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Xing, Lantao, Mishra, Yateendra, Tian, Glen, Ledwich, Gerard, Zhou,Chunjie, Du, Wenli, & Qian, Feng(2019)Distributed state-of-charge balance control with event-triggered signaltransmissions for multiple energy storage systems in smart grid.IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49(8),Article number: 8727724 1601-1611.

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https://doi.org/10.1109/TSMC.2019.2916152

1

Distributed State-of-Charge Balance Control with

Event-triggered Signal Transmissions for Multiple

Energy Storage Systems in Smart GridLantao Xing, Yateendra Mishra, Member, IEEE, Yu-Chu Tian, Member, IEEE,

Gerard Ledwich, Senior Member, IEEE, Chunjie Zhou, Wenli Du, and Feng Qian

Abstract—Modern power grid is increasingly integrated withbattery energy storage systems (BESSs). This paper deals withthe problem of state-of-charge (SoC) balance control for multipledistributed BESSs in smart grid. The BESSs are expected towork cooperatively to not only fulfil the overall power re-quirement, but also meet the constraints of the same relativeSoC variation rate. To achieve this objective, a distributed SoCbalance control approach is presented with event-triggered signaltransmissions. It is designed with the dynamic average consensus(DAC) mechanism for parameter estimations. The DAC enablesdistributed control of each BESS through communicating with itsneighbouring BESSs. Different from traditional periodic signaltransmission, the event-triggered signal transmission embeddedin our approach allows each BESS to transmit signal to its neigh-bouring BESSs only when needed, thus reducing communicationtraffic. Theoretical lower bounds are established for consecutiveinter-event intervals such that the Zeno behaviour is excluded.Case studies are conducted to demonstrate the effectiveness ofthe presented approach.

Index Terms—Battery energy storage systems; distributedcontrol; state-of-charge; event-triggered control; smart grid

I. INTRODUCTION

Battery energy storage systems are playing increasingly

important roles in modern smart grid [1][2]. By absorbing

power from the grid during off-peak time or supplying power

to the grid in peak time, BESSs enable the grid to have

the ability of peak-shaving/shifting, power quality enhance-

ment, and congestion relief [3]. Also, BESSs can provide

fast active power compensation for any mismatch between

power generation and consumption. Moreover, BESSs possess

other inherent advantages such as fast response, high energy

efficiency, long cycle life, and high charging and discharging

rates. Therefore, more and more BESSs of different types are

being integrated into modern energy systems to support smart

grid functions [4][5].

This work was supported by the Australian Research Council throughthe Discovery Project Scheme under Grant DP160102571 and Grant D-P170103305, the National Natural Science Foundation of China (NSFC) underGrant 61833011, and the Ministry of Education of China through the 111Project Scheme under Grant D18003. (Corresponding author: Yu-Chu Tian).L. Xing, Y. Mishra, Y.-C. Tian and G. Ledwitch are with the School

of Electrical Engineering and Computer Science, Queensland Universityof Technology, GPO Box 2434, Brisbane QLD 4001, Australia (e-mail:y.tian@qut.edu.au).C. Zhou is with the School of Automation, Huazhong University of Science

and Technology, 2037 Luoyu Road, Wuhan 430074, China.W. Du and F. Qian are with with the College of Information Science and

Engineering, East China University of Science and Technology, 130 MeilongRoad, Shanghai 200237, China.

Fig. 1: Block diagram of distributed BESSs

As shown in Fig. 1, the established BESSs in modern power

systems tend to be distributed for the following main reasons:

1) Installing BESSs next to the load center is more efficient

and flexible in energy delivery; 2) An increasing number

of companies are establishing their own BESSs to provide

grid support services [6]; and 3) An increasing number of

residents are installing BESSs combining with PVs for daily

use. Therefore, for power systems with multiple distributed

BESSs, it is crucial to make all these BESSs work in a

cooperative way so that the grid could be operated reliably.

Tremendous efforts have been made to investigate BESS

SoC balance control. Examples include centralized control

[7], decentralized control [8], and distributed control [9][10].

Since centralized control is sensitive to single-point failure

and decentralized control suffers from insufficient information,

multi-agent based distributed control becomes more suitable

for BESS coordination in smart grid [11]. In [9], a distributed

control approach is proposed for package level SoC balance

with multiple grid-connected BESSs. It is designed with a

distributed observer and energy coordinator to determine the

power output of each BESS. In [10], a distributed sliding mode

control scheme is reported for BESS SoC balance control in

DC microgrids. It is able to exclude circulating current and

guarantee fast SoC balancing. The work in [11] proposes a

two-level discrete control strategy for BESS SoC balance. The

upper-level determines the power reference for each BESS,

while the lower level forces each BESS to track the power

reference accurately. More results about distributed BESS SoC

balance control can be found in two recent papers [12][13].

Under distributed control, each BESS needs to trans-

mit/receive necessary information to/from its neighbouring

IEEE Transactions on Systems, Man, and Cybernetics: Systemsonline published on 31 May 2019. DOI: 10.1109/TSMC.2019.2916152

B17017

DOI: 10.1109/TSMC.2019.2916152

2

BESSs. The traditional signal transmission mechanism is time-triggered, i.e. the signal transmission is conducted periodicallywith a fixed time interval. In practice, the time interval tendsto be set conservatively small in order to address the worstsystem operation situations. However, when the number ofBESSs gets larger and the BESSs are geographically dispersed,a huge amount of data needs to be transmitted. This mayrapidly exhaust the network resources and imbalance thenetwork load [14]. If the communication network is congested,the distributed controllers may fail to work as the essentialinformation exchange is not always available.

Therefore, event-triggered control has received considerableattention in networked control systems [15][16][17]. Com-pared with periodical signal transmission, event-triggered con-trol signals transmit aperiodically only when some pre-definedconditions are violated. Specifically, the event-triggering con-dition threshold can be either some constants independent ofthe system states [18][19], or some variables relying on thesystem states [20][21][22]. Two latest surveys [23][24] havecovered some recent progress of event-triggered control.

The event-triggered signal transmission mechanism has alsobeen introduced into power system control, such as loadfrequency control [25][26][27] and economic dispatch [28].However, a report has not been found on event-triggeredcontrol for maintaining appropriate BESS SoC levels. Toprovide a solution to this problem, a distributed SoC balancecontrol approach with event-triggered signal transmission ispresented in this paper for multiple distributed BESSs in smartgrid. The main contributions of this paper include:

• To ensure the SoC values of all the BESSs to convergerather than diverge during operation, the constraints ofthe same relative SoC variation rate are established andformulated. It is proved that under these constraints,all the BESS SoC values will keep converging, and noBESSs will be fully discharged (or charged) ahead ofothers;

• To tackle the effects of the BESS parameter differenceson SoC balance, two lumped parameters are definedwhich lump both static BESS intrinsic parameters andtime-varying SoC values. The DAC mechanism for es-timating the average of the lumped parameters enablesall the BESSs to meet the relative SoC variation rateconstraints regardless of their parameter differences;

• A DAC-based distributed control approach with event-triggered signal transmissions is proposed. It enables allBESSs to work cooperatively to not only fulfill the overallpower requirement, but also guarantee appropriate SoCbalance. The event-triggered signal transmission allowseach BESS to transmit signals to its neighbouring BESSsonly when needed, thus reducing communication traffic.Theoretical lower bounds are established for inter-eventintervals such that the Zeno behaviour [29] is excluded;

The rest of this paper is organized as follows: Notationsand symbols used in the paper are listed in Table I. SectionII formulates the problem, including the control objectiveand constraints. Section III presents a control strategy withcomplete global information. Our distributed control approach

TABLE I: Notations and symbols.

i index for BESSs, i = 1, · · · , NL Laplacian matrix of a graphN The number of BESSst TimeV Lyapunov candidatez1, z2 Intermediate variablesBattery parametersQi BESS capacitySi BESS SoC valueρi, ηi Coulombic efficiencyVi, Ii Output voltage and current, respectivelyPi BESS power input/outputPref Power requirement from grid operatorsPavg Pavg , Pref/N

P P ,∑N

i=1 Pi

Notations for DAC (Dynamic Average Consensus)d Event condition threshold in case studydi Event condition threshold for the ith BESSd d , maxdiei Error caused by event condition , ei , xi − xi

mi mi , QiVi/ρitik Time instant at which an event is triggeredxi Estimation of τavg or µavg

xi Updated and transmitted signalyi DAC estimation errorµi A lumped parameter in charging modeτi A lumped parameter in discharging modeτavg , µavg Average values of all τi and µi, respectivelyυi, xi, Internal state of DACυi Transformed version of υiα, β, Positive design parametersω, ϖ, λ Bounded constantsΩy The set of DAC estimation errorsA1, A2, A3, B1, B2, R Matrices for the proof of DAC convergenceτ An intermediate variable∆1,Φ,ΠN Intermediate matrices

with event-triggered signal transmission is presented in SectionIV. Case studies are conducted in Section V. Finally, SectionVI concludes the paper.

II. PROBLEM FORMULATION

A. SoC Estimation

The basic Coulomb counting method [30] is used in thispaper to estimate the SoC of each BESS, as shown in (1)

Si = Si(0)−ρiQi

∫Iidt, i = 1, ..., N (1)

where Si denotes the SoC status, Si(0) is the initial SoC value,Qi denotes the BESS capacity, Ii is the BESS output current,and ρi is defined as

ρi =

1, dischargingηi, charging

(2)

in which ηi is the Coulombic efficiency. Differentiating bothsides of (1) yields

Si = − ρiQi

Ii, i = 1, ..., N (3)

Let Vi be the output voltage of BESS i, then the poweroutput Pi of BESS i can be obtained as

Pi = ViIi (4)

3

1

Δt

t t+Δt

1

0

0

+Δ

+Δ

Δ

Δ

of B

ES

S j

of B

ES

S i

1

Δt

t t+Δt

1

0

0

+Δ

+Δ

of B

ES

S i

Δ

Δ

of B

ES

S j

(a) Charging mode

1

Δt

t t+Δt

1

0

0

+Δ

+Δ

Δ

Δ

of B

ES

S j

of B

ES

S i

1

Δt

t t+Δt

1

0

0

+Δ

+Δ

of B

ES

S i

Δ

Δ

of B

ES

S j

(b) Discharging mode

Fig. 2: Formulation of SoC constraints

Similar to [8][12], it is assumed that the output voltage ofeach BESS can remain constant in a large range of SoC. Thus,substituting (4) into (3) gives

Si = − ρiPi

QiVi, i = 1, ..., N (5)

B. Overall power requirement

Assume that there are N BESSs, and the overall powerrequirement for the N BESSs are Pref , where Pref is givenby the grid operators based on specific task requirements.Therefore, the power output summation of all the BESSs Pshould meet the overall power requirement, i.e.,

P =

N∑i=1

Pi = Pref (6)

where Pi is the power output of the i-th BESS.

C. Formulation of SoC Constraints

To meet the above power requirement, one simple option isto make the power output of each BESS be Pi = Pref/N =Pavg . However, in this case, as can be seen from (5), sincethe BESS parameters are different due to manufacture flaws orworking conditions, even if initially Si(0) is the same, Si(t)tends to diverge. In order to maintain the overall maximumcapability of all the BESSs in supplying or absorbing power,it is desired that none of the BESSs are disconnected from thenetwork ahead of others. One reason for the early disconnec-tion of some BESSs is that they are already fully dischargedor charged while the others are still discharging or charging.Therefore, to prevent this from happening while fulfilling theoverall power requirement (6), Si needs to keep converginginstead of diverging.

One way to meet the SoC converging constraint is tomake every BESS have the same relative SoC variation rate.Specifically, as shown in Fig.2, in the discharging mode, allthe BESSs should have the same relative SoC variation rate,i.e.

lim∆t→0

∆Si/∆t

Si= lim

∆t→0

∆Sj/∆t

Sj, i, j = 1, ..., N. (7)

which is

Si

Si=

Sj

Sj, i, j = 1, ..., N (8)

Similarly, in the charging mode, all the BESSs should alsohave the same relative SoC variation rate, namely

lim∆t→0

∆Si/∆t

1− Si= lim

∆t→0

∆Sj/∆t

1− Sj, i, j = 1, ..., N (9)

which indicates

Si

1− Si=

Sj

1− Sj, i, j = 1, ..., N (10)

Lemma 1 is given below to show the necessity of the abovetwo constraints (8) and (10):

Lemma 1. The relative SoC variation rate constraint (8) ( or(10)) can guarantee that in discharging (or charging) mode, allthe BESS SoC values will keep converging, and with sufficienttime all the BESSs will be fully discharged (or charged) at thesame time.

Proof: The proof is given in Appendix.

D. Control objective with constraints

As formulated above, the control objective of this paperis to design a distributed control approach, which can makeall BESSs work cooperatively to not only fulfill the overallpower requirement (6), but also meet the relative SoC variationrate constraint (8) for discharging and (10) for charging,respectively.

III. CONTROL STRATEGY WITH GLOBAL INFORMATION

In this section, a general theory is developed to achieve theabove control objective with global information available. Thiswill facilitate our further development of the event-triggereddistributed control approach in the next Section.

A. Discharging mode

Define mi = QiVi/ρi, then (5) becomes

Si = −Pi/mi, i = 1, ..., N (11)

Let

τi = miSi, i = 1, ..., N (12)

be the lumped parameter which contains not only the BESSintrinsic parameter mi but also the time-varying SoC value Si.Then if the global information is available to each BESS, theaverage value τavg of all the BESSs can be computed as

τavg =1

N

N∑i=1

τi (13)

With τavg in hand, the power output reference of the i-thBESS can be set as

Pi =τiτavg

Pavg, i = 1, ..., N (14)

4

With the power output reference defined in (14), we havethe following Theorem:

Theorem 1. In discharging mode, the control strategy (12)-(14) can fulfill the overall power requirement (6) with con-straint (8) met.

Proof: With Eq. (14), the overall power output of allBESSs is

P =

N∑i=1

Pi =

N∑i=1

τiτavg

Pavg = Pref (15)

Thus, the control objective (6) is achieved. By combining(11) (12) and (14), it has

Si

Si=

Sj

Sj, i, j = 1, ..., N (16)

which indicates constraint (8) is met. This completes the proof.

B. Charging mode

In the charging mode, firstly define

µi = mi(1− Si), (17)

as a new lumped parameter. Then if all the global informationis available to every BESS, the average value µavg of all theBESSs can be computed as

µavg =1

N

N∑i=1

µi (18)

Then, the power output reference of the ith BESS can beset as

Pi =µi

µavgPavg, i = 1, ..., N (19)

Now we have the following Theorem 2:

Theorem 2. In charging mode, the control strategy (17)-(19)can guarantee objective (6) with constraint (10) satisfied.

Proof: With Eq. (19), the overall power output of allBESSs is

P =N∑i=1

Pi =N∑i=1

µi

µavgPavg = Pref (20)

Thus, control objective (6) is achieved. Moreover, combin-ing (11) (17) and (19) gives

Si

Sj

=1− Si

1− Sj, i, j = 1, ..., N (21)

which proves constraint (10) is satisfied.

Remark 1. In this paper, two new lumped parameters τi andµi are introduced. These two parameters contain not only theinformation of the BESS state-of-charge status Si but also theBESS parameter mi. With the help of the average values τavgand µavg , the effects of the different BESS parameters mi arewell compensated as indicated in the derivation of (16) and(21). This is essential to meet the SoC constraints (8)-(10).

Remark 2. The control strategies (12)-(14) and (17)-(19)require global information of τi and µi. While such globalinformation may be accessible in small-scale systems, it isgenerally unavailable in large-scale systems. Therefore, forlarge-scale systems with a large-number of geographicallydistributed BESSs, the control strategy is difficult to implementand susceptible to single-point failures [31]. To tackle theseconstraints, an event-triggered distributed control approachbased on the DAC algorithm is proposed in the next Section.

IV. EVENT-TRIGGERED DISTRIBUTED CONTROL

When global information is unavailable, the average valuesτavg and µavg are not accessible to each BESS any more.As a result, the control strategies in (12)-(14) and (17)-(19)become infeasible. Inspired by [32]-[33], an event-triggeredDAC algorithm is adopted to enable each BESS to estimatethe time-varying τavg and µavg . Then a proper power outputreference is generated based on the estimated values. In thispaper, the communication graph of the distributed BESSs isundirected.

A. Discharging mode

In the discharging mode, the key point is to make eachBESS estimate τavg . Thus, let xi (i = 1, ..., N) be the estimateof τavg in BESS i, then xi is designed as

υi = αβ∑j∈Ni

aij(xi − xj) (22)

˙xi = −αxi − β∑j∈Ni

aij(xi − xj)− υi (23)

xi = xi + τi, i = 1, ..., N (24)

where υi is an internal state, α and β are positive constants,xi(t) = xi(t

ik), tik (k ∈ z) is the time instant where xi is

updated and transmitted to its neighbours. During the timeinterval [tik, t

ik+1), xi holds as a constant xi(t

ik). The time

instant is defined by the following event-triggering condition:ei = xi(t)− xi(t)tik+1 = inft > tik| |ei| ≥ di, i = 1, ..., N

(25)

where di is a positive constant denoting the threshold of theevent condition.

Define yi = xi − τavg as the estimation error, τ =ΠN (τi + ατi), ΠN = IN − 1

N 1N1TN , y = [y1, ..., yN ]T ,υ = [υ1, ..., υN ]T and e = [e1, ..., en], from (22)-(25) is has[

yυ

]=

[−αIN − βL −IN

αβL 0N

]︸ ︷︷ ︸

A1

[yυ

]+

[τ − βLeαβLe

]

(26)

Because τi and τi must be bounded, it is concluded that

||τ || ≤ ω (27)

where ω is a bounded constant.Next it will show that the proposed algorithm in (22)-(25)

can make xi track τavg exponentially. Firstly, we give thefollowing Lemma 2:

5

Lemma 2. For matrix A1 defined in (26), it has only one zeroeigenvalue, while the other eigenvalues are negative.

Proof: The proof is given in Appendix.We make the following change of coordinates:

υ =

[1√N

1N×1, R

]︸ ︷︷ ︸

B1

υ, υ =

[ 1√N11×N

RT

]︸ ︷︷ ︸

B2

υ (28)

where R is a matrix making[

1√N1N×1 R

]an orthonormal

matrix. Moreover, it has RTR = IN−1 and RRT = ΠN

Subsequently, with the new coordinates, it has[y˙υ

]=

[−αIN − βL −B1

αβB2L 0N

] [yυ

]+

[τ − βLeαβB2Le

]

=

−αIN − βL −B1

01×N 01×N

αβRTL 0(N−1)×N

︸ ︷︷ ︸

A2

[yυ

]+

τ − βLe01×N

αβRTLe

(29)

As can be seen from (29), the (N +1)-th row of matrix A2

are all zero. Thus, it has

˙υ1 = 0 (30)

i.e. υ1 = υ1(0) will remain as a constant. By defining ¯υ =[υ2, ..., υN ]T , it has[

y˙υ

]=

[−αIN − βL −RαβRTL 0N−1

]︸ ︷︷ ︸

A3

[y¯υ

]

+

[τ − βLe− 1N×1√

Nυ1(0)

αβRTLe

]︸ ︷︷ ︸

∆1

(31)

The following Lemma is presented to show the character-istics of A3.

Lemma 3. For matrix A3 defined in (31), all its eigenvaluesare negative.

Proof: The proof is given in Appendix.Now it is ready to show that xi in the DAC algorithm shown

in (22)-(25) will exponentially converge to τavg .

Lemma 4. When the communication graph is undirected,let d = maxdi, i = 1, ..., N , the DAC algorithm (22)-(25) can guarantee that the estimation error |y| will expo-nentially converge to the set Ωy = y | ||y|| ≤ cϖ

λ ata rate of λ = minα, βλ2, c is a positive constant, andϖ = ω+ |υ1(0)|+

√Nβ||L||d+

√Nαβ||L||d. Moreover, the

Zeno behavior can be avoided.

Proof: The proof is presented in Appendix.

Remark 3 (The size of Ωy). As can be seen from Lemma 4,the size of Ωy can be adjusted by tuning the design parameters.Specifically, the size of Ωy can be reduced by adjusting α, βand decreasing d.

Equation (22)

Equation (23)+ Event

condition (25)

Power output

reference (32)Lumped

parameter (12)

+

BESS #i

BESS #j BESS #k

Fig. 3: The distributed control strategy in discharging mode

Remark 4 (Inter-event interval). The lower bound of twosuccessive events is established in (53) in Appendix. As can beseen from (53), with other parameters fixed, by increasing dithe lower bounds for the inter-event intervals can be increasedand thus the number of signal transmissions can be reduced.However, this is at the price of increasing the size of Ωy whichwill affect the control performance. Thus, there is a trade-offbetween the number of signal transmission and the controlperformance in setting the threshold di. This will be illustratedin the case studies in Section V.

From Lemma 4, it is obtained that xi can exponentiallyconverge to τavg . Thus, the power output reference for thei-th BESS can be set as

Pi =τixi

Pavg, i = 1, ..., N (32)

Equations (22)-(25) and (32) form our proposed distributedcontrol approach based on event-triggered mechanism, asshown in Fig. 3. Then, the following Theorem 3 is given:

Theorem 3. With the distributed controller (22)-(25) and (32)in discharging mode, the control objective (6) with constraint(8) can be ensured.

Proof: Since yi = xi − τavg can exponentially convergeto an adjustable small set, and yi ≪ xi, after quick transitionit has xi ≈ τavg . Then, we have

P =N∑i=1

τixi

Pavg ≈N∑i=1

τiτavg

Pavg = Pref (33)

Thus, the control objective (6) is achieved.From (11) and (32), it is obtained that

Si/Sj = (Si/Sj)(xj/xi), i, j = 1, ..., N (34)

Since the converge speed of xi is much faster than Si, (i =1, ..., N), it is safe to have

Si/Si ≈ Sj/Sj , i, j = 1, ..., N (35)

This completes the proof.

B. Charging mode

For the charging mode, we have the following Theorem 4:

Theorem 4. In charging mode, by replacing τi with µi in thedistributed controller (22)-(25) and (32), the control objective(6) with constraint (10) can also be ensured.

6

BESS #2 BESS #3 BESS #4 BESS #5

BESS #6BESS #7BESS #8BESS #9BESS #10

BESS #1

Fig. 4: Communication graph among distributed BESSs

0 1000 2000 3000 4000Time (s)

0

0.2

0.4

0.6

0.8

1

1.2

S i and

P/P

ref

S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

P/Pref

(a) Without the proposed approach.

0 1000 2000 3000Time (s)

0

0.2

0.4

0.6

0.8

1

1.2

S i and

P/P ref

S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

P/Pref

(b) With the proposed approach.

0 1000 2000 3000 4000Time (s)

-2.5

-2

-1.5

-1

-0.5

010-3

(c) With the proposed approach.

Fig. 5: Simulation results for discharging mode.

Proof: This theorem can be proved by following theproofs in Lemma 4 and Theorem 3.

Remark 5. In implementing the control approach (22)-(25)and (32), only xi needs to be transmitted. Different from thetraditional periodical signal transmission with a conservative-ly small period, in this paper, the event condition (25) is usedto transmit xi. In this way, the number of signal transmissionscan be significantly reduced. This will be illustrated later inour case studies.

V. CASE STUDIES

The effectiveness of the proposed distributed control ap-proach is demonstrated through four case studies in this

0 1000 2000 3000 4000Time (s)

0

0.2

0.4

0.6

0.8

1

1.2

S i and

P/P ref

S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

P/Pref

(a) Without the proposed approach.

0 1000 2000 3000 4000Time (s)

0

0.2

0.4

0.6

0.8

1

1.2

S i and

P/P ref

S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

P/Pref

(b) With the proposed approach.

0 1000 2000 3000 4000Time (s)

0

0.5

1

1.5

210-3

(c) With the proposed approach.

Fig. 6: Simulation results for charging mode.

Section. The first two case studies compare the control per-formance with and without the proposed approach in bothdischarging and charging modes. The third demonstrates theeffects of the event threshold value, while the fourth comparesthe proposed event-triggered signal transmission mechanismwith the traditional periodical one. In all these case studies, 10BESSs are grouped together to fulfill the power requirement.Their communication graph is shown in Fig.4. The allowableSoC range for the 10 BESSs is [0.1, 0.9], and the otherparameters are given in Table III in Appendix.

A. Case 1: Discharging mode

In this case study, the desired overall output for the 10BESSs is Pref = 200KW, i.e. Pavg = 200/10 = 20KW.The initial SoC values of the 10 BESSs are set as S1 = 0.9,S2 = 0.8, S3 = 0.75, S4 = 0.85, S5 = 0.88, S6 = 0.76,S7 = 0.83, S8 = 0.81, S9 = 0.79, S10 = 0.77. The otherparameters are set as di = 1e4 (i = 1, ..., 10), α = β = 5.

The simulation results are shown in Fig.5(a)-Fig.5(c). It isseen from Fig.5(a), without the proposed approach (i.e. theoutput reference is simply set as Pavg = 200/10 = 20KW),the SoC of different BESSs cannot converge because of theirparameter differences. Therefore, the SoC convergence con-straint cannot be met. As a result, some BESSs hit the lower

7

0 1000 2000 3000Time (s)

0

0.5

1

1.5

Si a

nd P

/Pre

f

d=10000

S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

P/Pref

0 1000 2000 3000Time (s)

0

0.5

1

1.5

Si a

nd P

/Pre

f

d=1000

Fig. 7: Control performance under different di values.

SoC limit 10% and thus are disconnected from the networkahead of others, after which the overall power requirementcannot be fulfilled. However, with our proposed approach,from Fig.5(b) and Fig.5(c), it is observed that while the BESSscan fulfill the power requirement, the SoC constraint (8) is alsomet. As a result, the SoC of each BESS can converge towardsthe same value and thus the overall power requirement can besatisfied for a longer time. Specifically, without the proposedapproach, the overall power requirement is met for around2800s, while with the proposed approach the time is extendedby about 20% to around 3350s. Clearly, this illustrates theadvantages of the proposed approach in discharging mode.

B. Case 2: Charging modeIn this case, the 10 BESSs work in the charging mode,

and the overall power requirement is Pref = −200KW, i.e.Pavg = −20KW. The initial SoC values of the 10 BESSs are:S1 = 0.2, S2 = 0.1, S3 = 0.15, S4 = 0.23, S5 = 0.18,S6 = 0.12, S7 = 0.25, S8 = 0.3, S9 = 0.28, S10 = 0.26. Allthe other parameters are set the same as in Case 1.

The simulation results are shown in Fig.6(a)-6(c). Again,with the proposed approach, the SoC of all BESSs canconverge toward the same value, and the time the controlobjective holds is extended from about 3025s without the pro-posed approach to around 3484s with the proposed approach,implying a 15% improvement. This illustrates the advantagesof the proposed approach in charging mode.

C. Case 3: Effects of event threshold di

In this case, we investigate the effects of the event thresholddi on the control performance. Here we consider the discharg-

Fig. 8: BESS output under different di values.

Fig. 9: xi(t) under different di values.

ing mode. For comparison, while the other parameters are setthe same as in Case 1, the following two sets of thresholdsare considered: 1). di = d = 1e3 and 2). di = d = 1e4.

The case study results are shown in Fig.7-10. From Fig.7,

8

Fig. 10: Triggering time of BESS 2 under different di values.

TABLE II: The number of triggering events.

BESS i 1 2 3 4 5d=1e3 65491 65487 65533 66114 66215d=1e4 6950 10224 6928 6995 7137

BESS i 6 7 8 9 10d=1e3 66085 65419 65647 66009 66268d=1e4 6923 6804 6834 8721 7082

it is observed that with different values of d, the controlperformance with respect to the overall power output and SoCconvergence are almost the same. However, as can be seenfrom Fig.9, with a bigger d, the bound of the estimation errorbetween xi and τavg becomes bigger. As a result, the poweroutput of each BESS fluctuates more severely as indicatedin Fig.8. The time intervals between two consequent eventsfor BESS 2 are presented in Fig.10, while the counterpartsof the other BESSs are omitted here for their similarities.The total number of triggering events for each BESS withdifferent d within 3300s is shown in Table II. As can be seen,by increasing d, the time intervals of two consecutive eventsbecome longer and thus the number of signal transmissionsis significantly reduced. Therefore, there exists a trade-offbetween the control performance and the number of triggeringevents in setting the threshold di. The trade-off is consistentwith the theoretical analyses.

D. Case 4: Comparison with periodical signal transmission

In this case, we compare the proposed event-triggeredsignal transmission mechanism with the traditional periodicalone. Here we consider the discharging case in Case 1. For

0 1000 2000 30000

0.5

1

1.5

P/P re

f

T=0.05s

0 1000 2000 30000

1

2

P/P re

f

T=0.08s

0 20 40 60 80 100Time (s)

-100

0

100

P/P re

f

T=0.1s

Fig. 11: P/Pref versus t under different interval T values.

comparison, we set the signal transmission time interval T as0.05s, 0.08s and 0.1s, respectively.

The overall power output of the 10 BESSs are shownin Fig.11. It is seen that when T = 0.05s, the controlperformance is satisfactory. When T increases to 0.1s, largeimpulsive spikes are observed in the starting period. Thismay damage the power electronic interfaces of the BESSs.Therefore, to meet the overall power requirement, the timeinterval for signal transmissions must be set smaller than 0.1s,implying that in 3, 300s each BESS has to conduct at least33, 000 signal transmissions. In comparison, as shown in Case3, by setting the threshold as d = 1e4, the overall powerrequirement is ensured with each BESS conducting less than10, 224 signal transmissions. Therefore, the number of signaltransmissions is significantly reduced with the proposed event-triggered transmission mechanism.

VI. CONCLUSION AND FUTURE WORK.

This paper has investigated the problem of distributed SoCbalance control for multiple BESSs in smart grid. A distributedSoC balance control approach with an event-triggered signaltransmission mechanism is proposed. This control approachenables all BESSs to not only fulfill the overall power re-quirement, but also meet constraints of the same relative SoCvariation rate. By adopting the event-triggering mechanism,each BESS decides when to transmit signal to its neighboursbased on a specified condition rather than a fixed time interval.In this way, the communication traffic can be significantlyreduced. Positive lower bounds for the inter-event intervalshave been established to exclude the Zeno behavior. Four

9

case study results have been provided to demonstrate theeffectiveness of the proposed approach.

There are several potential ways to extend the present-ed results. For example, it is interesting yet challengingto analyse the case where the communication links amongBESSs undergo random failures. Also, it is envisaged thatthe presented results can be integrated into power systems toprovide ancillary service, such as frequency regulation supportand voltage profile improvement.

VII. APPENDIX

A. BESS parameters.

TABLE III: BESS Parameters.

BESS i Pmax,i(KW ) ηi Qi(Ah) Vi(V )1 25 0.981 120 2202 25 0.982 118 2153 25 0.99 110 2184 25 0.993 125 2255 25 0.85 128 2236 25 0.87 130 2197 25 0.86 112 2218 25 0.95 122 2259 25 0.92 116 21510 25 0.9 118 216

B. Proof of Lemma 1.

1). In the discharging mode, let z1 = Si−Sj and V = 12z

21 .

From (8) it follows that

V = z21 Sj/Sj (36)

Since Sj < 0 and 0 < Sj < 1, it is obtained V ≤ 0. Thus,z1 will converge towards 0, implying Si and Sj will convergetowards the same value. Moreover, define z2 = Si/Sj . Itfollows from (8) that

z2 = (SiSj − SiSj)/S2j = 0 (37)

Thus, z2 will remain constant, which indicates that withsufficient discharging time both Si and Sj will hit zero at thesame time, i.e. all the BESSs are fully discharged at the sametime.

2). In the charging mode, similarly with z1 = Si − Sj andV = 1

2z21 , it has

V = − Sj

1− Sjz21 (38)

Since Sj > 0 and 0 < 1 − Sj < 1, it is obtained V ≤ 0.Thus, Si−Sj will converge towards the same value. Moreover,define z2 = 1−Si

1−Sjand it again has z2 = 0. This indicates that

z2 will remain constant and both 1 − Si and 1 − Sj will hitzero at the same time. Thus, with sufficient charging time allBESSs will be fully charged at the same time.

C. Proof of Lemma 2.

By elementary column operations, it is obtained that thecharacteristic equation of A1 is

0 = |sI −A1| = |s2IN + αsIN + βLs+ αβL| (39)

By comparing the characteristic equation of A1 with theone of L: 0 = |λiIn − L|, it is obtained that

s2 + (α+ βλi)s+ αβλi = 0 (40)

Since the communication graph is undirected, L only hasone zero eigenvalue, i.e. λ1 = 0, and all the other eigenvaluesλi > 0, (i = 2, ..., N). When λ1 = 0, it is obtained that s1 = 0or s1 = −α; when λi > 0, (i = 2, ..., N), it is obtained thatsi = −α or si = −βλi. Therefore, A1 has only one zeroeigenvalue and all the other eigenvalues are negative, i.e. either−α or −βλi, (i = 2, ..., N). This completes the proof.

D. Proof of Lemma 3.

Assume s is an eigenvalue of A2 in (29), since A3 is amatrix with A2’s (N + 1)-th row and (N + 1)-th columncrossed out, it has

|sI2N −A2| = s|sI2N−1 −A3| = 0. (41)

It is obtained from (41) that s = 0 or |sI2N−1 − A3| = 0.Thus, A3 have the same eigenvalues with A2 except s = 0.Since the coordinate transformation (28) is invertible, A1 andA2 must have the same eigenvalues. Therefore, from Lemma1, it is obtained that the eigenvalues of A3 are either −α or−βλi, (i = 2, ..., N). This completes the proof.

E. Proof of Lemma 4.

Let z = [y, ¯υ]T . Then, from (31) is is obtained that

z(t) = Φ(t, 0)z(0) +

∫ t

0

Φ(t, s)∆1ds (42)

where Φ(t, s) = eA3(t−s). From Lemma 2, all the eigenvaluesof A3 are negative, then based on [34] it has

||Φ(t, s)|| = ||eA3(t−s)|| ≤ ce−λ(t−s) (43)

in which c is a positive constant. By combining (42) and (43)yields

||y|| ≤ ||z(t)|| ≤ cz(0)e−λt +

∫ t

0

c||∆1||e−λ(t−s)ds (44)

Since

||αβRTLe|| = αβ√eTLRRTLe = αβ

√eTLΠNLe

= αβ√eTLLe ≤

√Nαβ||L||d (45)

it has ||∆1|| ≤ ϖ and

||y|| ≤ ||z(t)|| ≤ cz(0)e−λt +cϖ

λ(1− e−λt) (46)

Therefore, ||y|| will exponentially converge to the set Ωy ata rate of λ = minα, βλ2.

10

Now we show that the proposed event-triggered algorithm(22)-(25) can exclude the Zeno behavior.

d

dt|xi − xi| = − (xi − xi)xi

|xi − xi|≤ |xi|

≤| − α(xi − τi) + τi − υi − β∑j∈Ni

aij(xi − xj)|

≤| − α(xi −1

N

N∑j=1

τi)−α

N

N∑j=1

τi + ατi|

+ |τi|+ |υi|+ β∑j∈Ni

aij(|yi|+ |yj |+ 2d)|

≤α|xi − xi|+ α(|yi|+ d) + | αN

N∑j=1

τi + ατi|

+ |τi|+ |υi|+ β∑j∈Ni

aij(|yi|+ |yj |+ 2d)| (47)

From (46), we have

sup||y||, ||¯υ|| ≤ sup ||z|| ≤ cz(0) + cϖ/λ (48)

Thus,

||υ|| = ||B1υ|| ≤ ||B1||(|υ1(0)|+ ||¯υ||)≤ ||B1||(|υ1(0)|+ cz(0) + cϖ/λ) (49)

Bearing in mind |yi| ≤ ||y||, |υi| ≤ ||υ||, |τi| ≤ mi

and |τi| ≤ ki and defining ai as the number of BESS i′sneighbours , it has

d

dt|xi − xi| ≤ α|xi − xi|+ σi (50)

where

σi =(α+ 2aiβ)(cz(0) +cϖ

λ+ d) + | α

N

N∑i=1

mi + αmi|

+ ki + ||B1||(|υ1(0)|+ cz(0) + cϖ/λ) (51)

Noting that xi = xi(tik) and using the Comparison Lemma

[35], it has

|xi − xi| ≤σi

α(eα(t−tik) − 1) (52)

Thus, the time it takes for |xi − xi| to increase from 0 todi is

∆ti = tik+1 − tik ≥ 1

αln(

αdiσi

+ 1) (53)

Clearly, ∆ti > 0 is a positive constant, which indicates thatthe Zeno behavior cannot occur. This completes the proof.

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Lantao Xing received the B.E. degree in automationengineering from China University of Petroleum(East China), China, in 2009, and the Ph.D. de-gree in control science and engineering from Zhe-jiang University, China, in 2018. He is currentlya Research Fellow in Electrical Engineering withthe School of Electrical Engineering and ComputerScience, Queensland University of Technology, Bris-bane QLD, Australia. His research interests includenonlinear system control, event-triggered control,distributed frequency control, and smart grid.

Yateendra Mishra (S’06, M’09) received the Ph.D.degree from the University of Queensland, Bris-bane, Australia, in 2009. He is currently a SeniorLecturer and Advanced QLD Research Fellow withthe School of Electrical Engineering and ComputerScience, Queensland University of Technology, Bris-bane QLD, Australia. His research interests includedistributed generation and distributed energy storage,and power system stability and control and theirapplications in smart grid.

Yu-Chu Tian (M’00) received the Ph.D. degree incomputer and software engineering from the Uni-versity of Sydney, Sydney NSW, Australia, in 2009and the Ph.D. degree in industrial automation fromZhejiang University, Hangzhou, China, in 1993. Heis currently a Professor in Computer Science withthe School of Electrical Engineering and ComputerScience, Queensland University of Technology, Bris-bane QLD, Australia. His research interests includebig data computing, distributed computing, cloudcomputing, computer networks, and control systems.

Gerard Ledwich (SM’89) received the Ph.D. de-gree in electrical engineering from the Universityof Newcastle, Newcastle, Australia, in 1976. He isa Chair Professor in Electrical Asset Managementwith the School of Electrical Engineering and Com-puter Science, Queensland University of Technology,Brisbane QLD, Australia. His research interests arein the areas of power systems, power electronics,and wide area control of smart grid.

Chunjie Zhou received the Ph.D. degree in controlscience and engineering from Huazhong Universityof Science and Technology, Wuhan, China, in 2001,where he is currently a professor at the School ofAutomation. His research interests include industri-al communication, artificial intelligence, networkedcontrol systems, and security of industrial controlsystems.

Wenli Du received the Ph.D. degree in controltheory and control engineering from East ChinaUniversity of Science and Technology, Shanghai,China, in 2005, where she is currently the Dean andProfessor at the College of Information Science andEngineering. Her research interests include controltheory and applications, system modeling, advancedcontrol, and process optimization.

Feng Qian received the Ph.D. degree in automationfrom East China Institute of Chemical Technology,Shanghai, China, in 1995. He is currently a VicePresident and Professor with the East China Uni-versity of Science and Technology, and a mem-ber of the Chinese Academy of Engineering. Hiscurrent research interests include modeling, control,optimization, and integration of complex industrialprocesses, neural networks, and real-time intelligentcontrol with industrial applications.