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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
Modeling and Simulation ofPower Electronic Circuits
using Piecewise Smooth Differential Algebraic Equations
Jeroen Tant24 February 2017
3
Modeling and Simulation ofPower Electronic Circuits
using Piecewise Smooth Differential Algebraic Equations
Jeroen Tant24 February 2017
4
Introduction: Power Electronic Circuits
Power electronic circuitsform a
power transfer interface between electrical systems
4
Introduction: Power Electronic Circuits
Power electronic circuitsform a
power transfer interface between electrical systems
vdc
For example:
ACsystem
DCsystem
vac
PEcircuit
4
Introduction: Power Electronic Circuits
Power electronic circuitsform a
power transfer interface between electrical systems
vdc,2
For example:
PEcircuit
DCsystem
DCsystem
vdc,1
4
Introduction: Power Electronic Circuits
Power electronic circuitsform a
power transfer interface between electrical systems
vac,2
For example:
PEcircuit
ACsystem
ACsystem
vac,1
5
Introduction: Power Electronic Circuits
Power electronic circuitsuse
semiconductor devices as switches
MOSFET
IGBT
Thyristor
Diode
5
Introduction: Power Electronic Circuits
Power electronic circuitsuse
semiconductor devices as switches
MOSFET
IGBT
Thyristor
Diode is
vs
σs
used as switches: on or off
6
Introduction: Power Electronic Circuits
vin vs
For instance pulse width modulation (PWM):vin
t
vs
t
Power electronic circuitsemploy
specialized switching control techniques
6
Introduction: Power Electronic Circuits
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
6
Introduction: Power Electronic Circuits
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
6
Introduction: Power Electronic Circuits
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
6
Introduction: Power Electronic Circuits
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
6
Introduction: Power Electronic Circuits
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
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
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
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
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
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
Introduction: Power Electronic Circuits
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
12
Modeling and Simulation ofPower Electronic Circuits
using Piecewise Smooth Differential Algebraic Equations
Jeroen Tant24 February 2017
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
14
Introduction: Modeling and SimulationExample simulation tool: PLECS
14
Introduction: Modeling and SimulationExample simulation tool: PLECS
15
Outline• Introduction
– Power electronic circuits (NL)– Modeling and simulation (NL)– 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
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
Power electronic circuit simulationwith idealized switches
• in the field of power electronics
– transient simulation of switched converter circuits
• in the field of power systems
– electromagnetic transient simulation (EMT)
17
Scope
Power electronic circuit simulationwith idealized switches
• 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
Outline• Introduction
– Power electronic circuits (NL)– Modeling and simulation (NL)– 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
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
Model Composition (Chapter 3)
+
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
Model Composition (Chapter 3)
maxmin |u|RR
RR
RRR R
Piecewise linear operators (continuous)
Continuous-time model: building blocks
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
Model Composition (Chapter 3)
≥ 0R Z2
Continuous-time model: building blocks
u
ϕstep(u)
1
0
Piecewise linear step operator ϕstep (discontinuous)
28
Model Composition (Chapter 3)
≥ 0u
Continuous-time model: building blocks
Example: controlled switches
is
vs
s = 1
s = 0vs
iss
0 = svs + (1−s)iss = ϕstep(u)
Piecewise linear step operator ϕstep (discontinuous)
29
Model Composition (Chapter 3)
∫ Rn
Rn
reset
action function
execute
x
x
Rm ux(t, k+1)
=F (u(t, k))
Z2
Hybrid extension
one conceptual block
29
Model Composition (Chapter 3)
∫ 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
Model Composition (Chapter 3)Hybrid extension
one conceptual block enables the modeling of:
• periodic clock
30
Model Composition (Chapter 3)Hybrid extension
one conceptual block enables the modeling of:
• discrete state-space model• memory element
30
Model Composition (Chapter 3)Hybrid extension
one conceptual block enables the modeling of:
• latches and flip-flops
30
Model Composition (Chapter 3)Hybrid extension
one conceptual block enables the modeling of:
• low-level switch control
PWM with latch symmetric PWM
peak current mode hysteresis
30
Model Composition (Chapter 3)Hybrid extension
one conceptual block enables the modeling of:
• digital three-phase inverter control system
31
Model Composition (Chapter 3)
• 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
Model Composition (Chapter 3)
• 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
Piecewise Smooth DAEs (Chapter 4)
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
Piecewise Smooth DAEs (Chapter 4)
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
Solution Approaches (Chapter 6)
39
Solution Approaches (Chapter 6)
40
Solution Approaches (Chapter 6)
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
Solution Approaches (Chapter 6)
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
Solution Approaches (Chapter 6)
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
Solution Approaches (Chapter 6)
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
Solution Approaches (Chapter 6)
Mode selection and reinitialization algorithm
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
Solution Approaches (Chapter 6)
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
Mode selection and reinitialization algorithm
46
Solution Approaches (Chapter 6)
Solution approaches for power electronic circuits
• Constant structure approach: Ron / Roff switches
• Variable structure approach: on/off switches
47
Solution Approaches (Chapter 6)
RCC
s
RCC
Ron
s
impulsive configurationspossible
constant structure(Ron / Roff switches)(on / off switches)
variable structure
47
Solution Approaches (Chapter 6)
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
Solution Approaches (Chapter 6)
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
Solution Approaches (Chapter 6)
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
fast transientsdue to Ron / Roff
constant structurevariable structure(Ron / Roff switches)(on / off switches)
50
Solution Approaches (Chapter 6)
L
Roff R
IDEALL
R
IDEAL
avoid fast transientsin model
fast transientsdue to Ron / Roff
constant structurevariable structure(Ron / Roff switches)(on / off switches)
10.9 11 11.1
-50
0
t10.9 11 11.1
-50
0
t[ms]
vL [V] vL [V]
[ms]
50
Solution Approaches (Chapter 6)
avoid fast transientsin model
fast transientsdue to Ron / Roff
constant structurevariable structure(Ron / Roff switches)(on / off switches)
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
Outline• Introduction
– Power electronic circuits (NL)– Modeling and simulation (NL)– 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
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
Numerical Integration and Interpolation (Chapter 7)
Interpolation preserves:
• second order accuracy
• damping of fast transients
• suppression of numerical oscillations
New method for integration and interpolation
56
Numerical Integration and Interpolation (Chapter 7)
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
Numerical Integration and Interpolation (Chapter 7)
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
• Objective 2: improving computational efficiency
→ new method for integration and interpolation→ circuit partitioning technique