Distributed Cooperative Control of Micro-grids talks/2013 coop ctrl of...Distributed Cooperative Control of Micro-grids Supported by NSF, ARO, AFOSR Work of Ali Bidram with Dr. A

Distributed Cooperative Control of Micro-grids

Invited by Gary Feng

Thanks to Wen J. Li, Yong LiuBo YumingT.J. Tarn

Jie Huang Ning Xi

4

Professor Shengyuan XuVice Dean of School of Automation

6

7

Shi nian su muBai nian su ren

Wu nian su shueshang

UTA Research Institute (UTARI)The University of Texas at Arlington

F.L. LewisMoncrief-O’Donnell Endowed Chair

Head, Controls & Sensors Group

Distributed Cooperative Control of Micro-grids

Supported by NSF, ARO, AFOSR

Work of Ali Bidram with Dr. A. Davoudi

He who exerts his mind to the utmost knows nature’s pattern.

The way of learning is none other than finding the lost mind.

Meng Tz500 BC

Man’s task is to understand patterns in natureand society.

Mencius

Sun Tz bin fa孙子兵法

500 BC

Books Coming

F.L. Lewis, H. Zhang, A. Das, K. Hengster‐Movric, Cooperative Control of Multi‐Agent Systems: Optimal Design and Adaptive Control, Springer‐Verlag, 2013, to appear.

Key Point

Lyapunov Functions and Performance IndicesMust depend on graph topology

Hongwei Zhang, F.L. Lewis, and Abhijit Das“Optimal design for synchronization of cooperative systems: state feedback, observer and output feedback,”IEEE Trans. Automatic Control, vol. 56, no. 8, pp. 1948‐1952, August 2011.

OutlineCooperative ControlElectric Power MicrogridsCooperative Control for Synchronization in Microgrids

Synchronized Motion of biological groups

Fishschool

Birdsflock

Locustsswarm

Firefliessynchronize

The Power of Synchronization Coupled OscillatorsDiurnal Rhythm

1

2

3

4

56

Diameter= length of longest path between two nodes

Volume = sum of in-degrees1

N

ii

Vol d

Spanning treeRoot node

Strongly connected if for all nodes i and j there is a path from i to j.

Tree- every node has in-degree=1Leader or root node

Followers

Communication Graph

Communication Graph1

2

3

4

56

N nodes

[ ]ijA a

0 ( , )ij j i

i

a if v v E

if j N

oN1

Noi ji

jd a

Out-neighbors of node iCol sum= out-degree

42a

Adjacency matrix

0 0 1 0 0 01 0 0 0 0 11 1 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 1 0

A

iN1

N

i ijj

d a

In-neighbors of node iRow sum= in-degreei

(V,E)

i

Dynamic Graph- the Distributed Structure of ControlEach node has an associated state i ix u

Standard local voting protocol ( )i

i ij j ij N

u a x x

1

1i i

i i ij ij j i i i iNj N j N

N

xu x a a x d x a a

x

( )u Dx Ax D A x Lx L=D-A = graph Laplacian matrix

x Lx

If x is an n-vector then ( )nx L I x

x

1

N

uu

u

1

N

dD

d

Closed-loop dynamics

i

j

[ ]ijA a

1

2

3

4

56

Theorem. Graph contains a spanning tree iff e-val of L at is simple.

Graph strongly connected implies exists a spanning tree

Then 2 0

Then -L has one e-val at zero and all the rest stable

1 0

1 1

( ) (0) (0) (0)i i

N Nt tLt T T

i i i ij j

x t e x v e w x w x e v

Consensus Value and Convergence Rate

x Lx Closed-loop system with local voting protocol

Modal decomposition

Let be simple. Then for large t1 0

2 1 22 2 1 1 2 2

1( ) (0) (0) (0) 1 (0)

Nt t tT T T

j jj

x t v e w x v e w x v e w x x

2 determines the rate of convergence and is called the FIEDLER e-value

1 0

and the Fiedler e-val 2There is a big push to find expressions for the left e-vector for

Let graph have a spanning tree. Then all nodes reach consensus.

1

2 3

4 5 6

12

3

4 5

6

Graph Eigenvalues for Different Communication Topologies

Directed Tree-Chain of command

Directed Ring-Gossip networkOSCILLATIONS

Graph Eigenvalues for Different Communication Topologies

Directed graph-Better conditioned

Undirected graph-More ill-conditioned

65

34

2

1

4

5

6

2

3

1

Synchronization on Good Graphs

Chris Elliott fast video

65

34

2

1

1

2 3

4 5 6

Synchronization on Gossip Rings

Chris Elliott weird video

12

3

4 5

6

Controlled Consensus: Cooperative Tracker

Node state i ix uDistributed Local voting protocol with control node v

( ) ( )i

i ij j i i ij N

u a x x b v x

( ) 1x L B x B v i i

i ij i ij j ij N j N

u a x a x b v

0ib If control v is in the neighborhood of node i

{ }iB diag b

Theorem. Let graph have a spanning tree and for at least one root node. Then L+B is nonsingular with all e-vals positiveand -(L+B) is asymptotically stable

0ib

control node v

Ron Chen – pinning control

Local Neighborhood Tracking Error

These beautiful pictures are from a lecture by Ron Chen, City U. Hong KongPinning Control of Graphs

Natural and biological structures

26

What is a micro‐grid?• Micro-grid is a small-scale power system that provides the power for

a group of consumers.• Micro-grid enables

local power support for local and critical loads.

• Micro-grid has the ability to work in both grid-connected and islanded modes.

• Micro-grid facilitates the integration of Distributed EnergyResources (DER).

Photo from: http://www.horizonenergygroup.com

• The main building block of smart-grids

• Rural plants

• Business buildings, hospitals,and factories

Smart‐grid photo from: http://www.sustainable‐sphere.com

An introduction to micro‐grids: Micro‐grid applications

Distributed Generators (DG)Distributed Energy Resources (DER)

• Non-renewables Internal combustion engine Micro-turbines Fuel cells

• Renewables Photovoltaic Wind Biomass

29

Micro‐grid Advantages

• Micro-grid provides high quality and reliable power to the criticalconsumers

• During main grid disturbances, micro-grid can quickly disconnectform the main grid and provide reliable power for its local loads

• DGs can be simply installed close to the loads which significantlyreduces the power transmission line losses

• By using renewable energy resources, a micro-grid reduces CO2emissions

30

Micro‐grid Control Challenges

• In Grid-connected mode, the main grid has rotating synchronousgenerators that provide a frequency reference

• In Grid-connected mode, the main grid provides voltage support andpower quality

• During grid disturbances, micro-grid goes to islanded mode toprovide the power for its local loads

• In islanded mode, there is no frequency reference• In islanded mode, microgrid controller must provide voltage support

and power quality

31

• Voltage and frequency control for both grid-connected and islanded operating modes

• Proper load sharing and DG coordination• Power flow control between the microgrid and the main grid• Optimizing the microgrid operating cost

Hierarchical control structure32

An introduction to micro‐grids: Micro‐grid Objectives

Micro‐grid Hierarchical Control Structure

Tertiary ControlmodesOptimal operation in both operating

modes

Secondary Control

Primary Control

MicrogridTie

Power flow control in grid-tied mode

Voltage deviation mitigationFrequency deviation alleviation

Voltage stability provision

preservingFrequency stability

preservingPlug and play capability for DGs

Main grid

Do coop. ctrl. here toSynchronize frequencyand voltage

Bidram, A., & Davoudi, A. (2012). Hierarchical structure of microgrids control system. IEEE Transactions on Smart Grid, DOI: 10.1109/TSG.2012.2197425.

Maintains Stability

Micro‐grid Primary Control

Primary control: The primary control maintains voltage andfrequency stability

Conventional primary control: Droop techniques

n P

mag n Q

m PE v V n Q

Power calculator

vo

io

Q

P

ω

E E

ω

*Reference

voltage

Esin(ωt)

vo

P

Q

2

1 max1 maxP PN Nm P m P

1 max1 maxQ QN Nn Q n Q

Microgrid load conditions Resulting

Power

Droop Control

Required voltage and frequencyTo maintain stability

35

P

n

DG1 DG2min

max1P max 2P

2mP1mP

min max11 ( ) / PnmP min max 22 ( ) / PnmP

Design of Droop Control Parameters

Pick slopes so that

Then

1 max1 maxP PN Nm P m P

Balanced Load sharing

n Pm P

• Primary control (frequency droop)• Before islanding occurs

36

DG1 nominal power, Pmax1= 4 kW

DG2 nominal power, Pmax2= 6 kW

DG1

DG2

1ov

2ov

1bv

2bv

1cR 1cL

2cL2cR

3.5kW 1lineR

1lineL

Maingrid

3.5kW

1kW2.5kW

3.5kW

1 max1 2 max2P Pm P m P

P

ref

n

2.5kW 3.5kW

DG1 DG2min

Micro‐grid primary control1. Connected to Main Power Grid

Load is 7kWGrid supplies 1kW

n Pm P

37

( k W )P

re f

n

DG1DG2

m in

n e w

2 n ewP1n e wP1o ldP 2 o ldP2.5 2.8 3.5 4.2

Micro‐grid Primary Control2. After Islanding

DGs must make up an extra 1kW Pload= 7kWNew P1 + P2= 2.8kW + 4.2kW

Makes frequency decrease

Increase to restore frequency to ref valuen

i

iP

ni

maxP i

Pim

New Secondary Control Input for Frequency Synchronization

i ni Pi im P

Change

To synchronize

ni

i

Secondary Control input

Secondary Frequency Control

40

Secondary control: The secondary control restores the voltage andfrequency of the micro-grid to their nominal value.

Current Secondary control implementation: Centralized structure

Low reliability Requires a Central control authority Requires too many communication links

We want to develop a new Distributed Control structure Highly reliable Uses sparse communication network

41

( ) ( )

( ) ( )

n PE ref mag IE ref mag

n P ref I ref

V K v v K v v dt

K K dt

Micro‐grid secondary control

New Decentralized Secondary Control

• Decentralized control can beimplemented through multi-agent cooperative control.

• DG are assumed to be on acommunication graph.

• The decentralized controller oneach DG accesses the voltageand frequency information ofother neighbor DGs based onthe graph topology.

42

DG 1

DG 2

DG 3

ref and Vref

1 , V1

2 , V2

3 , V3

43

Microgrid

DG 1DG 2 DG 3

DG 4

DG 5

DG 6DG 7

DG 8

DG 1DG 2 DG 3

DG 4

DG 5

DG 6

DG 8

DG 7

Communication link

Cybercommunication

framework

Micro‐grid secondary control:Distributed CPS structure

Physical LayerThe interconnect structure of the power grid

Cyber layerA sparse, efficient communication network to allow

cooperative control for synchronization ofvoltage and frequency

Work of Ali BidramWith Dr. A. Davoudi

Cyber Physical System (CPS)

Dynamical model of a DG

44

vo io

VSC

vod*

iLd*

Currentcontroller

Voltagecontroller

LC filteriL

Power controller

vb

, voq*

vod , voq

iod , ioq

, iLq*

ω

ωn Vn

Outputconnector

Rc Lc

abc/dq

iLd , iLq

Rf Lf Cf

VSC‐ Voltage source converter Power electronics

Renewable DERProvides DC voltage

Primary ControlDroop control is here

MicrogirdNetworkLoad disturbances

Given load conditions ‐ pick using Droop to maintain stability0 0,v i * *, ,od oqv v

Primary Control Structure

iv


Power controller dynamics:

45

( ) ( ) ( )( )

i i i i i i i i i

i i i i i

uy h d u

x f x k x D g xx

13i x

abc/dqvoi

ioi

vodivoqiiodiioqi

vodi iodi + voqi ioqi

voqi iodi - vodi ioqi

Low-passfilter

ωni - mPi PiPi

Low-passfilter

Vni - nQi Qi

Qi

ωi

vodi*

voqi*0

ωni

Vni

[ ]Ti i i i di qi di qi ldi lqi odi oqi odi oqiP Q i i v v i i x

i i com

( )i ci i ci odi odi oqi oqiP P v i v i

( )i ci i ci oqi odi odi oqiQ Q v i v i

Pogaku, N., Prodanovic, M., & Green, T. C. (2007). Modeling, analysis and testing of autonomous operation of an inverter‐based microgrid. IEEE Transactions on Power Electronics, 22(2), 613–625.

Droop control is here

Heterogeneous agent dynamics


Voltage controller dynamics

46

Σ

Σ

vodi*

voqi*

vodi

voqi

KPViKIVi

+ s+

+

_

_

KPViKIVi

+ s

ωbCfi

ωbCfi

Fi

Fi

Σ

Σ

vodi

voqi

+

+

_Σ

Σ

+

+

iodi

ioqi

+

++

ildi*

ilqi*

* ,di odi odiv v

* ,qi oqi oqiv v

* *( ) ,ldi i odi b fi oqi PVi odi odi IVi dii Fi C v K v v K

* *( ) ,lqi i oqi b fi odi PVi oqi oqi IVi qii Fi C v K v v K


Current controller dynamics

47

*di ldi ldii i

*qi lqi lqii i

* *( )idi b fi lqi PCi ldi ldi ICi div L i K i i K

* *( )iqi b fi ldi PCi lqi lqi ICi qiv L i K i i K

Σ

Σ

vidi*

viqi*

KPCiKICi

+ s+

+

_

_

KPCiKICi

+ s

ωbLfi

ωbLfi

Σ

Σ

+

+

_

ilqiildi

+

ildi*

ilqi*

ildi

ilqi


Output filter dynamics

48

Output connector dynamics

1 1fildi ldi i lqi idi odi

fi fi fi

Ri i i v v

L L L

1 1filqi lqi i ldi iqi oqi

fi fi fi

Ri i i v v

L L L

1 1odi i oqi ldi odi

fi fiv v i i

C C

1 1oqi i odi lqi oqi

fi fiv v i i

C C

1 1ciodi odi i oqi odi bdi

ci ci ci

Ri i i v vL L L

1 1cioqi oqi i odi oqi bqi

ci ci ci

Ri i i v vL L L

Depends on microgrid conditions and loads

Voltage disturbances

Synchronization in Microgrid of Interconnected DG

DER 8 DER 6

DER 4

Rline1 Lline1

Pload1+jQload1

Rline2 Lline2Rline3 Lline3

Rc4Lc4

Rc3Lc3

Rc2Lc2

Rc1Lc1

vo4vo3vo2vo1

Pload2+jQload2

DER 3DER 2DER 1

DER 5DER 7

Pload3+jQload3Pload4+jQload4

Rline7 Lline7 Rline6 Lline6 Rline5 Lline5

Rline4

Lline4

vo5vo6vo7vo8

Lc5Lc6Lc7Lc8Rc8 Rc7 Rc6 Rc5

DG 1 DG 2 DG 3 DG 4

DG 8 DG 7 DG 6 DG 5

2 2,o magi odi oqiv v v Voltage synchronization (per unit)

i i ni Pi iy m P Frequency synchronization

Voltage synchronization

Secondary Control Synchronization Objectives

vo io

VSC

vod*

iLd*

Currentcontroller

Voltagecontroller

LC filteriL

Power controller

vb

, voq*

vod , voq

iod , ioq

, iLq*

ω

ωn Vn

Outputconnector

Rc Lc

abc/dq

iLd , iLq

Rf Lf Cf

VSC‐ Voltage source converter Power electronics



Voltage synchronization (per unit)

Frequency synchronization

Secondary Control InputsChange Droop control parameters to get synchronization

1. For secondary frequency control:

2. For secondary voltage control:

51

( ) ( ) ( )( )

i i i i i i i i i

i i i i i

uy h d u

x f x k x D g xx

13i x

i odiy v

i niu V

i i ni Pi iy m P

i niu

2 2,o magi odi oqiv v v 0id

0id

DG Microgrid Model and Synchronization Control Objectives

Heterogeneous Agent Dynamics – different dynamics

1. Distributed secondary frequency control of micro-grids2. Distributed secondary voltage control of micro-grids


1. Secondary Frequency Control

Secondary frequency control objective

DG 1

DG 2

DG 3

ref

,i ref i

The frequency of each DG synchronizes to a command nominal value

50ref Hz

For example

i ni Pi im P Droop control relationship


i

iP

ni

maxP i

Pim

New Secondary Control Input for Frequency Synchronization

i ni Pi im P

Change

To synchronize

ni

i

Secondary Control input


1. Secondary frequency control

i ni Pi im P

i ni Pi i im P u

i i iu c e

( ) ( )i

i ij i j i i refj N

e a g

[ ] N Nija

DG 1

DG 2

DG 3

ref

Droop control relationship

1 1 1 1

2 2 2 2

P

P

N PN N N

m P um P u

m P u

For all N DGs

Local Neighborhood Tracking Error

Differentiate Droop Control relationshipFeedback linearization


56

i ni Pi im P

i ni Pi i im P u

i i iu c e

( ) ( )i

i ij i j i i refj N

e a g

Theorem . Let the digraph of the communication network have a spanning tree and the pinning gain be nonzero for at least one DG placed on a root node.

Let the auxiliary control be chosen as above.

Then, the global neighborhood error is asymptotically stable. Moreover, the DG frequencies synchronize to

iu

ref

Droop control relationship

Using input-output feedback linearization

Books Coming

F.L. Lewis, H. Zhang, A. Das, K. Hengster‐Movric, Cooperative Control of Multi‐Agent Systems: Optimal Design and Adaptive Control, Springer‐Verlag, 2013, to appear.

Key Point

Lyapunov Functions and Performance IndicesMust depend on graph topology

Hongwei Zhang, F.L. Lewis, and Abhijit Das“Optimal design for synchronization of cooperative systems: state feedback, observer and output feedback,”IEEE Trans. Automatic Control, vol. 56, no. 8, pp. 1948‐1952, August 2011.


Lyapunov function: 1 1, where2

T

iV e Pe P diag w

1 2T N

Ne e e e

( ) NL G w 1satisfiesiw

Differentiating V

( )( ) ( )( )T TV e P L G e P L G u

,refe L G u c e

( ( ) ( ) )2

T TcV e P L G L G P e

0V

{ }ic diag c

Theorem of Zhihua Qu

Proof. Lyapunov Functions for Cooperative Control on GraphsGraph Laplacian Matrix L D

( ) ( ) 0TQ P L G L G P Coop ctrl Lyapunov equationQu, Z., (2009).Cooperative control of dynamical systems: Applications to autonomous vehicles. New York: Springer‐Verlag.


59

ref

ij N

( ) ( )ij i j j i refj

a g ie iu nii

ix

pim

1s

( ) ( )( )

i i i i i ii i i i i

uy h x du

x f x g xic

j

calculating iP

Restores Frequency Synchronization after islanding

i ni Pi i im P u

Feedback Linearization Inner Loop


DER 8 DER 6

DER 4

Rline1 Lline1

Pload1+jQload1


Rc4Lc4

Rc3Lc3

Rc2Lc2

Rc1Lc1

vo4vo3vo2vo1

Pload2+jQload2

DER 3DER 2DER 1

DER 5DER 7



Rline4

Lline4

vo5vo6vo7vo8


DER 1DER 2DER 3DER 4 LeaderDER 5DER 6DER 7DER 8

DG 1 DG 2 DG 3 DG 4

DG 8 DG 7 DG 6 DG 5

DG 5DG 6DG 7DG 8 DG 4 DG 3 DG 2 DG 1

Simulation Example

Physical MicrogridNetwork

Cyber communication network‐ sparse


61

0 0.5 1 1.5 2 2.5 3

49.6

49.8

50

50.2

t (s)

f (H

z)

DER1DER2DER3DER4DER5DER6DER7DER8

DG 1

DG 2

DG 3

DG 4

DG 5

DG 6

DG 7

DG 8

Islanding Turn onCoop secondary control

Ref. FrequencyIs 50 Hz

64

2. Secondary Voltage ControlMicrogrid of Interconnected DG

DER 8 DER 6

DER 4

Rline1 Lline1

Pload1+jQload1


Rc4Lc4

Rc3Lc3

Rc2Lc2

Rc1Lc1

vo4vo3vo2vo1

Pload2+jQload2

DER 3DER 2DER 1

DER 5DER 7



Rline4

Lline4

vo5vo6vo7vo8


DG 1 DG 2 DG 3 DG 4

DG 8 DG 7 DG 6 DG 5

2 2,o magi odi oqiv v v 2. Voltage synchronization (per unit)

i i ni Pi iy m P 1. Frequency synchronization


Synchronize per‐unitvoltages

2. Secondary Voltage controlSecondary Control Synchronization Objectives

vo io

VSC

vod*

iLd*

Currentcontroller

Voltagecontroller

LC filteriL

Power controller

vb

, voq*

vod , voq

iod , ioq

, iLq*

ω

ωn Vn

Outputconnector

Rc Lc

abc/dq

iLd , iLq

Rf Lf Cf

VSC‐ Voltage source converter Power electronics‐ DC to AC



Voltage synchronization (per unit)

Frequency synchronization

Secondary Control InputsChange Droop control parameters to get synchronization

2. Secondary Voltage Control

67

Secondary Voltage control objective

The per unit voltage of each DG synchronizes to a command nominal value

2 2,o magi odi oqiv v v Voltage synchronization (per unit)

n P

mag n Q

m PE v V n Q

Droop Control

Use this for frequency synchronization

Use this for voltage synchronization

2. Secondary Voltage Control

68

If , there is no direct relationship between the output and

input .

i odiy v

i niu V

Input-output feedback linearization for a heterogeneous nonlinear agents

( ) 1i i i

r r ri i i iy L h L L h u F g F

1i i i

r ri i i iv L h L L h u F g F

1 1( ) ( )i i i

r ri i i iu L L h L h v g F F

( ) ,ri iy v i

,1

,1 ,2

, 1

,

i i

i i

i r i

y yy y

i

y v

( ) ( ) ( )( )

i i i i i i i i i

i i i

uy h

x f x k x D g xx

( ) ( ) ( )i i i i i i i F x f x k x D

, ,i i iv i BA

Assume relative degree r is the same for all agentsZero dynamics can be different, but assume they are stable

Must use Lie derivatives

2. Secondary voltage control

69

0 1 0 0 00 0 1 0 00 0 0 1 0

0 0 0 0 10 0 0 0 0 0 r r

A

1[0,0, ,1]TrB

Leader node dynamics

The synchronization problem is to find a distributed such that iv

0 0 0

0 0 0

( )( )y h

x f xx

, ,i i iv i BA ,1 , 1[ ]Ti i i i ry y y

( )0 0 0 ,ry BY AY ( 1)

0 0 0 0[ ]r Ty y y Y

0, .i i Y

Assumption. The vector is bounded so that , with a finite but generally unknown bound.

( ) ( )0 0 ,r r

N y r y 1 ( )0r r

MYy

DG Agent Dynamics

is the first eigenvalue of

2. Secondary voltage controlTheorem. Let the digraph of the multi-agent system have a spanning tree and the pinning gain be nonzero for at least one root node.

Let all agents have stable zero dynamics

Let the auxiliary control be chosen as

i iv cK e

where is the coupling gain, and is the feedback control gain.

Then, are cooperative UUB with respect to and all nodes synchronize to if is chosen as

c R 1 rK R

1 rK R 1

1,TK R P B 1

1 1 1 1 0.T TP P Q P R P BA BA

andmin

1 ,2

c

min min ( )i iRe i L G

0( ) ( )i

iiN

ii jj ij

a g

e Y

i 0Y

0Y

Zhang, H., Lewis, F. L., & Das, A. (2011).Optimal design for synchronization of cooperative systems: State feedback, observer, and output feedback. IEEE Transactions on Automatic Control, 56(8), 1948–1952.

S. Tuna, “LQR‐based coupling gain for synchronization of linear systems,” Arxiv preprint arXiv:0801.3390, 2008.


71

Proof:

0( ) ( )i

iiN

ii jj ij

a g

e Y 0r rL G I L G I e δ

( ) ( )N NI I v A B

1 2TT T T

N 1 2TT T T

N e e e e

)00(

0( ) ( ) rN NI I A B y

00 N 1 Y

( ) ( )N rv c I K L G I δi iv cK e

i i iv BA ( )0 0 0

ry BY AY

Global Dynamics


Differentiating ( )2 2 2 00 ( ) ( )( )rT T T

N NV P P P I I v δ δ δ δ A δ B y

( )2 2 0

( )2 2 0

( ) ( )

( ) .

rT TN N

rT TN

V P I c L G K P I

P P I

δ A B δ δ B y

δ Hδ δ B y

Lemma 1. Let (A,B) be stabilizable. Let the digraph have a spanning tree and for at least one root node. Let be the eigenvalues of . The matrix is Hurwitz if and only if all the matrices are Hurwitz. [Fax and Murray 2004]

0ig i L G

( )NI c L G K H A B, ic K i A B

Lemma 2. Let (A,B) be stabilizable and matrices and be positive definite. Let feedback gain K be chosen aswhere is the unique positive definite solution of the control algebraic Riccati equation Then, all the matrices are Hurwitz if where .

TQ Q TR R1

1,TK R P B

1P1

1 1 1 1 0.T TP P Q P R P A A B B ic KA B

min

1 ,2

c

min min ( )i iRe

Lyapunov function2 2 2 2

1 , , 0,2

δ δT TV P P P P

73

Secondary voltage control

Lemma 1

Lemma 2 H is Hurwitz.2 2 .T

NrP P I H H

( ) ( )2 2 20 02

1 ( ) ( ) ( )2 2

r rT T T TN

TNr NV P P IP I P I

δ H H δ δ B y δ δ δ B y

22( ( ))

2r

N MV P I Y δ δ B

0V if 22 ( ( )) .r

N MP I Y

Bδ

Prove Uniform Ultimately Bounded

Secondary voltage control

74

Σ DG i

Mi (xi)

cKVni

xi-vi

_aij ( yi -yj )+gi ( yi -y0)

j

Nij∈

ei_Σ 1

Ni

y0 =vref

0

vodjvodj

yj =

vodivodi

yi =

(2) 2 1i i ii i i iy L h L L h u F g F

2 1i i ii i i iv L h L L h u F g F

1 1 2( ) ( )i i ii i i iu L L h L h v g F F

( ), .i i i

nii

v MV i

N

x

Synchronizes Output voltages after Islanding

iu

Feedback Linearization Inner Loop


DER 8 DER 6

DER 4

Rline1 Lline1

Pload1+jQload1


Rc4Lc4

Rc3Lc3

Rc2Lc2

Rc1Lc1

vo4vo3vo2vo1

Pload2+jQload2

DER 3DER 2DER 1

DER 5DER 7



Rline4

Lline4

vo5vo6vo7vo8


DER 1DER 2DER 3DER 4 LeaderDER 5DER 6DER 7DER 8

DG 1 DG 2 DG 3 DG 4

DG 8 DG 7 DG 6 DG 5

DG 5DG 6DG 7DG 8 DG 4 DG 3 DG 2 DG 1

Simulation Example

Physical MicrogridNetwork

Cyber communication network‐ sparse

Simulation results

76

1 1.2 1.4 1.6 1.8 2350

360

370

380

390

t (s)

v o,m

ag (V

)

DER1DER2DER3DER4DER5DER6DER7DER8

DG 1

DG 2

DG 3

DG 4

DG 5

DG 6

DG 7

DG 8

Islanding Turn onCoop secondary control

Ref. Per‐unitVoltageIs 380 V


78

79

80

81

82

Documents

Distributed Cooperative Control of Micro-grids talks/2013 coop ctrl of...Distributed Cooperative Control of Micro-grids Supported by NSF, ARO, AFOSR Work of Ali Bidram with Dr. A