power electronic circuits modeling and simulation ofesm/wp-admin/... · 2017-03-01 · 2 outline...

1

Modeling and Simulation ofPower Electronic Circuits

using Piecewise Smooth Differential Algebraic Equations

Jeroen Tant24 February 2017

2

Outline• Introduction

– Power electronic circuits– Modeling and simulation– Scope– Objectives

• Objective 1: solving power electronic circuits

– Model Composition(Chapter 3)– Piecewise smooth DAEs (Chapter 4)– Solution approaches (Chapter 6)

• Objective 2: improving computational efficiency

– Integration and interpolation method (Chapter 7)– Circuit partitioning technique (Chapter 8)

• Conclusions

3




4

Introduction: Power Electronic Circuits

Power electronic circuitsform a

power transfer interface between electrical systems

4




vdc

For example:

ACsystem

DCsystem

vac

PEcircuit

4




vdc,2

For example:

PEcircuit

DCsystem

DCsystem

vdc,1

4




vac,2

For example:

PEcircuit

ACsystem

ACsystem

vac,1

5


Power electronic circuitsuse

semiconductor devices as switches

MOSFET

IGBT

Thyristor

Diode

5


Power electronic circuitsuse

semiconductor devices as switches

MOSFET

IGBT

Thyristor

Diode is

vs

σs

used as switches: on or off

6


vin vs

For instance pulse width modulation (PWM):vin

t

vs

t

Power electronic circuitsemploy

specialized switching control techniques

6


vin vs

For instance pulse width modulation (PWM):vin

t

vs

t

vout

vs

t

vout

t

vs

Power electronic circuitsemploy

specialized switching control techniques

7

Introduction: Power Electronic CircuitsExample applications:• Consumer electronics

– mobile phone charger– laptop power supply– LED lamp driver

• Industry– motor drives

• Transportation– electric vehicles– trains

• Electric power system:– grid connection of wind turbines, photovoltaics,

battery storage, ...– high-voltage dc (HVDC) connections

8


DC cable170 km400 kV DC600 MW

AC-DCconverterstations

Bentwisch

Bjaeverskov

Example: high-voltage dc (HVDC) connection betweenDenmark and Germany

9 Photo: Jeroen TantAC Side

Bjaeverskov HVDC converter station, Denmark

10 Photo: Jeroen TantDC Side

HVDC converter station, Bjaeverskov, DenmarkHVDC converter station, Bjaeverskov, Denmark

11By Marshelec (Own work) [CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

HVDC example: Inside (an other converter station)HVDC converter station, valve hall

Thyristor valve hall

12




13

Introduction: Modeling and Simulation

realsystem

measurementstesting

mathematicalmodel

of systemtesting

$$$$$

$

Real system tests:

Computer modeling and simulation:

{approximation of real system

13

Introduction: Modeling and Simulation

realsystem

measurementstesting

computed results

computeralgorithm

mathematicalmodel

of systemtesting

$$$$$

$

Real system tests:

Computer modeling and simulation:approximation

14

Introduction: Modeling and SimulationExample simulation tool: PLECS

15


– Power electronic circuits (NL)– Modeling and simulation (NL)– Scope– Objectives


– Model Composition(Chapter 3)– Piecewise smooth DAEs (Chapter 4)– Solution approaches (Chapter 6)



• Conclusions

16

Scope

• idealized switches instead of detailed semiconducterdevice models

• to obtain workable model with fewer variables,equations, parameters

Power electronic circuit simulationwith idealized switches

is

vs

vs = 0

OFF:ON:

vs

iss

id = 0

vs = Ronis

is = Yoffvs

vs

is

OFF:ON:

or

17

Scope


• in the field of power electronics

– transient simulation of switched converter circuits

• in the field of power systems

– electromagnetic transient simulation (EMT)

17

Scope


• in the field of power electronics

– transient simulation of switched converter circuits

• in the field of power systems

– electromagnetic transient simulation (EMT)

−→ methods are mathematically equivalent

−→ treated separately in literature

18

Initial Latest Latest ImplementationSimulator Releasea Update Version Aspects

ATOSEC 1975 1988 5 [205–207]

PSPICE 1984 2016 17.2 (see Section 2.3.2)

Saber 1987 2015 2015.12 [186–188]

Krean 1990 1995 4.1 [208,209]

SIMPLORER 1991 2015 R16.2 [169,203,204]

SIMPLIS 1992 2015 8.0 [210–213]

Caspoc 1993 2009 2009 [214,215]

PSIM 1994 2015 10.0.4 [52,216,217]

LTSPICE 1999 2016 4.23k (see Section 2.3.2)

PLECS 2002 2015 3.7.4 [118,119,141,218]

GeckoCIRCUITS 2008 2016 1.72 [219]

NL5 2009 2016 2.2 [220]

aCirca

Tools specialized for power electronic circuits

18

Initial Latest Latest ImplementationSimulator Releasea Update Version Aspects

ATOSEC 1975 1988 5 [205–207]

PSPICE 1984 2016 17.2 (see Section 2.3.2)

Saber 1987 2015 2015.12 [186–188]

Krean 1990 1995 4.1 [208,209]

SIMPLORER 1991 2015 R16.2 [169,203,204]

SIMPLIS 1992 2015 8.0 [210–213]

Caspoc 1993 2009 2009 [214,215]

PSIM 1994 2015 10.0.4 [52,216,217]

LTSPICE 1999 2016 4.23k (see Section 2.3.2)

PLECS 2002 2015 3.7.4 [118,119,141,218]

GeckoCIRCUITS 2008 2016 1.72 [219]

NL5 2009 2016 2.2 [220]

aCirca use idealized switches

Tools specialized for power electronic circuits

19

Initial ImplementationSimulator Version Development Aspects

EMTP by BPA 1968 BPA until 1984 [27,234,235]

NETOMAC 1973 Siemens [30,236–238]

PSCAD/EMTDC 1976 Manitoba HVDC [30,96,111,239–243]

ATP 1984 Non-commercial [235,244–246]

EMTP by DCG 1987 EPRI until 1998 [235,247–249]

MicroTran 1987 UBC [235,250–252]

RTDS 1993 Manitoba HVDC / RTDS [253–262]

PowerFactory 1998 DIgSILENT [263–268]

SimPowerSystems 1998 Hydro-Quebec [140,231,269]

EMTP-RV 2003 Hydro-Quebec [270–276]

HYPERSIM 2003 Hydro-Quebec / OPAL-RT [277–282]

XTAP 2006 CRIEPI (Japan) [283–287]

eMEGAsim 2007 Opal-RT [32,261,288–292]

Tools specialized for EMT simulation of power systems

20

Objectives

1. Solving circuits with idealized switches

• clarify mathematical treatment• solve difficulties with idealized switches• develop numerical solution procedure

2. Improving computational efficiency

• for increased switching frequencies• for topologies with a large number of switches

21




– Model composition (Chapter 3)– Piecewise smooth DAEs (Chapter 4)– Solution approaches (Chapter 6)



• Conclusions

22

Model Composition (Chapter 3)

• Continuous-time model:differential algebraic equations (DAEs)

• Event-driven extension:hybrid event action block

Minimal set of fundamental building blocksrequired to model power electronic circuits

23


+

u−v

×

u÷v

√RR

RR

RR

RR

RR

R

R

R

R

f( )Rn Rm f( )Zn2 Zm

2

andZ2

Z2

Z2

or

not

expR R

sinR R

Z2

Z2

Z2

Z2Z2

......

∑Rn R

Rn RmK

c Rm/Zm

2

LogicOperators Linear Nonlinear

Constant

Continuous-time model: building blocks

24

Model Composition (Chapter 3)Continuous-time model: building blocks

ib

+

vb

− =0

ibvb

RRR

Aib = 0

ATe = vb

y=0

RR

∫Rn Rn

IntegratorEquation

Circuit Branch Kirchoff’s Laws

ii ij

ik

ei ej

vb

25


maxmin |u|RR

RR

RRR R

Piecewise linear operators (continuous)


y

x

z = xz = y

z

xy

z

xy

z = yz = x

26

Model Composition (Chapter 3)Continuous-time model: building blocks

Example: ideal diode models

0 = min (id,−vd)

id

vd

id

vd

id = max(

vdRON

, vdROFF

)

id

vd

0 = min(id, Vf − vd

)

id

vdVf

Piecewise linear operators (continuous)

27


≥ 0R Z2


u

ϕstep(u)

1

0

Piecewise linear step operator ϕstep (discontinuous)

28


≥ 0u


Example: controlled switches

is

vs

s = 1

s = 0vs

iss

0 = svs + (1−s)iss = ϕstep(u)

Piecewise linear step operator ϕstep (discontinuous)

29


∫ Rn

Rn

reset

action function

execute

x

x

Rm ux(t, k+1)

=F (u(t, k))

Z2

Hybrid extension

one conceptual block

30

Model Composition (Chapter 3)Hybrid extension

one conceptual block enables the modeling of:

• resettable timer

30



• periodic clock

30



• discrete state-space model• memory element

30



• latches and flip-flops

30



• low-level switch control

PWM with latch symmetric PWM

peak current mode hysteresis

30



• digital three-phase inverter control system

31


• Continuous-time model: semi-explicit DAE

• Hybrid extension: event action blocks

x = f(ib, vb, x, y)

0 = g(ib, vb, x, y)

0 = h(ib, vb, x, y)

0 = Aib,

vb = ATe

at triggered events:x(t, k + 1) = F

(ib(t, k), vb(t, k), x(t, k), y(t, k)

)

31


• Continuous-time model: semi-explicit DAE

• Hybrid extension: event action blocks

x = f(ib, vb, x, y)

0 = g(ib, vb, x, y)

0 = h(ib, vb, x, y)

0 = Aib,

vb = ATe

at triggered events:x(t, k + 1) = F

(ib(t, k), vb(t, k), x(t, k), y(t, k)

)piecewise smooth DAE

32

Piecewise Smooth DAEs (Chapter 4)

New model classfor DAEs with piecewise defined equations

x = f(x, y)

0 = g(x, y)

f and g defined withpiecewise defined operators,such as max, min, abs, step

f and g possibly discontinuous!

33

Piecewise Smooth DAEs (Chapter 4)Existing model classes not applicable:

• regular DAEs [77,80]– theory and methods assume smooth equations

• piecewise smooth dynamical systems [357,387]– requires transformation to ODE

• complementarity systems [389, 390]– continuous equations only (see Chapter 5)

• switching DAEs [394]– switching instants are predetermined

• hybrid DAEs / hybrid systems [137, 151, 395]– requires enumeration of modes and transitions

34

Piecewise Smooth DAEs (Chapter 4)New model class:

• piecewise smooth DAEs

– consider DAEs with piecewise smooth equations

– compact representation with all information includedin the equation definition

– more in line with existing circuit simulation tools

35


x = f(x, y)

0 = g(x, y)for (x, y) ∈ Rσ

max(u, v) =

{u for u > v

v for u ≤ v

Validity regions Rσ for each mode bounded by hyperplanesin intermediate variable space

regular DAE for each mode: ”mode-DAE”

→

ϕstep(u) =

{0 for u < 0

1 for u ≥ 1

x = fσ(x, y)

0 = gσ(x, y)

Each piecewise linear operator has two modes

⇒ 2n modes in total

36


37

Solution Approaches (Chapter 6)

Solution approaches for power electronic circuits

Essentially:

• solve until boundary of Rσ reached

• find new valid Rσ in which the solution can continue

x = fσ(x, y)

0 = gσ(x, y)

38


39


40


solve = solve regular DAEx = fσ(x, y)

0 = gσ(x, y)

if 0 = gσ(x, y) defines y uniquely, given x

• use ODE integration method

• solve 0 = gσ(x, y) simultaneously

xk+1 = xk +h

2fσ(xk, yk) +

h

2fσ(xk+1, yk+1),

0 = gσ(xk+1, yk+1)

e.g. trapezoidal method:

40


solve = solve regular DAEx = fσ(x, y)

0 = gσ(x, y)

if 0 = gσ(x, y) defines y uniquely, given x

• use ODE integration method

• solve 0 = gσ(x, y) simultaneously

otherwise

• DAE index ≥ 2

• use index reduction method to obtain index 1

41

Method Classa Order Stability for x = λx, Re(λ) < 0

Forward Euler 1-stage ERK 1 if |hλ| is sufficiently small

Backward Euler 1-stage IRK 1 L-stable

Trapezoidal rule 2-stage IRK 2 A-stable, not L-stable

Adams–Bashforth family [78] s-step ELM s if |hλ| is sufficiently small

Adams–Moulton family [78] s-step ILM s+1 s = 1: same as trapezoidal rules ≥ 2: if |hλ| is sufficiently small

BDF family [78]b s-step ILM s s = 1: same as backward Eulers ≤ 2 L-stables ≤ 6: A(α)-stable

Dormand–Prince 5(4) [78] 7-stage ERK 5 if |hλ| is sufficiently small

Lobatto IIIA family [78] s-stage IRK 2s−2 s = 2: same as trapezoidal rules ≥ 2: A-stable, not L-stable

Radau IIA family [78] s-stage IRK 2s−1 L-stable

DIRK(2,2) [84]c 2-stage IRK 2 L-stable

TR-BDF2 [85,86] 3-stage IRK 2 L-stable

aERK/IRK: explicit/implicit Runge–Kutta; ELM/ILM: explicit/implicit linear multistepbBDF: backward differentiation formulacDIRK(s,p): diagonally implicit Runge–Kutta method with s stages and order p. For a review, see [87].

ODE Integration Methods

42


solve until boundary of Rσ reachedx = fσ(x, y)

0 = gσ(x, y)

• detect when (xk+1, yk+1) /∈ Rσ

• interpolation between (xk, yk) and (xk+1, yk+1)

Rσ(xk, yk)

(xk+1, yk+1)

43


find new valid Rσ in which the solution can continue=

Mode selection and reinitialization algorithm

given initial values x∗:

find y∗ and σ

such that gσ(x∗, y∗) = 0 and (x∗, y∗) ∈ Rσ

= find valid configuration for all switches and diodes

44



given initial values x∗:

find y∗ and σ

such that gσ(x∗, y∗) = 0 and (x∗, y∗) ∈ Rσ

→ equations possibly discontinuous!

→ existing algorithms assume continuous equations

equivalent to solving g(x, y) = 0 as a piecewise smoothsystem of equations

45


For practical circuits:

g(x, y) = 0

g1(x, y1, y2, y3 . . . , yq) = 0

g2(x, y2, y3 . . . , yq) = 0

...gq−1(x, yq−1, yq) = 0

gq(x, yq) = 0

piecewise smooth system with discontinuities

decomposes into continuous subproblems


46


Solution approaches for power electronic circuits

• Constant structure approach: Ron / Roff switches

• Variable structure approach: on/off switches

47


RCC

s

RCC

Ron

s

impulsive configurationspossible

constant structure(Ron / Roff switches)(on / off switches)

variable structure

47


impulsive configurationspossible

also at intermediate invalidconfigurations

IDEAL

L

R IDEAL

L

Roff

constant structure(Ron / Roff switches)(on / off switches)

variable structure

48

Solution Approaches (Chapter 6)constant structurevariable structure

(Ron / Roff switches)(on / off switches)

underdeterminedconfigurations possible

?

?

10 V

10 A ??

5 V

5 V

10 V ROFF

ROFF

10 A

RONRON

5 A5 A

IDEAL

IDEAL

IDEALIDEAL

49


RCC RCC

Ron

DAE structure constantDAE index can change

at discontinuous conductionmodes

index reduction once at thebeginning

index reduction after everyswitch event

fast transientsdue to Ron / Roff

IDEAL IDEAL

constant structurevariable structure(Ron / Roff switches)(on / off switches)

49


L

Roff R

IDEALL

R

IDEAL

DAE structure constantDAE index can change

at discontinuous conductionmodes

index reduction once at thebeginning

index reduction after everyswitch event



50


L

Roff R

IDEALL

R

IDEAL

avoid fast transientsin model



10.9 11 11.1

-50

0

t10.9 11 11.1

-50

0

t[ms]

vL [V] vL [V]

[ms]

50


avoid fast transientsin model



10.9 11 11.1

-50

0

t10.9 11 11.1

-50

0

t[ms]

vL [V] vL [V]

[ms]

L-stable methodrecommended

Any integration method

51Figure 6.6 – Flowchart for the constant structure approach.

52Figure 6.12 – Flowchart for the variable structure approach.

53




– Model Composition (Chapter 3)– Piecewise smooth DAEs (Chapter 4)– Solution approaches (Chapter 6)



• Conclusions

54

Numerical Integration and Interpolation (Chapter 7)

New method for integration and interpolation forsimulation with the constant structure approach

Integration method recommended which:

• preservessystem stability (A-stability)

• damps fast transients caused by Ron/Roff (L-stability)

• avoids numerical oscillations (L-stability)

But these properties are not always preserved withinterpolation!

55


Interpolation preserves:

• second order accuracy

• damping of fast transients

• suppression of numerical oscillations

New method for integration and interpolation

56


Figure 7.14 – Test circuit consisting of a boost converter combinedwith an independent RLC circuit.

57

Numerical Integration and Interpolation (Chapter 7)New method

58

Numerical Integration and Interpolation (Chapter 7)EMTP-RV

59

Numerical Integration and Interpolation (Chapter 7)PSCAD

60Figure 7.21 – Comparison with EMTP-RV and PSCAD/EMTDC.

61

Numerical Integration and Interpolation (Chapter 7)PSIM

62Figure 7.25 – Comparison with PSIM.

63


64

Circuit Partitioning Technique (Chapter 8)

Reduce number of floating-point operations formatrix refactorization after switch events

• Full LU refactorization during simulation = bottleneck

• Partitioning with state variables as separators

• Reconstruct LU factor from partitions

65

Circuit Partitioning Technique (Chapter 8)

AC GridDC Side{

200×

Example: modular multilevel converter

Nodal analysis Branch analysis Branch analysis

Partitioning Method No No Yes

Single LU Refactorization (No. of FLOPs) 178 004 126 158 19 201

Triangular LU Solve (No. of FLOPs) 86 730 67 310 70 474

66

Conclusions• Objective 1: solving circuits with idealized switches

→ model composition framework→ piecewise smooth DAE framework→ solution approaches→ mode selection and reinitialization algorithm


→ new method for integration and interpolation→ circuit partitioning technique

power electronic circuits modeling and simulation ofesm/wp-admin/... · 2017-03-01 · 2 outline...

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