COMPLEXITY SCIENCE WORKSHOP 18, 19 June 2015

Systems & Control Research Centre

School of Mathematics, Computer Science and Engineering

CITY UNIVERSITY LONDON

Modeling and control of distributed generation power systems as complex nonlinear Hamiltonian systems

Presentation: Antonio T. Alexandridis

The deployment of modern power systems is mainly based on

• the high penetration of renewable energy sources (RES)• a decentralized structure known as distributed generation (DG).

In DG the power electronic interfaces used to connect the different parts, play an important role that:

• exploits the local capabilities of the system on managing energy in the scheme of a microgrid

• enables a cooperated and fast self-controlled local implementation

Note: The microgrid is defined as an integrated energy system consisting of distributed energy sources and multiple electrical loads operating as a single, local grid either in grid connected-mode or in islanded-mode with respect to the existing utility power system.

Introduction

The fast response of power electronic devices provide the possibility of new

control strategies in a wide area of DG sources and loadsAs a result, DG should be planned and analyzed as a whole (at least on a

microgrid level): All system dynamics should be taken into account

To this end, it is needed:• a complete dynamic analysis of the DG system• a systematic methodology of obtaining the complete DG system

nonlinear model

It is shown that this model• is in Hamiltonian form with certain damping properties• can be systematically obtained from the integration of all the DG

components, i.e., from their individual Hamiltonian models• damping and other structural properties can be effectively used for stable

control designs through Lyapunov analysis.

Problem formulation

Based on the Hamiltonian modeling, the entire system is described as

a large nonlinear system with external uncontrolled inputsAdvantages:• The Hamiltonian formulation provides an immediate construction

of energy based Lyapunov functions (!)Drawbacks:• The error dynamic model is not Hamiltonian• The entire system is nonautonomous with f=f(x,u(t))• To prove stability is not an easy task (a particular sequence of

stages for the analysis and design are needed: This is our current work)

• The integrated method is very efficient and complete

Why Hamiltonian modeling?

Hamiltonian systems

In the case where the system is given in the form:

where is the system Hamiltonian function, is a semi-positive definite matrix and is skew-symmetric, the system is called generalized Hamiltonian system.

In many cases, such as almost all DG system components can be written as

Where matrix is symmetric and positive definite, is semi-positive definite, is a skew-symmetric matrix and . represents the external input vector.

( )( ) ( ) ( )

H xx J x R x g x u

x

( )J x( )R x( )H x

( ) ( )Mx J x R x x E

M( )J x

( )R x

E

DG components in Hamiltonian form

• Almost all Distributed Generators (PV-arrays, Wind generators, Storage batteries, Power converters: dc/dc or ac/dc) are modeled as Hamiltonian systems

• The transmission line, local load and capacitor bank models can also be written in the general nonlinear Hamiltonian form

Thus, the complete DG network can be integrated and modeled as Hamiltonian system.

DG components in Hamiltonian form

Passivity Preliminaries

Let the nonlinear system

where are smooth.

Theorem 1. [Khalil] Assume that there is a continuous function

such that

for all functions , for all and all . Then the system with input and output is passive.

( ) ( ) ,

( )

x f x g x u

y h x

, , , , ,n mx R u y R f g h

0( ) (0) ( ) ( )

t TV t V y u d ( ) 0V t

( )u t 0t (0)V( )u t ( )y t

DG systems and components

Passivity analysis of Hamiltonian systems

Considering the storage function:

The derivative of the storage function is calculated as:

Assuming as output the vector and input the vector the above inequality becomes . Integrating from zero to , according to Theorem 1, proves that the nonlinear Hamiltonian system is passive.

1

2TV x Mx

1 1T T T

T T T T T T

V x M M J R x M E x J R x x E

x Jx x Rx x E x Rx x E x E

y x u ETV y u t

Typical configuration of a DG network

A DG example

Nonlinear Hamiltonian form of the system:

where:

with:

and

with:

1 8 8 1 1 8 8 1 8 8 1 1 8 1( ) ( ) ( ) ( )x x x xM x J R x E

1 11 1 1

TTdf qfx x I I

11 1

T

ds qs dr qr r dcx I I V

1 8 1 11 6 1( ) ( )TT

x x dgrid qgridE E V V

11 6 1

2( ) 0 0 0 0 0

3

T

m

x

TE

Hamiltonian model of a wind turbine induction generator

and

11 6 6 6 2

1 8 8

1

2 6

1

( ) | 0

( ) ___ | ___

00 |

0

x x

x

f

x

f

M

M

L

L

11 1 1 21 1 16 6 6 2

1 8 8

1

21 1 1 2 61

( , ) | ( , )

( ) _____ | ____________

0( , ) |

0

ds qs df qfx x

x

f f

df qf xf f

T

J m m J m m

J

LJ m m

L

11 11 6 26 6

1 8 8

1

2 61

( ) | 0

( ) ___ | ___

00 |

0

xx

x

f

xf

R x

R

R

R

Hamiltonian model of a wind turbine induction generator

Hamiltonian model of a Photovoltaic Generator

Nonlinear Hamiltonian form of the system:

where:

with:

and

2 4 4 2 2 4 4 2 4 4 2 2 4 1( ) {( ) ( ) } ( )x x x xM x J R x E

2 12 2 2

TTdf qfx x I I

12 2

T

pv dcx I V

2 4 1

2( ) 0 0 0

3x PV

T

E V

and

12 2 2 2 2

2 4 4

2

2 2

2

( ) | 0

( ) ___ | ___

00 |

0

x x

x

f

x

f

M

M

L

L

12 22 2 22 2

2 4 4

2

22 2 2 2 22

2 2( ) | ( , )

( ) ___ | ___

0( , ) |

0

df qfx

x

T f f

df qf xf f

xJ J m m

J

LJ m m

L

12 12 2 22 2

2 4 4

2

2 2

2

( ) | 0

( ) ___ | ___

00 |

0

xx

x

f

x

f

R x

R

R

R

Hamiltonian model of a Photovoltaic Generator

• Transmission line:

• Local load:

• Capacitor bank:

f df f df f f qf dbus dgrid

f qf f f df f qf qbus qgrid

L I R I L I V V

L I L I R I V V

L dL L dL f L qL dbus

L qL f L dL L qL qbus

L I R I L I V

L I L I R I V

1 2

1 2

...

...

dbus df df df dfn dL f qbus

qbus qf qf qf qL f dbusqfn

CV I I I I I CV

CV I I I I I CV

Hamiltonian model of a transmission line, a local load and a capacitor bank

The complete DG system can be integrated in the form of:

the state vector is:

( ) ( )Mx J x R x x E

11 12 1 1 2 2... ...

...

T Tdf qf df qf

T

df qf dL qL dbus qbus

x x x I I I I

I I I I V V

Obtaining the complete Hamiltonian Dynamic Model of a DG System

The external vector is:

and

211 6 1 4 1( ) ( ) ... 0 0 0 0 ...

... 0 0 0 0T

dgrid qgrid

T Tx xE E E

U U

2 211 6 6 12 1 1 2 2{( ) ,( ) ,..., , , , ,...

..., , , , , , }xx f f f f

f f L L

M diag M M L L L L

L L L L C C

11 6 6 12 2 2 1 1 2 2{( ) , ( ) ,..., , , , ,...

..., , , , ,0,0}

x x f f f f

f f L L

R diag R R R R R R

R R R R

Obtaining the complete Hamiltonian Dynamic Model of a DG System

and

11 6 6 6 2 21 6 2

2 6 12 2 2 2 2 22 2 2

1

21 2 6 2 2 2 2

1

2

22 2 2

2

( ) | ( ) |

___ | ___ ___ ___ | ___ ___ ___ ___ ___ ___ ___

| ( ) | ( )

| |

0( ) | | |

0

___ | ___ ___ | ___ ___ ___ ___ ___ | ___

0| ( ) |

0

x x x

x x x x

f fT

x x x

f f

f fT

x

f f

L

L

L

LJ

J 0 J 0

0 J 0 J 0

J 0 0 1

J 0

2 2

2 2 2 2

2 2 2 2

2 2 2 2 2 2 2 2

|

| | |

0| | | |

0

| | ___ | ___ | ___

0| | | |

0

| ___ | ___ ___ | ___ | ___

0| | | |

0

x

f f

x x

f f

f L

x x

f L

f

x x x x

f

L

L

L

L

C

C

1

0 1

0 0

0 0 0 1

1 1 1 1

Obtaining the complete Hamiltonian Dynamic Model of a DG System

Hamiltonian Dynamic Model of a DG System

Basic features

Matrix continues to be a symmetric, positive definite matrix is a semi-positive definite matrix. is skew-symmetric and depends from the duty-ratio control inputs of the generator-side converters, the grid-side converters and the boost converters of the complete DG system.

Since matrix does not appear in the passivity inequality, the complete DG system is passive for any non dynamic control scheme applied at the control inputs.

M

R

J

J

The Example

Simulations results

• We consider a squirrel-cage induction generator (SCIG) wind turbine and a Photovoltaic (PV) system. At time . the power of the PV system drops by and at time . the wind power increases almost by .

• Main task in a DG system is to extract the maximum amount of energy from the generators and achieve unity power factor.

• Passivity-based control is applied.

20kW10kW

5sect 9%15sect 10%

Simulations results

0 5 10 15 20 251.3

1.4

1.5

1.6

1.7

1.8

1.9

2x 10

4

time (sec)

Act

ive

pow

er o

f w

ind

gene

rato

r on

line

1 (

kW)

0 5 10 15 20 256000

6500

7000

7500

8000

8500

9000

9500

10000

time (sec)

Act

ive

pow

er o

f P

V s

yste

m o

n lin

e 2

(W)

Simulations results

0 5 10 15 20 252

2.1

2.2

2.3

2.4

2.5

2.6x 10

4

time (sec)

Act

ive

pow

er in

ject

ed t

o th

e gr

id (

kW)

0 5 10 15 20 25-6000

-4000

-2000

0

2000

4000

6000

time (sec)

Rea

ctiv

e po

wer

inje

cted

to

the

grid

(V

ar)

Conclusions

• A new approach that integrates the partial individual unit models of a distributed power system in a common Hamiltonian form has been presented.

• The Hamiltonian form with certain damping properties can be effectively used for stable control designs.*

• Passivity and stability analysis can be obtained from the complete nonlinear Hamiltonian model.*

• The interactions between the individual units are not ignored. Thus, contradictory performances can be avoided.

*Issues in DG control

Next steps constructing on the proposed model

A new suitable DG control is applied with main goals:

• The most possible simple decentralized structure

• Controllers independent from system parameters and operating conditions

• For example, local cascaded PI controllers are examined

• The stability analysis includes Current Inner-loop controllers

• Special stability analysis for external-loop controllers

A hard analysis is needed in order to obtain simple and efficient controllers guaranteeing stability

Thank you for your attention