, the evaluation of branch currents in more complex battery systems, which include different SP and PS battery packs connected in series and/or parallel connections, could be investigated to extend the applicability of the proposed reduced order model. FRANCESCO PORPORA (Member, IEEE) received the master's degree in electrical engineering and the Ph.D. degree in methods, models, and technologies for engineering from the University of Cassino and Southern Lazio, in 2017 and 2021, respectively. He is currently a Research Associate with the University of Cassino and Southern Lazio. His research activities mainly focus on the development, prototyping, and control of battery management systems as well as passive and active equalization techniques for Li-ion packs. His research is also related to modeling, optimization, and control of electrochemical energy storage systems and powertrain systems. GIORGIO RIZZONI (Life Fellow, IEEE) received the B.S., M.S., and Ph.D. degrees in ECE from the University of Michigan, in 1980Michigan, in , 1982Michigan, in , and 1986, respectively. He is the Ford Motor Company Chair of ElectroMechanical Systems and is a Professor of mechanical and aerospace engineering and of electrical and computer engineering with The Ohio State University (OSU). Since 1999, he has been with the Director of the Ohio State University Center for Automotive Research (CAR), an interdisciplinary university research center at the OSU College of Engineering. His research activities are related to modeling, control, and diagnosis of advanced propulsion systems, vehicle fault diagnosis and prognosis, electrified powertrains and energy storage systems, vehicle safety and intelligence, and sustainable mobility. He has contributed to the development of graduate curricula in these areas and was the Director of three U.S. Department of Energy Graduate Automotive Technology Education Centers of Excellence. During his career at Ohio State, he has directed externally sponsored research projects funded by major government agencies and by the automotive industry in approximately equal proportion. He is a fellow of SAE, in 2005. He was a recipient of the 1991 National Science Foundation Presidential Young Investigator Award and many other technical and teaching awards.
GIUSEPPE TOMASSO (Member, IEEE) received the master's degree in electrical engineering and the Ph.D. degree in industrial engineering, in 1994 and 1999, respectively. Since 2009, he has been the Chief of the Industrial Automation Laboratory with the University of Cassino and Southern Lazio, Italy. He is a Full Professor of power electronics and electric and hybrid vehicles with the University of Cassino and Southern Lazio. He is the Founder of four start-ups and the President of E-Lectra company, developing advanced technologies in the field of electric vehicles and energy storage systems. Since 2009, he has been also the Chairperson of the European Ph.D. School: Power Electronics, Electrical Machines, Energy Control and Power Systems. He is a coauthor of more than 130 publications in conference proceedings and international transactions. His main research interests are high-performance power converters, advanced modulation techniques, industrial automation and electrical drives, and electric and hybrid vehicle powertrains. He is also a promoter of a racing initiative related to electric go-karts for motorsport.
Open Access funding provided by 'Università degli Studi di Cassino e del Lazio Meridionale' within the CRUI CARE Agreement