# Scalable Optimal Power Management for Large-Scale Battery Energy Storage
Systems

Amir Farakhor, , Di Wu, , Yebin Wang, , and Huazhen Fang 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

Large-scale battery energy storage systems (BESS) are helping transition the
world towards sustainability with their broad use, among others, in
electrified transportation, power grid, and renewables. However, optimal power
management for them is often computationally formidable. To overcome this
challenge, we develop a scalable approach in the paper. The proposed approach
partitions the constituting cells of a large-scale BESS into clusters based on
their state-of-charge (SoC), temperature, and internal resistance. Each
cluster is characterized by a representative model that approximately captures
its collective SoC and temperature dynamics, as well as its overall power
losses in charging/discharging. Based on the clusters, we then formulate a
problem of receding-horizon optimal power control to minimize the power losses
while promoting SoC and temperature balancing. The cluster-based power
optimization will decide the power quota for each cluster, and then every
cluster will split the quota among the constituent cells. Since the number of
clusters is much fewer than the number of cells, the proposed approach
significantly reduces the computational costs, allowing optimal power
management to scale up to large-scale BESS. Extensive simulations are
performed to evaluate the proposed approach. The obtained results highlight a
significant computational overhead reduction by more than 60% for a small-
scale and 98% for a large-scale BESS compared to the conventional cell-level
optimization. Experimental validation based on a 20-cell prototype further
demonstrates its effectiveness and utility.

###### Index Terms:

Battery energy storage systems, advanced battery management, optimal control.

## Nomenclature

Variables

$v$

Cell terminal voltage

$u$, $\bar{u}$

Cell open-circuit voltage (OCV), cluster OCV

$i_{L}$, $\bar{i}_{L}$

Cell current, cluster current

$P_{b}$, $\bar{P}_{S}$

Cell internal power, cluster internal power

$P_{l}$, $\bar{P}_{l}$

Cell power loss, cluster power loss

$P$

Cell output power

$E$, $\bar{E}$

Cell energy, lumped cluster energy

$P_{\textrm{out}}$

BESS output power demand

$L$, $\bar{L}$

Total power losses of the cells, total power losses of the clusters

$\xi$, $\bar{\xi}$

Intra- and inter-cluster slack variables

$z$, $\bar{z}$

Intra- and inter-cluster optimization variables

Parameters

$n$

Number of battery cells

$n_{S}$

Number of cells within cluster $S$

$k$

Number of clusters

$m$, $\bar{m}$

Mass of a cell, lumped mass of a cluster

$Q$, $\bar{Q}$

Cell capacity, lumped cluster capacity

$q$, $\bar{q}$

SoC of a cell, lumped SoC of a cluster

$q_{\textrm{avg}}$

Average SoC

$\Delta q$, $\Delta\bar{q}$

SoC imbalance tolerance among cells, SoC imbalance tolerance among clusters

$\Delta E$, $\Delta\bar{E}$

Energy balancing threshold among cells, energy balancing threshold among
clusters

$\alpha$

Intercept coefficient of the SoC/OCV line segment

$\beta$

Slope coefficient of the SoC/OCV line segment

$R$, $\bar{R}$

Cell internal resistance, lumped cluster internal resistance

$R_{C}$

Resistance to capture the power losses of DC/DC converters

$R_{\textrm{conv}}$, $\bar{R}_{\textrm{conv}}$

Cell convective thermal resistance, cluster convective thermal resistance

$A$

Cell external surface.

$h$

Conductive heat transfer coefficient between the cell’s surface and the
environment

$C_{\textrm{th}}$, $\bar{C}_{\textrm{th}}$

Cell thermal capacitance, lumped cluster thermal capacitance

$T$, $\bar{T}$

Cell temperature, lumped cluster temperature

$T_{\textrm{avg}}$, $\bar{T}_{\textrm{avg}}$

Average cells’ temperature, average clusters’ temperature

$T_{\textrm{env}}$

Environmental temperature

$\Delta T$, $\Delta\bar{T}$

Temperature imbalance tolerance among cells, temperature imbalance tolerance
among clusters

$\lambda$, $\bar{\lambda}$

Penalty weight for the intra- and inter-cluster optimization

$\Delta t$

Sampling time

$H$

Optimization horizon

##  I Introduction

Battery energy storage systems (BESS) have emerged as an enabler for various
applications ranging from electric vehicles (passenger cars, semi trucks,
etc.), electric aircraft, smart grid, and renewable facilities [1, 2, 3, 4].
Whether small or large in size, BESS need power management strategies to
ensure their safe and proper operation, which provide various functions
including charging/discharging control, cell balancing, and power loss
minimization. While some simplistic approaches have gained wide use, there is
a growing demand for more sophisticated optimal power management to maximize
the performance and fully utilize the capabilities of BESS [5, 6]. However,
optimal power management faces the challenge of high computational expenses,
due to the use of numerical optimization. The challenge is especially intense
for large-scale BESS comprising great numbers of cells. We broadly define
large-scale BESS as those that offer high energy or power capacity and that
comprise many constituent entities like cells or modules, with examples
including grid- and vehicle-scale BESS. Optimal power management for them will
involve large numbers of decision variables as well as complex high-
dimensional optimization landscapes. Despite an increasing body of study on
optimal BESS power management, only limited effort has been dedicated to
overcoming the computational bottleneck in optimization.

###  I-A Literature Review

Early optimal power management strategies in the literature mainly focused on
cell balancing for BESS. The studies in [7, 8] use linear programming to
analyze and evaluate the performance of different balancing circuit topologies
for battery packs. However, the balancing considers only the SoC while
neglecting some other important factors like temperature and internal
resistance, and its computation can be heavy for large battery packs.

The use of power electronic converters in BESS circuit structures has enabled
cell-level bidirectional power control. Leveraging this capability, the study
in [9] pursues optimal power management to achieve SoC/temperature balancing
and terminal voltage regulation. The work in [10] further aims to minimize the
total power losses of a BESS while making the cells satisfy safety and
balancing requirements. It further convexifies the non-convex power
optimization problem using a technique in [11] for the benefit of computation.
The idea of convexified power optimization also finds success in a hybrid BESS
consisting of battery cells and supercapacitors [12] and a reconfigurable BESS
[13].

Looking back, the existing optimal power management strategies rely on
numerical optimization methods, and the need for computational resources
increases fast with the number of cells of the concerned BESS. The literature
has explored two methods to alleviate the computational burden, namely,
hierarchical control and distributed control. A hierarchical model predictive
control is proposed in [14] with different time scales and model complexities.
The framework considers both charge and temperature imbalances while
minimizing the total power losses. The study in [15] attempts to decompose an
optimal power management task into separate voltage and balancing control
subtasks. These methods can reduce the computation to a certain extent but
still lack scalability for large-scale BESS.

Distributed control represents another means to accelerate large-scale BESS
power management, which distributes computation among different computing
units for higher efficiency [16, 17]. The work in [16] employs a dynamic
average consensus protocol to achieve SoC balancing for a reconfigurable BESS.
An average consensus protocol is similarly used in [18] to balance the SoC of
serially connected BESS. Despite their fast computation, the average
consensus-based power management strategies only focus on cell balancing,
mostly in terms of SoC, without consideration of optimality in some metrics
such as balancing time and power loss. The study in [19] proposes a
distributed approach for cell balancing and power loss minimization for large-
scale BESS. However, it requires the use of distributed computing units and
communication networks, thus increasing the complexity of implementation.

The notion of optimal power management extends to control of distributed
energy resources, which often appears as the problem of optimal power dispatch
among the resources. For instance, the studies in [20, 21, 22] formulates
different optimal power management problems and solutions for shipboard
microgrid and all-electric ships. Optimal power management also finds
applications in the domain of hybrid electric vehicles [23]. In these
applications, the computational costs is also a topic of concern and interest.

###  I-B Statement of Contributions

Departing from the literature, this paper proposes clustering-based
optimization to achieve computationally efficient and scalable optimal power
management for large-scale BESS. The key notion lies in grouping the cells of
a BESS into clusters based on their characteristics, then performing inter-
cluster power optimization, and finally running intra-cluster power allocation
for individual cells. Centering around the notion, our main contribution are
as follows:

We develop a systematic design of a clustering-based power management
approach. The approach, which is illustrated in Fig 1, includes the following
crucial elements.

  * •

We leverage the $k$-means clustering method to partition the cells into
clusters according to their SoC, temperature, and internal resistance. The
cells within a cluster will share similar characteristics while differing
distinctly from those in another cluster.

  * •

We develop a representative electro-thermal model for each cluster to capture
the cluster’s aggregated dynamics in SoC, temperature, and power loss in
operation. Then, using the cluster models, we formulate and solve an optimal
power management problem.

  * •

Following the cluster-level optimization, we design three power split schemes
to determine the power to be assigned for the cells within each cluster.

The proposed approach is hierarchical—it breaks down a large-scale optimal
power management management problem into two levels, inter-cluster power
optimization and intra-cluster power allocation. The subproblems at each level
are considerably smaller in size and simpler in structure to allow fast
computation, and the hierarchical design readily scales up to large BESS.
Meanwhile, the clustering based on the cells’ characteristics upholds the
fidelity of the proposed approach in solving the original problem, thus
ensuring the overall performance in power management.

We develop a 20-cell battery pack and conduct a series of experiments, along
with extensive simulations, to validate and assess the performance of the
proposed approach. The results demonstrate the effectiveness of the proposed
optimal power management approach in significantly reducing the computation
while maintaining good accuracy.

Figure 1: The proposed clustering-based power management approach.

###  I-C Organization

The rest of the paper is organized as follows. Section II describes the
circuit structure of the considered large-scale BESS and the corresponding
optimal power management problem. Section III presents the cell clustering
approach, shows how to develop a representative electro-thermal model for a
cluster, and implements an inter-cluster power optimization problem for the
clusters. We then derive three schemes to handle cluster-to-cell power split.
In Sections IV and V, the simulation and experimental results demonstrate the
effectiveness of the proposed scalable optimal power management approach.
Finally, Section VI concludes the paper with final remarks.

##  II Overview of Optimal Power Management

In this section, we first introduce the circuit structure of a large-scale
BESS, and then present the overarching control objectives for the large-scale
BESS for optimal power management.

###  II-A Circuit Structure

Figure 2: The circuit structure of a large-scale BESS.

Fig. 2 depicts the circuit structure of a large-scale BESS. The BESS comprises
$n$ modules, each consisting of a cell and a DC/DC converter. The modules are
configured arbitrarily in series, parallel, or a mix of both to meet the
output power, capacity, or voltage requirements. For the purpose of
illustration, modules 1 and 2 are connected in parallel, and then connected
serially with the other modules. The circuit structure is taken from our
earlier study in [24, 13], in which the module connections are reconfigurable
via power switches. Here, we consider hardwired connections among the modules
so as to focus on the optimal power management design. Here, we use
synchronous DC/DC converters, but other types of bidirectional DC/DC converter
topologies are allowed in the circuit structure. The DC/DC converter consists
of an inductor, a capacitor, and two power switches. They allow controlled
bidirectional power flow through cells in charging/discharging. Because of the
DC/DC converters, one can independently regulate each cell’s
charging/discharging power.

The cell-level power control capability of the circuit structure brings about
several system-level advantages. First, this capability can be leveraged to
promote the balanced use of the cells in terms of their SoC and temperatures.
Second, unlike conventional structures, the circuit structure requires no
external power electronic devices to regulate its output voltage—the embedded
DC/DC converters can adjust the modules’ output voltages to supply the load
adequately. Finally, the circuit structure can accommodate heterogeneities
among the constituent cells, charging or discharging them based on their
individual conditions. In an extreme case, one can even leverage it to
construct a large-scale BESS using cells from different manufacturers with
different internal characteristics. It is worth mentioning that the circuit
structure can be extended to modules or battery packs rather than cells. In
this case, each battery pack will be equipped with a DC/DC converter to allow
regulated charging and discharging.

Next, we define the control objectives in the optimal power management of this
BESS circuit structure to fully take advantage of its capability.

###  II-B Control Objectives

Fig. 2 also illustrates how the circuit structure interacts with the battery
management system (BMS). The BMS comprises two types of controllers. At the
higher level, the optimal power management block collects the real-time
measurements of the cells and computes the optimal power allocation among
them; at the lower level, the local controllers generate control signals for
the DC/DC converters to regulate the cells’ charging/discharging currents.
This control architecture decouples power management and local power control,
and this paper focuses only on the study of the former problem, with mature
technologies available for the latter.

As discussed above in Section II.A, the circuit structure of the BESS allows
cell-level power control, making it possible to achieve various functions,
such as cell balancing and power loss minimization. To leverage this
capability, we consider the following optimal power management problem in the
paper:

######  Problem 1.

Find the reference values for the cells’ charging/discharging power to
minimize the system-wide power losses while ensuring the cells to comply with
the physical, safety, and balancing constraints and supplying the demanded
output power.

Problems of a similar kind have attracted various studies [14, 12, 13] for
different circuit structures. Existing studies generally adopt numerical
optimization frameworks. Their computational complexity depends on the number
of optimization variables, causing extremely heavy computational burdens for
large-scale BESS. To tackle the challenge, this paper proposes a scalable
optimal power management approach in the sequel.

##  III The Proposed Scalable Optimal Power Management

This section presents the proposed scalable optimal power management approach,
elaborating its four main elements: cell clustering, cluster model
development, inter-cluster power optimization, and intra-cluster power split.
To begin with, we partition the cells into clusters. We then introduce a
representative electro-thermal model for the clusters. We formulate the inter-
cluster optimal power management. Finally, we propose three schemes to split
the power quota of clusters among their constituent cells.

###  III-A Clustering

Figure 3: Illustration of cell clustering for a large-scale BESS.

Consider the BESS shown in Fig. 2 comprising $n$ cells, where $n$ is a large
number. As is common in practice, the cells are heterogeneous in SoC,
temperature, and internal resistance. An important objective in power
management is to overcome the heterogeneity, or in other words, to balance the
use of the cells via optimizing the power allocation among them based on their
conditions. Running optimization over many heterogeneous cells, however, is
computationally expensive. To treat this issue, we can group the cells into
clusters. The clusters are much fewer in number, and every cluster includes
cells with similar characteristics. We are then able to deal with cluster-
level power optimization for computational benefits.

To this end, we leverage the $k$-means clustering method to partition the $n$
cells into $k$ $(k\ll n)$ clusters $S_{1},...,S_{k}$ [25]. This method is a
favorable choice here because of its computational efficiency and capability
to handle large datasets, even though alternative clustering techniques, e.g.,
density-based spatial clustering, are also applicable [26]. For cell $i$, its
condition is characterized by the tuple $x_{i}=\\{q_{i},T_{i},R_{i}\\}$ for
$i=1,2,...,n$, where $q_{i}$, $T_{i}$, and $R_{i}$ are the SoC, temperature
and internal resistance, respectively. The clustering problem can be
translated into the following optimization problem:

$\begin{split}&\min_{r_{ij},c_{j}}\quad\sum_{i=1}^{n}\sum_{j=1}^{k}r_{ij}\left\|x_{i}-c_{j}\right\|^{2},\\\
&\textrm{s.t.}\quad\quad\sum_{j=1}^{k}r_{ij}=1\quad\forall i=1,...,n,\\\
\end{split}$ (1)

where $c_{j}$ is the centroid of cluster $S_{j}$, and $r_{ij}\in\\{0,1\\}$
with $r_{ij}=0$ if $x_{i}\notin S_{j}$ and $r_{ij}=1$ if $x_{i}\in S_{j}$. The
problem in (1) is NP-hard, often defying closed-form solution. Many heuristic
algorithms have been proposed in the literature to solve it. Here, we use the
naive $k$-means algorithm because of its effectiveness and efficiency. The
algorithm starts with an arbitrary set of centroids $c_{1}(0),...,c_{k}(0)$
and follows an alternate two-step procedure [27]. First, $x_{i}$ for
$i=1,2,...,n$ are each assigned to its nearest centroid at the $\ell$-th
iteration, i.e.,

$r_{ik}(\ell)=\begin{cases}1\quad\textrm{if}\;k=\textrm{argmin}_{j}\left\|x_{i}-c_{j}(\ell)\right\|^{2}\\\
0\quad\textrm{else}\end{cases}.$ (2)

Then, the centroids are refined and updated as follows:

$c_{j}(\ell+1)=\frac{\sum_{i=1}^{n}r_{ij}(\ell)x_{i}}{\sum_{i=1}^{n}r_{ij}(\ell)}.$
(3)

The $k$-means algorithm iterates these two steps until convergence when the
cluster assignments stop changing. Note that the algorithm requires to pre-
specify the number of clusters $k$. Clearly, the more diverse the cells, the
more clusters are needed to effectively categorize them. There are some useful
techniques, e.g., elbow curve [28] or gap statistics [29], to determine the
optimal number of clusters, and gap statistics is the choice in this paper.

Fig. 3 illustrates the clustering of a 400-cell BESS for $k=4$. The clusters
are labeled in different colors, and the cells within a cluster are closer to
each other in terms of SoC, temperature, and internal resistance. Next, we
will develop a representative model to capture the virtual collective dynamics
of a cluster.

###  III-B Electro-thermal Modeling

We first introduce a cell-level electro-thermal model and then aggregate the
cells to formulate a cluster-level model.

####  III-B1 Electrical Modeling

The electrical model of module $j$ is shown in Fig. 4 (a). We use the Rint
Model to describe the electrical dynamics of cell $j$, which consists of an
open-circuit voltage (OCV) and a series internal resistor [30]. The Rint model
offers a simple yet accurate enough representation of the cells. The model’s
governing equations are as follows:

$\displaystyle\dot{q}_{j}(t)$ $\displaystyle=-\frac{1}{Q_{j}}i_{L_{j}}(t),$
(4a) $\displaystyle v_{j}(t)$
$\displaystyle=u_{j}(q_{j}(t))-R_{j}i_{L_{j}}(t),$ (4b)

where $Q_{j}$, $u_{j}$, $v_{j}$, and $i_{L_{j}}$ are the cell’s capacity, OCV,
terminal voltage, and charging/discharging current, respectively. Following
[13], this paper assumes a piecewise linear approximation for the cell’s
SoC/OCV relationship as follows:

$u_{j}(q_{j}(t))=\alpha_{j}^{i}(q_{j}(t))+\beta_{j}^{i}(q_{j}(t))q_{j}(t),$
(5)

where $\alpha_{j}^{i}$ and $\beta_{j}^{i}$ are the intercept and slope
coefficients of the $i$th line segment. The cell’s charging/discharging power
can be expressed as

$P_{b_{j}}=u_{j}(q_{j}(t))i_{L_{j}}(t).$ (6)

Further, we idealize the DC/DC converter as a DC transformer with a series
resistance $R_{c}$ to capture its power loss. The output power of the module
is then given by

$P_{j}(t)=u_{j}(q_{j}(t))i_{L_{j}}(t)-(R_{j}+R_{C})i_{L_{j}}^{2}(t),$ (7)

where $R_{j}i_{L_{j}}^{2}(t)$ and $R_{C}i_{L_{j}}^{2}(t)$ represent the power
losses on the cell and converter, respectively. Next, we extend this cell-
level model to the cluster level.

(a)  (b)

Figure 4: The cell-level electro-thermal model. (a) The electrical model of
the module $j$. (b) The thermal model of the cell $j$.

Consider cluster $S_{j}$ with $n_{S_{j}}$ cells numbered from 1 to
$n_{S_{j}}$, with $\sum_{j=1}^{k}n_{S_{j}}=n$. We intend to aggregate the cell
models to derive a cluster model for $S_{j}$. Here, $S_{j}$ can be viewed as a
virtual module, though the cells within it may not share hardwired connection.
Then, we can assume the constituent cells to be connected virtually either in
parallel or series. We choose virtual parallel connection here to derive the
aggregate model for $S_{j}$. Fig. 5 illustrates the idea of lumping the cell-
level models into a single model for cluster $S_{j}$.

Figure 5: Representative cluster-level electrical model.

For cluster $S_{j}$, the aggregated capacity $\bar{Q}_{j}$ and applied current
$\bar{i}_{L_{j}}$ can be expressed as follows:

$\bar{Q}_{j}=\sum_{i=1}^{n_{S_{j}}}Q_{i},\quad\bar{i}_{L_{j}}=\sum_{i=1}^{n_{S_{j}}}i_{L_{i}}.$
(8)

The cluster’s SoC $\bar{q}_{j}$ is governed by

$\dot{\bar{q}}_{j}(t)=-\frac{1}{\bar{Q}_{j}}\bar{i}_{L_{j}}(t).$ (9)

Further, the cluster’s internal resistance $\bar{R}_{j}$ and OCV $\bar{u}_{j}$
are as follows:

$\displaystyle\bar{R}_{j}$
$\displaystyle=\frac{R_{n_{S_{j}}}+R_{C}}{1+\sum_{i=1}^{n_{S_{j}}-1}\frac{R_{n_{S_{j}}}+R_{C}}{R_{i}+R_{C}}},$
(10a) $\displaystyle\bar{u}_{j}$
$\displaystyle=\frac{1}{1+\sum_{i=1}^{n_{S_{j}}-1}\frac{R_{n_{S_{j}}}+R_{C}}{R_{i}+R_{C}}}\left(\sum_{i=1}^{n_{S_{j}}}\frac{R_{n_{S_{j}}}+R_{C}}{R_{i}+R_{C}}u_{i}\right).$
(10b)

The cluster’s SoC/OCV relationship is approximated as

$\bar{u}_{j}(\bar{q}_{j}(t))=\bar{\alpha}_{j}^{i}(\bar{q}_{j}(t))+\bar{\beta}_{j}^{i}(\bar{q}_{j}(t))\bar{q}_{j}(t),$
(11)

where

$\bar{\alpha}_{j}^{i}=\frac{1}{n_{S_{j}}}\sum_{j=1}^{n_{S_{j}}}\alpha_{j}^{i},\quad\bar{\beta}_{j}^{i}=\frac{1}{n_{S_{j}}}\sum_{j=1}^{n_{S_{j}}}\beta_{j}^{i}.$
(12)

The cluster’s internal charging/discharging power can be expressed by

$\bar{P}_{S_{j}}=\bar{u}_{j}(\bar{q}_{j}(t))\bar{i}_{L_{j}}(t).$ (13)

Its total power losses can be calculated as

$\bar{L}_{S_{j}}=\bar{R}_{j}\bar{i}_{L_{j}}^{2}(t).$ (14)

Putting together (8)-(14), we obtain an electrical model for cluster $S_{j}$
to grasp the aggregated dynamics of the constituent cells.

####  III-B2 Thermal Modeling

This paper uses a lumped thermal model to describe the cells’ thermal dynamics
[10]. This model assumes that the cell’s temperature is concentrated at a
single point. The simplification makes it tractable to deal with the cell-to-
cluster thermal modeling. Fig. 4 (b) depicts the model for cell j. It
considers the cell’s internal power loss, $R_{j}i_{L_{j}}^{2}(t)$, as the
source of heat generation and concentrates on the convection between the cell
and the environment. The cell-level thermal model can then be expressed by

$m_{j}C_{\textrm{th}}\dot{T}_{j}(t)=R_{j}i_{L_{j}}^{2}(t)-(T_{j}(t)-T_{\textrm{env}})/R_{\textrm{conv}},$
(15)

where $T_{j}$ and $T_{\textrm{env}}$ are the temperature of cell $j$ and the
environmental temperature, respectively. Further, $m_{j}$ and
$C_{\textrm{th}}$ are the mass of cell $j$ and specific heat capacity,
respectively; $R_{\textrm{conv}}$ is the convective thermal resistance between
cell $j$ and the environment, specified by

$R_{\textrm{conv}}=\frac{1}{hA_{j}},$

where $h$ and $A_{j}$ are the heat transfer coefficient between the cell’s
surface and the environment, and the external surface area of cell $j$,
respectively.

Given (15), we approximate the thermal dynamics of cluster $S_{j}$ as follows:

$\bar{m}_{j}C_{\textrm{th}}\dot{\bar{T}}_{j}(t)=\bar{R}_{j}\bar{i}_{L_{j}}^{2}(t)-(\bar{T}_{j}(t)-T_{\textrm{env}})/\bar{R}_{\textrm{conv},j}.$
(16)

Here, $\bar{T}_{j}$, $\bar{m}_{j}$, and $\bar{R}_{\textrm{conv},j}$ are
respectively the lumped temperature, mass, and thermal resistance of cluster
$S_{j}$, which are calculated as

$\bar{T}_{j}=\frac{1}{n_{S_{j}}}\sum_{j=1}^{n_{S_{j}}}T_{j},\
\bar{m}_{j}=\sum_{j=1}^{n_{S_{j}}}m_{j},\
\bar{R}_{\textrm{conv},j}=\frac{1}{h\sum_{j=1}^{n_{S_{j}}}A_{j}}.$

From above, we now have a cluster-level coupled electro-thermal model. The
model has a low-order, compact structure and is computationally amenable to
subsequent power optimization.

###  III-C Inter-Cluster Optimal Control

Based on the cell clustering and cluster modeling, we are now in a good
position to deal with the optimal power management for the clusters.

####  III-C1 Problem formulation

For the considered large-scale BESS, our goal is to minimize the total power
losses in operation, while satisfying the physical, safety, and balancing
constraints and ensuring the continuous power supply to the load. In pursuit
of the goal, we develop an inter-cluster optimization problem. The total power
losses for the clusters within the time horizon $[t,t+H)$ are

$\int_{t}^{t+H}\bar{L}(\tau)d\tau,$ (17)

where $H$ is the horizon length and
$\bar{L}(t)=\sum_{j=1}^{k}\bar{R}_{j}\bar{i}_{L_{j}}^{2}(t)$. We further
impose constraints on the charging/discharging currents, temperatures, and SoC
of the clusters to guarantee their safe operation as follows:

$\displaystyle\bar{i}_{L_{j}}^{\min}$
$\displaystyle\leq\bar{i}_{L_{j}}\leq\bar{i}_{L_{j}}^{\max},$ (18a)
$\displaystyle\bar{T}_{j}^{\min}$
$\displaystyle\leq\bar{T}_{j}\leq\bar{T}_{j}^{\max},$ (18b)
$\displaystyle\bar{q}_{j}^{\min}$
$\displaystyle\leq\bar{q}_{j}\leq\bar{q}_{j}^{\max},$ (18c)

where $\bar{i}_{L_{j}}^{\textrm{min}/\textrm{max}}$,
$\bar{T}_{{}_{j}}^{\textrm{min}/\textrm{max}}$, and
$\bar{q}_{{}_{j}}^{\textrm{min}/\textrm{max}}$ are the upper/lower safety
bounds for the current, temperature, and SoC, respectively. Note that the
bounds for $\bar{i}_{L_{j}}$ depend on $n_{S_{j}}$, i.e.,
$\bar{i}_{L_{j}}^{\textrm{min}/\textrm{max}}=\sum_{i=1}^{n_{S_{j}}}i_{L_{i}}^{\textrm{min}/\textrm{max}}$,
where $i_{L_{i}}^{\textrm{min}/\textrm{max}}$ is the upper/lower current
bounds for cell $i$. We also enforce cluster-level SoC and temperature
balancing constraints as follows:

$\displaystyle\left|\bar{q}_{j}-\bar{q}_{\textrm{avg}}\right|$
$\displaystyle\leq\Delta\bar{q},$ (19a)
$\displaystyle\left|\bar{T}_{j}-\bar{T}_{\textrm{avg}}\right|$
$\displaystyle\leq\Delta\bar{T},$ (19b)

where $\Delta\bar{q}$ and $\Delta\bar{T}$ are the maximum allowed SoC and
temperature deviations among the clusters, respectively. The terms
$\bar{q}_{\textrm{avg}}$ and $\bar{T}_{\textrm{avg}}$ represent the average
SoC and temperature of all the clusters, respectively. Further, the following
constraint is introduced to ensure the power supply and demand balance:

$\sum_{j=1}^{k}\bar{P}_{S_{j}}-\bar{R}_{j}\bar{i}_{L_{j}}^{2}=P_{\textrm{out}},$
(20)

where $P_{\textrm{out}}$ is the output power demand. Collecting the cost
function and constraints, one can compactly express the optimal power
management problem as

$\begin{split}\min_{\bar{i}_{L_{j}},j=1,...,k}&\quad\int_{t}^{t+H}\bar{L}(t)dt,\\\
\textrm{s.t.}&\quad\eqref{ClusterLevelSoCDynamic},\eqref{ClusterLevelTempDynamic},\eqref{SafetyConstraints}\mbox{-}\eqref{PowerBalanceConstraint}.\end{split}$
(21)

This problem seeks to find out the best $\bar{i}_{L_{j}}$ for cluster $j$ for
$j=1,...k$ in a predictive manner over a receding horizon. However, it is a
non-convex optimization problem due to the nonlinearity of the equality
constraint in (20), thus resisting the search for the global optimum. We adopt
and modify the convexificaiton technique in [11] to overcome this issue.

####  III-C2 Convexification of (21)

To start with, we define

$\bar{E}_{j}(t)=\frac{1}{2}\bar{C}_{j}\bar{u}_{j}^{2}(\bar{q}_{j}(t))-\bar{E}_{j}^{0},$
(22)

where $\bar{E}_{j}(t)$ is the remaining energy of cluster $S_{j}$,
$\bar{C}_{j}=\bar{Q}_{j}/\bar{\beta_{j}}$, and
$\bar{E}_{j}^{0}=\frac{1}{2}\bar{C}_{j}\bar{u}_{j}^{2}(\bar{q}_{j}(0))$ is the
initial energy. Here, $\bar{E}_{j}$ is introduced to replace SoC for the
purpose of convexificaiton, as will be seen later. Given (9), (11) and (22),
the evolution of $\bar{E}_{j}$ is governed by

$\dot{\bar{E}}_{j}(t)=-\bar{P}_{S_{j}}.$ (23)

The power losses of cluster $S_{j}$ can also be expressed in terms of
$\bar{E}_{j}$ and $\bar{P}_{S_{j}}$ as

$\bar{P}_{l_{j}}=\frac{\bar{R}_{j}\bar{C}_{j}\bar{P}_{S_{j}}^{2}}{2(\bar{E}_{j}+\bar{E}_{j}^{0})}.$
(24)

In the convex formulation of the optimal power management problem, we control
$\bar{P}_{S_{j}}$ to minimize the total power losses of the clusters, so (24)
serves as a nonlinear equality constraint resulting in a non-convex problem.
Since the power loss term appears in the cost function of the problem, the
following relaxation can be considered:

$\bar{P}_{l_{j}}\geq\frac{\bar{R}_{j}\bar{C}_{j}\bar{P}_{S_{j}}^{2}}{2(\bar{E}_{j}+\bar{E}_{j}^{0})},$
(25)

where the optimization problem will reduce $\bar{P}_{l_{j}}$ to its lower
bound.

Proceeding forward, we can reformulate the safety constraints (18a) and (18c)
in terms of $\bar{E}_{j}$ and $\bar{P}_{S_{j}}$ as follows:

$\displaystyle\sqrt{\frac{2}{\bar{C}_{j}}(\bar{E}_{j}+\bar{E}_{j}^{0})}\bar{i}_{L_{j}}^{\textrm{min}}$
$\displaystyle\leq\bar{P}_{S_{j}}\leq\sqrt{\frac{2}{\bar{C}_{j}}(\bar{E}_{j}+\bar{E}_{j}^{0})}\bar{i}_{L_{j}}^{\textrm{max}},$
(26a)
$\displaystyle\frac{1}{2}\bar{C}_{j}\bar{u}_{j}^{2}(\bar{q}_{j}^{\textrm{min}}(t))$
$\displaystyle\leq\bar{E}_{j}+\bar{E}_{j}^{0}\leq\frac{1}{2}\bar{C}_{j}\bar{u}_{j}^{2}(\bar{q}_{j}^{\textrm{max}}(t)).$
(26b)

We also rewrite and modify the constraints in (19) as

$\displaystyle\Bigg{|}\frac{2}{\bar{C}_{j}}\bar{E}_{j}(t)-\frac{1}{k}\sum_{l=1}^{k}\frac{2}{\bar{C}_{l}}\bar{E}_{l}(t)\Bigg{|}$
$\displaystyle\leq\Delta\bar{E}_{j}+\bar{\xi}^{(E)}_{j},$ (27a)
$\displaystyle\left|\bar{T}_{j}(t)-\bar{T}_{\textrm{avg}}(t)\right|$
$\displaystyle\leq\Delta\bar{T}+\bar{\xi}^{(T)}_{j},$ (27b)

where
$\Delta\bar{E}_{j}=(\bar{\alpha}_{j}+\bar{\beta}_{j}\Delta\bar{q})^{2}-\bar{\alpha}_{j}^{2}$,
and $\bar{\xi}^{(E)}_{j}$ and $\bar{\xi}^{(T)}_{j}$ are respectively the
energy and temperature slack variables. The addition of the slack variables is
to help fix the potential infeasibility issue, which would happen when the
clusters face too large variations in their initial SoC and temperature [13].
The slack variables will also be included in the cost function to penalize
constraint violations.

Now, we are ready to introduce a convex inter-cluster power optimization
problem and present it in a discrete-time form for the sake of computation and
implementation. We denote by
$\bar{z}_{j}=\left[\begin{smallmatrix}\bar{P}_{S_{j}}&\bar{P}_{l_{j}}&\bar{E}_{j}&\bar{T}_{j}&\bar{\xi}_{j}^{(E)}&\bar{\xi}_{j}^{(T)}\end{smallmatrix}\right]^{\top}$
the vector of the optimization variables for $j=1,...,k$. We state the new
problem as

$\displaystyle\min_{\bar{z}_{j},j=1,...,k}\quad\sum_{t=0}^{H}\sum_{j=1}^{k}\bar{P}_{l_{j}}[t]+\bar{\lambda}^{(E)}\bar{\xi}^{(E)}_{j}[t]+\bar{\lambda}^{(T)}\bar{\xi}^{(T)}_{j}[t],$
(28) Safety constraints: $\displaystyle\
\sqrt{\frac{2}{\bar{C}_{j}}(\bar{E}_{j}[t]+\bar{E}_{j}^{0})}\bar{i}_{L_{j}}^{\textrm{min}}\leq\bar{P}_{S_{j}}[t]\leq\sqrt{\frac{2}{\bar{C}_{j}}(\bar{E}_{j}[t]+\bar{E}_{j}^{0})}\bar{i}_{L_{j}}^{\textrm{max}},$
$\displaystyle\bar{T}_{j}^{\textrm{min}}\leq\bar{T}_{j}\leq\bar{T}_{j}^{\textrm{max}}$
$\displaystyle\
\frac{1}{2}\bar{C}_{j}\bar{u}_{j}^{2}(\bar{q}_{j}^{\textrm{min}}[t])\leq\bar{E}_{j}[t]+\bar{E}_{j}^{0}\leq\frac{1}{2}\bar{C}_{j}\bar{u}_{j}^{2}(\bar{q}_{j}^{\textrm{max}}[t]),$
Balancing constraints: $\displaystyle\
\Bigg{|}\frac{2}{\bar{C}_{j}}\bar{E}_{j}[t]-\frac{1}{k}\sum_{l=1}^{t}\frac{2}{\bar{C}_{l}}\bar{E}_{l}[t]\Bigg{|}\leq\Delta\bar{E}_{j}+\bar{\xi}^{(E)}_{j}[t],$
$\displaystyle\
\left|\bar{T}_{j}[t]-\bar{T}_{\textrm{avg}}[t]\right|\leq\Delta\bar{T}+\bar{\xi}^{(T)}_{j}[t],$
Power loss constraint: $\displaystyle\
\bar{P}_{l_{j}}[t]\geq\frac{\bar{R}_{j}\bar{C}_{j}\bar{P}_{S_{j}}^{2}[t]}{2(\bar{E}_{j}[t]+\bar{E}_{j}^{0})},$
Energy dynamics: $\displaystyle\
\bar{E}_{j}[t+1]-\bar{E}_{j}[t]=-\bar{P}_{S_{j}}[t]\Delta t,$ Thermal
dynamics: $\displaystyle\ \bar{T}_{j}[t+1]=\bar{T}_{j}[t]+\frac{\Delta
t}{\bar{m}_{j}C_{\textrm{th}}}\Big{[}\bar{P}_{l_{j}}[t]-(\bar{T}_{j}[t]-T_{\textrm{env}})/\bar{R}_{\textrm{conv}}\Big{]},$
Power supply-demand balance: $\displaystyle\
\sum_{j=1}^{k}\bar{P}_{S_{j}}[t]-\bar{P}_{l_{j}}[t]=P_{\textrm{out}}[t],$

where $\bar{\lambda}^{(E)}$ and $\bar{\lambda}^{(T)}$ are the penalization
weights for $\bar{\xi}^{(E)}$ and $\bar{\xi}^{(T)}$, respectively. The
optimization problem in (28) is convex because the cost function and the
domain are both convex. The convexity would allow efficient computation of the
global optimum with well-known algorithms.

######  Remark 1.

Cell-based optimal power management for BESS has been investigated in
different studies, including our prior work [13], but the existing approach
face a significant computational bottleneck. For example, one must solve a
receding-horizon constrained optimization problem in [13] that is akin to (28)
but involves all cells. Its computation will quickly reach a formidable level
when the number of cells grows. Our cluster-based approach, as shown in (28),
will be computationally much cheaper. This is because it involves only $6Hk-k$
optimization variables and $11Hk+5k+H$ constraints with $k\ll n$, contrasting
with $6Hn-n$ variables and $11Hn+5n+H$ constraints in [13], as summarized in
Table I. While the clustering and power splitting will add some computation,
there is little compromise to the overall efficiency and scalability of our
approach.

TABLE I: Comparison of the Computational Complexity of (28)

Method | # of optimization variables | # of constraints
---|---|---
[13] | $6Hn-n$ | $11Hn+5n+H$
(28) in our approach | $6Hk-k$ | $11Hk+5k+H$
$k\ll n$ |  |

We emphasize that, while (28) attempts to balance the clusters in SoC and
temperature, it is of our interest to make the cells balanced as well. But
some cells might be slightly outside the desired balancing bounds due to the
aggregation in the cluster modeling. We address this issue by an adaptive
choice of $\Delta\bar{E}_{j}$ and $\Delta\bar{T}$, as discussed next.

####  III-C3 Adaptive cell balancing

The adaptive selection of balancing bounds has been examined within the
framework of two conditions: bound tightening and bound relaxing.

  * •

Bound tightening: To clarify why a cluster-level balancing constraint may not
effectively enforce cell-level balancing, Fig. 6 (a) illustrates an example in
which seven cells are grouped into two clusters, $S_{1}$ and $S_{2}$. The
aggregated temperatures of the clusters $S_{1}$ and $S_{2}$ are $\bar{T}_{1}$
and $\bar{T}_{2}$, respectively. The average of them is
$\bar{T}_{\textrm{avg}}$. Suppose that
$\left|\bar{T}_{i}-\bar{T}_{\textrm{avg}}\right|=\Delta\bar{T}$ for $i=1,2$ as
shown in Fig. 6. Even though the clusters are balanced in this case, some
cells (marked in red) are still outside of the desired balancing bounds. As
minor as the deviation might be, it is still preferable to maximize the cell-
level balance even in the cluster-level optimization procedure. The same
phenomenon may also appear in SoC balancing. To address this issue, we suggest
to dynamically modify $\Delta\bar{q}$ and $\Delta\bar{T}$ as follows:

$\displaystyle\Delta\bar{q}^{\prime}$
$\displaystyle=\Delta\bar{q}-\frac{1}{2}\max\\{d_{1}^{(q)},...,d_{k}^{(q)}\\},$
(29a) $\displaystyle\Delta\bar{T}^{\prime}$
$\displaystyle=\Delta\bar{T}-\frac{1}{2}\max\\{d_{1}^{(T)},...,d_{k}^{(T)}\\},$
(29b)

where $\Delta\bar{q}^{\prime}$ and $\Delta\bar{T}^{\prime}$ are the modified
cluster-level balancing bounds to replace $\Delta\bar{q}$ and $\Delta\bar{T}$
in (28), and $d_{j}^{(q/T)}$ is the maximum deviation of a cell’s
SoC/temperature from the centroid of cluster $S_{j}$. With (29), we
practically tighten the balancing bounds for the clusters so as to improve the
cell balancing.

  * •

Bound relaxing: This condition involves adjusting the bounds to expand the
permissible range of cluster-level balancing. As illustrated in Figure 6 (b),
the temperature distribution of cells within clusters $S_{1}$ and $S_{2}$
adheres to the desired bound of $\Delta\bar{T}$. To enhance power loss
minimization, the balancing bounds for the clusters are relaxed, providing the
optimization process with increased flexibility. The relaxation of bounds is
determined by utilizing (29), where $d_{j}^{(q,T)}\leq 0$.

Note that we do not update the balancing bounds at every time instant.
Instead, (29) is applied only when $\bar{\xi}_{j}^{(E)}$ and
$\bar{\xi}_{j}^{(T)}$ reach zero—at this moment, the clusters are balanced so
that it is tenable to narrow/relax the bounds. The adaptive cell balancing
procedure will be examined in the simulation results.

###  III-D Cluster-to-Cell Power Split

Suppose that the optimal power quota $\bar{P}_{S_{j}}^{*}$ is obtained for
cluster $j$ for $j=1,...k$ by solving the optimization problem in (28). The
remaining question is how to split $\bar{P}_{S_{j}}^{*}$ among the cells
within the cluster. We propose three schemes that derive from different
assumptions about the intra-cluster cell discrepancy.

Scheme #1: Equal power split. The first scheme assumes that the constituent
cells of the clusters are approximately identical, neglecting discrepancies
among them in terms of SoC, temperature and internal resistance. It is thus
sensible to equally divide the optimal power assigned to a cluster among the
constituent cells. For cluster $S_{j}$, we have

$P_{b_{1}}=...=P_{b_{n_{S_{j}}}}=\frac{\bar{P}_{S_{j}}^{*}}{n_{S_{j}}}.$ (30)

Scheme #2: Internal-resistance-based power split. The second scheme considers
the internal resistance variations among the cells and aims to reduce the
power losses by an uneven distribution of the cluster power. It divides
$\bar{P}_{S_{j}}^{*}$ by

$P_{b_{i}}=\frac{\frac{1}{R_{i}}\bar{P}_{S_{j}}^{*}}{\frac{1}{R_{1}}+\frac{1}{R_{2}}+...+\frac{1}{R_{n_{S_{j}}}}},\quad
i=1,...,n_{S_{j}}.$ (31)

From above, the less $R_{i}$ is, the larger $P_{b_{i}}$ is.

Compared to scheme #1, this scheme prioritizes loss minimization over SoC and
temperature balancing. However, both of them are based on heuristics, implying
suboptimal cell-level power assignment. One can perform intra-cluster
optimization to achieve optimal power split among the cells.

(a)  (b)

Figure 6: Adaptive cell balancing illustration. (a) Bound tightening. (b)
Bound relaxing.

Scheme #3: Optimal power split. We can apply the idea in (28) to achieve
optimal power split by minimizing the power losses of all the cells within the
cluster under some safety and SoC balancing constraints. Considering cluster
$S_{j}$, the total power losses can be expressed as

$L_{j}(t)=\sum_{l=1}^{n_{S_{j}}}(R_{l}+R_{C})i_{L_{l}}^{2}(t).$ (32)

We enforce the following constraints to ensure safety and intra-cluster
balancing:

$\displaystyle i_{L_{l}}^{\min}\leq i_{L_{l}}$ $\displaystyle\leq
i_{L_{l}}^{\max},$ (33a) $\displaystyle T_{l}^{\min}\leq T_{l}$
$\displaystyle\leq T_{l}^{\max},$ (33b) $\displaystyle q_{l}^{\min}\leq q_{l}$
$\displaystyle\leq q_{l}^{\max},$ (33c)
$\displaystyle\left|q_{l}-q_{j,\textrm{avg}}\right|$ $\displaystyle\leq\Delta
q,$ (33d) $\displaystyle\left|T_{l}-T_{j,\textrm{avg}}\right|$
$\displaystyle\leq\Delta T,$ (33e)

for $l=1,...,n_{S_{j}}$, where
$T_{l}^{\textrm{min}/\textrm{max}}=\bar{T}_{j}^{\textrm{min}/\textrm{max}}$
and
$q_{l}^{\textrm{min}/\textrm{max}}=\bar{q}_{j}^{\textrm{min}/\textrm{max}}$,
and $q_{j,\textrm{avg}}$ and $T_{j,\textrm{avg}}$ are the average SoC and
temperature for cluster $j$, respectively. Further, we must make the total
power of the cells equal to the cluster’s optimal power. That is,

$\sum_{l=1}^{n_{S_{j}}}\left(P_{b_{l}}-(R_{l}+R_{C})i_{L_{l}}^{2}\right)=\bar{P}_{S_{j}}^{*}-\bar{P}_{l_{j}}^{*}.$
(34)

We can then derive the intra-cluster optimization problem as:

$\begin{split}\min_{i_{L_{l}},l=1,...,n_{S_{j}}}&\quad L_{j}(t),\\\
\textrm{subject
to}&\quad\eqref{ClusterLevelSoCDynamic},\eqref{ClusterLevelTempDynamic},\eqref{IntraClusterSafety-
BalancingConstraints},\eqref{IntraClusterPowerBalanceConstraint}.\end{split}$
(35)

For the constituent cells of each cluster, the above nonlinear continuous
optimization problem aims to find out the best currents $i_{L_{l}}$ for
$l=1,\dots,n_{S_{j}}$. Note that (35), similar to (21), is non-convex due to
the nonlinearity of the equality constraint (34). We can follow the same
convexification procedure as in (28) to formulate an alternative convex
optimization problem to solve (35). Considering cluster $S_{j}$, the problem
is stated as below:

$\begin{split}&\min_{z_{l},l=1,...,n_{S_{j}}}\quad\sum_{l=1}^{n_{S_{j}}}P_{l_{l}}[t]+\lambda^{(E)}\xi^{(E)}_{l}[t]+\lambda^{(T)}\xi^{(T)}_{l}[t],\\\
&\textrm{Safety constraints:}\quad\\\ &\
\sqrt{\frac{2}{C_{l}}(E_{l}[t]+E_{l}^{0})}i_{L_{l}}^{\textrm{min}}\leq
P_{b_{l}}[t]\leq\sqrt{\frac{2}{C_{l}}(E_{l}[t]+E_{l}^{0})}i_{L_{l}}^{\textrm{max}},\\\
&T_{l}^{\textrm{min}}\leq T_{l}\leq T_{l}^{\textrm{max}}\\\ &\
\frac{1}{2}C_{l}u_{l}^{2}(q_{l}^{\textrm{min}}[t])\leq
E_{l}[k]+E_{l}^{0}\leq\frac{1}{2}C_{l}u_{l}^{2}(q_{l}^{\textrm{max}}[t]),\\\
&\textrm{Balancing constraints:}\quad\\\ &\
\Bigg{|}\frac{2}{C_{l}}E_{l}[t]-\frac{1}{n_{S_{j}}}\sum_{l=1}^{n_{S_{j}}}\frac{2}{C_{l}}E_{l}[t]\Bigg{|}\leq\Delta
E_{l}+\xi^{(E)}_{l}[t],\\\ &\
\left|T_{l}[t]-T_{\textrm{avg}}[t]\right|\leq\Delta T+\xi^{(T)}_{l}[t],\\\
&\textrm{Power loss constraint:}\quad\\\ &\
P_{l_{l}}[t]\geq\frac{(R_{l}+R_{C})C_{l}P_{b_{l}}^{2}[t]}{2(E_{l}[t]+E_{l}^{0})},\\\
&\textrm{Energy dynamics:}\quad\\\ &\ E_{l}[t+1]-E_{l}[t]=-P_{b_{l}}[t]\Delta
t,\\\ &\textrm{Thermal dynamics:}\quad\\\ &\ T_{l}[t+1]=T_{l}[t]+\frac{\Delta
t}{m_{l}C_{\textrm{th}}}\Big{[}P_{l_{l}}[t]-(T_{l}[t]-T_{\textrm{env}})/R_{\textrm{conv}}\Big{]},\\\
&\textrm{Power supply-demand balance:}\quad\\\ &\
\sum_{l=1}^{n_{S_{j}}}\left(P_{b_{l}}[t]-P_{l_{l}}[t]\right)=\bar{P}^{*}_{S_{j}}[t]-\bar{P}^{*}_{l_{j}}[t],\end{split}$
(36)

where
$z_{l}=\left[\begin{smallmatrix}P_{b_{l}}&P_{l_{l}}&E_{l}&T_{l}&\xi_{l}^{(E)}&\xi_{l}^{(T)}\end{smallmatrix}\right]^{\top}$
collects the optimization variables, $E_{l}$ denotes the remaining energy in
cell $l$, $\xi_{l}^{(E)}$ and $\xi_{l}^{(T)}$ represent the energy and
temperature slack variables with associated penalizing weights $\lambda^{(E)}$
and $\lambda^{(T)}$ in the cost function. We highlight that scheme #3 must
address the above intra-cluster optimization problems for each cluster and
thus requires more computation. However, each such problem involves only
$6n_{S_{j}}$ optimization variables, and independent of each other, they can
be parallelized in computing. Thus, scheme #3 can still be manageable if there
are enough computing resources.

##  IV Simulation Results

In this section, we carry out simulations for a BESS consisting of 400 cells.
Table II summarizes the specifications of the BESS and the parameters of the
proposed optimal power management approach. The battery cells are based on
Samsung INR18650-25R whose parameters have been identified and reported in
[31]. The profile of the output power $P_{\textrm{out}}$ is based on the Urban
Dynamometer Driving Schedule (UDDS) with the peak charging and discharging
power of 6 kW and 10 kW, respectively. The simulations simulate the BESS
operation for 2,400 seconds, in which the proposed approach runs with a
10-second predictive horizon. We use the CVX package [32, 33] with Matlab to
solve the convex optimization problems in (28) and (36) using a workstation
with a 3.5GHz Intel Core i9-10920X CPU and 128GB of RAM.

We initialize the cells by randomly sampling their initial SoC, temperature,
and internal resistance from the uniform distributions
$\mathcal{U}(0.7,0.75)$, $\mathcal{U}(301,305)\,\textrm{K}$, and
$\mathcal{U}(31.3,41.3)\,\textrm{m}\Omega$, respectively. The randomized
variations will lead to many possible combinations within the cells at the
clustering stage, allowing us to better investigate the efficacy of the
proposed approach in handling the cell heterogeneities. Further, we conduct
the simulations with schemes #1-3 for the cluster-to-cell power split to
compare their respective performances.

TABLE II: Specifications of the considered large-scale BESS

Symbol | Parameter | Value [Unit]
---|---|---
$n$ | Number of battery cells | 400
$v$ | Cell nominal voltage | 3.6 [V]
$Q$ | Cell nominal capacity | 2.5 [Ah]
$R$ | 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 | [-7.5,7.5] [A]
$C_{\textrm{th}}$ | Specific thermal capacitance | 918.49 [J/(K.Kg)]
$m$ | Cell Mass | 0.0438 [Kg]
$A$ | External surface area | 0.0042 [m2]
$h$ | Convection heat transfer coefficient | 5.8 [W/(K.m2)]
$R_{\textrm{conv}}$ | Convection thermal resistance | 41.05 [K/W]
$T_{\textrm{env}}$ | Environment temperature | 298 [K]
$\Delta\bar{q}$ | SoC balancing threshold | 0.5%
$\Delta\bar{T}$ | Temperature balancing threshold | 0.5 [K]
$\Delta t$ | Sampling time | 1 [s]
$H$ | Horizon length | 10 [s]

Fig. 7 depicts the SoC and temperature balancing performance of the proposed
approach. The proposed approach under three power split schemes effectively
alleviates the SoC and temperature unbalance among the cells and successfully
drives the cells within the desired bounds. Having a closer look at Figs. 7
(a)-(c), we see that schemes #1 and #2 show similar SoC balancing
performances, and the proposed approach drives the cells within the desired
bound after about 1,000 seconds. Scheme #3 shows superior performance, and the
cell balancing is achieved only after 700 seconds. A similar performance is
observed for the temperature balancing in Figs. 7 (d)-(f), where scheme #3
outperforms schemes #1 and #2 with a balancing time of 1,100 seconds versus
1,400 and 1,700 seconds. To conclude, schemes #1 and #2 provide acceptable
performance and high computational efficiency, and scheme #3 would deliver the
best balancing performance while requiring more computation. A user may choose
one of them in practice based on the performance expectations and computing
resources.

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

(e)  (f)

Figure 7: Simulation results of the SoC and temperature balancing. (a) The SoC
of the cells in scheme #1. (b) The SoC of the cells in scheme #2. (c) The SoC
of the cells in scheme #3. (d) The temperature of the cells in scheme #1. (e)
The temperature of the cells in scheme #2. (f) The temperature of the cells in
scheme #3. Figure 8: The power loss comparison. Figure 9: The total cumulative
power loss. Figure 10: Evolution in the number of clusters.

Fig. 8 shows the power loss profiles of the BESS under the proposed approach
with the three power split schemes, while comparing with the conventional case
of no cell-level power control. Overall, we observe the proposed approach
leads to lower power losses than in the conventional case. The magnified views
in the two time windows of Fig. 8 show that scheme #3 causes conspicuously
minimum power losses when the cells are unbalanced and that the three schemes
produce almost the same power losses when the cells are balanced. Fig. 9 shows
the total cumulative power losses. It is interesting to observe that the
proposed approach bears more power losses for the sake of cell balancing in
the early time interval of 0-200 seconds. However, its cumulative power loss
will become less as time goes by and lower by 4.6% than in the conventional
case at the end of the simulation run.

Fig. 10 illustrates the change in the number of clusters $k$ throughout the
simulation run. The value of $k$ depends on the level of variations in the
cells’ SOC, temperature, and internal resistance. The more balanced the cells,
the fewer the clusters. We can see that the cells are initially grouped into
17 clusters, and fewer clusters are formed as the cells become more uniform.
Note that the number of clusters also determines the computational costs of
the inter-cluster optimization in (28), so computation becomes less as the
simulation runs forward.

It is of our interest to observe the evolution of the cells’ SoC and
temperature. Fig. 11 provides snapshots from the cell’s SoC and temperature
under the three power split schemes at the time instants of 1,500, 1,000, and
2,000 seconds. The black boxes in the plots show the desired SoC and
temperature balancing bounds, and each color represents a cluster. The
proposed optimal power management approach effectively categorizes the cells
into clusters, performs inter-cluster optimization, and drives the cells into
the desired bound under all three power split schemes. Taking a closer look at
Fig. 11 (c) at the time instant of 2,000 seconds, one can see that the 400
cells form seven distinct clusters. However, the cells are more scattered at
the same time instant for schemes #1 and #2, while scheme #3 leads to better
intra-cluster convergence.

(a)  (b)  (c)

Figure 11: The evolution of cells’ SoC, temperature, and the clustering
procedure. (a) Scheme #1. (b) Scheme #2. (c) Scheme #3.

We further assess the roles of the slack variables and balancing bounds. Fig.
12 depicts the evolution of the slack variables over clusters
$\sum_{j=1}^{k}\bar{\xi}_{j}^{(E)}$ and $\sum_{j=1}^{k}\bar{\xi}_{j}^{(T)}$.
Note that the cells and the resulting clusters are not fully bounded by the
balancing constraints in (27). Because of this, the power optimization problem
in (28) would have been infeasible to solve even at the beginning time instant
without the introduction of the slack variables. However, the slack variables
help prevent the infeasibility issue from occurring. They are initially
nonzero to slightly relax the balancing constraints and then penalized to
decrease toward zero as the balancing constraints are increasingly satisfied.

While zero cluster-level slack variables suggest that the clusters should be
balanced, this does not guarantee that all the cells are within the desired
bounds, as aforementioned in Section III.C. Fig. 13 shows the utility of the
adaptive cell balancing bounds $\Delta\bar{q}$ and $\Delta\bar{T}$. We can see
that the original SoC and temperature balancing bounds are 0.5% and 0.5 K,
respectively. According to Fig. 12 (b), the temperature slack variables reach
zero at the time instant of 400 seconds. At this moment, the temperature
balancing bounds are tightened as shown in Fig. 13 (b), which leads to an
increase in the temperature slack variables. Fig. 12 (a) also indicates that
the SoC slack variables reach zero between the time steps of 750-900 seconds
for the three power split schemes. At the same time instants, we can see an
update in the SoC balancing bounds in Fig. 13 (a). We can conclude from the
evolution of the balancing bounds the usefulness of tighter bounds for the
cluster-level balancing to improve the uniformity at the cell level.

(a)  (b)

Figure 12: The evolution of the slack variables. (a) Energy slack variables.
(b) Temperature slack variables.

(a)  (b)

Figure 13: The evolution of the balancing bounds. (a) Energy balancing bounds.
(b) Temperature balancing bounds.

As a main objective of the study is to enable computationally efficient
optimal power management, we measure the computation time of the approach
under schemes #1 and #3 for different cell numbers and cluster numbers. Note
that scheme #2 is skipped as its computational performance is similar to that
of scheme #1. We also compare the proposed approach with cell-based power
optimization as in the works [13, 12, 10]. Table III summarizes the
computation time, with further illustration shown in Fig. 14 on a base-10
logarithmic scale. We observe several key findings. Firstly, the computation
time for scheme #1 remains independent of the cell number and depends only on
the number of clusters. Secondly, the proposed approach under scheme #3
requires more computation time than scheme #1 due to the optimal power split
among the constituent cells of the clusters. Finally, our proposed approach
exhibits substantially less computation time compared to the cell-level
optimization. The improvement is more significant when dealing with larger
numbers of cells. For instance, our proposed approach takes 6.76 and 19.43
seconds for schemes #1 and #3 with 15 clusters, less than 2% of the about 1200
seconds required by the cell-level optimization. The results validate the
computational efficiency and scalability of the proposed approach for large-
scale BESS.

TABLE III: Numerical comparison of the proposed approach and cell-level
optimization

Cell Number ($n$) |  Method |  Average Computation Time (s) |  Relative Computation Time Reduction (%)
---|---|---|---
50 | Cell-level optimization | 29.02 | —
Scheme #1 (15 clusters) | 6.69 | 76.94
Scheme #1 (10 clusters) | 4.75 | 83.63
Scheme #1 (5 clusters) | 2.74 | 90.55
Scheme #3 (15 clusters) | 10.49 | 63.85
Scheme #3 (10 clusters) | 7.55 | 73.98
Scheme #3 (5 clusters) | 5.99 | 79.35
100 | Cell-level optimization | 72.77 | —
Scheme #1 (15 clusters) | 6.67 | 90.83
Scheme #1 (10 clusters) | 4.75 | 93.47
Scheme #1 (5 clusters) | 2.69 | 96.30
Scheme #3 (15 clusters) | 11.71 | 83.90
Scheme #3 (10 clusters) | 8.61 | 88.16
Scheme #3 (5 clusters) | 7.65 | 89.48
200 | Cell-level optimization | 274.97 | —
Scheme #1 (15 clusters) | 6.70 | 97.56
Scheme #1 (10 clusters) | 4.75 | 98.27
Scheme #1 (5 clusters) | 2.76 | 98.99
Scheme #3 (15 clusters) | 14.13 | 94.86
Scheme #3 (10 clusters) | 10.61 | 96.14
Scheme #3 (5 clusters) | 10.68 | 96.11
400 | Cell-level optimization | 1200.45 | —
Scheme #1 (15 clusters) | 6.76 | 99.43
Scheme #1 (10 clusters) | 4.83 | 99.59
Scheme #1 (5 clusters) | 2.79 | 99.76
Scheme #3 (15 clusters) | 19.43 | 98.38
Scheme #3 (10 clusters) | 14.46 | 98.79
Scheme #3 (5 clusters) | 16.80 | 98.60

Figure 14: Comparative analysis of computation times: the proposed approach
with schemes #1 and #3 versus the cell-level optimization.

##  V Experimental Results

We develop a lab-scale prototype to validate the proposed optimal power
management approach. The experimental validation encompasses various aspects,
including the evaluation of cell clustering, model aggregation, optimization,
and cluster-to-cell power split. Fig. 15 (a) depicts the experimental setup,
and Fig. 15 (b) shows a 20-cell 4s5p battery pack. Table IV lists the
specifications of the key components of the battery pack. We use K-type
thermocouples to measure the surface temperatures of the cells. The
thermocouples are connected to a National Instruments PCIe-6321 DAQ board, and
the measurements are collected via LabVIEW. We solve the optimization problem
using the CVX package [32, 33], and the optimal control decisions are
transferred to local controllers through DSP TMS320F28335. The local
controllers generate 200-kHz switching signals for the DC/DC converters. We
use a 150-W power load to discharge the battery pack. The experiment lasts
thirty minutes with a step size of thirty seconds for the optimization, i.e.,
$\Delta t=30$ s.

The cells’ initial SoC ranges from 75% to 80%, whereas their initial
temperature is 21.7°C. The cells’ output discharging current is limited to be
no more than 5 A. In the experiment, we also examine the performance of the
three cluster-to-cell power split schemes. Because the constituent cells are
identical, scheme #2 will reduce to scheme #1, and both will generate the same
results. Therefore, we will apply only schemes #1 and #3 to do the power split
in the experiment. The experimental results are given in Figs. 16-17.

(a)  (b)

Figure 15: The lab-scale prototype. (a) The experimental setup. (b) The circuit. TABLE IV: List of Key Components in the Experimental Setup Device | Model (Value)
---|---
MOSFET | CSD86356Q5D
Gate driver | TPS28225
Inductor | SER2915H-333KL (33 $\mu$H)
Capacitor | (10 $\mu$F)
Local controller | STM8S003F3P6
Main controller | TMS320f28335
Battery cell | Samsung INR18650-25R

Fig. 16 (a) shows the cells’ SoC profiles under scheme #1. We can observe that
the initial SoC values of the cells are not balanced and do not lie within the
desired bound. However, the proposed approach successfully distributes the
output power among different cells to drive the cells’ SoC to enter the
tolerable bounds. Fig. 16 (b) illustrates the SoC difference of the cells from
the average for a better visualization of the SoC balancing trend. It can be
seen that scheme #1 manages to balance cells’ SoC after about 900 seconds.
Fig. 16 (c) and Fig. 16 (d) depict the cells’ temperature profiles and the
differences from the average, respectively. The cells see their temperatures
drift away from each other due to the uneven power allocation among the cells,
but the deviation is bounded within the tolerance range. Note that, under
scheme #1, the cells are categorized to three to six clusters during the
experiment.

Fig. 16 (e) and Fig. 16 (f) show the cells’ SoC profiles and the deviation
from the average under scheme #3. In this case, the SoC is balanced among the
cells after about 750 seconds, faster than when scheme #1 is applied, because
of the optimal intra-cluster power split. Fig. 16 (g) and Fig. 16 (h) also
depict the temperature of the cells and the temperature difference from the
average, respectively. Similar to Figs. 16 (c)-(d), the cells’ temperatures
diverge mildly from the same initial point, but the divergence is almost
constantly bounded throughout the experiment. There is a minor violation in
the temperature balancing constraint around 70-80 seconds which is allowed by
the slack variables for the sake of the feasibility of the optimization
problem. This experiment finds a maximum of four clusters, fewer than under
scheme #1.

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

(e)  (f)  (g)  (h)

Figure 16: Experimental results of the SoC and temperature balancing. (a) The
SoC of the cells in scheme #1. (b) The SoC difference of the cells from the
average in scheme #1. (c) The temperature of the cells in scheme #1. (d) The
temperature difference of the cells from the average in scheme #1. (e) The SoC
of the cells in scheme #3. (f) The SoC difference of the cells from the
average in scheme #3. (g) The temperature of the cells in scheme #3. (h) The
temperature difference of the cells from the average in scheme #3.

(a)  (b)

Figure 17: The output power profiles of the cells. (a) Scheme #1. (b) Scheme
#3.

Figs. 17 (a)-(b) illustrate the output profiles of the cells under schemes #1
and #3, respectively. Both show the cells are assigned different power levels
based on their conditions, and after the balancing is achieved eventually,
they are assigned the same power. Yet, there is an interesting difference in
Figs. 17 (a)-(b). Scheme #1 distributes the total power load equally among the
cells within a cluster; by contrast, scheme #3 performs intra-cluster power
split in an optimal way, leading to more variability in power allocation, but
faster adjustment, among the cells.

##  VI Conclusion and Future Work

The rapidly growing use of large-scale BESS is transforming grid,
transportation, and more sectors towards decarbonization and a sustainable
future. However, an open challenge for this technology is optimal power
management, which struggles with high computational complexity in
optimization. Thus, this paper proposes a computationally efficient and
scalable approach to deal with the challenge. The approach distinguishes
itself from the literature by enabling clustering-based optimization. It
partitions a large number of cells into just a much smaller number of clusters
based on their characteristics and develops aggregated electro-thermal models
for the clusters. Inter-cluster power management then naturally arises to
compute the power quota among the clusters at considerably high computational
efficiency. Intra-cluster power split then divides the power quota among the
cells within each cluster under three different schemes. We further propose
adaptive cell balancing bounds to improve the cell-level control performance.
The proposed approach is designed to perform cell balancing in terms of SoC
and temperature while minimizing the total power losses. We conduct
simulations for a 400-cell BESS to validate the computational efficiency and
scalability of the proposed approach. The results show a substantial reduction
in computational overhead, by margins as large as 98% for a 400-cell BESS when
compared to conventional cell-level optimization. Further, we develop a lab-
scale 20-cell prototype for practical validation. The results show that the
approach can effectively reduce power losses, enhance cell balancing, and
require much less computational costs. The study can be expanded in several
directions. Our future work will include extending the proposed approach to
second-life battery systems, incorporating more sophisticated battery models,
and studying multi-level clustering for stronger scalability.

## References

  * [1] W. Zhou, N. Zhang, and H. Zhai, “Enhanced battery power constraint handling in MPC-based HEV energy management: A two-phase dual-model approach,” _IEEE Transactions on Transportation Electrification_ , vol. 7, no. 3, pp. 1236–1248, 2021.
  * [2] M. Starke, R. S. K. Moorthy, A. Adib, B. Dean, M. Chinthavali, B. Xiao, and S. Campbell, “A MW scale charging architecture for supporting extreme fast charging of heavy-duty electric vehicles,” in _IEEE Transportation Electrification Conference and Expo_ , 2022, pp. 485–490.
  * [3] A. Bills, S. Sripad, W. L. Fredericks, M. Guttenberg, D. Charles, E. Frank, and V. Viswanathan, “Universal battery performance and degradation model for electric aircraft,” _arXiv preprint arXiv:2008.01527_ , 2020.
  * [4] Y. Liu, Z. Yang, X. Wu, D. Sha, F. Lin, and X. Fang, “An adaptive energy management strategy of stationary hybrid energy storage system,” _IEEE Transactions on Transportation Electrification_ , vol. 8, no. 2, pp. 2261–2272, 2022.
  * [5] Y. Wang, F. Xu, S. Mao, S. Yang, and Y. Shen, “Adaptive online power management for more electric aircraft with hybrid energy storage systems,” _IEEE Transactions on Transportation Electrification_ , vol. 6, no. 4, pp. 1780–1790, 2020.
  * [6] M. Shen and Q. Gao, “A review on battery management system from the modeling efforts to its multiapplication and integration,” _International Journal of Energy Research_ , vol. 43, no. 10, pp. 5042–5075, 2019.
  * [7] M. Preindl, C. Danielson, and F. Borrelli, “Performance evaluation of battery balancing hardware,” in _European Control Conference_ , 2013, pp. 4065–4070.
  * [8] R. Gu, P. Malysz, M. Preindl, H. Yang, and A. Emadi, “Linear programming based design and analysis of battery pack balancing topologies,” in _IEEE Transportation Electrification Conference and Expo_ , 2015, pp. 1–7.
  * [9] J. V. Barreras, C. Pinto, R. de Castro, E. Schaltz, S. J. Andreasen, and R. E. Araújo, “Multi-objective control of balancing systems for li-ion battery packs: A paradigm shift?” in _IEEE Vehicle Power and Propulsion Conference_ , 2014, pp. 1–7.
  * [10] C. Pinto, J. V. Barreras, E. Schaltz, and R. E. Araújo, “Evaluation of advanced control for Li-ion battery balancing systems using convex optimization,” _IEEE Transactions on Sustainable Energy_ , vol. 7, no. 4, pp. 1703–1717, 2016.
  * [11] N. Murgovski, L. Johannesson, and J. Sj$\ddot{\textrm{o}}$berg, “Convex modeling of energy buffers in power control applications,” _IFAC Proceedings Volumes_ , vol. 45, no. 30, pp. 92–99, 2012, 3rd IFAC Workshop on Engine and Powertrain Control, Simulation and Modeling.
  * [12] R. de Castro, C. Pinto, J. Varela Barreras, R. E. Araújo, and D. A. Howey, “Smart and hybrid balancing system: Design, modeling, and experimental demonstration,” _IEEE Transactions on Vehicular Technology_ , vol. 68, no. 12, pp. 11 449–11 461, 2019.
  * [13] A. Farakhor, D. Wu, Y. Wang, and H. Fang, “A novel modular, reconfigurable battery energy storage system: Design, control, and experimentation,” _IEEE Transactions on Transportation Electrification_ , vol. 9, no. 2, pp. 2878–2890, 2023.
  * [14] R. de Castro, H. Pereira, R. E. Araújo, J. V. Barreras, and H. C. Pangborn, “qTSL: A multilayer control framework for managing capacity, temperature, stress, and losses in hybrid balancing systems,” _IEEE Transactions on Control Systems Technology_ , vol. 30, no. 3, pp. 1228–1243, 2022\.
  * [15] F. Altaf, B. Egardt, and L. Johannesson Mårdh, “Load management of modular battery using model predictive control: Thermal and state-of-charge balancing,” _IEEE Transactions on Control Systems Technology_ , vol. 25, no. 1, pp. 47–62, 2017.
  * [16] T. Morstyn, M. Momayyezan, B. Hredzak, and V. G. Agelidis, “Distributed control for state-of-charge balancing between the modules of a reconfigurable battery energy storage system,” _IEEE Transactions on Power Electronics_ , vol. 31, no. 11, pp. 7986–7995, 2016.
  * [17] Q. Ouyang, Z. Wang, K. Liu, G. Xu, and Y. Li, “Optimal charging control for lithium-ion battery packs: A distributed average tracking approach,” _IEEE Transactions on Industrial Informatics_ , vol. 16, no. 5, pp. 3430–3438, 2020.
  * [18] Q. Ouyang, J. Chen, J. Zheng, and H. Fang, “Optimal cell-to-cell balancing topology design for serially connected lithium-ion battery packs,” _IEEE Transactions on Sustainable Energy_ , vol. 9, no. 1, pp. 350–360, 2018\.
  * [19] A. Farakhor, Y. Wang, D. Wu, and H. Fang, “Distributed optimal power management for battery energy storage systems: A novel accelerated tracking ADMM approach,” in _American Control Conference_ , 2023, pp. 3106–3112.
  * [20] F. D’Agostino, M. Gallo, M. Saviozzi, and F. Silvestro, “A security-constrained optimal power management algorithm for shipboard microgrids with battery energy storage system,” in _IEEE International Conference on Electrical Systems for Aircraft, Railway, Ship Propulsion and Road Vehicles & International Transportation Electrification Conference_, 2023, pp. 1–6.
  * [21] S. Nasiri, M. Parniani, and S. Peyghami, “A multi-objective optimal power management strategy for enhancement of battery and propellers lifespan in all-electric ships,” _Journal of Energy Storage_ , vol. 65, p. 107183, 2023\.
  * [22] S. Wen, X. Jin, Y. Zheng, and M. Wang, “Probabilistic coordination of optimal power management and voyage scheduling for all-electric ships,” _IEEE Transactions on Transportation Electrification_ , pp. 1–1, 2023.
  * [23] S. F. da Silva, J. J. Eckert, F. L. Silva, F. C. Corrêa, L. C. Silva, A. V. Bueno, and F. G. Dedini, “Aging-aware optimal power management control and component sizing of a fuel cell hybrid electric vehicle powertrain,” _Energy Conversion and Management_ , vol. 292, p. 117330, 2023.
  * [24] A. Farakhor and H. Fang, “A novel modular, reconfigurable battery energy storage system design,” in _47th Annual Conference of the IEEE Industrial Electronics Society_ , 2021, pp. 1–6.
  * [25] A. Likas, N. Vlassis, and J. J. Verbeek, “The global k-means clustering algorithm,” _Pattern Recognition_ , vol. 36, no. 2, pp. 451–461, 2003.
  * [26] L. Duan, L. Xu, F. Guo, J. Lee, and B. Yan, “A local-density based spatial clustering algorithm with noise,” _Information Systems_ , vol. 32, no. 7, pp. 978–986, 2007.
  * [27] K. P. Sinaga and M.-S. Yang, “Unsupervised k-means clustering algorithm,” _IEEE Access_ , vol. 8, pp. 80 716–80 727, 2020.
  * [28] A. J. Onumanyi, D. N. Molokomme, S. J. Isaac, and A. M. Abu-Mahfouz, “Autoelbow: An automatic elbow detection method for estimating the number of clusters in a dataset,” _Applied Sciences_ , vol. 12, no. 15, 2022.
  * [29] R. Tibshirani, G. Walther, and T. Hastie, “Estimating the number of clusters in a data set via the gap statistic,” _Journal of the Royal Statistical Society: Series B (Statistical Methodology)_ , vol. 63, no. 2, pp. 411–423, 2001\.
  * [30] H. He, R. Xiong, and J. Fan, “Evaluation of lithium-ion battery equivalent circuit models for state of charge estimation by an experimental approach,” _Energies_ , vol. 4, no. 4, pp. 582–598, 2011.
  * [31] N. Tian, Y. Wang, J. Chen, and H. Fang, “One-shot parameter identification of the Thevenin’s model for batteries: Methods and validation,” _Journal of Energy Storage_ , vol. 29, p. 101282, 2020.
  * [32] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.1,” http://cvxr.com/cvx, Mar. 2014.
  * [33] ——, “Graph implementations for nonsmooth convex programs,” in _Recent Advances in Learning and Control_ , ser. Lecture Notes in Control and Information Sciences, V. Blondel, S. Boyd, and H. Kimura, Eds. Springer-Verlag Limited, 2008, pp. 95–110.

|  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.
candidate 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.
---|---