Passivity based control applied to power converters Marco Liserre [email protected] Passivity based...

Marco Liserre [email protected]

Passivity based control applied to power converters


Marco Liserre

[email protected]



Passivity based control history

70’s definition of dissipative systems (Willems) 1981 application to rigid robots (Arimoto e Takagi)

in power electronics . . .

1991 first theoretical paper (Ortega, Espinoza & others) 1996 first experimental paper (Cecati, & others, IAS Annual

Meeting) 1998 first book “Passivity-Based Control of Euler-Lagrange

Systems” (Ortega and Sira-Ramirez, Springer, ISBN 1852330163)

1999 application to active filters (Mattavelli and Stankovic, ISCAS 99)

2002 Brayton-Moser formulation (Jeltsema and Scherpen, Am. Control Conf.)

2002 application to multilevel converters (Cecati, Dell’Aquila, Liserre, Monopoli, IECON 2002)



Contribution of my research group on the topic

collaborations with:

University of L’Aquila (Prof. Cecati)University of Delft (Prof. Scherpen)

main papers:

C. Cecati, A. Dell’Aquila, M. Liserre, V. G. Monopoli “A passivity-based multilevel active rectifier with adaptive compensation for traction applications” IEEE Transactions on Industry Applications, Sep./Oct. 2003, vol. 39, no. 5.

A. Dell’Aquila, M. Liserre, V. G. Monopoli, P. Rotondo “An Energy-Based Control for an n-H-Bridges Multilevel Active Rectifier” IEEE Transactions on Industrial Electronics, June 2005, vol. 52, no. 3.



Basic idea of the Passivity-based approach

The basic idea of the PBC is to use the energy to describe the state of the system

Since the main goal of any controller is to feed a dynamic system through a desired evolution as well as to guarantee its steady state behavior, an energy-based controller shapes the energy of the system and its variations according to the desired state trajectory

If the controller is designed aiming at obtaining the minimum energy transformation, optimum control is achieved

The PBC offers a method to design controllers that make the system Lyapunov-stable

The “energy approach” is particularly suitable when dealing with:

electromechanical systems as electrical machinesgrid connected converters (non-linear model)



The introduction of damping

The control objective is usually achieved through an energy reshaping process and by injecting damping to modify the dissipation structure of the system

From a circuit theoretic perspective, a PBC forces the closed-loop dynamics to behave as if there are artificial resistors — the control parameters — connected in series or in parallel to the real circuit elements

When the PBC is applied to grid connected converters, harmonic rejection is one of the main task, hence the passive damping can be substituted by a dynamic damping (i.e. virtual inductive and capacitive elements should be added)

The point of view is always the energy reshaping (i.e. the energy associated to the harmonics)



The Eulero-Lagrange formulation

Passivity-based control has been firstly developed on the basis of Eulero-Lagrange formulation

One of the major advantages of using the EL approach is that the physical structure (e.g., energy, dissipation, and interconnection), including the nonlinear phenomena and features, is explicitly incorporated in the model, and thus in the corresponding PBC

This in contrast to conventional techniques that are mainly based on linearized dynamics and corresponding proportional-integral–derivative (PID) or lead–lag control



The Passivity Based Controller design

In the context of EL-based PBC designs for power converters, two fundamental questions arise:

which variables have to be stabilized to a certain value in order to regulate the output(s) of interest toward a desired equilibrium value? In other words, are the zero-dynamics of the output(s) to be controlled stable with respect to the available control input(s), and if not, for which state variables are they stable?

where to inject the damping and how to tune the various parameters associated to the energy modification and to the damping assignment stage?



Dissipativity definition

dissipativity definition



Passivity definition



Definitions

Supply Rate: speed of the energy flow from a source to the system Storage function: energy accumulated in a system Dissipative systems: systems verifying dissipation inequality:

“Along time trajectories of dissipative systems the following relationship holds:energy flow ≥ storage function”

(In other words, dissipative systems can accumulate less energy than that supplied by external sources)

The basic idea of PBC is to shape the energy of the system according to a desired state trajectory, leaving uncontrolled those parts of the system not involved in energy transformations, this result can be obtained only working on “strictly passive” systems



Feedback systems decomposition

dividing the system into simpler subsystems, each one identifying those parts of the system actively involved in energy transformations

each subsystem has to be passive introducing energy balances, expressed in terms of the Eulero-Lagrange equations

passivity invariance




The full order model describing the system is divided into simpler subsystems identifying those parts actively involved in energy transformation

Hence, energy balances, expressed in terms of the Eulero-Lagrange equations (based on the variational method and energy functions expressed in terms of generalised coordinates), are introduced

The system goes in the direction where the integral of the Lagrangian is minimized (Hamilton's principle)




This formulation highlights active, dissipative and workless forces i.e. the active parts of the system (those which energy can be modified by external forces), those passive (i.e. dissipating energy, e.g. thermal energy), and those parts which do not contribute in any form to control actions and can be neglected during controller design

Because of the energy approach, it is quite straightforward to obtain fast response under condition that the control "moves" the minimum amount of energy inside the system

Moreover, because global stability is ensured by passivity properties, a simple a effective controller can be designed



Eulero-Lagrange formulation

The eulero-lagrange formulation is particularly suited for the control of electromechanical systems as electrical motors

In fact different subsystems are related by their ability to transform energy, therefore it is a good thing to define energy functions for each one, expressed in terms of generalised coordinates qi.

In electric motor case:qm mechanical position (for mechanical subsystems)qe electric charge (for electrical subsystems)

Using variational approch we can introduce Lagrangian equations of the system and apply Hamilton's principle. This method highlights subsystems interconnections and their various energies: dissipated, stored and supplied




The mechanical subsystem does not take an active part in control actions, i.e. it doesn't produce energy but only transforms and dissipates the input energy, for design purposes its contribution can be considered as an external disturbance for the electrical subsystem and the controller has to compensate for this disturbance, in order to maintain electrical equation balance. In “passivity terms”, it defines a passive mapping around the electrical subsystem, it can be neglected during controller design and the attention can be focused on the electrical subsystem.

induction motor formulation




The electrical subsystem is simply passive, then its evolution can be corrupted by any external disturbance leading to instability. Therefore, in order to obtain global stability, it is an important step of the approach to make it strictly passive by means of the addition of a suitable dissipative term (damping injection)



Passivity-based control of the H-bridge converter

PBC has been successfully applied to d.c./d.c. converters, active rectifiers and multilevel topologies

Particularly the single-phase Voltage Source Converter (VSC) also called H-bridge or full bridge can be used as universal converter due to the possibility to perform dc/dc, dc/ac or ac/dc conversion

Moreover it can be used as basic cell of the cascade multilevel converters

In the following it will be reviewed the application of the PBC to H-bridge single phase inverters (one-stage and multilevel) using the Brayton-Moser formulation which is the most suitable for the converter control



Passivity-based control of the H-bridge converter

Control of one H-bridge-based active rectifierG. Escobar, D. Chevreau, R. Ortega, E. Mendes, “An adaptive passivity-based controller for a unity power factor rectifier”, IEEE Trans. on Cont. Syst. Techn., vol. 9, no. 4, July 2001, pp. 637 –644

Control of two (multilevel) H-bridge-based active rectifierC. Cecati, A. Dell'Aquila, M. Liserre and V. G. Monopoli, "A passivity-based multilevel active rectifier with adaptive compensation for traction applications", IEEE Trans. on Ind. Applicat., vol. 39, Sept./Oct. 2003 pp. 1404-1413the two dc-links are not independent !

Control of n (multilevel) H-bridge-based active rectifierA. Dell’Aquila, M. Liserre, V. G. Monopoli, P. Rotondo “An Energy-Based Control for an n-H-Bridges Multilevel Active Rectifier” IEEE Transactions on Industrial Electronics, June 2005, vol. 52, no. 3.the n dc-links are independent !



Brayton-Moser Equations

Brayton and Moser, introduced in 1964 a scalar function of the voltages across capacitors and the currents through inductors in order to characterize a given network

This function was called the Mixed-Potential Function P(iL, vC) and it allows to analyze the dynamics and the stability of a broad class of RLC networks

These equations can be considered an effective alternative to Euler-Lagrange formulation

This formulation has a main advantage over the counterpart in case of power converter control: it allows the controllers to be implemented using measurable quantities such as voltages and currents.



Topologically Complete Networks = networks which state variables form a complete set of variables

Complete Set of Variables = set of variables that can be chosen independently without violating Kirchhoff’s laws and determining either the current or voltage (or both) in every branch of the network

Additionally for Topologically Complete Networks it is possible to define two subnetworks

One subnetwork has to contain all inductors and current-controlled resistors

The other has to contain all capacitors and voltage controlled resistors




R(iL) is the Current Potential (Content) and is related with the current-controlled resistors and voltage sources

G(vC) is the Voltage Potential (Co-content) and is related with the voltage-controlled resistors and current sources

N(iL,vC) is related to the internal power circulating across the dynamic elements


For the class of topologically complete networks it is possible to construct the mixed-potential function directly. For this class it is known that the mixed potential is of the form:

L C L C L CP( i ,v ) R( i ) G( v ) N( i ,v )



PR is the Dissipative Current Potential

PG is the Dissipative Voltage Potential


The components of the Mixed-Potential Function can be analysed in more detail as follows:

L C R L E L G C J C T L C

R G N

P( i ,v ) P ( i ) P ( i ) P ( v ) P ( v ) P ( i ,v )




The dissipative current and voltage potentials can be calculated as follows:

Li

R L R L L0P ( i ) v ( i' )di'

Cv

G C G C C0P ( v ) i ( v' )dv'

In case of linear resistor PR is half the dissipated power expressed in terms of inductor current, and PG is half the dissipated power expressed in terms of capacitor voltages.

PE is the total supplied power to the voltage sources E

PJ is the total supplied power to the current sources J


R G N

P( i ,v ) P ( i ) P ( i ) P ( v ) P ( v ) P ( i ,v )




PT is the internal power circulating across the dynamic elements and is represented by:


R G N

P( i ,v ) P ( i ) P ( i ) P ( v ) P ( v ) P ( i ,v )

In this representation denotes the interconnection matrix and it is determined by KVL

and KCL

TT L CP ( x ) i v



Finally the expression of the mixed-potential function



R G N

P( i ,v ) P ( i ) P ( i ) P ( v ) P ( v ) P ( i ,v )

can be rewritten as follows:

D T FP( x ) P ( x ) P ( x ) P ( x )

PD(x)= PR(x)- PG(x) is the Dissipative Potential

PF(x)= PJ(x)- PE(x) is the Total Supplied Power



iL = (iL1 , . . . , iL )T are the currents through the inductors

vC = (vC1 , . . . , vC )T are the voltages across the capacitors.


L CLL

L

C L CC

C

P( i ,v )diL( i )

dt i

dv P( i ,v )C( v )

dt v

These differential equations correspond with Kirchhoff’s voltage and current laws

The dynamic behaviour of topologically complete networks is governed by the following differential equations :




and with the diagonal square matrix Q(x) R(+)x(+) defined as

The previous equations can be expressed in a more compact way as follows:

TT

x1 n

P( x ) P( x ) P( x )Q( x )x P( x ) ...

x x x

with the state vector xRn = R+ defined as

L

C

ix

v

( )( )

( )

L

C

L i 0Q x

0 C v




When a circuit contains only linear passive inductors and capacitors, then the diagonal matrices L(iL) Rx and C(vC) Rx are of the form:

1

2L

L 0 0 0

0 L 0 0L( i )

. . . .

0 0 0 L

1

2C

C 0 0 0

0 C 0 0C( v )

. . . .

0 0 0 C

The Brayton-Moser equations are closely related to the co-Hamiltonian H*(iL, vC) (that represents the total co-energy stored in the network).

If the co-Hamiltonian is known, then the matrices L(iL) and C(vC) can be easily found as follows

* *( ) ( , ), ( ) ( , ) L C

2 2L i L C C v L CL i H i v C v H i v



Switched Brayton-Moser Equations

For a circuit with one or more switches it is possible to obtain a single Switched Mixed-Potential Function by properly combining the individual mixed-potential

functions associated to each operating mode. u=0

P0(x)

u=1P1(x)

Then it is possible to obtain one Switched Mixed-Potential Function parameterized by

u as

( , ) ( ) ( ) ( ), , 0 1P u x 1 u P x uP x u 0 1The Switched Mixed-Potential Function is consistent with the individual Mixed-

Potential Functions



Switched Brayton-Moser Equations

It is worth to notice that the only difference between each individual Mixed-Potential

Function and the Switched Mixed-Potential Function will be in the term

and in particular in the interconnection matrix which becomes a function of u, (u)

TT L CP ( x ) i v



Average State Model

When the switching frequency is sufficiently high, it is possible to prove that the average state model of a circuit with a single switch can be derived from the discrete model by only replacing the discrete variable u{0,1} with the continuously varying duty-cycle variable μ[0,1] . Additionally, to show that the model is a state average model, the state vector x is replaced by the state average vector z

Discrete

Model

Average

State

Model

uμ

xz



Average State Model

The former result can be extended to circuits with multiple switches. In this case the matrix (u) assumes as many configurations as the possible combinations of the status of the switches are (e. g. for an H-bridge converter is a mono-dimensional matrix and may assume three distinct values {-1,0,1})

Discrete

Model

Average

State

Model

uiμi

xizi



Passivity Based Control – ProcedureTo design a Passivity Based controller the average co-energy function H*(z) and the

dissipative potential PD(z) have to be modified. To this purpose the Brayton-Moser equations can be rewritten as:

The first two derivative terms are still function of z, in the sense that the partial derivatives of PT(z) and PD(z) are still dependent on z

The third term is constant meaning that the partial derivative of PF(z) is not dependent on z anymore

T D FP ( z ) P ( z ) P ( z )

Qzz z z

f(z)constant

The following step is to rewrite the previous equations by replacing the state variables z with an auxiliary system of variables ξ which represent the desired state trajectories for

inductor currents and capacitor voltages:

f(ξ) constant

T D FP ( ) P ( ) P ( )

QThe first two derivative terms are still function of ξ

The third term is constant and is obviously equal to the partial derivative of PF(z)



Passivity Based Control – Procedurebeing z = z − ξ the average state errors, it is possible to write:

Assuming that the first two derivatives are linear functions of z and the last two derivatives are linear functions of ξ, yields:

T D T DP ( z ) P ( z ) P ( ) P ( )

Qz Qz Q ( z ) ( z ) ( ) ( )z z

T D T DP ( z ) P ( z ) P ( z ) P ( z )

Qz( z ) ( z ) ( z ) ( z )

The previous expression represents the error dynamics and it could be obtained from

T D FP ( z ) P ( z ) P ( z )

Qzz z z

by simply replacing the variable z with the error variable z and eliminating the derivative of PF



Passivity Based Control – ProcedureThe next step is to add a damping term to the error dynamics to ensure asymptotic

stability

This injection can be seen as an

expansion of the dissipative potential ( )( ) ( )

VT D P zP z P zQz

z z z

Considering z = (iL, vC)T where

iL = (z1 . . . z)T are the error-currents through the inductors

vC= (z+1 . . . z+)T are the error-voltages across the capacitors The injected dissipation can be decomposed as follows:

V Vr L Vg CP ( z ) P ( i ) P ( v )

The injected dissipation together with the dissipative potential of the system, gives the Total Modified Dissipation Potential PM

M D Vz z

P ( z ) P ( z ) P ( z )



Passivity Based Control – Procedure

the controller dynamics are obtained

VF T D P ( z )P ( z ) P ( ) P ( )

Qz z

Subtracting

( )( ) ( )

VT D P zP z P z

Qzz z z

from

T D FP ( z ) P ( z ) P ( z )

Qzz z z




THEOREM 1 (R-Stability)

If RS is a positive-definite and constant matrix, and

Two theorems ensure the stability of the closed loop system.

1 112 2

SL R ( )C 1

with 0 << 1, then for all (iL, vC) the solutions of

tend to zero as t → ∞

( )( ) ( )

VT D P zP z P z

Qzz z z

where closed-loop resistance matrix RS is

2M

S L 2L

P ( z )R ( i )

i




THEOREM 2 (G-Stability)

If GP is a positive-definite and constant matrix, and

with 0 << 1, then for all (iL, vC) the solutions of

tend to zero as t → ∞.

( )( ) ( )

VT D P zP z P z

Qzz z z

1 11 T2 2

PC G ( )L 1

where closed-loop conductance matrix GP is

2M

P C 2C

P ( z )G ( v )

v



Passivity Based Control – Procedure With these theorems lower bounds are found for the control parameters (RS

and/or GP )

These lower bounds ensure a ”reasonably nice” response in terms of overshoot, settling-time, etc

If just one of these theorems is satisfied, the system is stable. This means there

are two damping injection strategies that can be selected:

Although it is sufficient to use only one of these strategies, the equations could contain both the series damping injection term and the parallel damping injection term

Series Damping (damping on inductor currents)

M D Vr Lz z

P ( z ) P ( z ) P ( i )

Parallel Damping (damping on capacitor voltages)

M D Vg Cz z

P ( z ) P ( z ) P ( v )



Passivity Based Control – ProcedureFinally, if n is the number of minimum phase states it is possible to modify n equations of the + differential equations in

To this purpose n minimum phase states have to be found. Consequently the remaining +-n state variables will be indirectly controlled through the control of the n selected states

VF T D P ( z )P ( z ) P ( ) P ( )

Qz z

For the n selected variables it is possible to set the derivative of reference value to zero obtaining n algebraic equations:

and +-n differential equations:

VF T D

i i i i

P ( z )P ( z ) P ( ) P ( )0

z z

VF T D

ii i i i

P ( z )P ( z ) P ( ) P ( )Q

z z

Controller

Equations




At the beginning initial values of the n control inputs have to be set

Controller Implementation

Using these values the differential equations

can be solved to obtain the time evolution of the auxiliary variables for the indirectly controlled variables.

VF T D

ii i i i

P ( z )P ( z ) P ( ) P ( )Q

z z

The former references are needed to solve the algebraic equations

VF T D

i i i i

P ( z )P ( z ) P ( ) P ( )0

z z

which solutions are the set of values for the control inputs to be applied in the next switching period.



PBC of an H-bridge

a

C Ri

e

L

b

+DC

-DC

grid

T1

T2

T3

T4

The passivity-based controller will be designed by inspection, identifying the potential functions



PBC of an H-bridge

D R GP P P

a

C Rie

L

b

+DC

-DC

T1

T2

T3

T4

RP 022

G

zP

2R

22

D

zP

2R

1z

2z

F J EP P P

JP 0 E 1P ez

F 1P ez

T 1 2P z z



PBC of an H-bridge

a

C Rie

L

b

+DC

-DC

T1

T2

T3

T4

1z

2z

22

D

zP

2R

F 1P e z

T 1 2P z z

22

1 2 1

zP z z e z

2R

21

21

2

Pz e

z

P zz

z R



PBC of an H-bridge

a

C Rie

L

b

+DC

-DC

T1

T2

T3

T4

1z

2z1 2

22 1

Lz z e

zCz z

R

LKT

LKC

1 2

22 1

L e

CR

controller

1 2

22 1

Lz z e

zCz z

R



PBC of an H-bridge: damping injection

a

C Rie

L

b

+DC

-DC

T1

T2

T3

T4

1z

2z

2 2DS 1 2

VDP

R z zP ( z )

2 2R

22 2DS 12 2

MDP

R zz zP ( z )

2R 2 2R

1 DS

2DP

R( z ) R

1 1G( z )

R R

1 2

22 1

Lz z e

zCz z

R

1 2 DS 1

2 22 1

DP

Lz z e R z

z zCz z

R R



PBC of an H-bridge: damping injection

a

C Rie

L

b

+DC

-DC

T1

T2

T3

T4

1z

2z

1 2 DS 1

2 22 1

DP

Lz z e R z

z zCz z

R R

1 2

22 1

Lz z e

zCz z

R

1 2 DS 1

2 22 1

DP

L e R z

zC

R R



PBC of an H-bridge: control variables

a

C Rie

L

b

+DC

-DC

T1

T2

T3

T4

1z

2z

1 2 DS 1

2 22 1

DP

L e R z

zC

R R

which variables have to be stabilized to a certain value in order to regulate the output(s) of interest toward a desired equilibrium value?

in other words, are the zero-dynamics of the output(s) to be controlled stable with respect to the available control input(s), and if not, for which state variables are they stable?



PBC of an H-bridge: zero-dynamics The steady-state solution in case of direct control of the dc-voltage or in

case of indirect control of the dc-voltage (through the grid current) should be found

Switching function in case of direct control

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Funz

ione

di s

witc

hing

s

Tempo [s] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-1.5

-1

-0.5

0

0.5

1

1.5

Fu

nzi

on

e d

i sw

itch

ing

s

Tempo [s]

Switching function in case of indirect control

21 1 1

1

Rz Lz RLCz e RCe

CR Lz e

2

d

1 eRR

L V

A stable system can be obtained only by indirectly controlling the dc voltages through the ac current i*



as i i*, vc ξ2 Vd

PBC of an H-bridge

reference voltage Vd

and load conductance θ

power balancereference current i*

internally generated ξ2

ODEswitching function µ

algebraic

A stable system can be obtained only by indirectly controlling the dc voltages through the ac current i*

This means that

E

VI d

d

22

and *di I sin t

1

1

*2 DS 1

*2 2 DP 2

Lz e R z

C z G z

load conductancedc voltage reference

grid voltage amplitudecontroller

R

1

From the power balance it results that



PBC of an H-bridge Since it is possible to control directly the grid current,

The d.c. voltage is controllable only indirectly, through an internal variable ξ2

1 1 21

( )z e Rz zL

average KVL on the a.c. side

*1 1 1

2

1DSe Rz Lz R z

predictive action

+

damping injection

average KCL on the d.c. side

differential eq. +

damping injection

Then it is necessary to estimate the d.c. load

2 1 21 z z zC *

2 1 2 21 ˆ

DPz G zC

2 1 2 2ˆ ˆ ˆ c c z



PBC of an H-bridge: damping tuning

1 112 2

DSL R ( )C 1

1 11 T2 2

DPC G ( )L 1

DS

L1

C R

DP

C1

L G

a

C Ri

e

L

b

+DC

-DC

T1

T2

T3

T4

DS

LR

1 C

DP

CG

1 L

RDS

GDP



Passivity Control of the Multilevel Converter

The use of the passivity-based control (PBC) properly fits stability problems related to this type of converter

Two approaches for the PBC design have been considered

the first is developed considering the overall multilevel converter

the second is developed by splitting the system into n subsystems and controlling them independently

As regards the second, the partition of the multilevel converter is done on the basis of energy considerations

The main advantage of the second approach is the separate control of the different DC-links and a flexible loading capability



Mathematical model of the system

The converter is controlled with a discrete switching function ui (i=1,2,...,n) for each H-bridge

1 3i i iu T T

1 1 2,1

1( )

n

i ii

x e Rx u xL

2, 1 2,1

i i i ii

x u x xC

1 equation

n equations

n+1 equations

e

R L

+

_

T11 T31

T31 T41

R1

C1

vc1

+

_

T1i T3i

T3i T4i

Ri

Ci

vci

+

_

T1n T3n

T3n T4n

Rn

Cn

vcn

a

b

i



PBC of an H-bridge Since it is possible to control directly the grid current,

The d.c. voltage is controllable only indirectly, through an internal variable ξ2

1 1 21

( )z e Rz zL

average KVL on the a.c. side

*1 1 1

2

1DSe Rz Lz R z

predictive action

+

damping injection

average KCL on the d.c. side

differential eq. +

damping injection

Then it is necessary to estimate the d.c. load

2 1 21 z z zC *

2 1 2 21 ˆ

DPz G zC

2 1 2 2ˆ ˆ ˆ c c z



PBC of an n-H-bridge converter

It is not possible a simple extension of the PBC of one H-bridge active rectifier to n-bridge active rectifier

In fact the PBC of an H-bridge active rectifier needs one algebraic equation and one differential equation





Thus “a simple extension” of this control needs n algebraic equations and n differential equations. However this is not possible since the n H-bridges are connected in series on the grid side and the ac voltage equation results in only one algebraic equation


*1 1 1

2

1DSe Rz Lz R z






Thus “a simple extension” of this control needs n algebraic equations and n differential equations. However this is not possible since the n H-bridges are connected in series on the grid side and the ac voltage equation results in only one algebraic equation

We have proposed two PBC approaches:

1. one algebraic eq. plus n differential eq. (with ξ2,1= .. = ξ2,n)

2. n algebraic eq. (based on n virtual KVL’s) plus n differential eq. (with ξ2,1≠ .. ≠ ξ2,n)

*1 1 1

2

1DSe Rz Lz R z



Passivity-based control approach 1

* sindi I t

E

V2I

n

1ii

2d

d

Indirect control of output voltages

E

Vn2I

2d

d

2 1 2 2ˆ ˆ ˆc c z

1 2 2,1 2,2 2, 2... & ...n n

reference voltage Vd and load

conductance θequal for all the bridges

power balancereference current i*

internally generated ξ2, equal

for all the bridges

ODEswitching function µ

algebraic

2, 1, 2, 2,ˆ ˆ ˆ i i i i i i i ic c z

*1 1 1

21

*2 1 2 2,

2 1 2 2

1

1 ˆ for 1, 2,....,

ˆ ˆ ˆ

n

i DSi

DP i

e Rz Lz R z

z G z i nC

c c z



PBC 2: Model formulation in subsystems

n21 i....ii

1n

1ii

i1

e

H-Bridge 1+

Load 1 1 e

a2

n e

i

in

H-Bridge i+

Load i

H-Bridge n+

Load n

an

an+1

Rn Ln

ii

Ri

i e

ai+1

ai

Li

R1 a1 L1

i1

R1

1 e

a2

ai

i e

n e

ii

in

a1

ai+1

an

an+1

L1

Ri Li

Rn Ln

H-Bridge 1 +

Load 1

H-Bridge i +

Load i

H-Bridge n +

Load n



i1

R1

1 e

a2

ai

i e

n e

ii

in

a1

ai+1

an

an+1

L1

Ri Li

Rn Ln

H-Bridge 1 +

Load 1

H-Bridge i +

Load i

H-Bridge n +

Load n

PBC 2: Model formulation in subsystems

n21 i....ii

1n

1ii

i1

e

H-Bridge 1+

Load 1 1 e

a2

n e

i

in

H-Bridge i+

Load i

H-Bridge n+

Load n

an

an+1

Rn Ln

ii

Ri

i e

ai+1

ai

Li

R1 a1 L1

1 1 2,

2, 1 2,

i i i i i

i i i i i

L x e R x u x

C x u x x

n KVL




Indirect control of each output voltage achieved via the separate control of each bridge leading to n passivity-based controllers related through i

n

1ii

2i,d

i2

i,di

V

V

supervisor n controllers for n H-bridges

*1 1 , 1

2,

*2, 1 2, , 2,

2, 1, 2, 2,

1 for 1,2,....,

1 ˆ for 1, 2,....,

ˆ ˆ ˆ for 1, 2,....,

i i i i DS ii

i i i i DP i ii

i i i i i i i i

e R z L z R z i n

z G z i nC

c c z i n




only changing the parameters of the controllers

only changing the parameters of the controllers



Harmonic compensation

energy function includes energy function includes energy related to harmonicsenergy related to harmonics

In case of harmonics, the design results in the use of an RLC active damping branch very effective in harmonic rejection

The damping is made by a resistance and a band-pass filter




bandpass filters



Set-up for the multilevel active rectifier

Rated rms grid voltage 220 [V] Rated power 1 [kW] Reference dc bus voltage 400-560 [V] Ac inductance 10 [mH] Dc capacitors 2330 [F]

CONTROLLER (Dspace card)

driving signal

& enable

GRID

VLT 5006

vc1

VLT 5006

vc2ie



PBC tuning: voltage error damping GDP

3 4 5 6 7 8

240

250

260

270

280

290

300

310

3 4 5 6 7 8

240

250

260

270

280

290

300

310

GDP = 0.1 GDP= 1 GDP = 10

dc-link voltage due to a laod step change



0 2 4 6220

240

260

280

300

Time [s ]

Ca

pa

cito

r v

olt

ag

es

[V

] γ = 0.01

PBC tuning: estimate parameter γ

γ = 0.01

0 2 4 650

100

150

200

250

300

350

Time [s ]

Esti

ma

te R

1=

R2

[O

hm

]

estimate dc-link load resistance due to a load step change

it has a strong influence on the dc-link dynamic



Steady-state (PBC 1 & 2)

grid voltage

grid current

dc-link voltage

dc-link voltage



Dynamical test

Measured DC voltages [10 V/div] consequent to a load step change from half to full load on both the DC-links (330 F)

PBC 1 PBC 2



Dynamical test (PBC 2)

Measured DC voltages [50 V/div] and grid current [4 A/div] (2330 F)

dc voltage reference step on one bus

dc load steps on the two buses leading to different loads



Dynamical test for active load (PBC 2)

a dc motor has been used as active load



Unbalance loads on the two dc-links (PBC 2)

Steady-state behavior of PBC2 with full load on DC bus 1 and half load on DC bus 2

multilevel ac voltage [200 V/div] grid voltage [100V/div] grid current [10 A/div]



Computational efforts comparison

5 7 9 11

4567

9

12

15

nr. of de sire d le ve ls

nr.

of

con

tro

l eq

ua

tion

s

P BC 2

P BC 1

PBC 2 needs p-3 equations more than PBC 1

However PBC 1 employs a division by the reference current that leads to computational problems

with p = number of desired levels




R-dampingR-damping

RLC-dampingRLC-damping



Conclusions

Passivity-based theory offers a straightforward approach to design controllers without linearazing the system: physical and intuitive representation of the control problem design method to make the system Lyapunov-stable feedback decomposition useful for electromechanical systems

Eulero-Lagrange formulation more suitable for electrical motors

Brayton-Moser formulation more suitable for power converters (tuning procedure)

Optimal results can be obtained with RLC damping of harmonics (similar to those obtained with generalized integrators – resonant controllers – linear approach)

Passivity based control applied to power converters Marco Liserre [email protected] Passivity based...

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Transcript of Passivity based control applied to power converters Marco Liserre [email protected] Passivity based...