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IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 7, NO. 1, JANUARY 2016 189

Balanced Control Strategies for InterconnectedHeterogeneous Battery Systems

Le Yi Wang, Fellow, IEEE, Caisheng Wang, Senior Member, IEEE, George Yin, Fellow, IEEE,Feng Lin, Fellow, IEEE, Michael P. Polis, Life Senior Member, IEEE, Caiping Zhang, Senior Member, IEEE,

and Jiuchun Jiang, Senior Member, IEEE

Abstract—This paper develops new balanced charge/dischargestrategies that distribute charge or discharge currents prop-erly so that during operation, battery systems maintain uni-form state-of-charge (SOC) all the time. The proposed balancedcharge/discharge control strategies are useful for interconnectedheterogeneous battery systems that can be built from battery mod-ules with different types, ages, and power/capacity ratings. Bothvoltage-based and SOC-based balanced charge/discharge strate-gies are developed. Their convergence properties are rigorouslyestablished, and illustrative examples using production batteriesdemonstrate their convergence behavior under different charg-ing current profiles. The approach will be especially useful forbattery storage systems to support power grids with renewableenergy sources where the battery systems are required to operatecontinuously.

Index Terms—Balanced charge/discharge strategy, battery sys-tem, smart grid, sustainable energy development.

I. INTRODUCTION

A LL ELECTRIC or hybrid electric vehicles (EVs orHEVs, which will be collectively called electric drive

vehicles or EDVs) employ batteries that are composed of indi-vidual cells being strung together to form the battery pack forthe vehicle. It is well understood that unbalanced cells in astring limit the system’s useful capacity [1]. Thus, almost allcurrent EDV battery management systems (BMSs) employ bal-ancing/equalization circuits that use either passive energy dis-posal or active energy shuffling methods [2], [3]. Reference [4]contains a survey of cell balancing methods in production bat-tery systems. Since EDV usage inherently involves idle time(parking), battery cell balancing can be performed when thevehicle is not in use. Furthermore, to avoid power loss and

Manuscript received March 29, 2015; revised August 16, 2015; acceptedSeptember 28, 2015. Date of publication October 26, 2015; date of currentversion December 11, 2015. This work was supported in part by the NationalScience Foundation under Grant ECCS 1507096. Paper no. TSTE-00221-2015.

L. Y. Wang, C. Wang, and F. Lin are with the Department of Electricaland Computer Engineering, Wayne State University, Detroit, MI 48202 USA(e-mail: [email protected]; [email protected]; [email protected]).

G. Yin is with the Department of Mathematics, Wayne State University,Detroit, MI 48202 USA (e-mail: [email protected]).

M. P. Polis is with the Department of Industrial and Systems Engineering,Oakland University, Rochester, MI 48309 USA (e-mail: [email protected]).

C. Zhang and J. Jiang are with the National Active Distribution NetworkTechnology Research Center, Beijing Jiaotong University, Beijing 100044,China (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSTE.2015.2487223

reduce costs, usually battery cells are selected to be nearly uni-form and cell balancing currents are small. Consequently, cellbalancing usually only needs to correct small cell imbalances.

As EDVs gain more market penetration, old EDV batter-ies that are no longer suitable for vehicle applications canstill have substantial (up to 75%) capacity left [5]. One mil-lion retired 15-kWh/40-kW EDV batteries with an averageof 50% remaining power and energy capability can provide7500 MWh of energy capacity and 20 000 MW of power capac-ity, a huge waste if not utilized in secondary applications. Asdistributed renewable energy sources such as solar and wind areincreasingly being integrated into power distribution systems,integrating energy storage devices into the grid becomes highlydesirable and, in some cases, mandatory. It is well recognizedthat used EDV batteries will become a major part of stationaryenergy storage devices for power distribution grid support [6],[7]. Consequently, for distribution network power quality man-agement, battery systems will play an increasingly importantrole in microgrids and distribution networks with renewablegenerators such as photovoltaic generation systems and windfarms [8]. Furthermore, the introduction of dc microgrids [9],[10], [11] will promote more and wider applications of batterysystems in future electric energy systems.

However, unlike EDVs, in stationary usage to support gridoperations, battery systems are used continuously without idletime. Also, old battery packs have much larger variations intheir capacities and internal parameters, and as such their bal-anced operation is far more critical and challenging. Desirablemanagement schemes must provide the following vital func-tions: battery modules must be charged or discharged in abalanced manner all the time so that battery capacities can befully utilized and overcharge/overdischarge is avoided.

Consider the typical scenario of battery systems that supportmicrogrid operations [1], [12], [13], shown in Fig. 1. The bat-tery modules are mostly stationary and stored in distributedlocations, but involve EDV batteries as part of the intercon-nected battery systems. In such applications, battery modulesare naturally of different types, capacities, and ages. These bat-tery modules are interconnected to form a supporting energystorage system. Charge/discharge control and coordination ofsuch systems are the main functions studied in this paper.

This paper develops new balanced charge/discharge strate-gies that distribute charge or discharge currents properly sothat during operations battery modules maintain balanced state-of-charge (SOC) continuously without any energy shuffling

1949-3029 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

190 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 7, NO. 1, JANUARY 2016

Fig. 1. Distributed battery system for supporting microgrid operation.

among the batteries or energy disposal. We term such strate-gies balanced charge/discharge control strategies for inter-connected battery systems. The basic building blocks for theinterconnected battery system are (new or used) battery mod-ules. Each module may have its own BMS including internalcell balancing, charge/discharge protection, thermal manage-ment, charge/discharge rate control, etc. Here, we focus oncharge/discharge coordination of battery modules in their inter-connected system networks. Implementation of the proposedbalanced charge/discharge control strategies requires reliableestimation of the internal parameters and SOC of each bat-tery module. We assume that the joint estimation methodologiesintroduced and fully developed in our recent papers [14]–[17],are employed to obtain this information.

Cell balancing is an essential BMS function, especially forLi-ion batteries [2], [18], which aims to reduce SOC imbal-ances within a string of cells by controlling the SOCs of thecells so that they become approximately equal. This can beachieved by dissipating energy from the cells of higher SOCs toshunt resistors, or shuffling energy from the highest SOC cell tothe lowest SOC cell, or by incremental cell balancing throughpaired cells in stages [1]–[3]. Methods for cell equalizationor battery balancing operation have been studied extensivelyand many have been commercially employed in EV batteries[19], [20]. Reference [21] developed a power electronics topol-ogy and its switching control method to achieve cell voltagebalancing. Reference [22] introduced switched capacitors toenhance balancing performance. Reference [23] introduced acell equalization method based on sequential polling of batterycells with SOC estimators based on an extended Kalman filterand Coulomb counting. The method enjoys low-computationalcomplexity. Reference [24] considered parallel-connected bat-tery modules and studied impact of unequal charging strategiesthrough switches controlled by deterministic dynamic pro-gramming (DDP) and DDP-inspired heuristic algorithms onanode-side film formation. This paper targets serial-connectedbattery modules of highly diversified capacities and demandscontinuous balanced operations. As a result, a coordinatedcontrol strategy for networked battery systems is required. A

Fig. 2. Battery system with heterogeneous battery modules.

coordination-type control for networked systems is developedin this paper.

Subsystem coordination algorithms have been investigatedand applied in various aspects of power and energy manage-ment, including power system scheduling and optimal powerdispatch [25], power system estimation [26], voltage regula-tion [27]–[29], distributed power management [30], [31], andvoltage management of distributed solar energy generators [32].

The main contributions of this paper are as follows.1) A new framework for continuous balanced charge/

discharge operation is developed for interconnected het-erogeneous battery systems.

2) New feedback control strategies of average consensustypes are introduced which can achieve and maintainSOC balance continuously during operation under anycharge/discharge profiles.

3) The strategies are verified on both voltage-based andSOC-based models, by combining them with model iden-tification and SOC estimation algorithms.

This paper is organized as follows. Section II sets upthe main configurations of battery systems under study.Section III presents control and coordination strategies for mod-ule charge/discharge in a string. Owing to the serial connectiontopology, open-loop control strategies are shown to be non-convergent. A feedback scheme is introduced that achievesconvergence to the balanced state and maintains such a stateafterward. These results are then extended to more desirableSOC-based balanced charge/discharge under more accuratenonlinear models for battery systems. Section IV contains acase study on the control strategies by using some produc-tion battery systems. Finally, Section V summarizes the mainfindings of this paper, discusses some potential implementationissues, and points out a few worthy research directions.

II. PRELIMINARIES

A. Systems of Interconnected Heterogeneous Battery Modules

In this paper, we consider a networked battery system thatis built from battery modules, shown in Fig. 2. The system

WANG et al.: BALANCED CONTROL STRATEGIES FOR INTERCONNECTED HETEROGENEOUS BATTERY SYSTEMS 191

Fig. 3. String model structure. For the jth battery module, Rj and Cj are theequivalent parameters of the circuit model; vj is the internal voltage, which

represents the SOC; voutj is the battery terminal voltage; and SWjb is the bypass

switch that can be realized by power electronics.

Fig. 4. Typical battery characterizing charge/discharge curves between theOCV and SOC during a charge operation or between the OCV and depth ofdischarge (DOD) during a discharge operation, and their circuit representation.

consists of K strings. Each string has its own bi-directionaldc/dc converters to match the dc bus voltage specification. Eachstring consists of m modules which are serially connected,shown in Fig. 3. The number m may take different values fromstring to string. In consideration of battery modules of differenttypes, ages, and capacities, they are heterogeneous.

B. Simplified Battery Models and Their CircuitRepresentations

For simplicity and clarity, we start with simplified circuitmodels for battery modules, although our methods are not con-fined to such representations. Typical charge/discharge char-acteristic curves are shown in Fig. 4. Such curves representrelationships between the level of charge and the open-circuitvoltage v0 in a battery module [1], [18], [33], [34].

Suppose that the battery maximum capacity is Q (Ah). TheSOC, denoted by s(t), is

s(t) = i(t)/(3600Q). (1)

To avoid overcharge or overdischarge, the normal batteryoperations are in the middle range. In this range, the mappingis approximated by a straight line. Referring to the symbols inFig. 4, in the nominal operating range

1

Ev0 = i/Q, v = v0 +Ri. (2)

In other words, the voltage–current dynamic relationship canbe represented by a capacitor of capacitance C = Q/E (Ah/V)or equivalently C = 3600Q/E (F), leading to

v0 = i/C. (3)

During operation, Q and R depend on SOC and temperature,and can be identified in real time [16], [17].

In more realistic and accurate battery modeling, nonlinearand higher order models can be used and can be identifiedwith parameter and SOC estimation algorithms to update modelcharacteristics and obtain SOC estimates. These will be coveredin Sections III-D and IV. By comparing (1) and (3), voltage-based algorithms can be converted to SOC-based algorithms byreplacing v0 by s and C by Q (after a unit conversion).

Charge/discharge control for modules within a string isachieved by PWM control of bypass switches, shown in Fig. 3.When a bypass switch SWj

b is ON, the jth module is OFF

line. We shall use pj to indicate the state of SWjb: If pj = 1,

SWjb is ON; and if pj = 0, SWj

b is OFF. The string dc/dc con-verter serves the purpose of power conditioning so that theconverter output is controlled to satisfy the quality requirementsfor dc-bus connection.

In our methodology development, switches are assumed tobe ideal and instantaneous. In practical implementation, fastswitching will introduce power losses and other issues. Thecontrol structure of this paper does not require fast switching.As long as it is comparable to the charge/discharge time scale(which is in the scale of minutes and hours), it is sufficient.

III. BALANCED CHARGE/DISCHARGE CONTROL

STRATEGIES

Under a charge/discharge operation, the dynamic models forthe modules in a string are{

Cj vj = (1− pj)i, j = 1, . . . ,m

voutj = (1− pj)(vj +Rji).

Let the complementary duty cycle (namely, 1− duty cycle) forSWj

b be denoted by δj = 1− pj . By the standard averagingmethod [35] and by using the same notation for the voltages,we obtain

Cj vj = δji, voutj = δj(vj +Rji), j = 1, . . . ,m.

Let δ = [δ1, . . . , δm]′ be the control variables with the con-straints 0 ≤ δj ≤ 1, j = 1, . . . ,m, the state x = [v1, . . . , vm]′,and M = diag[C1, . . . , Cm]. Then

x = M−1δi. (4)


Denote 1l = [1, 1, . . . , 1]′ and the state average λ(t) =1m

∑mi=1 vi =

1m1l′x(t), where z′ denotes the transpose for a

matrix or vector z. The goal of balanced control strategiesis to achieve convergence x(t)− λ(t)1l → 0 during a balancereaching phase from an unbalanced initial condition, or sustainx(t) = λ(t)1l during a balance maintenance phase with a bal-anced initial condition. This control problem may be viewedas a coordination or consensus-type control problem [30], [31],[36], [37]. However, due to its special control constraints andnonlinear structures with respect to the duty cycles, the resultsof this paper cannot be derived from the existing literature onconsensus control problems for linear networked systems.

A. Balance Maintenance Phase

Suppose that the state is balanced at t0, x(t0) = λ(t0)1l.Control strategies in this phase aim to maintain x(t) =λ(t)1l ∀t ≥ t0. We first consider an open-loop control strategy:Let Cmax = maxj=1,...,m Cj , and

δ(t) ≡ δ∗ =1

Cmax[C1, . . . , Cm]′. (5)

Proposition 1: Suppose that x(t0) = λ(t0)1l. Under the con-stant control strategy (5), the state balance is maintained forall t ≥ t0, namely x(t) = λ(t)1l, t ≥ t0 and λ(t) = λ(t0) +∫ t

t0

i(τ)Cmax

dτ .

Proof: Under (5), vj =δjCj

i = iCmax

, j = 1, . . . ,m.

This implies that x(t) = x(t0) +∫ t

t0

i(τ)Cmax

dτ1l = (λ(t0)+∫ t

t0

i(τ)Cmax

dτ)1l. This proves the theorem. �

B. Balance Reaching Phase

The goal of this phase is to design a control strategy forδ(t) in (4) such that starting from an unbalanced state x(t0) �=λ(t0)1l, the convergence to consensus can be achieved: x(t)−λ(t)1l → 0, as t → ∞.

We proceed to show that the open-loop strategy (5) is not aconvergent strategy for the balance reaching phase. Define thebalance error ε(t) = x(t)− λ(t)1l = [ε1(t), . . . , εm(t)]′. Notethat λ(t) = 1

m1l′x(t) is the average of x(t).Proposition 2: Under the constant control strategy (5), if the

initial balance error ε(t0) �= 0, then ε(t) = ε(t0) �= 0, t ≥ t0.

Proof: Under (5), x(t) = x(t0) +∫ t

t0

i(τ)Cmax

dτ1l. This impliesthat

λ(t) =1

m1l′x(t)

=1

m1l′(x(t0) +

∫ t

t0

i(τ)

Cmaxdτ1l

)=

1

m1l′x(t0) +

1

m1l′∫ t

t0

i(τ)

Cmaxdτ1l

= λ(t0) +

∫ t

t0

i(τ)

Cmaxdτ.

Consequently

ε(t) = x(t)− λ(t)1l

= x(t0) +

∫ t

t0

i(τ)

Cmaxdτ1l−

(λ(t0) +

∫ t

t0

i(τ)

Cmaxdτ

)1l

= x(t0)− λ(t0)1l

= ε(t0).

We now introduce a modified nonlinear feedback con-trol strategy. In the following derivations, for a vector x =[x1, x2, . . . , xm]′, denote ‖x‖∞ = maxi |xi|. Since the con-trol variables are duty cycles, they must be confined in therange [0, 1]. A projection (component-wise bounding) opera-tor Π is used to ensure that this constraint is always met: Forx = [x1, x2, . . . , xm]′

Π(xj) =

⎧⎪⎨⎪⎩xj , 0 ≤ xj ≤ 1

0, xj < 0

1, xj > 1.

The second projection (also component-wise bounding) oper-ator is on the error signal εj to limit control action. For somepreselected 0 < b < 1

Πb(εj) =

⎧⎪⎨⎪⎩εj , |εj | ≤ b

−b, εj < −b

b, εj > b.

The modified feedback control1 is

δ(t) = Π [(1− β(t))δ∗ − κ sgn(i(t))MΠb(ε(t))] (6)

where β(t) = min{b, ‖ε(t)‖∞}, and

0 < κ ≤ κmax = min

{1− b

bCmax,

1

Cmax

}.

The gain κ in the feedback control law (6) has a unit pro-portional to 1/Ah and can be treated as a scaling factor in thealgorithm. A relatively larger value of κ can lead to a fasterconvergence to the balanced phase. It can be seen from (6) thata battery module with a lower open-circuit voltage will have ahigher duty cycle δ during the charge process (i.e., to be chargedmore), or a lower value of the duty cycle during the dischargeprocess (i.e., to be less discharged).

In this control strategy, Π is to ensure that the duty cyclesare bounded in [0, 1]. β(t) is included to ensure that during thetransient period, the target (1− β(t))δ∗ is an interior point. Onthe other hand, as ε(t) becomes small, this target moves towardthe ideal balanced control δ(t) = δ∗ from (5). sgn(i(t)) accom-modates both charge and discharge operations, i.e., positivefor charging and negative for discharging. The projection Πb

and relatively small κ values ensure that the duty cycle controlaction remains in [0, 1], which will be established in Lemma 1.

Lemma 1: Under the feedback control law (6)

0 ≤ (1− β(t))δ∗ − κ sgn(i(t))MΠb(ε(t)) ≤ 1.

1Although it is possible to employ more sophisticated dynamic feedbackfor this networked coordination problem, the saturation from duty cycles mayintroduce integration winding-up issues and other difficulties in convergenceanalysis. The guaranteed performance of this algorithm renders it a simple andpreferred control structure.


Proof. When κ ≤ κmax, for the jth element of δ(t), we have

(1− β(t))Cj

Cmax− κ sgn(i(t))CjΠb(εj(t))

≥ (1− b)Cj

Cmax− κCjb, since Πb(εj(t)) ≤ b

=Cj

Cmax(1− b− κCmaxb)

≥ Cj

Cmax

(1− b− 1− b

bCmaxCmaxb

)= 0.

To prove the upper bound, we consider two cases.1) If ‖ε(t)‖∞ ≤ b, we have β(t) = ‖ε(t)‖∞, Πb(εj(t)) =

εj(t), and

(1− β(t))Cj


=Cj

Cmax(1− ‖ε(t)‖∞ − κ sgn(i(t))Cmaxεj(t))

≤ Cj

Cmax(1− ‖ε(t)‖∞ + κCmax|εj(t)|)

≤ Cj

Cmax(1− ‖ε(t)‖∞ + ‖ε(t)‖∞),

since κCmax ≤ 1

≤ 1.

2) If ‖ε(t)‖∞ > b, then β(t) = b. It follows that

(1− β(t))Cj


=Cj

Cmax(1− b− κ sgn(i(t))CmaxΠb(εj(t)))

≤ Cj

Cmax(1− b+ κCmaxb)

≤ Cj

Cmax(1− b+ b), since κCmax ≤ 1

≤ 1. �

Denote q(t) =∫ t

t0|i(τ)|dτ and z(t) = ε(t)′ε(t) =

‖ε(t)‖2.Theorem 1: Under the feedback control law (6), a) if q(t) →

∞ as t → ∞, then z(t) → 0, t → ∞. b) If for some t0 ≥ 0 andα > 0, q(t) ≥ αt, t ≥ t0, then there exists a finite t1 > t0 suchthat z(t) ≤ e−2κα(t−t1)z(t1), t ≥ t1, namely the convergenceis exponentially fast.

Proof: By Lemma 1, (6) is simplified to

δ(t) = (1− β(t))δ∗ − κ sgn(i(t))MΠb(ε(t)).

1) From ε(t) = x(t)− λ(t)1l and λ(t) = 1m1l′x(t), we

obtain

ε(t) = (Im − 1

m1l1l′)x(t).

From the state equation, x(t) = M−1δ(t)i(t), we have

ε(t) =

(Im− 1

m1l1l′

)x(t) =

(Im− 1

m1l1l′

)M−1δ(t)i(t).

This, together with the feedback control (6), implies

ε(t) =

(Im − 1

m1l1l′

)M−1

× [(1− β(t))δ∗ − κ sgn(i(t))MΠb(ε(t))] i(t)

= f(ε(t)).

By the equalities M−1δ∗ = 1Cmax

1l and (Im − 1m1l1l′)

1l = 0, we obtain

f(ε) =

(Im − 1

m1l1l′

)M−1

× [(1− β(t))δ∗ − κ sgn(i(t))MΠb(ε(t))] i(t)

=

(Im − 1

m1l1l′

)[−κ sgn(i(t))Πb(ε(t))]i(t)

=

(Im − 1

m1l1l′

)(−κ|i(t)|Πb(ε(t))).

Then, by ε′(t)1l = 0, we have ε′f = ε′[−κ|i(t)|Πb(ε(t))]. It follows that

z(t) = ε′(t)ε(t) + ε′(t)ε(t)= f ′(ε(t))ε(t) + ε′(t)f(ε(t))= −2κ|i(t)|ε′(t)Πb(ε(t)).

We note that

ε′(t)Πb(ε(t)) =m∑j=1

ε′j(t)Πb(εj(t))

and

ε′j(t)Πb(εj(t)) =

{b|εj(t)|, |εj(t)| > b

ε2j (t), |εj(t)| ≤ b.

Consequently, if |εj(t)| ≤ b, for all j = 1, . . . ,m, thenΠb is not active and ε′(t)Πb(ε(t)) = z(t).2 In this case,z(t) = −2κ|i(t)|z(t), and

z(t) = z(t0)e−2κq(t) (7)

which is monotone decreasing. By hypothesis, q(t) → ∞as t → ∞, which implies that

z(t) = z(t0)e−2κq(t) → 0, t → ∞.

On the other hand, if for at least one j, |εj(t)| > b, then

ε′(t)Πb(ε(t)) =

m∑j=1

ε′j(t)Πb(εj(t))

=m∑j=1

min{b, |εj(t)|}|εj(t)|

≥ b2.

It implies that

z(t) ≤ −2κ|i(t)|b2

2We will use the fact that if z(t) ≤ b2, then |εj(t)| ≤ b, for all j =1, . . . ,m in the following derivation steps.


or

z(t) ≤ z(t0)− 2κb2q(t), t ≥ t0.

Under the hypothesis q(t) → ∞ as t → ∞, thereexists t0 < t1 < ∞ at which z(t1) ≤ b2, which implies|εj(t1)| ≤ b, for all j = 1, . . . ,m and (7) becomes validwith t0 replaced by t1. In fact, by the monotonicity of zunder (7), for all t ≥ t1, |εj(t)| ≤ b, for all j = 1, . . . ,m.Consequently, by (7)

z(t) = z(t1)e−2κ(q(t)−q(t1)), t ≥ t1

which converges to 0 under the hypothesis q(t) → ∞.2) From (7), if q(t) ≥ αt, then for t ≥ t1

z(t) = z(t1)e−2κ(q(t)−q(t1)) ≤ z(t1)e

−2κα(t−t1)

which goes to 0 exponentially fast. ��Remark 1: Theorem 1 covers any current profiles, includ-

ing alternating among charge, discharge, idle, pulse types, etc.The condition q(t) → ∞ may be interpreted as a condition of“input persistency” which guarantees convergence to the bal-anced state. On the other hand, if the input current is uniformlypersistent q(t) ≥ αt, exponential convergence is guaranteed.The explicit exponent of convergence is also obtained.

In this theoretical development, the estimation of vj for con-trol implementation is not discussed. This will be covered inSection III-D on SOC estimation.

C. Examples

Example 1: Consider a string of five battery modules withthe following parameters: C1 = 25 000 F, R1 = 0.052 Ω; C2 =20 000 F, R2 = 0.1 Ω; C3 = 35 000 F, R3 = 0.032 Ω; C4 =26710 F, R4 = 0.07 Ω; C5 = 45 500 F, R5 = 0.022 Ω.

The states are updated every second, and the simulation isrun over 24 000 updating points, namely 400 min. The initialvoltages (Volt) are not balanced: v1(0) = 24.5, v2(0) = 23.5,v3(0) = 24.2, v4(0) = 24.1, v5(0) = 23. The charging currentfor the entire module is a constant 10 (A). At each updating timek, the average voltage λ(k) = (v1(k) + · · ·+ v5(k))/5 is cal-culated and then the balancing errors are computed as ej(k) =vj(k)− λ(k), j = 1, . . . , 5. Then, δk is updated according toeither (5) in the open-loop strategy or (6) under the feedbackcontrol.

The module voltages, balancing errors, and duty cyclesare plotted in Fig. 5. The open-loop constant balanc-ing strategy uses δ∗ From (5). For this example, δ∗ =[0.5495, 0.4396, 0.7692, 0.5870, 1]′. The plots show that bal-ancing is not achieved, consistent with Proposition 2.

Then, the control strategy is modified according to (6). In thisexample, Cmax = 45 500. If we choose b = 0.5, then κmax ={

1−0.10.1×45500 ,

145500

}= 0.000022. Select κ = 0.00002. Fig. 6

shows that the balanced state is asymptotically achieved. It canbe seen that it takes about 300 min to complete the reachingphase. After that the balanced charging strategy maintains thebattery modules in a balanced state. It is also noted that sincethe feedback gain κ is relatively small, the duty cycles do notreach the boundaries, as established in Lemma 1.

Fig. 5. Open-loop control strategy: If a constant strategy is used, balancing isnot achieved.

Fig. 6. Feedback control strategy: A modified feedback control strategy basedon (6) achieves convergence.

D. SOC-Based Balanced Charge/Discharge Strategies

We now consider more general problems of SOC-based bal-anced control strategies under nonlinear battery models. Thebattery dynamic system can be accurately modeled by themodel structure in Fig. 7 [17].3 It is easy to derive its state spacerepresentation as ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

vp = − 1CpRp

vp +1Cp

i

s = 1Q(s) i

v = vocv + iR+ vp

= f(s; θ) + iR+ vp

(8)

where s is the SOC, Q is the capacity, vocv = f(s; θ) is a non-linear output mapping from the SOC s to the open-circuit volt-age vocv, and θ contains the model parameters of the mapping.Since the battery capacity Q typically depends on environment

3To be concrete, this model structure is selected for its wide usage in the fieldof battery modeling. However, other battery models can be used also.


Fig. 7. Nonlinear battery model.

conditions and also SOC, it introduces nonlinearity and must beestimated also.

When the battery is operated over a large range, the rela-tionship between the OCV and SOC becomes highly nonlin-ear, expressed generically as vocv(t) = f (s(t), θ) where f isassumed to be continuously differentiable and invertible withrespect to s in the operating range. For charge/discharge con-trol strategy development, the model parameters in (8) and inthe nonlinear function f(·) need to be estimated.

Suppose that the SOC sj(t) has been estimated as sj(t)[17]. The goal of the balanced strategy is to achieve con-vergence sj(t) → λ(t), j = 1, . . . ,m for the average λ(t) =1m

∑mj=1 sj(t) by using the estimates. The main idea to link

a voltage balancing algorithm (6) and an SOC balancing algo-rithm is to devise a reliable and convergent SOC and parameterestimation algorithm for the SOCs and capacities of the mod-ules and combine them with (6).

To be more specific, whenever SOC and capacity estimatess and Q are obtained, our balancing strategies can be directlymodified to become an SOC balancing strategy as follows:

1) replace Cj by Qj and Cmax by Qmax =

max{Q1, . . . , Qm}.2) then, the open-loop strategy becomes

δ(t) = δ∗ =1

Qmax

[Q1, . . . , Qm];

3) for the feedback control, replace the state vector x by theSOC estimate vector x = [s1, . . . , sm]′;

4) then, the estimated balancing errors become ε(t) =

x(t)− λ(t)1l with λ(t) = 1m x′(t)1l;

5) the feedback control (6) is modified to

δ(t) = Π((1− β(t))δ∗ − κ sgn(i(t))MΠb(ε(t))

)(9)

where β(t) = min{b, ‖ε(t)‖∞}, M = diag[Q1, . . . ,

Qm].The following theorem states the impact of estimation errors

from x and Qj on the balancing errors. In the following erroranalysis, the true capacities Qj , j = 1, . . . ,m are assumed tobe constant but unknown. Let ex(t) = x(t)− x(t) and eQ(t) =

Qj(t)−Qj . The actual balance errors are still ε(t) = x(t)−λ(t)1l and z(t) = ε′(t)ε(t).

Suppose that ex∞ = supt≥t0 ‖ex(t)‖∞ and eQ∞ = supt≥t0

‖eQ(t)‖∞.

Theorem 2: Suppose q(t) → ∞ as t → ∞ and the feedbackcontrol (9) is applied.

a) There exists estimation error bounds μx > 0 and μQ > 0such that if ex∞ ≤ μx and eQ∞ ≤ μQ, we have

lim sup z(t) = O(ex∞, eQ∞)

where O(ex∞, eQ∞) means bounded by c1ex∞ + c2e

Q∞ for

some c1, c2 ≥ 0.b) If in addition, ‖ex(t)‖∞ → 0 and ‖eQ(t)‖∞ → 0, then

lim sup z(t) = 0.

Proof: Since the convergence properties and proofs arevery similar to Section III-C, we will only outline the main stepsand ideas that involve the perturbation errors.

From ε(t) = x(t)− λ(t)1l, λ(t) = 1m1l′x(t), x(t) =

M−1δ(t)i(t), we have

ε(t) = (Im − 1

m1l1l′)M−1δ(t)i(t)

= (Im − 1

m1l1l′)M−1

× ((1− β(t))δ∗ − κ sgn(i(t))MΠb(ε(t))) i(t)

= f(ε(t))

= f(ε) + Δ(ex(t), eQ(t)).

1) Under the assumptions on the estimation error bounds,Δ∞(t0) := supt≥t0 ‖Δ(ex(t), eQ(t))‖ is a continuousfunction of μx and μQ. Since the nominal system ε(t) =f(ε) has been shown to be asymptotically Lyapunovstable, for sufficiently small Δ∞, the perturbed systemwill also be convergent to a set whose bounds dependon Δ∞ continuously. This implies that lim sup z(t) is acontinuous function of (ex∞, eQ∞). Hence, lim sup z(t) =O(ex∞, eQ∞).

2) If ‖ex(t)‖∞ → 0 and ‖eQ(t)‖∞ → 0, we apply the con-clusion of (a) with a shifted starting time t0 → ∞. Thisimplies that Δ∞(t0) goes to 0 as t0 goes to ∞. It followsthat lim sup z(t) = 0. �

In Section IV, we use a production battery system and ourestimation algorithms to demonstrate how the balanced controlstrategies developed here can be combined with system identi-fication and SOC estimation algorithms to form an SOC-basedcontrol strategy for balanced operation.

Remark 2: It is essential that SOC estimation be reliablesince the control strategy relies on such estimation to balancebattery operations. For SOC estimation and system identifica-tion, we refer the reader to the available methods [14], [15],[16], [17], [38], [39]. Since these are established results, wewill not cover algorithm details.

During the algorithm implementation, for a battery module’sSOC and parameter estimation, we need its terminal current andvoltage data. In our previous paper [16], we have shown that byusing certain scheduled switching control, one terminal voltagemeasurement suffices. On the other hand, for practical batterysupport of grids, used EV battery modules typically have theirown voltage and current measurements.

In implementation of the control strategy, the control struc-ture consists of two levels: the higher level (global) coordinator


is a consensus control for coordination which will determinethe “desired” charge/discharge currents for the battery modules,and the lower level (local) controllers are a servo-mechanismthat delivers the desired currents by controlling switching dutycycles on power electronic circuits. The local dynamics ofpower electronics are nonlinear and the feedback design of thelocal nonlinear control systems must take into consideration ofnonlinear system structures and parameters. This local servo-mechanism is a standard power electronics control problem. Asa result, we do not detail it in this paper.

IV. CASE STUDIES

A. Battery Systems and Models

We now use some production battery systems to evaluate themethods introduced in this paper. The battery modules usedin this study are lithium-ion batteries with LiMn2O4 cathodeand graphite anode. Its nominal rated capacity is Q = 90 Ahwith a weight of 2.8 kg. The dimension of the battery is346 mm ×255 mm×18 mm. The system, its parameter esti-mation, and the SOC estimation have been presented in ourrecent paper [17]. The rated OCV range within typical SOCis 3.8–4.1 V.

The model structure and parameters are validated on anexperimental battery testing platform. The instrumentationsused to perform the experiments include an Arbin testing sys-tem, a thermal chamber, and a data-acquisition computer. AnArbin instrument BT2000 battery testing system was used tocarry out the charge and discharge test.

The battery model structure (8) is used here. The SOC–OCVrelationship is plotted as the solid line in Fig. 8. The experi-mental data show three characteristic sections. 1) For the highSOC values over 0.92, the curve is fast rising. 2) In the middlerange of the SOC in [0.1, 0.92], the curve is nearly linear. (3)When the SOC drops below 0.1, the curve shows a fast voltagedrop. Based on these observations, our model structure [17] isrepresented by

y = v − (R+Rp)i = a− b(− ln(s))2.1 + cs+ de30(s−1)

(10)

where y is the OCV; s is the SOC; and a, b, c, and d are modelparameters to be determined. For the data in Fig. 8, a least-square data fitting algorithm results in model parameters a =3.81, b = 0.022, c = 0.31, d = 0.07. The model is illustratedas the point curve in Fig. 8. In the typical operating range of theSOC in [0.15, 1], the fitting error is about 0.005 V (5 mV).

B. SOC Estimation and Control Strategy Evaluation

In this study, 24 battery units are packaged into 1 batterymodule to form a battery module of rated 90 Ah and 95 Vat 50% SOC. The modules have the same model structurebut with different capacities. For this evaluation, ten heteroge-neous battery modules are used for this evaluation. Their modelparameters are R = 0.01 Ω, Rp = 0.012 Ω, Cp = 15 000 F,a = 38.1, b = 0.22, c = 3.1, d = 0.7. The capacities (Ah) areQ1 = 90, Q2 = 87, Q3 = 84, Q4 = 81, Q5 = 78, Q6 = 75,

Fig. 8. SOC–OCV mapping: experimental data (solid line) versus model(points).

Fig. 9. SOC estimation by the observer design.

Q7 = 72, Q8 = 69, Q9 = 66, Q10 = 63Ah. In this case study,for simplicity, we focus on SOC estimation. Hence, the truevalues of the battery capacities are used in the SOC estimationalgorithms and battery balancing algorithms.

Each battery module has its own terminal voltage and currentmeasurements and its SOC is estimated by an observer-basedestimator developed in [17]. The 10 battery modules have dif-ferent initial SOC values 0.33, 0.35, 0.41, 0.34, 0.415, 0.32,0.422, 0.432, 0.409, 0.36, but the observers for all battery mod-ules start with the same (but incorrect) initial SOC estimate 0.4.The charging current is 20 A, or 0.2222 C. Under the accuratebattery model, Fig. 9 shows that after a relatively short transientperiod, SOC estimation becomes accurate. In practice, whenmodel parameters are unknown, integrated parameter estima-tion and SOC estimation can be performed using the algorithmsfrom our previous work [15], [17], which will not be furtherdetailed.

Without active balancing management, the SOCs among thebattery modules do not converge, even under a simple con-stant charging current, depicted in Fig. 10. In this case whenone battery module reaches its SOC limit, the entire stringmust stop, even though most battery modules still have substan-tial capacities left for further energy storage. In this example,


Fig. 10. SOC balancing errors without active balancing control: the differencebetween the maximum SOC and the minimum SOC within the ten batterymodules indicates that the SOCs of the batteries do not converge.

Fig. 11. Charge current and SOC profiles with active balancing control.

Fig. 12. Each switch’s complementary duty cycle.

without balanced control, the total realized capacity storagefor the entire string is only 38.8889 Ah, which is derived bythe time interval up to the time when one battery module

reaches sj = 1. It is noted that under the given testing condi-tions, the total available capacity for the string from the giveninitial SOCs is 47.6634 Ah, which can be calculated from∑10

j=1(1− sj(0))Qj .We now apply the control strategy (6) to control and bal-

ance battery SOCs in real time. The resulting SOC trajectoriesare shown in Fig. 11. The individual switch’s complemen-tary duty cycle δj(t) is plotted in Fig. 12. In comparison toFig. 10, SOC balanced operation performs well even undertime-varying charge/discharge currents. By using the balancedcontrol strategy, the total available capacity is fully utilized.

V. CONCLUDING REMARKS

Departing from currently used battery balancing strategies,this work offers new balanced control strategies that aim toachieve and maintain balanced states during routine batteryoperations. These new strategies offer distinct advantages.

1) They accommodate heterogeneous battery modules ofdifferent types, ages, and capacities. This is of criti-cal importance in implementing large grid-scale batterystorage systems using retired EDV batteries.

2) Rather than dumping energy, or shuffling energyamong battery modules, persistent balanced operationis achieved by tuning charge/discharge rates differentlyfrom one battery module to another.

3) When battery modules in a string are modified, thestrategies are easily adapted to accommodate changes inbattery system structures and parameters.

As a first attempt in this direction, this work focuses onthe introduction of the methodology, and the validation of itsfundamentals. This paper does not address potential implemen-tation issues, some of which are enumerated here. First, themethods require new power electronics topologies. Althoughthe switches are standard power electronics components, theirselection and costs must be taken into consideration. Second,when implementing control strategies, there are different waysof coordinating all switches for the same duty cycles. Some ofthem will result in lower current and voltage ratings, and as suchlower power losses. This topic is an interesting research area topursue in the future. Finally, cost/benefit analysis is of primaryimportance and requires collaboration between battery pro-ducers, utilities, and automotive companies so that economic,technological, and environmental impact can be clearly under-stood to implement a better strategy for sustainable energydevelopment.

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Le Yi Wang (S’85–M’89–SM’01–F’12) received thePh.D. degree in electrical engineering from McGillUniversity, Montreal, QC, Canada, in 1990.

Since 1990, he has been with Wayne StateUniversity, Detroit, MI, USA, where he is currentlya Professor with the Department of Electrical andComputer Engineering. He was a Visiting Faculty atthe University of Michigan, Ann Arbor, MI, USA, in1996, and a Visiting Faculty Fellow at the Universityof Western Sydney, Sydney, NSW, Australia, in 2009and 2013, respectively. His research interests include

complexity and information, system identification, robust control, H-infinityoptimization, time-varying systems, adaptive systems, hybrid and nonlinearsystems, information processing and learning, as well as medical, automo-tive, communications, power systems, and computer applications of controlmethodologies.

Dr. Wang was a Keynote Speaker at several international conferences. Heserves on the IFAC Technical Committee on Modeling, Identification, andSignal Processing. He was an Associate Editor of the IEEE TRANSACTIONS

ON AUTOMATIC CONTROL and several other journals, and currently is anEditor of the Journal of System Sciences and Complexity, and an AssociateEditor of Journal of Control Theory and Applications. He is a member ofthe Core International Expert Group, Academy of Mathematics and SystemsScience, Chinese Academy of Sciences, Beijing, China, and an InternationalExpert Adviser at Beijing Jiao Tong University, Beijing, China.


Caisheng Wang (S’02–M’06–SM’08) received theB.S. and M.S. degrees from Chongqing University,Chongqing, China, in 1994 and 1997 respectively,and the Ph.D. degree from Montana State University,Bozeman, MT, USA, in 2006, all in electricalengineering.

From August 1997 to May 2002, he worked as anElectrical Engineer and then a Vice Department Chairwith Zhejiang Electric Power Test and ResearchInstitute, Hangzhou, China. Since August 2006, hehas been with Wayne State University, Detroit, MI,

USA, where he is currently an Associate Professor with the Department ofElectrical and Computer Engineering. His research interests include modelingand control of power systems and electric vehicles, energy storage devices, dis-tributed generation and microgrids, alternative/hybrid energy power generationsystems, and fault diagnosis and online monitoring of electric apparatus.

Dr. Wang is an Associate Editor of IEEE TRANSACTIONS ON SMART GRID.

George Yin (S’87–M’87–SM’96–F’02) received theB.S. degree in mathematics from the University ofDelaware, Newark, DE, USA, in 1983, and the M.S.degree in electrical engineering and the Ph.D. degreein applied mathematics from Brown University,Providence, RI, USA, in 1987.

He joined the Department of Mathematics, WayneState University, Detroit, MI, USA, in 1987, andbecame a Professor in 1996. His research interestsinclude stochastic processes, stochastic systems the-ory and applications, recursive algorithms, identifica-

tion, and signal processing.Dr. Yin is a Fellow of SIAM. He served as Cochair for a number of con-

ferences and served on many committees for IFAC, IEEE, and SIAM. He isan Associate Editor of SIAM Journal on Control and Optimization, AppliedMathematics and Optimization, and a number of other journals. He was anAssociate Editor of Automatica 1995–2011 and the IEEE TRANSACTIONS ON

AUTOMATIC CONTROL 1994–1998.

Feng Lin (S’85–M’88–SM’07–F’09) receivedthe B.Eng. degree in electrical engineering fromShanghai Jiao-Tong University, Shanghai, China, in1982, and the M.A.Sc. and Ph.D. degrees in electricalengineering from the University of Toronto, Toronto,ON, Canada, in 1984 and 1988, respectively.

From 1987 to 1988, he was a Postdoctoral Fellowwith Harvard University, Cambridge, MA, USA.Since 1988, he has been with the Department ofElectrical and Computer Engineering, Wayne StateUniversity, Detroit, MI, USA, where he is currently

a Professor. He was a Consultant for GM, Ford, Hitachi, and other auto com-panies. He is the author of a book entitled Robust Control Design: An OptimalControl Approach (Wiley, 2007). His research interests include discrete-eventsystems, hybrid systems, robust control, and their applications in alternativeenergy, biomedical systems, and automotive control.

Dr. Lin was an Associate Editor of the IEEE TRANSACTIONS ON

AUTOMATIC CONTROL. He has coauthored a paper that received a GeorgeAxelby Outstanding Paper Award from the IEEE Control Systems Society.

Michael P. Polis (S’69–M’72–SM’82–LSM’09)received the B.S. degree in electrical engineeringfrom the University of Florida, Gainesville, FL, USA,in 1966, and the M.S.E.E. and Ph.D. degrees fromPurdue University, West Lafayette, IN, USA, in 1968and 1972, respectively.

From 1972 to 1983, he was a Faculty Memberof Electrical Engineering with Ecole Polytechniquede Montreal, Montreal, QC, Canada. From 1983to 1987, he directed the Systems Theory andOperations Research Program at the National Science

Foundation (NSF). In 1987, he joined Wayne State University, Detroit, MI,USA, as a Chair of Electrical and Computer Engineering. From 1993 to 2001,he was the Dean of the School of Engineering and Computer Science, OaklandUniversity, Rochester, MI, USA, where he is currently a Professor with theDepartment of Industrial and Systems Engineering. He has been a consul-tant for several companies, and an expert witness for numerous law firms.His research interests include energy systems, transportation systems and theidentification, and control of distributed parameter systems.

Dr. Polis has been an Associate Editor of the IEEE TRANSACTIONS ON

AUTOMATIC CONTROL. He is a coauthor of the paper that was named the “BestPaper IEEE TRANSACTIONS ON AUTOMATIC CONTROL 1974–1975.”

Caiping Zhang (M’13–SM’14) was born in HenanProvince, China. She received the B.S. degreefrom Henan University of Science and Technology,Luoyang, China, in 2004, and the Ph.D. degree fromBeijing Institute of Technology, Beijing, China, in2010, both in vehicle engineering.

From 2010 to 2012, she was a PostdoctoralResearcher at Beijing Jiaotong University, Beijing,China. Currently, she is an Associate Professor withBeijing Jiaotong University. Her research interestsinclude battery modeling, states estimation, optimal

charging, battery second use technology, and battery energy storage system.

Jiuchun Jiang (M’12–SM’14) was born in JilinProvince, China. He received the B.S. degree in elec-trical engineering and the Ph.D. degree in power sys-tem automation from Northern Jiaotong University,Beijing, China, in 1993 and 1999, respectively.

Currently, he is a Professor with the School ofElectrical Engineering, Beijing Jiaong University,Beijing, China. His research interests include batteryapplication technology in electric vehicles and energystorage system, electric vehicle charging stations, andmicrogrid technology.

Dr. Jiang received the National Science and Technology Progress 2nd Awardfor his work on EV bus system, and the Beijing Science and TechnologyProgress 2nd Award for his work on EV charging system.

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