# A Novel Modular, Reconfigurable Battery Energy Storage System: Design,
Control, and Experimentation

Amir Farakhor, , Di Wu, , Yebin Wang, , and Huazhen Fang This work was
supported in part by the U.S. National Science Foundation under Awards
CMMI-1763093 and CMMI-1847651, and in part by the U.S. Department of Energy,
Office of Electricity through the Energy Storage Program.A. Farakhor and H.
Fang (corresponding author) are with the Department of Mechanical Engineering,
University of Kansas, Lawrence, KS, USA (Email<EMAIL_ADDRESS>a.farakhor@ku.edu).D. Wu is with the Pacific Northwest National Laboratory,
Richland, WA, USA (Email: di.wu@pnnl.gov).Y. Wang is with the Mitsubishi
Electric Research Laboratories, Cambridge, MA, USA (Email:
yebinwang@merl.com).

###### Abstract

This paper presents a novel modular, reconfigurable battery energy storage
system. The proposed design is characterized by a tight integration of
reconfigurable power switches and DC/DC converters. This characteristic
enables isolation of faulty cells from the system and allows fine power
control for individual cells toward optimal system-level performance. An
optimal power management approach is developed to extensively exploit the
merits of the proposed design. Based on receding-horizon convex optimization,
this approach aims to minimize the total power losses in charging/discharging
while allocating the power in line with each cell’s condition to achieve
state-of-charge (SoC) and temperature balancing. By appropriate design, the
approach manages to regulate the power of a cell across its full SoC range and
guarantees the feasibility of the optimization problem. We perform extensive
simulations and further develop a lab-scale prototype to validate the proposed
system design and power management approach.

###### Index Terms:

Battery management systems, cell balancing, convex optimization,
reconfigurable battery energy storage systems (RBESSs).

## Nomenclature

Variables

$S$

Reconfiguration switches–binary (1/0) variable

$V_{t}^{*}$

RBESS reference output voltage

$V_{C}^{\textrm{max}}$

Maximum output voltage of DC/DC converters

$I_{C}^{\textrm{max}}$

Maximum output current of DC/DC converters

$v$

Cell voltage

$u$

Cell open-circuit voltage

$i_{L}$

Cell current

$P_{b}$

Cell internal power

$P$

Cell output power

$E$

Cell energy

$P_{l}$

Module power losses

$J$

Total power losses

$P_{\textrm{out}}$

RBESS output power

$I_{\textrm{out}}$

RBESS output charging/discharging current

$\dot{Q}_{\textrm{cnd}}$

Conductive heat transfer rate

$\dot{Q}_{\textrm{conv}}$

Convective heat transfer rate

$\xi$

Slack variable

$z$

Optimization variables

Parameters

$n$

Number of battery cells

$n_{s}$

Number of cells in series

$n_{p}$

Number of cells in parallel

$\cal{J}$

Set of in-service cells

$L$

Inductor for DC/DC converters

$C$

Output capacitor for DC/DC converters

$\bar{Q}$

Cell capacity

$q,q^{\textrm{min}},q^{\textrm{max}}$

SoC of a cell, and its lower and upper limits

$q_{\textrm{avg}}$

Average SoC

$\Delta q$

SoC imbalance tolerance

$\Delta E$

Energy imbalance tolerance

$\alpha$

Intercept coefficient of the SoC/OCV line segment

$\beta$

Slope coefficient of the SoC/OCV line segment

$R$

Cell internal resistance

$R_{C}$

Resistance to capture the power losses of DC/DC converters

$R_{\textrm{cnd}}$

Conductive thermal resistance

$R_{\textrm{conv}}$

Convective thermal resistance

$C_{\textrm{th}}$

Thermal capacitance

$T,T^{\textrm{max}}$

Cell temperature and its upper limit

$T_{\textrm{avg}}$

Average temperature

$T_{\textrm{env}}$

Environmental temperature

$\Delta T$

Temperature imbalance tolerance

$\lambda$

Penalty weight for the multi-objective optimization

$\Delta t$

Sampling time

$H$

Optimization horizon

##  I Introduction

Lithium-ion battery energy storage systems (BESSs) have proven themselves as
an enabling technology for various applications, including electric cars,
electric aircraft, smart grid, and space systems[1, 2, 3, 4]. Despite their
high energy density and long cycle life, lithium-ion batteries suffer from
safety risks, which trace to the high reactivity of lithium and flammability
of the commonly used electrolyte solutions and are exacerbated by side
reactions, aging, and degradation [5]. Hence, it is imperative to ensure their
safe and reliable operation, particularly in safety-critical applications [6].
Reconfigurable BESS (RBESS) have attracted much attention as a promising means
to achieve this end. An RBESS characteristically uses power electronics
switches to make the connection among the constituent cells reconfigurable,
providing the capability to bypass faulty cells without interrupting the
operation of the system [7]. This feature overcomes the vulnerability of
conventional hardwired BESS to single-cell failures due to the fixed
configuration [8]. This paper proposes a novel RBESS design that integrates
reconfigurable switches with DC/DC converters. The new design uses the
switches to reconfigure the connectivity of the cells for the sake of safety
and meanwhile, leverages the converters to achieve robust power management and
supply from cell to system level. Further, we present an optimal power
management algorithm and develop a lab-scale prototype to validate the
proposed RBESS design and control.

###  I-A Literature Review

This paper centers around the RBESS circuit design and power management.
Therefore, we survey the literature on the two dimensions one by one. The
review will also encompass some recent studies about hardwired BESS due to the
relevance.

####  I-A1 Review of RBESS circuit design

The literature has presented two main ways to design RBESS circuit
architectures. The first one builds and integrates a circuit of controllable
power electronics switches with the cells. By controlling the switches, a cell
can be put into or cut off from the connection with other cells when a fault
occurs. The circuit topology plays a key role in the level of
reconfigurability, functional flexibility, and circuit complexity. The study
in [9] shows a switching circuit for a series-connected battery pack, which
uses only two switches for every cell. More sophisticated topologies can
provide more versatile reconfiguration, though at the expense of using larger
numbers of switches. The work in [10] considers a string of modules of
parallel-connected cells in series and allows switch-based bypass of any cell
or module. The circuit topologies proposed in [11, 12] place five and six
switches, respectively, around each cell to realize arbitrary series and
parallel connection among the cells, and the one shown in [13] uses only four
to achieve the same end. It is interesting to point out that reconfigurable
switching circuits can enable more functions than just isolating faulty cells.
A battery system designed in [14] dynamically connects some battery modules in
series or parallel to produce multi-level and even AC voltage output. In [15],
an inverter based on switching circuits combines batteries and supercapacitors
for hybrid energy storage. While easily reconfigurable, switching circuits are
unable to do cell-level charging/discharging power regulation. This limitation
will lead to unbalanced use of the cells following the reconfiguration-based
bypass of a cell.

Another important way for RBESS design uses converters. A converter can not
only connect a cell into the circuit, but also charge or discharge it at a
controlled current, voltage, or power. This capability offers a higher degree
of freedom for cell-level control to enhance the balanced use of the cells. In
[16], a centralized converter interfaces with a few cells, and it contains a
selector to select and put the cells into operation. The work in [17] pairs
converters, which by design include power switches to offer a bypass mode,
with individual cells one by one to form a reconfigurable pack. However,
converter-based RBESS architectures are unable to offer flexible topology
changes, compared to the switching circuits. As such, only series-connected
RBESS is considered in [16, 17]. Converters have also found their way into
hardwired BESS. The studies in [18, 19, 20] integrate DC/DC converters with
the cells to enable cell balancing and power loss minimization. However, the
fixed hardwired connections among the cells make them susceptible to single-
cell faults.

####  I-A2 Review of RBESS control and power management

For a given switching-circuit-based RBESS, an essential question is how to
control its topology. A main approach lies in finding out a connection
topology to meet load requirements. For example, a heuristic method in [10]
groups battery cells into modules and chooses a minimum set of modules on a
load request to perform charging/discharging. The method of dynamic
programming finds use in identifying an optimal configuration topology in
[11], and rule-based bypassing mechanisms are proposed in [12] to control the
switches for stable or responsive voltage supply. The study in [13] exploits
the switching circuit to achieve cell equalization through a hierarchical
strategy that combines intra-module equalization and system-level
reconfiguration. Considering the switching circuits in [9, 10], the study in
[21] proposes to sort cells according to their capacity and then reconfigure
them into serial strings to restrain the cell imbalance.

On a related note, the literature has included a few studies about power
management for hardwired BESS based on converters. For converter-based RBESS,
the main subject of investigation is developing control approaches to manage
the cell-to-system-level operation. Optimal control has shown as a useful
solution in this regard. However, due to a lithium-ion battery’s nonlinear
characteristics, optimization problems posed for the power management are
often non-convex, and thus defy the computation of global optima. To mitigate
the issue, the studies in [18, 19] choose to leverage a battery model
convexification approach proposed in [22] to formulate convex optimal power
control problems. The convexification therein involves a linear approximation
of the relationship between the state-of-charge (SoC) and the open-circuit-
voltage (OCV), thus restricting the proposed methods to only limited operating
ranges. It is also possible to use a linear battery model to achieve optimal
control that even admits closed-form solutions [23], but the method is also
just applicable to operating ranges in which the linear model is accurate.

As another valuable approach, distributed power management treats the cells as
independent agents and makes them perform individual control toward a global
goal. One can decompose a global optimization task and distribute it among
individual cells for charging or load sharing, as shown in [20]. As an
alternative way, distributed consensus control is applied in [24] to achieve
SoC balancing among the cells. Not requiring numerical optimization, this
method offers fast computation. But the lack of optimality makes it in need of
more time to converge.

TABLE I: Characteristics of different BESS structures. |  Switch-based RBESS |  Converter-based RBESS |  Converter-based hardwired BESS | Proposed design
---|---|---|---|---
| [9] | [10] | [11] | [12] | [13] | [14] | [15] |  [16] | [17] | [18] | [19] | [20]
Safety (Bypass of faulty cells) | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |  ✓ |  ✓ |  $\times$ |  $\times$ | $\hskip 3.0pt\times$ | ✓
Output voltage regulation | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  ✓ |  ✓ |  $\times$ |  $\times$ |  $\times$ | ✓
SoC balancing | ✓ | ✓ | ✓ | ✓ | ✓ | $\times$ | ✓ |  ✓ |  ✓ |  ✓ |  ✓ |  ✓ | ✓
Temperature balancing | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  $\times$ |  $\times$ |  ✓ |  ✓ |  ✓ | ✓
State-of-health balancing | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  $\times$ |  $\times$ |  $\times$ |  $\times$ |  $\times$ | ✓
Optimal cell-level power control | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |  $\times$ |  $\times$ |  ✓ |  ✓ |  ✓ | ✓
Compatibility with cells of different types | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | ✓ |  $\times$ |  ✓ |  $\times$ |  ✓ |  ✓ | ✓
Flexibility in series/parallel connection | L | M | H | H | H | M | L |  L |  L |  $\times$ |  $\times$ |  $\times$ | H
L: Low, M: Medium, H: High |  |  |  |  |  |  |  |  |  |  |  |  |

Table I summarizes the main BESS architectures in the literature for
comparison. In this context, we propose the presented work to improve the
state of the art.

###  I-B Statement of Contributions

Despite recent advances, the RBESS technology remains far from reaching a
level of maturity in both design and control. The existing RBESS architectures
use either only switches or only converters to enable good reconfigurability
or good power regulation capability, but not both. There is also a lack of
power control approaches sophisticated enough to maximize the operating
performance of RBESS. To overcome these substantial shortcomings, our study
presents the following specific contributions.

  1. 1)

We propose a new modular, reconfigurable power electronics architecture for
RBESS. Differing from the existing ones, the proposed architecture, for the
first time, integrates power switches with DC/DC converters for a combination
of their respective merits, while using the fewest number of switches for each
cell to our best knowledge. Among the various benefits that the proposed
design brings are high reconfigurability—arbitrary bypass and parallel or
series connection among the adjacent cells—and high regulatability in power
supply from cell to system level to satisfy exogenous power demands, even
under the occurrence of cell failures, and ensure equal use of the cells
simultaneously.

  2. 2)

We develop a power management approach based on optimal control for the
proposed RBESS to minimize the system-wide energy loss while supplying
demanded power and equalizing the cells in SoC and temperature. Compared to
the prior methods, our approach is different in two aspects. First, we adopt
piecewise linear SoC/OCV approximation in the battery model convexification to
enable control across low to high SoC regions. Second, previous methods may be
subject to infeasibility when the cells’ actual conditions make the pre-set
operating constraints unsatisfiable. Here, we introduce slack variables to
relax the optimization problem to ensure the feasibility and practical
applicability of our approach. Finally, we present a reconfiguration method to
modify the switching circuit topology after a faulty cell is isolated.

  3. 3)

We develop an experimental prototype and conduct a series of experiments to
validate and assess the performance of the proposed RBESS. The experiments
involve various scenarios under non-uniform cell conditions, fault occurrence,
and reconfiguration. The results demonstrate the effectiveness of the proposed
architecture and power management approach.

Based on the above contributions, our RBESS design presents significant
advantages to distinguish itself from the literature, as shown in Table I. A
preliminary conference version of this work appeared in [25] to report the
RBESS architecture design. Here, we introduce substantial expansions in
optimal power management and experimental validation.

###  I-C Organization

The remainder of the paper is organized as follows. Section II presents the
power electronics architecture of the proposed RBESS and discusses its unique
features. Section III develops the optimal power management approach, which
covers the modeling, optimization problem formulation, and convexification.
This section also presents a switching circuit reconfiguration method. Section
IV provides extensive simulation results, and Section V proceeds to develop an
experimental prototype to validate the proposed RBESS design and control
approach. Finally, Section VI concludes the study.

##  II Architecture of the Proposed RBESS

Figure 1: The proposed modular DC/DC converter-integrated RBESS.

This section elaborates the proposed RBESS. Fig. 1 shows the power electronics
architecture. As illustrated, every cell is connected with a DC/DC converter
to make up a module. Here, we employ synchronous DC/DC converters, even though
other types of converter topologies are also allowed. The converter, which
comprises two power switches, an inductor, and an output capacitor, provides
bi-directional power processing to control the charging and discharging of the
cell. As such, the module has the capability of cell-level power control. The
modules are connected via a switching circuit, which places three power
switches between every two adjacent modules. With an appropriate
reconfiguration of the switches, the switching circuit can bypass a module
subject to faults and achieve arbitrary series or parallel connection among
adjacent modules. We label the three switches connecting modules $i$ and $i+1$
as $S_{ij}$ for $j=1,2,3$. For switch $S_{ij}$, $S_{ij}=1$ when it is on, and
$S_{ij}=0$ when it is off. To bypass and isolate cell $i$ for $1\leq i\leq
n-1$ from the battery pack, we let $S_{i1}=1$ and $S_{i2}=S_{i3}=0$. To bypass
cell $n$, we let $S_{(n-1)2}=1$ and $S_{(n-1)3}=S_{(n-1)1}=0$. Modules $i$ and
$i+1$ are configured in series when $S_{i1}=S_{i2}=0$ and $S_{i3}=1$, and in
parallel when $S_{i1}=S_{i2}=1$ and $S_{i3}=0$.

The proposed design uses only $3(n-1)$ power switches for $n$ cells. To our
knowledge, this is more economical than any other RBESS design in the
literature to provide the same level of reconfigurability. The circuit
simplicity further results in convenient operation and reconfiguration.
Specifically, a cell requires only one switch to be on for the series and
bypass configurations, and a parallel configuration needs only two switches to
be on, as seen from above.

Fig. 1 illustrates how the proposed design dovetails with the battery
management system (BMS). The BMS adopts a two-layer control strategy. At the
higher level, the optimal power management generates optimal
charging/discharging currents for the cells; at the lower level, PI-based
local current mode controllers perform reference tracking for the cell
currents, as is common in control of DC/DC converters [26].

Our proposed RBESS can provide high reconfigurability and control flexibility,
which lead to distinct benefits for practical applications. A summary of the
major ones is as follows.

  1. 1)

The proposed design allows to bypass and isolate any faulty module. Hence, the
battery pack can continue to operate rather than shut down as a whole, despite
safety threats and anomalies. Further, following the bypass of a module, the
switching circuit can reconfigure to redirect the power flow and share the
load equally among the remaining in-service cells to promote balanced use of
them.

  2. 2)

In the proposed design, the embedded DC/DC converters take on responsibility
for the external power electronic devices in the conventional designs to
control the charging/discharging of the cells. The DC/DC converters would
yield useful functions with their capability of power conversion and control.
First, they can regulate their output voltage so that the RBESS can supply
desired or reference voltage. The voltage supply can remain consistent before
and after a fault-induced reconfiguration. Second, the converters apply
individual current or voltage control to the cells, thus making it possible to
customize and optimize the charging/discharging for each cell based on its
present condition. One can translate this strength into cell balancing, e.g.,
by charging (resp., discharging) the cells with high SoC less (resp., more)
relative to the cells with low SoC. It is also viable to balance the cells in
temperature and state-of-health.

  3. 3)

Even though beyond the scope of this paper, the proposed design can
accommodate the heterogeneity of the cells. For example, one can leverage it
to integrate heterogeneous cells from different manufacturers or even based on
different electrochemistries to form a workable system. In a similar vein, the
design can potentially enable hybrid energy storage consisting of battery
cells, supercapacitors, and even solar cells.

Note that the use of the embedded DC/DC converters can mitigate the use of
exogenous power electronics devices, and that one can connect a string or pack
of cells, rather than a single cell, to each DC/DC converter in practical
adoptions. These factors would help make the overall implementation cost of
the proposed RBESS manageable. Finally, the diverse new functions and safety
improvements brought by the design are valuable especially for high-stakes,
safety-critical applications including electric vehicles and aircraft.

##  III Modeling and Optimal Control of the Proposed RBESS

This section investigates optimal power management for the proposed RBESS. We
will begin with the electrical and thermal modeling for the modules. We then
proceed to present a model-based optimal control problem for the RBESS and
convexify it for computational tractability. Here, the modeling and optimal
control formulation are a refinement of the methodology in [18, 19], and
improvements are introduced to expand the operating SoC range of the RBESS and
ensure the feasibility of the optimization problem. Finally, we present a
switching circuit reconfiguration mechanism that dovetails with the power
management method.

###  III-A Electrical and Thermal Modeling

Consider the proposed RBESS consisting of $n$ modules. Each module includes a
cell, a DC/DC converter, and three power switches connected in cascade, as
shown in Fig. 2 (a). We denote the set of all the in-service modules as
$\mathcal{J}$. The electrical model of module $j$ for $j\in\mathcal{J}$ is
schematically shown in Fig. 2 (b). It includes two parts. The first part is
the Rint model to describe the cell’s electrical dynamics [27], which
comprises an OCV source $u_{j}$ in series with an internal resistor $R_{j}$.
The model’s governing equations are:

$\displaystyle\dot{q}_{j}(t)$
$\displaystyle=-\frac{1}{\bar{Q}_{j}}i_{L_{j}}(t),$ (1a) $\displaystyle
v_{j}(t)$ $\displaystyle=u_{j}(q_{j}(t))-R_{j}i_{L_{j}}(t),$ (1b)

where $v_{j}$, $i_{L_{j}}$, $u_{j}$, $\bar{Q}_{j}$, and $q_{j}$ are the
terminal voltage of the cell, applied current in Ampere, OCV, capacity, and
SoC, respectively. The internal power of the battery cell is given by

$P_{b_{j}}=u_{j}(q_{j}(t))i_{L_{j}}(t).$ (2)

The DC/DC converter is modeled as an ideal DC/DC transformer along with a
series resistor $R_{C}$ to capture power losses. The reconfiguration switches
are also modeled by the ideal switches with series resistors $R_{S_{ji}}$. For
module $j$, we assume that we can collect the power losses on $R_{S_{ji}}$ for
$i=1,2,3$ in a single resistor $R_{S_{j}}$. Thus, the module’s output power
$P_{j}$ can be calculated as

$P_{j}=u_{j}(q_{j}(t))i_{L_{j}}(t)-(R_{j}+R_{C}+R_{S_{j}})i_{L_{j}}^{2}(t),$
(3)

where $R_{j}i_{L_{j}}^{2}(t)$, $R_{C}i_{L_{j}}^{2}(t)$, and
$R_{S_{j}}i_{L_{j}}^{2}(t)$ represent the internal power losses of the cell,
the converter, and the reconfiguration switches, respectively.

We adopt a lumped thermal model in [18] to describe the thermal dynamics of
module $j$. The thermal model is depicted in Fig. 2 (c). The model captures
the heat transfer due to the convection between module $j$ and the
environment, $\dot{Q}_{\textrm{conv},j}$, and the conduction between module
$j$ and its adjacent cells, $\dot{Q}_{\textrm{cnd},j}$. Meanwhile, the power
loss caused by the internal resistor, $R_{j}i_{L_{j}}^{2}$, translates into
heat generation, which becomes the main heating source. Combining all, the
thermal model is governed by

$\displaystyle C_{\textrm{th},j}\dot{T}_{j}(t)$
$\displaystyle=R_{j}i_{L_{j}}^{2}(t)-\dot{Q}_{\textrm{cnd},j}-\dot{Q}_{\textrm{conv},j},$
(4a) $\displaystyle\dot{Q}_{\textrm{conv},j}(t)$
$\displaystyle=(T_{j}(t)-T_{\textrm{env}})/R_{\textrm{conv}},$ (4b)
$\displaystyle\dot{Q}_{\textrm{cnd},j}(t)$
$\displaystyle=(2T_{j}(t)-T_{j+1}(t)-T_{j-1}(t))/R_{\textrm{cnd}},$ (4c)

where $T_{j}$ and $T_{\textrm{env}}$ are the cell’s and environmental
temperatures, respectively. In addition, the term $C_{\textrm{th},j}$
represents the thermal capacitance of the cell; $R_{\textrm{cnd}}$ and
$R_{\textrm{conv}}$ are the thermal resistances between neighboring cells and
between cell $j$ and the environment, respectively. Here, $R_{\textrm{conv}}$
depends inversely on the external surface area of the cell $A_{j}$ and the
convective heat transfer coefficient between the cell’s surface and the
environment $h$. For instance, one can consider a parallel forced air cooling
approach for the proposed design [28], which allows every cell to experience
the same amount of cooling air.

The above electro-thermal model is concise but expressive and computationally
efficient. Putting them together for all the modules, one can obtain a
complete description of the dynamics of the RBESS, which allows us to perform
optimal power management design subsequently.

(a)  (b)  (c)

Figure 2: The proposed DC/DC converter-integrated cell module. (a) The circuit
structure of the proposed module. (b) The electrical model of the proposed
module. (c) The thermal model of the cell $j$.

###  III-B Problem Formulation

The aim of our RBESS power management is to distribute the power load among
the cells so that the power losses can be minimized under some key safety,
balancing and power demand satisfaction constraints. To begin with, we will
formulate the optimization problem.

The total power losses of the RBESS can be expressed by

$J(t)=\sum_{j\in\mathcal{J}}(R_{j}+R_{C}+R_{S_{j}})i_{L_{j}}^{2}(t).$ (5)

We use the following objective function to encompass the total power losses
over a horizon:

$\int_{0}^{H}J(t)dt,$ (6)

where $H$ is the planning horizon length. For the sake of safety, we require
each cell to operate within some favorable current, SoC, and temperature
ranges:

$\displaystyle i_{L_{j}}^{\textrm{min}}\leq i_{L_{j}}$ $\displaystyle\leq
i_{L_{j}}^{\textrm{max}},$ (7a) $\displaystyle q_{j}^{\textrm{min}}\leq q_{j}$
$\displaystyle\leq q_{j}^{\textrm{max}},$ (7b) $\displaystyle T_{j}$
$\displaystyle\leq T_{j}^{\textrm{max}},$ (7c)

where $i_{L_{j}}^{\textrm{min/max}}$, $q_{j}^{\textrm{min/max}}$, and
$T_{j}^{\textrm{max}}$ are the lower/upper safety bounds for the current, SoC,
and temperature of cell $j$, respectively. It is important to note that
$i_{L_{j}}^{\textrm{min}}$ can be set to be zero as the zero current means the
bypass of the module. Further, we impose the following SoC and temperature
balancing constraints to equalize the cells and make an even usage of them:

$\displaystyle|q_{j}(t)-q_{\textrm{avg}}(t)|$ $\displaystyle\leq\Delta q,$ (8)
$\displaystyle|T_{j}(t)-T_{\textrm{avg}}(t)|$ $\displaystyle\leq\Delta T,$ (9)

Here, $q_{\textrm{avg}}(t)$ and $T_{\textrm{avg}}(t)$ represent the average
SoC and temperature of all the cells that belong to $\mathcal{J}$. They are
calculated as

$X_{\textrm{avg}}(t)=\frac{1}{\textrm{card}(\mathcal{J})}\sum_{j\in\mathcal{J}}X_{j}(t),$

where $X=q$ and $T$, and $\textrm{card}(\mathcal{J})$ is the cardinality of
$\mathcal{J}$. The SoC and temperature thresholds $\Delta q$ and $\Delta T$
determine the tolerated deviation of each cell’s SoC and temperature from the
average. Here, $\Delta q$ and $\Delta T$ are tunable parameters, and one can
tune them to meet the SoC and temperature balancing requirements for a
specifically considered application. While lower $\Delta q$ and $\Delta T$
values force a more balanced SoC and temperature distribution among the cells,
higher values allow more deviation for the cells’ SoC and temperature from the
average. To make the RBESS meet the power demands, we present the following
output power satisfaction constraint:

$\sum_{j\in\mathcal{J}}P_{b_{j}}-(R_{j}+R_{C}+R_{S_{j}})i_{L_{j}}^{2}=P_{\textrm{out}},$
(10)

where $P_{\textrm{out}}$ is the total power demanded of the RBESS.

Summing up the above, our power management approach is based on addressing the
constrained nonlinear optimization problem as follows:

$\displaystyle\min_{i_{L_{j}},j\in\mathcal{J}}$
$\displaystyle\int_{0}^{H}J(t)dt,$ (11) s.t.
$\displaystyle\eqref{SoCDynamic},\eqref{TempDynamic},\eqref{SafetyConstraints}-\eqref{DemandSatisfaction}.$

This optimization problem pursues predictive minimization of the power losses
while complying with the constraints that promote safety, SoC and temperature
balancing, and power supply-demand match. Note that the optimization problem
(11) is non-convex due to the nonlinearity of the equality constraint (10).
Thus, the solution to this problem is neither trivial nor computationally
cheap. To overcome the issue, we relax the problem slightly to formulate a
convex optimization problem, as suggested in [22]. The convexification is
described in detail as follows.

###  III-C Convex Problem Formulation

For the sake of convexification, we begin with linearizing the SoC/OCV curve.
The existing studies, e.g., [18, 19] perform the linearization for only the
medium SoC range, where the OCV is closely linear with SoC for lithium-ion
batteries. However, this treatment excludes the use of the low and high SoC
ranges. To address the issue, we introduce multi-segment linearization based
on different SoC ranges to approximate the complete SoC/OCV curve:

$u_{j}(q_{j}(t))=\alpha^{i}_{j}(q_{j}(t))+\beta^{i}_{j}(q_{j}(t))q_{j}(t),$
(12)

where $\alpha^{i}_{j}$ and $\beta^{i}_{j}$ are the intercept and slope
coefficients of the $i$-th line segment for cell $j$. Fig. 3 illustrates an
example of the linearization, where the SoC/OCV curve taken from a real cell
is approximated by three line segments. Differing from the literature, the
$\alpha^{i}_{j}$ and $\beta^{i}_{j}$ values are SoC-dependent, and the multi-
segment linear approximation spans the SoC/OCV curve from 0 to 100% SoC. To
ease the notation, we will drop the superscript $i$ from $\alpha^{i}_{j}$ and
$\beta^{i}_{j}$ in the sequel without causing confusion. Next, we present a
convex model by introducing the notion of accumulated energy $E_{j}$ to take
the place of SoC. The accumulated energy of a battery cell can be expressed as

$E_{j}(t)=\frac{1}{2}C_{j}u_{j}^{2}(q_{j}(t))-E_{j}^{0},$ (13)

where $C_{j}=\bar{Q}_{j}/\beta_{j}$ and
$E_{j}^{0}=\frac{1}{2}C_{j}u_{j}^{2}(q_{j}(0))$ is the initial energy.
Inserting (12) to (13) and using (1b), the dynamic equation of the cell’s
accumulated energy can be derived as

$\dot{E}_{j}(t)=-P_{b_{j}}.$ (14)

In the above, we extract a desirably linear dynamic model to represent the
evolution of $E_{j}(t)$ driven by $P_{b_{j}}$. Based on (14), we will
reformulate the optimization problem to be one with respect to $P_{b_{j}}$, as
will be seen later.

Figure 3: The SoC/OCV curve of the simulated cells and the multi-segment
linearization.

Proceeding forward, we consider module $j$’s power loss, $P_{l_{j}}(t)$, which
can be expressed in terms of $P_{b_{j}}$ as

$P_{l_{j}}(t)=\frac{(R_{j}+R_{C}+R_{S_{j}})C_{j}P_{b_{j}}^{2}(t)}{2(E_{j}(t)+E_{j}^{0})}.$
(15)

As our optimization goal is to minimize the total power loss, (15) serves as
an equality constraint. Since $P_{l_{j}}(t)$ is not a linear function of
$P_{b_{j}}$, the resulting optimization problem would be non-convex due to the
nonlinearity of (15). Thanks to the fact that the objective function minimizes
the total loss of the battery pack, we can relax (15) to comply with the
convexity requirement

$P_{l_{j}}(t)\geq\frac{(R_{j}+R_{C}+R_{S_{j}})C_{j}P_{b_{j}}^{2}(t)}{2(E_{j}(t)+E_{j}^{0})},$
(16)

by which the optimization problem will practically reduce $P_{l_{j}}(t)$ to
its lower bound. The safety constraints (7a)-(7b) can also be reformulated in
terms of the $P_{b_{j}}$ and $E_{j}$ as follows:

$\displaystyle\sqrt{\frac{2}{C_{j}}(E_{j}+E_{j}^{0})}i_{L_{j}}^{\textrm{min}}$
$\displaystyle\leq
P_{b_{j}}\leq\sqrt{\frac{2}{C_{j}}(E_{j}+E_{j}^{0})}i_{L_{j}}^{\textrm{max}},$
(17a) $\displaystyle\frac{1}{2}C_{j}u_{j}^{2}(q_{j}^{\textrm{min}}(t))$
$\displaystyle\leq
E_{j}+E_{j}^{0}\leq\frac{1}{2}C_{j}u_{j}^{2}(q_{j}^{\textrm{max}}(t)).$ (17b)

Similarly, the SoC balancing constraint (8) translates into the following one:

$\Bigg{|}\frac{2}{C_{j}}E_{j}(t)-\frac{1}{\textrm{card}(\mathcal{J})}\sum_{i\in\mathcal{J}}\frac{2}{C_{i}}E_{i}(t)\Bigg{|}\leq\Delta
E_{j},$ (18)

where $\Delta E_{j}=(\alpha_{j}+\beta_{j}\Delta q)^{2}-\alpha_{j}^{2}$. It is
worth noting that the SoC balancing constraint, either (8) or (18), may result
in infeasibility for the optimization problem, when $\Delta q$ or $\Delta E$
fails to bound the cells’ initial difference in SoC. The same issue applies to
the temperature balancing constraint (9). Once happening, the infeasibility
will cause the power optimization procedure to abort. While it is possible to
make $\Delta E$ and $\Delta T$ large enough to forestall the issue, this will
sacrifice the achievable performance in both power loss minimization and cell
balancing. To guarantee the feasibility, we introduce slack variables to
modify the constraints in (18) and (9) as follows:

$\displaystyle\Bigg{|}\frac{2}{C_{j}}E_{j}(t)-\frac{1}{\textrm{card}(\mathcal{J})}\sum_{l\in\mathcal{J}}\frac{2}{C_{l}}E_{l}(t)\Bigg{|}$
$\displaystyle\leq\Delta E_{j}+\xi^{(E)}_{j},$ (19)
$\displaystyle|T_{j}(t)-T_{\textrm{avg}}(t)|$ $\displaystyle\leq\Delta
T+\xi^{(T)}_{j},$ (20)

where $\xi^{(E)}_{j},\xi^{(T)}_{j}\geq 0$ denote the SoC and temperature slack
variables, respectively. The slack variables will be included into the
objective function to penalize potential constraint violations. As such, if a
cell’s SoC or temperature is beyond the constraints, it will be driven close
to the constraints by heavily penalizing the corresponding slack variables,
without compromising the feasibility. Besides, the use of the slack variables
will improve the power control flexibility. This will be discussed in the
simulation study in Section IV.

Based on the above, we are now ready to make a convex relaxation of the
problem in (11). Here, we also turn our focus to discrete-time optimization
for the sake of computation, through applying the forward Euler method to (4)
and (14) with the sampling time of $\Delta t$. We use the vector
$z_{j}=[P_{b_{j}},P_{l_{j}},E_{j},T_{j},\xi_{j}^{(E)},\xi_{j}^{(T)}]^{\top},j\in\cal{J}$
to collect all the optimization variables and propose the following convex
optimization problem for the RBESS power management:

$\displaystyle\min_{z_{j},j\in\mathcal{J}}\quad\sum_{k=0}^{H}\sum_{j\in\mathcal{J}}P_{l_{j}}[k]+\lambda^{(E)}\xi^{(E)}_{j}[k]+\lambda^{(T)}\xi^{(T)}_{j}[k],$
(21) Safety constraints: $\displaystyle\
\sqrt{\frac{2}{C_{j}}(E_{j}[k]+E_{j}^{0})}i_{L_{j}}^{\textrm{min}}\leq
P_{b_{j}}[k]\leq\sqrt{\frac{2}{C_{j}}(E_{j}[k]+E_{j}^{0})}i_{L_{j}}^{\textrm{max}},$
$\displaystyle T_{j}\leq T_{j}^{\textrm{max}}$ $\displaystyle\
\frac{1}{2}C_{j}u_{j}^{2}(q_{j}^{\textrm{min}}[k])\leq
E_{j}[k]+E_{j}^{0}\leq\frac{1}{2}C_{j}u_{j}^{2}(q_{j}^{\textrm{max}}[k]),$
Balancing constraints: $\displaystyle\
\Bigg{|}\frac{2}{C_{j}}E_{j}[k]-\frac{1}{\textrm{card}(\mathcal{J})}\sum_{l\in\mathcal{J}}\frac{2}{C_{l}}E_{l}[k]\Bigg{|}\leq\Delta
E_{j}+\xi^{(E)}_{j}[k],$ $\displaystyle\
|T_{j}[k]-T_{\textrm{avg}}[k]|\leq\Delta T+\xi^{(T)}_{j}[k],$ Power loss
constraint: $\displaystyle\
P_{l_{j}}[k]\geq\frac{(R_{j}+R_{C}+R_{S_{j}})C_{j}P_{b_{j}}^{2}[k]}{2(E_{j}[k]+E_{j}^{0})},$
Energy dynamics: $\displaystyle\ E_{j}[k+1]-E_{j}[k]=-P_{b_{j}}[k]\Delta t,$
Thermal dynamics: $\displaystyle\ T_{j}[k+1]=T_{j}[k]+\frac{\Delta
t}{C_{\textrm{th},j}}\Big{[}P_{l_{j}}[k]-(T_{j}[k]-T_{\textrm{env}})/R_{\textrm{conv}}$
$\displaystyle\quad-(2T_{j}[k]-T_{j+1}[k]-T_{j-1}[k])/R_{\textrm{cnd}}\Big{]},$
Power supply-demand balance: $\displaystyle\
\sum_{j\in\mathcal{J}}P_{b_{j}}[k]-P_{l_{j}}[k]=P_{\textrm{out}}[k],$

where $\lambda^{(E)}$ and $\lambda^{(T)}$ are the respective penalty weights
for $\xi^{(E)}$ and $\xi^{(T)}$. The above problem is verifiably convex as a
result of the convex cost function and constraints. The convexity makes it
advantageous in practice as robust algorithms are available to find out its
global optimum with efficient computation. The problem setup is similar to
[18, 19]. However, we crucially introduce the slack variables here to make the
problem always feasible. This improvement eliminates the risk of no solution
to satisfy the hard constraints, further enhancing the practical aspects of
power management.

The problem is designed to be implemented in a receding-horizon manner. This
will bring three benefits. First, predictive optimization over a limited time
horizon rather than the whole mission duration will make the computation more
manageable. Second, the receding-horizon power control can better respond to
changes that occur to the RBESS in operation, e.g., fault-triggered cell
bypass and switching circuit reconfiguration. Finally, the SoC change in each
receding horizon is slight, so the optimization only needs to consider a
single SoC/OCV linear segment and hence runs more efficiently.

###  III-D Reconfiguration

Algorithm 1 Power management of the proposed RBESS

1: for Run-time do

2: if a fault occurs to cell $i$ then

3: Bypass cell $i$

4: Determine the set of the in-service cells $\mathcal{J}$

5: Calculate $n_{s}$ and $n_{p}$ from (22)

6: Reconfigure the switch circuit $S_{i1:3},i=1,2,...n$

7: end if

8: Run the optimal power management strategy (21)

9: return $P_{b_{j}}$

10: if $P_{b_{j}}==0$ for any $j$ then

11: Bypass cell $j$

12: end if

13: end for

The proposed RBESS allows dynamic switching of the power switches to bypass
faulty cells, ensuring continuous system operation. Following the bypass, an
important question is how to reconfigure the connection topology among the
cells. However, it is not easy to identify a complete answer, as the large
discrete reconfiguration decision space due to the use of switches would defy
an exhaustive search for an optimal topology. In addition, inappropriate
reconfiguration may produce poor topologies to cause short circuits or other
issues. Note that the power management approach in (21) determines the optimal
charging/discharging power of the cells individually and is not affected by
any arbitrary series or parallel connection among them. This makes its run and
the reconfiguration procedure separable but contiguous.

Here, we leverage an efficient heuristic to address the question and outline
it as below. Suppose that all the remaining cells are approximately uniform in
SoC and temperature at the time of the reconfiguration, since the power
management based on (21) has driven cell balance. The reconfiguration then
should yield a topology that facilitates a balanced use of the cells and makes
every cell take an even power load. A straightforward topology design to
fulfill this need is one based on $n_{s}$ serially connected modules with each
module consisting of $n_{p}$ cells in parallel connection. We denote this
topology as $n_{p}{\mathrm{P}}n_{s}{\mathrm{S}}$. We can determine $n_{s}$ and
$n_{p}$ by

$n_{s}=\frac{V_{t}^{*}}{V_{C}^{\textrm{max}}},\qquad
n_{p}=\frac{I_{\textrm{out}}}{i_{C}^{\textrm{max}}},$ (22)

where $V_{t}^{*}$ is the desired terminal voltage; $V_{C}^{\textrm{max}}$ and
$i_{C}^{\textrm{max}}$ are the maximum output voltage and current stresses of
the DC/DC converters, and $I_{\textrm{out}}=P_{\textrm{out}}/V_{t}^{*}$ is the
output charging/discharging current of the battery pack. Subsequently, the
RBESS can follow the series/parallel switching analysis in Section II to
reconfigure the switch circuit.

This heuristic-based reconfiguration mechanism is computationally fast, fail-
safe, and easy to implement. Further, it promotes system-wide cell balance and
fits together with the power management in (21). This leads us to the overall
RBESS management approach as shown in Algorithm 1.

##  IV Simulation Results

TABLE II: Specifications of the Proposed RBESS

Symbol | Parameter | Value [Unit]
---|---|---
$n$ | Number of battery cells | 15
$v$ | Cell nominal voltage | 3.6 [V]
$\bar{Q}$ | Cell nominal capacity | 2.5 [A.h]
$R$ | Cell internal resistance | 31.3 [m$\Omega$]
$[q^{\textrm{min}},q^{\textrm{max}}]$ | Cell SoC limits | [0.05,0.95]
$[i^{\textrm{min}},i^{\textrm{max}}]$ | Cell current limits | [-10,10] [A]
$v_{\textrm{cut-off}}$ | Cell cut-off voltage | 3.3 [V]
$C_{\textrm{th}}$ | Thermal capacitance | 40.23 [J/K]
$R_{\textrm{conv}}$ | Convection thermal resistance | 41.05 [K/W]
$R_{\textrm{cnd}}$ | Conductance thermal resistance | 26.6 [K/W]
$T_{\textrm{env}}$ | Environment temperature | 298 [K]
$\Delta q$ | SoC balancing threshold | 1%
$\Delta T$ | Temperature balancing threshold | 0.5 [K]
$\Delta t$ | Sampling time | 1 [s]

This section presents simulation results to evaluate the proposed RBESS design
and power management approach. Table II summarizes the specifications of the
RBESS under simulation. The battery cells are assumed to be Samsung
INR18650-25R, and we have identified their parameters (see Table II) and
SoC/OCV relationship (see Fig. 3) from experiments using the approach in [29].
We approximate the SoC/OCV curve using a piecewise linear function with three
segments that together span from zero to 100% SoC. The power load profile for
$P_{\textrm{out}}$ is obtained by repeating the scaled Urban Dynamometer
Driving Schedule (UDDS). We use the CVX package [30] to configure and solve
the convex optimization problem in (21) to compute $P_{b}$. The optimization
runs over a receding horizon of 20 seconds, i.e., $H=20$.

The initial SoC of the cells is drawn from a normal distribution with mean of
90% and variance of 3%. Similarly, the initial temperature of the cells
follows a normal distribution with mean of 308 K and variance of 3 K. In order
to investigate whether the power management can handle the cells’
heterogeneity, a white Gaussian noise with variance of 4 m$\Omega$ is added to
the internal resistance value of each cell. Furthermore, it is assumed that
cells 4, 8, and 14 are bypassed and isolated from the battery pack at the
2,000th, 4,000th, and 6,000th seconds, respectively.

Fig. 4 depicts the SoC and temperature balancing performance of the proposed
power management approach. The tolerated SoC and temperature deviation bounds,
$\Delta q$ and $\Delta T$, are 1% and 0.5 K, respectively. According to Fig. 4
(a), the cells are different in their initial SoC. Among them, cell 2 has the
lowest initial SoC of 86.48%, and cell 15 has the highest SoC of 92.61%. The
difference is beyond the desired error bounds. However, the power management
approach successfully drives the SoC of the cells to reach within the bounds
after 200 seconds and continues to regulate the charging/discharging power of
the cells to ensure SoC balance in the battery pack. Both cells 2 and 15 end
up with the same SoC of 8.2% when the simulation is finished. It is important
to note the key of incorporating the slack variables in guaranteeing the
feasibility of the power optimization. Fig. 4 (b) illustrates the deviation of
the cells’ SoC from the average. The tolerance bound is set to be 1%. Some of
the cells initially are beyond this bound—for example, cell 15 deviates from
the average SoC by 3.5%. In this case, the optimization problem would have
been infeasible, but this issue is avoided as the slack variable $\xi^{(E)}$
permits slight violation of the SoC balancing constraints with a negligible
compromise to physical safety of the cells. Meanwhile, the penalization of
$\xi^{(E)}$ as in (21) in the cost function forces the cells to remain within
the tolerated error bound once after they enter the bound, keeping the SoC
balanced. The SoC of the bypassed cells remains unchanged after isolation as
the cells are no longer used.

(a)  (b)  (c)  (d)

Figure 4: Simulation results of the SoC and temperature balancing. (a) The SoC
of the cells. (b) The SoC difference of the cells from the average. (c) The
temperature of the cells. (d) The temperature difference of the cells from the
average.

Fig. 4 (c) shows the evolution of the cells’ temperature. Similar to the SoC
initialization, the initial temperatures of the cells stretch beyond the
desired bounds where cells 1 and 2 have the highest and lowest temperature of
37.92$\tccentigrade$ and 29.73$\tccentigrade$, respectively. The power
management approach effectively controls the cell temperatures to reach a
balanced temperature after 500 seconds. Note that a cell’s temperature is
still affected by the temperature of its adjacent cells and the environment
after it is bypassed. As shown in Fig. 4 (c), when the average temperature of
the battery pack increases, the temperature of the bypassed cells also rises
due to the conductive heat transfer among adjacent cells. Here, even though
the cells’ initial temperature difference exceeds the bound of
0.5$\tccentigrade$ (see Fig. 4 (d)), the optimization for temperature
balancing maintains feasibility as a result of introducing the slack variable
$\xi^{(T)}$.

To further investigate the role of the slack variables in the formulated
optimization problem, Fig. 5 depicts their evolution through time. When the
cells’ SoC or temperature lies outside the balancing constraints, the slack
variables will take nonzero values to relax the balancing constraints gently,
thus turning the nominally infeasible optimization problem into a feasible
one. The slack variables will decrease and approach zero as the cells are
increasingly balanced in SoC and temperature. When they are zero, the SoC and
temperature balancing constraints are fully satisfied. Penalizing the slack
variables restricts over-relaxation of the constraints and tightens the bounds
as the SoC and temperature get closer to or into the constraints. The
penalization weights associated with the slack variables are subject to tuning
so as to achieve the performance desired by a user. In general, heavier
penalization will lead to less constraint relaxation and more time to achieve
balancing.

Figure 5: The change of the slack variables.

The output power profiles of the cells are shown in Fig. 6. We can see that
the power of the individual cells is regulated to vary from one to another.
This is because the cells have different conditions in SoC, temperature and
internal resistances and must collectively minimize the overall power losses
while complying with safety and balancing constraints. The cells can also
adjust their own output on the bypass of a faulty cell. The peak power of
battery cells are around 28 W for $1,560<t<1,570$ s before any cells are
bypassed from the pack. However, when three cells are bypassed from the pack,
the peak power of the remaining cells is increased to around 33 W for
$5,750<t<5,760$ s to compensate for the bypassed cells and to ensure a
continuous power supply to the load.

It is of our interest to investigate whether the proposed RBESS is more
capable of reducing the total power losses than conventional hardwired battery
systems. Fig. 7 shows a comparison of the resultant power losses, which
focuses on $1,000<t<2,000$ s for the purpose of visual illustration. The
hardwired pack is found to constantly suffer more power dissipation, because
the it neglects that the cells have different internal resistances (associated
with the higher power losses, the pack also faces higher operating
temperatures as well as significant SoC and temperature imbalance). By
contrast, the proposed RBESS is able to optimally allocate the
charging/discharging power among the cells to gain more power efficiency.

We further assess whether the proposed power management approach can
distribute power among the cells relative to their state-of-health (SoH),
which is important to reduce the cell aging and degradation. To this end, we
consider the root-mean-square (RMS) of the output power of the cells, and use
the internal resistance as the SoH indicator—overall, the higher the internal
resistance, the more degraded the cell is. Fig. 8 illustrates the normalized
RMS of the output power of the battery cells in comparison to their internal
resistance values. We observe that the cells with lower resistances are
allocated more power load overall, see the groups of cells 1-4, cells 6-7, and
cells 9-10. While the pattern is obvious, the power distribution also depends
on each cell’s SoC and temperature and thus shows certain perturbations. We
can argue that the power management approach contributes to a balanced use of
the battery cells in terms of SoH.

Figure 6: The output power profiles of the cells. Figure 7: The power loss
comparison. Figure 8: The normalized internal resistance and RMS of the output
power of the cells.

##  V Experimental Results

We develop a lab-scale prototype of the proposed RBESS for experimental
validation. Fig. 9 (a) shows the experimental setup, and Figs. 9 (b)-(c)
illustrate the circuit boards of the RBESS prototype based on the design in
Fig. 1. The RBESS is a pack of five cells integrated with five converters
(Fig. 9 (b)) and 12 relay switches for reconfigurable connection (Fig. 9 (c)).
Table III lays out the specifications of the key components of the prototype.
Type K thermocouples are attached on the surface of each cell to measure their
temperature. A National Instruments PCIe-6321 DAQ board with LabVIEW is used
to collect the cells’ voltages, temperatures, and output power data. Using the
CVX package, we then solve the optimal power management problem using MATLAB
every minute (i.e., $\Delta t=60$ s). The optimal power values of the cells
are then fed to local controllers using DSP TMS320F28335. The local
controllers based on STM8S003F3P6 microcontrollers, generate 250 kHz PWM
signals to DC/DC converters. The prototype is connected to a 20 $\Omega$
resistance load with a total output discharge power of 50 W.

(a)  (b)  (c)

Figure 9: Lab-scale prototype of the proposed RBESS. (a) The experimental setup. (b) Top circuit view. (c) Bottom circuit view TABLE III: List of Key Components Device | Model (Value)
---|---
MOSFET | CSD86356Q5D
Relay switch | TE OJT-SS-105HM
Gate driver | TPS28225
Inductor |  SER2915H-333KL (33 $\mu$H)
Capacitor |  (10 $\mu$F)
Local controller | STM8S003F3P6
Main controller | TMS320f28335
Battery cell | Samsung INR18650-25R

(a)  (b)  (c)  (d)

Figure 10: Experimental results of the proposed RBESS. (a) The SoC of the
cells. (b) The temperature of the cells. (c) The SoC difference of the cells
from the average. (d) The temperature difference of the cells from the
average. Figure 11: The output power profiles of the cells.

The cells, labeled from 1 to 5 in order, have an initial SoC of 87%, 89%, 82%,
91%, and 93%, respectively. The experiment lasts for 30 minutes with a
sampling time of $\Delta t=60$ s. Each cell’s output current is limited to 5
A. To investigate the effect of fault occurrence, a fault is assumed for cell
3 after 15 minutes of discharging in the experiment. The obtained results are
shown in Figs. 10-11.

Fig. 10 (a) shows the SoC of the battery cells. The initial SoC values of the
cells are not within the desired tolerance bound. However, the optimal power
management approach successfully distributes the discharging power among the
cells such that the cells reach the SoC balancing bounds after about four
minutes. The corresponding output power profiles of the cells are shown in
Fig. 11. It can be seen that cell 3, with the lowest initial SoC, is assigned
zero power load (and thus bypassed by reconfiguration) in the first two
minutes, while cell 5, with the highest SoC, delivers the maximum allowed
power. Not only does the proper distribution of the output power lead to cell
balancing, but the reconfiguration capability of the proposed design also
helps cell balancing. Fig. 10 (b) also shows the SoC deviation of the cells
from their average value. We point out that, without the inclusion of the
slack variables, the optimization would have been infeasible at the very
initial moment when the SoC deviation goes beyond the SoC balancing
constraint. Fig. 10 (c) depicts the temperature of the cells. The initial
temperature of all the cells is 20.7$\tccentigrade$. Due to the uneven power
distribution among the cells for SoC balancing, the cells will see their
temperature rise and slightly drift away from each other. However, the
deviation remains within the desired bound without violating the temperature
balancing constraint. Fig. 10 (d) also shows the temperature deviation of the
cells from the average value. The difference increases from zero to the pre-
specified bound of 0.5$\tccentigrade$ in the beginning. But afterwards, it
shows a declining trend and is well bounded.

When a fault occurs to cell 3 after 15 minutes of discharging, the cell is
bypassed and isolated, as indicated in Fig. 11. Right after this happens, the
other four cells that remain in service increase their discharging power
accordingly, continuing to supply a total output power of 50 W as demanded.
This highlights the benefit of the proposed RBESS in ensuring robust and
consistent operation despite cell faults.

##  VI Conclusion and Future Work

The RBESS technology offers an important way to enhance the safe use of
lithium-ion batteries. In this paper, we proposed a novel modular RBESS
design, which distinguishes itself by the integration of reconfigurable power
switches and DC/DC converters. The design harnesses the switching circuit
reconfiguration to bypass any defective cells, and exploits the DC/DC
converters to facilitate optimal power distribution at the cell level and
ensure consistent power storage/supply at the system level. Based on the
design, we developed a power management approach to achieve power-loss-
minimized operation of the RBESS along with SoC and temperature balancing
among the cells. Compared to existing methods, this approach allows wide-SoC-
range operation of the cells by multi-segment SoC/OCV approximation and
guarantees the feasibility of the optimization problem via mild relaxation. We
conducted extensive simulations and then developed a lab-scale prototype of
the RBESS design to perform validation experiments. The results substantiate
the effectiveness of the proposed design and the power management approach.
The study can benefit and potentially drive the use of lithium-ion batteries
for safety-critical applications.

Based on the study, several interesting research questions are worth pursing
further. One is how to quantify and assess the reliability of the proposed
RBESS given the failure rates of the batteries, switches, and converters. A
subsequent question is how to optimize the RBESS architecture and size under
specified reliability metrics and power requirements. Finally, it is important
to enable fast computation in large-size RBESS power management, and how to
scale up the optimization approach in this paper is open to exploration.

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|  Amir Farakhor (Student Member, IEEE) received the B.Sc. and M.Sc. degrees
in electrical engineering from the Azarbaijan Shahid Madani University,
Tabriz, Iran in 2012 and 2014, respectively, and Ph.D. degree in Power
Electronics from the University of Tabriz in Feb 2019. He is currently a Ph.D.
student in mechanical engineering at the University of Kansas, Lawrence, KS,
USA. His research interests include power electronics, battery management
systems, renewable energies, and distributed generation.
---|---
|  Di Wu (Senior Member, IEEE) is a Chief Research Engineer and a Team Leader
within the Optimization and Control Group at the Pacific Northwest National
Laboratory (PNNL). He received the B.S. and M.S. degrees in electrical
engineering from Shanghai Jiao Tong University, China, in 2003 and 2006,
respectively, and the Ph.D. in electrical and computer engineering from Iowa
State University, Ames, in 2012. At PNNL, Dr. Wu leads research work in the
areas of energy storage analytics, building-to-grid integration, microgrid
design, and hybrid energy systems. Dr. Wu is a Senior Member of IEEE and a
member of the IEEE Power and Energy Society and the Control System Society. He
serves as an Editor for the IEEE Open Access Journal of Power and Energy and
IEEE Transactions on Energy Markets, Policy and Regulation.
---|---
|  Yebin Wang (Senior Member, IEEE) received the B.Eng. degree in mechatronics
engineering from Zhejiang University, Hangzhou, China, in 1997, the M.Eng.
degree in control theory and control engineering from Tsinghua University,
Beijing, China, in 2001, and the Ph.D. degree in electrical engineering from
the University of Alberta, Edmonton, AB, Canada, in 2008. He has been with
Mitsubishi Electric Research Laboratories, Cambridge, MA, USA, since 2009,
where he is currently a Senior Principal Research Scientist. From 2001 to
2003, he was a Software Engineer, the Project Manager, and the Manager of the
Research and Development Department in Industries, Beijing, China. His current
research interests include nonlinear control and estimation, optimal control,
adaptive systems, and their applications, including mechatronic systems.
---|---
|  Huazhen Fang (Member, IEEE) received the B.Eng. degree in computer science
and technology from Northwestern Polytechnic University, Xi’an, China, in
2006, the M.Sc. degree in mechanical engineering from the University of
Saskatchewan, Saskatoon, Canada, in 2009, and the Ph.D. degree in mechanical
engineering from the Department of Mechanical and Aerospace Engineering,
University of California, San Diego, La Jolla, CA, USA, in 2014. He is an
Associate Professor of mechanical engineering with the University of Kansas,
Lawrence, KS, USA. His research interests include control and estimation
theory with application to energy management and robotics. Dr. Fang received
the National Science Foundation CAREER Award in 2019. He currently serves as
an Associate Editor for IEEE Transactions on Industrial Electronics, IEEE Open
Journal of the Industrial Electronics Society, IEEE Control Systems Letters,
IEEE Open Journal of Control Systems, and Information Sciences.
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