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

Modeling and Simulation ofPower Electronic Circuits

using Piecewise Smooth Differential Algebraic Equations

Jeroen Tant24 February 2017

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

Introduction: Power Electronic Circuits

Power electronic circuitsform a

power transfer interface between electrical systems

For example:

ACsystem

DCsystem

PEcircuit

For example:

PEcircuit

DCsystem

For example:

PEcircuit

ACsystem

Power electronic circuitsuse

semiconductor devices as switches

MOSFET

Thyristor

Power electronic circuitsuse

semiconductor devices as switches

MOSFET

Thyristor

Diode is

used as switches: on or off

vin vs

For instance pulse width modulation (PWM):vin

Power electronic circuitsemploy

specialized switching control techniques

vin vs

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

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

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

Introduction: Modeling and Simulation

realsystem

measurementstesting

mathematicalmodel

of systemtesting

Real system tests:

Computer modeling and simulation:

{approximation of real system

Introduction: Modeling and Simulation

realsystem

measurementstesting

computed results

computeralgorithm

mathematicalmodel

of systemtesting

Real system tests:

Computer modeling and simulation:approximation

Introduction: Modeling and SimulationExample simulation tool: PLECS

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

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

• Conclusions

• idealized switches instead of detailed semiconducterdevice models

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

Power electronic circuit simulationwith idealized switches

vs = 0

OFF:ON:

id = 0

vs = Ronis

is = Yoffvs

OFF:ON:

• in the field of power electronics

– transient simulation of switched converter circuits

• in the field of power systems

– electromagnetic transient simulation (EMT)

• 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

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

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

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

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

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

• Conclusions

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

f( )Rn Rm f( )Zn2 Zm

expR R

sinR R

......

∑Rn R

Rn RmK

c Rm/Zm

LogicOperators Linear Nonlinear

Constant

Continuous-time model: building blocks

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

− =0

Aib = 0

ATe = vb

∫Rn Rn

IntegratorEquation

Circuit Branch Kirchoff’s Laws

maxmin |u|RR

Piecewise linear operators (continuous)

z = xz = y

z = yz = x

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

Example: ideal diode models

0 = min (id,−vd)

id = max(

, vdROFF

0 = min(id, Vf − vd

Piecewise linear operators (continuous)

≥ 0R Z2

ϕstep(u)

Piecewise linear step operator ϕstep (discontinuous)

≥ 0u

Example: controlled switches

s = 0vs

0 = svs + (1−s)iss = ϕstep(u)

Piecewise linear step operator ϕstep (discontinuous)

∫ Rn

action function

execute

Rm ux(t, k+1)

=F (u(t, k))

Hybrid extension

one conceptual block

∫ Rn

action function

execute

Rm ux(t, k+1)

=F (u(t, k))

Hybrid extension

one conceptual block

Model Composition (Chapter 3)Hybrid extension

one conceptual block enables the modeling of:

• resettable timer

• periodic clock

• discrete state-space model• memory element

• latches and flip-flops

• low-level switch control

PWM with latch symmetric PWM

peak current mode hysteresis

• digital three-phase inverter control system

• 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)

• 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

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!

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

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

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

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)

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) +

2fσ(xk+1, yk+1),

0 = gσ(xk+1, yk+1)

e.g. trapezoidal method:

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

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

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)

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

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

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

Solution approaches for power electronic circuits

• Constant structure approach: Ron / Roff switches

• Variable structure approach: on/off switches

impulsive configurationspossible

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

variable structure

impulsive configurationspossible

also at intermediate invalidconfigurations

R IDEAL

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

variable structure

Solution Approaches (Chapter 6)constant structurevariable structure

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

underdeterminedconfigurations possible

10 A ??

10 V ROFF

RONRON

5 A5 A

IDEALIDEAL

RCC RCC

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)

Roff R

IDEALL

DAE structure constantDAE index can change

at discontinuous conductionmodes

index reduction once at thebeginning

index reduction after everyswitch event

Roff R

IDEALL

avoid fast transientsin model

10.9 11 11.1

t10.9 11 11.1

vL [V] vL [V]

avoid fast transientsin model

10.9 11 11.1

t10.9 11 11.1

vL [V] vL [V]

L-stable methodrecommended

Any integration method

51Figure 6.6 – Flowchart for the constant structure approach.

52Figure 6.12 – Flowchart for the variable structure approach.

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

• Conclusions

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!

Interpolation preserves:

• second order accuracy

• damping of fast transients

• suppression of numerical oscillations

New method for integration and interpolation

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

Numerical Integration and Interpolation (Chapter 7)New method

Numerical Integration and Interpolation (Chapter 7)EMTP-RV

Numerical Integration and Interpolation (Chapter 7)PSCAD

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

Numerical Integration and Interpolation (Chapter 7)PSIM

62Figure 7.25 – Comparison with PSIM.

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

Circuit Partitioning Technique (Chapter 8)

AC GridDC Side{

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

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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