Performance Analysis of Reduced CommunicationNetwork in DC Microgrid

A. B. Shyam, Anoop Ingle, Soumya Ranjan Sahoo and Sandeep Anand

Department of Electrical Engineering, Indian Institute of Technology Kanpur, India.

Email: shyam.ab33@gmail.com, anoopingle9@gmail.com, srsahoo@iitk.ac.in and asandeep@iitk.ac.in

Abstract—Distributed secondary controllers for dc microgridshave proven to be more effective and reliable as comparedto decentralized and centralized controllers, respectively. Thedistributed secondary controller relies on information exchangebetween the distributed units. Conventionally, a full communica-tion network topology is used, wherein all controllers communi-cate with each other. Recently, adoption of consensus control inDC microgrids have enabled achieving good performance usingreduced communication network. This work aims at comparingthe performance of the secondary controller with reduced andfull communication topologies. It is shown that with propertuning of gains, the secondary controller under reduced commu-nication has similar performance as under full communication.The results are substantiated with numerical simulations usingMATLAB/Simulink..

Index Terms—DC microgrid, dominant poles, full communi-cation, reduced communication, secondary control.

I. INTRODUCTION

Ever increasing energy demand has led to an increase in

deployment of distributed generation sources in the electrical

system. Less capital cost, reduction in transmission losses and

remote electrification are other important advantages of a dis-

tributed generation system. Microgrid refers to a distribution

system with distributed storage, loads and generation, which is

capable of operating in islanded mode or in standalone mode.

The term microgrid is used to refer those sources, storage

units and loads which can operate either in standalone mode

or in grid-connected mode. As most of the power electronic

loads are dc in nature, conversion losses are less in case of dc

microgrid as compared to ac microgrid. Absence of harmonics,

reactive power and need for synchronization is making dc

microgrid popular in remote electrification.

With multiple sources available in a dc microgrid, each

source is required to supply current in proportion to their

scheduled values, which in turn are determined by microgrid

central controller (MGCC). In case of absence of MGCC,

current supplied by storage units is typically shared in pro-

portion to their rated values. This ensures increase in the

availability and reliability of supply for longer duration. To

achieve this, in literature, voltage droop control is suggested,

where the terminal voltage of a source decreases as the current

supplied by it increases. As a result, proportional current

sharing is achieved at the expense of poor voltage regulation.

To overcome this problem, secondary controllers are used, as

per the hierarchical control structure defined in [1]–[3], which

generate the reference signal for the primary droop controller.

Based on the communication network involved, secondary

controllers are classified into three categories, namely, central-

ized, decentralized and distributed secondary controllers [3]–

[5]. In centralized secondary controllers given in [2], [6], a

central controller receives the network data of the microgrid,

processes it and sets reference signal for the primary con-

trollers of different sources. The shortcomings of centralized

controllers includes high bandwidth communication channel

between each source and the centralized controller and low

reliability due to single point of failure.

Decentralized secondary controllers are available in each

distributed sources, and make use of the local information

available with the source to generate reference signal for the

inner primary control loop [7]. As only local information is

used, they do not require communication links between the

distributed sources and are thus are more reliable and require

less cost for realization. However, their performance suffered

due to unavailability of communication network.

Similar to decentralized controllers, distributed secondary

controllers are designed to work with individual sources,

but share information with controllers of other sources [8],

[9]. Thus, the distributed controllers are free from single

point of failure as well as has better performance compared

to decentralized controller. Due to these advantages, these

controller are one of the most popular in literature.

Distributed controller can be implemented with full (all-to-

all) communication or reduced communication. In distributed

controller with full communication, each controller receives

information from every source in the network. With this, a

system with n sources requires (n−1)×(n−2) · · · 2×1 com-

munication links. Thus, large number of communication links

are required. Further, adding a new source requires adding nmore communication links, thereby reducing the scalability.

On the contrary, with reduced communication network, each

controller exchanges information with some of the sources,

which are referred as cyber-neighbours. Moreover, addition of

a new source requires only a few links to be added, and thus

is easily scalable.

Some of the drawbacks of reduced communication topology

based distributed secondary controller are as follows. As the

controller works with local averaging of quantities, the time

976978-1-5386-4291-7/18/$31.00 c©2018 IEEE

domain responses are poor compared to that of one with full

communication. In this paper a comparison is done between

distributed secondary controller employing full communica-

tion [10] and reduced communication [11]. State equations

are derived for both the cases and and conclusions on time

domain specifications are drawn from the eigenvalues of the

system matrix derived.

Rest of the paper is organised as follows: Section II dis-

cusses the secondary control algorithm used for the analysis.

Necessary mathematical model of the dc microgrid common

to both full and reduced communication, is also presented.

Section III and IV give the complete model of the dc mi-

crogrid system employing full communication and reduced

communication, respectively. Dominant poles of the system

which determine the settling time of the system in both the

cases is discussed. Section V discusses the simulation studies

carried out on a dc microgrid system to validate the findings.

Section VI summarizes this paper.

II. SECONDARY CONTROLLER UNDER CONSIDERATION

The secondary controllers used for the study are illustrated

in Fig. 1. The basic control algorithm used for the analysis

introduces a vertical shift in droop characteristics to ensure

low voltage regulation while keeping a constant high value of

droop gain to ensure proportional current sharing. With the

assumption that the inner control loops are faster, the output

voltage vi is assumed equal to the reference voltage vrefi . The

control equation for source-i is given by

vi = vnomi +Δvi − diii (1)

Δvi = kiii (2)

where vnomi is the nominal voltage, di is the droop coefficient,

ii is the converter current, Δvi is the vertical shift in droop

characteristics, ki is a positive constant and ii is the average

current calculated by source-i. All quantities mentioned are in

per unit. The value of droop dj is chosen to be very high so

as to make currents nearly equal in all converters. Now that

the converter currents are nearly equal, poor voltage regulation

due to high droop is compensated by having ki a little less

than di. With the assumption d1 = d2 = · · · = dn = d and

k1 = k2 = · · · = kn = k, the controller equation (1) and (2)

are written in matrix form as

v = vnom +Δv − di (3)

Δv = ki (4)

where v, vnom, i and i are column vectors of n dimension,

and d and k are scalars. On combining the above equations

and by applying small-signal analysis, we have

v = ki− di. (5)

Interconnecting cable between source-i and j is modelled

as a combination of resistor rij and inductor lij . The equation

for this cable is

vij = lijdiijdt

+ rijiij . (6)

The branch equations for the entire microgrid is combined and

expressed as

vbr = Lbrdibrdt

+Rbribr. (7)

where vbr and ibr are column vectors of b dimension, and

Lbr and Rbr are diagonal matrices of b × b dimension. The

small-signal equation around the operating point is given by

vbr = Lbrdibrdt

+Rbribr. (8)

In similar way, the small-signal equations for load capacitor

is obtained as

ic = Cdv

dt(9)

where ic is a column vector of n dimension representing

output capacitor currents, and C is a diagonal matrix of n×ndimension containing values of output capacitors.

The branch voltages and currents are related to output

voltages of sources as

vbr = Mv (10)

i− ic − iL = MT ibr (11)

where iL is the perturbed load current vector of dimension nand M is the incidence matrix of dimension b× n. For each

branch-ij in the system, corresponding row of M has +1 at

ith column, −1 at jth column and 0 at remaining columns.

On substituting (10) in (8) and rearranging leads to

dibrdt

= −L−1br Rbribr + L−1

br Mv. (12)

Similarly, from (9) and (11), we have

dv

dt= C−1

(i− iL −MT ibr

). (13)

Laplace transforms of (5), (12) and (13) gives

V(s) = kI(s)− dI(s) (14)

(sIb + L−1br Rbr)Ibr(s) = −L−1

br MV(s) (15)

C−1(I(s)− IL(s)−MTIbr(s)

)= V(s) (16)

where V(s), Ibr(s), I(s) and IL(s) are the Laplace transforms

of v, ibr,ˆi and iL, respectively. Equations (14)–(16) will be

used in the subsequent section for analysing the controller

under study with full communication and reduced communi-

cation.

III. SYSTEM EQUATION FOR CONTROLLER WITH

FULL COMMUNICATION

For controller with full communication, source-i calculates

the average current by adding all the source currents and

dividing with the total number of sources, ie.,

ii(t) =1

n

n∑j=1

ij(t). (17)

2018 IEEE Innovative Smart Grid Technologies - Asia (ISGT Asia) 977

ii =1n

n∑j=1

ij ki ++−

To

inn

er

con

trol

loo

p

vnomi

diii

ij ii

Δvi

COMMUNICATION CHANNEL

(a)

∑j∈ni

maij (ij − ii)∫

++

ki ++−

To

inn

er

con

trol

loo

p

vnomi

diii

ijΔviii

COMMUNICATION CHANNEL

(b)

Fig. 1. Secondary controllers under study based on (a) full communication [10] and (b) reduced communication [11] network.

The small-signal equation of average current calculated by all

sources is represented in matrix form as

ˆi =1

n1 1T i (18)

where 1 ∈ Rn×1 is a vector whose all elements are equal to

1. Substituting for ˆi in (5), we have

i = −1

d

(In − k

nd1 1T

)−1

v (19)

where In is an identity matrix of size n. On taking Laplace

transform and using VF(s) to represent Laplace transform of

v with full communication, we have

I(s) = −1

d

(In − k

nd1 1T

)−1

VF(s) (20)

The matrix formed by 1 1T has the eigenvalues 0 with

multiplicity (n− 1), and n with multiplicity 1. Thus,

det

(In − k

nd1 1T

)= 1(n−1)

(d− k

d

)(21)

Thus,

I(s) =

(d− k

d

)adj

(In − k

nd1 1T

)(−VF(s)

d

)(22)

Hence, for full communication, the settling time of the system

is given by the dominant poles of VF(s). Equations (15), (16)

and (22) are rewritten to form the state equation as⎡⎢⎣(sIb + L−1

br Rbr) −L−1br M

C−1MT

(sIn +

C−1

d

(In − k

nd1 1T

)−1)⎤⎥⎦

[Ibr(s)VF(s)

]=

[0

−C−1

]IL(s) (23)

IV. SYSTEM EQUATION FOR CONTROLLER WITH

REDUCED COMMUNICATION

For controller with reduced communication, source-i re-

ceives information of estimated average currents from neigh-

bouring sources and updates its estimated average current as

ii(t) = ii(t) +

∫ t

0

∑j∈ni

maij (ij(τ)− ii(τ)) dτ . (24)

where, aij is the adjacency matrix entry corresponding to

communication from source-j to source-i and m is a pos-

itive constant. On perturbing, taking Laplace transform and

representing in matrix form, we have

I(s) = s(sIn +mL )−1I(s). (25)

where L is the Laplacian matrix of the communication graph.

Substituting the above expression in (14) and using VR(s) to

represent Laplace transform of v with reduced communication,

we have

I(s) = −VR(s)

d+

k

ds(sIn +mL )−1I(s) (26)

which on rearranging gives

I(s) =(d− k

dsIn +mL

)−1

(sIn +mL )

(−VR(s)

d

)

= B−1A

(−VR(s)

d

)(27)

where A = [sIn +mL ] and B = [((d− k)/d)sIn +mL ].Lemma 1: For a connected undirected graph, mL has one

eigenvalue at 0. Let the other eigenvalues of mL be 0 <λ1 ≤ λ2 ≤ · · · ≤ λn−1. The poles added by the determinant

of B are s (d− k) /d, [s(d − k)/d] + λ1, [s(d − k)/d] + λ2,

· · · , [s(d− k)/d] + λn−1.

Proof: See the Appendix. �Lemma 2: The product of adjoint of B and A cancels out

the pole at origin.

Proof: See the Appendix. �Thus (27), takes the form

I(s) =dn

(d− k)

1

(s(d− k) + dλ1)

1

(s(d− k) + dλ2)

· · · 1

(s(d− k) + dλn−1)G

(−VR(s)

d

)(28)

where G is a matrix with each of its element being a

polynomial with degree (n−1) and hence does not contribute

to the system poles.

From (28), the poles of the system other than those of

VR(s) are −dλ1/(d−k),−dλ2/(d−k), · · · ,−dλn−1/(d−k).Since d > k ≥ 0 and k and d are chosen to be close to

each other, we have dλ1/(d − k) > λ1, dλ2/(d − k) >λ2, · · · , dλn−1/(d − k) > λn−1. By selecting m to be large

we have λ1 to be significantly large as well. If, VR(s) has

978 2018 IEEE Innovative Smart Grid Technologies - Asia (ISGT Asia)

Fig. 2. Four bus dc microgrid system used for simulation study.

(a) (b)

Fig. 3. Communication topology used for simulation study. (a) Full commu-nication, (b) Reduced communication.

Fig. 4. Locus of dominant poles of the system with reduced communicationon varying m.

the same dominant poles as VF(s), then we can conclude

that the settling time of the system for both full and reduced

communication will be similar.

State equation for reduced communication is written using

(15), (16) and (27) as

⎡⎣(sIb + L−1

br Rbr) −L−1br M

C−1MT

(sIn +

C−1

dB−1A

)⎤⎦[

Ibr(s)VR(s)

]

=

[0

−C−1

]IL(s) (29)

Fig. 5. Simulation results for the system with full communication. Convertercurrents; Yaxis: 0.05 A/div, Y-min: 0.5 A, Y-max: 0.75 A; output voltages;Y-axis: 20 V/div, Y-min: 340 V, Y-max: 440 V, X-axis: 0.05 s/div.

V. SIMULATION STUDIES

In order to compare the dominant poles and settling time of

a system with full and reduced communication, a simulation

study is carried out for a 400 V dc microgrid system of four

converters shown in Fig. 2. For the given circuit, the incidence

matrix which relates node and branch voltages is given by

M =

⎡⎢⎢⎣

1 −1 0 00 1 −1 00 0 1 −1−1 0 0 1

⎤⎥⎥⎦

All the branch resistances are 205 mΩ. Hence, Rbr = 0.205I4.

Branch inductance is taken to be 0.463 mH. So, that gives

Lbr = 0.463 × 10−3I4. The loads connected at each bus are

4 Ω, 6 Ω, 6.5 Ω and 7 Ω, respectively. The value of droop gain

is set to 1.9 Ω. This droop gain is high enough as compared

to transmission line resistances. The value of k is 1.8 Ω. For

such a system, with full communication, the most dominant

pole for estimated perturbed average current of the system for

each converter is at −442.8. The other dominant poles have

real part −747.7 as can be seen from Fig. 4.For reduced communication, the communication graph is

shown in Fig. 3(b). For such a graph, the Laplacian matrix is,

L =

⎡⎢⎢⎣

2 −1 0 −1−1 2 2 −10 −1 2 −1−1 0 −1 2

⎤⎥⎥⎦

The value of m is first set to 40. The dominant pole

corresponding to source 1 is located at −442.8 which is

the same as that for full communication. This is the pole

contributed by sIb + L−1br Rbr. Additionally there are three

pair of complex conjugate poles with their real parts ranging

from −650.6 and −695.6. These are the poles contributed by(sIn + (C−1/d)B−1A

). The dominant poles of the reduced

2018 IEEE Innovative Smart Grid Technologies - Asia (ISGT Asia) 979

(a)

(b)

(c)

Fig. 6. Simulation results for the system with reduced communication. (a)m = 40, (b) m = 100, (c) m = 240. Converter currents; Y-axis: 0.05 A/div,Y-min: 0.5 A, Y-max: 0.75 A; output voltages; Y-axis: 20 V/div, Y-min: 340V, Y-max: 440 V, X-axis: 0.05 s/div.

communication system for m = 40 is shown in Fig. 4. These

poles are close to −442.8 and significantly increase the settling

time of the system.

The trend for the dominant poles for increasing values

of m is shown in Fig. 4. Value of m is varied from 40

to 240 in steps of 40. For m = 40, poles contributed by(sIn + (C−1/d)B−1A

)are very close to the poles con-

tributed by sIb + L−1br Rbr. As m increases, the pole con-

tributed by sIb + L−1br Rbr remains at −442.8, whereas, the

poles contributed by(sIn + (C−1/d)B−1A

)move towards

−747.7. The conclusion drawn from such phenomena is that

the settling time decreases with increasing value of m.

In order to find the settling time of the system, load

connected at bus-1 is changed to 2.5 Ω at t = 0.1 s and

again back to 4 Ω at t = 0.3 s. The plots for voltage and

current of the converters with full communication are shown

in Fig. 5. The settling time of the system for both instances of

load change is 0.0746 s. The same is repeated with reduced

communication for m = 40, 100 and 240, and the voltage and

current waveforms are shown in Fig. 6. The system settles to

±2% band of steady state value after 0.1536 s for both load

change when m = 40. The settling time in the case where

m = 100 is 0.075 s which is better than for m = 40. For

m = 240, settling time decreases to 0.0746 s which is same as

in full communication. Thus, for the given system, choosing

a high value of m for reduced communication gives results

similar to full communication.

VI. CONCLUSION

In this paper, performance of distributed secondary con-

troller with full and reduced communication network are

compared. System model are developed for both the cases and

dominant poles, which determine the settling time of the sys-

tem are found out. For full communication, the dominant poles

of the system are determined by the matrix sIb + L−1br Rbr. For

reduced communication, the same matrix gives the dominant

poles for a high m. This makes the decay rate same for

both full and reduced communication. Hence, it is concluded

that for higher values of m, the performance of the system

with reduced communication is same as that of one with full

communication. Reduced number of links does not affect the

system performance, given that the value of m is high. This

finding is also observed during the simulation studies. The

settling time for the system employing reduced communication

improves with increasing m and for a significantly high m,

settling time for reduced and full communication becomes

same.

APPENDIX

A. Proof of Lemma 1

In the matrix B, s[(d−k)/d]In which is a diagonal matrix is

added to mL . Hence, factors of determinant of B are s[(d−

980 2018 IEEE Innovative Smart Grid Technologies - Asia (ISGT Asia)

k)/d], s[(d−k)/d]+λ1, s[(d−k)/d]+λ2, · · · , s[(d−k)/d]+λn−1, and thus

det(B) =s(d− k)

d

s(d− k) + dλ1

d

s(d− k) + dλ2

d· · ·

s(d− k) + dλn−1

d(30)

�B. Proof of Lemma 2

Let the matrix mL be

mL =

⎡⎢⎢⎢⎣l11 l12 · · · l1nl21 l22 · · · l2n...

.... . .

...

ln1 ln2 · · · lnn

⎤⎥⎥⎥⎦

Then the B matrix is of the form of

B =

⎡⎢⎢⎢⎣sr + l11 l12 · · · l1n

l21 sr + l22 · · · l2n...

.... . .

...

ln1 ln2 · · · sr + lnn

⎤⎥⎥⎥⎦

where r is a constant. Kirchhoff’s Matrix-Tree Theorem states

that for a connected graph Laplacian mL , all its cofactors are

equal and the common value is the number of spanning trees

in mL . From this information, it is easy to see that for B,

adj(B) =

⎡⎢⎢⎢⎣sf11(s) + p sf12(s) + p · · · sf1n(s) + psf21(s) + p sf22(s) + p · · · sf2n(s) + p

......

. . ....

sfn1(s) + p sfn2(s) + p · · · sfnn(s) + p

⎤⎥⎥⎥⎦

where, p is the spanning tree of mL and fij(s) is a polynomial

function of degree n − 2 corresponding to ith row and jth

column. Now, adj(B) is multiplied by A. The element in ithrow and jth column for the multiplied matrix looks like

[adj(B)A]ij =n∑

k=1

(sfik(s) + p)(s+ lkj)

=n∑

k=1

(s2fik(s) + s(fik(s)lkj + p) + plkj)

For a connected graph,∑n

k=1 lkj = 0. Therefore,

[adj(B)A]ij = sGij

where, Gij =∑n

k=1(sfik + (fik(slkj + p)). Thus adj(B)Acancels out one pole at origin. �

ACKNOWLEDGMENT

This work was supported in part by INDO-UK Center for

Education and Research in Clean Energy, Govt. of India and in

part by UI-ASSIST: US-India collAborative for smart diStribu-

tion System wIth STorage, Indo-US Science and Technology

Forum, Government of India.

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2018 IEEE Innovative Smart Grid Technologies - Asia (ISGT Asia) 981