Post on 25-Mar-2018
transcript
Voltage Recovery and Optimal Allocation of VAr Support via Quadratic Power System Modeling and Simulation
George StefopoulosPh.D. CandidateSchool of Electrical and Computer EngineeringGeorgia Institute of Technology
PSERC Tele-seminar
May 5th, 2009
2
Presentation outline Introduction
Understanding basic voltage phenomena
Modeling of factors affecting voltage problems Electric-load modeling
Power-system analysis and simulation Quasi-steady-state analysis Full transient analysis
Optimization and reactive support Control of voltage and reactive power Optimal allocation and operation of static and
dynamic VAR sources
Conclusions
3
Basic Concepts: Voltage Phenomena
Voltage recovery following faults Rate of return to normal voltage level after a
disturbance, fault, etc.
Voltage stability Ability of a power system to maintain
acceptable voltages at all system buses under normal conditions and after disturbances
Voltage collapse Phenomenon in which a relatively fast
sequence of events after voltage instability leads to a voltage decay to unacceptably low values – in general a non-recoverable situation
Introduction → Modeling → Simulation → Optimization → Synopsis
4
Voltage Recovery following faults: Effects of Load
Typically motors will stall if their terminal voltage sags below 90% for too long (e.g. more than 20 cycles)
The voltage recovery, followingthe clearing of a fault, maybe slow for weak systems withheavy induction motor loads
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.00 0.50 1.00 1.50Seconds
2.00
Volta
ge (p
u)
Motors will tripif voltage sagsfor too long
-0.50-1.00
FaultFault Cleared
Introduction → Modeling → Simulation → Optimization → Synopsis
5
Voltage Recovery: Field Recordings
Phase A (V) Phase B (V) Phase C (V)
Voltage Recovery SlowLoad SheddingOvervoltageLoad RestorationVoltage SagVoltage collapse
Introduction → Modeling → Simulation → Optimization → Synopsis
There exist many field recording of fault delayed voltage recovery events
Duration of events may last from less than a second up to several minutes
6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Active Power P (p.u.)
Vol
tage
V (p
.u.)
Voltage Stability: P-V Curves
Point of voltage collapse
Stable region
Unstable region
Introduction → Modeling → Simulation → Optimization → Synopsis
7
Voltage Stability: System Oscillations
During transient swings voltage collapse may occur near the center of oscillation. Below
example shows a snapshot of a transient swing
Introduction → Modeling → Simulation → Optimization → Synopsis
8
Voltage Collapse: Effects of Reactive Support
Introduction → Modeling → Simulation → Optimization → Synopsis
9
Analysis of Voltage Related Phenomena Dynamic in nature phenomena
Small-disturbance stability Static steady-state analysis (PV curves) Continuation power flow Linearizations
V-Q sensitivity analysis Q-V modal analysis
Large-disturbance stability Non-linear dynamic analysis Dynamic simulation
Transient time frame Long-term time frame
Introduction → Modeling → Simulation → Optimization → Synopsis
10
Presentation outline
IntroductionModeling of factors affecting
voltage problems Power-system analysis and
simulationOptimization and reactive
supportConclusions
Network/Load Modeling Existing power-flow analysis tools typically do not
represent load dynamics Load dynamics may introduces asymmetries and
imbalances Three-phase models vs. single-phase equivalents
Positive sequence analysis – Balanced and symmetric operation
Symmetrical component analysis – Unbalanced operation symmetric conditions
Phase domain analysis – Unbalanced operation, asymmetric conditions, physically-based analysis
Proposed approach Power system modeling, with emphasis on dynamic loads
(induction motor loads) Physically-based, three-phase, high fidelity system
representation Quadratized component modeling
11Introduction → Modeling → Simulation → Optimization → Synopsis
12
Electric Load Modeling
Induction motors VAR requirements vary drastically with
operating conditions (motor speed) Dynamics
Cold load pick-up High currents Motor restart/stalling Transformer inrush currents
Introduction → Modeling → Simulation → Optimization → Synopsis
13
Traditional Electric Load Modeling
Static load representation: Constant impedance load Constant current load Constant power load Voltage/frequency-dependent load
models
Cannot capture all the phenomena
Introduction → Modeling → Simulation → Optimization → Synopsis
14
Characteristics of Induction Motor Loads
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100
Speed (% of synchronous)
Pow
er F
acto
r (%
)
Induction motor operating conditions for different operating speed values
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6
7
8
Speed (% of synchronous)
Torq
ue, P
ower
, Cur
rent
(p.u
.)
Reactive power
Motor current
Active power
Mechanica l loadSlip-torque characteristic
Operating po in t
Introduction → Modeling → Simulation → Optimization → Synopsis
15
Effects of Induction Motor Loads(Steady State)
Voltage profile of the 24-bus RTS after a line contingency
(a) constant power load representation
(b) induction motors (50%)
(a) (b)
Introduction → Modeling → Simulation → Optimization → Synopsis
16
Contingency simulation:
Effects of load dynamics 50% Induction motors2% Slowdown during fault
Vmax=1.01, Vmin=0.8250% Induction motors
Vmax=1.046, Vmin=0.908
Effects of Induction Motor Loads (Transient)
Introduction → Modeling → Simulation → Optimization → Synopsis
17
Electric Load Modeling Static load representation:
Constant impedance load Constant current load Constant power load Voltage/frequency-dependent load models
Dynamic load representation Induction motors Generalized dynamic load models
Three-phase and single-phase modelsIssue: In general the compositionof the load is not known. Need realdata to define model
Introduction → Modeling → Simulation → Optimization → Synopsis
18
Estimation of Electric Load Composition
Obtain time recordings of transient events
Identify load composition Specific signature of each load type
Identify load model parameters
Issue:
Identification of load parameters
Introduction → Modeling → Simulation → Optimization → Synopsis
19
3-Phase Induction Motor Models:NEMA Design Motor Models
Design A Design B
Design C Design D
Introduction → Modeling → Simulation → Optimization → Synopsis
20
Slip-Dependent Rotor Impedance
0 20 40 60 80 1000
0.5
1
1.5
2
2.5
3
3.5
4
Torq
ue (p
.u.)
Speed (% of rated)
NEMA DESIGN A, B, C, D for AC INDUCTION MOTORSDesign A
Design B
Design C
Design D Deep-bar squirrel-cage motors
Double-cage rotors
Using slip-dependentmotor parameters the torque-speed motor characteristics are accurately represented
Slip-dependentrotor parameters
Introduction → Modeling → Simulation → Optimization → Synopsis
21
Slip-Dependent Rotor Impedance
Idk~
r1 jx1 r2(s) jx2(s)
r2(s)( 1- s )
sjxmEn
~
BUS k
sedsxscsbasr
⋅+=⋅+⋅+=
)()(
2
22
This model can capture the behavior of any motor type by appropriate selection of the model parameters
Figure shows single-phase equivalent
Introduction → Modeling → Simulation → Optimization → Synopsis
22
Induction Motor Model Estimation Realistic representation of motor loads Accurate representation of motors of
various classes Correct identification of motor equivalent
circuit Practical estimation of slip-dependent
rotor models
No-load test Blocked-rotor test Field measurements
Introduction → Modeling → Simulation → Optimization → Synopsis
23
Induction Motor Model Estimation
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
2
2.5
3
3.5
Speed (% of synchronous)
Torq
ue (p
.u.)
WrrrwJ Tm
iii == ∑
=1
2min
Measured speed-torque curve(m measurement points)
Least-squares estimation
−−
=.)(
,)(
,
,,
imeasuredi
imeasurediemi IpI
TpTr
Introduction → Modeling → Simulation → Optimization → Synopsis
Solution process Gauss-Newton-type method
( ) )()()()( 11 nTnnTnnn pWrpHpWHpHpp −+ −=
[ ]npp
TTTem
n pIpTpH=
∂∂∂∂= //)(
Need for global convergence strategies (line search, trust region) Need for proper state and equation scaling
Single-Phase Induction Motor Model: Operational Characteristics
24Introduction → Modeling → Simulation → Optimization → Synopsis
Split phase
Capacitor start
Capacitor start, capacitor run
Permanent split capacitor
Model Quadratization Description of a system using linear or quadratic
equations without introducing any approximations Introduction of additional state variables
25
)),(),(( 0 )),(),(()(
ttytxgttytxftx
==
[ ] )(...)()()(
)()()(00
)( 1
321 tbXFX
XFX
tztytx
tAtAtAtx
nT
T
+
+
⋅=
( )
( )
+
++=
Tnn
T
TT
FFX
FFXAJ ...
11
Introduction → Modeling → Simulation → Optimization → Synopsis
26
Presentation outline
IntroductionModeling of factors affecting
voltage problems Power-system analysis and
simulationOptimization and reactive
supportConclusions
27
Types of Analysis
Steady-state analysis
Quasi-steady-state analysis
Full transient analysis
Introduction → Modeling → Simulation → Optimization → Synopsis
28
Quasi-steady-state analysis
Analysis through time using simplified, yet realistic, dynamic models
Consideration of only essential dynamic characteristics of power systems components (ignore fast electric phenomena)
Sinusoidal steady-state network conditions Computation of RMS values of electrical
quantities at fundamental frequency Consideration of motor dynamics only
Introduction → Modeling → Simulation → Optimization → Synopsis
Three-Phase Induction Motor Model: Steady-State, Quasi-Steady-State
Augmentation of the steady-state equation set with the swing equation of the rotor motion
Constant torque mode or slip-dependent torque mode
29
jxsrs jxr
jxm
rr
1-ss
( )gm rrV1 E1
I1
jxsrs jxr
jxm
rr
s-12-s
( )gm rrV2 E2
I2
j(xs+xr)rs+rr
V0
I0
Introduction → Modeling → Simulation → Optimization → Synopsis
)()()(
tTtTdt
tdJ Lm
n −=ω
ssnn s ωωω −−=0
constTL =2
nnL cbaT ωω ++=
Single-Phase Induction Motor Model
30
+
_+
+ +
+
_
_
_
_C CsV2
V1
Vmain
VauxIauxImain
Vc VcsVs
I1
I2Ic Ics
Steady-state Frequency-domain analysis of equivalent circuit
Quasi-steady-state Frequency-domain analysis of equivalent circuit Inclusion of mechanical equations for rotor movement Switching model
Introduction → Modeling → Simulation → Optimization → Synopsis
Distribution Feeder Voltage Recovery: Quasi-Steady-State Analysis
31Introduction → Modeling → Simulation → Optimization → Synopsis
Distribution Feeder Voltage Recovery: Quasi-Steady-State Analysis
32Introduction → Modeling → Simulation → Optimization → Synopsis
0.0 0.5 1.0 1.5
17.92
493.0
968.1
1.443 k
1.918 k
2.393 k
2.869 k
3.344 k
3.819 k
4.294 k
4.769 k
5.244 k
5.719 k
6.194 k
6.669 k
7.145 k
7.620 k
8.095 k PXFMR__Bus_SUB2__Voltage_Phase_A (V)
0.0 0.5 1.0 1.5
243.9 m
38.79
77.34
115.9
154.4
193.0
231.5
270.1
308.6
347.2
385.7
424.3
462.8
501.4
539.9
578.5
617.0
655.6 Current__Line_FDR1-POLE13__Phase_A (A)Current__Line_FDR2-POLE1__Phase_A (A)PXFMR__Bus_SUB2__Current_Phase_A (A)
Distribution Feeder Voltage Recovery: Quasi-Steady-State Analysis
33Introduction → Modeling → Simulation → Optimization → Synopsis
0.0 0.5 1.0 1.5
386.2 m
16.63
32.87
49.12
65.36
81.60
97.85
114.1
130.3
146.6
162.8
179.1
195.3
211.6
227.8
244.0
260.3
276.5 IndMotor_MCC-P2:_Voltage_Phase_B (V)IndMotor_MCC-P3:_Voltage_Phase_B (V)IndMotor_MCC-P8A:_Voltage__Phase_B (V)IndMotor_MCC1:_Voltage_Phase_B (V)IndMotor_MCC3:_Voltage_Phase_B (V)IndMotor_MCC5:_Voltage_Phase_B (V)IndMotor_MCC6:_Voltage__Phase_B (V)IndMotor_MCC8:_Voltage__Phase_B (V)
0.0 0.5 1.0 1.5
1.501
7.251
13.00
18.75
24.50
30.25
36.00
41.75
47.50
53.25
59.00
64.75
70.50
76.25
82.00
87.75
93.50
99.25 IndMotor_MCC-P2:_Speed (%)IndMotor_MCC-P3:_Speed (%)IndMotor_MCC-P8A:_Speed (%)IndMotor_MCC1:_Speed (%)IndMotor_MCC3:_Speed (%)IndMotor_MCC5:_Speed (%)IndMotor_MCC6:_Speed (%)IndMotor_MCC8:_Speed (%)
Distribution Feeder Voltage Recovery: Quasi-Steady-State Analysis
34Introduction → Modeling → Simulation → Optimization → Synopsis
0.0 0.5 1.0 1.5
2.857
92.98 k
186.0 k
278.9 k
371.9 k
464.9 k
557.9 k
650.9 k
743.8 k
836.8 k
929.8 k
1.023 M
1.116 M
1.209 M
1.302 M
1.395 M
1.488 M
1.581 M IndMotor_MCC-P2:_Real_Power (W)IndMotor_MCC-P3:_Real_Power (W)IndMotor_MCC-P8A:_Real_Power (W)IndMotor_MCC1:_Real_Power (W)IndMotor_MCC3:_Real_Power (W)IndMotor_MCC5:_Real_Power (W)IndMotor_MCC6:_Real_Power (W)IndMotor_MCC8:_Real_Power (W)
0.0 0.5 1.0 1.5
-3.748
111.3 k
222.6 k
333.9 k
445.2 k
556.6 k
667.9 k
779.2 k
890.5 k
1.002 M
1.113 M
1.224 M
1.336 M
1.447 M
1.558 M
1.670 M
1.781 M
1.892 M IndMotor_MCC-P2:_Reactive_Power (VA)IndMotor_MCC-P3:_Reactive_Power (VA)IndMotor_MCC-P8A:_Reactive_Power (VA)IndMotor_MCC1:_Reactive_Power (VA)IndMotor_MCC3:_Reactive_Power (VA)IndMotor_MCC5:_Reactive_Power (VA)IndMotor_MCC6:_Reactive_Power (VA)IndMotor_MCC8:_Reactive_Power (VA)
35
Full transient analysis
Time domain analysis using high-fidelity models
Inclusion of fast dynamics Computation of actual time domain
waveform Consideration of waveform harmonic
distortion Consideration of motor dynamics and
transformer inrush currents
Introduction → Modeling → Simulation → Optimization → Synopsis
Three-Phase Induction Motor Model: Transient Analysis
36
ReferenceStator phase A magnetic axis
vAs(t)
vNs(t)
iAs(t)
iNs(t)
vBs(t)iBs(t)
vCs(t)iCs(t)
Rotor phase A magnetic axis
vAr(t)
θm(t)
ωm(t)
iAr(t)
iBr(t)vBr(t)
vCr(t)
vNr(t)
iCr(t)
iNr(t)
ias(t)
ibs(t)ics(t)
ibr(t)
iar(t)
icr(t)
Reference
Stator phase A magnetic axis
vAs(t)iAs(t)
vBs(t)iBs(t)
vCs(t)iCs(t)
Rotor phase A magnetic axis
vAr(t)
θm(t)
ωm(t)
iAr(t)
iBr(t)vBr(t)
vCr(t)iCr(t)
ias(t)
ibs(t)
ics(t)
ibr(t)
iar(t)
icr(t)
Introduction → Modeling → Simulation → Optimization → Synopsis
Time domain analysis of mutually coupled windings Inclusion of mechanical equations for rotor movement
Single-Phase Induction Motor Model
37
+
_+
+ +
+
_
_
_
_C CsV2
V1
Vmain
VauxIauxImain
Vc VcsVs
I1
I2Ic Ics
Full transient Time domain analysis of mutually coupled windings Inclusion of mechanical equations for rotor movement Switching model
Introduction → Modeling → Simulation → Optimization → Synopsis
Modeling of Inrush Currents: Saturable-Core Reactor Model
38
v1(t)
v2(t)
i1(t)
i2(t)
iL(t)
)()()(21 tvtv
dttd
−=λ
( ))()()(0
01 tsigntitin
λλλ
⋅⋅=
)()( titi 12 −=
Introduction → Modeling → Simulation → Optimization → Synopsis
Modeling of Inrush Currents: Saturable-Core Transformer Model
39
v1(t)
v2(t)
i1(t)
i2(t)
gL
igL(t)iL(t)
r1 L1 r2L2
v3(t)
i3(t)
v4(t)
i4(t)
+
-
e1(t) te1(t)
1:t
Im(t)
)()(1 te
dttd=
λ
)()()()()( 11
11121 tedt
tdiLtirtvtv ++=−
)()(
)()()( 13
23243 ttedt
tdiLtirtvtv ++=−
( ) )()()()( 10
0 tegtsigntiti L
n
m ⋅+⋅⋅= λλλ
0)()()( 31 =−+ tittiti m
)()( titi 12 −=
)()( 34 titi −=
Introduction → Modeling → Simulation → Optimization → Synopsis
Distribution Feeder Voltage Recovery: Full Transient Analysis
40Introduction → Modeling → Simulation → Optimization → Synopsis
Distribution Feeder Voltage Recovery: Full Transient Analysis
41Introduction → Modeling → Simulation → Optimization → Synopsis
42
Presentation outline
IntroductionModeling of factors affecting
voltage problems Power-system analysis and
simulationOptimization and reactive
supportConclusions
43
Control of Voltage and Reactive Power
Generator VAR outputs Regulating transformers (ULTC) Shunt capacitors/reactors
Fixed capacitor banks (FC) Mechanically switched capacitor banks
(MSC)
Series capacitors Synchronous condensers
Introduction → Modeling → Simulation → Optimization → Synopsis
44
Control of Voltage and Reactive Power Static VAR (shunt) compensators (SVC)
Thyristor-switched reactor (TSR) Thyristor-controlled reactor (TCR) Thyristor-switched capacitor (TSC) Static synchronous compensator (STATCOM)
Static series compensators Thyristor-switched series capacitor (TSSC) Thyristor-controlled series capacitor (TCSC) GTO Thyristor-controlled series capacitor (GCSC) Static synchronous series compensator (SSSC) Phase angle regulator (PAR)
Combined compensators Unified power flow controller (UPFC)
Introduction → Modeling → Simulation → Optimization → Synopsis
45
Static VAR Systems
Static VAR compensator at CERN laboratories, Switzerland (www.triumf.ca)
Combination of SVC and switched shunt capacitors/reactors with coordinated outputs
Introduction → Modeling → Simulation → Optimization → Synopsis
46
SuperVAR Machine: Superconducting Synchronous Condenser
American Superconductor Corp. and Tennessee Valley Authority
Installed in Nov. 2005, in the TVA network, at Gallatin, TN (Hoeganaes Corp. steel plant)
(IEEE Spectrum, Jan. 2006)
www.amsuper.com
Introduction → Modeling → Simulation → Optimization → Synopsis
47
Mitigation of Voltage Recovery Problems
How can voltage problems be controlled Planning for adequate VAR support Addition of dynamic VAR sources for fast
response
Develop methodology for the selection of the optimal mix and placement of static and dynamic VAR resources applicable in large power systems, to improve voltage recovery and dynamic performance
Introduction → Modeling → Simulation → Optimization → Synopsis
Mitigation of Voltage Recovery Problems
Optimal allocation of new static and dynamic reactive resources (planning problem) Selection of location Selection of size Determination of optimal mix
(static/dynamic)
Optimal operation of existing static and dynamic reactive resources (operational problem)
48Introduction → Modeling → Simulation → Optimization → Synopsis
Performance Indices
49
Voltage Security Index
∑
−=
j
n
j,step
j,avejjV V
VVWJ
2
( )dttzztzJt
tthreshold∫ −=
2
1
)())((
Voltage Recovery Index
dtdu
tdzdu
tzdJ t
t∫−=2
1
)())((
( )
+
++= ∑∑
==
n
iiRR
n
iidipdipiVV JfWJfWJfWXCWXJ
1,4
1,3,21 )()(
12))(( tttvJdip −=Voltage Dip Index
Annualized cost of additions
Performance penalties
Introduction → Modeling → Simulation → Optimization → Synopsis
Reactive Resource Allocation –Planning Problem
Selection of a limited number if states at each stage
Evaluation of cost for each state at each stage
Application of dynamic programming recurrence to define optimal transitions
Determination of optimal trajectory by backtracking
50
Cost*(X0,k+3)Cost*(X0,k+2)Cost*(X0,k+1)
Cost*(X1,k+3)Cost*(X1,k+2)Cost*(X1,k+1)Cost*(X1,k)
Cost*(XN,k)
State 0X0,k
Cost0,k
State 1X1,k
Cost1,k
State 2X2,k
Cost2,k
.
.
.
State NXN,k
CostN,k
States
State 0X0,k+1
Cost0,k+1
State 1X1,k+1
Cost1,k+1
State 2X2,k+1
Cost2,k+1
.
.
.
State NXN,k+1
CostN,k+1
Stages(years in the
planning horizon)
...
0 Timek k+1 k+2 k+3
State 0X0,k+2
Cost0,k+2
State 1X1,k+2
Cost1,k+2
State 2X2,k+2
Cost2,k+2
.
.
.
State NXN,k+2
CostN,k+2
State 0X0,k+3
Cost0,k+3
State 1X1,k+3
Cost1,k+3
State 2X2,k+3
Cost2,k+3
.
.
.
State NXN,k+3
CostN,k+3
Base Case
Cost*(XN,k+1) Cost*(XN,k+2) Cost*(XN,k+3)
Cost*(X2,k+3)Cost*(X2,k+2)Cost*(X2,k+1)Cost*(X2,k)
Cost*(X0,k)
[ ])()(min)( 1,,,*
1,*
++ →+= kikjkjjki XXTXCostXCost
Introduction → Modeling → Simulation → Optimization → Synopsis
Evaluation of Cost for Specific State at Specific Stage
Solution of base case optimal power flow
Selection of critical contingencies Optimal simulation of each critical
contingencies (operational problem) Computation of performance
51Introduction → Modeling → Simulation → Optimization → Synopsis
Reactive Resource Allocation
Assume N resources (VAr modules or financial amount) and P candidate locations
Allocate the resources in the specific locations such that the combined performance index is minimized
Total number of cases
52
−−+
11
PPN
Introduction → Modeling → Simulation → Optimization → Synopsis
Reactive Resource Allocation Solution via
dynamic programming
Stage: Location State: Remaining
units of resource to be allocated
Optimal value function
Recurrence relation
Boundary conditions
Answer:53
n = Np1(N)
f1(N)
States(remaining units
of resource)
n = 0pk(0)
fk(0)
n = 1pk(1)
fk(1)
n = 2pk(2)
fk(2)
.
.
.
n = Npk(N)
fk(N)
Stages(location)
...
2 k k+1 P
n = 0pk+1(0)
fk+1(0)
n = 1pk+1(1)
fk+1(1)
n = 2pk+1(2)
fk+1(2)
.
.
.
n = Npk+1(N)
fk+1(N)
n = 0pp(0)
fP(0)
n = 1pp(1)
fP(1)
n = 2pp(2)
fP(2)
.
.
.
n = Npp(N)
fP(N)
...
1
n = 0pk(0)
fk(0)
n = 1pk(1)
fk(1)
n = 2pk(2)
fk(2)
.
.
.
n = Npk(N)
fk(N)
Introduction → Modeling → Simulation → Optimization → Synopsis
NiPN
i2)2(
1+− ∑
=
)(),( npnf kk
( )[ ])](|[min)( 1,...,1,0 kkknnk nNpnJnfk
−= +=
)(),( 11 NpNf
Operational Problem Assume specific dynamic VAr sources in a system Compute the optimal reactive injections--------------------------------------------------------------------- Classical optimal control problem
54
( )( )tptutytxg
tptutytxftx,),(),(),( 0,),(),(),()(
==
[ ]FFF ttytxJ ),(),( ϕ=
[ ]FFF tptytx ,),(),( 0 ψ=
[ ] [ ]{ }dtfxgvJ F
IF
t
t
TTt
T ∫ −+−+= λµψϕˆ
( ) ul htptutytxhh ≤≤ ,),(),(),(
Direct Transcription ( )
0)( ..,)(=
=Xcts
yxXF MMφ
Introduction → Modeling → Simulation → Optimization → Synopsis
Static and Dynamic Sensitivity Analysis
Static sensitivity Determine favorable locations based on
steady state criteria Rank system contingencies
Trajectory sensitivity Determine favorable locations based on
dynamic performance penalties
55Introduction → Modeling → Simulation → Optimization → Synopsis
Static Sensitivity Analysis
56
)0.0,(),( 0 =−=∆ iinew uxJuxJJ
: initial state0xnewx : post-installation state
Complete nonlinear approach
PI Sensitivity Approach
ii
ududJJ ∆⋅=∆
( )0.0,, oo =−
∆+=∆ iii
i
uxJuududxxJJ
State Sensitivity Approach
∂
∂−
∂∂
=i
iT
i
i
i uuxgx
uuxJ
dudJ ),(ˆ),(
1),(),(ˆ−
∂∂
∂∂
=x
uxgx
uxJx iiT
PI Sensitivity
∂
∂
∂∂
−=−
i
ii
i uuxg
xuxg
dudx ),(),( 1
State Sensitivity
Introduction → Modeling → Simulation → Optimization → Synopsis
Trajectory Sensitivity Analysis
57
)()()()(tDxtCytBxtAx
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uu
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Numerical
Integration
),(0 uXG= 1
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Introduction → Modeling → Simulation → Optimization → Synopsis
Synopsis
Understanding and characterizing voltage problems and develop mitigation techniques
Simulation of voltage recovery phenomena in transmission and distribution networks
Optimal allocation and operation of static and dynamic reactive support resources
Development and implementation of appropriate simulation tools Modeling Simulation Decision making
58Introduction → Modeling → Simulation → Optimization → Synopsis